LMFDB - Embedded newform 208.4.a.l.1.1

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [208,4,Mod(1,208)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(208, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([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("208.1");

S:= CuspForms(chi, 4);

N := Newforms(S);

Level:	$ N $	$=$	$ 208 = 2^{4} \cdot 13 $
Weight:	$ k $	$=$	$ 4 $
Character orbit:	$[\chi]$	$=$	208.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:	$12.2723972812$
Analytic rank:	$1$
Dimension:	$3$
Coefficient field:	3.3.18257.1
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{3} - x^{2} - 26x + 8 $ x^3 - x^2 - 26*x + 8
Coefficient ring:	$\Z[a_1, \ldots, a_{5}]$
Coefficient ring index:	$ 2 $
Twist minimal:	no (minimal twist has level 104)
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.1
Root			$0.305203$ of defining polynomial
Character	$\chi$	$=$	208.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-8.80082 q^{3} +6.19042 q^{5} -3.19918 q^{7} +50.4545 q^{9} +O(q^{10})$	$q-8.80082 q^{3} +6.19042 q^{5} -3.19918 q^{7} +50.4545 q^{9} +35.2466 q^{11} +13.0000 q^{13} -54.4808 q^{15} -99.5410 q^{17} -23.9567 q^{19} +28.1554 q^{21} -147.290 q^{23} -86.6787 q^{25} -206.419 q^{27} +199.870 q^{29} +24.9348 q^{31} -310.199 q^{33} -19.8042 q^{35} -325.116 q^{37} -114.411 q^{39} -389.479 q^{41} -422.973 q^{43} +312.334 q^{45} +391.432 q^{47} -332.765 q^{49} +876.043 q^{51} -356.250 q^{53} +218.191 q^{55} +210.839 q^{57} +473.323 q^{59} +312.979 q^{61} -161.413 q^{63} +80.4754 q^{65} +99.3166 q^{67} +1296.27 q^{69} -811.986 q^{71} +960.819 q^{73} +762.844 q^{75} -112.760 q^{77} -701.850 q^{79} +454.385 q^{81} +452.741 q^{83} -616.201 q^{85} -1759.02 q^{87} -849.529 q^{89} -41.5893 q^{91} -219.446 q^{93} -148.302 q^{95} +1253.84 q^{97} +1778.35 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 3 q - 8 q^{5} - 36 q^{7} + 73 q^{9}+O(q^{10}) $ 3 * q - 8 * q^5 - 36 * q^7 + 73 * q^9	$ 3 q - 8 q^{5} - 36 q^{7} + 73 q^{9} - 52 q^{11} + 39 q^{13} - 244 q^{15} - 116 q^{17} - 124 q^{19} - 154 q^{21} - 232 q^{23} + 191 q^{25} - 12 q^{27} - 30 q^{29} - 240 q^{31} - 564 q^{33} + 340 q^{35} - 264 q^{37} - 374 q^{41} - 248 q^{43} - 966 q^{45} + 412 q^{47} - 443 q^{49} - 92 q^{51} - 386 q^{53} + 400 q^{55} + 52 q^{57} + 940 q^{59} + 1206 q^{61} - 864 q^{63} - 104 q^{65} + 564 q^{67} + 512 q^{69} - 1260 q^{71} + 142 q^{73} + 3788 q^{75} + 1188 q^{77} - 1040 q^{79} + 1571 q^{81} - 756 q^{83} + 2510 q^{85} - 1536 q^{87} - 18 q^{89} - 468 q^{91} - 768 q^{93} - 384 q^{95} + 2174 q^{97} + 1940 q^{99}+O(q^{100}) $ 3 * q - 8 * q^5 - 36 * q^7 + 73 * q^9 - 52 * q^11 + 39 * q^13 - 244 * q^15 - 116 * q^17 - 124 * q^19 - 154 * q^21 - 232 * q^23 + 191 * q^25 - 12 * q^27 - 30 * q^29 - 240 * q^31 - 564 * q^33 + 340 * q^35 - 264 * q^37 - 374 * q^41 - 248 * q^43 - 966 * q^45 + 412 * q^47 - 443 * q^49 - 92 * q^51 - 386 * q^53 + 400 * q^55 + 52 * q^57 + 940 * q^59 + 1206 * q^61 - 864 * q^63 - 104 * q^65 + 564 * q^67 + 512 * q^69 - 1260 * q^71 + 142 * q^73 + 3788 * q^75 + 1188 * q^77 - 1040 * q^79 + 1571 * q^81 - 756 * q^83 + 2510 * q^85 - 1536 * q^87 - 18 * q^89 - 468 * q^91 - 768 * q^93 - 384 * q^95 + 2174 * q^97 + 1940 * 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
$3$	−8.80082	−1.69372	−0.846860	−	0.531816i	$-0.821510\pi$
$3$	−8.80082	−1.69372	−0.846860	+	0.531816i	$0.821510\pi$
$4$	0	0
$5$	6.19042	0.553688	0.276844	−	0.960915i	$-0.410712\pi$
$5$	6.19042	0.553688	0.276844	+	0.960915i	$0.410712\pi$
$6$	0	0
$7$	−3.19918	−0.172739	−0.0863696	−	0.996263i	$-0.527527\pi$
$7$	−3.19918	−0.172739	−0.0863696	+	0.996263i	$0.527527\pi$
$8$	0	0
$9$	50.4545	1.86869
$10$	0	0
$11$	35.2466	0.966113	0.483056	−	0.875589i	$-0.339527\pi$
$11$	35.2466	0.966113	0.483056	+	0.875589i	$0.339527\pi$
$12$	0	0
$13$	13.0000	0.277350
$13$	13.0000	0.277350
$14$	0	0
$15$	−54.4808	−0.937792
$16$	0	0
$17$	−99.5410	−1.42013	−0.710066	−	0.704135i	$-0.751335\pi$
$17$	−99.5410	−1.42013	−0.710066	+	0.704135i	$0.751335\pi$
$18$	0	0
$19$	−23.9567	−0.289266	−0.144633	−	0.989485i	$-0.546200\pi$
$19$	−23.9567	−0.289266	−0.144633	+	0.989485i	$0.546200\pi$
$20$	0	0
$21$	28.1554	0.292572
$22$	0	0
$23$	−147.290	−1.33531	−0.667653	−	0.744472i	$-0.732701\pi$
$23$	−147.290	−1.33531	−0.667653	+	0.744472i	$0.732701\pi$
$24$	0	0
$25$	−86.6787	−0.693430
$26$	0	0
$27$	−206.419	−1.47131
$28$	0	0
$29$	199.870	1.27982	0.639911	−	0.768449i	$-0.278971\pi$
$29$	199.870	1.27982	0.639911	+	0.768449i	$0.278971\pi$
$30$	0	0
$31$	24.9348	0.144465	0.0722325	−	0.997388i	$-0.476988\pi$
$31$	24.9348	0.144465	0.0722325	+	0.997388i	$0.476988\pi$
$32$	0	0
$33$	−310.199	−1.63632
$34$	0	0
$35$	−19.8042	−0.0956436
$36$	0	0
$37$	−325.116	−1.44456	−0.722280	−	0.691601i	$-0.756906\pi$
$37$	−325.116	−1.44456	−0.722280	+	0.691601i	$0.756906\pi$
$38$	0	0
$39$	−114.411	−0.469753
$40$	0	0
$41$	−389.479	−1.48357	−0.741785	−	0.670637i	$-0.766021\pi$
$41$	−389.479	−1.48357	−0.741785	+	0.670637i	$0.766021\pi$
$42$	0	0
$43$	−422.973	−1.50007	−0.750033	−	0.661401i	$-0.769962\pi$
$43$	−422.973	−1.50007	−0.750033	+	0.661401i	$0.769962\pi$
$44$	0	0
$45$	312.334	1.03467
$46$	0	0
$47$	391.432	1.21481	0.607407	−	0.794391i	$-0.292210\pi$
$47$	391.432	1.21481	0.607407	+	0.794391i	$0.292210\pi$
$48$	0	0
$49$	−332.765	−0.970161
$50$	0	0
$51$	876.043	2.40531
$52$	0	0
$53$	−356.250	−0.923297	−0.461649	−	0.887063i	$-0.652742\pi$
$53$	−356.250	−0.923297	−0.461649	+	0.887063i	$0.652742\pi$
$54$	0	0
$55$	218.191	0.534925
$56$	0	0
$57$	210.839	0.489935
$58$	0	0
$59$	473.323	1.04443	0.522216	−	0.852813i	$-0.325106\pi$
$59$	473.323	1.04443	0.522216	+	0.852813i	$0.325106\pi$
$60$	0	0
$61$	312.979	0.656932	0.328466	−	0.944516i	$-0.393468\pi$
$61$	312.979	0.656932	0.328466	+	0.944516i	$0.393468\pi$
$62$	0	0
$63$	−161.413	−0.322795
$64$	0	0
$65$	80.4754	0.153565
$66$	0	0
$67$	99.3166	0.181096	0.0905482	−	0.995892i	$-0.471138\pi$
$67$	99.3166	0.181096	0.0905482	+	0.995892i	$0.471138\pi$
$68$	0	0
$69$	1296.27	2.26163
$70$	0	0
$71$	−811.986	−1.35725	−0.678627	−	0.734483i	$-0.737425\pi$
$71$	−811.986	−1.35725	−0.678627	+	0.734483i	$0.737425\pi$
$72$	0	0
$73$	960.819	1.54048	0.770242	−	0.637752i	$-0.220135\pi$
$73$	960.819	1.54048	0.770242	+	0.637752i	$0.220135\pi$
$74$	0	0
$75$	762.844	1.17448
$76$	0	0
$77$	−112.760	−0.166886
$78$	0	0
$79$	−701.850	−0.999548	−0.499774	−	0.866156i	$-0.666584\pi$
$79$	−701.850	−0.999548	−0.499774	+	0.866156i	$0.666584\pi$
$80$	0	0
$81$	454.385	0.623299
$82$	0	0
$83$	452.741	0.598732	0.299366	−	0.954138i	$-0.403225\pi$
$83$	452.741	0.598732	0.299366	+	0.954138i	$0.403225\pi$
$84$	0	0
$85$	−616.201	−0.786310
$86$	0	0
$87$	−1759.02	−2.16766
$88$	0	0
$89$	−849.529	−1.01180	−0.505898	−	0.862593i	$-0.668839\pi$
$89$	−849.529	−1.01180	−0.505898	+	0.862593i	$0.668839\pi$
$90$	0	0
$91$	−41.5893	−0.0479093
$92$	0	0
$93$	−219.446	−0.244683
$94$	0	0
$95$	−148.302	−0.160163
$96$	0	0
$97$	1253.84	1.31246	0.656228	−	0.754562i	$-0.272151\pi$
$97$	1253.84	1.31246	0.656228	+	0.754562i	$0.272151\pi$
$98$	0	0
$99$	1778.35	1.80536
$100$	0	0
$101$	−869.390	−0.856510	−0.428255	−	0.903658i	$-0.640871\pi$
$101$	−869.390	−0.856510	−0.428255	+	0.903658i	$0.640871\pi$
$102$	0	0
$103$	−390.254	−0.373329	−0.186664	−	0.982424i	$-0.559768\pi$
$103$	−390.254	−0.373329	−0.186664	+	0.982424i	$0.559768\pi$
$104$	0	0
$105$	174.294	0.161993
$106$	0	0
$107$	−1820.10	−1.64444	−0.822221	−	0.569168i	$-0.807265\pi$
$107$	−1820.10	−1.64444	−0.822221	+	0.569168i	$0.807265\pi$
$108$	0	0
$109$	817.048	0.717972	0.358986	−	0.933343i	$-0.383123\pi$
$109$	817.048	0.717972	0.358986	+	0.933343i	$0.383123\pi$
$110$	0	0
$111$	2861.29	2.44668
$112$	0	0
$113$	−589.515	−0.490769	−0.245385	−	0.969426i	$-0.578914\pi$
$113$	−589.515	−0.490769	−0.245385	+	0.969426i	$0.578914\pi$
$114$	0	0
$115$	−911.786	−0.739343
$116$	0	0
$117$	655.909	0.518280
$118$	0	0
$119$	318.449	0.245313
$120$	0	0
$121$	−88.6794	−0.0666261
$122$	0	0
$123$	3427.74	2.51275
$124$	0	0
$125$	−1310.38	−0.937631
$126$	0	0
$127$	−2236.41	−1.56259	−0.781297	−	0.624159i	$-0.785442\pi$
$127$	−2236.41	−1.56259	−0.781297	+	0.624159i	$0.785442\pi$
$128$	0	0
$129$	3722.51	2.54069
$130$	0	0
$131$	−830.159	−0.553674	−0.276837	−	0.960917i	$-0.589286\pi$
$131$	−830.159	−0.553674	−0.276837	+	0.960917i	$0.589286\pi$
$132$	0	0
$133$	76.6418	0.0499676
$134$	0	0
$135$	−1277.82	−0.814646
$136$	0	0
$137$	1637.94	1.02145	0.510724	−	0.859745i	$-0.329378\pi$
$137$	1637.94	1.02145	0.510724	+	0.859745i	$0.329378\pi$
$138$	0	0
$139$	−1132.13	−0.690836	−0.345418	−	0.938449i	$-0.612263\pi$
$139$	−1132.13	−0.690836	−0.345418	+	0.938449i	$0.612263\pi$
$140$	0	0
$141$	−3444.92	−2.05755
$142$	0	0
$143$	458.205	0.267951
$144$	0	0
$145$	1237.28	0.708622
$146$	0	0
$147$	2928.61	1.64318
$148$	0	0
$149$	−376.443	−0.206976	−0.103488	−	0.994631i	$-0.533000\pi$
$149$	−376.443	−0.206976	−0.103488	+	0.994631i	$0.533000\pi$
$150$	0	0
$151$	1948.35	1.05003	0.525015	−	0.851093i	$-0.324060\pi$
$151$	1948.35	1.05003	0.525015	+	0.851093i	$0.324060\pi$
$152$	0	0
$153$	−5022.29	−2.65378
$154$	0	0
$155$	154.357	0.0799885
$156$	0	0
$157$	−23.0852	−0.0117350	−0.00586750	−	0.999983i	$-0.501868\pi$
$157$	−23.0852	−0.0117350	−0.00586750	+	0.999983i	$0.501868\pi$
$158$	0	0
$159$	3135.30	1.56381
$160$	0	0
$161$	471.206	0.230660
$162$	0	0
$163$	587.764	0.282437	0.141219	−	0.989978i	$-0.454898\pi$
$163$	587.764	0.282437	0.141219	+	0.989978i	$0.454898\pi$
$164$	0	0
$165$	−1920.26	−0.906013
$166$	0	0
$167$	−3791.14	−1.75669	−0.878344	−	0.478028i	$-0.841352\pi$
$167$	−3791.14	−1.75669	−0.878344	+	0.478028i	$0.841352\pi$
$168$	0	0
$169$	169.000	0.0769231
$170$	0	0
$171$	−1208.72	−0.540547
$172$	0	0
$173$	−228.425	−0.100386	−0.0501931	−	0.998740i	$-0.515984\pi$
$173$	−228.425	−0.100386	−0.0501931	+	0.998740i	$0.515984\pi$
$174$	0	0
$175$	277.301	0.119783
$176$	0	0
$177$	−4165.64	−1.76897
$178$	0	0
$179$	3941.65	1.64588	0.822941	−	0.568127i	$-0.192332\pi$
$179$	3941.65	1.64588	0.822941	+	0.568127i	$0.192332\pi$
$180$	0	0
$181$	3595.43	1.47650	0.738250	−	0.674528i	$-0.235653\pi$
$181$	3595.43	1.47650	0.738250	+	0.674528i	$0.235653\pi$
$182$	0	0
$183$	−2754.47	−1.11266
$184$	0	0
$185$	−2012.60	−0.799835
$186$	0	0
$187$	−3508.48	−1.37201
$188$	0	0
$189$	660.371	0.254153
$190$	0	0
$191$	−2495.41	−0.945347	−0.472674	−	0.881238i	$-0.656711\pi$
$191$	−2495.41	−0.945347	−0.472674	+	0.881238i	$0.656711\pi$
$192$	0	0
$193$	3192.61	1.19072	0.595360	−	0.803459i	$-0.297010\pi$
$193$	3192.61	1.19072	0.595360	+	0.803459i	$0.297010\pi$
$194$	0	0
$195$	−708.250	−0.260097
$196$	0	0
$197$	−3892.00	−1.40758	−0.703791	−	0.710408i	$-0.748511\pi$
$197$	−3892.00	−1.40758	−0.703791	+	0.710408i	$0.748511\pi$
$198$	0	0
$199$	−2573.51	−0.916740	−0.458370	−	0.888761i	$-0.651567\pi$
$199$	−2573.51	−0.916740	−0.458370	+	0.888761i	$0.651567\pi$
$200$	0	0
$201$	−874.068	−0.306726
$202$	0	0
$203$	−639.418	−0.221076
$204$	0	0
$205$	−2411.04	−0.821435
$206$	0	0
$207$	−7431.44	−2.49527
$208$	0	0
$209$	−844.392	−0.279463
$210$	0	0
$211$	3325.12	1.08488	0.542442	−	0.840093i	$-0.317500\pi$
$211$	3325.12	1.08488	0.542442	+	0.840093i	$0.317500\pi$
$212$	0	0
$213$	7146.15	2.29881
$214$	0	0
$215$	−2618.38	−0.830568
$216$	0	0
$217$	−79.7707	−0.0249548
$218$	0	0
$219$	−8456.00	−2.60915
$220$	0	0
$221$	−1294.03	−0.393874
$222$	0	0
$223$	4248.65	1.27583	0.637917	−	0.770106i	$-0.279796\pi$
$223$	4248.65	1.27583	0.637917	+	0.770106i	$0.279796\pi$
$224$	0	0
$225$	−4373.33	−1.29580
$226$	0	0
$227$	782.051	0.228663	0.114332	−	0.993443i	$-0.463527\pi$
$227$	782.051	0.228663	0.114332	+	0.993443i	$0.463527\pi$
$228$	0	0
$229$	3870.91	1.11702	0.558509	−	0.829499i	$-0.311374\pi$
$229$	3870.91	1.11702	0.558509	+	0.829499i	$0.311374\pi$
$230$	0	0
$231$	992.381	0.282657
$232$	0	0
$233$	3978.49	1.11863	0.559313	−	0.828957i	$-0.311065\pi$
$233$	3978.49	1.11863	0.559313	+	0.828957i	$0.311065\pi$
$234$	0	0
$235$	2423.13	0.672627
$236$	0	0
$237$	6176.86	1.69295
$238$	0	0
$239$	286.445	0.0775255	0.0387627	−	0.999248i	$-0.487658\pi$
$239$	286.445	0.0775255	0.0387627	+	0.999248i	$0.487658\pi$
$240$	0	0
$241$	3175.60	0.848790	0.424395	−	0.905477i	$-0.360487\pi$
$241$	3175.60	0.848790	0.424395	+	0.905477i	$0.360487\pi$
$242$	0	0
$243$	1574.35	0.415615
$244$	0	0
$245$	−2059.96	−0.537166
$246$	0	0
$247$	−311.437	−0.0802279
$248$	0	0
$249$	−3984.49	−1.01408
$250$	0	0
$251$	−736.290	−0.185156	−0.0925781	−	0.995705i	$-0.529511\pi$
$251$	−736.290	−0.185156	−0.0925781	+	0.995705i	$0.529511\pi$
$252$	0	0
$253$	−5191.46	−1.29006
$254$	0	0
$255$	5423.07	1.33179
$256$	0	0
$257$	−4326.05	−1.05001	−0.525003	−	0.851100i	$-0.675936\pi$
$257$	−4326.05	−1.05001	−0.525003	+	0.851100i	$0.675936\pi$
$258$	0	0
$259$	1040.10	0.249532
$260$	0	0
$261$	10084.3	2.39158
$262$	0	0
$263$	6938.31	1.62675	0.813374	−	0.581741i	$-0.197628\pi$
$263$	6938.31	1.62675	0.813374	+	0.581741i	$0.197628\pi$
$264$	0	0
$265$	−2205.34	−0.511218
$266$	0	0
$267$	7476.55	1.71370
$268$	0	0
$269$	1852.50	0.419883	0.209942	−	0.977714i	$-0.432673\pi$
$269$	1852.50	0.419883	0.209942	+	0.977714i	$0.432673\pi$
$270$	0	0
$271$	3309.60	0.741860	0.370930	−	0.928661i	$-0.379039\pi$
$271$	3309.60	0.741860	0.370930	+	0.928661i	$0.379039\pi$
$272$	0	0
$273$	366.020	0.0811448
$274$	0	0
$275$	−3055.13	−0.669931
$276$	0	0
$277$	1313.10	0.284825	0.142413	−	0.989807i	$-0.454514\pi$
$277$	1313.10	0.284825	0.142413	+	0.989807i	$0.454514\pi$
$278$	0	0
$279$	1258.07	0.269960
$280$	0	0
$281$	618.244	0.131250	0.0656251	−	0.997844i	$-0.479096\pi$
$281$	618.244	0.131250	0.0656251	+	0.997844i	$0.479096\pi$
$282$	0	0
$283$	2186.30	0.459230	0.229615	−	0.973282i	$-0.426253\pi$
$283$	2186.30	0.459230	0.229615	+	0.973282i	$0.426253\pi$
$284$	0	0
$285$	1305.18	0.271271
$286$	0	0
$287$	1246.01	0.256271
$288$	0	0
$289$	4995.42	1.01678
$290$	0	0
$291$	−11034.8	−2.22293
$292$	0	0
$293$	−2811.16	−0.560511	−0.280255	−	0.959926i	$-0.590419\pi$
$293$	−2811.16	−0.560511	−0.280255	+	0.959926i	$0.590419\pi$
$294$	0	0
$295$	2930.07	0.578289
$296$	0	0
$297$	−7275.56	−1.42145
$298$	0	0
$299$	−1914.77	−0.370347
$300$	0	0
$301$	1353.17	0.259120
$302$	0	0
$303$	7651.35	1.45069
$304$	0	0
$305$	1937.47	0.363735
$306$	0	0
$307$	3372.60	0.626985	0.313492	−	0.949591i	$-0.398501\pi$
$307$	3372.60	0.626985	0.313492	+	0.949591i	$0.398501\pi$
$308$	0	0
$309$	3434.56	0.632314
$310$	0	0
$311$	1192.89	0.217501	0.108750	−	0.994069i	$-0.465315\pi$
$311$	1192.89	0.217501	0.108750	+	0.994069i	$0.465315\pi$
$312$	0	0
$313$	2424.92	0.437905	0.218953	−	0.975735i	$-0.429736\pi$
$313$	2424.92	0.437905	0.218953	+	0.975735i	$0.429736\pi$
$314$	0	0
$315$	−999.213	−0.178728
$316$	0	0
$317$	−5306.75	−0.940243	−0.470122	−	0.882602i	$-0.655790\pi$
$317$	−5306.75	−0.940243	−0.470122	+	0.882602i	$0.655790\pi$
$318$	0	0
$319$	7044.71	1.23645
$320$	0	0
$321$	16018.3	2.78522
$322$	0	0
$323$	2384.68	0.410796
$324$	0	0
$325$	−1126.82	−0.192323
$326$	0	0
$327$	−7190.69	−1.21604
$328$	0	0
$329$	−1252.26	−0.209846
$330$	0	0
$331$	−6906.83	−1.14693	−0.573465	−	0.819230i	$-0.694401\pi$
$331$	−6906.83	−1.14693	−0.573465	+	0.819230i	$0.694401\pi$
$332$	0	0
$333$	−16403.6	−2.69943
$334$	0	0
$335$	614.811	0.100271
$336$	0	0
$337$	−7410.03	−1.19777	−0.598887	−	0.800834i	$-0.704390\pi$
$337$	−7410.03	−1.19777	−0.598887	+	0.800834i	$0.704390\pi$
$338$	0	0
$339$	5188.22	0.831225
$340$	0	0
$341$	878.865	0.139569
$342$	0	0
$343$	2161.89	0.340324
$344$	0	0
$345$	8024.47	1.25224
$346$	0	0
$347$	1921.92	0.297331	0.148665	−	0.988888i	$-0.452502\pi$
$347$	1921.92	0.297331	0.148665	+	0.988888i	$0.452502\pi$
$348$	0	0
$349$	−231.773	−0.0355488	−0.0177744	−	0.999842i	$-0.505658\pi$
$349$	−231.773	−0.0355488	−0.0177744	+	0.999842i	$0.505658\pi$
$350$	0	0
$351$	−2683.45	−0.408068
$352$	0	0
$353$	−1111.06	−0.167523	−0.0837614	−	0.996486i	$-0.526693\pi$
$353$	−1111.06	−0.167523	−0.0837614	+	0.996486i	$0.526693\pi$
$354$	0	0
$355$	−5026.53	−0.751495
$356$	0	0
$357$	−2802.62	−0.415491
$358$	0	0
$359$	−9272.97	−1.36326	−0.681628	−	0.731699i	$-0.738728\pi$
$359$	−9272.97	−1.36326	−0.681628	+	0.731699i	$0.738728\pi$
$360$	0	0
$361$	−6285.08	−0.916325
$362$	0	0
$363$	780.452	0.112846
$364$	0	0
$365$	5947.87	0.852947
$366$	0	0
$367$	−5130.98	−0.729796	−0.364898	−	0.931048i	$-0.618896\pi$
$367$	−5130.98	−0.729796	−0.364898	+	0.931048i	$0.618896\pi$
$368$	0	0
$369$	−19651.0	−2.77233
$370$	0	0
$371$	1139.71	0.159490
$372$	0	0
$373$	−9431.54	−1.30924	−0.654620	−	0.755958i	$-0.727171\pi$
$373$	−9431.54	−1.30924	−0.654620	+	0.755958i	$0.727171\pi$
$374$	0	0
$375$	11532.4	1.58808
$376$	0	0
$377$	2598.30	0.354959
$378$	0	0
$379$	5172.60	0.701052	0.350526	−	0.936553i	$-0.386003\pi$
$379$	5172.60	0.701052	0.350526	+	0.936553i	$0.386003\pi$
$380$	0	0
$381$	19682.3	2.64660
$382$	0	0
$383$	−8335.31	−1.11205	−0.556024	−	0.831166i	$-0.687674\pi$
$383$	−8335.31	−1.11205	−0.556024	+	0.831166i	$0.687674\pi$
$384$	0	0
$385$	−698.031	−0.0924025
$386$	0	0
$387$	−21340.9	−2.80315
$388$	0	0
$389$	−5438.62	−0.708866	−0.354433	−	0.935081i	$-0.615326\pi$
$389$	−5438.62	−0.708866	−0.354433	+	0.935081i	$0.615326\pi$
$390$	0	0
$391$	14661.4	1.89631
$392$	0	0
$393$	7306.08	0.937769
$394$	0	0
$395$	−4344.75	−0.553438
$396$	0	0
$397$	−9517.97	−1.20326	−0.601629	−	0.798776i	$-0.705481\pi$
$397$	−9517.97	−1.20326	−0.601629	+	0.798776i	$0.705481\pi$
$398$	0	0
$399$	−674.511	−0.0846310
$400$	0	0
$401$	16033.7	1.99672	0.998362	−	0.0572096i	$-0.0182203\pi$
$401$	16033.7	1.99672	0.998362	+	0.0572096i	$0.0182203\pi$
$402$	0	0
$403$	324.152	0.0400674
$404$	0	0
$405$	2812.83	0.345113
$406$	0	0
$407$	−11459.2	−1.39561
$408$	0	0
$409$	14160.6	1.71198	0.855989	−	0.516994i	$-0.172949\pi$
$409$	14160.6	1.71198	0.855989	+	0.516994i	$0.172949\pi$
$410$	0	0
$411$	−14415.2	−1.73005
$412$	0	0
$413$	−1514.25	−0.180414
$414$	0	0
$415$	2802.66	0.331511
$416$	0	0
$417$	9963.69	1.17008
$418$	0	0
$419$	−1854.53	−0.216228	−0.108114	−	0.994138i	$-0.534481\pi$
$419$	−1854.53	−0.216228	−0.108114	+	0.994138i	$0.534481\pi$
$420$	0	0
$421$	10335.0	1.19643	0.598213	−	0.801337i	$-0.295878\pi$
$421$	10335.0	1.19643	0.598213	+	0.801337i	$0.295878\pi$
$422$	0	0
$423$	19749.5	2.27010
$424$	0	0
$425$	8628.09	0.984762
$426$	0	0
$427$	−1001.28	−0.113478
$428$	0	0
$429$	−4032.59	−0.453835
$430$	0	0
$431$	−6956.25	−0.777426	−0.388713	−	0.921359i	$-0.627080\pi$
$431$	−6956.25	−0.777426	−0.388713	+	0.921359i	$0.627080\pi$
$432$	0	0
$433$	13139.0	1.45825	0.729123	−	0.684383i	$-0.239928\pi$
$433$	13139.0	1.45825	0.729123	+	0.684383i	$0.239928\pi$
$434$	0	0
$435$	−10889.0	−1.20021
$436$	0	0
$437$	3528.58	0.386259
$438$	0	0
$439$	11358.1	1.23484	0.617420	−	0.786634i	$-0.288178\pi$
$439$	11358.1	1.23484	0.617420	+	0.786634i	$0.288178\pi$
$440$	0	0
$441$	−16789.5	−1.81293
$442$	0	0
$443$	6949.61	0.745340	0.372670	−	0.927964i	$-0.378442\pi$
$443$	6949.61	0.745340	0.372670	+	0.927964i	$0.378442\pi$
$444$	0	0
$445$	−5258.94	−0.560219
$446$	0	0
$447$	3313.01	0.350559
$448$	0	0
$449$	1531.95	0.161018	0.0805091	−	0.996754i	$-0.474345\pi$
$449$	1531.95	0.161018	0.0805091	+	0.996754i	$0.474345\pi$
$450$	0	0
$451$	−13727.8	−1.43330
$452$	0	0
$453$	−17147.1	−1.77846
$454$	0	0
$455$	−257.455	−0.0265268
$456$	0	0
$457$	−13047.0	−1.33548	−0.667738	−	0.744396i	$-0.732737\pi$
$457$	−13047.0	−1.33548	−0.667738	+	0.744396i	$0.732737\pi$
$458$	0	0
$459$	20547.2	2.08945
$460$	0	0
$461$	−14339.8	−1.44875	−0.724374	−	0.689408i	$-0.757871\pi$
$461$	−14339.8	−1.44875	−0.724374	+	0.689408i	$0.757871\pi$
$462$	0	0
$463$	−14943.2	−1.49993	−0.749964	−	0.661478i	$-0.769929\pi$
$463$	−14943.2	−1.49993	−0.749964	+	0.661478i	$0.769929\pi$
$464$	0	0
$465$	−1358.47	−0.135478
$466$	0	0
$467$	−4520.86	−0.447967	−0.223983	−	0.974593i	$-0.571906\pi$
$467$	−4520.86	−0.447967	−0.223983	+	0.974593i	$0.571906\pi$
$468$	0	0
$469$	−317.731	−0.0312825
$470$	0	0
$471$	203.168	0.0198758
$472$	0	0
$473$	−14908.4	−1.44923
$474$	0	0
$475$	2076.54	0.200586
$476$	0	0
$477$	−17974.4	−1.72535
$478$	0	0
$479$	−12784.8	−1.21952	−0.609761	−	0.792585i	$-0.708735\pi$
$479$	−12784.8	−1.21952	−0.609761	+	0.792585i	$0.708735\pi$
$480$	0	0
$481$	−4226.51	−0.400649
$482$	0	0
$483$	−4147.00	−0.390673
$484$	0	0
$485$	7761.80	0.726691
$486$	0	0
$487$	12878.7	1.19834	0.599168	−	0.800623i	$-0.295498\pi$
$487$	12878.7	1.19834	0.599168	+	0.800623i	$0.295498\pi$
$488$	0	0
$489$	−5172.81	−0.478369
$490$	0	0
$491$	−6736.08	−0.619135	−0.309567	−	0.950878i	$-0.600184\pi$
$491$	−6736.08	−0.619135	−0.309567	+	0.950878i	$0.600184\pi$
$492$	0	0
$493$	−19895.2	−1.81752
$494$	0	0
$495$	11008.7	0.999606
$496$	0	0
$497$	2597.69	0.234451
$498$	0	0
$499$	3870.03	0.347187	0.173594	−	0.984817i	$-0.444462\pi$
$499$	3870.03	0.347187	0.173594	+	0.984817i	$0.444462\pi$
$500$	0	0
$501$	33365.1	2.97534
$502$	0	0
$503$	7742.60	0.686333	0.343167	−	0.939275i	$-0.388500\pi$
$503$	7742.60	0.686333	0.343167	+	0.939275i	$0.388500\pi$
$504$	0	0
$505$	−5381.89	−0.474239
$506$	0	0
$507$	−1487.34	−0.130286
$508$	0	0
$509$	−12222.2	−1.06432	−0.532159	−	0.846645i	$-0.678619\pi$
$509$	−12222.2	−1.06432	−0.532159	+	0.846645i	$0.678619\pi$
$510$	0	0
$511$	−3073.83	−0.266102
$512$	0	0
$513$	4945.12	0.425599
$514$	0	0
$515$	−2415.84	−0.206708
$516$	0	0
$517$	13796.6	1.17365
$518$	0	0
$519$	2010.32	0.170026
$520$	0	0
$521$	−17268.2	−1.45208	−0.726041	−	0.687651i	$-0.758642\pi$
$521$	−17268.2	−1.45208	−0.726041	+	0.687651i	$0.758642\pi$
$522$	0	0
$523$	16048.6	1.34179	0.670897	−	0.741551i	$-0.265909\pi$
$523$	16048.6	1.34179	0.670897	+	0.741551i	$0.265909\pi$
$524$	0	0
$525$	−2440.47	−0.202878
$526$	0	0
$527$	−2482.03	−0.205159
$528$	0	0
$529$	9527.30	0.783044
$530$	0	0
$531$	23881.3	1.95171
$532$	0	0
$533$	−5063.23	−0.411469
$534$	0	0
$535$	−11267.2	−0.910508
$536$	0	0
$537$	−34689.8	−2.78766
$538$	0	0
$539$	−11728.8	−0.937285
$540$	0	0
$541$	7244.68	0.575736	0.287868	−	0.957670i	$-0.407054\pi$
$541$	7244.68	0.575736	0.287868	+	0.957670i	$0.407054\pi$
$542$	0	0
$543$	−31642.8	−2.50077
$544$	0	0
$545$	5057.87	0.397532
$546$	0	0
$547$	−11855.6	−0.926706	−0.463353	−	0.886174i	$-0.653354\pi$
$547$	−11855.6	−0.926706	−0.463353	+	0.886174i	$0.653354\pi$
$548$	0	0
$549$	15791.2	1.22760
$550$	0	0
$551$	−4788.22	−0.370209
$552$	0	0
$553$	2245.34	0.172661
$554$	0	0
$555$	17712.6	1.35470
$556$	0	0
$557$	15990.8	1.21643	0.608213	−	0.793773i	$-0.291886\pi$
$557$	15990.8	1.21643	0.608213	+	0.793773i	$0.291886\pi$
$558$	0	0
$559$	−5498.65	−0.416043
$560$	0	0
$561$	30877.5	2.32380
$562$	0	0
$563$	14379.6	1.07642	0.538212	−	0.842810i	$-0.319100\pi$
$563$	14379.6	1.07642	0.538212	+	0.842810i	$0.319100\pi$
$564$	0	0
$565$	−3649.35	−0.271733
$566$	0	0
$567$	−1453.66	−0.107668
$568$	0	0
$569$	−7067.43	−0.520707	−0.260353	−	0.965513i	$-0.583839\pi$
$569$	−7067.43	−0.520707	−0.260353	+	0.965513i	$0.583839\pi$
$570$	0	0
$571$	6121.43	0.448640	0.224320	−	0.974515i	$-0.427984\pi$
$571$	6121.43	0.448640	0.224320	+	0.974515i	$0.427984\pi$
$572$	0	0
$573$	21961.6	1.60115
$574$	0	0
$575$	12766.9	0.925942
$576$	0	0
$577$	−27602.2	−1.99150	−0.995749	−	0.0921068i	$-0.970640\pi$
$577$	−27602.2	−1.99150	−0.995749	+	0.0921068i	$0.970640\pi$
$578$	0	0
$579$	−28097.6	−2.01674
$580$	0	0
$581$	−1448.40	−0.103425
$582$	0	0
$583$	−12556.6	−0.892009
$584$	0	0
$585$	4060.35	0.286965
$586$	0	0
$587$	18603.3	1.30807	0.654037	−	0.756462i	$-0.273074\pi$
$587$	18603.3	1.30807	0.654037	+	0.756462i	$0.273074\pi$
$588$	0	0
$589$	−597.355	−0.0417888
$590$	0	0
$591$	34252.8	2.38405
$592$	0	0
$593$	10274.8	0.711530	0.355765	−	0.934575i	$-0.384220\pi$
$593$	10274.8	0.711530	0.355765	+	0.934575i	$0.384220\pi$
$594$	0	0
$595$	1971.33	0.135827
$596$	0	0
$597$	22649.0	1.55270
$598$	0	0
$599$	6509.85	0.444049	0.222024	−	0.975041i	$-0.428734\pi$
$599$	6509.85	0.444049	0.222024	+	0.975041i	$0.428734\pi$
$600$	0	0
$601$	1967.19	0.133516	0.0667582	−	0.997769i	$-0.478734\pi$
$601$	1967.19	0.133516	0.0667582	+	0.997769i	$0.478734\pi$
$602$	0	0
$603$	5010.97	0.338412
$604$	0	0
$605$	−548.963	−0.0368901
$606$	0	0
$607$	25146.4	1.68148	0.840742	−	0.541436i	$-0.182119\pi$
$607$	25146.4	1.68148	0.840742	+	0.541436i	$0.182119\pi$
$608$	0	0
$609$	5627.40	0.374440
$610$	0	0
$611$	5088.62	0.336929
$612$	0	0
$613$	−7539.58	−0.496771	−0.248386	−	0.968661i	$-0.579900\pi$
$613$	−7539.58	−0.496771	−0.248386	+	0.968661i	$0.579900\pi$
$614$	0	0
$615$	21219.1	1.39128
$616$	0	0
$617$	14505.6	0.946473	0.473236	−	0.880935i	$-0.343086\pi$
$617$	14505.6	0.946473	0.473236	+	0.880935i	$0.343086\pi$
$618$	0	0
$619$	19448.2	1.26282	0.631412	−	0.775447i	$-0.282476\pi$
$619$	19448.2	1.26282	0.631412	+	0.775447i	$0.282476\pi$
$620$	0	0
$621$	30403.4	1.96465
$622$	0	0
$623$	2717.79	0.174777
$624$	0	0
$625$	2723.04	0.174275
$626$	0	0
$627$	7431.35	0.473333
$628$	0	0
$629$	32362.4	2.05147
$630$	0	0
$631$	−23478.4	−1.48124	−0.740618	−	0.671926i	$-0.765467\pi$
$631$	−23478.4	−1.48124	−0.740618	+	0.671926i	$0.765467\pi$
$632$	0	0
$633$	−29263.8	−1.83749
$634$	0	0
$635$	−13844.3	−0.865190
$636$	0	0
$637$	−4325.95	−0.269074
$638$	0	0
$639$	−40968.4	−2.53628
$640$	0	0
$641$	−15711.7	−0.968137	−0.484068	−	0.875030i	$-0.660841\pi$
$641$	−15711.7	−0.968137	−0.484068	+	0.875030i	$0.660841\pi$
$642$	0	0
$643$	−7224.65	−0.443099	−0.221549	−	0.975149i	$-0.571111\pi$
$643$	−7224.65	−0.443099	−0.221549	+	0.975149i	$0.571111\pi$
$644$	0	0
$645$	23043.9	1.40675
$646$	0	0
$647$	−2265.62	−0.137667	−0.0688336	−	0.997628i	$-0.521928\pi$
$647$	−2265.62	−0.137667	−0.0688336	+	0.997628i	$0.521928\pi$
$648$	0	0
$649$	16683.0	1.00904
$650$	0	0
$651$	702.048	0.0422664
$652$	0	0
$653$	−3173.28	−0.190169	−0.0950843	−	0.995469i	$-0.530312\pi$
$653$	−3173.28	−0.190169	−0.0950843	+	0.995469i	$0.530312\pi$
$654$	0	0
$655$	−5139.03	−0.306563
$656$	0	0
$657$	48477.6	2.87868
$658$	0	0
$659$	14953.8	0.883942	0.441971	−	0.897029i	$-0.354279\pi$
$659$	14953.8	0.883942	0.441971	+	0.897029i	$0.354279\pi$
$660$	0	0
$661$	−21140.5	−1.24398	−0.621988	−	0.783027i	$-0.713675\pi$
$661$	−21140.5	−1.24398	−0.621988	+	0.783027i	$0.713675\pi$
$662$	0	0
$663$	11388.6	0.667112
$664$	0	0
$665$	474.445	0.0276664
$666$	0	0
$667$	−29438.8	−1.70895
$668$	0	0
$669$	−37391.6	−2.16090
$670$	0	0
$671$	11031.4	0.634671
$672$	0	0
$673$	−10502.4	−0.601543	−0.300772	−	0.953696i	$-0.597244\pi$
$673$	−10502.4	−0.601543	−0.300772	+	0.953696i	$0.597244\pi$
$674$	0	0
$675$	17892.1	1.02025
$676$	0	0
$677$	−22024.6	−1.25033	−0.625165	−	0.780493i	$-0.714968\pi$
$677$	−22024.6	−1.25033	−0.625165	+	0.780493i	$0.714968\pi$
$678$	0	0
$679$	−4011.26	−0.226713
$680$	0	0
$681$	−6882.69	−0.387291
$682$	0	0
$683$	−5722.51	−0.320594	−0.160297	−	0.987069i	$-0.551245\pi$
$683$	−5722.51	−0.320594	−0.160297	+	0.987069i	$0.551245\pi$
$684$	0	0
$685$	10139.5	0.565563
$686$	0	0
$687$	−34067.2	−1.89191
$688$	0	0
$689$	−4631.25	−0.256077
$690$	0	0
$691$	17576.6	0.967647	0.483824	−	0.875166i	$-0.339248\pi$
$691$	17576.6	0.967647	0.483824	+	0.875166i	$0.339248\pi$
$692$	0	0
$693$	−5689.25	−0.311857
$694$	0	0
$695$	−7008.37	−0.382507
$696$	0	0
$697$	38769.2	2.10687
$698$	0	0
$699$	−35014.0	−1.89464
$700$	0	0
$701$	1552.10	0.0836265	0.0418132	−	0.999125i	$-0.486687\pi$
$701$	1552.10	0.0836265	0.0418132	+	0.999125i	$0.486687\pi$
$702$	0	0
$703$	7788.71	0.417862
$704$	0	0
$705$	−21325.5	−1.13924
$706$	0	0
$707$	2781.33	0.147953
$708$	0	0
$709$	−15218.8	−0.806143	−0.403072	−	0.915168i	$-0.632057\pi$
$709$	−15218.8	−0.806143	−0.403072	+	0.915168i	$0.632057\pi$
$710$	0	0
$711$	−35411.5	−1.86784
$712$	0	0
$713$	−3672.64	−0.192905
$714$	0	0
$715$	2836.48	0.148361
$716$	0	0
$717$	−2520.95	−0.131306
$718$	0	0
$719$	−11326.7	−0.587503	−0.293751	−	0.955882i	$-0.594904\pi$
$719$	−11326.7	−0.587503	−0.293751	+	0.955882i	$0.594904\pi$
$720$	0	0
$721$	1248.49	0.0644886
$722$	0	0
$723$	−27947.9	−1.43761
$724$	0	0
$725$	−17324.4	−0.887467
$726$	0	0
$727$	1784.64	0.0910435	0.0455218	−	0.998963i	$-0.485505\pi$
$727$	1784.64	0.0910435	0.0455218	+	0.998963i	$0.485505\pi$
$728$	0	0
$729$	−26124.0	−1.32723
$730$	0	0
$731$	42103.2	2.13029
$732$	0	0
$733$	−2732.60	−0.137695	−0.0688477	−	0.997627i	$-0.521932\pi$
$733$	−2732.60	−0.137695	−0.0688477	+	0.997627i	$0.521932\pi$
$734$	0	0
$735$	18129.3	0.909809
$736$	0	0
$737$	3500.57	0.174960
$738$	0	0
$739$	−31860.0	−1.58591	−0.792957	−	0.609277i	$-0.791460\pi$
$739$	−31860.0	−1.58591	−0.792957	+	0.609277i	$0.791460\pi$
$740$	0	0
$741$	2740.91	0.135884
$742$	0	0
$743$	−10493.2	−0.518116	−0.259058	−	0.965862i	$-0.583412\pi$
$743$	−10493.2	−0.518116	−0.259058	+	0.965862i	$0.583412\pi$
$744$	0	0
$745$	−2330.34	−0.114600
$746$	0	0
$747$	22842.8	1.11884
$748$	0	0
$749$	5822.81	0.284060
$750$	0	0
$751$	10411.3	0.505879	0.252939	−	0.967482i	$-0.418603\pi$
$751$	10411.3	0.505879	0.252939	+	0.967482i	$0.418603\pi$
$752$	0	0
$753$	6479.96	0.313603
$754$	0	0
$755$	12061.1	0.581389
$756$	0	0
$757$	3634.03	0.174479	0.0872397	−	0.996187i	$-0.472195\pi$
$757$	3634.03	0.174479	0.0872397	+	0.996187i	$0.472195\pi$
$758$	0	0
$759$	45689.1	2.18499
$760$	0	0
$761$	2193.80	0.104501	0.0522504	−	0.998634i	$-0.483361\pi$
$761$	2193.80	0.104501	0.0522504	+	0.998634i	$0.483361\pi$
$762$	0	0
$763$	−2613.88	−0.124022
$764$	0	0
$765$	−31090.1	−1.46937
$766$	0	0
$767$	6153.20	0.289673
$768$	0	0
$769$	−19576.2	−0.917992	−0.458996	−	0.888438i	$-0.651791\pi$
$769$	−19576.2	−0.917992	−0.458996	+	0.888438i	$0.651791\pi$
$770$	0	0
$771$	38072.8	1.77842
$772$	0	0
$773$	4932.43	0.229505	0.114752	−	0.993394i	$-0.463393\pi$
$773$	4932.43	0.229505	0.114752	+	0.993394i	$0.463393\pi$
$774$	0	0
$775$	−2161.31	−0.100176
$776$	0	0
$777$	−9153.76	−0.422638
$778$	0	0
$779$	9330.64	0.429146
$780$	0	0
$781$	−28619.7	−1.31126
$782$	0	0
$783$	−41256.9	−1.88301
$784$	0	0
$785$	−142.907	−0.00649753
$786$	0	0
$787$	13840.0	0.626864	0.313432	−	0.949611i	$-0.398521\pi$
$787$	13840.0	0.626864	0.313432	+	0.949611i	$0.398521\pi$
$788$	0	0
$789$	−61062.9	−2.75525
$790$	0	0
$791$	1885.96	0.0847751
$792$	0	0
$793$	4068.73	0.182200
$794$	0	0
$795$	19408.8	0.865860
$796$	0	0
$797$	27660.1	1.22932	0.614662	−	0.788791i	$-0.289293\pi$
$797$	27660.1	1.22932	0.614662	+	0.788791i	$0.289293\pi$
$798$	0	0
$799$	−38963.6	−1.72520
$800$	0	0
$801$	−42862.6	−1.89073
$802$	0	0
$803$	33865.6	1.48828
$804$	0	0
$805$	2916.96	0.127714
$806$	0	0
$807$	−16303.5	−0.711165
$808$	0	0
$809$	1392.86	0.0605320	0.0302660	−	0.999542i	$-0.490365\pi$
$809$	1392.86	0.0605320	0.0302660	+	0.999542i	$0.490365\pi$
$810$	0	0
$811$	−5203.92	−0.225320	−0.112660	−	0.993634i	$-0.535937\pi$
$811$	−5203.92	−0.225320	−0.112660	+	0.993634i	$0.535937\pi$
$812$	0	0
$813$	−29127.2	−1.25650
$814$	0	0
$815$	3638.51	0.156382
$816$	0	0
$817$	10133.1	0.433918
$818$	0	0
$819$	−2098.37	−0.0895273
$820$	0	0
$821$	14644.3	0.622520	0.311260	−	0.950325i	$-0.399249\pi$
$821$	14644.3	0.622520	0.311260	+	0.950325i	$0.399249\pi$
$822$	0	0
$823$	−31478.4	−1.33326	−0.666628	−	0.745391i	$-0.732263\pi$
$823$	−31478.4	−1.33326	−0.666628	+	0.745391i	$0.732263\pi$
$824$	0	0
$825$	26887.6	1.13468
$826$	0	0
$827$	−1480.74	−0.0622618	−0.0311309	−	0.999515i	$-0.509911\pi$
$827$	−1480.74	−0.0622618	−0.0311309	+	0.999515i	$0.509911\pi$
$828$	0	0
$829$	31312.5	1.31185	0.655927	−	0.754824i	$-0.272278\pi$
$829$	31312.5	1.31185	0.655927	+	0.754824i	$0.272278\pi$
$830$	0	0
$831$	−11556.4	−0.482414
$832$	0	0
$833$	33123.8	1.37776
$834$	0	0
$835$	−23468.7	−0.972657
$836$	0	0
$837$	−5147.01	−0.212553
$838$	0	0
$839$	−26560.0	−1.09291	−0.546456	−	0.837488i	$-0.684024\pi$
$839$	−26560.0	−1.09291	−0.546456	+	0.837488i	$0.684024\pi$
$840$	0	0
$841$	15558.8	0.637944
$842$	0	0
$843$	−5441.05	−0.222301
$844$	0	0
$845$	1046.18	0.0425914
$846$	0	0
$847$	283.701	0.0115090
$848$	0	0
$849$	−19241.2	−0.777807
$850$	0	0
$851$	47886.3	1.92893
$852$	0	0
$853$	7079.10	0.284155	0.142077	−	0.989856i	$-0.454622\pi$
$853$	7079.10	0.284155	0.142077	+	0.989856i	$0.454622\pi$
$854$	0	0
$855$	−7482.51	−0.299294
$856$	0	0
$857$	3154.29	0.125728	0.0628638	−	0.998022i	$-0.479977\pi$
$857$	3154.29	0.125728	0.0628638	+	0.998022i	$0.479977\pi$
$858$	0	0
$859$	−42297.3	−1.68005	−0.840027	−	0.542544i	$-0.817461\pi$
$859$	−42297.3	−1.68005	−0.840027	+	0.542544i	$0.817461\pi$
$860$	0	0
$861$	−10965.9	−0.434051
$862$	0	0
$863$	19500.7	0.769190	0.384595	−	0.923086i	$-0.374341\pi$
$863$	19500.7	0.769190	0.384595	+	0.923086i	$0.374341\pi$
$864$	0	0
$865$	−1414.04	−0.0555826
$866$	0	0
$867$	−43963.8	−1.72213
$868$	0	0
$869$	−24737.8	−0.965676
$870$	0	0
$871$	1291.12	0.0502271
$872$	0	0
$873$	63261.9	2.45257
$874$	0	0
$875$	4192.14	0.161966
$876$	0	0
$877$	44030.7	1.69534	0.847669	−	0.530525i	$-0.178005\pi$
$877$	44030.7	1.69534	0.847669	+	0.530525i	$0.178005\pi$
$878$	0	0
$879$	24740.5	0.949348
$880$	0	0
$881$	34313.2	1.31219	0.656096	−	0.754678i	$-0.272207\pi$
$881$	34313.2	1.31219	0.656096	+	0.754678i	$0.272207\pi$
$882$	0	0
$883$	51045.1	1.94542	0.972709	−	0.232029i	$-0.0745363\pi$
$883$	51045.1	1.94542	0.972709	+	0.232029i	$0.0745363\pi$
$884$	0	0
$885$	−25787.0	−0.979459
$886$	0	0
$887$	−25721.7	−0.973676	−0.486838	−	0.873492i	$-0.661850\pi$
$887$	−25721.7	−0.973676	−0.486838	+	0.873492i	$0.661850\pi$
$888$	0	0
$889$	7154.68	0.269921
$890$	0	0
$891$	16015.5	0.602178
$892$	0	0
$893$	−9377.43	−0.351404
$894$	0	0
$895$	24400.5	0.911305
$896$	0	0
$897$	16851.5	0.627265
$898$	0	0
$899$	4983.70	0.184890
$900$	0	0
$901$	35461.5	1.31120
$902$	0	0
$903$	−11909.0	−0.438877
$904$	0	0
$905$	22257.2	0.817519
$906$	0	0
$907$	−29028.6	−1.06271	−0.531355	−	0.847149i	$-0.678317\pi$
$907$	−29028.6	−1.06271	−0.531355	+	0.847149i	$0.678317\pi$
$908$	0	0
$909$	−43864.6	−1.60055
$910$	0	0
$911$	4983.04	0.181224	0.0906122	−	0.995886i	$-0.471118\pi$
$911$	4983.04	0.181224	0.0906122	+	0.995886i	$0.471118\pi$
$912$	0	0
$913$	15957.6	0.578443
$914$	0	0
$915$	−17051.3	−0.616066
$916$	0	0
$917$	2655.82	0.0956413
$918$	0	0
$919$	−38302.5	−1.37485	−0.687423	−	0.726257i	$-0.741258\pi$
$919$	−38302.5	−1.37485	−0.687423	+	0.726257i	$0.741258\pi$
$920$	0	0
$921$	−29681.6	−1.06194
$922$	0	0
$923$	−10555.8	−0.376435
$924$	0	0
$925$	28180.6	1.00170
$926$	0	0
$927$	−19690.1	−0.697634
$928$	0	0
$929$	−21853.0	−0.771768	−0.385884	−	0.922547i	$-0.626103\pi$
$929$	−21853.0	−0.771768	−0.385884	+	0.922547i	$0.626103\pi$
$930$	0	0
$931$	7971.97	0.280634
$932$	0	0
$933$	−10498.4	−0.368385
$934$	0	0
$935$	−21719.0	−0.759664
$936$	0	0
$937$	−54479.5	−1.89943	−0.949716	−	0.313113i	$-0.898628\pi$
$937$	−54479.5	−1.89943	−0.949716	+	0.313113i	$0.898628\pi$
$938$	0	0
$939$	−21341.3	−0.741689
$940$	0	0
$941$	6789.26	0.235200	0.117600	−	0.993061i	$-0.462480\pi$
$941$	6789.26	0.235200	0.117600	+	0.993061i	$0.462480\pi$
$942$	0	0
$943$	57366.3	1.98102
$944$	0	0
$945$	4087.97	0.140721
$946$	0	0
$947$	43766.5	1.50182	0.750908	−	0.660407i	$-0.229616\pi$
$947$	43766.5	1.50182	0.750908	+	0.660407i	$0.229616\pi$
$948$	0	0
$949$	12490.6	0.427253
$950$	0	0
$951$	46703.8	1.59251
$952$	0	0
$953$	−2670.49	−0.0907718	−0.0453859	−	0.998970i	$-0.514452\pi$
$953$	−2670.49	−0.0907718	−0.0453859	+	0.998970i	$0.514452\pi$
$954$	0	0
$955$	−15447.6	−0.523427
$956$	0	0
$957$	−61999.3	−2.09420
$958$	0	0
$959$	−5240.05	−0.176444
$960$	0	0
$961$	−29169.3	−0.979130
$962$	0	0
$963$	−91832.1	−3.07295
$964$	0	0
$965$	19763.6	0.659287
$966$	0	0
$967$	2520.14	0.0838080	0.0419040	−	0.999122i	$-0.486658\pi$
$967$	2520.14	0.0838080	0.0419040	+	0.999122i	$0.486658\pi$
$968$	0	0
$969$	−20987.1	−0.695773
$970$	0	0
$971$	8581.10	0.283605	0.141803	−	0.989895i	$-0.454710\pi$
$971$	8581.10	0.283605	0.141803	+	0.989895i	$0.454710\pi$
$972$	0	0
$973$	3621.89	0.119334
$974$	0	0
$975$	9916.97	0.325741
$976$	0	0
$977$	−51537.0	−1.68763	−0.843815	−	0.536634i	$-0.819696\pi$
$977$	−51537.0	−1.68763	−0.843815	+	0.536634i	$0.819696\pi$
$978$	0	0
$979$	−29943.0	−0.977509
$980$	0	0
$981$	41223.7	1.34166
$982$	0	0
$983$	31601.3	1.02536	0.512678	−	0.858581i	$-0.328654\pi$
$983$	31601.3	1.02536	0.512678	+	0.858581i	$0.328654\pi$
$984$	0	0
$985$	−24093.1	−0.779361
$986$	0	0
$987$	11020.9	0.355420
$988$	0	0
$989$	62299.7	2.00305
$990$	0	0
$991$	28216.3	0.904461	0.452230	−	0.891901i	$-0.350629\pi$
$991$	28216.3	0.904461	0.452230	+	0.891901i	$0.350629\pi$
$992$	0	0
$993$	60785.8	1.94258
$994$	0	0
$995$	−15931.1	−0.507588
$996$	0	0
$997$	48017.5	1.52530	0.762652	−	0.646809i	$-0.223897\pi$
$997$	48017.5	1.52530	0.762652	+	0.646809i	$0.223897\pi$
$998$	0	0
$999$	67110.1	2.12539

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	208.4.a.l.1.1		3
3.2	odd	2		1872.4.a.bm.1.2		3
4.3	odd	2		104.4.a.e.1.3	✓	3
8.3	odd	2		832.4.a.bc.1.1		3
8.5	even	2		832.4.a.bb.1.3		3
12.11	even	2		936.4.a.m.1.2		3
52.51	odd	2		1352.4.a.h.1.3		3

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
104.4.a.e.1.3	✓	3	4.3	odd	2
208.4.a.l.1.1		3	1.1	even	1	trivial
832.4.a.bb.1.3		3	8.5	even	2
832.4.a.bc.1.1		3	8.3	odd	2
936.4.a.m.1.2		3	12.11	even	2
1352.4.a.h.1.3		3	52.51	odd	2
1872.4.a.bm.1.2		3	3.2	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:	\( N \)	\(=\)	\( 208 = 2^{4} \cdot 13 \)
Weight:	\( k \)	\(=\)	\( 4 \)
Character orbit:	\([\chi]\)	\(=\)	208.a (trivial)

\(f(q)\)	\(=\)	\(q-8.80082 q^{3} +6.19042 q^{5} -3.19918 q^{7} +50.4545 q^{9} +O(q^{10})\)	\(q-8.80082 q^{3} +6.19042 q^{5} -3.19918 q^{7} +50.4545 q^{9} +35.2466 q^{11} +13.0000 q^{13} -54.4808 q^{15} -99.5410 q^{17} -23.9567 q^{19} +28.1554 q^{21} -147.290 q^{23} -86.6787 q^{25} -206.419 q^{27} +199.870 q^{29} +24.9348 q^{31} -310.199 q^{33} -19.8042 q^{35} -325.116 q^{37} -114.411 q^{39} -389.479 q^{41} -422.973 q^{43} +312.334 q^{45} +391.432 q^{47} -332.765 q^{49} +876.043 q^{51} -356.250 q^{53} +218.191 q^{55} +210.839 q^{57} +473.323 q^{59} +312.979 q^{61} -161.413 q^{63} +80.4754 q^{65} +99.3166 q^{67} +1296.27 q^{69} -811.986 q^{71} +960.819 q^{73} +762.844 q^{75} -112.760 q^{77} -701.850 q^{79} +454.385 q^{81} +452.741 q^{83} -616.201 q^{85} -1759.02 q^{87} -849.529 q^{89} -41.5893 q^{91} -219.446 q^{93} -148.302 q^{95} +1253.84 q^{97} +1778.35 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 3 q - 8 q^{5} - 36 q^{7} + 73 q^{9}+O(q^{10}) \) 3 * q - 8 * q^5 - 36 * q^7 + 73 * q^9	\( 3 q - 8 q^{5} - 36 q^{7} + 73 q^{9} - 52 q^{11} + 39 q^{13} - 244 q^{15} - 116 q^{17} - 124 q^{19} - 154 q^{21} - 232 q^{23} + 191 q^{25} - 12 q^{27} - 30 q^{29} - 240 q^{31} - 564 q^{33} + 340 q^{35} - 264 q^{37} - 374 q^{41} - 248 q^{43} - 966 q^{45} + 412 q^{47} - 443 q^{49} - 92 q^{51} - 386 q^{53} + 400 q^{55} + 52 q^{57} + 940 q^{59} + 1206 q^{61} - 864 q^{63} - 104 q^{65} + 564 q^{67} + 512 q^{69} - 1260 q^{71} + 142 q^{73} + 3788 q^{75} + 1188 q^{77} - 1040 q^{79} + 1571 q^{81} - 756 q^{83} + 2510 q^{85} - 1536 q^{87} - 18 q^{89} - 468 q^{91} - 768 q^{93} - 384 q^{95} + 2174 q^{97} + 1940 q^{99}+O(q^{100}) \) 3 * q - 8 * q^5 - 36 * q^7 + 73 * q^9 - 52 * q^11 + 39 * q^13 - 244 * q^15 - 116 * q^17 - 124 * q^19 - 154 * q^21 - 232 * q^23 + 191 * q^25 - 12 * q^27 - 30 * q^29 - 240 * q^31 - 564 * q^33 + 340 * q^35 - 264 * q^37 - 374 * q^41 - 248 * q^43 - 966 * q^45 + 412 * q^47 - 443 * q^49 - 92 * q^51 - 386 * q^53 + 400 * q^55 + 52 * q^57 + 940 * q^59 + 1206 * q^61 - 864 * q^63 - 104 * q^65 + 564 * q^67 + 512 * q^69 - 1260 * q^71 + 142 * q^73 + 3788 * q^75 + 1188 * q^77 - 1040 * q^79 + 1571 * q^81 - 756 * q^83 + 2510 * q^85 - 1536 * q^87 - 18 * q^89 - 468 * q^91 - 768 * q^93 - 384 * q^95 + 2174 * q^97 + 1940 * q^99

Introduction

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