LMFDB - Embedded newform 2496.4.a.bl.1.2

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [2496,4,Mod(1,2496)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(2496, base_ring=CyclotomicField(2))

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

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

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

chi := DirichletCharacter("2496.1");

S:= CuspForms(chi, 4);

N := Newforms(S);

Level:	$ N $	$=$	$ 2496 = 2^{6} \cdot 3 \cdot 13 $
Weight:	$ k $	$=$	$ 4 $
Character orbit:	$[\chi]$	$=$	2496.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:	$147.268767374$
Analytic rank:	$0$
Dimension:	$3$
Coefficient field:	3.3.3144.1
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{3} - x^{2} - 16x - 8 $ x^3 - x^2 - 16*x - 8
Coefficient ring:	$\Z[a_1, \ldots, a_{11}]$
Coefficient ring index:	$ 2^{3} $
Twist minimal:	no (minimal twist has level 39)
Fricke sign:	$1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.2
Root			$4.73549$ of defining polynomial
Character	$\chi$	$=$	2496.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-3.00000 q^{3} +3.90776 q^{5} +36.4129 q^{7} +9.00000 q^{9} +O(q^{10})$	$q-3.00000 q^{3} +3.90776 q^{5} +36.4129 q^{7} +9.00000 q^{9} -19.1943 q^{11} -13.0000 q^{13} -11.7233 q^{15} -83.8839 q^{17} -46.8492 q^{19} -109.239 q^{21} +103.905 q^{23} -109.729 q^{25} -27.0000 q^{27} -108.341 q^{29} -147.532 q^{31} +57.5828 q^{33} +142.293 q^{35} +160.012 q^{37} +39.0000 q^{39} +231.490 q^{41} +340.314 q^{43} +35.1699 q^{45} +119.653 q^{47} +982.902 q^{49} +251.652 q^{51} +732.879 q^{53} -75.0067 q^{55} +140.548 q^{57} +229.782 q^{59} -108.943 q^{61} +327.716 q^{63} -50.8009 q^{65} -10.3955 q^{67} -311.714 q^{69} -869.201 q^{71} -1099.07 q^{73} +329.188 q^{75} -698.920 q^{77} +140.410 q^{79} +81.0000 q^{81} +159.474 q^{83} -327.799 q^{85} +325.023 q^{87} +1067.93 q^{89} -473.368 q^{91} +442.596 q^{93} -183.075 q^{95} +858.881 q^{97} -172.748 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 3 q - 9 q^{3} - 4 q^{5} + 30 q^{7} + 27 q^{9}+O(q^{10}) $ 3 * q - 9 * q^3 - 4 * q^5 + 30 * q^7 + 27 * q^9	$ 3 q - 9 q^{3} - 4 q^{5} + 30 q^{7} + 27 q^{9} + 16 q^{11} - 39 q^{13} + 12 q^{15} - 146 q^{17} - 94 q^{19} - 90 q^{21} - 48 q^{23} + 145 q^{25} - 81 q^{27} + 2 q^{29} + 302 q^{31} - 48 q^{33} - 80 q^{35} - 374 q^{37} + 117 q^{39} + 480 q^{41} + 260 q^{43} - 36 q^{45} - 24 q^{47} + 447 q^{49} + 438 q^{51} + 678 q^{53} - 1552 q^{55} + 282 q^{57} + 1788 q^{59} - 230 q^{61} + 270 q^{63} + 52 q^{65} - 74 q^{67} + 144 q^{69} - 948 q^{71} - 222 q^{73} - 435 q^{75} - 112 q^{77} - 24 q^{79} + 243 q^{81} + 796 q^{83} + 248 q^{85} - 6 q^{87} + 1436 q^{89} - 390 q^{91} - 906 q^{93} - 4032 q^{95} + 3242 q^{97} + 144 q^{99}+O(q^{100}) $ 3 * q - 9 * q^3 - 4 * q^5 + 30 * q^7 + 27 * q^9 + 16 * q^11 - 39 * q^13 + 12 * q^15 - 146 * q^17 - 94 * q^19 - 90 * q^21 - 48 * q^23 + 145 * q^25 - 81 * q^27 + 2 * q^29 + 302 * q^31 - 48 * q^33 - 80 * q^35 - 374 * q^37 + 117 * q^39 + 480 * q^41 + 260 * q^43 - 36 * q^45 - 24 * q^47 + 447 * q^49 + 438 * q^51 + 678 * q^53 - 1552 * q^55 + 282 * q^57 + 1788 * q^59 - 230 * q^61 + 270 * q^63 + 52 * q^65 - 74 * q^67 + 144 * q^69 - 948 * q^71 - 222 * q^73 - 435 * q^75 - 112 * q^77 - 24 * q^79 + 243 * q^81 + 796 * q^83 + 248 * q^85 - 6 * q^87 + 1436 * q^89 - 390 * q^91 - 906 * q^93 - 4032 * q^95 + 3242 * q^97 + 144 * 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 $

$2$	0	0
$2$	0	0
$3$	−3.00000	−0.577350
$3$	−3.00000	−0.577350
$4$	0	0
$5$	3.90776	0.349521	0.174761	−	0.984611i	$-0.444085\pi$
$5$	3.90776	0.349521	0.174761	+	0.984611i	$0.444085\pi$
$6$	0	0
$7$	36.4129	1.96611	0.983057	−	0.183301i	$-0.0586782\pi$
$7$	36.4129	1.96611	0.983057	+	0.183301i	$0.0586782\pi$
$8$	0	0
$9$	9.00000	0.333333
$10$	0	0
$11$	−19.1943	−0.526117	−0.263059	−	0.964780i	$-0.584731\pi$
$11$	−19.1943	−0.526117	−0.263059	+	0.964780i	$0.584731\pi$
$12$	0	0
$13$	−13.0000	−0.277350
$13$	−13.0000	−0.277350
$14$	0	0
$15$	−11.7233	−0.201796
$16$	0	0
$17$	−83.8839	−1.19676	−0.598378	−	0.801214i	$-0.704188\pi$
$17$	−83.8839	−1.19676	−0.598378	+	0.801214i	$0.704188\pi$
$18$	0	0
$19$	−46.8492	−0.565681	−0.282840	−	0.959167i	$-0.591277\pi$
$19$	−46.8492	−0.565681	−0.282840	+	0.959167i	$0.591277\pi$
$20$	0	0
$21$	−109.239	−1.13514
$22$	0	0
$23$	103.905	0.941983	0.470991	−	0.882138i	$-0.343896\pi$
$23$	103.905	0.941983	0.470991	+	0.882138i	$0.343896\pi$
$24$	0	0
$25$	−109.729	−0.877835
$26$	0	0
$27$	−27.0000	−0.192450
$28$	0	0
$29$	−108.341	−0.693738	−0.346869	−	0.937914i	$-0.612755\pi$
$29$	−108.341	−0.693738	−0.346869	+	0.937914i	$0.612755\pi$
$30$	0	0
$31$	−147.532	−0.854759	−0.427379	−	0.904072i	$-0.640563\pi$
$31$	−147.532	−0.854759	−0.427379	+	0.904072i	$0.640563\pi$
$32$	0	0
$33$	57.5828	0.303754
$34$	0	0
$35$	142.293	0.687198
$36$	0	0
$37$	160.012	0.710969	0.355484	−	0.934682i	$-0.384316\pi$
$37$	160.012	0.710969	0.355484	+	0.934682i	$0.384316\pi$
$38$	0	0
$39$	39.0000	0.160128
$40$	0	0
$41$	231.490	0.881772	0.440886	−	0.897563i	$-0.354664\pi$
$41$	231.490	0.881772	0.440886	+	0.897563i	$0.354664\pi$
$42$	0	0
$43$	340.314	1.20692	0.603458	−	0.797395i	$-0.293789\pi$
$43$	340.314	1.20692	0.603458	+	0.797395i	$0.293789\pi$
$44$	0	0
$45$	35.1699	0.116507
$46$	0	0
$47$	119.653	0.371346	0.185673	−	0.982612i	$-0.440554\pi$
$47$	119.653	0.371346	0.185673	+	0.982612i	$0.440554\pi$
$48$	0	0
$49$	982.902	2.86560
$50$	0	0
$51$	251.652	0.690947
$52$	0	0
$53$	732.879	1.89941	0.949705	−	0.313146i	$-0.101383\pi$
$53$	732.879	1.89941	0.949705	+	0.313146i	$0.101383\pi$
$54$	0	0
$55$	−75.0067	−0.183889
$56$	0	0
$57$	140.548	0.326596
$58$	0	0
$59$	229.782	0.507035	0.253518	−	0.967331i	$-0.418412\pi$
$59$	229.782	0.507035	0.253518	+	0.967331i	$0.418412\pi$
$60$	0	0
$61$	−108.943	−0.228668	−0.114334	−	0.993442i	$-0.536473\pi$
$61$	−108.943	−0.228668	−0.114334	+	0.993442i	$0.536473\pi$
$62$	0	0
$63$	327.716	0.655371
$64$	0	0
$65$	−50.8009	−0.0969397
$66$	0	0
$67$	−10.3955	−0.0189555	−0.00947774	−	0.999955i	$-0.503017\pi$
$67$	−10.3955	−0.0189555	−0.00947774	+	0.999955i	$0.503017\pi$
$68$	0	0
$69$	−311.714	−0.543854
$70$	0	0
$71$	−869.201	−1.45289	−0.726445	−	0.687224i	$-0.758829\pi$
$71$	−869.201	−1.45289	−0.726445	+	0.687224i	$0.758829\pi$
$72$	0	0
$73$	−1099.07	−1.76214	−0.881072	−	0.472982i	$-0.843178\pi$
$73$	−1099.07	−1.76214	−0.881072	+	0.472982i	$0.843178\pi$
$74$	0	0
$75$	329.188	0.506818
$76$	0	0
$77$	−698.920	−1.03441
$78$	0	0
$79$	140.410	0.199967	0.0999835	−	0.994989i	$-0.468121\pi$
$79$	140.410	0.199967	0.0999835	+	0.994989i	$0.468121\pi$
$80$	0	0
$81$	81.0000	0.111111
$82$	0	0
$83$	159.474	0.210898	0.105449	−	0.994425i	$-0.466372\pi$
$83$	159.474	0.210898	0.105449	+	0.994425i	$0.466372\pi$
$84$	0	0
$85$	−327.799	−0.418291
$86$	0	0
$87$	325.023	0.400530
$88$	0	0
$89$	1067.93	1.27192	0.635959	−	0.771723i	$-0.280605\pi$
$89$	1067.93	1.27192	0.635959	+	0.771723i	$0.280605\pi$
$90$	0	0
$91$	−473.368	−0.545302
$92$	0	0
$93$	442.596	0.493495
$94$	0	0
$95$	−183.075	−0.197717
$96$	0	0
$97$	858.881	0.899032	0.449516	−	0.893272i	$-0.351596\pi$
$97$	858.881	0.899032	0.449516	+	0.893272i	$0.351596\pi$
$98$	0	0
$99$	−172.748	−0.175372
$100$	0	0
$101$	1574.16	1.55084	0.775421	−	0.631444i	$-0.217538\pi$
$101$	1574.16	1.55084	0.775421	+	0.631444i	$0.217538\pi$
$102$	0	0
$103$	−129.724	−0.124098	−0.0620489	−	0.998073i	$-0.519763\pi$
$103$	−129.724	−0.124098	−0.0620489	+	0.998073i	$0.519763\pi$
$104$	0	0
$105$	−426.879	−0.396754
$106$	0	0
$107$	1957.43	1.76853	0.884263	−	0.466990i	$-0.154661\pi$
$107$	1957.43	1.76853	0.884263	+	0.466990i	$0.154661\pi$
$108$	0	0
$109$	−1228.77	−1.07977	−0.539886	−	0.841738i	$-0.681533\pi$
$109$	−1228.77	−1.07977	−0.539886	+	0.841738i	$0.681533\pi$
$110$	0	0
$111$	−480.037	−0.410478
$112$	0	0
$113$	1629.50	1.35655	0.678275	−	0.734808i	$-0.262728\pi$
$113$	1629.50	1.35655	0.678275	+	0.734808i	$0.262728\pi$
$114$	0	0
$115$	406.035	0.329243
$116$	0	0
$117$	−117.000	−0.0924500
$118$	0	0
$119$	−3054.46	−2.35296
$120$	0	0
$121$	−962.580	−0.723201
$122$	0	0
$123$	−694.470	−0.509092
$124$	0	0
$125$	−917.267	−0.656343
$126$	0	0
$127$	276.112	0.192921	0.0964607	−	0.995337i	$-0.469248\pi$
$127$	276.112	0.192921	0.0964607	+	0.995337i	$0.469248\pi$
$128$	0	0
$129$	−1020.94	−0.696813
$130$	0	0
$131$	96.2240	0.0641765	0.0320883	−	0.999485i	$-0.489784\pi$
$131$	96.2240	0.0641765	0.0320883	+	0.999485i	$0.489784\pi$
$132$	0	0
$133$	−1705.92	−1.11219
$134$	0	0
$135$	−105.510	−0.0672653
$136$	0	0
$137$	2618.38	1.63287	0.816435	−	0.577438i	$-0.195947\pi$
$137$	2618.38	1.63287	0.816435	+	0.577438i	$0.195947\pi$
$138$	0	0
$139$	−1963.34	−1.19805	−0.599023	−	0.800732i	$-0.704444\pi$
$139$	−1963.34	−1.19805	−0.599023	+	0.800732i	$0.704444\pi$
$140$	0	0
$141$	−358.960	−0.214396
$142$	0	0
$143$	249.526	0.145919
$144$	0	0
$145$	−423.370	−0.242476
$146$	0	0
$147$	−2948.71	−1.65446
$148$	0	0
$149$	301.111	0.165557	0.0827784	−	0.996568i	$-0.473621\pi$
$149$	301.111	0.165557	0.0827784	+	0.996568i	$0.473621\pi$
$150$	0	0
$151$	342.973	0.184839	0.0924197	−	0.995720i	$-0.470540\pi$
$151$	342.973	0.184839	0.0924197	+	0.995720i	$0.470540\pi$
$152$	0	0
$153$	−754.955	−0.398918
$154$	0	0
$155$	−576.520	−0.298756
$156$	0	0
$157$	1286.97	0.654211	0.327106	−	0.944988i	$-0.393927\pi$
$157$	1286.97	0.654211	0.327106	+	0.944988i	$0.393927\pi$
$158$	0	0
$159$	−2198.64	−1.09662
$160$	0	0
$161$	3783.47	1.85205
$162$	0	0
$163$	−532.561	−0.255910	−0.127955	−	0.991780i	$-0.540841\pi$
$163$	−532.561	−0.255910	−0.127955	+	0.991780i	$0.540841\pi$
$164$	0	0
$165$	225.020	0.106168
$166$	0	0
$167$	41.9542	0.0194402	0.00972011	−	0.999953i	$-0.496906\pi$
$167$	41.9542	0.0194402	0.00972011	+	0.999953i	$0.496906\pi$
$168$	0	0
$169$	169.000	0.0769231
$170$	0	0
$171$	−421.643	−0.188560
$172$	0	0
$173$	1066.50	0.468694	0.234347	−	0.972153i	$-0.424705\pi$
$173$	1066.50	0.468694	0.234347	+	0.972153i	$0.424705\pi$
$174$	0	0
$175$	−3995.57	−1.72592
$176$	0	0
$177$	−689.346	−0.292737
$178$	0	0
$179$	−3174.61	−1.32559	−0.662797	−	0.748799i	$-0.730631\pi$
$179$	−3174.61	−1.32559	−0.662797	+	0.748799i	$0.730631\pi$
$180$	0	0
$181$	2725.43	1.11923	0.559613	−	0.828754i	$-0.310950\pi$
$181$	2725.43	1.11923	0.559613	+	0.828754i	$0.310950\pi$
$182$	0	0
$183$	326.829	0.132021
$184$	0	0
$185$	625.290	0.248498
$186$	0	0
$187$	1610.09	0.629634
$188$	0	0
$189$	−983.149	−0.378379
$190$	0	0
$191$	784.888	0.297343	0.148672	−	0.988887i	$-0.452500\pi$
$191$	784.888	0.297343	0.148672	+	0.988887i	$0.452500\pi$
$192$	0	0
$193$	−1255.87	−0.468391	−0.234195	−	0.972190i	$-0.575246\pi$
$193$	−1255.87	−0.468391	−0.234195	+	0.972190i	$0.575246\pi$
$194$	0	0
$195$	152.403	0.0559682
$196$	0	0
$197$	2777.35	1.00446	0.502229	−	0.864734i	$-0.332513\pi$
$197$	2777.35	1.00446	0.502229	+	0.864734i	$0.332513\pi$
$198$	0	0
$199$	1490.43	0.530924	0.265462	−	0.964121i	$-0.414476\pi$
$199$	1490.43	0.530924	0.265462	+	0.964121i	$0.414476\pi$
$200$	0	0
$201$	31.1866	0.0109440
$202$	0	0
$203$	−3945.01	−1.36397
$204$	0	0
$205$	904.608	0.308198
$206$	0	0
$207$	935.141	0.313994
$208$	0	0
$209$	899.236	0.297614
$210$	0	0
$211$	−2305.63	−0.752255	−0.376127	−	0.926568i	$-0.622745\pi$
$211$	−2305.63	−0.752255	−0.376127	+	0.926568i	$0.622745\pi$
$212$	0	0
$213$	2607.60	0.838827
$214$	0	0
$215$	1329.87	0.421842
$216$	0	0
$217$	−5372.07	−1.68055
$218$	0	0
$219$	3297.21	1.01737
$220$	0	0
$221$	1090.49	0.331920
$222$	0	0
$223$	1241.98	0.372956	0.186478	−	0.982459i	$-0.440293\pi$
$223$	1241.98	0.372956	0.186478	+	0.982459i	$0.440293\pi$
$224$	0	0
$225$	−987.564	−0.292612
$226$	0	0
$227$	1724.76	0.504300	0.252150	−	0.967688i	$-0.418862\pi$
$227$	1724.76	0.504300	0.252150	+	0.967688i	$0.418862\pi$
$228$	0	0
$229$	3273.72	0.944688	0.472344	−	0.881414i	$-0.343408\pi$
$229$	3273.72	0.944688	0.472344	+	0.881414i	$0.343408\pi$
$230$	0	0
$231$	2096.76	0.597215
$232$	0	0
$233$	−2129.52	−0.598752	−0.299376	−	0.954135i	$-0.596778\pi$
$233$	−2129.52	−0.598752	−0.299376	+	0.954135i	$0.596778\pi$
$234$	0	0
$235$	467.577	0.129793
$236$	0	0
$237$	−421.231	−0.115451
$238$	0	0
$239$	−5082.38	−1.37553	−0.687765	−	0.725933i	$-0.741408\pi$
$239$	−5082.38	−1.37553	−0.687765	+	0.725933i	$0.741408\pi$
$240$	0	0
$241$	4765.65	1.27379	0.636893	−	0.770953i	$-0.280219\pi$
$241$	4765.65	1.27379	0.636893	+	0.770953i	$0.280219\pi$
$242$	0	0
$243$	−243.000	−0.0641500
$244$	0	0
$245$	3840.95	1.00159
$246$	0	0
$247$	609.039	0.156892
$248$	0	0
$249$	−478.422	−0.121762
$250$	0	0
$251$	4339.96	1.09138	0.545689	−	0.837988i	$-0.316268\pi$
$251$	4339.96	1.09138	0.545689	+	0.837988i	$0.316268\pi$
$252$	0	0
$253$	−1994.37	−0.495594
$254$	0	0
$255$	983.396	0.241500
$256$	0	0
$257$	4359.49	1.05812	0.529062	−	0.848583i	$-0.322544\pi$
$257$	4359.49	1.05812	0.529062	+	0.848583i	$0.322544\pi$
$258$	0	0
$259$	5826.51	1.39785
$260$	0	0
$261$	−975.068	−0.231246
$262$	0	0
$263$	608.077	0.142569	0.0712844	−	0.997456i	$-0.477290\pi$
$263$	608.077	0.142569	0.0712844	+	0.997456i	$0.477290\pi$
$264$	0	0
$265$	2863.92	0.663884
$266$	0	0
$267$	−3203.80	−0.734342
$268$	0	0
$269$	−3454.29	−0.782942	−0.391471	−	0.920190i	$-0.628034\pi$
$269$	−3454.29	−0.782942	−0.391471	+	0.920190i	$0.628034\pi$
$270$	0	0
$271$	3703.72	0.830204	0.415102	−	0.909775i	$-0.363746\pi$
$271$	3703.72	0.830204	0.415102	+	0.909775i	$0.363746\pi$
$272$	0	0
$273$	1420.10	0.314830
$274$	0	0
$275$	2106.18	0.461844
$276$	0	0
$277$	3566.89	0.773696	0.386848	−	0.922144i	$-0.373564\pi$
$277$	3566.89	0.773696	0.386848	+	0.922144i	$0.373564\pi$
$278$	0	0
$279$	−1327.79	−0.284920
$280$	0	0
$281$	−117.474	−0.0249392	−0.0124696	−	0.999922i	$-0.503969\pi$
$281$	−117.474	−0.0249392	−0.0124696	+	0.999922i	$0.503969\pi$
$282$	0	0
$283$	1737.62	0.364984	0.182492	−	0.983207i	$-0.441584\pi$
$283$	1737.62	0.364984	0.182492	+	0.983207i	$0.441584\pi$
$284$	0	0
$285$	549.226	0.114152
$286$	0	0
$287$	8429.23	1.73366
$288$	0	0
$289$	2123.51	0.432223
$290$	0	0
$291$	−2576.64	−0.519057
$292$	0	0
$293$	−1904.05	−0.379643	−0.189822	−	0.981819i	$-0.560791\pi$
$293$	−1904.05	−0.379643	−0.189822	+	0.981819i	$0.560791\pi$
$294$	0	0
$295$	897.934	0.177219
$296$	0	0
$297$	518.245	0.101251
$298$	0	0
$299$	−1350.76	−0.261259
$300$	0	0
$301$	12391.8	2.37293
$302$	0	0
$303$	−4722.49	−0.895379
$304$	0	0
$305$	−425.724	−0.0799242
$306$	0	0
$307$	−2862.39	−0.532134	−0.266067	−	0.963955i	$-0.585724\pi$
$307$	−2862.39	−0.532134	−0.266067	+	0.963955i	$0.585724\pi$
$308$	0	0
$309$	389.172	0.0716479
$310$	0	0
$311$	4201.55	0.766071	0.383036	−	0.923734i	$-0.374879\pi$
$311$	4201.55	0.766071	0.383036	+	0.923734i	$0.374879\pi$
$312$	0	0
$313$	3427.74	0.619002	0.309501	−	0.950899i	$-0.399838\pi$
$313$	3427.74	0.619002	0.309501	+	0.950899i	$0.399838\pi$
$314$	0	0
$315$	1280.64	0.229066
$316$	0	0
$317$	−1676.09	−0.296966	−0.148483	−	0.988915i	$-0.547439\pi$
$317$	−1676.09	−0.296966	−0.148483	+	0.988915i	$0.547439\pi$
$318$	0	0
$319$	2079.52	0.364987
$320$	0	0
$321$	−5872.30	−1.02106
$322$	0	0
$323$	3929.89	0.676982
$324$	0	0
$325$	1426.48	0.243468
$326$	0	0
$327$	3686.32	0.623406
$328$	0	0
$329$	4356.93	0.730108
$330$	0	0
$331$	−11156.6	−1.85264	−0.926319	−	0.376740i	$-0.877045\pi$
$331$	−11156.6	−1.85264	−0.926319	+	0.376740i	$0.877045\pi$
$332$	0	0
$333$	1440.11	0.236990
$334$	0	0
$335$	−40.6233	−0.00662534
$336$	0	0
$337$	1636.44	0.264517	0.132259	−	0.991215i	$-0.457777\pi$
$337$	1636.44	0.264517	0.132259	+	0.991215i	$0.457777\pi$
$338$	0	0
$339$	−4888.49	−0.783205
$340$	0	0
$341$	2831.77	0.449703
$342$	0	0
$343$	23300.7	3.66799
$344$	0	0
$345$	−1218.10	−0.190088
$346$	0	0
$347$	2977.87	0.460693	0.230347	−	0.973109i	$-0.426014\pi$
$347$	2977.87	0.460693	0.230347	+	0.973109i	$0.426014\pi$
$348$	0	0
$349$	−9847.29	−1.51035	−0.755177	−	0.655521i	$-0.772449\pi$
$349$	−9847.29	−1.51035	−0.755177	+	0.655521i	$0.772449\pi$
$350$	0	0
$351$	351.000	0.0533761
$352$	0	0
$353$	4687.34	0.706747	0.353374	−	0.935482i	$-0.385034\pi$
$353$	4687.34	0.706747	0.353374	+	0.935482i	$0.385034\pi$
$354$	0	0
$355$	−3396.63	−0.507816
$356$	0	0
$357$	9163.38	1.35848
$358$	0	0
$359$	−2069.88	−0.304301	−0.152151	−	0.988357i	$-0.548620\pi$
$359$	−2069.88	−0.304301	−0.152151	+	0.988357i	$0.548620\pi$
$360$	0	0
$361$	−4664.16	−0.680005
$362$	0	0
$363$	2887.74	0.417540
$364$	0	0
$365$	−4294.91	−0.615906
$366$	0	0
$367$	−7299.16	−1.03818	−0.519092	−	0.854719i	$-0.673730\pi$
$367$	−7299.16	−1.03818	−0.519092	+	0.854719i	$0.673730\pi$
$368$	0	0
$369$	2083.41	0.293924
$370$	0	0
$371$	26686.3	3.73446
$372$	0	0
$373$	8964.32	1.24438	0.622192	−	0.782865i	$-0.286242\pi$
$373$	8964.32	1.24438	0.622192	+	0.782865i	$0.286242\pi$
$374$	0	0
$375$	2751.80	0.378940
$376$	0	0
$377$	1408.43	0.192408
$378$	0	0
$379$	4399.26	0.596239	0.298120	−	0.954529i	$-0.403641\pi$
$379$	4399.26	0.596239	0.298120	+	0.954529i	$0.403641\pi$
$380$	0	0
$381$	−828.337	−0.111383
$382$	0	0
$383$	3529.74	0.470917	0.235459	−	0.971884i	$-0.424341\pi$
$383$	3529.74	0.470917	0.235459	+	0.971884i	$0.424341\pi$
$384$	0	0
$385$	−2731.21	−0.361547
$386$	0	0
$387$	3062.82	0.402305
$388$	0	0
$389$	3034.77	0.395549	0.197775	−	0.980248i	$-0.436629\pi$
$389$	3034.77	0.395549	0.197775	+	0.980248i	$0.436629\pi$
$390$	0	0
$391$	−8715.93	−1.12732
$392$	0	0
$393$	−288.672	−0.0370523
$394$	0	0
$395$	548.690	0.0698927
$396$	0	0
$397$	3997.36	0.505344	0.252672	−	0.967552i	$-0.418691\pi$
$397$	3997.36	0.505344	0.252672	+	0.967552i	$0.418691\pi$
$398$	0	0
$399$	5117.75	0.642125
$400$	0	0
$401$	−9092.88	−1.13236	−0.566181	−	0.824281i	$-0.691580\pi$
$401$	−9092.88	−1.13236	−0.566181	+	0.824281i	$0.691580\pi$
$402$	0	0
$403$	1917.92	0.237067
$404$	0	0
$405$	316.529	0.0388357
$406$	0	0
$407$	−3071.32	−0.374053
$408$	0	0
$409$	7143.54	0.863631	0.431816	−	0.901962i	$-0.357873\pi$
$409$	7143.54	0.863631	0.431816	+	0.901962i	$0.357873\pi$
$410$	0	0
$411$	−7855.13	−0.942738
$412$	0	0
$413$	8367.04	0.996889
$414$	0	0
$415$	623.187	0.0737134
$416$	0	0
$417$	5890.02	0.691692
$418$	0	0
$419$	−8213.84	−0.957691	−0.478845	−	0.877899i	$-0.658945\pi$
$419$	−8213.84	−0.957691	−0.478845	+	0.877899i	$0.658945\pi$
$420$	0	0
$421$	7997.40	0.925818	0.462909	−	0.886406i	$-0.346806\pi$
$421$	7997.40	0.925818	0.462909	+	0.886406i	$0.346806\pi$
$422$	0	0
$423$	1076.88	0.123782
$424$	0	0
$425$	9204.53	1.05055
$426$	0	0
$427$	−3966.94	−0.449587
$428$	0	0
$429$	−748.577	−0.0842462
$430$	0	0
$431$	−13694.8	−1.53053	−0.765263	−	0.643718i	$-0.777391\pi$
$431$	−13694.8	−1.53053	−0.765263	+	0.643718i	$0.777391\pi$
$432$	0	0
$433$	6716.57	0.745445	0.372722	−	0.927943i	$-0.378424\pi$
$433$	6716.57	0.745445	0.372722	+	0.927943i	$0.378424\pi$
$434$	0	0
$435$	1270.11	0.139993
$436$	0	0
$437$	−4867.84	−0.532862
$438$	0	0
$439$	−5933.32	−0.645061	−0.322531	−	0.946559i	$-0.604534\pi$
$439$	−5933.32	−0.645061	−0.322531	+	0.946559i	$0.604534\pi$
$440$	0	0
$441$	8846.12	0.955201
$442$	0	0
$443$	6923.40	0.742530	0.371265	−	0.928527i	$-0.378924\pi$
$443$	6923.40	0.742530	0.371265	+	0.928527i	$0.378924\pi$
$444$	0	0
$445$	4173.23	0.444562
$446$	0	0
$447$	−903.333	−0.0955843
$448$	0	0
$449$	8886.78	0.934061	0.467030	−	0.884241i	$-0.345324\pi$
$449$	8886.78	0.934061	0.467030	+	0.884241i	$0.345324\pi$
$450$	0	0
$451$	−4443.28	−0.463916
$452$	0	0
$453$	−1028.92	−0.106717
$454$	0	0
$455$	−1849.81	−0.190594
$456$	0	0
$457$	10965.0	1.12237	0.561184	−	0.827691i	$-0.310346\pi$
$457$	10965.0	1.12237	0.561184	+	0.827691i	$0.310346\pi$
$458$	0	0
$459$	2264.87	0.230316
$460$	0	0
$461$	−10069.2	−1.01729	−0.508644	−	0.860977i	$-0.669853\pi$
$461$	−10069.2	−1.01729	−0.508644	+	0.860977i	$0.669853\pi$
$462$	0	0
$463$	5599.72	0.562076	0.281038	−	0.959697i	$-0.409321\pi$
$463$	5599.72	0.562076	0.281038	+	0.959697i	$0.409321\pi$
$464$	0	0
$465$	1729.56	0.172487
$466$	0	0
$467$	13247.8	1.31271	0.656355	−	0.754452i	$-0.272097\pi$
$467$	13247.8	1.31271	0.656355	+	0.754452i	$0.272097\pi$
$468$	0	0
$469$	−378.532	−0.0372686
$470$	0	0
$471$	−3860.90	−0.377709
$472$	0	0
$473$	−6532.08	−0.634979
$474$	0	0
$475$	5140.73	0.496575
$476$	0	0
$477$	6595.92	0.633137
$478$	0	0
$479$	16725.4	1.59541	0.797707	−	0.603045i	$-0.206046\pi$
$479$	16725.4	1.59541	0.797707	+	0.603045i	$0.206046\pi$
$480$	0	0
$481$	−2080.16	−0.197187
$482$	0	0
$483$	−11350.4	−1.06928
$484$	0	0
$485$	3356.30	0.314231
$486$	0	0
$487$	−5305.86	−0.493699	−0.246850	−	0.969054i	$-0.579395\pi$
$487$	−5305.86	−0.493699	−0.246850	+	0.969054i	$0.579395\pi$
$488$	0	0
$489$	1597.68	0.147750
$490$	0	0
$491$	−16200.2	−1.48901	−0.744506	−	0.667616i	$-0.767315\pi$
$491$	−16200.2	−1.48901	−0.744506	+	0.667616i	$0.767315\pi$
$492$	0	0
$493$	9088.05	0.830234
$494$	0	0
$495$	−675.060	−0.0612963
$496$	0	0
$497$	−31650.2	−2.85655
$498$	0	0
$499$	−4392.70	−0.394076	−0.197038	−	0.980396i	$-0.563132\pi$
$499$	−4392.70	−0.394076	−0.197038	+	0.980396i	$0.563132\pi$
$500$	0	0
$501$	−125.863	−0.0112238
$502$	0	0
$503$	−14955.2	−1.32568	−0.662841	−	0.748760i	$-0.730650\pi$
$503$	−14955.2	−1.32568	−0.662841	+	0.748760i	$0.730650\pi$
$504$	0	0
$505$	6151.46	0.542052
$506$	0	0
$507$	−507.000	−0.0444116
$508$	0	0
$509$	−13403.4	−1.16719	−0.583593	−	0.812047i	$-0.698353\pi$
$509$	−13403.4	−1.16719	−0.583593	+	0.812047i	$0.698353\pi$
$510$	0	0
$511$	−40020.4	−3.46458
$512$	0	0
$513$	1264.93	0.108865
$514$	0	0
$515$	−506.930	−0.0433748
$516$	0	0
$517$	−2296.66	−0.195371
$518$	0	0
$519$	−3199.49	−0.270601
$520$	0	0
$521$	19643.0	1.65178	0.825888	−	0.563834i	$-0.190674\pi$
$521$	19643.0	1.65178	0.825888	+	0.563834i	$0.190674\pi$
$522$	0	0
$523$	−14657.4	−1.22548	−0.612738	−	0.790286i	$-0.709932\pi$
$523$	−14657.4	−1.22548	−0.612738	+	0.790286i	$0.709932\pi$
$524$	0	0
$525$	11986.7	0.996462
$526$	0	0
$527$	12375.6	1.02294
$528$	0	0
$529$	−1370.83	−0.112668
$530$	0	0
$531$	2068.04	0.169012
$532$	0	0
$533$	−3009.37	−0.244560
$534$	0	0
$535$	7649.19	0.618137
$536$	0	0
$537$	9523.82	0.765332
$538$	0	0
$539$	−18866.1	−1.50764
$540$	0	0
$541$	−13921.3	−1.10633	−0.553164	−	0.833072i	$-0.686580\pi$
$541$	−13921.3	−1.10633	−0.553164	+	0.833072i	$0.686580\pi$
$542$	0	0
$543$	−8176.30	−0.646185
$544$	0	0
$545$	−4801.75	−0.377403
$546$	0	0
$547$	2324.11	0.181667	0.0908335	−	0.995866i	$-0.471047\pi$
$547$	2324.11	0.181667	0.0908335	+	0.995866i	$0.471047\pi$
$548$	0	0
$549$	−980.488	−0.0762226
$550$	0	0
$551$	5075.68	0.392434
$552$	0	0
$553$	5112.75	0.393158
$554$	0	0
$555$	−1875.87	−0.143471
$556$	0	0
$557$	16962.8	1.29037	0.645185	−	0.764027i	$-0.276780\pi$
$557$	16962.8	1.29037	0.645185	+	0.764027i	$0.276780\pi$
$558$	0	0
$559$	−4424.08	−0.334738
$560$	0	0
$561$	−4830.27	−0.363519
$562$	0	0
$563$	389.000	0.0291197	0.0145599	−	0.999894i	$-0.495365\pi$
$563$	389.000	0.0291197	0.0145599	+	0.999894i	$0.495365\pi$
$564$	0	0
$565$	6367.69	0.474143
$566$	0	0
$567$	2949.45	0.218457
$568$	0	0
$569$	2217.56	0.163383	0.0816914	−	0.996658i	$-0.473968\pi$
$569$	2217.56	0.163383	0.0816914	+	0.996658i	$0.473968\pi$
$570$	0	0
$571$	17087.3	1.25233	0.626167	−	0.779689i	$-0.284623\pi$
$571$	17087.3	1.25233	0.626167	+	0.779689i	$0.284623\pi$
$572$	0	0
$573$	−2354.67	−0.171671
$574$	0	0
$575$	−11401.4	−0.826906
$576$	0	0
$577$	−3977.26	−0.286959	−0.143480	−	0.989653i	$-0.545829\pi$
$577$	−3977.26	−0.286959	−0.143480	+	0.989653i	$0.545829\pi$
$578$	0	0
$579$	3767.61	0.270425
$580$	0	0
$581$	5806.92	0.414650
$582$	0	0
$583$	−14067.1	−0.999312
$584$	0	0
$585$	−457.208	−0.0323132
$586$	0	0
$587$	16880.3	1.18693	0.593463	−	0.804861i	$-0.297760\pi$
$587$	16880.3	1.18693	0.593463	+	0.804861i	$0.297760\pi$
$588$	0	0
$589$	6911.75	0.483521
$590$	0	0
$591$	−8332.06	−0.579924
$592$	0	0
$593$	2423.25	0.167810	0.0839048	−	0.996474i	$-0.473261\pi$
$593$	2423.25	0.167810	0.0839048	+	0.996474i	$0.473261\pi$
$594$	0	0
$595$	−11936.1	−0.822408
$596$	0	0
$597$	−4471.29	−0.306529
$598$	0	0
$599$	−3900.55	−0.266064	−0.133032	−	0.991112i	$-0.542471\pi$
$599$	−3900.55	−0.266064	−0.133032	+	0.991112i	$0.542471\pi$
$600$	0	0
$601$	28653.4	1.94476	0.972378	−	0.233413i	$-0.0749893\pi$
$601$	28653.4	1.94476	0.972378	+	0.233413i	$0.0749893\pi$
$602$	0	0
$603$	−93.5599	−0.00631850
$604$	0	0
$605$	−3761.54	−0.252774
$606$	0	0
$607$	214.736	0.0143589	0.00717946	−	0.999974i	$-0.497715\pi$
$607$	214.736	0.0143589	0.00717946	+	0.999974i	$0.497715\pi$
$608$	0	0
$609$	11835.0	0.787487
$610$	0	0
$611$	−1555.49	−0.102993
$612$	0	0
$613$	−26438.5	−1.74199	−0.870996	−	0.491290i	$-0.836525\pi$
$613$	−26438.5	−1.74199	−0.870996	+	0.491290i	$0.836525\pi$
$614$	0	0
$615$	−2713.82	−0.177938
$616$	0	0
$617$	6700.96	0.437229	0.218615	−	0.975811i	$-0.429846\pi$
$617$	6700.96	0.437229	0.218615	+	0.975811i	$0.429846\pi$
$618$	0	0
$619$	27319.1	1.77391	0.886953	−	0.461860i	$-0.152818\pi$
$619$	27319.1	1.77391	0.886953	+	0.461860i	$0.152818\pi$
$620$	0	0
$621$	−2805.42	−0.181285
$622$	0	0
$623$	38886.6	2.50074
$624$	0	0
$625$	10131.7	0.648429
$626$	0	0
$627$	−2697.71	−0.171828
$628$	0	0
$629$	−13422.4	−0.850855
$630$	0	0
$631$	−7126.87	−0.449629	−0.224815	−	0.974402i	$-0.572178\pi$
$631$	−7126.87	−0.449629	−0.224815	+	0.974402i	$0.572178\pi$
$632$	0	0
$633$	6916.88	0.434315
$634$	0	0
$635$	1078.98	0.0674301
$636$	0	0
$637$	−12777.7	−0.794775
$638$	0	0
$639$	−7822.81	−0.484297
$640$	0	0
$641$	23615.0	1.45513	0.727565	−	0.686039i	$-0.240652\pi$
$641$	23615.0	1.45513	0.727565	+	0.686039i	$0.240652\pi$
$642$	0	0
$643$	8144.41	0.499509	0.249755	−	0.968309i	$-0.419650\pi$
$643$	8144.41	0.499509	0.249755	+	0.968309i	$0.419650\pi$
$644$	0	0
$645$	−3989.60	−0.243551
$646$	0	0
$647$	9682.00	0.588313	0.294157	−	0.955757i	$-0.404961\pi$
$647$	9682.00	0.588313	0.294157	+	0.955757i	$0.404961\pi$
$648$	0	0
$649$	−4410.50	−0.266760
$650$	0	0
$651$	16116.2	0.970268
$652$	0	0
$653$	18193.6	1.09030	0.545152	−	0.838337i	$-0.316472\pi$
$653$	18193.6	1.09030	0.545152	+	0.838337i	$0.316472\pi$
$654$	0	0
$655$	376.020	0.0224310
$656$	0	0
$657$	−9891.64	−0.587381
$658$	0	0
$659$	−9300.88	−0.549789	−0.274895	−	0.961474i	$-0.588643\pi$
$659$	−9300.88	−0.549789	−0.274895	+	0.961474i	$0.588643\pi$
$660$	0	0
$661$	5437.29	0.319949	0.159974	−	0.987121i	$-0.448859\pi$
$661$	5437.29	0.319949	0.159974	+	0.987121i	$0.448859\pi$
$662$	0	0
$663$	−3271.47	−0.191634
$664$	0	0
$665$	−6666.32	−0.388735
$666$	0	0
$667$	−11257.1	−0.653489
$668$	0	0
$669$	−3725.94	−0.215326
$670$	0	0
$671$	2091.08	0.120306
$672$	0	0
$673$	−8682.75	−0.497319	−0.248659	−	0.968591i	$-0.579990\pi$
$673$	−8682.75	−0.497319	−0.248659	+	0.968591i	$0.579990\pi$
$674$	0	0
$675$	2962.69	0.168939
$676$	0	0
$677$	−13300.1	−0.755041	−0.377521	−	0.926001i	$-0.623223\pi$
$677$	−13300.1	−0.755041	−0.377521	+	0.926001i	$0.623223\pi$
$678$	0	0
$679$	31274.4	1.76760
$680$	0	0
$681$	−5174.27	−0.291158
$682$	0	0
$683$	504.175	0.0282455	0.0141228	−	0.999900i	$-0.495504\pi$
$683$	504.175	0.0282455	0.0141228	+	0.999900i	$0.495504\pi$
$684$	0	0
$685$	10232.0	0.570722
$686$	0	0
$687$	−9821.16	−0.545416
$688$	0	0
$689$	−9527.43	−0.526802
$690$	0	0
$691$	−13443.8	−0.740124	−0.370062	−	0.929007i	$-0.620664\pi$
$691$	−13443.8	−0.740124	−0.370062	+	0.929007i	$0.620664\pi$
$692$	0	0
$693$	−6290.28	−0.344802
$694$	0	0
$695$	−7672.27	−0.418742
$696$	0	0
$697$	−19418.3	−1.05527
$698$	0	0
$699$	6388.55	0.345690
$700$	0	0
$701$	−28735.6	−1.54826	−0.774128	−	0.633030i	$-0.781811\pi$
$701$	−28735.6	−1.54826	−0.774128	+	0.633030i	$0.781811\pi$
$702$	0	0
$703$	−7496.44	−0.402181
$704$	0	0
$705$	−1402.73	−0.0749361
$706$	0	0
$707$	57319.9	3.04913
$708$	0	0
$709$	−17610.2	−0.932812	−0.466406	−	0.884571i	$-0.654451\pi$
$709$	−17610.2	−0.932812	−0.466406	+	0.884571i	$0.654451\pi$
$710$	0	0
$711$	1263.69	0.0666557
$712$	0	0
$713$	−15329.3	−0.805168
$714$	0	0
$715$	975.087	0.0510016
$716$	0	0
$717$	15247.1	0.794163
$718$	0	0
$719$	−9226.04	−0.478544	−0.239272	−	0.970953i	$-0.576909\pi$
$719$	−9226.04	−0.478544	−0.239272	+	0.970953i	$0.576909\pi$
$720$	0	0
$721$	−4723.63	−0.243990
$722$	0	0
$723$	−14296.9	−0.735420
$724$	0	0
$725$	11888.2	0.608987
$726$	0	0
$727$	33246.0	1.69604	0.848022	−	0.529961i	$-0.177793\pi$
$727$	33246.0	1.69604	0.848022	+	0.529961i	$0.177793\pi$
$728$	0	0
$729$	729.000	0.0370370
$730$	0	0
$731$	−28546.9	−1.44438
$732$	0	0
$733$	4423.26	0.222888	0.111444	−	0.993771i	$-0.464452\pi$
$733$	4423.26	0.222888	0.111444	+	0.993771i	$0.464452\pi$
$734$	0	0
$735$	−11522.8	−0.578267
$736$	0	0
$737$	199.535	0.00997281
$738$	0	0
$739$	4529.56	0.225470	0.112735	−	0.993625i	$-0.464039\pi$
$739$	4529.56	0.225470	0.112735	+	0.993625i	$0.464039\pi$
$740$	0	0
$741$	−1827.12	−0.0905814
$742$	0	0
$743$	10851.5	0.535803	0.267901	−	0.963446i	$-0.413670\pi$
$743$	10851.5	0.535803	0.267901	+	0.963446i	$0.413670\pi$
$744$	0	0
$745$	1176.67	0.0578656
$746$	0	0
$747$	1435.27	0.0702994
$748$	0	0
$749$	71275.9	3.47712
$750$	0	0
$751$	−33022.6	−1.60454	−0.802272	−	0.596958i	$-0.796376\pi$
$751$	−33022.6	−1.60454	−0.802272	+	0.596958i	$0.796376\pi$
$752$	0	0
$753$	−13019.9	−0.630107
$754$	0	0
$755$	1340.26	0.0646053
$756$	0	0
$757$	3443.77	0.165345	0.0826724	−	0.996577i	$-0.473654\pi$
$757$	3443.77	0.165345	0.0826724	+	0.996577i	$0.473654\pi$
$758$	0	0
$759$	5983.12	0.286131
$760$	0	0
$761$	19562.6	0.931858	0.465929	−	0.884822i	$-0.345720\pi$
$761$	19562.6	0.931858	0.465929	+	0.884822i	$0.345720\pi$
$762$	0	0
$763$	−44743.2	−2.12295
$764$	0	0
$765$	−2950.19	−0.139430
$766$	0	0
$767$	−2987.17	−0.140626
$768$	0	0
$769$	−17061.1	−0.800049	−0.400025	−	0.916504i	$-0.630998\pi$
$769$	−17061.1	−0.800049	−0.400025	+	0.916504i	$0.630998\pi$
$770$	0	0
$771$	−13078.5	−0.610908
$772$	0	0
$773$	−10798.4	−0.502448	−0.251224	−	0.967929i	$-0.580833\pi$
$773$	−10798.4	−0.502448	−0.251224	+	0.967929i	$0.580833\pi$
$774$	0	0
$775$	16188.6	0.750337
$776$	0	0
$777$	−17479.5	−0.807046
$778$	0	0
$779$	−10845.1	−0.498802
$780$	0	0
$781$	16683.7	0.764391
$782$	0	0
$783$	2925.20	0.133510
$784$	0	0
$785$	5029.16	0.228661
$786$	0	0
$787$	35607.0	1.61277	0.806386	−	0.591390i	$-0.201420\pi$
$787$	35607.0	1.61277	0.806386	+	0.591390i	$0.201420\pi$
$788$	0	0
$789$	−1824.23	−0.0823122
$790$	0	0
$791$	59334.8	2.66713
$792$	0	0
$793$	1416.26	0.0634210
$794$	0	0
$795$	−8591.76	−0.383293
$796$	0	0
$797$	−22155.3	−0.984668	−0.492334	−	0.870406i	$-0.663856\pi$
$797$	−22155.3	−0.984668	−0.492334	+	0.870406i	$0.663856\pi$
$798$	0	0
$799$	−10037.0	−0.444410
$800$	0	0
$801$	9611.40	0.423973
$802$	0	0
$803$	21095.9	0.927094
$804$	0	0
$805$	14784.9	0.647329
$806$	0	0
$807$	10362.9	0.452032
$808$	0	0
$809$	−22524.6	−0.978889	−0.489445	−	0.872034i	$-0.662800\pi$
$809$	−22524.6	−0.978889	−0.489445	+	0.872034i	$0.662800\pi$
$810$	0	0
$811$	−4452.39	−0.192780	−0.0963900	−	0.995344i	$-0.530730\pi$
$811$	−4452.39	−0.192780	−0.0963900	+	0.995344i	$0.530730\pi$
$812$	0	0
$813$	−11111.2	−0.479318
$814$	0	0
$815$	−2081.12	−0.0894460
$816$	0	0
$817$	−15943.4	−0.682729
$818$	0	0
$819$	−4260.31	−0.181767
$820$	0	0
$821$	7097.27	0.301701	0.150851	−	0.988557i	$-0.451799\pi$
$821$	7097.27	0.301701	0.150851	+	0.988557i	$0.451799\pi$
$822$	0	0
$823$	−12193.8	−0.516463	−0.258231	−	0.966083i	$-0.583140\pi$
$823$	−12193.8	−0.516463	−0.258231	+	0.966083i	$0.583140\pi$
$824$	0	0
$825$	−6318.53	−0.266646
$826$	0	0
$827$	−7427.97	−0.312329	−0.156164	−	0.987731i	$-0.549913\pi$
$827$	−7427.97	−0.312329	−0.156164	+	0.987731i	$0.549913\pi$
$828$	0	0
$829$	−16966.2	−0.710810	−0.355405	−	0.934712i	$-0.615657\pi$
$829$	−16966.2	−0.710810	−0.355405	+	0.934712i	$0.615657\pi$
$830$	0	0
$831$	−10700.7	−0.446693
$832$	0	0
$833$	−82449.7	−3.42943
$834$	0	0
$835$	163.947	0.00679477
$836$	0	0
$837$	3983.36	0.164498
$838$	0	0
$839$	12025.7	0.494844	0.247422	−	0.968908i	$-0.420417\pi$
$839$	12025.7	0.494844	0.247422	+	0.968908i	$0.420417\pi$
$840$	0	0
$841$	−12651.3	−0.518728
$842$	0	0
$843$	352.422	0.0143986
$844$	0	0
$845$	660.412	0.0268862
$846$	0	0
$847$	−35050.4	−1.42189
$848$	0	0
$849$	−5212.85	−0.210724
$850$	0	0
$851$	16626.0	0.669720
$852$	0	0
$853$	22187.2	0.890593	0.445297	−	0.895383i	$-0.353098\pi$
$853$	22187.2	0.890593	0.445297	+	0.895383i	$0.353098\pi$
$854$	0	0
$855$	−1647.68	−0.0659058
$856$	0	0
$857$	5746.19	0.229038	0.114519	−	0.993421i	$-0.463467\pi$
$857$	5746.19	0.229038	0.114519	+	0.993421i	$0.463467\pi$
$858$	0	0
$859$	−8305.66	−0.329902	−0.164951	−	0.986302i	$-0.552747\pi$
$859$	−8305.66	−0.329902	−0.164951	+	0.986302i	$0.552747\pi$
$860$	0	0
$861$	−25287.7	−1.00093
$862$	0	0
$863$	38086.4	1.50229	0.751146	−	0.660137i	$-0.229502\pi$
$863$	38086.4	1.50229	0.751146	+	0.660137i	$0.229502\pi$
$864$	0	0
$865$	4167.61	0.163819
$866$	0	0
$867$	−6370.53	−0.249544
$868$	0	0
$869$	−2695.07	−0.105206
$870$	0	0
$871$	135.142	0.00525731
$872$	0	0
$873$	7729.93	0.299677
$874$	0	0
$875$	−33400.4	−1.29044
$876$	0	0
$877$	2098.53	0.0808009	0.0404005	−	0.999184i	$-0.487137\pi$
$877$	2098.53	0.0808009	0.0404005	+	0.999184i	$0.487137\pi$
$878$	0	0
$879$	5712.14	0.219187
$880$	0	0
$881$	14555.3	0.556619	0.278309	−	0.960491i	$-0.410226\pi$
$881$	14555.3	0.556619	0.278309	+	0.960491i	$0.410226\pi$
$882$	0	0
$883$	−2122.88	−0.0809066	−0.0404533	−	0.999181i	$-0.512880\pi$
$883$	−2122.88	−0.0809066	−0.0404533	+	0.999181i	$0.512880\pi$
$884$	0	0
$885$	−2693.80	−0.102318
$886$	0	0
$887$	−12487.3	−0.472696	−0.236348	−	0.971668i	$-0.575951\pi$
$887$	−12487.3	−0.472696	−0.236348	+	0.971668i	$0.575951\pi$
$888$	0	0
$889$	10054.1	0.379305
$890$	0	0
$891$	−1554.74	−0.0584575
$892$	0	0
$893$	−5605.66	−0.210063
$894$	0	0
$895$	−12405.6	−0.463323
$896$	0	0
$897$	4052.28	0.150838
$898$	0	0
$899$	15983.7	0.592978
$900$	0	0
$901$	−61476.8	−2.27313
$902$	0	0
$903$	−37175.5	−1.37001
$904$	0	0
$905$	10650.3	0.391193
$906$	0	0
$907$	−29679.8	−1.08655	−0.543275	−	0.839555i	$-0.682816\pi$
$907$	−29679.8	−1.08655	−0.543275	+	0.839555i	$0.682816\pi$
$908$	0	0
$909$	14167.5	0.516948
$910$	0	0
$911$	24800.0	0.901934	0.450967	−	0.892541i	$-0.351079\pi$
$911$	24800.0	0.901934	0.450967	+	0.892541i	$0.351079\pi$
$912$	0	0
$913$	−3060.99	−0.110957
$914$	0	0
$915$	1277.17	0.0461442
$916$	0	0
$917$	3503.80	0.126178
$918$	0	0
$919$	−6597.90	−0.236828	−0.118414	−	0.992964i	$-0.537781\pi$
$919$	−6597.90	−0.236828	−0.118414	+	0.992964i	$0.537781\pi$
$920$	0	0
$921$	8587.17	0.307228
$922$	0	0
$923$	11299.6	0.402959
$924$	0	0
$925$	−17558.0	−0.624113
$926$	0	0
$927$	−1167.51	−0.0413659
$928$	0	0
$929$	15056.0	0.531724	0.265862	−	0.964011i	$-0.414344\pi$
$929$	15056.0	0.531724	0.265862	+	0.964011i	$0.414344\pi$
$930$	0	0
$931$	−46048.1	−1.62102
$932$	0	0
$933$	−12604.7	−0.442291
$934$	0	0
$935$	6291.85	0.220070
$936$	0	0
$937$	−35777.0	−1.24737	−0.623683	−	0.781677i	$-0.714365\pi$
$937$	−35777.0	−1.24737	−0.623683	+	0.781677i	$0.714365\pi$
$938$	0	0
$939$	−10283.2	−0.357381
$940$	0	0
$941$	22973.6	0.795873	0.397937	−	0.917413i	$-0.369726\pi$
$941$	22973.6	0.795873	0.397937	+	0.917413i	$0.369726\pi$
$942$	0	0
$943$	24052.9	0.830615
$944$	0	0
$945$	−3841.92	−0.132251
$946$	0	0
$947$	51038.3	1.75134	0.875671	−	0.482908i	$-0.160420\pi$
$947$	51038.3	1.75134	0.875671	+	0.482908i	$0.160420\pi$
$948$	0	0
$949$	14287.9	0.488731
$950$	0	0
$951$	5028.26	0.171454
$952$	0	0
$953$	−22586.5	−0.767733	−0.383866	−	0.923389i	$-0.625408\pi$
$953$	−22586.5	−0.767733	−0.383866	+	0.923389i	$0.625408\pi$
$954$	0	0
$955$	3067.16	0.103928
$956$	0	0
$957$	−6238.57	−0.210726
$958$	0	0
$959$	95342.8	3.21041
$960$	0	0
$961$	−8025.32	−0.269387
$962$	0	0
$963$	17616.9	0.589509
$964$	0	0
$965$	−4907.64	−0.163712
$966$	0	0
$967$	−36678.7	−1.21976	−0.609880	−	0.792494i	$-0.708782\pi$
$967$	−36678.7	−1.21976	−0.609880	+	0.792494i	$0.708782\pi$
$968$	0	0
$969$	−11789.7	−0.390855
$970$	0	0
$971$	−32635.2	−1.07859	−0.539296	−	0.842116i	$-0.681310\pi$
$971$	−32635.2	−1.07859	−0.539296	+	0.842116i	$0.681310\pi$
$972$	0	0
$973$	−71491.0	−2.35549
$974$	0	0
$975$	−4279.45	−0.140566
$976$	0	0
$977$	44432.6	1.45499	0.727496	−	0.686112i	$-0.240684\pi$
$977$	44432.6	1.45499	0.727496	+	0.686112i	$0.240684\pi$
$978$	0	0
$979$	−20498.2	−0.669178
$980$	0	0
$981$	−11059.0	−0.359924
$982$	0	0
$983$	484.485	0.0157199	0.00785996	−	0.999969i	$-0.497498\pi$
$983$	484.485	0.0157199	0.00785996	+	0.999969i	$0.497498\pi$
$984$	0	0
$985$	10853.2	0.351079
$986$	0	0
$987$	−13070.8	−0.421528
$988$	0	0
$989$	35360.2	1.13689
$990$	0	0
$991$	−48017.1	−1.53917	−0.769583	−	0.638546i	$-0.779536\pi$
$991$	−48017.1	−1.53917	−0.769583	+	0.638546i	$0.779536\pi$
$992$	0	0
$993$	33469.8	1.06962
$994$	0	0
$995$	5824.25	0.185569
$996$	0	0
$997$	26561.9	0.843755	0.421877	−	0.906653i	$-0.361371\pi$
$997$	26561.9	0.843755	0.421877	+	0.906653i	$0.361371\pi$
$998$	0	0
$999$	−4320.33	−0.136826

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	2496.4.a.bl.1.2		3
4.3	odd	2		2496.4.a.bp.1.2		3
8.3	odd	2		624.4.a.t.1.2		3
8.5	even	2		39.4.a.c.1.1	✓	3
24.5	odd	2		117.4.a.f.1.3		3
24.11	even	2		1872.4.a.bk.1.2		3
40.29	even	2		975.4.a.l.1.3		3
56.13	odd	2		1911.4.a.k.1.1		3
104.5	odd	4		507.4.b.g.337.5		6
104.21	odd	4		507.4.b.g.337.2		6
104.77	even	2		507.4.a.h.1.3		3
312.77	odd	2		1521.4.a.u.1.1		3

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
39.4.a.c.1.1	✓	3	8.5	even	2
117.4.a.f.1.3		3	24.5	odd	2
507.4.a.h.1.3		3	104.77	even	2
507.4.b.g.337.2		6	104.21	odd	4
507.4.b.g.337.5		6	104.5	odd	4
624.4.a.t.1.2		3	8.3	odd	2
975.4.a.l.1.3		3	40.29	even	2
1521.4.a.u.1.1		3	312.77	odd	2
1872.4.a.bk.1.2		3	24.11	even	2
1911.4.a.k.1.1		3	56.13	odd	2
2496.4.a.bl.1.2		3	1.1	even	1	trivial
2496.4.a.bp.1.2		3	4.3	odd	2

Overview	Random
Universe	Knowledge

Classical	Maass
Hilbert	Bianchi

Elliptic curves over $\Q$
Elliptic curves over $\Q(\alpha)$
Genus 2 curves over $\Q$
Higher genus families
Abelian varieties over $\F_{q}$

Level:	\( N \)	\(=\)	\( 2496 = 2^{6} \cdot 3 \cdot 13 \)
Weight:	\( k \)	\(=\)	\( 4 \)
Character orbit:	\([\chi]\)	\(=\)	2496.a (trivial)

\(f(q)\)	\(=\)	\(q-3.00000 q^{3} +3.90776 q^{5} +36.4129 q^{7} +9.00000 q^{9} +O(q^{10})\)	\(q-3.00000 q^{3} +3.90776 q^{5} +36.4129 q^{7} +9.00000 q^{9} -19.1943 q^{11} -13.0000 q^{13} -11.7233 q^{15} -83.8839 q^{17} -46.8492 q^{19} -109.239 q^{21} +103.905 q^{23} -109.729 q^{25} -27.0000 q^{27} -108.341 q^{29} -147.532 q^{31} +57.5828 q^{33} +142.293 q^{35} +160.012 q^{37} +39.0000 q^{39} +231.490 q^{41} +340.314 q^{43} +35.1699 q^{45} +119.653 q^{47} +982.902 q^{49} +251.652 q^{51} +732.879 q^{53} -75.0067 q^{55} +140.548 q^{57} +229.782 q^{59} -108.943 q^{61} +327.716 q^{63} -50.8009 q^{65} -10.3955 q^{67} -311.714 q^{69} -869.201 q^{71} -1099.07 q^{73} +329.188 q^{75} -698.920 q^{77} +140.410 q^{79} +81.0000 q^{81} +159.474 q^{83} -327.799 q^{85} +325.023 q^{87} +1067.93 q^{89} -473.368 q^{91} +442.596 q^{93} -183.075 q^{95} +858.881 q^{97} -172.748 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 3 q - 9 q^{3} - 4 q^{5} + 30 q^{7} + 27 q^{9}+O(q^{10}) \) 3 * q - 9 * q^3 - 4 * q^5 + 30 * q^7 + 27 * q^9	\( 3 q - 9 q^{3} - 4 q^{5} + 30 q^{7} + 27 q^{9} + 16 q^{11} - 39 q^{13} + 12 q^{15} - 146 q^{17} - 94 q^{19} - 90 q^{21} - 48 q^{23} + 145 q^{25} - 81 q^{27} + 2 q^{29} + 302 q^{31} - 48 q^{33} - 80 q^{35} - 374 q^{37} + 117 q^{39} + 480 q^{41} + 260 q^{43} - 36 q^{45} - 24 q^{47} + 447 q^{49} + 438 q^{51} + 678 q^{53} - 1552 q^{55} + 282 q^{57} + 1788 q^{59} - 230 q^{61} + 270 q^{63} + 52 q^{65} - 74 q^{67} + 144 q^{69} - 948 q^{71} - 222 q^{73} - 435 q^{75} - 112 q^{77} - 24 q^{79} + 243 q^{81} + 796 q^{83} + 248 q^{85} - 6 q^{87} + 1436 q^{89} - 390 q^{91} - 906 q^{93} - 4032 q^{95} + 3242 q^{97} + 144 q^{99}+O(q^{100}) \) 3 * q - 9 * q^3 - 4 * q^5 + 30 * q^7 + 27 * q^9 + 16 * q^11 - 39 * q^13 + 12 * q^15 - 146 * q^17 - 94 * q^19 - 90 * q^21 - 48 * q^23 + 145 * q^25 - 81 * q^27 + 2 * q^29 + 302 * q^31 - 48 * q^33 - 80 * q^35 - 374 * q^37 + 117 * q^39 + 480 * q^41 + 260 * q^43 - 36 * q^45 - 24 * q^47 + 447 * q^49 + 438 * q^51 + 678 * q^53 - 1552 * q^55 + 282 * q^57 + 1788 * q^59 - 230 * q^61 + 270 * q^63 + 52 * q^65 - 74 * q^67 + 144 * q^69 - 948 * q^71 - 222 * q^73 - 435 * q^75 - 112 * q^77 - 24 * q^79 + 243 * q^81 + 796 * q^83 + 248 * q^85 - 6 * q^87 + 1436 * q^89 - 390 * q^91 - 906 * q^93 - 4032 * q^95 + 3242 * q^97 + 144 * q^99

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