LMFDB - Embedded newform 208.4.f.e.129.10

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [208,4,Mod(129,208)]

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, 1]))

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

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

chi := DirichletCharacter("208.129");

S:= CuspForms(chi, 4);

N := Newforms(S);

Level:	$ N $	$=$	$ 208 = 2^{4} \cdot 13 $
Weight:	$ k $	$=$	$ 4 $
Character orbit:	$[\chi]$	$=$	208.f (of order $2$, degree $1$, not minimal)

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:	no
Analytic conductor:	$12.2723972812$
Analytic rank:	$0$
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} + 170x^{8} + 8945x^{6} + 145432x^{4} + 614160x^{2} + 20736 $ x^10 + 170x^8 + 8945x^6 + 145432x^4 + 614160x^2 + 20736
Coefficient ring:	$\Z[a_1, \ldots, a_{13}]$
Coefficient ring index:	$ 2^{17} $
Twist minimal:	no (minimal twist has level 104)
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label			129.10
Root			$-8.78626i$ of defining polynomial
Character	$\chi$	$=$	208.129
Dual form			208.4.f.e.129.9

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+7.78626 q^{3} +7.31325i q^{5} +0.135778i q^{7} +33.6258 q^{9} +O(q^{10})$	$q+7.78626 q^{3} +7.31325i q^{5} +0.135778i q^{7} +33.6258 q^{9} +48.8071i q^{11} +(36.8433 - 28.9753i) q^{13} +56.9429i q^{15} +68.4219 q^{17} +83.8596i q^{19} +1.05720i q^{21} -11.7885 q^{23} +71.5163 q^{25} +51.5900 q^{27} -177.612 q^{29} -197.881i q^{31} +380.024i q^{33} -0.992978 q^{35} -283.473i q^{37} +(286.872 - 225.609i) q^{39} -70.5556i q^{41} -28.3737 q^{43} +245.914i q^{45} +536.222i q^{47} +342.982 q^{49} +532.750 q^{51} -5.84895 q^{53} -356.938 q^{55} +652.952i q^{57} -304.585i q^{59} -378.273 q^{61} +4.56564i q^{63} +(211.904 + 269.445i) q^{65} +698.973i q^{67} -91.7881 q^{69} -922.474i q^{71} -1144.97i q^{73} +556.845 q^{75} -6.62692 q^{77} -464.652 q^{79} -506.203 q^{81} -328.257i q^{83} +500.387i q^{85} -1382.94 q^{87} -737.841i q^{89} +(3.93420 + 5.00251i) q^{91} -1540.75i q^{93} -613.287 q^{95} -944.961i q^{97} +1641.18i q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 10 q - 6 q^{3} + 72 q^{9}+O(q^{10}) $ 10 * q - 6 * q^3 + 72 * q^9	$ 10 q - 6 q^{3} + 72 q^{9} - 24 q^{13} + 58 q^{17} - 180 q^{23} + 28 q^{25} - 354 q^{27} - 392 q^{29} - 154 q^{35} + 532 q^{39} + 234 q^{43} - 128 q^{49} + 510 q^{51} - 1244 q^{53} + 576 q^{55} - 56 q^{61} + 566 q^{65} + 1748 q^{69} - 472 q^{75} - 304 q^{77} - 1908 q^{79} + 2282 q^{81} + 64 q^{87} - 582 q^{91} + 2340 q^{95}+O(q^{100}) $ 10 * q - 6 * q^3 + 72 * q^9 - 24 * q^13 + 58 * q^17 - 180 * q^23 + 28 * q^25 - 354 * q^27 - 392 * q^29 - 154 * q^35 + 532 * q^39 + 234 * q^43 - 128 * q^49 + 510 * q^51 - 1244 * q^53 + 576 * q^55 - 56 * q^61 + 566 * q^65 + 1748 * q^69 - 472 * q^75 - 304 * q^77 - 1908 * q^79 + 2282 * q^81 + 64 * q^87 - 582 * q^91 + 2340 * q^95

Display 100 coefficients 10 coefficients

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/208\mathbb{Z}\right)^\times$.

$n$	$53$	$79$	$145$
$\chi(n)$	$1$	$1$	$-1$

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
$3$	7.78626			1.49847			0.749233	−	0.662307i	$-0.230422\pi$
$3$	7.78626			1.49847			0.749233	+	0.662307i	$0.230422\pi$
$4$		0			0
$5$			7.31325i			0.654117i	0.945004	+	0.327059i	$0.106057\pi$
$5$			7.31325i			0.654117i	−0.945004	+	0.327059i	$0.893943\pi$
$6$		0			0
$7$			0.135778i			0.00733131i	0.999993	+	0.00366566i	$0.00116682\pi$
$7$			0.135778i			0.00733131i	−0.999993	+	0.00366566i	$0.998833\pi$
$8$		0			0
$9$	33.6258			1.24540
$10$		0			0
$11$			48.8071i			1.33781i	0.743349	+	0.668904i	$0.233236\pi$
$11$			48.8071i			1.33781i	−0.743349	+	0.668904i	$0.766764\pi$
$12$		0			0
$13$	36.8433	−	28.9753i	0.786039	−	0.618177i
$13$	36.8433	−	28.9753i	0.786039	−	0.618177i
$14$		0			0
$15$			56.9429i			0.980172i
$16$		0			0
$17$	68.4219			0.976162			0.488081	−	0.872798i	$-0.337697\pi$
$17$	68.4219			0.976162			0.488081	+	0.872798i	$0.337697\pi$
$18$		0			0
$19$			83.8596i			1.01256i	0.862368	+	0.506282i	$0.168981\pi$
$19$			83.8596i			1.01256i	−0.862368	+	0.506282i	$0.831019\pi$
$20$		0			0
$21$			1.05720i			0.0109857i
$22$		0			0
$23$	−11.7885			−0.106873			−0.0534363	−	0.998571i	$-0.517017\pi$
$23$	−11.7885			−0.106873			−0.0534363	+	0.998571i	$0.517017\pi$
$24$		0			0
$25$	71.5163			0.572131
$26$		0			0
$27$	51.5900			0.367722
$28$		0			0
$29$	−177.612			−1.13730			−0.568652	−	0.822578i	$-0.692535\pi$
$29$	−177.612			−1.13730			−0.568652	+	0.822578i	$0.692535\pi$
$30$		0			0
$31$		−	197.881i		−	1.14647i	−0.819392	−	0.573234i	$-0.805689\pi$
$31$		−	197.881i		−	1.14647i	0.819392	−	0.573234i	$-0.194311\pi$
$32$		0			0
$33$			380.024i			2.00466i
$34$		0			0
$35$	−0.992978			−0.00479554
$36$		0			0
$37$		−	283.473i		−	1.25953i	−0.776784	−	0.629767i	$-0.783150\pi$
$37$		−	283.473i		−	1.25953i	0.776784	−	0.629767i	$-0.216850\pi$
$38$		0			0
$39$	286.872	−	225.609i	1.17785	−	0.926317i
$40$		0			0
$41$		−	70.5556i		−	0.268755i	−0.990930	−	0.134377i	$-0.957097\pi$
$41$		−	70.5556i		−	0.268755i	0.990930	−	0.134377i	$-0.0429034\pi$
$42$		0			0
$43$	−28.3737			−0.100627			−0.0503134	−	0.998733i	$-0.516022\pi$
$43$	−28.3737			−0.100627			−0.0503134	+	0.998733i	$0.516022\pi$
$44$		0			0
$45$			245.914i			0.814637i
$46$		0			0
$47$			536.222i			1.66417i	0.554648	+	0.832085i	$0.312853\pi$
$47$			536.222i			1.66417i	−0.554648	+	0.832085i	$0.687147\pi$
$48$		0			0
$49$	342.982			0.999946
$50$		0			0
$51$	532.750			1.46274
$52$		0			0
$53$	−5.84895			−0.0151588			−0.00757938	−	0.999971i	$-0.502413\pi$
$53$	−5.84895			−0.0151588			−0.00757938	+	0.999971i	$0.502413\pi$
$54$		0			0
$55$	−356.938			−0.875083
$56$		0			0
$57$			652.952i			1.51729i
$58$		0			0
$59$		−	304.585i		−	0.672094i	−0.941845	−	0.336047i	$-0.890910\pi$
$59$		−	304.585i		−	0.672094i	0.941845	−	0.336047i	$-0.109090\pi$
$60$		0			0
$61$	−378.273			−0.793981			−0.396991	−	0.917823i	$-0.629945\pi$
$61$	−378.273			−0.793981			−0.396991	+	0.917823i	$0.629945\pi$
$62$		0			0
$63$			4.56564i			0.00913041i
$64$		0			0
$65$	211.904	+	269.445i	0.404360	+	0.514161i
$66$		0			0
$67$			698.973i			1.27452i	0.770647	+	0.637262i	$0.219933\pi$
$67$			698.973i			1.27452i	−0.770647	+	0.637262i	$0.780067\pi$
$68$		0			0
$69$	−91.7881			−0.160145
$70$		0			0
$71$		−	922.474i		−	1.54194i	−0.636873	−	0.770969i	$-0.719773\pi$
$71$		−	922.474i		−	1.54194i	0.636873	−	0.770969i	$-0.280227\pi$
$72$		0			0
$73$		−	1144.97i		−	1.83573i	−0.396896	−	0.917864i	$-0.629912\pi$
$73$		−	1144.97i		−	1.83573i	0.396896	−	0.917864i	$-0.370088\pi$
$74$		0			0
$75$	556.845			0.857318
$76$		0			0
$77$	−6.62692			−0.00980789
$78$		0			0
$79$	−464.652			−0.661740			−0.330870	−	0.943676i	$-0.607342\pi$
$79$	−464.652			−0.661740			−0.330870	+	0.943676i	$0.607342\pi$
$80$		0			0
$81$	−506.203			−0.694380
$82$		0			0
$83$		−	328.257i		−	0.434107i	−0.976160	−	0.217054i	$-0.930355\pi$
$83$		−	328.257i		−	0.434107i	0.976160	−	0.217054i	$-0.0696447\pi$
$84$		0			0
$85$			500.387i			0.638524i
$86$		0			0
$87$	−1382.94			−1.70421
$88$		0			0
$89$		−	737.841i		−	0.878775i	−0.898298	−	0.439387i	$-0.855196\pi$
$89$		−	737.841i		−	0.878775i	0.898298	−	0.439387i	$-0.144804\pi$
$90$		0			0
$91$	3.93420	+	5.00251i	0.00453205	+	0.00576270i
$92$		0			0
$93$		−	1540.75i		−	1.71794i
$94$		0			0
$95$	−613.287			−0.662335
$96$		0			0
$97$		−	944.961i		−	0.989137i	−0.869139	−	0.494568i	$-0.835326\pi$
$97$		−	944.961i		−	0.989137i	0.869139	−	0.494568i	$-0.164674\pi$
$98$		0			0
$99$			1641.18i			1.66610i
$100$		0			0
$101$	476.464			0.469406			0.234703	−	0.972067i	$-0.424588\pi$
$101$	476.464			0.469406			0.234703	+	0.972067i	$0.424588\pi$
$102$		0			0
$103$	−1935.85			−1.85190			−0.925948	−	0.377652i	$-0.876732\pi$
$103$	−1935.85			−1.85190			−0.925948	+	0.377652i	$0.876732\pi$
$104$		0			0
$105$	−7.73158			−0.00718595
$106$		0			0
$107$	−621.442			−0.561468			−0.280734	−	0.959786i	$-0.590578\pi$
$107$	−621.442			−0.561468			−0.280734	+	0.959786i	$0.590578\pi$
$108$		0			0
$109$		−	182.483i		−	0.160355i	−0.996781	−	0.0801775i	$-0.974451\pi$
$109$		−	182.483i		−	0.160355i	0.996781	−	0.0801775i	$-0.0255487\pi$
$110$		0			0
$111$		−	2207.20i		−	1.88737i
$112$		0			0
$113$	−2328.76			−1.93868			−0.969341	−	0.245719i	$-0.920976\pi$
$113$	−2328.76			−1.93868			−0.969341	+	0.245719i	$0.920976\pi$
$114$		0			0
$115$		−	86.2121i		−	0.0699072i
$116$		0			0
$117$	1238.89	−	974.317i	0.978932	−	0.769877i
$118$		0			0
$119$			9.29018i			0.00715655i
$120$		0			0
$121$	−1051.13			−0.789730
$122$		0			0
$123$		−	549.364i		−	0.402719i
$124$		0			0
$125$			1437.17i			1.02836i
$126$		0			0
$127$	1994.85			1.39381			0.696906	−	0.717162i	$-0.254559\pi$
$127$	1994.85			1.39381			0.696906	+	0.717162i	$0.254559\pi$
$128$		0			0
$129$	−220.925			−0.150786
$130$		0			0
$131$	2712.93			1.80939			0.904694	−	0.426062i	$-0.140100\pi$
$131$	2712.93			1.80939			0.904694	+	0.426062i	$0.140100\pi$
$132$		0			0
$133$	−11.3863			−0.00742342
$134$		0			0
$135$			377.291i			0.240533i
$136$		0			0
$137$			2714.11i			1.69257i	0.532729	+	0.846286i	$0.321167\pi$
$137$			2714.11i			1.69257i	−0.532729	+	0.846286i	$0.678833\pi$
$138$		0			0
$139$	−2576.46			−1.57217			−0.786087	−	0.618116i	$-0.787896\pi$
$139$	−2576.46			−1.57217			−0.786087	+	0.618116i	$0.787896\pi$
$140$		0			0
$141$			4175.16i			2.49370i
$142$		0			0
$143$	1414.20	+	1798.22i	0.827002	+	1.05157i
$144$		0			0
$145$		−	1298.92i		−	0.743930i
$146$		0			0
$147$	2670.54			1.49839
$148$		0			0
$149$			623.998i			0.343086i	0.985177	+	0.171543i	$0.0548753\pi$
$149$			623.998i			0.343086i	−0.985177	+	0.171543i	$0.945125\pi$
$150$		0			0
$151$		−	1625.53i		−	0.876049i	−0.898963	−	0.438025i	$-0.855678\pi$
$151$		−	1625.53i		−	0.876049i	0.898963	−	0.438025i	$-0.144322\pi$
$152$		0			0
$153$	2300.74			1.21571
$154$		0			0
$155$	1447.15			0.749924
$156$		0			0
$157$	2614.41			1.32900			0.664500	−	0.747288i	$-0.268645\pi$
$157$	2614.41			1.32900			0.664500	+	0.747288i	$0.268645\pi$
$158$		0			0
$159$	−45.5414			−0.0227149
$160$		0			0
$161$		−	1.60061i		−	0.000783516i
$162$		0			0
$163$			379.462i			0.182342i	0.995835	+	0.0911710i	$0.0290610\pi$
$163$			379.462i			0.182342i	−0.995835	+	0.0911710i	$0.970939\pi$
$164$		0			0
$165$	−2779.21			−1.31128
$166$		0			0
$167$		−	2832.36i		−	1.31242i	−0.754577	−	0.656212i	$-0.772158\pi$
$167$		−	2832.36i		−	1.31242i	0.754577	−	0.656212i	$-0.227842\pi$
$168$		0			0
$169$	517.864	−	2135.09i	0.235714	−	0.971822i
$170$		0			0
$171$			2819.84i			1.26105i
$172$		0			0
$173$	2263.45			0.994720			0.497360	−	0.867544i	$-0.334303\pi$
$173$	2263.45			0.994720			0.497360	+	0.867544i	$0.334303\pi$
$174$		0			0
$175$			9.71034i			0.00419447i
$176$		0			0
$177$		−	2371.57i		−	1.00711i
$178$		0			0
$179$	1451.51			0.606096			0.303048	−	0.952975i	$-0.401996\pi$
$179$	1451.51			0.606096			0.303048	+	0.952975i	$0.401996\pi$
$180$		0			0
$181$	−2219.59			−0.911495			−0.455747	−	0.890109i	$-0.650628\pi$
$181$	−2219.59			−0.911495			−0.455747	+	0.890109i	$0.650628\pi$
$182$		0			0
$183$	−2945.33			−1.18975
$184$		0			0
$185$	2073.11			0.823882
$186$		0			0
$187$			3339.47i			1.30592i
$188$		0			0
$189$			7.00478i			0.00269589i
$190$		0			0
$191$	−50.7725			−0.0192344			−0.00961719	−	0.999954i	$-0.503061\pi$
$191$	−50.7725			−0.0192344			−0.00961719	+	0.999954i	$0.503061\pi$
$192$		0			0
$193$			3155.03i			1.17671i	0.808604	+	0.588353i	$0.200223\pi$
$193$			3155.03i			1.17671i	−0.808604	+	0.588353i	$0.799777\pi$
$194$		0			0
$195$	1649.94	+	2097.96i	0.605920	+	0.770453i
$196$		0			0
$197$			418.624i			0.151400i	0.997131	+	0.0756999i	$0.0241191\pi$
$197$			418.624i			0.151400i	−0.997131	+	0.0756999i	$0.975881\pi$
$198$		0			0
$199$	−2308.41			−0.822307			−0.411153	−	0.911566i	$-0.634874\pi$
$199$	−2308.41			−0.822307			−0.411153	+	0.911566i	$0.634874\pi$
$200$		0			0
$201$			5442.38i			1.90983i
$202$		0			0
$203$		−	24.1158i		−	0.00833793i
$204$		0			0
$205$	515.991			0.175797
$206$		0			0
$207$	−396.397			−0.133099
$208$		0			0
$209$	−4092.94			−1.35462
$210$		0			0
$211$	969.591			0.316348			0.158174	−	0.987411i	$-0.449439\pi$
$211$	969.591			0.316348			0.158174	+	0.987411i	$0.449439\pi$
$212$		0			0
$213$		−	7182.62i		−	2.31054i
$214$		0			0
$215$		−	207.504i		−	0.0658217i
$216$		0			0
$217$	26.8679			0.00840511
$218$		0			0
$219$		−	8915.00i		−	2.75077i
$220$		0			0
$221$	2520.89	−	1982.55i	0.767301	−	0.603441i
$222$		0			0
$223$			1972.73i			0.592392i	0.955127	+	0.296196i	$0.0957182\pi$
$223$			1972.73i			0.592392i	−0.955127	+	0.296196i	$0.904282\pi$
$224$		0			0
$225$	2404.79			0.712531
$226$		0			0
$227$			3899.14i			1.14007i	0.821622	+	0.570033i	$0.193070\pi$
$227$			3899.14i			1.14007i	−0.821622	+	0.570033i	$0.806930\pi$
$228$		0			0
$229$		−	1949.05i		−	0.562431i	−0.959645	−	0.281216i	$-0.909262\pi$
$229$		−	1949.05i		−	0.562431i	0.959645	−	0.281216i	$-0.0907376\pi$
$230$		0			0
$231$	−51.5989			−0.0146968
$232$		0			0
$233$	4682.21			1.31649			0.658245	−	0.752804i	$-0.271299\pi$
$233$	4682.21			1.31649			0.658245	+	0.752804i	$0.271299\pi$
$234$		0			0
$235$	−3921.52			−1.08856
$236$		0			0
$237$	−3617.90			−0.991594
$238$		0			0
$239$			2759.41i			0.746825i	0.927665	+	0.373412i	$0.121812\pi$
$239$			2759.41i			0.746825i	−0.927665	+	0.373412i	$0.878188\pi$
$240$		0			0
$241$		−	560.962i		−	0.149937i	−0.997186	−	0.0749683i	$-0.976114\pi$
$241$		−	560.962i		−	0.149937i	0.997186	−	0.0749683i	$-0.0238856\pi$
$242$		0			0
$243$	−5334.36			−1.40823
$244$		0			0
$245$			2508.31i			0.654082i
$246$		0			0
$247$	2429.86	+	3089.67i	0.625944	+	0.795915i
$248$		0			0
$249$		−	2555.90i		−	0.650495i
$250$		0			0
$251$	4031.51			1.01381			0.506906	−	0.862001i	$-0.330789\pi$
$251$	4031.51			1.01381			0.506906	+	0.862001i	$0.330789\pi$
$252$		0			0
$253$		−	575.361i		−	0.142975i
$254$		0			0
$255$			3896.14i			0.956806i
$256$		0			0
$257$	5017.70			1.21788			0.608940	−	0.793216i	$-0.291595\pi$
$257$	5017.70			1.21788			0.608940	+	0.793216i	$0.291595\pi$
$258$		0			0
$259$	38.4894			0.00923404
$260$		0			0
$261$	−5972.36			−1.41640
$262$		0			0
$263$	2726.36			0.639219			0.319610	−	0.947549i	$-0.396448\pi$
$263$	2726.36			0.639219			0.319610	+	0.947549i	$0.396448\pi$
$264$		0			0
$265$		−	42.7748i		−	0.00991561i
$266$		0			0
$267$		−	5745.02i		−	1.31681i
$268$		0			0
$269$	558.197			0.126520			0.0632600	−	0.997997i	$-0.479850\pi$
$269$	558.197			0.126520			0.0632600	+	0.997997i	$0.479850\pi$
$270$		0			0
$271$			933.345i			0.209213i	0.994514	+	0.104606i	$0.0333583\pi$
$271$			933.345i			0.209213i	−0.994514	+	0.104606i	$0.966642\pi$
$272$		0			0
$273$	30.6327	+	38.9508i	0.00679112	+	0.00863520i
$274$		0			0
$275$			3490.50i			0.765401i
$276$		0			0
$277$	4099.76			0.889280			0.444640	−	0.895709i	$-0.353332\pi$
$277$	4099.76			0.889280			0.444640	+	0.895709i	$0.353332\pi$
$278$		0			0
$279$		−	6653.91i		−	1.42781i
$280$		0			0
$281$			3507.02i			0.744524i	0.928128	+	0.372262i	$0.121418\pi$
$281$			3507.02i			0.744524i	−0.928128	+	0.372262i	$0.878582\pi$
$282$		0			0
$283$	−3410.06			−0.716279			−0.358140	−	0.933668i	$-0.616589\pi$
$283$	−3410.06			−0.716279			−0.358140	+	0.933668i	$0.616589\pi$
$284$		0			0
$285$	−4775.21			−0.992487
$286$		0			0
$287$	9.57989			0.00197032
$288$		0			0
$289$	−231.444			−0.0471084
$290$		0			0
$291$		−	7357.71i		−	1.48219i
$292$		0			0
$293$			3634.90i			0.724754i	0.932032	+	0.362377i	$0.118035\pi$
$293$			3634.90i			0.724754i	−0.932032	+	0.362377i	$0.881965\pi$
$294$		0			0
$295$	2227.50			0.439628
$296$		0			0
$297$			2517.96i			0.491942i
$298$		0			0
$299$	−434.327	+	341.575i	−0.0840060	+	0.0660662i
$300$		0			0
$301$		−	3.85252i		−	0.000737726i
$302$		0			0
$303$	3709.87			0.703388
$304$		0			0
$305$		−	2766.40i		−	0.519357i
$306$		0			0
$307$		−	8640.88i		−	1.60639i	−0.595718	−	0.803194i	$-0.703132\pi$
$307$		−	8640.88i		−	1.60639i	0.595718	−	0.803194i	$-0.296868\pi$
$308$		0			0
$309$	−15073.0			−2.77500
$310$		0			0
$311$	−1012.08			−0.184533			−0.0922667	−	0.995734i	$-0.529411\pi$
$311$	−1012.08			−0.184533			−0.0922667	+	0.995734i	$0.529411\pi$
$312$		0			0
$313$	−10193.2			−1.84074			−0.920371	−	0.391047i	$-0.872113\pi$
$313$	−10193.2			−1.84074			−0.920371	+	0.391047i	$0.872113\pi$
$314$		0			0
$315$	−33.3896			−0.00597236
$316$		0			0
$317$			7968.03i			1.41176i	0.708330	+	0.705882i	$0.249449\pi$
$317$			7968.03i			1.41176i	−0.708330	+	0.705882i	$0.750551\pi$
$318$		0			0
$319$		−	8668.75i		−	1.52149i
$320$		0			0
$321$	−4838.71			−0.841340
$322$		0			0
$323$			5737.83i			0.988426i
$324$		0			0
$325$	2634.90	−	2072.21i	0.449717	−	0.353678i
$326$		0			0
$327$		−	1420.86i		−	0.240286i
$328$		0			0
$329$	−72.8070			−0.0122006
$330$		0			0
$331$			258.905i			0.0429932i	0.999769	+	0.0214966i	$0.00684310\pi$
$331$			258.905i			0.0429932i	−0.999769	+	0.0214966i	$0.993157\pi$
$332$		0			0
$333$		−	9532.01i		−	1.56862i
$334$		0			0
$335$	−5111.76			−0.833688
$336$		0			0
$337$	−3462.77			−0.559731			−0.279865	−	0.960039i	$-0.590290\pi$
$337$	−3462.77			−0.559731			−0.279865	+	0.960039i	$0.590290\pi$
$338$		0			0
$339$	−18132.3			−2.90505
$340$		0			0
$341$	9658.00			1.53375
$342$		0			0
$343$			93.1411i			0.0146622i
$344$		0			0
$345$		−	671.270i		−	0.104753i
$346$		0			0
$347$	2096.77			0.324382			0.162191	−	0.986759i	$-0.448144\pi$
$347$	2096.77			0.324382			0.162191	+	0.986759i	$0.448144\pi$
$348$		0			0
$349$		−	4336.67i		−	0.665148i	−0.943077	−	0.332574i	$-0.892083\pi$
$349$		−	4336.67i		−	0.665148i	0.943077	−	0.332574i	$-0.107917\pi$
$350$		0			0
$351$	1900.75	−	1494.84i	0.289044	−	0.227318i
$352$		0			0
$353$		−	8900.16i		−	1.34195i	−0.741481	−	0.670974i	$-0.765876\pi$
$353$		−	8900.16i		−	1.34195i	0.741481	−	0.670974i	$-0.234124\pi$
$354$		0			0
$355$	6746.29			1.00861
$356$		0			0
$357$			72.3357i			0.0107238i
$358$		0			0
$359$			6481.04i			0.952803i	0.879228	+	0.476402i	$0.158059\pi$
$359$			6481.04i			0.952803i	−0.879228	+	0.476402i	$0.841941\pi$
$360$		0			0
$361$	−173.435			−0.0252858
$362$		0			0
$363$	−8184.37			−1.18338
$364$		0			0
$365$	8373.42			1.20078
$366$		0			0
$367$	8068.01			1.14754			0.573769	−	0.819017i	$-0.305481\pi$
$367$	8068.01			1.14754			0.573769	+	0.819017i	$0.305481\pi$
$368$		0			0
$369$		−	2372.49i		−	0.334707i
$370$		0			0
$371$		−	0.794157i		−	0.000111134i
$372$		0			0
$373$	1747.04			0.242516			0.121258	−	0.992621i	$-0.461307\pi$
$373$	1747.04			0.242516			0.121258	+	0.992621i	$0.461307\pi$
$374$		0			0
$375$			11190.2i			1.54096i
$376$		0			0
$377$	−6543.84	+	5146.38i	−0.893965	+	0.703055i
$378$		0			0
$379$		−	2594.48i		−	0.351634i	−0.984423	−	0.175817i	$-0.943743\pi$
$379$		−	2594.48i		−	0.351634i	0.984423	−	0.175817i	$-0.0562567\pi$
$380$		0			0
$381$	15532.4			2.08858
$382$		0			0
$383$			3998.61i			0.533471i	0.963770	+	0.266736i	$0.0859451\pi$
$383$			3998.61i			0.533471i	−0.963770	+	0.266736i	$0.914055\pi$
$384$		0			0
$385$		−	48.4643i		−	0.00641551i
$386$		0			0
$387$	−954.088			−0.125320
$388$		0			0
$389$	−7516.57			−0.979705			−0.489853	−	0.871805i	$-0.662949\pi$
$389$	−7516.57			−0.979705			−0.489853	+	0.871805i	$0.662949\pi$
$390$		0			0
$391$	−806.590			−0.104325
$392$		0			0
$393$	21123.6			2.71131
$394$		0			0
$395$		−	3398.12i		−	0.432855i
$396$		0			0
$397$			10396.0i			1.31426i	0.753776	+	0.657132i	$0.228230\pi$
$397$			10396.0i			1.31426i	−0.753776	+	0.657132i	$0.771770\pi$
$398$		0			0
$399$	−88.6565			−0.0111237
$400$		0			0
$401$		−	7101.74i		−	0.884399i	−0.896917	−	0.442199i	$-0.854198\pi$
$401$		−	7101.74i		−	0.884399i	0.896917	−	0.442199i	$-0.145802\pi$
$402$		0			0
$403$	−5733.66	−	7290.60i	−0.708720	−	0.901168i
$404$		0			0
$405$		−	3701.99i		−	0.454206i
$406$		0			0
$407$	13835.5			1.68501
$408$		0			0
$409$		−	2783.48i		−	0.336514i	−0.985743	−	0.168257i	$-0.946186\pi$
$409$		−	2783.48i		−	0.336514i	0.985743	−	0.168257i	$-0.0538139\pi$
$410$		0			0
$411$			21132.8i			2.53626i
$412$		0			0
$413$	41.3558			0.00492733
$414$		0			0
$415$	2400.63			0.283957
$416$		0			0
$417$	−20061.0			−2.35585
$418$		0			0
$419$	8997.41			1.04905			0.524525	−	0.851395i	$-0.324243\pi$
$419$	8997.41			1.04905			0.524525	+	0.851395i	$0.324243\pi$
$420$		0			0
$421$			15874.4i			1.83770i	0.394608	+	0.918850i	$0.370880\pi$
$421$			15874.4i			1.83770i	−0.394608	+	0.918850i	$0.629120\pi$
$422$		0			0
$423$			18030.9i			2.07256i
$424$		0			0
$425$	4893.28			0.558492
$426$		0			0
$427$		−	51.3611i		−	0.00582093i
$428$		0			0
$429$	11011.3	+	14001.4i	1.23923	+	1.57574i
$430$		0			0
$431$			304.502i			0.0340309i	0.999855	+	0.0170155i	$0.00541645\pi$
$431$			304.502i			0.0340309i	−0.999855	+	0.0170155i	$0.994584\pi$
$432$		0			0
$433$	−3265.88			−0.362467			−0.181234	−	0.983440i	$-0.558009\pi$
$433$	−3265.88			−0.362467			−0.181234	+	0.983440i	$0.558009\pi$
$434$		0			0
$435$		−	10113.8i		−	1.11475i
$436$		0			0
$437$		−	988.577i		−	0.108215i
$438$		0			0
$439$	2313.24			0.251492			0.125746	−	0.992062i	$-0.459868\pi$
$439$	2313.24			0.251492			0.125746	+	0.992062i	$0.459868\pi$
$440$		0			0
$441$	11533.0			1.24533
$442$		0			0
$443$	1890.20			0.202723			0.101362	−	0.994850i	$-0.467680\pi$
$443$	1890.20			0.202723			0.101362	+	0.994850i	$0.467680\pi$
$444$		0			0
$445$	5396.02			0.574822
$446$		0			0
$447$			4858.60i			0.514103i
$448$		0			0
$449$		−	33.0543i		−	0.00347423i	−0.999998	−	0.00173711i	$-0.999447\pi$
$449$		−	33.0543i		−	0.00347423i	0.999998	−	0.00173711i	$-0.000552941\pi$
$450$		0			0
$451$	3443.61			0.359542
$452$		0			0
$453$		−	12656.8i		−	1.31273i
$454$		0			0
$455$	−36.5846	+	28.7718i	−0.00376948	+	0.00296449i
$456$		0			0
$457$			14656.9i			1.50027i	0.661285	+	0.750134i	$0.270011\pi$
$457$			14656.9i			1.50027i	−0.661285	+	0.750134i	$0.729989\pi$
$458$		0			0
$459$	3529.89			0.358956
$460$		0			0
$461$			4103.72i			0.414598i	0.978278	+	0.207299i	$0.0664672\pi$
$461$			4103.72i			0.414598i	−0.978278	+	0.207299i	$0.933533\pi$
$462$		0			0
$463$		−	8874.95i		−	0.890829i	−0.895324	−	0.445414i	$-0.853056\pi$
$463$		−	8874.95i		−	0.890829i	0.895324	−	0.445414i	$-0.146944\pi$
$464$		0			0
$465$	11267.9			1.12374
$466$		0			0
$467$	−18695.1			−1.85248			−0.926239	−	0.376937i	$-0.876977\pi$
$467$	−18695.1			−1.85248			−0.926239	+	0.376937i	$0.876977\pi$
$468$		0			0
$469$	−94.9050			−0.00934394
$470$		0			0
$471$	20356.5			1.99146
$472$		0			0
$473$		−	1384.84i		−	0.134619i
$474$		0			0
$475$			5997.33i			0.579319i
$476$		0			0
$477$	−196.675			−0.0188787
$478$		0			0
$479$			5105.34i			0.486992i	0.969902	+	0.243496i	$0.0782942\pi$
$479$			5105.34i			0.486992i	−0.969902	+	0.243496i	$0.921706\pi$
$480$		0			0
$481$	−8213.73	−	10444.1i	−0.778615	−	0.990042i
$482$		0			0
$483$		−	12.4628i		−	0.00117407i
$484$		0			0
$485$	6910.74			0.647011
$486$		0			0
$487$			1654.66i			0.153962i	0.997033	+	0.0769812i	$0.0245282\pi$
$487$			1654.66i			0.153962i	−0.997033	+	0.0769812i	$0.975472\pi$
$488$		0			0
$489$			2954.59i			0.273233i
$490$		0			0
$491$	6034.82			0.554679			0.277340	−	0.960772i	$-0.410547\pi$
$491$	6034.82			0.554679			0.277340	+	0.960772i	$0.410547\pi$
$492$		0			0
$493$	−12152.6			−1.11019
$494$		0			0
$495$	−12002.3			−1.08983
$496$		0			0
$497$	125.252			0.0113044
$498$		0			0
$499$		−	6772.30i		−	0.607555i	−0.952743	−	0.303777i	$-0.901752\pi$
$499$		−	6772.30i		−	0.607555i	0.952743	−	0.303777i	$-0.0982479\pi$
$500$		0			0
$501$		−	22053.5i		−	1.96662i
$502$		0			0
$503$	−15983.1			−1.41680			−0.708402	−	0.705809i	$-0.750584\pi$
$503$	−15983.1			−1.41680			−0.708402	+	0.705809i	$0.750584\pi$
$504$		0			0
$505$			3484.50i			0.307046i
$506$		0			0
$507$	4032.22	−	16624.4i	0.353209	−	1.45624i
$508$		0			0
$509$			10450.5i			0.910042i	0.890481	+	0.455021i	$0.150368\pi$
$509$			10450.5i			0.910042i	−0.890481	+	0.455021i	$0.849632\pi$
$510$		0			0
$511$	155.461			0.0134583
$512$		0			0
$513$			4326.32i			0.372342i
$514$		0			0
$515$		−	14157.4i		−	1.21136i
$516$		0			0
$517$	−26171.4			−2.22634
$518$		0			0
$519$	17623.8			1.49055
$520$		0			0
$521$	−14201.7			−1.19422			−0.597109	−	0.802160i	$-0.703684\pi$
$521$	−14201.7			−1.19422			−0.597109	+	0.802160i	$0.703684\pi$
$522$		0			0
$523$	75.4417			0.00630752			0.00315376	−	0.999995i	$-0.498996\pi$
$523$	75.4417			0.00630752			0.00315376	+	0.999995i	$0.498996\pi$
$524$		0			0
$525$			75.6072i			0.00628527i
$526$		0			0
$527$		−	13539.4i		−	1.11914i
$528$		0			0
$529$	−12028.0			−0.988578
$530$		0			0
$531$		−	10241.9i		−	0.837025i
$532$		0			0
$533$	−2044.37	−	2599.51i	−0.166138	−	0.211252i
$534$		0			0
$535$		−	4544.76i		−	0.367266i
$536$		0			0
$537$	11301.9			0.908214
$538$		0			0
$539$			16739.9i			1.33774i
$540$		0			0
$541$			23842.8i			1.89479i	0.320069	+	0.947394i	$0.396294\pi$
$541$			23842.8i			1.89479i	−0.320069	+	0.947394i	$0.603706\pi$
$542$		0			0
$543$	−17282.3			−1.36584
$544$		0			0
$545$	1334.54			0.104891
$546$		0			0
$547$	−5324.81			−0.416220			−0.208110	−	0.978105i	$-0.566731\pi$
$547$	−5324.81			−0.416220			−0.208110	+	0.978105i	$0.566731\pi$
$548$		0			0
$549$	−12719.7			−0.988824
$550$		0			0
$551$		−	14894.5i		−	1.15159i
$552$		0			0
$553$		−	63.0895i		−	0.00485142i
$554$		0			0
$555$	16141.8			1.23456
$556$		0			0
$557$		−	18886.7i		−	1.43672i	−0.695671	−	0.718361i	$-0.744893\pi$
$557$		−	18886.7i		−	1.43672i	0.695671	−	0.718361i	$-0.255107\pi$
$558$		0			0
$559$	−1045.38	+	822.136i	−0.0790965	+	0.0622051i
$560$		0			0
$561$			26002.0i			1.95687i
$562$		0			0
$563$	241.043			0.0180440			0.00902198	−	0.999959i	$-0.497128\pi$
$563$	241.043			0.0180440			0.00902198	+	0.999959i	$0.497128\pi$
$564$		0			0
$565$		−	17030.8i		−	1.26813i
$566$		0			0
$567$		−	68.7312i		−	0.00509072i
$568$		0			0
$569$	12286.2			0.905209			0.452604	−	0.891711i	$-0.350495\pi$
$569$	12286.2			0.905209			0.452604	+	0.891711i	$0.350495\pi$
$570$		0			0
$571$	−2108.67			−0.154545			−0.0772724	−	0.997010i	$-0.524621\pi$
$571$	−2108.67			−0.154545			−0.0772724	+	0.997010i	$0.524621\pi$
$572$		0			0
$573$	−395.328			−0.0288221
$574$		0			0
$575$	−843.069			−0.0611451
$576$		0			0
$577$		−	21593.9i		−	1.55800i	−0.627024	−	0.779000i	$-0.715727\pi$
$577$		−	21593.9i		−	1.55800i	0.627024	−	0.779000i	$-0.284273\pi$
$578$		0			0
$579$			24565.9i			1.76325i
$580$		0			0
$581$	44.5701			0.00318258
$582$		0			0
$583$		−	285.470i		−	0.0202795i
$584$		0			0
$585$	7125.43	+	9060.29i	0.503590	+	0.640336i
$586$		0			0
$587$		−	11252.6i		−	0.791218i	−0.918419	−	0.395609i	$-0.870534\pi$
$587$		−	11252.6i		−	0.791218i	0.918419	−	0.395609i	$-0.129466\pi$
$588$		0			0
$589$	16594.2			1.16087
$590$		0			0
$591$			3259.52i			0.226867i
$592$		0			0
$593$			19019.5i			1.31709i	0.752540	+	0.658547i	$0.228829\pi$
$593$			19019.5i			1.31709i	−0.752540	+	0.658547i	$0.771171\pi$
$594$		0			0
$595$	−67.9414			−0.00468122
$596$		0			0
$597$	−17973.9			−1.23220
$598$		0			0
$599$	−18023.8			−1.22944			−0.614718	−	0.788747i	$-0.710730\pi$
$599$	−18023.8			−1.22944			−0.614718	+	0.788747i	$0.710730\pi$
$600$		0			0
$601$	13917.0			0.944573			0.472286	−	0.881445i	$-0.343429\pi$
$601$	13917.0			0.944573			0.472286	+	0.881445i	$0.343429\pi$
$602$		0			0
$603$			23503.5i			1.58729i
$604$		0			0
$605$		−	7687.18i		−	0.516576i
$606$		0			0
$607$	−18666.7			−1.24820			−0.624101	−	0.781344i	$-0.714535\pi$
$607$	−18666.7			−1.24820			−0.624101	+	0.781344i	$0.714535\pi$
$608$		0			0
$609$		−	187.772i		−	0.0124941i
$610$		0			0
$611$	15537.2	+	19756.2i	1.02875	+	1.30810i
$612$		0			0
$613$		−	13906.2i		−	0.916259i	−0.888885	−	0.458130i	$-0.848520\pi$
$613$		−	13906.2i		−	0.916259i	0.888885	−	0.458130i	$-0.151480\pi$
$614$		0			0
$615$	4017.64			0.263426
$616$		0			0
$617$		−	23475.9i		−	1.53177i	−0.642975	−	0.765887i	$-0.722300\pi$
$617$		−	23475.9i		−	1.53177i	0.642975	−	0.765887i	$-0.277700\pi$
$618$		0			0
$619$		−	23948.7i		−	1.55505i	−0.628850	−	0.777527i	$-0.716474\pi$
$619$		−	23948.7i		−	1.55505i	0.628850	−	0.777527i	$-0.283526\pi$
$620$		0			0
$621$	−608.168			−0.0392994
$622$		0			0
$623$	100.182			0.00644258
$624$		0			0
$625$	−1570.87			−0.100536
$626$		0			0
$627$	−31868.7			−2.02985
$628$		0			0
$629$		−	19395.8i		−	1.22951i
$630$		0			0
$631$			27232.2i			1.71806i	0.511922	+	0.859032i	$0.328934\pi$
$631$			27232.2i			1.71806i	−0.511922	+	0.859032i	$0.671066\pi$
$632$		0			0
$633$	7549.49			0.474037
$634$		0			0
$635$			14588.8i			0.911717i
$636$		0			0
$637$	12636.6	−	9937.99i	0.785997	−	0.618144i
$638$		0			0
$639$		−	31018.9i		−	1.92033i
$640$		0			0
$641$	−22438.4			−1.38262			−0.691312	−	0.722556i	$-0.742967\pi$
$641$	−22438.4			−1.38262			−0.691312	+	0.722556i	$0.742967\pi$
$642$		0			0
$643$			10428.7i			0.639608i	0.947484	+	0.319804i	$0.103617\pi$
$643$			10428.7i			0.639608i	−0.947484	+	0.319804i	$0.896383\pi$
$644$		0			0
$645$		−	1615.68i		−	0.0986315i
$646$		0			0
$647$	−15871.6			−0.964419			−0.482209	−	0.876056i	$-0.660166\pi$
$647$	−15871.6			−0.964419			−0.482209	+	0.876056i	$0.660166\pi$
$648$		0			0
$649$	14865.9			0.899133
$650$		0			0
$651$	209.200			0.0125948
$652$		0			0
$653$	−15914.4			−0.953719			−0.476859	−	0.878980i	$-0.658225\pi$
$653$	−15914.4			−0.953719			−0.476859	+	0.878980i	$0.658225\pi$
$654$		0			0
$655$			19840.3i			1.18355i
$656$		0			0
$657$		−	38500.4i		−	2.28621i
$658$		0			0
$659$	−6216.03			−0.367439			−0.183719	−	0.982979i	$-0.558814\pi$
$659$	−6216.03			−0.367439			−0.183719	+	0.982979i	$0.558814\pi$
$660$		0			0
$661$			24368.2i			1.43391i	0.697120	+	0.716955i	$0.254465\pi$
$661$			24368.2i			1.43391i	−0.697120	+	0.716955i	$0.745535\pi$
$662$		0			0
$663$	19628.3	−	15436.6i	1.14977	−	0.904235i
$664$		0			0
$665$		−	83.2707i		−	0.00485579i
$666$		0			0
$667$	2093.78			0.121547
$668$		0			0
$669$			15360.1i			0.887679i
$670$		0			0
$671$		−	18462.4i		−	1.06219i
$672$		0			0
$673$	28636.5			1.64020			0.820102	−	0.572217i	$-0.193917\pi$
$673$	28636.5			1.64020			0.820102	+	0.572217i	$0.193917\pi$
$674$		0			0
$675$	3689.53			0.210385
$676$		0			0
$677$	10393.7			0.590047			0.295024	−	0.955490i	$-0.404672\pi$
$677$	10393.7			0.590047			0.295024	+	0.955490i	$0.404672\pi$
$678$		0			0
$679$	128.305			0.00725167
$680$		0			0
$681$			30359.7i			1.70835i
$682$		0			0
$683$		−	821.792i		−	0.0460395i	−0.999735	−	0.0230198i	$-0.992672\pi$
$683$		−	821.792i		−	0.0460395i	0.999735	−	0.0230198i	$-0.00732806\pi$
$684$		0			0
$685$	−19849.0			−1.10714
$686$		0			0
$687$		−	15175.8i		−	0.842784i
$688$		0			0
$689$	−215.495	+	169.475i	−0.0119154	+	0.00937080i
$690$		0			0
$691$		−	16582.9i		−	0.912944i	−0.889738	−	0.456472i	$-0.849113\pi$
$691$		−	16582.9i		−	0.912944i	0.889738	−	0.456472i	$-0.150887\pi$
$692$		0			0
$693$	−222.835			−0.0122147
$694$		0			0
$695$		−	18842.3i		−	1.02839i
$696$		0			0
$697$		−	4827.55i		−	0.262348i
$698$		0			0
$699$	36456.9			1.97271
$700$		0			0
$701$	1226.22			0.0660680			0.0330340	−	0.999454i	$-0.489483\pi$
$701$	1226.22			0.0660680			0.0330340	+	0.999454i	$0.489483\pi$
$702$		0			0
$703$	23772.0			1.27536
$704$		0			0
$705$	−30534.0			−1.63117
$706$		0			0
$707$			64.6933i			0.00344136i
$708$		0			0
$709$		−	9919.13i		−	0.525417i	−0.964875	−	0.262709i	$-0.915384\pi$
$709$		−	9919.13i		−	0.525417i	0.964875	−	0.262709i	$-0.0846158\pi$
$710$		0			0
$711$	−15624.3			−0.824130
$712$		0			0
$713$			2332.72i			0.122526i
$714$		0			0
$715$	−13150.8	+	10342.4i	−0.687849	+	0.540956i
$716$		0			0
$717$			21485.4i			1.11909i
$718$		0			0
$719$	−32459.0			−1.68361			−0.841806	−	0.539781i	$-0.818507\pi$
$719$	−32459.0			−1.68361			−0.841806	+	0.539781i	$0.818507\pi$
$720$		0			0
$721$		−	262.846i		−	0.0135768i
$722$		0			0
$723$		−	4367.79i		−	0.224675i
$724$		0			0
$725$	−12702.2			−0.650687
$726$		0			0
$727$	−12625.9			−0.644113			−0.322056	−	0.946721i	$-0.604374\pi$
$727$	−12625.9			−0.644113			−0.322056	+	0.946721i	$0.604374\pi$
$728$		0			0
$729$	−27867.2			−1.41580
$730$		0			0
$731$	−1941.38			−0.0982279
$732$		0			0
$733$		−	20596.5i		−	1.03786i	−0.854818	−	0.518929i	$-0.826331\pi$
$733$		−	20596.5i		−	1.03786i	0.854818	−	0.518929i	$-0.173669\pi$
$734$		0			0
$735$			19530.3i			0.980119i
$736$		0			0
$737$	−34114.8			−1.70507
$738$		0			0
$739$			19323.8i			0.961892i	0.876750	+	0.480946i	$0.159707\pi$
$739$			19323.8i			0.961892i	−0.876750	+	0.480946i	$0.840293\pi$
$740$		0			0
$741$	18919.5	+	24056.9i	0.937955	+	1.19265i
$742$		0			0
$743$			6351.69i			0.313622i	0.987629	+	0.156811i	$0.0501213\pi$
$743$			6351.69i			0.313622i	−0.987629	+	0.156811i	$0.949879\pi$
$744$		0			0
$745$	−4563.45			−0.224419
$746$		0			0
$747$		−	11037.9i		−	0.540637i
$748$		0			0
$749$		−	84.3781i		−	0.00411630i
$750$		0			0
$751$	15458.6			0.751123			0.375561	−	0.926797i	$-0.377450\pi$
$751$	15458.6			0.751123			0.375561	+	0.926797i	$0.377450\pi$
$752$		0			0
$753$	31390.4			1.51916
$754$		0			0
$755$	11887.9			0.573039
$756$		0			0
$757$	3811.28			0.182990			0.0914950	−	0.995806i	$-0.470835\pi$
$757$	3811.28			0.182990			0.0914950	+	0.995806i	$0.470835\pi$
$758$		0			0
$759$		−	4479.91i		−	0.214243i
$760$		0			0
$761$		−	2482.24i		−	0.118241i	−0.998251	−	0.0591203i	$-0.981170\pi$
$761$		−	2482.24i		−	0.118241i	0.998251	−	0.0591203i	$-0.0188296\pi$
$762$		0			0
$763$	24.7771			0.00117561
$764$		0			0
$765$			16825.9i			0.795217i
$766$		0			0
$767$	−8825.43	−	11221.9i	−0.415473	−	0.528292i
$768$		0			0
$769$		−	3824.97i		−	0.179365i	−0.995970	−	0.0896827i	$-0.971415\pi$
$769$		−	3824.97i		−	0.179365i	0.995970	−	0.0896827i	$-0.0285853\pi$
$770$		0			0
$771$	39069.1			1.82495
$772$		0			0
$773$			4521.07i			0.210364i	0.994453	+	0.105182i	$0.0335426\pi$
$773$			4521.07i			0.210364i	−0.994453	+	0.105182i	$0.966457\pi$
$774$		0			0
$775$		−	14151.7i		−	0.655929i
$776$		0			0
$777$	299.688			0.0138369
$778$		0			0
$779$	5916.77			0.272131
$780$		0			0
$781$	45023.3			2.06282
$782$		0			0
$783$	−9163.03			−0.418212
$784$		0			0
$785$			19119.9i			0.869321i
$786$		0			0
$787$			3882.30i			0.175844i	0.996127	+	0.0879218i	$0.0280226\pi$
$787$			3882.30i			0.175844i	−0.996127	+	0.0879218i	$0.971977\pi$
$788$		0			0
$789$	21228.1			0.957848
$790$		0			0
$791$		−	316.194i		−	0.0142131i
$792$		0			0
$793$	−13936.8	+	10960.6i	−0.624100	+	0.490821i
$794$		0			0
$795$		−	333.056i		−	0.0148582i
$796$		0			0
$797$	−7406.75			−0.329185			−0.164593	−	0.986362i	$-0.552631\pi$
$797$	−7406.75			−0.329185			−0.164593	+	0.986362i	$0.552631\pi$
$798$		0			0
$799$			36689.3i			1.62450i
$800$		0			0
$801$		−	24810.5i		−	1.09443i
$802$		0			0
$803$	55882.4			2.45585
$804$		0			0
$805$	11.7057			0.000512511
$806$		0			0
$807$	4346.27			0.189586
$808$		0			0
$809$	15770.5			0.685368			0.342684	−	0.939451i	$-0.388664\pi$
$809$	15770.5			0.685368			0.342684	+	0.939451i	$0.388664\pi$
$810$		0			0
$811$			2376.73i			0.102908i	0.998675	+	0.0514540i	$0.0163855\pi$
$811$			2376.73i			0.102908i	−0.998675	+	0.0514540i	$0.983614\pi$
$812$		0			0
$813$			7267.26i			0.313498i
$814$		0			0
$815$	−2775.10			−0.119273
$816$		0			0
$817$		−	2379.41i		−	0.101891i
$818$		0			0
$819$	132.291	+	168.213i	0.00564421	+	0.00717686i
$820$		0			0
$821$			32219.8i			1.36964i	0.728711	+	0.684822i	$0.240120\pi$
$821$			32219.8i			1.36964i	−0.728711	+	0.684822i	$0.759880\pi$
$822$		0			0
$823$	−15094.3			−0.639312			−0.319656	−	0.947534i	$-0.603567\pi$
$823$	−15094.3			−0.639312			−0.319656	+	0.947534i	$0.603567\pi$
$824$		0			0
$825$			27178.0i			1.14693i
$826$		0			0
$827$			24384.3i			1.02530i	0.858597	+	0.512651i	$0.171337\pi$
$827$			24384.3i			1.02530i	−0.858597	+	0.512651i	$0.828663\pi$
$828$		0			0
$829$	31375.8			1.31451			0.657254	−	0.753669i	$-0.271718\pi$
$829$	31375.8			1.31451			0.657254	+	0.753669i	$0.271718\pi$
$830$		0			0
$831$	31921.8			1.33256
$832$		0			0
$833$	23467.4			0.976109
$834$		0			0
$835$	20713.8			0.858479
$836$		0			0
$837$		−	10208.7i		−	0.421582i
$838$		0			0
$839$		−	42182.5i		−	1.73576i	−0.496775	−	0.867879i	$-0.665483\pi$
$839$		−	42182.5i		−	1.73576i	0.496775	−	0.867879i	$-0.334517\pi$
$840$		0			0
$841$	7157.20			0.293460
$842$		0			0
$843$			27306.6i			1.11564i
$844$		0			0
$845$	15614.5	+	3787.27i	0.635686	+	0.154185i
$846$		0			0
$847$		−	142.720i		−	0.00578976i
$848$		0			0
$849$	−26551.6			−1.07332
$850$		0			0
$851$			3341.72i			0.134610i
$852$		0			0
$853$			47639.4i			1.91224i	0.292973	+	0.956121i	$0.405355\pi$
$853$			47639.4i			1.91224i	−0.292973	+	0.956121i	$0.594645\pi$
$854$		0			0
$855$	−20622.2			−0.824872
$856$		0			0
$857$	−34662.4			−1.38162			−0.690809	−	0.723037i	$-0.742745\pi$
$857$	−34662.4			−1.38162			−0.690809	+	0.723037i	$0.742745\pi$
$858$		0			0
$859$	19396.4			0.770426			0.385213	−	0.922828i	$-0.374128\pi$
$859$	19396.4			0.770426			0.385213	+	0.922828i	$0.374128\pi$
$860$		0			0
$861$	74.5915			0.00295246
$862$		0			0
$863$		−	27157.8i		−	1.07122i	−0.844465	−	0.535610i	$-0.820082\pi$
$863$		−	27157.8i		−	1.07122i	0.844465	−	0.535610i	$-0.179918\pi$
$864$		0			0
$865$			16553.1i			0.650663i
$866$		0			0
$867$	−1802.08			−0.0705904
$868$		0			0
$869$		−	22678.3i		−	0.885281i
$870$		0			0
$871$	20252.9	+	25752.5i	0.787882	+	1.00183i
$872$		0			0
$873$		−	31775.1i		−	1.23187i
$874$		0			0
$875$	−195.136			−0.00753921
$876$		0			0
$877$		−	18267.6i		−	0.703367i	−0.936119	−	0.351683i	$-0.885609\pi$
$877$		−	18267.6i		−	0.703367i	0.936119	−	0.351683i	$-0.114391\pi$
$878$		0			0
$879$			28302.2i			1.08602i
$880$		0			0
$881$	24390.0			0.932713			0.466357	−	0.884597i	$-0.345566\pi$
$881$	24390.0			0.932713			0.466357	+	0.884597i	$0.345566\pi$
$882$		0			0
$883$	−19285.6			−0.735007			−0.367503	−	0.930022i	$-0.619787\pi$
$883$	−19285.6			−0.735007			−0.367503	+	0.930022i	$0.619787\pi$
$884$		0			0
$885$	17343.9			0.658768
$886$		0			0
$887$	−12668.4			−0.479552			−0.239776	−	0.970828i	$-0.577074\pi$
$887$	−12668.4			−0.479552			−0.239776	+	0.970828i	$0.577074\pi$
$888$		0			0
$889$			270.856i			0.0102185i
$890$		0			0
$891$		−	24706.3i		−	0.928947i
$892$		0			0
$893$	−44967.3			−1.68508
$894$		0			0
$895$			10615.3i			0.396458i
$896$		0			0
$897$	−3381.78	+	2659.59i	−0.125880	+	0.0989979i
$898$		0			0
$899$			35146.2i			1.30388i
$900$		0			0
$901$	−400.196			−0.0147974
$902$		0			0
$903$		−	29.9967i		−	0.00110546i
$904$		0			0
$905$		−	16232.4i		−	0.596224i
$906$		0			0
$907$	−18984.1			−0.694993			−0.347496	−	0.937681i	$-0.612968\pi$
$907$	−18984.1			−0.694993			−0.347496	+	0.937681i	$0.612968\pi$
$908$		0			0
$909$	16021.5			0.584598
$910$		0			0
$911$	43842.9			1.59449			0.797246	−	0.603655i	$-0.206290\pi$
$911$	43842.9			1.59449			0.797246	+	0.603655i	$0.206290\pi$
$912$		0			0
$913$	16021.3			0.580752
$914$		0			0
$915$		−	21539.9i		−	0.778238i
$916$		0			0
$917$			368.356i			0.0132652i
$918$		0			0
$919$	−16856.3			−0.605046			−0.302523	−	0.953142i	$-0.597829\pi$
$919$	−16856.3			−0.605046			−0.302523	+	0.953142i	$0.597829\pi$
$920$		0			0
$921$		−	67280.1i		−	2.40712i
$922$		0			0
$923$	−26729.0	−	33987.0i	−0.953190	−	1.21202i
$924$		0			0
$925$		−	20273.0i		−	0.720618i
$926$		0			0
$927$	−65094.5			−2.30635
$928$		0			0
$929$		−	11638.3i		−	0.411025i	−0.978655	−	0.205512i	$-0.934114\pi$
$929$		−	11638.3i		−	0.411025i	0.978655	−	0.205512i	$-0.0658861\pi$
$930$		0			0
$931$			28762.3i			1.01251i
$932$		0			0
$933$	−7880.33			−0.276517
$934$		0			0
$935$	−24422.4			−0.854222
$936$		0			0
$937$	26226.8			0.914399			0.457199	−	0.889364i	$-0.348853\pi$
$937$	26226.8			0.914399			0.457199	+	0.889364i	$0.348853\pi$
$938$		0			0
$939$	−79366.6			−2.75829
$940$		0			0
$941$			47829.4i			1.65695i	0.560023	+	0.828477i	$0.310792\pi$
$941$			47829.4i			1.65695i	−0.560023	+	0.828477i	$0.689208\pi$
$942$		0			0
$943$			831.744i			0.0287225i
$944$		0			0
$945$	−51.2277			−0.00176343
$946$		0			0
$947$			458.924i			0.0157477i	0.999969	+	0.00787383i	$0.00250634\pi$
$947$			458.924i			0.0157477i	−0.999969	+	0.00787383i	$0.997494\pi$
$948$		0			0
$949$	−33175.7	−	42184.4i	−1.13480	−	1.44295i
$950$		0			0
$951$			62041.1i			2.11548i
$952$		0			0
$953$	3592.12			0.122099			0.0610494	−	0.998135i	$-0.480555\pi$
$953$	3592.12			0.122099			0.0610494	+	0.998135i	$0.480555\pi$
$954$		0			0
$955$		−	371.312i		−	0.0125815i
$956$		0			0
$957$		−	67497.1i		−	2.27991i
$958$		0			0
$959$	−368.516			−0.0124088
$960$		0			0
$961$	−9365.93			−0.314388
$962$		0			0
$963$	−20896.5			−0.699252
$964$		0			0
$965$	−23073.5			−0.769703
$966$		0			0
$967$		−	28648.2i		−	0.952703i	−0.879255	−	0.476352i	$-0.841959\pi$
$967$		−	28648.2i		−	0.952703i	0.879255	−	0.476352i	$-0.158041\pi$
$968$		0			0
$969$			44676.2i			1.48112i
$970$		0			0
$971$	−18717.4			−0.618611			−0.309305	−	0.950963i	$-0.600096\pi$
$971$	−18717.4			−0.618611			−0.309305	+	0.950963i	$0.600096\pi$
$972$		0			0
$973$		−	349.826i		−	0.0115261i
$974$		0			0
$975$	20516.0	−	16134.7i	0.673885	−	0.529975i
$976$		0			0
$977$		−	13854.2i		−	0.453669i	−0.973933	−	0.226835i	$-0.927162\pi$
$977$		−	13854.2i		−	0.453669i	0.973933	−	0.226835i	$-0.0728377\pi$
$978$		0			0
$979$	36011.9			1.17563
$980$		0			0
$981$		−	6136.13i		−	0.199706i
$982$		0			0
$983$		−	30197.3i		−	0.979800i	−0.871779	−	0.489900i	$-0.837033\pi$
$983$		−	30197.3i		−	0.979800i	0.871779	−	0.489900i	$-0.162967\pi$
$984$		0			0
$985$	−3061.50			−0.0990331
$986$		0			0
$987$	−566.894			−0.0182821
$988$		0			0
$989$	334.483			0.0107542
$990$		0			0
$991$	31398.1			1.00645			0.503227	−	0.864155i	$-0.332146\pi$
$991$	31398.1			1.00645			0.503227	+	0.864155i	$0.332146\pi$
$992$		0			0
$993$			2015.90i			0.0644238i
$994$		0			0
$995$		−	16882.0i		−	0.537885i
$996$		0			0
$997$	30961.1			0.983497			0.491749	−	0.870737i	$-0.336358\pi$
$997$	30961.1			0.983497			0.491749	+	0.870737i	$0.336358\pi$
$998$		0			0
$999$		−	14624.4i		−	0.463159i

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	208.4.f.e.129.10		10
4.3	odd	2		104.4.f.a.25.2	yes	10
8.3	odd	2		832.4.f.k.129.9		10
8.5	even	2		832.4.f.l.129.1		10
12.11	even	2		936.4.c.a.649.4		10
13.12	even	2	inner	208.4.f.e.129.9		10
52.31	even	4		1352.4.a.l.1.1		5
52.47	even	4		1352.4.a.k.1.1		5
52.51	odd	2		104.4.f.a.25.1	✓	10
104.51	odd	2		832.4.f.k.129.10		10
104.77	even	2		832.4.f.l.129.2		10
156.155	even	2		936.4.c.a.649.7		10

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
104.4.f.a.25.1	✓	10	52.51	odd	2
104.4.f.a.25.2	yes	10	4.3	odd	2
208.4.f.e.129.9		10	13.12	even	2	inner
208.4.f.e.129.10		10	1.1	even	1	trivial
832.4.f.k.129.9		10	8.3	odd	2
832.4.f.k.129.10		10	104.51	odd	2
832.4.f.l.129.1		10	8.5	even	2
832.4.f.l.129.2		10	104.77	even	2
936.4.c.a.649.4		10	12.11	even	2
936.4.c.a.649.7		10	156.155	even	2
1352.4.a.k.1.1		5	52.47	even	4
1352.4.a.l.1.1		5	52.31	even	4

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.f (of order \(2\), degree \(1\), not minimal)

\(f(q)\)	\(=\)	\(q+7.78626 q^{3} +7.31325i q^{5} +0.135778i q^{7} +33.6258 q^{9} +O(q^{10})\)	\(q+7.78626 q^{3} +7.31325i q^{5} +0.135778i q^{7} +33.6258 q^{9} +48.8071i q^{11} +(36.8433 - 28.9753i) q^{13} +56.9429i q^{15} +68.4219 q^{17} +83.8596i q^{19} +1.05720i q^{21} -11.7885 q^{23} +71.5163 q^{25} +51.5900 q^{27} -177.612 q^{29} -197.881i q^{31} +380.024i q^{33} -0.992978 q^{35} -283.473i q^{37} +(286.872 - 225.609i) q^{39} -70.5556i q^{41} -28.3737 q^{43} +245.914i q^{45} +536.222i q^{47} +342.982 q^{49} +532.750 q^{51} -5.84895 q^{53} -356.938 q^{55} +652.952i q^{57} -304.585i q^{59} -378.273 q^{61} +4.56564i q^{63} +(211.904 + 269.445i) q^{65} +698.973i q^{67} -91.7881 q^{69} -922.474i q^{71} -1144.97i q^{73} +556.845 q^{75} -6.62692 q^{77} -464.652 q^{79} -506.203 q^{81} -328.257i q^{83} +500.387i q^{85} -1382.94 q^{87} -737.841i q^{89} +(3.93420 + 5.00251i) q^{91} -1540.75i q^{93} -613.287 q^{95} -944.961i q^{97} +1641.18i q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 10 q - 6 q^{3} + 72 q^{9}+O(q^{10}) \) 10 * q - 6 * q^3 + 72 * q^9	\( 10 q - 6 q^{3} + 72 q^{9} - 24 q^{13} + 58 q^{17} - 180 q^{23} + 28 q^{25} - 354 q^{27} - 392 q^{29} - 154 q^{35} + 532 q^{39} + 234 q^{43} - 128 q^{49} + 510 q^{51} - 1244 q^{53} + 576 q^{55} - 56 q^{61} + 566 q^{65} + 1748 q^{69} - 472 q^{75} - 304 q^{77} - 1908 q^{79} + 2282 q^{81} + 64 q^{87} - 582 q^{91} + 2340 q^{95}+O(q^{100}) \) 10 * q - 6 * q^3 + 72 * q^9 - 24 * q^13 + 58 * q^17 - 180 * q^23 + 28 * q^25 - 354 * q^27 - 392 * q^29 - 154 * q^35 + 532 * q^39 + 234 * q^43 - 128 * q^49 + 510 * q^51 - 1244 * q^53 + 576 * q^55 - 56 * q^61 + 566 * q^65 + 1748 * q^69 - 472 * q^75 - 304 * q^77 - 1908 * q^79 + 2282 * q^81 + 64 * q^87 - 582 * q^91 + 2340 * q^95

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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