LMFDB - Embedded newform 8048.2.a.q.1.8

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [8048,2,Mod(1,8048)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(8048, base_ring=CyclotomicField(2))

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

N = Newforms(chi, 2, names="a")

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

chi := DirichletCharacter("8048.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 8048 = 2^{4} \cdot 503 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	8048.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:	$64.2636035467$
Analytic rank:	$1$
Dimension:	$12$
Coefficient field:	$\mathbb{Q}[x]/(x^{12} - \cdots)$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{12} - 5 x^{11} - 13 x^{10} + 94 x^{9} + 15 x^{8} - 616 x^{7} + 368 x^{6} + 1643 x^{5} - 1463 x^{4} + \cdots - 208 $ x^12 - 5x^11 - 13x^10 + 94x^9 + 15x^8 - 616x^7 + 368x^6 + 1643x^5 - 1463x^4 - 1556x^3 + 1284x^2 + 576*x - 208
Coefficient ring:	$\Z[a_1, \ldots, a_{5}]$
Coefficient ring index:	$ 2^{2} $
Twist minimal:	no (minimal twist has level 1006)
Fricke sign:	$1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.8
Root			$-0.665271$ of defining polynomial
Character	$\chi$	$=$	8048.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+0.665271 q^{3} +2.52521 q^{5} +3.86663 q^{7} -2.55741 q^{9} +O(q^{10})$	$q+0.665271 q^{3} +2.52521 q^{5} +3.86663 q^{7} -2.55741 q^{9} -4.92422 q^{11} +2.85653 q^{13} +1.67995 q^{15} -1.32149 q^{17} -5.94123 q^{19} +2.57236 q^{21} -4.15434 q^{23} +1.37669 q^{25} -3.69719 q^{27} +6.92173 q^{29} -9.15427 q^{31} -3.27594 q^{33} +9.76407 q^{35} -7.83364 q^{37} +1.90036 q^{39} -6.05568 q^{41} -1.21697 q^{43} -6.45801 q^{45} +2.72919 q^{47} +7.95086 q^{49} -0.879149 q^{51} -7.03708 q^{53} -12.4347 q^{55} -3.95253 q^{57} -3.47826 q^{59} -11.1235 q^{61} -9.88859 q^{63} +7.21333 q^{65} +1.48776 q^{67} -2.76376 q^{69} +0.0164425 q^{71} +10.2937 q^{73} +0.915871 q^{75} -19.0401 q^{77} +1.08665 q^{79} +5.21261 q^{81} +8.66142 q^{83} -3.33704 q^{85} +4.60482 q^{87} +11.4741 q^{89} +11.0451 q^{91} -6.09007 q^{93} -15.0028 q^{95} -13.8759 q^{97} +12.5933 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 12 q - 5 q^{3} + 5 q^{5} - 8 q^{7} + 15 q^{9}+O(q^{10}) $ 12 * q - 5 * q^3 + 5 * q^5 - 8 * q^7 + 15 * q^9	$ 12 q - 5 q^{3} + 5 q^{5} - 8 q^{7} + 15 q^{9} - 18 q^{11} + 4 q^{13} - 2 q^{15} - 2 q^{17} - 6 q^{19} + q^{21} - 13 q^{23} + q^{25} - 8 q^{27} + 20 q^{29} - 7 q^{31} - 8 q^{33} - q^{35} + 10 q^{37} - 7 q^{39} + 2 q^{41} - 8 q^{43} - 7 q^{45} - 12 q^{47} + 4 q^{49} - 2 q^{51} + 12 q^{53} - 8 q^{55} - 10 q^{57} - 6 q^{59} - 10 q^{61} - 7 q^{63} + 4 q^{65} - 7 q^{67} - 12 q^{69} - 22 q^{71} - 23 q^{73} + 34 q^{75} - 19 q^{77} - 13 q^{79} - 28 q^{81} - q^{83} - 28 q^{85} + 22 q^{87} - 3 q^{89} + 21 q^{91} - 33 q^{93} + 2 q^{95} - 70 q^{97} + 6 q^{99}+O(q^{100}) $ 12 * q - 5 * q^3 + 5 * q^5 - 8 * q^7 + 15 * q^9 - 18 * q^11 + 4 * q^13 - 2 * q^15 - 2 * q^17 - 6 * q^19 + q^21 - 13 * q^23 + q^25 - 8 * q^27 + 20 * q^29 - 7 * q^31 - 8 * q^33 - q^35 + 10 * q^37 - 7 * q^39 + 2 * q^41 - 8 * q^43 - 7 * q^45 - 12 * q^47 + 4 * q^49 - 2 * q^51 + 12 * q^53 - 8 * q^55 - 10 * q^57 - 6 * q^59 - 10 * q^61 - 7 * q^63 + 4 * q^65 - 7 * q^67 - 12 * q^69 - 22 * q^71 - 23 * q^73 + 34 * q^75 - 19 * q^77 - 13 * q^79 - 28 * q^81 - q^83 - 28 * q^85 + 22 * q^87 - 3 * q^89 + 21 * q^91 - 33 * q^93 + 2 * q^95 - 70 * q^97 + 6 * 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
$3$	0.665271	0.384094	0.192047	−	0.981386i	$-0.438487\pi$
$3$	0.665271	0.384094	0.192047	+	0.981386i	$0.438487\pi$
$4$	0	0
$5$	2.52521	1.12931	0.564654	−	0.825328i	$-0.309010\pi$
$5$	2.52521	1.12931	0.564654	+	0.825328i	$0.309010\pi$
$6$	0	0
$7$	3.86663	1.46145	0.730725	−	0.682672i	$-0.239182\pi$
$7$	3.86663	1.46145	0.730725	+	0.682672i	$0.239182\pi$
$8$	0	0
$9$	−2.55741	−0.852472
$10$	0	0
$11$	−4.92422	−1.48471	−0.742354	−	0.670008i	$-0.766291\pi$
$11$	−4.92422	−1.48471	−0.742354	+	0.670008i	$0.766291\pi$
$12$	0	0
$13$	2.85653	0.792258	0.396129	−	0.918195i	$-0.370353\pi$
$13$	2.85653	0.792258	0.396129	+	0.918195i	$0.370353\pi$
$14$	0	0
$15$	1.67995	0.433761
$16$	0	0
$17$	−1.32149	−0.320509	−0.160254	−	0.987076i	$-0.551231\pi$
$17$	−1.32149	−0.320509	−0.160254	+	0.987076i	$0.551231\pi$
$18$	0	0
$19$	−5.94123	−1.36301	−0.681506	−	0.731813i	$-0.738674\pi$
$19$	−5.94123	−1.36301	−0.681506	+	0.731813i	$0.738674\pi$
$20$	0	0
$21$	2.57236	0.561335
$22$	0	0
$23$	−4.15434	−0.866241	−0.433120	−	0.901336i	$-0.642587\pi$
$23$	−4.15434	−0.866241	−0.433120	+	0.901336i	$0.642587\pi$
$24$	0	0
$25$	1.37669	0.275338
$26$	0	0
$27$	−3.69719	−0.711524
$28$	0	0
$29$	6.92173	1.28533	0.642666	−	0.766146i	$-0.277828\pi$
$29$	6.92173	1.28533	0.642666	+	0.766146i	$0.277828\pi$
$30$	0	0
$31$	−9.15427	−1.64416	−0.822078	−	0.569375i	$-0.807185\pi$
$31$	−9.15427	−1.64416	−0.822078	+	0.569375i	$0.807185\pi$
$32$	0	0
$33$	−3.27594	−0.570268
$34$	0	0
$35$	9.76407	1.65043
$36$	0	0
$37$	−7.83364	−1.28784	−0.643921	−	0.765092i	$-0.722694\pi$
$37$	−7.83364	−1.28784	−0.643921	+	0.765092i	$0.722694\pi$
$38$	0	0
$39$	1.90036	0.304302
$40$	0	0
$41$	−6.05568	−0.945739	−0.472869	−	0.881132i	$-0.656782\pi$
$41$	−6.05568	−0.945739	−0.472869	+	0.881132i	$0.656782\pi$
$42$	0	0
$43$	−1.21697	−0.185587	−0.0927934	−	0.995685i	$-0.529580\pi$
$43$	−1.21697	−0.185587	−0.0927934	+	0.995685i	$0.529580\pi$
$44$	0	0
$45$	−6.45801	−0.962703
$46$	0	0
$47$	2.72919	0.398094	0.199047	−	0.979990i	$-0.436215\pi$
$47$	2.72919	0.398094	0.199047	+	0.979990i	$0.436215\pi$
$48$	0	0
$49$	7.95086	1.13584
$50$	0	0
$51$	−0.879149	−0.123106
$52$	0	0
$53$	−7.03708	−0.966617	−0.483308	−	0.875450i	$-0.660565\pi$
$53$	−7.03708	−0.966617	−0.483308	+	0.875450i	$0.660565\pi$
$54$	0	0
$55$	−12.4347	−1.67669
$56$	0	0
$57$	−3.95253	−0.523525
$58$	0	0
$59$	−3.47826	−0.452830	−0.226415	−	0.974031i	$-0.572701\pi$
$59$	−3.47826	−0.452830	−0.226415	+	0.974031i	$0.572701\pi$
$60$	0	0
$61$	−11.1235	−1.42422	−0.712108	−	0.702070i	$-0.752259\pi$
$61$	−11.1235	−1.42422	−0.712108	+	0.702070i	$0.752259\pi$
$62$	0	0
$63$	−9.88859	−1.24584
$64$	0	0
$65$	7.21333	0.894703
$66$	0	0
$67$	1.48776	0.181759	0.0908795	−	0.995862i	$-0.471032\pi$
$67$	1.48776	0.181759	0.0908795	+	0.995862i	$0.471032\pi$
$68$	0	0
$69$	−2.76376	−0.332718
$70$	0	0
$71$	0.0164425	0.00195137	0.000975685	−	1.00000i	$-0.499689\pi$
$71$	0.0164425	0.00195137	0.000975685		1.00000i	$0.499689\pi$
$72$	0	0
$73$	10.2937	1.20478	0.602391	−	0.798201i	$-0.294215\pi$
$73$	10.2937	1.20478	0.602391	+	0.798201i	$0.294215\pi$
$74$	0	0
$75$	0.915871	0.105756
$76$	0	0
$77$	−19.0401	−2.16983
$78$	0	0
$79$	1.08665	0.122258	0.0611290	−	0.998130i	$-0.480530\pi$
$79$	1.08665	0.122258	0.0611290	+	0.998130i	$0.480530\pi$
$80$	0	0
$81$	5.21261	0.579179
$82$	0	0
$83$	8.66142	0.950714	0.475357	−	0.879793i	$-0.342319\pi$
$83$	8.66142	0.950714	0.475357	+	0.879793i	$0.342319\pi$
$84$	0	0
$85$	−3.33704	−0.361953
$86$	0	0
$87$	4.60482	0.493689
$88$	0	0
$89$	11.4741	1.21625	0.608125	−	0.793842i	$-0.291922\pi$
$89$	11.4741	1.21625	0.608125	+	0.793842i	$0.291922\pi$
$90$	0	0
$91$	11.0451	1.15785
$92$	0	0
$93$	−6.09007	−0.631511
$94$	0	0
$95$	−15.0028	−1.53926
$96$	0	0
$97$	−13.8759	−1.40889	−0.704444	−	0.709760i	$-0.748803\pi$
$97$	−13.8759	−1.40889	−0.704444	+	0.709760i	$0.748803\pi$
$98$	0	0
$99$	12.5933	1.26567
$100$	0	0
$101$	−19.1406	−1.90456	−0.952281	−	0.305222i	$-0.901269\pi$
$101$	−19.1406	−1.90456	−0.952281	+	0.305222i	$0.901269\pi$
$102$	0	0
$103$	−12.5801	−1.23955	−0.619777	−	0.784778i	$-0.712777\pi$
$103$	−12.5801	−1.23955	−0.619777	+	0.784778i	$0.712777\pi$
$104$	0	0
$105$	6.49575	0.633920
$106$	0	0
$107$	9.09048	0.878809	0.439405	−	0.898289i	$-0.355189\pi$
$107$	9.09048	0.878809	0.439405	+	0.898289i	$0.355189\pi$
$108$	0	0
$109$	19.8985	1.90593	0.952965	−	0.303081i	$-0.0980152\pi$
$109$	19.8985	1.90593	0.952965	+	0.303081i	$0.0980152\pi$
$110$	0	0
$111$	−5.21149	−0.494653
$112$	0	0
$113$	−15.5894	−1.46653	−0.733265	−	0.679943i	$-0.762005\pi$
$113$	−15.5894	−1.46653	−0.733265	+	0.679943i	$0.762005\pi$
$114$	0	0
$115$	−10.4906	−0.978253
$116$	0	0
$117$	−7.30532	−0.675377
$118$	0	0
$119$	−5.10972	−0.468407
$120$	0	0
$121$	13.2479	1.20436
$122$	0	0
$123$	−4.02867	−0.363253
$124$	0	0
$125$	−9.14962	−0.818367
$126$	0	0
$127$	−6.70321	−0.594814	−0.297407	−	0.954751i	$-0.596122\pi$
$127$	−6.70321	−0.594814	−0.297407	+	0.954751i	$0.596122\pi$
$128$	0	0
$129$	−0.809617	−0.0712828
$130$	0	0
$131$	−16.0756	−1.40453	−0.702265	−	0.711915i	$-0.747828\pi$
$131$	−16.0756	−1.40453	−0.702265	+	0.711915i	$0.747828\pi$
$132$	0	0
$133$	−22.9726	−1.99197
$134$	0	0
$135$	−9.33617	−0.803530
$136$	0	0
$137$	−2.91705	−0.249221	−0.124610	−	0.992206i	$-0.539768\pi$
$137$	−2.91705	−0.249221	−0.124610	+	0.992206i	$0.539768\pi$
$138$	0	0
$139$	13.4352	1.13956	0.569780	−	0.821797i	$-0.307028\pi$
$139$	13.4352	1.13956	0.569780	+	0.821797i	$0.307028\pi$
$140$	0	0
$141$	1.81565	0.152905
$142$	0	0
$143$	−14.0662	−1.17627
$144$	0	0
$145$	17.4788	1.45154
$146$	0	0
$147$	5.28948	0.436269
$148$	0	0
$149$	−13.4810	−1.10440	−0.552202	−	0.833710i	$-0.686212\pi$
$149$	−13.4810	−1.10440	−0.552202	+	0.833710i	$0.686212\pi$
$150$	0	0
$151$	1.47209	0.119797	0.0598987	−	0.998204i	$-0.480922\pi$
$151$	1.47209	0.119797	0.0598987	+	0.998204i	$0.480922\pi$
$152$	0	0
$153$	3.37960	0.273224
$154$	0	0
$155$	−23.1165	−1.85676
$156$	0	0
$157$	21.5694	1.72143	0.860713	−	0.509090i	$-0.170018\pi$
$157$	21.5694	1.72143	0.860713	+	0.509090i	$0.170018\pi$
$158$	0	0
$159$	−4.68156	−0.371272
$160$	0	0
$161$	−16.0633	−1.26597
$162$	0	0
$163$	−4.36985	−0.342273	−0.171136	−	0.985247i	$-0.554744\pi$
$163$	−4.36985	−0.342273	−0.171136	+	0.985247i	$0.554744\pi$
$164$	0	0
$165$	−8.27243	−0.644008
$166$	0	0
$167$	10.3303	0.799380	0.399690	−	0.916650i	$-0.369118\pi$
$167$	10.3303	0.799380	0.399690	+	0.916650i	$0.369118\pi$
$168$	0	0
$169$	−4.84026	−0.372328
$170$	0	0
$171$	15.1942	1.16193
$172$	0	0
$173$	4.85469	0.369095	0.184548	−	0.982824i	$-0.440918\pi$
$173$	4.85469	0.369095	0.184548	+	0.982824i	$0.440918\pi$
$174$	0	0
$175$	5.32315	0.402392
$176$	0	0
$177$	−2.31398	−0.173930
$178$	0	0
$179$	7.28750	0.544693	0.272347	−	0.962199i	$-0.412200\pi$
$179$	7.28750	0.544693	0.272347	+	0.962199i	$0.412200\pi$
$180$	0	0
$181$	5.19186	0.385908	0.192954	−	0.981208i	$-0.438193\pi$
$181$	5.19186	0.385908	0.192954	+	0.981208i	$0.438193\pi$
$182$	0	0
$183$	−7.40013	−0.547033
$184$	0	0
$185$	−19.7816	−1.45437
$186$	0	0
$187$	6.50731	0.475861
$188$	0	0
$189$	−14.2957	−1.03986
$190$	0	0
$191$	26.8109	1.93997	0.969986	−	0.243162i	$-0.0781846\pi$
$191$	26.8109	1.93997	0.969986	+	0.243162i	$0.0781846\pi$
$192$	0	0
$193$	−24.4711	−1.76147	−0.880734	−	0.473610i	$-0.842950\pi$
$193$	−24.4711	−1.76147	−0.880734	+	0.473610i	$0.842950\pi$
$194$	0	0
$195$	4.79882	0.343651
$196$	0	0
$197$	8.45247	0.602214	0.301107	−	0.953590i	$-0.402644\pi$
$197$	8.45247	0.602214	0.301107	+	0.953590i	$0.402644\pi$
$198$	0	0
$199$	20.3178	1.44029	0.720146	−	0.693823i	$-0.244075\pi$
$199$	20.3178	1.44029	0.720146	+	0.693823i	$0.244075\pi$
$200$	0	0
$201$	0.989765	0.0698126
$202$	0	0
$203$	26.7638	1.87845
$204$	0	0
$205$	−15.2919	−1.06803
$206$	0	0
$207$	10.6244	0.738445
$208$	0	0
$209$	29.2559	2.02367
$210$	0	0
$211$	−9.92169	−0.683038	−0.341519	−	0.939875i	$-0.610941\pi$
$211$	−9.92169	−0.683038	−0.341519	+	0.939875i	$0.610941\pi$
$212$	0	0
$213$	0.0109387	0.000749510 0
$214$	0	0
$215$	−3.07312	−0.209585
$216$	0	0
$217$	−35.3962	−2.40285
$218$	0	0
$219$	6.84807	0.462750
$220$	0	0
$221$	−3.77487	−0.253925
$222$	0	0
$223$	9.58927	0.642145	0.321072	−	0.947055i	$-0.395957\pi$
$223$	9.58927	0.642145	0.321072	+	0.947055i	$0.395957\pi$
$224$	0	0
$225$	−3.52076	−0.234718
$226$	0	0
$227$	10.4251	0.691941	0.345970	−	0.938245i	$-0.387550\pi$
$227$	10.4251	0.691941	0.345970	+	0.938245i	$0.387550\pi$
$228$	0	0
$229$	−11.2200	−0.741437	−0.370719	−	0.928745i	$-0.620889\pi$
$229$	−11.2200	−0.741437	−0.370719	+	0.928745i	$0.620889\pi$
$230$	0	0
$231$	−12.6669	−0.833418
$232$	0	0
$233$	16.5287	1.08283	0.541416	−	0.840755i	$-0.317889\pi$
$233$	16.5287	1.08283	0.541416	+	0.840755i	$0.317889\pi$
$234$	0	0
$235$	6.89178	0.449570
$236$	0	0
$237$	0.722918	0.0469586
$238$	0	0
$239$	−22.3890	−1.44822	−0.724111	−	0.689684i	$-0.757750\pi$
$239$	−22.3890	−1.44822	−0.724111	+	0.689684i	$0.757750\pi$
$240$	0	0
$241$	−28.5194	−1.83709	−0.918547	−	0.395311i	$-0.870637\pi$
$241$	−28.5194	−1.83709	−0.918547	+	0.395311i	$0.870637\pi$
$242$	0	0
$243$	14.5594	0.933983
$244$	0	0
$245$	20.0776	1.28271
$246$	0	0
$247$	−16.9713	−1.07986
$248$	0	0
$249$	5.76219	0.365164
$250$	0	0
$251$	2.71652	0.171465	0.0857327	−	0.996318i	$-0.472677\pi$
$251$	2.71652	0.171465	0.0857327	+	0.996318i	$0.472677\pi$
$252$	0	0
$253$	20.4569	1.28611
$254$	0	0
$255$	−2.22004	−0.139024
$256$	0	0
$257$	21.6780	1.35224	0.676119	−	0.736792i	$-0.263660\pi$
$257$	21.6780	1.35224	0.676119	+	0.736792i	$0.263660\pi$
$258$	0	0
$259$	−30.2898	−1.88212
$260$	0	0
$261$	−17.7017	−1.09571
$262$	0	0
$263$	−2.91152	−0.179532	−0.0897659	−	0.995963i	$-0.528612\pi$
$263$	−2.91152	−0.179532	−0.0897659	+	0.995963i	$0.528612\pi$
$264$	0	0
$265$	−17.7701	−1.09161
$266$	0	0
$267$	7.63337	0.467154
$268$	0	0
$269$	26.9869	1.64542	0.822709	−	0.568463i	$-0.192462\pi$
$269$	26.9869	1.64542	0.822709	+	0.568463i	$0.192462\pi$
$270$	0	0
$271$	−5.38472	−0.327099	−0.163549	−	0.986535i	$-0.552294\pi$
$271$	−5.38472	−0.327099	−0.163549	+	0.986535i	$0.552294\pi$
$272$	0	0
$273$	7.34801	0.444722
$274$	0	0
$275$	−6.77911	−0.408796
$276$	0	0
$277$	−0.213276	−0.0128145	−0.00640725	−	0.999979i	$-0.502040\pi$
$277$	−0.213276	−0.0128145	−0.00640725	+	0.999979i	$0.502040\pi$
$278$	0	0
$279$	23.4113	1.40160
$280$	0	0
$281$	19.7130	1.17598	0.587990	−	0.808868i	$-0.299919\pi$
$281$	19.7130	1.17598	0.587990	+	0.808868i	$0.299919\pi$
$282$	0	0
$283$	22.1411	1.31615	0.658077	−	0.752951i	$-0.271370\pi$
$283$	22.1411	1.31615	0.658077	+	0.752951i	$0.271370\pi$
$284$	0	0
$285$	−9.98096	−0.591221
$286$	0	0
$287$	−23.4151	−1.38215
$288$	0	0
$289$	−15.2537	−0.897274
$290$	0	0
$291$	−9.23125	−0.541146
$292$	0	0
$293$	7.99933	0.467326	0.233663	−	0.972318i	$-0.424929\pi$
$293$	7.99933	0.467326	0.233663	+	0.972318i	$0.424929\pi$
$294$	0	0
$295$	−8.78333	−0.511385
$296$	0	0
$297$	18.2057	1.05640
$298$	0	0
$299$	−11.8670	−0.686286
$300$	0	0
$301$	−4.70559	−0.271226
$302$	0	0
$303$	−12.7337	−0.731532
$304$	0	0
$305$	−28.0891	−1.60838
$306$	0	0
$307$	15.0436	0.858581	0.429291	−	0.903166i	$-0.358764\pi$
$307$	15.0436	0.858581	0.429291	+	0.903166i	$0.358764\pi$
$308$	0	0
$309$	−8.36917	−0.476106
$310$	0	0
$311$	17.6598	1.00140	0.500698	−	0.865622i	$-0.333077\pi$
$311$	17.6598	1.00140	0.500698	+	0.865622i	$0.333077\pi$
$312$	0	0
$313$	−24.4413	−1.38150	−0.690752	−	0.723092i	$-0.742720\pi$
$313$	−24.4413	−1.38150	−0.690752	+	0.723092i	$0.742720\pi$
$314$	0	0
$315$	−24.9708	−1.40694
$316$	0	0
$317$	0.546761	0.0307092	0.0153546	−	0.999882i	$-0.495112\pi$
$317$	0.546761	0.0307092	0.0153546	+	0.999882i	$0.495112\pi$
$318$	0	0
$319$	−34.0841	−1.90834
$320$	0	0
$321$	6.04763	0.337546
$322$	0	0
$323$	7.85128	0.436857
$324$	0	0
$325$	3.93255	0.218138
$326$	0	0
$327$	13.2379	0.732057
$328$	0	0
$329$	10.5528	0.581794
$330$	0	0
$331$	−7.99286	−0.439327	−0.219664	−	0.975576i	$-0.570496\pi$
$331$	−7.99286	−0.439327	−0.219664	+	0.975576i	$0.570496\pi$
$332$	0	0
$333$	20.0339	1.09785
$334$	0	0
$335$	3.75691	0.205262
$336$	0	0
$337$	7.84276	0.427222	0.213611	−	0.976919i	$-0.431477\pi$
$337$	7.84276	0.427222	0.213611	+	0.976919i	$0.431477\pi$
$338$	0	0
$339$	−10.3712	−0.563286
$340$	0	0
$341$	45.0776	2.44109
$342$	0	0
$343$	3.67663	0.198520
$344$	0	0
$345$	−6.97909	−0.375741
$346$	0	0
$347$	2.63749	0.141588	0.0707940	−	0.997491i	$-0.477447\pi$
$347$	2.63749	0.141588	0.0707940	+	0.997491i	$0.477447\pi$
$348$	0	0
$349$	−8.86543	−0.474555	−0.237278	−	0.971442i	$-0.576255\pi$
$349$	−8.86543	−0.474555	−0.237278	+	0.971442i	$0.576255\pi$
$350$	0	0
$351$	−10.5611	−0.563710
$352$	0	0
$353$	17.6867	0.941369	0.470684	−	0.882302i	$-0.344007\pi$
$353$	17.6867	0.941369	0.470684	+	0.882302i	$0.344007\pi$
$354$	0	0
$355$	0.0415209	0.00220370
$356$	0	0
$357$	−3.39935	−0.179913
$358$	0	0
$359$	−14.9935	−0.791328	−0.395664	−	0.918395i	$-0.629485\pi$
$359$	−14.9935	−0.791328	−0.395664	+	0.918395i	$0.629485\pi$
$360$	0	0
$361$	16.2982	0.857799
$362$	0	0
$363$	8.81345	0.462586
$364$	0	0
$365$	25.9937	1.36057
$366$	0	0
$367$	22.7244	1.18620	0.593102	−	0.805128i	$-0.297903\pi$
$367$	22.7244	1.18620	0.593102	+	0.805128i	$0.297903\pi$
$368$	0	0
$369$	15.4869	0.806216
$370$	0	0
$371$	−27.2098	−1.41266
$372$	0	0
$373$	5.38740	0.278949	0.139475	−	0.990226i	$-0.455459\pi$
$373$	5.38740	0.278949	0.139475	+	0.990226i	$0.455459\pi$
$374$	0	0
$375$	−6.08698	−0.314330
$376$	0	0
$377$	19.7721	1.01831
$378$	0	0
$379$	−4.84935	−0.249094	−0.124547	−	0.992214i	$-0.539748\pi$
$379$	−4.84935	−0.249094	−0.124547	+	0.992214i	$0.539748\pi$
$380$	0	0
$381$	−4.45945	−0.228465
$382$	0	0
$383$	13.7964	0.704963	0.352481	−	0.935819i	$-0.385338\pi$
$383$	13.7964	0.704963	0.352481	+	0.935819i	$0.385338\pi$
$384$	0	0
$385$	−48.0804	−2.45040
$386$	0	0
$387$	3.11231	0.158207
$388$	0	0
$389$	−3.81099	−0.193225	−0.0966125	−	0.995322i	$-0.530801\pi$
$389$	−3.81099	−0.193225	−0.0966125	+	0.995322i	$0.530801\pi$
$390$	0	0
$391$	5.48993	0.277638
$392$	0	0
$393$	−10.6946	−0.539472
$394$	0	0
$395$	2.74403	0.138067
$396$	0	0
$397$	17.0753	0.856983	0.428492	−	0.903546i	$-0.359045\pi$
$397$	17.0753	0.856983	0.428492	+	0.903546i	$0.359045\pi$
$398$	0	0
$399$	−15.2830	−0.765106
$400$	0	0
$401$	16.6744	0.832678	0.416339	−	0.909209i	$-0.363313\pi$
$401$	16.6744	0.832678	0.416339	+	0.909209i	$0.363313\pi$
$402$	0	0
$403$	−26.1494	−1.30260
$404$	0	0
$405$	13.1629	0.654072
$406$	0	0
$407$	38.5746	1.91207
$408$	0	0
$409$	11.7701	0.581996	0.290998	−	0.956724i	$-0.406013\pi$
$409$	11.7701	0.581996	0.290998	+	0.956724i	$0.406013\pi$
$410$	0	0
$411$	−1.94063	−0.0957242
$412$	0	0
$413$	−13.4491	−0.661789
$414$	0	0
$415$	21.8719	1.07365
$416$	0	0
$417$	8.93806	0.437699
$418$	0	0
$419$	−31.7618	−1.55166	−0.775832	−	0.630939i	$-0.782670\pi$
$419$	−31.7618	−1.55166	−0.775832	+	0.630939i	$0.782670\pi$
$420$	0	0
$421$	−28.4937	−1.38870	−0.694348	−	0.719639i	$-0.744307\pi$
$421$	−28.4937	−1.38870	−0.694348	+	0.719639i	$0.744307\pi$
$422$	0	0
$423$	−6.97967	−0.339363
$424$	0	0
$425$	−1.81928	−0.0882481
$426$	0	0
$427$	−43.0104	−2.08142
$428$	0	0
$429$	−9.35780	−0.451799
$430$	0	0
$431$	5.59399	0.269453	0.134726	−	0.990883i	$-0.456984\pi$
$431$	5.59399	0.269453	0.134726	+	0.990883i	$0.456984\pi$
$432$	0	0
$433$	3.81276	0.183230	0.0916148	−	0.995795i	$-0.470797\pi$
$433$	3.81276	0.183230	0.0916148	+	0.995795i	$0.470797\pi$
$434$	0	0
$435$	11.6282	0.557527
$436$	0	0
$437$	24.6819	1.18070
$438$	0	0
$439$	−19.5628	−0.933679	−0.466840	−	0.884342i	$-0.654607\pi$
$439$	−19.5628	−0.933679	−0.466840	+	0.884342i	$0.654607\pi$
$440$	0	0
$441$	−20.3337	−0.968269
$442$	0	0
$443$	6.80653	0.323388	0.161694	−	0.986841i	$-0.448304\pi$
$443$	6.80653	0.323388	0.161694	+	0.986841i	$0.448304\pi$
$444$	0	0
$445$	28.9744	1.37352
$446$	0	0
$447$	−8.96850	−0.424195
$448$	0	0
$449$	28.1146	1.32681	0.663404	−	0.748261i	$-0.269111\pi$
$449$	28.1146	1.32681	0.663404	+	0.748261i	$0.269111\pi$
$450$	0	0
$451$	29.8195	1.40415
$452$	0	0
$453$	0.979342	0.0460135
$454$	0	0
$455$	27.8913	1.30756
$456$	0	0
$457$	39.2925	1.83803	0.919013	−	0.394228i	$-0.128988\pi$
$457$	39.2925	1.83803	0.919013	+	0.394228i	$0.128988\pi$
$458$	0	0
$459$	4.88580	0.228049
$460$	0	0
$461$	−16.3528	−0.761626	−0.380813	−	0.924652i	$-0.624356\pi$
$461$	−16.3528	−0.761626	−0.380813	+	0.924652i	$0.624356\pi$
$462$	0	0
$463$	−3.32027	−0.154306	−0.0771530	−	0.997019i	$-0.524583\pi$
$463$	−3.32027	−0.154306	−0.0771530	+	0.997019i	$0.524583\pi$
$464$	0	0
$465$	−15.3787	−0.713171
$466$	0	0
$467$	7.12740	0.329817	0.164908	−	0.986309i	$-0.447267\pi$
$467$	7.12740	0.329817	0.164908	+	0.986309i	$0.447267\pi$
$468$	0	0
$469$	5.75263	0.265632
$470$	0	0
$471$	14.3495	0.661190
$472$	0	0
$473$	5.99264	0.275542
$474$	0	0
$475$	−8.17922	−0.375288
$476$	0	0
$477$	17.9967	0.824013
$478$	0	0
$479$	−26.4983	−1.21074	−0.605368	−	0.795946i	$-0.706974\pi$
$479$	−26.4983	−1.21074	−0.605368	+	0.795946i	$0.706974\pi$
$480$	0	0
$481$	−22.3770	−1.02030
$482$	0	0
$483$	−10.6865	−0.486251
$484$	0	0
$485$	−35.0397	−1.59107
$486$	0	0
$487$	−8.57166	−0.388419	−0.194209	−	0.980960i	$-0.562214\pi$
$487$	−8.57166	−0.388419	−0.194209	+	0.980960i	$0.562214\pi$
$488$	0	0
$489$	−2.90713	−0.131465
$490$	0	0
$491$	−30.6031	−1.38110	−0.690549	−	0.723286i	$-0.742631\pi$
$491$	−30.6031	−1.38110	−0.690549	+	0.723286i	$0.742631\pi$
$492$	0	0
$493$	−9.14700	−0.411960
$494$	0	0
$495$	31.8006	1.42933
$496$	0	0
$497$	0.0635773	0.00285183
$498$	0	0
$499$	29.2732	1.31045	0.655223	−	0.755435i	$-0.272575\pi$
$499$	29.2732	1.31045	0.655223	+	0.755435i	$0.272575\pi$
$500$	0	0
$501$	6.87243	0.307037
$502$	0	0
$503$	1.00000	0.0445878
$503$	1.00000	0.0445878
$504$	0	0
$505$	−48.3341	−2.15084
$506$	0	0
$507$	−3.22008	−0.143009
$508$	0	0
$509$	−28.2411	−1.25177	−0.625883	−	0.779917i	$-0.715261\pi$
$509$	−28.2411	−1.25177	−0.625883	+	0.779917i	$0.715261\pi$
$510$	0	0
$511$	39.8018	1.76073
$512$	0	0
$513$	21.9658	0.969815
$514$	0	0
$515$	−31.7674	−1.39984
$516$	0	0
$517$	−13.4391	−0.591052
$518$	0	0
$519$	3.22968	0.141767
$520$	0	0
$521$	−34.1075	−1.49427	−0.747137	−	0.664670i	$-0.768572\pi$
$521$	−34.1075	−1.49427	−0.747137	+	0.664670i	$0.768572\pi$
$522$	0	0
$523$	−10.3227	−0.451382	−0.225691	−	0.974199i	$-0.572464\pi$
$523$	−10.3227	−0.451382	−0.225691	+	0.974199i	$0.572464\pi$
$524$	0	0
$525$	3.54134	0.154557
$526$	0	0
$527$	12.0973	0.526966
$528$	0	0
$529$	−5.74143	−0.249627
$530$	0	0
$531$	8.89534	0.386025
$532$	0	0
$533$	−17.2982	−0.749269
$534$	0	0
$535$	22.9554	0.992447
$536$	0	0
$537$	4.84816	0.209214
$538$	0	0
$539$	−39.1518	−1.68639
$540$	0	0
$541$	−3.02834	−0.130199	−0.0650993	−	0.997879i	$-0.520736\pi$
$541$	−3.02834	−0.130199	−0.0650993	+	0.997879i	$0.520736\pi$
$542$	0	0
$543$	3.45399	0.148225
$544$	0	0
$545$	50.2479	2.15238
$546$	0	0
$547$	−22.9168	−0.979850	−0.489925	−	0.871765i	$-0.662976\pi$
$547$	−22.9168	−0.979850	−0.489925	+	0.871765i	$0.662976\pi$
$548$	0	0
$549$	28.4474	1.21410
$550$	0	0
$551$	−41.1236	−1.75192
$552$	0	0
$553$	4.20169	0.178674
$554$	0	0
$555$	−13.1601	−0.558616
$556$	0	0
$557$	−42.7509	−1.81142	−0.905708	−	0.423902i	$-0.860660\pi$
$557$	−42.7509	−1.81142	−0.905708	+	0.423902i	$0.860660\pi$
$558$	0	0
$559$	−3.47632	−0.147033
$560$	0	0
$561$	4.32912	0.182776
$562$	0	0
$563$	21.1292	0.890489	0.445245	−	0.895409i	$-0.353117\pi$
$563$	21.1292	0.890489	0.445245	+	0.895409i	$0.353117\pi$
$564$	0	0
$565$	−39.3666	−1.65617
$566$	0	0
$567$	20.1553	0.846442
$568$	0	0
$569$	−16.9149	−0.709111	−0.354556	−	0.935035i	$-0.615368\pi$
$569$	−16.9149	−0.709111	−0.354556	+	0.935035i	$0.615368\pi$
$570$	0	0
$571$	−19.8900	−0.832371	−0.416185	−	0.909280i	$-0.636633\pi$
$571$	−19.8900	−0.832371	−0.416185	+	0.909280i	$0.636633\pi$
$572$	0	0
$573$	17.8365	0.745132
$574$	0	0
$575$	−5.71924	−0.238509
$576$	0	0
$577$	−23.7705	−0.989578	−0.494789	−	0.869013i	$-0.664755\pi$
$577$	−23.7705	−0.989578	−0.494789	+	0.869013i	$0.664755\pi$
$578$	0	0
$579$	−16.2799	−0.676570
$580$	0	0
$581$	33.4905	1.38942
$582$	0	0
$583$	34.6521	1.43514
$584$	0	0
$585$	−18.4475	−0.762709
$586$	0	0
$587$	2.47355	0.102094	0.0510472	−	0.998696i	$-0.483744\pi$
$587$	2.47355	0.102094	0.0510472	+	0.998696i	$0.483744\pi$
$588$	0	0
$589$	54.3876	2.24100
$590$	0	0
$591$	5.62318	0.231307
$592$	0	0
$593$	14.3476	0.589186	0.294593	−	0.955623i	$-0.404816\pi$
$593$	14.3476	0.589186	0.294593	+	0.955623i	$0.404816\pi$
$594$	0	0
$595$	−12.9031	−0.528976
$596$	0	0
$597$	13.5169	0.553208
$598$	0	0
$599$	−40.3074	−1.64692	−0.823459	−	0.567376i	$-0.807959\pi$
$599$	−40.3074	−1.64692	−0.823459	+	0.567376i	$0.807959\pi$
$600$	0	0
$601$	−26.8974	−1.09717	−0.548584	−	0.836095i	$-0.684833\pi$
$601$	−26.8974	−1.09717	−0.548584	+	0.836095i	$0.684833\pi$
$602$	0	0
$603$	−3.80482	−0.154944
$604$	0	0
$605$	33.4538	1.36009
$606$	0	0
$607$	−37.8462	−1.53613	−0.768065	−	0.640372i	$-0.778780\pi$
$607$	−37.8462	−1.53613	−0.768065	+	0.640372i	$0.778780\pi$
$608$	0	0
$609$	17.8052	0.721502
$610$	0	0
$611$	7.79601	0.315393
$612$	0	0
$613$	−40.9962	−1.65582	−0.827911	−	0.560860i	$-0.810471\pi$
$613$	−40.9962	−1.65582	−0.827911	+	0.560860i	$0.810471\pi$
$614$	0	0
$615$	−10.1732	−0.410225
$616$	0	0
$617$	35.5557	1.43142	0.715710	−	0.698398i	$-0.246103\pi$
$617$	35.5557	1.43142	0.715710	+	0.698398i	$0.246103\pi$
$618$	0	0
$619$	5.76781	0.231828	0.115914	−	0.993259i	$-0.463020\pi$
$619$	5.76781	0.231828	0.115914	+	0.993259i	$0.463020\pi$
$620$	0	0
$621$	15.3594	0.616351
$622$	0	0
$623$	44.3660	1.77749
$624$	0	0
$625$	−29.9882	−1.19953
$626$	0	0
$627$	19.4631	0.777281
$628$	0	0
$629$	10.3521	0.412765
$630$	0	0
$631$	−7.23876	−0.288171	−0.144085	−	0.989565i	$-0.546024\pi$
$631$	−7.23876	−0.288171	−0.144085	+	0.989565i	$0.546024\pi$
$632$	0	0
$633$	−6.60061	−0.262351
$634$	0	0
$635$	−16.9270	−0.671729
$636$	0	0
$637$	22.7118	0.899876
$638$	0	0
$639$	−0.0420504	−0.00166349
$640$	0	0
$641$	1.50143	0.0593030	0.0296515	−	0.999560i	$-0.490560\pi$
$641$	1.50143	0.0593030	0.0296515	+	0.999560i	$0.490560\pi$
$642$	0	0
$643$	33.6035	1.32519	0.662595	−	0.748977i	$-0.269455\pi$
$643$	33.6035	1.32519	0.662595	+	0.748977i	$0.269455\pi$
$644$	0	0
$645$	−2.04445	−0.0805003
$646$	0	0
$647$	−4.83217	−0.189972	−0.0949861	−	0.995479i	$-0.530281\pi$
$647$	−4.83217	−0.189972	−0.0949861	+	0.995479i	$0.530281\pi$
$648$	0	0
$649$	17.1277	0.672320
$650$	0	0
$651$	−23.5481	−0.922922
$652$	0	0
$653$	14.9757	0.586043	0.293021	−	0.956106i	$-0.405339\pi$
$653$	14.9757	0.586043	0.293021	+	0.956106i	$0.405339\pi$
$654$	0	0
$655$	−40.5942	−1.58615
$656$	0	0
$657$	−26.3252	−1.02704
$658$	0	0
$659$	3.90762	0.152219	0.0761096	−	0.997099i	$-0.475750\pi$
$659$	3.90762	0.152219	0.0761096	+	0.997099i	$0.475750\pi$
$660$	0	0
$661$	21.9155	0.852416	0.426208	−	0.904625i	$-0.359849\pi$
$661$	21.9155	0.852416	0.426208	+	0.904625i	$0.359849\pi$
$662$	0	0
$663$	−2.51131	−0.0975313
$664$	0	0
$665$	−58.0105	−2.24955
$666$	0	0
$667$	−28.7552	−1.11341
$668$	0	0
$669$	6.37946	0.246644
$670$	0	0
$671$	54.7745	2.11454
$672$	0	0
$673$	7.39904	0.285212	0.142606	−	0.989780i	$-0.454452\pi$
$673$	7.39904	0.285212	0.142606	+	0.989780i	$0.454452\pi$
$674$	0	0
$675$	−5.08987	−0.195909
$676$	0	0
$677$	−35.4034	−1.36066	−0.680331	−	0.732905i	$-0.738164\pi$
$677$	−35.4034	−1.36066	−0.680331	+	0.732905i	$0.738164\pi$
$678$	0	0
$679$	−53.6532	−2.05902
$680$	0	0
$681$	6.93554	0.265771
$682$	0	0
$683$	11.2912	0.432047	0.216023	−	0.976388i	$-0.430691\pi$
$683$	11.2912	0.432047	0.216023	+	0.976388i	$0.430691\pi$
$684$	0	0
$685$	−7.36617	−0.281447
$686$	0	0
$687$	−7.46433	−0.284782
$688$	0	0
$689$	−20.1016	−0.765810
$690$	0	0
$691$	−27.7284	−1.05484	−0.527420	−	0.849605i	$-0.676840\pi$
$691$	−27.7284	−1.05484	−0.527420	+	0.849605i	$0.676840\pi$
$692$	0	0
$693$	48.6936	1.84972
$694$	0	0
$695$	33.9268	1.28692
$696$	0	0
$697$	8.00253	0.303117
$698$	0	0
$699$	10.9961	0.415909
$700$	0	0
$701$	15.1510	0.572245	0.286122	−	0.958193i	$-0.407634\pi$
$701$	15.1510	0.572245	0.286122	+	0.958193i	$0.407634\pi$
$702$	0	0
$703$	46.5414	1.75534
$704$	0	0
$705$	4.58490	0.172677
$706$	0	0
$707$	−74.0098	−2.78342
$708$	0	0
$709$	−14.4405	−0.542323	−0.271162	−	0.962534i	$-0.587408\pi$
$709$	−14.4405	−0.542323	−0.271162	+	0.962534i	$0.587408\pi$
$710$	0	0
$711$	−2.77902	−0.104221
$712$	0	0
$713$	38.0300	1.42423
$714$	0	0
$715$	−35.5200	−1.32837
$716$	0	0
$717$	−14.8947	−0.556254
$718$	0	0
$719$	−2.01880	−0.0752886	−0.0376443	−	0.999291i	$-0.511985\pi$
$719$	−2.01880	−0.0752886	−0.0376443	+	0.999291i	$0.511985\pi$
$720$	0	0
$721$	−48.6427	−1.81155
$722$	0	0
$723$	−18.9731	−0.705618
$724$	0	0
$725$	9.52906	0.353901
$726$	0	0
$727$	−4.04132	−0.149884	−0.0749422	−	0.997188i	$-0.523877\pi$
$727$	−4.04132	−0.149884	−0.0749422	+	0.997188i	$0.523877\pi$
$728$	0	0
$729$	−5.95192	−0.220442
$730$	0	0
$731$	1.60822	0.0594822
$732$	0	0
$733$	−21.2760	−0.785846	−0.392923	−	0.919571i	$-0.628536\pi$
$733$	−21.2760	−0.785846	−0.392923	+	0.919571i	$0.628536\pi$
$734$	0	0
$735$	13.3570	0.492682
$736$	0	0
$737$	−7.32606	−0.269859
$738$	0	0
$739$	1.54987	0.0570130	0.0285065	−	0.999594i	$-0.490925\pi$
$739$	1.54987	0.0570130	0.0285065	+	0.999594i	$0.490925\pi$
$740$	0	0
$741$	−11.2905	−0.414767
$742$	0	0
$743$	32.5846	1.19541	0.597707	−	0.801715i	$-0.296079\pi$
$743$	32.5846	1.19541	0.597707	+	0.801715i	$0.296079\pi$
$744$	0	0
$745$	−34.0423	−1.24721
$746$	0	0
$747$	−22.1508	−0.810457
$748$	0	0
$749$	35.1495	1.28434
$750$	0	0
$751$	−31.3118	−1.14258	−0.571291	−	0.820747i	$-0.693557\pi$
$751$	−31.3118	−1.14258	−0.571291	+	0.820747i	$0.693557\pi$
$752$	0	0
$753$	1.80722	0.0658589
$754$	0	0
$755$	3.71735	0.135288
$756$	0	0
$757$	−3.25550	−0.118323	−0.0591615	−	0.998248i	$-0.518843\pi$
$757$	−3.25550	−0.118323	−0.0591615	+	0.998248i	$0.518843\pi$
$758$	0	0
$759$	13.6094	0.493989
$760$	0	0
$761$	−50.9362	−1.84643	−0.923217	−	0.384278i	$-0.874450\pi$
$761$	−50.9362	−1.84643	−0.923217	+	0.384278i	$0.874450\pi$
$762$	0	0
$763$	76.9402	2.78542
$764$	0	0
$765$	8.53420	0.308555
$766$	0	0
$767$	−9.93573	−0.358758
$768$	0	0
$769$	−39.9611	−1.44103	−0.720517	−	0.693438i	$-0.756095\pi$
$769$	−39.9611	−1.44103	−0.720517	+	0.693438i	$0.756095\pi$
$770$	0	0
$771$	14.4218	0.519387
$772$	0	0
$773$	−25.6506	−0.922586	−0.461293	−	0.887248i	$-0.652614\pi$
$773$	−25.6506	−0.922586	−0.461293	+	0.887248i	$0.652614\pi$
$774$	0	0
$775$	−12.6026	−0.452698
$776$	0	0
$777$	−20.1509	−0.722911
$778$	0	0
$779$	35.9782	1.28905
$780$	0	0
$781$	−0.0809666	−0.00289721
$782$	0	0
$783$	−25.5909	−0.914545
$784$	0	0
$785$	54.4673	1.94402
$786$	0	0
$787$	−40.5208	−1.44441	−0.722205	−	0.691679i	$-0.756871\pi$
$787$	−40.5208	−1.44441	−0.722205	+	0.691679i	$0.756871\pi$
$788$	0	0
$789$	−1.93695	−0.0689572
$790$	0	0
$791$	−60.2786	−2.14326
$792$	0	0
$793$	−31.7745	−1.12835
$794$	0	0
$795$	−11.8219	−0.419281
$796$	0	0
$797$	5.33460	0.188961	0.0944806	−	0.995527i	$-0.469881\pi$
$797$	5.33460	0.188961	0.0944806	+	0.995527i	$0.469881\pi$
$798$	0	0
$799$	−3.60660	−0.127592
$800$	0	0
$801$	−29.3440	−1.03682
$802$	0	0
$803$	−50.6882	−1.78875
$804$	0	0
$805$	−40.5633	−1.42967
$806$	0	0
$807$	17.9536	0.631996
$808$	0	0
$809$	−12.7029	−0.446611	−0.223305	−	0.974749i	$-0.571685\pi$
$809$	−12.7029	−0.446611	−0.223305	+	0.974749i	$0.571685\pi$
$810$	0	0
$811$	54.2073	1.90347	0.951737	−	0.306914i	$-0.0992962\pi$
$811$	54.2073	1.90347	0.951737	+	0.306914i	$0.0992962\pi$
$812$	0	0
$813$	−3.58230	−0.125637
$814$	0	0
$815$	−11.0348	−0.386532
$816$	0	0
$817$	7.23032	0.252957
$818$	0	0
$819$	−28.2470	−0.987030
$820$	0	0
$821$	36.3066	1.26711	0.633555	−	0.773697i	$-0.281595\pi$
$821$	36.3066	1.26711	0.633555	+	0.773697i	$0.281595\pi$
$822$	0	0
$823$	−37.1894	−1.29634	−0.648171	−	0.761495i	$-0.724466\pi$
$823$	−37.1894	−1.29634	−0.648171	+	0.761495i	$0.724466\pi$
$824$	0	0
$825$	−4.50995	−0.157016
$826$	0	0
$827$	19.3341	0.672314	0.336157	−	0.941806i	$-0.390873\pi$
$827$	19.3341	0.672314	0.336157	+	0.941806i	$0.390873\pi$
$828$	0	0
$829$	−39.8763	−1.38496	−0.692480	−	0.721437i	$-0.743482\pi$
$829$	−39.8763	−1.38496	−0.692480	+	0.721437i	$0.743482\pi$
$830$	0	0
$831$	−0.141886	−0.00492198
$832$	0	0
$833$	−10.5070	−0.364046
$834$	0	0
$835$	26.0861	0.902747
$836$	0	0
$837$	33.8450	1.16986
$838$	0	0
$839$	−14.3938	−0.496929	−0.248464	−	0.968641i	$-0.579926\pi$
$839$	−14.3938	−0.496929	−0.248464	+	0.968641i	$0.579926\pi$
$840$	0	0
$841$	18.9103	0.652080
$842$	0	0
$843$	13.1145	0.451688
$844$	0	0
$845$	−12.2227	−0.420473
$846$	0	0
$847$	51.2248	1.76011
$848$	0	0
$849$	14.7299	0.505527
$850$	0	0
$851$	32.5436	1.11558
$852$	0	0
$853$	38.2090	1.30825	0.654125	−	0.756386i	$-0.273037\pi$
$853$	38.2090	1.30825	0.654125	+	0.756386i	$0.273037\pi$
$854$	0	0
$855$	38.3685	1.31218
$856$	0	0
$857$	−50.0375	−1.70925	−0.854624	−	0.519248i	$-0.826212\pi$
$857$	−50.0375	−1.70925	−0.854624	+	0.519248i	$0.826212\pi$
$858$	0	0
$859$	39.4753	1.34688	0.673441	−	0.739241i	$-0.264816\pi$
$859$	39.4753	1.34688	0.673441	+	0.739241i	$0.264816\pi$
$860$	0	0
$861$	−15.5774	−0.530876
$862$	0	0
$863$	−45.2191	−1.53928	−0.769639	−	0.638479i	$-0.779564\pi$
$863$	−45.2191	−1.53928	−0.769639	+	0.638479i	$0.779564\pi$
$864$	0	0
$865$	12.2591	0.416822
$866$	0	0
$867$	−10.1478	−0.344638
$868$	0	0
$869$	−5.35091	−0.181517
$870$	0	0
$871$	4.24983	0.144000
$872$	0	0
$873$	35.4865	1.20104
$874$	0	0
$875$	−35.3783	−1.19600
$876$	0	0
$877$	−29.5389	−0.997459	−0.498729	−	0.866758i	$-0.666200\pi$
$877$	−29.5389	−0.997459	−0.498729	+	0.866758i	$0.666200\pi$
$878$	0	0
$879$	5.32172	0.179497
$880$	0	0
$881$	9.94031	0.334897	0.167449	−	0.985881i	$-0.446447\pi$
$881$	9.94031	0.334897	0.167449	+	0.985881i	$0.446447\pi$
$882$	0	0
$883$	13.6084	0.457958	0.228979	−	0.973431i	$-0.426461\pi$
$883$	13.6084	0.457958	0.228979	+	0.973431i	$0.426461\pi$
$884$	0	0
$885$	−5.84329	−0.196420
$886$	0	0
$887$	−31.4509	−1.05602	−0.528008	−	0.849239i	$-0.677061\pi$
$887$	−31.4509	−1.05602	−0.528008	+	0.849239i	$0.677061\pi$
$888$	0	0
$889$	−25.9189	−0.869291
$890$	0	0
$891$	−25.6680	−0.859912
$892$	0	0
$893$	−16.2147	−0.542606
$894$	0	0
$895$	18.4025	0.615127
$896$	0	0
$897$	−7.89476	−0.263598
$898$	0	0
$899$	−63.3634	−2.11329
$900$	0	0
$901$	9.29943	0.309809
$902$	0	0
$903$	−3.13049	−0.104176
$904$	0	0
$905$	13.1105	0.435809
$906$	0	0
$907$	23.3255	0.774512	0.387256	−	0.921972i	$-0.373423\pi$
$907$	23.3255	0.774512	0.387256	+	0.921972i	$0.373423\pi$
$908$	0	0
$909$	48.9505	1.62359
$910$	0	0
$911$	52.0312	1.72387	0.861935	−	0.507019i	$-0.169253\pi$
$911$	52.0312	1.72387	0.861935	+	0.507019i	$0.169253\pi$
$912$	0	0
$913$	−42.6507	−1.41153
$914$	0	0
$915$	−18.6869	−0.617769
$916$	0	0
$917$	−62.1584	−2.05265
$918$	0	0
$919$	−55.9000	−1.84397	−0.921985	−	0.387225i	$-0.873434\pi$
$919$	−55.9000	−1.84397	−0.921985	+	0.387225i	$0.873434\pi$
$920$	0	0
$921$	10.0080	0.329776
$922$	0	0
$923$	0.0469685	0.00154599
$924$	0	0
$925$	−10.7845	−0.354592
$926$	0	0
$927$	32.1725	1.05668
$928$	0	0
$929$	−15.0814	−0.494806	−0.247403	−	0.968913i	$-0.579577\pi$
$929$	−15.0814	−0.494806	−0.247403	+	0.968913i	$0.579577\pi$
$930$	0	0
$931$	−47.2379	−1.54816
$932$	0	0
$933$	11.7486	0.384631
$934$	0	0
$935$	16.4323	0.537394
$936$	0	0
$937$	17.9652	0.586898	0.293449	−	0.955975i	$-0.405197\pi$
$937$	17.9652	0.586898	0.293449	+	0.955975i	$0.405197\pi$
$938$	0	0
$939$	−16.2601	−0.530628
$940$	0	0
$941$	−6.68206	−0.217829	−0.108914	−	0.994051i	$-0.534737\pi$
$941$	−6.68206	−0.217829	−0.108914	+	0.994051i	$0.534737\pi$
$942$	0	0
$943$	25.1574	0.819237
$944$	0	0
$945$	−36.0996	−1.17432
$946$	0	0
$947$	31.5759	1.02608	0.513039	−	0.858365i	$-0.328520\pi$
$947$	31.5759	1.02608	0.513039	+	0.858365i	$0.328520\pi$
$948$	0	0
$949$	29.4041	0.954498
$950$	0	0
$951$	0.363744	0.0117952
$952$	0	0
$953$	42.5552	1.37850	0.689249	−	0.724525i	$-0.257941\pi$
$953$	42.5552	1.37850	0.689249	+	0.724525i	$0.257941\pi$
$954$	0	0
$955$	67.7033	2.19083
$956$	0	0
$957$	−22.6752	−0.732984
$958$	0	0
$959$	−11.2792	−0.364223
$960$	0	0
$961$	52.8007	1.70325
$962$	0	0
$963$	−23.2481	−0.749160
$964$	0	0
$965$	−61.7947	−1.98924
$966$	0	0
$967$	28.2512	0.908498	0.454249	−	0.890875i	$-0.349908\pi$
$967$	28.2512	0.908498	0.454249	+	0.890875i	$0.349908\pi$
$968$	0	0
$969$	5.22323	0.167794
$970$	0	0
$971$	26.1828	0.840245	0.420122	−	0.907467i	$-0.361987\pi$
$971$	26.1828	0.840245	0.420122	+	0.907467i	$0.361987\pi$
$972$	0	0
$973$	51.9491	1.66541
$974$	0	0
$975$	2.61621	0.0837857
$976$	0	0
$977$	−29.1324	−0.932027	−0.466013	−	0.884778i	$-0.654310\pi$
$977$	−29.1324	−0.932027	−0.466013	+	0.884778i	$0.654310\pi$
$978$	0	0
$979$	−56.5008	−1.80577
$980$	0	0
$981$	−50.8887	−1.62475
$982$	0	0
$983$	−15.2250	−0.485603	−0.242802	−	0.970076i	$-0.578066\pi$
$983$	−15.2250	−0.485603	−0.242802	+	0.970076i	$0.578066\pi$
$984$	0	0
$985$	21.3443	0.680085
$986$	0	0
$987$	7.02046	0.223464
$988$	0	0
$989$	5.05573	0.160763
$990$	0	0
$991$	−21.7553	−0.691081	−0.345541	−	0.938404i	$-0.612304\pi$
$991$	−21.7553	−0.691081	−0.345541	+	0.938404i	$0.612304\pi$
$992$	0	0
$993$	−5.31742	−0.168743
$994$	0	0
$995$	51.3068	1.62653
$996$	0	0
$997$	−8.08508	−0.256057	−0.128029	−	0.991770i	$-0.540865\pi$
$997$	−8.08508	−0.256057	−0.128029	+	0.991770i	$0.540865\pi$
$998$	0	0
$999$	28.9624	0.916331

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	8048.2.a.q.1.8		12
4.3	odd	2		1006.2.a.j.1.5	✓	12
12.11	even	2		9054.2.a.bi.1.3		12

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
1006.2.a.j.1.5	✓	12	4.3	odd	2
8048.2.a.q.1.8		12	1.1	even	1	trivial
9054.2.a.bi.1.3		12	12.11	even	2

Overview	Random
Universe	Knowledge

Classical	Maass
Hilbert	Bianchi

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

Level:	\( N \)	\(=\)	\( 8048 = 2^{4} \cdot 503 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	8048.a (trivial)

\(f(q)\)	\(=\)	\(q+0.665271 q^{3} +2.52521 q^{5} +3.86663 q^{7} -2.55741 q^{9} +O(q^{10})\)	\(q+0.665271 q^{3} +2.52521 q^{5} +3.86663 q^{7} -2.55741 q^{9} -4.92422 q^{11} +2.85653 q^{13} +1.67995 q^{15} -1.32149 q^{17} -5.94123 q^{19} +2.57236 q^{21} -4.15434 q^{23} +1.37669 q^{25} -3.69719 q^{27} +6.92173 q^{29} -9.15427 q^{31} -3.27594 q^{33} +9.76407 q^{35} -7.83364 q^{37} +1.90036 q^{39} -6.05568 q^{41} -1.21697 q^{43} -6.45801 q^{45} +2.72919 q^{47} +7.95086 q^{49} -0.879149 q^{51} -7.03708 q^{53} -12.4347 q^{55} -3.95253 q^{57} -3.47826 q^{59} -11.1235 q^{61} -9.88859 q^{63} +7.21333 q^{65} +1.48776 q^{67} -2.76376 q^{69} +0.0164425 q^{71} +10.2937 q^{73} +0.915871 q^{75} -19.0401 q^{77} +1.08665 q^{79} +5.21261 q^{81} +8.66142 q^{83} -3.33704 q^{85} +4.60482 q^{87} +11.4741 q^{89} +11.0451 q^{91} -6.09007 q^{93} -15.0028 q^{95} -13.8759 q^{97} +12.5933 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 12 q - 5 q^{3} + 5 q^{5} - 8 q^{7} + 15 q^{9}+O(q^{10}) \) 12 * q - 5 * q^3 + 5 * q^5 - 8 * q^7 + 15 * q^9	\( 12 q - 5 q^{3} + 5 q^{5} - 8 q^{7} + 15 q^{9} - 18 q^{11} + 4 q^{13} - 2 q^{15} - 2 q^{17} - 6 q^{19} + q^{21} - 13 q^{23} + q^{25} - 8 q^{27} + 20 q^{29} - 7 q^{31} - 8 q^{33} - q^{35} + 10 q^{37} - 7 q^{39} + 2 q^{41} - 8 q^{43} - 7 q^{45} - 12 q^{47} + 4 q^{49} - 2 q^{51} + 12 q^{53} - 8 q^{55} - 10 q^{57} - 6 q^{59} - 10 q^{61} - 7 q^{63} + 4 q^{65} - 7 q^{67} - 12 q^{69} - 22 q^{71} - 23 q^{73} + 34 q^{75} - 19 q^{77} - 13 q^{79} - 28 q^{81} - q^{83} - 28 q^{85} + 22 q^{87} - 3 q^{89} + 21 q^{91} - 33 q^{93} + 2 q^{95} - 70 q^{97} + 6 q^{99}+O(q^{100}) \) 12 * q - 5 * q^3 + 5 * q^5 - 8 * q^7 + 15 * q^9 - 18 * q^11 + 4 * q^13 - 2 * q^15 - 2 * q^17 - 6 * q^19 + q^21 - 13 * q^23 + q^25 - 8 * q^27 + 20 * q^29 - 7 * q^31 - 8 * q^33 - q^35 + 10 * q^37 - 7 * q^39 + 2 * q^41 - 8 * q^43 - 7 * q^45 - 12 * q^47 + 4 * q^49 - 2 * q^51 + 12 * q^53 - 8 * q^55 - 10 * q^57 - 6 * q^59 - 10 * q^61 - 7 * q^63 + 4 * q^65 - 7 * q^67 - 12 * q^69 - 22 * q^71 - 23 * q^73 + 34 * q^75 - 19 * q^77 - 13 * q^79 - 28 * q^81 - q^83 - 28 * q^85 + 22 * q^87 - 3 * q^89 + 21 * q^91 - 33 * q^93 + 2 * q^95 - 70 * q^97 + 6 * q^99

Introduction

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