LMFDB - Embedded newform 208.10.a.f.1.2

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [208,10,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, 10, names="a")

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

chi := DirichletCharacter("208.1");

S:= CuspForms(chi, 10);

N := Newforms(S);

Level:	$ N $	$=$	$ 208 = 2^{4} \cdot 13 $
Weight:	$ k $	$=$	$ 10 $
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:	$107.127453922$
Analytic rank:	$1$
Dimension:	$4$
Coefficient field:	$\mathbb{Q}[x]/(x^{4} - \cdots)$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{4} - 2x^{3} - 12623x^{2} - 303924x + 1814436 $ x^4 - 2x^3 - 12623x^2 - 303924*x + 1814436
Coefficient ring:	$\Z[a_1, \ldots, a_{5}]$
Coefficient ring index:	$ 2^{4} $
Twist minimal:	no (minimal twist has level 52)
Fricke sign:	$+1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.2
Root			$-31.2489$ 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)

$f(q)$	$=$	$q-62.9670 q^{3} +1026.55 q^{5} +4497.03 q^{7} -15718.2 q^{9} +33526.0 q^{11} +28561.0 q^{13} -64638.9 q^{15} -309188. q^{17} -343515. q^{19} -283164. q^{21} +635205. q^{23} -899315. q^{25} +2.22910e6 q^{27} -819974. q^{29} -2.76467e6 q^{31} -2.11103e6 q^{33} +4.61644e6 q^{35} -1.71160e7 q^{37} -1.79840e6 q^{39} -1.04193e7 q^{41} +3.30301e7 q^{43} -1.61355e7 q^{45} +2.06974e7 q^{47} -2.01303e7 q^{49} +1.94686e7 q^{51} +8.67661e7 q^{53} +3.44162e7 q^{55} +2.16301e7 q^{57} +2.54505e7 q^{59} +2.17931e7 q^{61} -7.06850e7 q^{63} +2.93194e7 q^{65} +6.00246e7 q^{67} -3.99970e7 q^{69} -1.65751e8 q^{71} -3.19379e7 q^{73} +5.66271e7 q^{75} +1.50767e8 q^{77} -4.12771e8 q^{79} +1.69020e8 q^{81} -5.84636e8 q^{83} -3.17398e8 q^{85} +5.16313e7 q^{87} +1.09589e8 q^{89} +1.28440e8 q^{91} +1.74083e8 q^{93} -3.52636e8 q^{95} +1.08922e9 q^{97} -5.26967e8 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 4 q + 147 q^{3} - 1947 q^{5} - 10251 q^{7} + 22199 q^{9} - 22038 q^{11} + 114244 q^{13} + 137363 q^{15} - 696135 q^{17} + 254502 q^{19} + 165205 q^{21} + 2038992 q^{23} + 16923 q^{25} + 9068877 q^{27} - 6437112 q^{29}+ \cdots - 1154490140 q^{99}+O(q^{100}) $ 4 * q + 147 * q^3 - 1947 * q^5 - 10251 * q^7 + 22199 * q^9 - 22038 * q^11 + 114244 * q^13 + 137363 * q^15 - 696135 * q^17 + 254502 * q^19 + 165205 * q^21 + 2038992 * q^23 + 16923 * q^25 + 9068877 * q^27 - 6437112 * q^29 + 10739008 * q^31 - 13872374 * q^33 + 30660837 * q^35 - 30452151 * q^37 + 4198467 * q^39 - 33969870 * q^41 + 13058901 * q^43 - 54565744 * q^45 + 44894985 * q^47 - 32869065 * q^49 + 83733323 * q^51 - 155596242 * q^53 + 13538650 * q^55 - 224212490 * q^57 - 61432458 * q^59 + 115267998 * q^61 - 203352774 * q^63 - 55608267 * q^65 - 406952918 * q^67 + 289997880 * q^69 - 51911697 * q^71 + 465992648 * q^73 - 1124268872 * q^75 + 198991878 * q^77 - 11428448 * q^79 + 584641484 * q^81 - 796564404 * q^83 + 1528724447 * q^85 - 419924838 * q^87 - 399435336 * q^89 - 292778811 * q^91 - 218300796 * q^93 - 939051978 * q^95 - 1151600100 * q^97 - 1154490140 * q^99

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
$3$	−62.9670	−0.448815	−0.224407	−	0.974495i	$-0.572045\pi$
$3$	−62.9670	−0.448815	−0.224407	+	0.974495i	$0.572045\pi$
$4$	0	0
$5$	1026.55	0.734541	0.367271	−	0.930114i	$-0.380292\pi$
$5$	1026.55	0.734541	0.367271	+	0.930114i	$0.380292\pi$
$6$	0	0
$7$	4497.03	0.707920	0.353960	−	0.935260i	$-0.384835\pi$
$7$	4497.03	0.707920	0.353960	+	0.935260i	$0.384835\pi$
$8$	0	0
$9$	−15718.2	−0.798565
$10$	0	0
$11$	33526.0	0.690422	0.345211	−	0.938525i	$-0.387807\pi$
$11$	33526.0	0.690422	0.345211	+	0.938525i	$0.387807\pi$
$12$	0	0
$13$	28561.0	0.277350
$13$	28561.0	0.277350
$14$	0	0
$15$	−64638.9	−0.329673
$16$	0	0
$17$	−309188.	−0.897847	−0.448924	−	0.893570i	$-0.648192\pi$
$17$	−309188.	−0.897847	−0.448924	+	0.893570i	$0.648192\pi$
$18$	0	0
$19$	−343515.	−0.604720	−0.302360	−	0.953194i	$-0.597775\pi$
$19$	−343515.	−0.604720	−0.302360	+	0.953194i	$0.597775\pi$
$20$	0	0
$21$	−283164.	−0.317725
$22$	0	0
$23$	635205.	0.473303	0.236651	−	0.971595i	$-0.423950\pi$
$23$	635205.	0.473303	0.236651	+	0.971595i	$0.423950\pi$
$24$	0	0
$25$	−899315.	−0.460449
$26$	0	0
$27$	2.22910e6	0.807223
$28$	0	0
$29$	−819974.	−0.215283	−0.107641	−	0.994190i	$-0.534330\pi$
$29$	−819974.	−0.215283	−0.107641	+	0.994190i	$0.534330\pi$
$30$	0	0
$31$	−2.76467e6	−0.537669	−0.268835	−	0.963186i	$-0.586638\pi$
$31$	−2.76467e6	−0.537669	−0.268835	+	0.963186i	$0.586638\pi$
$32$	0	0
$33$	−2.11103e6	−0.309872
$34$	0	0
$35$	4.61644e6	0.519997
$36$	0	0
$37$	−1.71160e7	−1.50140	−0.750698	−	0.660645i	$-0.770283\pi$
$37$	−1.71160e7	−1.50140	−0.750698	+	0.660645i	$0.770283\pi$
$38$	0	0
$39$	−1.79840e6	−0.124479
$40$	0	0
$41$	−1.04193e7	−0.575854	−0.287927	−	0.957652i	$-0.592966\pi$
$41$	−1.04193e7	−0.575854	−0.287927	+	0.957652i	$0.592966\pi$
$42$	0	0
$43$	3.30301e7	1.47333	0.736667	−	0.676256i	$-0.236399\pi$
$43$	3.30301e7	1.47333	0.736667	+	0.676256i	$0.236399\pi$
$44$	0	0
$45$	−1.61355e7	−0.586579
$46$	0	0
$47$	2.06974e7	0.618693	0.309347	−	0.950949i	$-0.399890\pi$
$47$	2.06974e7	0.618693	0.309347	+	0.950949i	$0.399890\pi$
$48$	0	0
$49$	−2.01303e7	−0.498849
$50$	0	0
$51$	1.94686e7	0.402967
$52$	0	0
$53$	8.67661e7	1.51046	0.755229	−	0.655461i	$-0.227526\pi$
$53$	8.67661e7	1.51046	0.755229	+	0.655461i	$0.227526\pi$
$54$	0	0
$55$	3.44162e7	0.507144
$56$	0	0
$57$	2.16301e7	0.271408
$58$	0	0
$59$	2.54505e7	0.273440	0.136720	−	0.990610i	$-0.456344\pi$
$59$	2.54505e7	0.273440	0.136720	+	0.990610i	$0.456344\pi$
$60$	0	0
$61$	2.17931e7	0.201528	0.100764	−	0.994910i	$-0.467871\pi$
$61$	2.17931e7	0.201528	0.100764	+	0.994910i	$0.467871\pi$
$62$	0	0
$63$	−7.06850e7	−0.565321
$64$	0	0
$65$	2.93194e7	0.203725
$66$	0	0
$67$	6.00246e7	0.363909	0.181954	−	0.983307i	$-0.441758\pi$
$67$	6.00246e7	0.363909	0.181954	+	0.983307i	$0.441758\pi$
$68$	0	0
$69$	−3.99970e7	−0.212425
$70$	0	0
$71$	−1.65751e8	−0.774094	−0.387047	−	0.922060i	$-0.626505\pi$
$71$	−1.65751e8	−0.774094	−0.387047	+	0.922060i	$0.626505\pi$
$72$	0	0
$73$	−3.19379e7	−0.131630	−0.0658148	−	0.997832i	$-0.520965\pi$
$73$	−3.19379e7	−0.131630	−0.0658148	+	0.997832i	$0.520965\pi$
$74$	0	0
$75$	5.66271e7	0.206656
$76$	0	0
$77$	1.50767e8	0.488764
$78$	0	0
$79$	−4.12771e8	−1.19231	−0.596153	−	0.802871i	$-0.703305\pi$
$79$	−4.12771e8	−1.19231	−0.596153	+	0.802871i	$0.703305\pi$
$80$	0	0
$81$	1.69020e8	0.436271
$82$	0	0
$83$	−5.84636e8	−1.35218	−0.676089	−	0.736820i	$-0.736327\pi$
$83$	−5.84636e8	−1.35218	−0.676089	+	0.736820i	$0.736327\pi$
$84$	0	0
$85$	−3.17398e8	−0.659506
$86$	0	0
$87$	5.16313e7	0.0966221
$88$	0	0
$89$	1.09589e8	0.185145	0.0925725	−	0.995706i	$-0.470491\pi$
$89$	1.09589e8	0.185145	0.0925725	+	0.995706i	$0.470491\pi$
$90$	0	0
$91$	1.28440e8	0.196342
$92$	0	0
$93$	1.74083e8	0.241314
$94$	0	0
$95$	−3.52636e8	−0.444192
$96$	0	0
$97$	1.08922e9	1.24923	0.624617	−	0.780931i	$-0.285255\pi$
$97$	1.08922e9	1.24923	0.624617	+	0.780931i	$0.285255\pi$
$98$	0	0
$99$	−5.26967e8	−0.551347
$100$	0	0
$101$	−1.48654e9	−1.42145	−0.710725	−	0.703470i	$-0.751633\pi$
$101$	−1.48654e9	−1.42145	−0.710725	+	0.703470i	$0.751633\pi$
$102$	0	0
$103$	−1.02890e9	−0.900752	−0.450376	−	0.892839i	$-0.648710\pi$
$103$	−1.02890e9	−0.900752	−0.450376	+	0.892839i	$0.648710\pi$
$104$	0	0
$105$	−2.90683e8	−0.233382
$106$	0	0
$107$	1.04055e9	0.767423	0.383712	−	0.923453i	$-0.374646\pi$
$107$	1.04055e9	0.767423	0.383712	+	0.923453i	$0.374646\pi$
$108$	0	0
$109$	−1.57399e9	−1.06803	−0.534016	−	0.845475i	$-0.679318\pi$
$109$	−1.57399e9	−1.06803	−0.534016	+	0.845475i	$0.679318\pi$
$110$	0	0
$111$	1.07775e9	0.673849
$112$	0	0
$113$	−1.04518e9	−0.603030	−0.301515	−	0.953461i	$-0.597492\pi$
$113$	−1.04518e9	−0.603030	−0.301515	+	0.953461i	$0.597492\pi$
$114$	0	0
$115$	6.52072e8	0.347660
$116$	0	0
$117$	−4.48926e8	−0.221482
$118$	0	0
$119$	−1.39043e9	−0.635604
$120$	0	0
$121$	−1.23395e9	−0.523317
$122$	0	0
$123$	6.56074e8	0.258452
$124$	0	0
$125$	−2.92818e9	−1.07276
$126$	0	0
$127$	−3.09838e9	−1.05686	−0.528430	−	0.848977i	$-0.677219\pi$
$127$	−3.09838e9	−1.05686	−0.528430	+	0.848977i	$0.677219\pi$
$128$	0	0
$129$	−2.07980e9	−0.661254
$130$	0	0
$131$	−2.06224e9	−0.611814	−0.305907	−	0.952061i	$-0.598960\pi$
$131$	−2.06224e9	−0.611814	−0.305907	+	0.952061i	$0.598960\pi$
$132$	0	0
$133$	−1.54480e9	−0.428094
$134$	0	0
$135$	2.28829e9	0.592939
$136$	0	0
$137$	−2.86422e9	−0.694647	−0.347323	−	0.937745i	$-0.612909\pi$
$137$	−2.86422e9	−0.694647	−0.347323	+	0.937745i	$0.612909\pi$
$138$	0	0
$139$	8.90327e8	0.202294	0.101147	−	0.994871i	$-0.467749\pi$
$139$	8.90327e8	0.202294	0.101147	+	0.994871i	$0.467749\pi$
$140$	0	0
$141$	−1.30325e9	−0.277679
$142$	0	0
$143$	9.57536e8	0.191489
$144$	0	0
$145$	−8.41747e8	−0.158134
$146$	0	0
$147$	1.26755e9	0.223891
$148$	0	0
$149$	5.73412e9	0.953079	0.476539	−	0.879153i	$-0.341891\pi$
$149$	5.73412e9	0.953079	0.476539	+	0.879153i	$0.341891\pi$
$150$	0	0
$151$	−4.86470e9	−0.761483	−0.380741	−	0.924682i	$-0.624331\pi$
$151$	−4.86470e9	−0.761483	−0.380741	+	0.924682i	$0.624331\pi$
$152$	0	0
$153$	4.85986e9	0.716989
$154$	0	0
$155$	−2.83808e9	−0.394940
$156$	0	0
$157$	−1.16202e9	−0.152639	−0.0763194	−	0.997083i	$-0.524317\pi$
$157$	−1.16202e9	−0.152639	−0.0763194	+	0.997083i	$0.524317\pi$
$158$	0	0
$159$	−5.46340e9	−0.677916
$160$	0	0
$161$	2.85654e9	0.335061
$162$	0	0
$163$	−1.13760e10	−1.26225	−0.631124	−	0.775682i	$-0.717406\pi$
$163$	−1.13760e10	−1.26225	−0.631124	+	0.775682i	$0.717406\pi$
$164$	0	0
$165$	−2.16709e9	−0.227614
$166$	0	0
$167$	3.27059e9	0.325388	0.162694	−	0.986677i	$-0.447982\pi$
$167$	3.27059e9	0.325388	0.162694	+	0.986677i	$0.447982\pi$
$168$	0	0
$169$	8.15731e8	0.0769231
$170$	0	0
$171$	5.39943e9	0.482909
$172$	0	0
$173$	−3.11020e9	−0.263986	−0.131993	−	0.991251i	$-0.542138\pi$
$173$	−3.11020e9	−0.263986	−0.131993	+	0.991251i	$0.542138\pi$
$174$	0	0
$175$	−4.04424e9	−0.325961
$176$	0	0
$177$	−1.60254e9	−0.122724
$178$	0	0
$179$	−1.52628e10	−1.11121	−0.555604	−	0.831447i	$-0.687513\pi$
$179$	−1.52628e10	−1.11121	−0.555604	+	0.831447i	$0.687513\pi$
$180$	0	0
$181$	−2.29573e7	−0.00158989	−0.000794946	−	1.00000i	$-0.500253\pi$
$181$	−2.29573e7	−0.00158989	−0.000794946		1.00000i	$0.500253\pi$
$182$	0	0
$183$	−1.37225e9	−0.0904488
$184$	0	0
$185$	−1.75705e10	−1.10284
$186$	0	0
$187$	−1.03658e10	−0.619893
$188$	0	0
$189$	1.00243e10	0.571450
$190$	0	0
$191$	−2.14494e10	−1.16618	−0.583090	−	0.812407i	$-0.698157\pi$
$191$	−2.14494e10	−1.16618	−0.583090	+	0.812407i	$0.698157\pi$
$192$	0	0
$193$	−1.53540e9	−0.0796553	−0.0398276	−	0.999207i	$-0.512681\pi$
$193$	−1.53540e9	−0.0796553	−0.0398276	+	0.999207i	$0.512681\pi$
$194$	0	0
$195$	−1.84615e9	−0.0914349
$196$	0	0
$197$	−2.28558e10	−1.08118	−0.540591	−	0.841286i	$-0.681799\pi$
$197$	−2.28558e10	−1.08118	−0.540591	+	0.841286i	$0.681799\pi$
$198$	0	0
$199$	−2.46711e10	−1.11519	−0.557596	−	0.830112i	$-0.688276\pi$
$199$	−2.46711e10	−1.11519	−0.557596	+	0.830112i	$0.688276\pi$
$200$	0	0
$201$	−3.77957e9	−0.163328
$202$	0	0
$203$	−3.68745e9	−0.152403
$204$	0	0
$205$	−1.06960e10	−0.422988
$206$	0	0
$207$	−9.98426e9	−0.377963
$208$	0	0
$209$	−1.15167e10	−0.417512
$210$	0	0
$211$	1.26469e10	0.439252	0.219626	−	0.975584i	$-0.429516\pi$
$211$	1.26469e10	0.439252	0.219626	+	0.975584i	$0.429516\pi$
$212$	0	0
$213$	1.04368e10	0.347425
$214$	0	0
$215$	3.39071e10	1.08222
$216$	0	0
$217$	−1.24328e10	−0.380627
$218$	0	0
$219$	2.01103e9	0.0590773
$220$	0	0
$221$	−8.83072e9	−0.249018
$222$	0	0
$223$	3.28365e10	0.889171	0.444585	−	0.895736i	$-0.353351\pi$
$223$	3.28365e10	0.889171	0.444585	+	0.895736i	$0.353351\pi$
$224$	0	0
$225$	1.41356e10	0.367699
$226$	0	0
$227$	4.31647e10	1.07898	0.539489	−	0.841993i	$-0.318617\pi$
$227$	4.31647e10	1.07898	0.539489	+	0.841993i	$0.318617\pi$
$228$	0	0
$229$	−7.44836e10	−1.78979	−0.894893	−	0.446280i	$-0.852748\pi$
$229$	−7.44836e10	−1.78979	−0.894893	+	0.446280i	$0.852748\pi$
$230$	0	0
$231$	−9.49337e9	−0.219365
$232$	0	0
$233$	−7.32589e9	−0.162839	−0.0814196	−	0.996680i	$-0.525945\pi$
$233$	−7.32589e9	−0.162839	−0.0814196	+	0.996680i	$0.525945\pi$
$234$	0	0
$235$	2.12470e10	0.454456
$236$	0	0
$237$	2.59910e10	0.535125
$238$	0	0
$239$	3.65913e10	0.725416	0.362708	−	0.931903i	$-0.381852\pi$
$239$	3.65913e10	0.725416	0.362708	+	0.931903i	$0.381852\pi$
$240$	0	0
$241$	−2.38135e9	−0.0454722	−0.0227361	−	0.999742i	$-0.507238\pi$
$241$	−2.38135e9	−0.0454722	−0.0227361	+	0.999742i	$0.507238\pi$
$242$	0	0
$243$	−5.45182e10	−1.00303
$244$	0	0
$245$	−2.06649e10	−0.366425
$246$	0	0
$247$	−9.81114e9	−0.167719
$248$	0	0
$249$	3.68128e10	0.606878
$250$	0	0
$251$	3.23994e10	0.515234	0.257617	−	0.966247i	$-0.417063\pi$
$251$	3.23994e10	0.515234	0.257617	+	0.966247i	$0.417063\pi$
$252$	0	0
$253$	2.12959e10	0.326779
$254$	0	0
$255$	1.99856e10	0.295996
$256$	0	0
$257$	1.32429e9	0.0189359	0.00946794	−	0.999955i	$-0.496986\pi$
$257$	1.32429e9	0.0189359	0.00946794	+	0.999955i	$0.496986\pi$
$258$	0	0
$259$	−7.69713e10	−1.06287
$260$	0	0
$261$	1.28885e10	0.171917
$262$	0	0
$263$	−2.90518e10	−0.374432	−0.187216	−	0.982319i	$-0.559946\pi$
$263$	−2.90518e10	−0.374432	−0.187216	+	0.982319i	$0.559946\pi$
$264$	0	0
$265$	8.90700e10	1.10949
$266$	0	0
$267$	−6.90049e9	−0.0830958
$268$	0	0
$269$	−1.34615e11	−1.56750	−0.783751	−	0.621075i	$-0.786696\pi$
$269$	−1.34615e11	−1.56750	−0.783751	+	0.621075i	$0.786696\pi$
$270$	0	0
$271$	6.20930e10	0.699328	0.349664	−	0.936875i	$-0.386296\pi$
$271$	6.20930e10	0.699328	0.349664	+	0.936875i	$0.386296\pi$
$272$	0	0
$273$	−8.08746e9	−0.0881211
$274$	0	0
$275$	−3.01504e10	−0.317904
$276$	0	0
$277$	1.35421e11	1.38206	0.691028	−	0.722828i	$-0.257158\pi$
$277$	1.35421e11	1.38206	0.691028	+	0.722828i	$0.257158\pi$
$278$	0	0
$279$	4.34555e10	0.429364
$280$	0	0
$281$	1.25698e11	1.20268	0.601338	−	0.798995i	$-0.294635\pi$
$281$	1.25698e11	1.20268	0.601338	+	0.798995i	$0.294635\pi$
$282$	0	0
$283$	−7.23802e10	−0.670781	−0.335390	−	0.942079i	$-0.608868\pi$
$283$	−7.23802e10	−0.670781	−0.335390	+	0.942079i	$0.608868\pi$
$284$	0	0
$285$	2.22045e10	0.199360
$286$	0	0
$287$	−4.68560e10	−0.407659
$288$	0	0
$289$	−2.29907e10	−0.193871
$290$	0	0
$291$	−6.85851e10	−0.560675
$292$	0	0
$293$	−8.33332e10	−0.660562	−0.330281	−	0.943883i	$-0.607144\pi$
$293$	−8.33332e10	−0.660562	−0.330281	+	0.943883i	$0.607144\pi$
$294$	0	0
$295$	2.61263e10	0.200853
$296$	0	0
$297$	7.47330e10	0.557325
$298$	0	0
$299$	1.81421e10	0.131271
$300$	0	0
$301$	1.48537e11	1.04300
$302$	0	0
$303$	9.36032e10	0.637968
$304$	0	0
$305$	2.23718e10	0.148031
$306$	0	0
$307$	−3.49416e10	−0.224502	−0.112251	−	0.993680i	$-0.535806\pi$
$307$	−3.49416e10	−0.224502	−0.112251	+	0.993680i	$0.535806\pi$
$308$	0	0
$309$	6.47867e10	0.404271
$310$	0	0
$311$	2.70140e11	1.63744	0.818722	−	0.574190i	$-0.194683\pi$
$311$	2.70140e11	1.63744	0.818722	+	0.574190i	$0.194683\pi$
$312$	0	0
$313$	2.49404e11	1.46877	0.734385	−	0.678734i	$-0.237471\pi$
$313$	2.49404e11	1.46877	0.734385	+	0.678734i	$0.237471\pi$
$314$	0	0
$315$	−7.25619e10	−0.415251
$316$	0	0
$317$	−1.51758e10	−0.0844083	−0.0422041	−	0.999109i	$-0.513438\pi$
$317$	−1.51758e10	−0.0844083	−0.0422041	+	0.999109i	$0.513438\pi$
$318$	0	0
$319$	−2.74905e10	−0.148636
$320$	0	0
$321$	−6.55202e10	−0.344431
$322$	0	0
$323$	1.06211e11	0.542946
$324$	0	0
$325$	−2.56853e10	−0.127706
$326$	0	0
$327$	9.91097e10	0.479348
$328$	0	0
$329$	9.30768e10	0.437986
$330$	0	0
$331$	2.44364e11	1.11895	0.559476	−	0.828846i	$-0.311002\pi$
$331$	2.44364e11	1.11895	0.559476	+	0.828846i	$0.311002\pi$
$332$	0	0
$333$	2.69032e11	1.19896
$334$	0	0
$335$	6.16184e10	0.267306
$336$	0	0
$337$	−1.88482e11	−0.796040	−0.398020	−	0.917377i	$-0.630303\pi$
$337$	−1.88482e11	−0.796040	−0.398020	+	0.917377i	$0.630303\pi$
$338$	0	0
$339$	6.58120e10	0.270649
$340$	0	0
$341$	−9.26882e10	−0.371219
$342$	0	0
$343$	−2.71998e11	−1.06107
$344$	0	0
$345$	−4.10590e10	−0.156035
$346$	0	0
$347$	4.71900e11	1.74730	0.873650	−	0.486554i	$-0.161746\pi$
$347$	4.71900e11	1.74730	0.873650	+	0.486554i	$0.161746\pi$
$348$	0	0
$349$	1.56715e11	0.565454	0.282727	−	0.959200i	$-0.408761\pi$
$349$	1.56715e11	0.565454	0.282727	+	0.959200i	$0.408761\pi$
$350$	0	0
$351$	6.36655e10	0.223883
$352$	0	0
$353$	−2.24245e11	−0.768664	−0.384332	−	0.923195i	$-0.625568\pi$
$353$	−2.24245e11	−0.768664	−0.384332	+	0.923195i	$0.625568\pi$
$354$	0	0
$355$	−1.70152e11	−0.568604
$356$	0	0
$357$	8.75510e10	0.285269
$358$	0	0
$359$	2.89494e11	0.919846	0.459923	−	0.887959i	$-0.347877\pi$
$359$	2.89494e11	0.919846	0.459923	+	0.887959i	$0.347877\pi$
$360$	0	0
$361$	−2.04685e11	−0.634313
$362$	0	0
$363$	7.76984e10	0.234873
$364$	0	0
$365$	−3.27859e10	−0.0966874
$366$	0	0
$367$	−9.74321e10	−0.280353	−0.140176	−	0.990127i	$-0.544767\pi$
$367$	−9.74321e10	−0.280353	−0.140176	+	0.990127i	$0.544767\pi$
$368$	0	0
$369$	1.63773e11	0.459857
$370$	0	0
$371$	3.90190e11	1.06928
$372$	0	0
$373$	−3.26746e11	−0.874019	−0.437010	−	0.899457i	$-0.643962\pi$
$373$	−3.26746e11	−0.874019	−0.437010	+	0.899457i	$0.643962\pi$
$374$	0	0
$375$	1.84379e11	0.481471
$376$	0	0
$377$	−2.34193e10	−0.0597087
$378$	0	0
$379$	−2.92912e11	−0.729223	−0.364611	−	0.931160i	$-0.618798\pi$
$379$	−2.92912e11	−0.729223	−0.364611	+	0.931160i	$0.618798\pi$
$380$	0	0
$381$	1.95095e11	0.474334
$382$	0	0
$383$	7.61159e11	1.80751	0.903755	−	0.428050i	$-0.140799\pi$
$383$	7.61159e11	1.80751	0.903755	+	0.428050i	$0.140799\pi$
$384$	0	0
$385$	1.54771e11	0.359017
$386$	0	0
$387$	−5.19172e11	−1.17655
$388$	0	0
$389$	2.07102e11	0.458576	0.229288	−	0.973359i	$-0.426360\pi$
$389$	2.07102e11	0.458576	0.229288	+	0.973359i	$0.426360\pi$
$390$	0	0
$391$	−1.96398e11	−0.424953
$392$	0	0
$393$	1.29853e11	0.274591
$394$	0	0
$395$	−4.23732e11	−0.875798
$396$	0	0
$397$	−3.70909e11	−0.749393	−0.374697	−	0.927147i	$-0.622253\pi$
$397$	−3.70909e11	−0.749393	−0.374697	+	0.927147i	$0.622253\pi$
$398$	0	0
$399$	9.72713e10	0.192135
$400$	0	0
$401$	−7.34806e11	−1.41913	−0.709567	−	0.704638i	$-0.751109\pi$
$401$	−7.34806e11	−1.41913	−0.709567	+	0.704638i	$0.751109\pi$
$402$	0	0
$403$	−7.89616e10	−0.149123
$404$	0	0
$405$	1.73508e11	0.320459
$406$	0	0
$407$	−5.73832e11	−1.03660
$408$	0	0
$409$	−6.21193e11	−1.09767	−0.548835	−	0.835931i	$-0.684928\pi$
$409$	−6.21193e11	−1.09767	−0.548835	+	0.835931i	$0.684928\pi$
$410$	0	0
$411$	1.80351e11	0.311768
$412$	0	0
$413$	1.14452e11	0.193574
$414$	0	0
$415$	−6.00159e11	−0.993231
$416$	0	0
$417$	−5.60612e10	−0.0907925
$418$	0	0
$419$	−4.78555e11	−0.758523	−0.379261	−	0.925290i	$-0.623822\pi$
$419$	−4.78555e11	−0.758523	−0.379261	+	0.925290i	$0.623822\pi$
$420$	0	0
$421$	9.40501e11	1.45912	0.729558	−	0.683919i	$-0.239726\pi$
$421$	9.40501e11	1.45912	0.729558	+	0.683919i	$0.239726\pi$
$422$	0	0
$423$	−3.25325e11	−0.494067
$424$	0	0
$425$	2.78057e11	0.413413
$426$	0	0
$427$	9.80044e10	0.142666
$428$	0	0
$429$	−6.02932e10	−0.0859430
$430$	0	0
$431$	−4.19824e10	−0.0586030	−0.0293015	−	0.999571i	$-0.509328\pi$
$431$	−4.19824e10	−0.0586030	−0.0293015	+	0.999571i	$0.509328\pi$
$432$	0	0
$433$	1.05931e12	1.44820	0.724098	−	0.689698i	$-0.242256\pi$
$433$	1.05931e12	1.44820	0.724098	+	0.689698i	$0.242256\pi$
$434$	0	0
$435$	5.30023e10	0.0709729
$436$	0	0
$437$	−2.18203e11	−0.286216
$438$	0	0
$439$	−8.67785e11	−1.11512	−0.557560	−	0.830137i	$-0.688262\pi$
$439$	−8.67785e11	−1.11512	−0.557560	+	0.830137i	$0.688262\pi$
$440$	0	0
$441$	3.16412e11	0.398363
$442$	0	0
$443$	−2.81834e11	−0.347677	−0.173839	−	0.984774i	$-0.555617\pi$
$443$	−2.81834e11	−0.347677	−0.173839	+	0.984774i	$0.555617\pi$
$444$	0	0
$445$	1.12499e11	0.135997
$446$	0	0
$447$	−3.61060e11	−0.427756
$448$	0	0
$449$	−5.32421e11	−0.618225	−0.309112	−	0.951026i	$-0.600032\pi$
$449$	−5.32421e11	−0.618225	−0.309112	+	0.951026i	$0.600032\pi$
$450$	0	0
$451$	−3.49318e11	−0.397582
$452$	0	0
$453$	3.06316e11	0.341765
$454$	0	0
$455$	1.31850e11	0.144221
$456$	0	0
$457$	−8.95427e11	−0.960300	−0.480150	−	0.877186i	$-0.659418\pi$
$457$	−8.95427e11	−0.960300	−0.480150	+	0.877186i	$0.659418\pi$
$458$	0	0
$459$	−6.89212e11	−0.724763
$460$	0	0
$461$	−1.39531e12	−1.43886	−0.719429	−	0.694566i	$-0.755597\pi$
$461$	−1.39531e12	−1.43886	−0.719429	+	0.694566i	$0.755597\pi$
$462$	0	0
$463$	1.38392e12	1.39957	0.699787	−	0.714352i	$-0.253278\pi$
$463$	1.38392e12	1.39957	0.699787	+	0.714352i	$0.253278\pi$
$464$	0	0
$465$	1.78705e11	0.177255
$466$	0	0
$467$	1.20010e12	1.16759	0.583797	−	0.811900i	$-0.301566\pi$
$467$	1.20010e12	1.16759	0.583797	+	0.811900i	$0.301566\pi$
$468$	0	0
$469$	2.69932e11	0.257619
$470$	0	0
$471$	7.31688e10	0.0685065
$472$	0	0
$473$	1.10737e12	1.01722
$474$	0	0
$475$	3.08928e11	0.278443
$476$	0	0
$477$	−1.36380e12	−1.20620
$478$	0	0
$479$	1.32451e12	1.14960	0.574798	−	0.818295i	$-0.305081\pi$
$479$	1.32451e12	1.14960	0.574798	+	0.818295i	$0.305081\pi$
$480$	0	0
$481$	−4.88851e11	−0.416412
$482$	0	0
$483$	−1.79868e11	−0.150380
$484$	0	0
$485$	1.11814e12	0.917614
$486$	0	0
$487$	7.55891e11	0.608946	0.304473	−	0.952521i	$-0.401520\pi$
$487$	7.55891e11	0.608946	0.304473	+	0.952521i	$0.401520\pi$
$488$	0	0
$489$	7.16311e11	0.566515
$490$	0	0
$491$	−2.32497e12	−1.80531	−0.902653	−	0.430369i	$-0.858384\pi$
$491$	−2.32497e12	−1.80531	−0.902653	+	0.430369i	$0.858384\pi$
$492$	0	0
$493$	2.53526e11	0.193291
$494$	0	0
$495$	−5.40959e11	−0.404987
$496$	0	0
$497$	−7.45387e11	−0.547997
$498$	0	0
$499$	−2.47822e12	−1.78932	−0.894659	−	0.446749i	$-0.852582\pi$
$499$	−2.47822e12	−1.78932	−0.894659	+	0.446749i	$0.852582\pi$
$500$	0	0
$501$	−2.05939e11	−0.146039
$502$	0	0
$503$	2.17502e12	1.51498	0.757490	−	0.652846i	$-0.226425\pi$
$503$	2.17502e12	1.51498	0.757490	+	0.652846i	$0.226425\pi$
$504$	0	0
$505$	−1.52602e12	−1.04411
$506$	0	0
$507$	−5.13641e10	−0.0345242
$508$	0	0
$509$	−2.69953e12	−1.78262	−0.891308	−	0.453398i	$-0.850212\pi$
$509$	−2.69953e12	−1.78262	−0.891308	+	0.453398i	$0.850212\pi$
$510$	0	0
$511$	−1.43626e11	−0.0931833
$512$	0	0
$513$	−7.65731e11	−0.488144
$514$	0	0
$515$	−1.05622e12	−0.661640
$516$	0	0
$517$	6.93901e11	0.427159
$518$	0	0
$519$	1.95840e11	0.118481
$520$	0	0
$521$	−1.99567e12	−1.18664	−0.593320	−	0.804966i	$-0.702183\pi$
$521$	−1.99567e12	−1.18664	−0.593320	+	0.804966i	$0.702183\pi$
$522$	0	0
$523$	9.68879e11	0.566255	0.283128	−	0.959082i	$-0.408628\pi$
$523$	9.68879e11	0.566255	0.283128	+	0.959082i	$0.408628\pi$
$524$	0	0
$525$	2.54654e11	0.146296
$526$	0	0
$527$	8.54801e11	0.482745
$528$	0	0
$529$	−1.39767e12	−0.775985
$530$	0	0
$531$	−4.00035e11	−0.218360
$532$	0	0
$533$	−2.97586e11	−0.159713
$534$	0	0
$535$	1.06818e12	0.563704
$536$	0	0
$537$	9.61053e11	0.498727
$538$	0	0
$539$	−6.74890e11	−0.344416
$540$	0	0
$541$	1.23932e12	0.622008	0.311004	−	0.950409i	$-0.399335\pi$
$541$	1.23932e12	0.622008	0.311004	+	0.950409i	$0.399335\pi$
$542$	0	0
$543$	1.44555e9	0.000713567 0
$544$	0	0
$545$	−1.61579e12	−0.784513
$546$	0	0
$547$	−3.42271e12	−1.63466	−0.817329	−	0.576171i	$-0.804546\pi$
$547$	−3.42271e12	−1.63466	−0.817329	+	0.576171i	$0.804546\pi$
$548$	0	0
$549$	−3.42548e11	−0.160933
$550$	0	0
$551$	2.81674e11	0.130186
$552$	0	0
$553$	−1.85624e12	−0.844058
$554$	0	0
$555$	1.10636e12	0.494970
$556$	0	0
$557$	−2.36878e12	−1.04274	−0.521371	−	0.853330i	$-0.674579\pi$
$557$	−2.36878e12	−1.04274	−0.521371	+	0.853330i	$0.674579\pi$
$558$	0	0
$559$	9.43371e11	0.408629
$560$	0	0
$561$	6.52705e11	0.278217
$562$	0	0
$563$	−3.01556e12	−1.26497	−0.632485	−	0.774573i	$-0.717965\pi$
$563$	−3.01556e12	−1.26497	−0.632485	+	0.774573i	$0.717965\pi$
$564$	0	0
$565$	−1.07293e12	−0.442951
$566$	0	0
$567$	7.60090e11	0.308845
$568$	0	0
$569$	4.53744e12	1.81470	0.907351	−	0.420374i	$-0.138101\pi$
$569$	4.53744e12	1.81470	0.907351	+	0.420374i	$0.138101\pi$
$570$	0	0
$571$	−1.49980e12	−0.590432	−0.295216	−	0.955431i	$-0.595392\pi$
$571$	−1.49980e12	−0.590432	−0.295216	+	0.955431i	$0.595392\pi$
$572$	0	0
$573$	1.35061e12	0.523399
$574$	0	0
$575$	−5.71249e11	−0.217932
$576$	0	0
$577$	4.72072e12	1.77303	0.886516	−	0.462697i	$-0.153118\pi$
$577$	4.72072e12	1.77303	0.886516	+	0.462697i	$0.153118\pi$
$578$	0	0
$579$	9.66797e10	0.0357505
$580$	0	0
$581$	−2.62912e12	−0.957235
$582$	0	0
$583$	2.90892e12	1.04285
$584$	0	0
$585$	−4.60846e11	−0.162688
$586$	0	0
$587$	−4.21328e12	−1.46470	−0.732350	−	0.680928i	$-0.761577\pi$
$587$	−4.21328e12	−1.46470	−0.732350	+	0.680928i	$0.761577\pi$
$588$	0	0
$589$	9.49705e11	0.325140
$590$	0	0
$591$	1.43916e12	0.485250
$592$	0	0
$593$	−2.20583e12	−0.732530	−0.366265	−	0.930511i	$-0.619364\pi$
$593$	−2.20583e12	−0.732530	−0.366265	+	0.930511i	$0.619364\pi$
$594$	0	0
$595$	−1.42735e12	−0.466878
$596$	0	0
$597$	1.55347e12	0.500515
$598$	0	0
$599$	−4.91855e12	−1.56105	−0.780525	−	0.625125i	$-0.785048\pi$
$599$	−4.91855e12	−1.56105	−0.780525	+	0.625125i	$0.785048\pi$
$600$	0	0
$601$	2.33329e12	0.729514	0.364757	−	0.931103i	$-0.381152\pi$
$601$	2.33329e12	0.729514	0.364757	+	0.931103i	$0.381152\pi$
$602$	0	0
$603$	−9.43476e11	−0.290605
$604$	0	0
$605$	−1.26672e12	−0.384398
$606$	0	0
$607$	1.21387e12	0.362930	0.181465	−	0.983397i	$-0.441916\pi$
$607$	1.21387e12	0.362930	0.181465	+	0.983397i	$0.441916\pi$
$608$	0	0
$609$	2.32188e11	0.0684008
$610$	0	0
$611$	5.91138e11	0.171595
$612$	0	0
$613$	1.95734e12	0.559878	0.279939	−	0.960018i	$-0.409686\pi$
$613$	1.95734e12	0.559878	0.279939	+	0.960018i	$0.409686\pi$
$614$	0	0
$615$	6.73494e11	0.189844
$616$	0	0
$617$	1.95450e12	0.542940	0.271470	−	0.962447i	$-0.412490\pi$
$617$	1.95450e12	0.542940	0.271470	+	0.962447i	$0.412490\pi$
$618$	0	0
$619$	−3.53916e11	−0.0968929	−0.0484465	−	0.998826i	$-0.515427\pi$
$619$	−3.53916e11	−0.0968929	−0.0484465	+	0.998826i	$0.515427\pi$
$620$	0	0
$621$	1.41594e12	0.382061
$622$	0	0
$623$	4.92825e11	0.131068
$624$	0	0
$625$	−1.24946e12	−0.327538
$626$	0	0
$627$	7.25171e11	0.187386
$628$	0	0
$629$	5.29207e12	1.34802
$630$	0	0
$631$	5.84963e12	1.46891	0.734457	−	0.678655i	$-0.237437\pi$
$631$	5.84963e12	1.46891	0.734457	+	0.678655i	$0.237437\pi$
$632$	0	0
$633$	−7.96339e11	−0.197143
$634$	0	0
$635$	−3.18065e12	−0.776307
$636$	0	0
$637$	−5.74943e11	−0.138356
$638$	0	0
$639$	2.60530e12	0.618165
$640$	0	0
$641$	7.50183e12	1.75512	0.877558	−	0.479470i	$-0.159171\pi$
$641$	7.50183e12	1.75512	0.877558	+	0.479470i	$0.159171\pi$
$642$	0	0
$643$	7.67238e12	1.77003	0.885015	−	0.465563i	$-0.154148\pi$
$643$	7.67238e12	1.77003	0.885015	+	0.465563i	$0.154148\pi$
$644$	0	0
$645$	−2.13503e12	−0.485719
$646$	0	0
$647$	4.37688e10	0.00981963	0.00490981	−	0.999988i	$-0.498437\pi$
$647$	4.37688e10	0.00981963	0.00490981	+	0.999988i	$0.498437\pi$
$648$	0	0
$649$	8.53254e11	0.188789
$650$	0	0
$651$	7.82855e11	0.170831
$652$	0	0
$653$	3.28232e12	0.706434	0.353217	−	0.935541i	$-0.385088\pi$
$653$	3.28232e12	0.706434	0.353217	+	0.935541i	$0.385088\pi$
$654$	0	0
$655$	−2.11700e12	−0.449402
$656$	0	0
$657$	5.02005e11	0.105115
$658$	0	0
$659$	1.10753e12	0.228755	0.114378	−	0.993437i	$-0.463513\pi$
$659$	1.10753e12	0.228755	0.114378	+	0.993437i	$0.463513\pi$
$660$	0	0
$661$	−4.63388e12	−0.944143	−0.472072	−	0.881560i	$-0.656494\pi$
$661$	−4.63388e12	−0.944143	−0.472072	+	0.881560i	$0.656494\pi$
$662$	0	0
$663$	5.56044e11	0.111763
$664$	0	0
$665$	−1.58582e12	−0.314453
$666$	0	0
$667$	−5.20852e11	−0.101894
$668$	0	0
$669$	−2.06762e12	−0.399073
$670$	0	0
$671$	7.30637e11	0.139139
$672$	0	0
$673$	−4.86961e12	−0.915011	−0.457506	−	0.889207i	$-0.651257\pi$
$673$	−4.86961e12	−0.915011	−0.457506	+	0.889207i	$0.651257\pi$
$674$	0	0
$675$	−2.00467e12	−0.371685
$676$	0	0
$677$	2.76031e12	0.505021	0.252511	−	0.967594i	$-0.418744\pi$
$677$	2.76031e12	0.505021	0.252511	+	0.967594i	$0.418744\pi$
$678$	0	0
$679$	4.89827e12	0.884359
$680$	0	0
$681$	−2.71795e12	−0.484262
$682$	0	0
$683$	8.09795e12	1.42391	0.711954	−	0.702226i	$-0.247810\pi$
$683$	8.09795e12	1.42391	0.711954	+	0.702226i	$0.247810\pi$
$684$	0	0
$685$	−2.94027e12	−0.510247
$686$	0	0
$687$	4.69001e12	0.803283
$688$	0	0
$689$	2.47813e12	0.418926
$690$	0	0
$691$	1.46112e12	0.243800	0.121900	−	0.992542i	$-0.461101\pi$
$691$	1.46112e12	0.243800	0.121900	+	0.992542i	$0.461101\pi$
$692$	0	0
$693$	−2.36979e12	−0.390310
$694$	0	0
$695$	9.13967e11	0.148593
$696$	0	0
$697$	3.22153e12	0.517029
$698$	0	0
$699$	4.61289e11	0.0730847
$700$	0	0
$701$	1.33493e12	0.208798	0.104399	−	0.994535i	$-0.466708\pi$
$701$	1.33493e12	0.208798	0.104399	+	0.994535i	$0.466708\pi$
$702$	0	0
$703$	5.87962e12	0.907925
$704$	0	0
$705$	−1.33786e12	−0.203967
$706$	0	0
$707$	−6.68503e12	−1.00627
$708$	0	0
$709$	9.92029e12	1.47440	0.737202	−	0.675673i	$-0.236147\pi$
$709$	9.92029e12	1.47440	0.737202	+	0.675673i	$0.236147\pi$
$710$	0	0
$711$	6.48801e12	0.952134
$712$	0	0
$713$	−1.75613e12	−0.254480
$714$	0	0
$715$	9.82961e11	0.140656
$716$	0	0
$717$	−2.30404e12	−0.325577
$718$	0	0
$719$	−6.63993e12	−0.926581	−0.463290	−	0.886207i	$-0.653331\pi$
$719$	−6.63993e12	−0.926581	−0.463290	+	0.886207i	$0.653331\pi$
$720$	0	0
$721$	−4.62699e12	−0.637661
$722$	0	0
$723$	1.49946e11	0.0204086
$724$	0	0
$725$	7.37415e11	0.0991267
$726$	0	0
$727$	1.29834e13	1.72379	0.861896	−	0.507085i	$-0.169277\pi$
$727$	1.29834e13	1.72379	0.861896	+	0.507085i	$0.169277\pi$
$728$	0	0
$729$	1.06016e11	0.0139026
$730$	0	0
$731$	−1.02125e13	−1.32283
$732$	0	0
$733$	5.25333e12	0.672151	0.336075	−	0.941835i	$-0.390900\pi$
$733$	5.25333e12	0.672151	0.336075	+	0.941835i	$0.390900\pi$
$734$	0	0
$735$	1.30120e12	0.164457
$736$	0	0
$737$	2.01238e12	0.251251
$738$	0	0
$739$	−6.73695e12	−0.830928	−0.415464	−	0.909610i	$-0.636381\pi$
$739$	−6.73695e12	−0.830928	−0.415464	+	0.909610i	$0.636381\pi$
$740$	0	0
$741$	6.17778e11	0.0752749
$742$	0	0
$743$	1.08892e13	1.31084	0.655418	−	0.755267i	$-0.272493\pi$
$743$	1.08892e13	1.31084	0.655418	+	0.755267i	$0.272493\pi$
$744$	0	0
$745$	5.88638e12	0.700076
$746$	0	0
$747$	9.18940e12	1.07980
$748$	0	0
$749$	4.67937e12	0.543275
$750$	0	0
$751$	−5.22656e12	−0.599565	−0.299783	−	0.954008i	$-0.596914\pi$
$751$	−5.22656e12	−0.599565	−0.299783	+	0.954008i	$0.596914\pi$
$752$	0	0
$753$	−2.04009e12	−0.231245
$754$	0	0
$755$	−4.99388e12	−0.559341
$756$	0	0
$757$	−6.87850e12	−0.761312	−0.380656	−	0.924717i	$-0.624302\pi$
$757$	−6.87850e12	−0.761312	−0.380656	+	0.924717i	$0.624302\pi$
$758$	0	0
$759$	−1.34094e12	−0.146663
$760$	0	0
$761$	1.67281e12	0.180807	0.0904033	−	0.995905i	$-0.471184\pi$
$761$	1.67281e12	0.180807	0.0904033	+	0.995905i	$0.471184\pi$
$762$	0	0
$763$	−7.07830e12	−0.756081
$764$	0	0
$765$	4.98891e12	0.526658
$766$	0	0
$767$	7.26892e11	0.0758387
$768$	0	0
$769$	8.95428e12	0.923341	0.461670	−	0.887052i	$-0.347250\pi$
$769$	8.95428e12	0.923341	0.461670	+	0.887052i	$0.347250\pi$
$770$	0	0
$771$	−8.33868e10	−0.00849870
$772$	0	0
$773$	−3.60927e12	−0.363589	−0.181795	−	0.983337i	$-0.558191\pi$
$773$	−3.60927e12	−0.363589	−0.181795	+	0.983337i	$0.558191\pi$
$774$	0	0
$775$	2.48631e12	0.247569
$776$	0	0
$777$	4.84665e12	0.477032
$778$	0	0
$779$	3.57920e12	0.348231
$780$	0	0
$781$	−5.55697e12	−0.534452
$782$	0	0
$783$	−1.82781e12	−0.173781
$784$	0	0
$785$	−1.19287e12	−0.112119
$786$	0	0
$787$	−1.64130e13	−1.52511	−0.762554	−	0.646924i	$-0.776055\pi$
$787$	−1.64130e13	−1.52511	−0.762554	+	0.646924i	$0.776055\pi$
$788$	0	0
$789$	1.82931e12	0.168051
$790$	0	0
$791$	−4.70021e12	−0.426897
$792$	0	0
$793$	6.22434e11	0.0558938
$794$	0	0
$795$	−5.60847e12	−0.497957
$796$	0	0
$797$	−1.25670e13	−1.10324	−0.551618	−	0.834097i	$-0.685989\pi$
$797$	−1.25670e13	−1.10324	−0.551618	+	0.834097i	$0.685989\pi$
$798$	0	0
$799$	−6.39938e12	−0.555492
$800$	0	0
$801$	−1.72254e12	−0.147850
$802$	0	0
$803$	−1.07075e12	−0.0908800
$804$	0	0
$805$	2.93239e12	0.246116
$806$	0	0
$807$	8.47630e12	0.703519
$808$	0	0
$809$	1.35050e13	1.10848	0.554239	−	0.832358i	$-0.313009\pi$
$809$	1.35050e13	1.10848	0.554239	+	0.832358i	$0.313009\pi$
$810$	0	0
$811$	7.73437e12	0.627814	0.313907	−	0.949454i	$-0.398362\pi$
$811$	7.73437e12	0.627814	0.313907	+	0.949454i	$0.398362\pi$
$812$	0	0
$813$	−3.90981e12	−0.313869
$814$	0	0
$815$	−1.16780e13	−0.927173
$816$	0	0
$817$	−1.13463e13	−0.890955
$818$	0	0
$819$	−2.01883e12	−0.156792
$820$	0	0
$821$	1.58629e13	1.21853	0.609267	−	0.792965i	$-0.291464\pi$
$821$	1.58629e13	1.21853	0.609267	+	0.792965i	$0.291464\pi$
$822$	0	0
$823$	8.41508e12	0.639380	0.319690	−	0.947522i	$-0.396421\pi$
$823$	8.41508e12	0.639380	0.319690	+	0.947522i	$0.396421\pi$
$824$	0	0
$825$	1.89848e12	0.142680
$826$	0	0
$827$	−1.58883e13	−1.18114	−0.590572	−	0.806985i	$-0.701098\pi$
$827$	−1.58883e13	−1.18114	−0.590572	+	0.806985i	$0.701098\pi$
$828$	0	0
$829$	5.92895e12	0.435996	0.217998	−	0.975949i	$-0.430047\pi$
$829$	5.92895e12	0.435996	0.217998	+	0.975949i	$0.430047\pi$
$830$	0	0
$831$	−8.52703e12	−0.620288
$832$	0	0
$833$	6.22406e12	0.447890
$834$	0	0
$835$	3.35743e12	0.239011
$836$	0	0
$837$	−6.16273e12	−0.434019
$838$	0	0
$839$	1.64277e10	0.00114459	0.000572294	−	1.00000i	$-0.499818\pi$
$839$	1.64277e10	0.00114459	0.000572294		1.00000i	$0.499818\pi$
$840$	0	0
$841$	−1.38348e13	−0.953653
$842$	0	0
$843$	−7.91480e12	−0.539779
$844$	0	0
$845$	8.37391e11	0.0565032
$846$	0	0
$847$	−5.54913e12	−0.370467
$848$	0	0
$849$	4.55756e12	0.301057
$850$	0	0
$851$	−1.08722e13	−0.710615
$852$	0	0
$853$	−9.54453e12	−0.617282	−0.308641	−	0.951179i	$-0.599874\pi$
$853$	−9.54453e12	−0.617282	−0.308641	+	0.951179i	$0.599874\pi$
$854$	0	0
$855$	5.54279e12	0.354716
$856$	0	0
$857$	2.04272e12	0.129358	0.0646791	−	0.997906i	$-0.479398\pi$
$857$	2.04272e12	0.129358	0.0646791	+	0.997906i	$0.479398\pi$
$858$	0	0
$859$	6.42844e11	0.0402844	0.0201422	−	0.999797i	$-0.493588\pi$
$859$	6.42844e11	0.0402844	0.0201422	+	0.999797i	$0.493588\pi$
$860$	0	0
$861$	2.95038e12	0.182963
$862$	0	0
$863$	1.95507e13	1.19981	0.599906	−	0.800071i	$-0.295205\pi$
$863$	1.95507e13	1.19981	0.599906	+	0.800071i	$0.295205\pi$
$864$	0	0
$865$	−3.19278e12	−0.193909
$866$	0	0
$867$	1.44766e12	0.0870121
$868$	0	0
$869$	−1.38386e13	−0.823194
$870$	0	0
$871$	1.71436e12	0.100930
$872$	0	0
$873$	−1.71206e13	−0.997595
$874$	0	0
$875$	−1.31681e13	−0.759429
$876$	0	0
$877$	7.79769e10	0.00445110	0.00222555	−	0.999998i	$-0.499292\pi$
$877$	7.79769e10	0.00445110	0.00222555	+	0.999998i	$0.499292\pi$
$878$	0	0
$879$	5.24724e12	0.296470
$880$	0	0
$881$	−1.91901e13	−1.07321	−0.536605	−	0.843833i	$-0.680294\pi$
$881$	−1.91901e13	−1.07321	−0.536605	+	0.843833i	$0.680294\pi$
$882$	0	0
$883$	1.12577e13	0.623198	0.311599	−	0.950214i	$-0.399135\pi$
$883$	1.12577e13	0.623198	0.311599	+	0.950214i	$0.399135\pi$
$884$	0	0
$885$	−1.64509e12	−0.0901459
$886$	0	0
$887$	1.53600e13	0.833171	0.416585	−	0.909097i	$-0.363227\pi$
$887$	1.53600e13	0.833171	0.416585	+	0.909097i	$0.363227\pi$
$888$	0	0
$889$	−1.39335e13	−0.748173
$890$	0	0
$891$	5.66658e12	0.301211
$892$	0	0
$893$	−7.10987e12	−0.374136
$894$	0	0
$895$	−1.56681e13	−0.816229
$896$	0	0
$897$	−1.14235e12	−0.0589162
$898$	0	0
$899$	2.26696e12	0.115751
$900$	0	0
$901$	−2.68270e13	−1.35616
$902$	0	0
$903$	−9.35293e12	−0.468115
$904$	0	0
$905$	−2.35669e10	−0.00116784
$906$	0	0
$907$	2.80870e13	1.37807	0.689037	−	0.724726i	$-0.258034\pi$
$907$	2.80870e13	1.37807	0.689037	+	0.724726i	$0.258034\pi$
$908$	0	0
$909$	2.33657e13	1.13512
$910$	0	0
$911$	2.32769e13	1.11967	0.559837	−	0.828603i	$-0.310864\pi$
$911$	2.32769e13	1.11967	0.559837	+	0.828603i	$0.310864\pi$
$912$	0	0
$913$	−1.96005e13	−0.933574
$914$	0	0
$915$	−1.40869e12	−0.0664384
$916$	0	0
$917$	−9.27397e12	−0.433115
$918$	0	0
$919$	−2.70538e13	−1.25115	−0.625573	−	0.780166i	$-0.715135\pi$
$919$	−2.70538e13	−1.25115	−0.625573	+	0.780166i	$0.715135\pi$
$920$	0	0
$921$	2.20017e12	0.100760
$922$	0	0
$923$	−4.73402e12	−0.214695
$924$	0	0
$925$	1.53927e13	0.691316
$926$	0	0
$927$	1.61724e13	0.719309
$928$	0	0
$929$	3.43871e13	1.51470	0.757348	−	0.653012i	$-0.226495\pi$
$929$	3.43871e13	1.51470	0.757348	+	0.653012i	$0.226495\pi$
$930$	0	0
$931$	6.91508e12	0.301664
$932$	0	0
$933$	−1.70099e13	−0.734910
$934$	0	0
$935$	−1.06411e13	−0.455337
$936$	0	0
$937$	1.33334e13	0.565086	0.282543	−	0.959255i	$-0.408822\pi$
$937$	1.33334e13	0.565086	0.282543	+	0.959255i	$0.408822\pi$
$938$	0	0
$939$	−1.57042e13	−0.659206
$940$	0	0
$941$	−2.40200e13	−0.998666	−0.499333	−	0.866410i	$-0.666421\pi$
$941$	−2.40200e13	−0.998666	−0.499333	+	0.866410i	$0.666421\pi$
$942$	0	0
$943$	−6.61841e12	−0.272553
$944$	0	0
$945$	1.02905e13	0.419753
$946$	0	0
$947$	−2.65641e13	−1.07330	−0.536649	−	0.843806i	$-0.680310\pi$
$947$	−2.65641e13	−1.07330	−0.536649	+	0.843806i	$0.680310\pi$
$948$	0	0
$949$	−9.12178e11	−0.0365075
$950$	0	0
$951$	9.55575e11	0.0378837
$952$	0	0
$953$	−3.67569e13	−1.44352	−0.721758	−	0.692146i	$-0.756665\pi$
$953$	−3.67569e13	−1.44352	−0.721758	+	0.692146i	$0.756665\pi$
$954$	0	0
$955$	−2.20190e13	−0.856608
$956$	0	0
$957$	1.73099e12	0.0667100
$958$	0	0
$959$	−1.28805e13	−0.491755
$960$	0	0
$961$	−1.87962e13	−0.710912
$962$	0	0
$963$	−1.63555e13	−0.612837
$964$	0	0
$965$	−1.57617e12	−0.0585101
$966$	0	0
$967$	−9.08923e12	−0.334278	−0.167139	−	0.985933i	$-0.553453\pi$
$967$	−9.08923e12	−0.334278	−0.167139	+	0.985933i	$0.553453\pi$
$968$	0	0
$969$	−6.68777e12	−0.243683
$970$	0	0
$971$	−1.45138e13	−0.523956	−0.261978	−	0.965074i	$-0.584375\pi$
$971$	−1.45138e13	−0.523956	−0.261978	+	0.965074i	$0.584375\pi$
$972$	0	0
$973$	4.00383e12	0.143208
$974$	0	0
$975$	1.61733e12	0.0573162
$976$	0	0
$977$	−3.54653e13	−1.24531	−0.622656	−	0.782496i	$-0.713946\pi$
$977$	−3.54653e13	−1.24531	−0.622656	+	0.782496i	$0.713946\pi$
$978$	0	0
$979$	3.67408e12	0.127828
$980$	0	0
$981$	2.47403e13	0.852892
$982$	0	0
$983$	3.17386e13	1.08417	0.542084	−	0.840324i	$-0.317635\pi$
$983$	3.17386e13	1.08417	0.542084	+	0.840324i	$0.317635\pi$
$984$	0	0
$985$	−2.34627e13	−0.794173
$986$	0	0
$987$	−5.86076e12	−0.196574
$988$	0	0
$989$	2.09809e13	0.697333
$990$	0	0
$991$	3.73377e13	1.22975	0.614874	−	0.788626i	$-0.289207\pi$
$991$	3.73377e13	1.22975	0.614874	+	0.788626i	$0.289207\pi$
$992$	0	0
$993$	−1.53869e13	−0.502203
$994$	0	0
$995$	−2.53262e13	−0.819155
$996$	0	0
$997$	5.39988e13	1.73084	0.865418	−	0.501051i	$-0.167053\pi$
$997$	5.39988e13	1.73084	0.865418	+	0.501051i	$0.167053\pi$
$998$	0	0
$999$	−3.81534e13	−1.21196

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	208.10.a.f.1.2		4
4.3	odd	2		52.10.a.a.1.3	✓	4

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
52.10.a.a.1.3	✓	4	4.3	odd	2
208.10.a.f.1.2		4	1.1	even	1	trivial

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:	\( k \)	\(=\)	\( 10 \)
Character orbit:	\([\chi]\)	\(=\)	208.a (trivial)

\(f(q)\)	\(=\)	\(q-62.9670 q^{3} +1026.55 q^{5} +4497.03 q^{7} -15718.2 q^{9} +33526.0 q^{11} +28561.0 q^{13} -64638.9 q^{15} -309188. q^{17} -343515. q^{19} -283164. q^{21} +635205. q^{23} -899315. q^{25} +2.22910e6 q^{27} -819974. q^{29} -2.76467e6 q^{31} -2.11103e6 q^{33} +4.61644e6 q^{35} -1.71160e7 q^{37} -1.79840e6 q^{39} -1.04193e7 q^{41} +3.30301e7 q^{43} -1.61355e7 q^{45} +2.06974e7 q^{47} -2.01303e7 q^{49} +1.94686e7 q^{51} +8.67661e7 q^{53} +3.44162e7 q^{55} +2.16301e7 q^{57} +2.54505e7 q^{59} +2.17931e7 q^{61} -7.06850e7 q^{63} +2.93194e7 q^{65} +6.00246e7 q^{67} -3.99970e7 q^{69} -1.65751e8 q^{71} -3.19379e7 q^{73} +5.66271e7 q^{75} +1.50767e8 q^{77} -4.12771e8 q^{79} +1.69020e8 q^{81} -5.84636e8 q^{83} -3.17398e8 q^{85} +5.16313e7 q^{87} +1.09589e8 q^{89} +1.28440e8 q^{91} +1.74083e8 q^{93} -3.52636e8 q^{95} +1.08922e9 q^{97} -5.26967e8 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 4 q + 147 q^{3} - 1947 q^{5} - 10251 q^{7} + 22199 q^{9} - 22038 q^{11} + 114244 q^{13} + 137363 q^{15} - 696135 q^{17} + 254502 q^{19} + 165205 q^{21} + 2038992 q^{23} + 16923 q^{25} + 9068877 q^{27} - 6437112 q^{29}+ \cdots - 1154490140 q^{99}+O(q^{100}) \) 4 * q + 147 * q^3 - 1947 * q^5 - 10251 * q^7 + 22199 * q^9 - 22038 * q^11 + 114244 * q^13 + 137363 * q^15 - 696135 * q^17 + 254502 * q^19 + 165205 * q^21 + 2038992 * q^23 + 16923 * q^25 + 9068877 * q^27 - 6437112 * q^29 + 10739008 * q^31 - 13872374 * q^33 + 30660837 * q^35 - 30452151 * q^37 + 4198467 * q^39 - 33969870 * q^41 + 13058901 * q^43 - 54565744 * q^45 + 44894985 * q^47 - 32869065 * q^49 + 83733323 * q^51 - 155596242 * q^53 + 13538650 * q^55 - 224212490 * q^57 - 61432458 * q^59 + 115267998 * q^61 - 203352774 * q^63 - 55608267 * q^65 - 406952918 * q^67 + 289997880 * q^69 - 51911697 * q^71 + 465992648 * q^73 - 1124268872 * q^75 + 198991878 * q^77 - 11428448 * q^79 + 584641484 * q^81 - 796564404 * q^83 + 1528724447 * q^85 - 419924838 * q^87 - 399435336 * q^89 - 292778811 * q^91 - 218300796 * q^93 - 939051978 * q^95 - 1151600100 * q^97 - 1154490140 * q^99

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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