LMFDB - Embedded newform 1004.2.a.a.1.2

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [1004,2,Mod(1,1004)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(1004, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([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("1004.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 1004 = 2^{2} \cdot 251 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	1004.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:	$8.01698036294$
Analytic rank:	$1$
Dimension:	$7$
Coefficient field:	$\mathbb{Q}[x]/(x^{7} - \cdots)$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{7} - 3x^{6} - 6x^{5} + 18x^{4} + 8x^{3} - 17x^{2} - 9x - 1 $ x^7 - 3x^6 - 6x^5 + 18x^4 + 8x^3 - 17x^2 - 9x - 1
Coefficient ring:	$\Z[a_1, \ldots, a_{5}]$
Coefficient ring index:	$ 1 $
Twist minimal:	yes
Fricke sign:	$1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.2
Root			$2.29157$ of defining polynomial
Character	$\chi$	$=$	1004.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-2.29157 q^{3} -1.78468 q^{5} +0.671050 q^{7} +2.25130 q^{9} +O(q^{10})$	$q-2.29157 q^{3} -1.78468 q^{5} +0.671050 q^{7} +2.25130 q^{9} +2.23015 q^{11} +0.0525243 q^{13} +4.08972 q^{15} +5.52610 q^{17} +1.49944 q^{19} -1.53776 q^{21} -7.13437 q^{23} -1.81492 q^{25} +1.71570 q^{27} +2.08515 q^{29} -9.61596 q^{31} -5.11055 q^{33} -1.19761 q^{35} +5.33025 q^{37} -0.120363 q^{39} -11.8708 q^{41} -9.09066 q^{43} -4.01785 q^{45} +5.21388 q^{47} -6.54969 q^{49} -12.6635 q^{51} +4.27480 q^{53} -3.98011 q^{55} -3.43607 q^{57} -0.163804 q^{59} +11.4800 q^{61} +1.51073 q^{63} -0.0937391 q^{65} -10.6984 q^{67} +16.3489 q^{69} -8.24538 q^{71} -12.8635 q^{73} +4.15901 q^{75} +1.49654 q^{77} -7.86983 q^{79} -10.6855 q^{81} -5.22811 q^{83} -9.86233 q^{85} -4.77826 q^{87} -2.70075 q^{89} +0.0352465 q^{91} +22.0357 q^{93} -2.67602 q^{95} +16.2733 q^{97} +5.02074 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 7 q - 3 q^{3} - 2 q^{5} - 6 q^{7}+O(q^{10}) $ 7 * q - 3 * q^3 - 2 * q^5 - 6 * q^7	$ 7 q - 3 q^{3} - 2 q^{5} - 6 q^{7} - 5 q^{11} - q^{13} - 6 q^{15} - 8 q^{17} - 15 q^{19} - 3 q^{21} - 5 q^{23} - 9 q^{25} - 9 q^{27} - 21 q^{31} - 7 q^{35} - q^{37} - 23 q^{39} - 10 q^{41} - 23 q^{43} - 4 q^{45} - 10 q^{47} - 13 q^{49} - 20 q^{51} - q^{53} - 23 q^{55} - 6 q^{57} - 4 q^{59} + 3 q^{61} - 4 q^{63} + 4 q^{65} - 28 q^{67} + 18 q^{69} - 18 q^{71} - 7 q^{73} + 11 q^{75} + 6 q^{77} - 30 q^{79} - 5 q^{81} + 13 q^{83} + q^{85} - 7 q^{87} - 18 q^{91} + 36 q^{93} - 2 q^{97} - 10 q^{99}+O(q^{100}) $ 7 * q - 3 * q^3 - 2 * q^5 - 6 * q^7 - 5 * q^11 - q^13 - 6 * q^15 - 8 * q^17 - 15 * q^19 - 3 * q^21 - 5 * q^23 - 9 * q^25 - 9 * q^27 - 21 * q^31 - 7 * q^35 - q^37 - 23 * q^39 - 10 * q^41 - 23 * q^43 - 4 * q^45 - 10 * q^47 - 13 * q^49 - 20 * q^51 - q^53 - 23 * q^55 - 6 * q^57 - 4 * q^59 + 3 * q^61 - 4 * q^63 + 4 * q^65 - 28 * q^67 + 18 * q^69 - 18 * q^71 - 7 * q^73 + 11 * q^75 + 6 * q^77 - 30 * q^79 - 5 * q^81 + 13 * q^83 + q^85 - 7 * q^87 - 18 * q^91 + 36 * q^93 - 2 * q^97 - 10 * 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
$3$	−2.29157	−1.32304	−0.661520	−	0.749928i	$-0.730088\pi$
$3$	−2.29157	−1.32304	−0.661520	+	0.749928i	$0.730088\pi$
$4$	0	0
$5$	−1.78468	−0.798133	−0.399067	−	0.916922i	$-0.630666\pi$
$5$	−1.78468	−0.798133	−0.399067	+	0.916922i	$0.630666\pi$
$6$	0	0
$7$	0.671050	0.253633	0.126817	−	0.991926i	$-0.459524\pi$
$7$	0.671050	0.253633	0.126817	+	0.991926i	$0.459524\pi$
$8$	0	0
$9$	2.25130	0.750433
$10$	0	0
$11$	2.23015	0.672416	0.336208	−	0.941788i	$-0.390855\pi$
$11$	2.23015	0.672416	0.336208	+	0.941788i	$0.390855\pi$
$12$	0	0
$13$	0.0525243	0.0145676	0.00728381	−	0.999973i	$-0.497681\pi$
$13$	0.0525243	0.0145676	0.00728381	+	0.999973i	$0.497681\pi$
$14$	0	0
$15$	4.08972	1.05596
$16$	0	0
$17$	5.52610	1.34028	0.670138	−	0.742236i	$-0.266235\pi$
$17$	5.52610	1.34028	0.670138	+	0.742236i	$0.266235\pi$
$18$	0	0
$19$	1.49944	0.343995	0.171997	−	0.985097i	$-0.444978\pi$
$19$	1.49944	0.343995	0.171997	+	0.985097i	$0.444978\pi$
$20$	0	0
$21$	−1.53776	−0.335567
$22$	0	0
$23$	−7.13437	−1.48762	−0.743810	−	0.668391i	$-0.766983\pi$
$23$	−7.13437	−1.48762	−0.743810	+	0.668391i	$0.766983\pi$
$24$	0	0
$25$	−1.81492	−0.362984
$26$	0	0
$27$	1.71570	0.330187
$28$	0	0
$29$	2.08515	0.387202	0.193601	−	0.981080i	$-0.437983\pi$
$29$	2.08515	0.387202	0.193601	+	0.981080i	$0.437983\pi$
$30$	0	0
$31$	−9.61596	−1.72708	−0.863539	−	0.504282i	$-0.831757\pi$
$31$	−9.61596	−1.72708	−0.863539	+	0.504282i	$0.831757\pi$
$32$	0	0
$33$	−5.11055	−0.889633
$34$	0	0
$35$	−1.19761	−0.202433
$36$	0	0
$37$	5.33025	0.876288	0.438144	−	0.898905i	$-0.355636\pi$
$37$	5.33025	0.876288	0.438144	+	0.898905i	$0.355636\pi$
$38$	0	0
$39$	−0.120363	−0.0192735
$40$	0	0
$41$	−11.8708	−1.85391	−0.926956	−	0.375170i	$-0.877584\pi$
$41$	−11.8708	−1.85391	−0.926956	+	0.375170i	$0.877584\pi$
$42$	0	0
$43$	−9.09066	−1.38631	−0.693157	−	0.720787i	$-0.743781\pi$
$43$	−9.09066	−1.38631	−0.693157	+	0.720787i	$0.743781\pi$
$44$	0	0
$45$	−4.01785	−0.598946
$46$	0	0
$47$	5.21388	0.760523	0.380261	−	0.924879i	$-0.375834\pi$
$47$	5.21388	0.760523	0.380261	+	0.924879i	$0.375834\pi$
$48$	0	0
$49$	−6.54969	−0.935670
$50$	0	0
$51$	−12.6635	−1.77324
$52$	0	0
$53$	4.27480	0.587189	0.293595	−	0.955930i	$-0.405148\pi$
$53$	4.27480	0.587189	0.293595	+	0.955930i	$0.405148\pi$
$54$	0	0
$55$	−3.98011	−0.536678
$56$	0	0
$57$	−3.43607	−0.455118
$58$	0	0
$59$	−0.163804	−0.0213254	−0.0106627	−	0.999943i	$-0.503394\pi$
$59$	−0.163804	−0.0213254	−0.0106627	+	0.999943i	$0.503394\pi$
$60$	0	0
$61$	11.4800	1.46987	0.734934	−	0.678138i	$-0.237213\pi$
$61$	11.4800	1.46987	0.734934	+	0.678138i	$0.237213\pi$
$62$	0	0
$63$	1.51073	0.190335
$64$	0	0
$65$	−0.0937391	−0.0116269
$66$	0	0
$67$	−10.6984	−1.30702	−0.653509	−	0.756918i	$-0.726704\pi$
$67$	−10.6984	−1.30702	−0.653509	+	0.756918i	$0.726704\pi$
$68$	0	0
$69$	16.3489	1.96818
$70$	0	0
$71$	−8.24538	−0.978546	−0.489273	−	0.872131i	$-0.662738\pi$
$71$	−8.24538	−0.978546	−0.489273	+	0.872131i	$0.662738\pi$
$72$	0	0
$73$	−12.8635	−1.50556	−0.752779	−	0.658273i	$-0.771287\pi$
$73$	−12.8635	−1.50556	−0.752779	+	0.658273i	$0.771287\pi$
$74$	0	0
$75$	4.15901	0.480241
$76$	0	0
$77$	1.49654	0.170547
$78$	0	0
$79$	−7.86983	−0.885425	−0.442712	−	0.896664i	$-0.645984\pi$
$79$	−7.86983	−0.885425	−0.442712	+	0.896664i	$0.645984\pi$
$80$	0	0
$81$	−10.6855	−1.18728
$82$	0	0
$83$	−5.22811	−0.573860	−0.286930	−	0.957952i	$-0.592635\pi$
$83$	−5.22811	−0.573860	−0.286930	+	0.957952i	$0.592635\pi$
$84$	0	0
$85$	−9.86233	−1.06972
$86$	0	0
$87$	−4.77826	−0.512283
$88$	0	0
$89$	−2.70075	−0.286279	−0.143140	−	0.989703i	$-0.545720\pi$
$89$	−2.70075	−0.286279	−0.143140	+	0.989703i	$0.545720\pi$
$90$	0	0
$91$	0.0352465	0.00369483
$92$	0	0
$93$	22.0357	2.28499
$94$	0	0
$95$	−2.67602	−0.274554
$96$	0	0
$97$	16.2733	1.65230	0.826150	−	0.563450i	$-0.190526\pi$
$97$	16.2733	1.65230	0.826150	+	0.563450i	$0.190526\pi$
$98$	0	0
$99$	5.02074	0.504604
$100$	0	0
$101$	7.31471	0.727841	0.363920	−	0.931430i	$-0.381438\pi$
$101$	7.31471	0.727841	0.363920	+	0.931430i	$0.381438\pi$
$102$	0	0
$103$	−8.43476	−0.831102	−0.415551	−	0.909570i	$-0.636411\pi$
$103$	−8.43476	−0.831102	−0.415551	+	0.909570i	$0.636411\pi$
$104$	0	0
$105$	2.74441	0.267827
$106$	0	0
$107$	−13.4235	−1.29770	−0.648851	−	0.760915i	$-0.724750\pi$
$107$	−13.4235	−1.29770	−0.648851	+	0.760915i	$0.724750\pi$
$108$	0	0
$109$	−10.8033	−1.03477	−0.517386	−	0.855752i	$-0.673095\pi$
$109$	−10.8033	−1.03477	−0.517386	+	0.855752i	$0.673095\pi$
$110$	0	0
$111$	−12.2147	−1.15936
$112$	0	0
$113$	6.62110	0.622861	0.311431	−	0.950269i	$-0.399192\pi$
$113$	6.62110	0.622861	0.311431	+	0.950269i	$0.399192\pi$
$114$	0	0
$115$	12.7326	1.18732
$116$	0	0
$117$	0.118248	0.0109320
$118$	0	0
$119$	3.70829	0.339939
$120$	0	0
$121$	−6.02642	−0.547856
$122$	0	0
$123$	27.2029	2.45280
$124$	0	0
$125$	12.1624	1.08784
$126$	0	0
$127$	−1.34857	−0.119666	−0.0598329	−	0.998208i	$-0.519057\pi$
$127$	−1.34857	−0.119666	−0.0598329	+	0.998208i	$0.519057\pi$
$128$	0	0
$129$	20.8319	1.83415
$130$	0	0
$131$	14.5213	1.26873	0.634365	−	0.773034i	$-0.281262\pi$
$131$	14.5213	1.26873	0.634365	+	0.773034i	$0.281262\pi$
$132$	0	0
$133$	1.00620	0.0872484
$134$	0	0
$135$	−3.06198	−0.263533
$136$	0	0
$137$	4.81237	0.411148	0.205574	−	0.978642i	$-0.434094\pi$
$137$	4.81237	0.411148	0.205574	+	0.978642i	$0.434094\pi$
$138$	0	0
$139$	−17.6736	−1.49905	−0.749527	−	0.661974i	$-0.769719\pi$
$139$	−17.6736	−1.49905	−0.749527	+	0.661974i	$0.769719\pi$
$140$	0	0
$141$	−11.9480	−1.00620
$142$	0	0
$143$	0.117137	0.00979551
$144$	0	0
$145$	−3.72132	−0.309039
$146$	0	0
$147$	15.0091	1.23793
$148$	0	0
$149$	3.49560	0.286371	0.143185	−	0.989696i	$-0.454265\pi$
$149$	3.49560	0.286371	0.143185	+	0.989696i	$0.454265\pi$
$150$	0	0
$151$	−3.05261	−0.248418	−0.124209	−	0.992256i	$-0.539639\pi$
$151$	−3.05261	−0.248418	−0.124209	+	0.992256i	$0.539639\pi$
$152$	0	0
$153$	12.4409	1.00579
$154$	0	0
$155$	17.1614	1.37844
$156$	0	0
$157$	−2.92795	−0.233676	−0.116838	−	0.993151i	$-0.537276\pi$
$157$	−2.92795	−0.233676	−0.116838	+	0.993151i	$0.537276\pi$
$158$	0	0
$159$	−9.79602	−0.776875
$160$	0	0
$161$	−4.78752	−0.377310
$162$	0	0
$163$	11.1100	0.870200	0.435100	−	0.900382i	$-0.356713\pi$
$163$	11.1100	0.870200	0.435100	+	0.900382i	$0.356713\pi$
$164$	0	0
$165$	9.12070	0.710046
$166$	0	0
$167$	6.79223	0.525599	0.262799	−	0.964851i	$-0.415354\pi$
$167$	6.79223	0.525599	0.262799	+	0.964851i	$0.415354\pi$
$168$	0	0
$169$	−12.9972	−0.999788
$170$	0	0
$171$	3.37568	0.258145
$172$	0	0
$173$	19.7703	1.50311	0.751553	−	0.659673i	$-0.229305\pi$
$173$	19.7703	1.50311	0.751553	+	0.659673i	$0.229305\pi$
$174$	0	0
$175$	−1.21790	−0.0920646
$176$	0	0
$177$	0.375368	0.0282144
$178$	0	0
$179$	−17.5575	−1.31231	−0.656154	−	0.754627i	$-0.727818\pi$
$179$	−17.5575	−1.31231	−0.656154	+	0.754627i	$0.727818\pi$
$180$	0	0
$181$	−3.21551	−0.239007	−0.119503	−	0.992834i	$-0.538130\pi$
$181$	−3.21551	−0.239007	−0.119503	+	0.992834i	$0.538130\pi$
$182$	0	0
$183$	−26.3073	−1.94469
$184$	0	0
$185$	−9.51280	−0.699395
$186$	0	0
$187$	12.3241	0.901224
$188$	0	0
$189$	1.15132	0.0837463
$190$	0	0
$191$	−12.1389	−0.878342	−0.439171	−	0.898404i	$-0.644728\pi$
$191$	−12.1389	−0.878342	−0.439171	+	0.898404i	$0.644728\pi$
$192$	0	0
$193$	11.5502	0.831399	0.415700	−	0.909502i	$-0.363537\pi$
$193$	11.5502	0.831399	0.415700	+	0.909502i	$0.363537\pi$
$194$	0	0
$195$	0.214810	0.0153829
$196$	0	0
$197$	−0.556439	−0.0396447	−0.0198223	−	0.999804i	$-0.506310\pi$
$197$	−0.556439	−0.0396447	−0.0198223	+	0.999804i	$0.506310\pi$
$198$	0	0
$199$	15.8849	1.12605	0.563025	−	0.826440i	$-0.309637\pi$
$199$	15.8849	1.12605	0.563025	+	0.826440i	$0.309637\pi$
$200$	0	0
$201$	24.5162	1.72924
$202$	0	0
$203$	1.39924	0.0982072
$204$	0	0
$205$	21.1856	1.47967
$206$	0	0
$207$	−16.0616	−1.11636
$208$	0	0
$209$	3.34398	0.231308
$210$	0	0
$211$	−23.8560	−1.64231	−0.821156	−	0.570703i	$-0.806671\pi$
$211$	−23.8560	−1.64231	−0.821156	+	0.570703i	$0.806671\pi$
$212$	0	0
$213$	18.8949	1.29466
$214$	0	0
$215$	16.2239	1.10646
$216$	0	0
$217$	−6.45279	−0.438044
$218$	0	0
$219$	29.4776	1.99191
$220$	0	0
$221$	0.290255	0.0195247
$222$	0	0
$223$	6.27008	0.419876	0.209938	−	0.977715i	$-0.432674\pi$
$223$	6.27008	0.419876	0.209938	+	0.977715i	$0.432674\pi$
$224$	0	0
$225$	−4.08592	−0.272395
$226$	0	0
$227$	5.69103	0.377727	0.188863	−	0.982003i	$-0.439520\pi$
$227$	5.69103	0.377727	0.188863	+	0.982003i	$0.439520\pi$
$228$	0	0
$229$	−21.9135	−1.44808	−0.724042	−	0.689756i	$-0.757718\pi$
$229$	−21.9135	−1.44808	−0.724042	+	0.689756i	$0.757718\pi$
$230$	0	0
$231$	−3.42944	−0.225640
$232$	0	0
$233$	11.6290	0.761841	0.380921	−	0.924608i	$-0.375607\pi$
$233$	11.6290	0.761841	0.380921	+	0.924608i	$0.375607\pi$
$234$	0	0
$235$	−9.30511	−0.606999
$236$	0	0
$237$	18.0343	1.17145
$238$	0	0
$239$	17.3971	1.12533	0.562664	−	0.826686i	$-0.309777\pi$
$239$	17.3971	1.12533	0.562664	+	0.826686i	$0.309777\pi$
$240$	0	0
$241$	−2.35929	−0.151975	−0.0759875	−	0.997109i	$-0.524211\pi$
$241$	−2.35929	−0.151975	−0.0759875	+	0.997109i	$0.524211\pi$
$242$	0	0
$243$	19.3396	1.24064
$244$	0	0
$245$	11.6891	0.746789
$246$	0	0
$247$	0.0787570	0.00501119
$248$	0	0
$249$	11.9806	0.759239
$250$	0	0
$251$	1.00000	0.0631194
$251$	1.00000	0.0631194
$252$	0	0
$253$	−15.9107	−1.00030
$254$	0	0
$255$	22.6002	1.41528
$256$	0	0
$257$	−18.2850	−1.14059	−0.570293	−	0.821442i	$-0.693170\pi$
$257$	−18.2850	−1.14059	−0.570293	+	0.821442i	$0.693170\pi$
$258$	0	0
$259$	3.57687	0.222256
$260$	0	0
$261$	4.69429	0.290569
$262$	0	0
$263$	−1.62067	−0.0999350	−0.0499675	−	0.998751i	$-0.515912\pi$
$263$	−1.62067	−0.0999350	−0.0499675	+	0.998751i	$0.515912\pi$
$264$	0	0
$265$	−7.62916	−0.468655
$266$	0	0
$267$	6.18896	0.378758
$268$	0	0
$269$	−26.9395	−1.64253	−0.821265	−	0.570547i	$-0.806731\pi$
$269$	−26.9395	−1.64253	−0.821265	+	0.570547i	$0.806731\pi$
$270$	0	0
$271$	1.70433	0.103531	0.0517653	−	0.998659i	$-0.483515\pi$
$271$	1.70433	0.103531	0.0517653	+	0.998659i	$0.483515\pi$
$272$	0	0
$273$	−0.0807698	−0.00488841
$274$	0	0
$275$	−4.04754	−0.244076
$276$	0	0
$277$	0.720069	0.0432648	0.0216324	−	0.999766i	$-0.493114\pi$
$277$	0.720069	0.0432648	0.0216324	+	0.999766i	$0.493114\pi$
$278$	0	0
$279$	−21.6484	−1.29606
$280$	0	0
$281$	−9.07112	−0.541138	−0.270569	−	0.962701i	$-0.587212\pi$
$281$	−9.07112	−0.541138	−0.270569	+	0.962701i	$0.587212\pi$
$282$	0	0
$283$	−10.0455	−0.597140	−0.298570	−	0.954388i	$-0.596510\pi$
$283$	−10.0455	−0.597140	−0.298570	+	0.954388i	$0.596510\pi$
$284$	0	0
$285$	6.13228	0.363245
$286$	0	0
$287$	−7.96592	−0.470213
$288$	0	0
$289$	13.5378	0.796342
$290$	0	0
$291$	−37.2914	−2.18606
$292$	0	0
$293$	−9.10277	−0.531790	−0.265895	−	0.964002i	$-0.585667\pi$
$293$	−9.10277	−0.531790	−0.265895	+	0.964002i	$0.585667\pi$
$294$	0	0
$295$	0.292337	0.0170205
$296$	0	0
$297$	3.82628	0.222023
$298$	0	0
$299$	−0.374728	−0.0216711
$300$	0	0
$301$	−6.10029	−0.351615
$302$	0	0
$303$	−16.7622	−0.962962
$304$	0	0
$305$	−20.4882	−1.17315
$306$	0	0
$307$	−9.11554	−0.520251	−0.260126	−	0.965575i	$-0.583764\pi$
$307$	−9.11554	−0.520251	−0.260126	+	0.965575i	$0.583764\pi$
$308$	0	0
$309$	19.3289	1.09958
$310$	0	0
$311$	29.5389	1.67500	0.837498	−	0.546440i	$-0.184018\pi$
$311$	29.5389	1.67500	0.837498	+	0.546440i	$0.184018\pi$
$312$	0	0
$313$	−1.02555	−0.0579676	−0.0289838	−	0.999580i	$-0.509227\pi$
$313$	−1.02555	−0.0579676	−0.0289838	+	0.999580i	$0.509227\pi$
$314$	0	0
$315$	−2.69618	−0.151912
$316$	0	0
$317$	−3.25854	−0.183018	−0.0915089	−	0.995804i	$-0.529169\pi$
$317$	−3.25854	−0.183018	−0.0915089	+	0.995804i	$0.529169\pi$
$318$	0	0
$319$	4.65019	0.260361
$320$	0	0
$321$	30.7610	1.71691
$322$	0	0
$323$	8.28605	0.461048
$324$	0	0
$325$	−0.0953273	−0.00528781
$326$	0	0
$327$	24.7566	1.36904
$328$	0	0
$329$	3.49878	0.192894
$330$	0	0
$331$	1.22129	0.0671281	0.0335640	−	0.999437i	$-0.489314\pi$
$331$	1.22129	0.0671281	0.0335640	+	0.999437i	$0.489314\pi$
$332$	0	0
$333$	12.0000	0.657596
$334$	0	0
$335$	19.0932	1.04318
$336$	0	0
$337$	28.7954	1.56859	0.784293	−	0.620391i	$-0.213026\pi$
$337$	28.7954	1.56859	0.784293	+	0.620391i	$0.213026\pi$
$338$	0	0
$339$	−15.1727	−0.824070
$340$	0	0
$341$	−21.4451	−1.16132
$342$	0	0
$343$	−9.09252	−0.490950
$344$	0	0
$345$	−29.1776	−1.57087
$346$	0	0
$347$	−13.0949	−0.702970	−0.351485	−	0.936194i	$-0.614323\pi$
$347$	−13.0949	−0.702970	−0.351485	+	0.936194i	$0.614323\pi$
$348$	0	0
$349$	−14.8887	−0.796974	−0.398487	−	0.917174i	$-0.630465\pi$
$349$	−14.8887	−0.796974	−0.398487	+	0.917174i	$0.630465\pi$
$350$	0	0
$351$	0.0901160	0.00481004
$352$	0	0
$353$	−29.2068	−1.55452	−0.777260	−	0.629179i	$-0.783391\pi$
$353$	−29.2068	−1.55452	−0.777260	+	0.629179i	$0.783391\pi$
$354$	0	0
$355$	14.7154	0.781010
$356$	0	0
$357$	−8.49782	−0.449752
$358$	0	0
$359$	27.4180	1.44707	0.723534	−	0.690288i	$-0.242516\pi$
$359$	27.4180	1.44707	0.723534	+	0.690288i	$0.242516\pi$
$360$	0	0
$361$	−16.7517	−0.881668
$362$	0	0
$363$	13.8100	0.724835
$364$	0	0
$365$	22.9572	1.20164
$366$	0	0
$367$	−31.9200	−1.66621	−0.833105	−	0.553115i	$-0.813439\pi$
$367$	−31.9200	−1.66621	−0.833105	+	0.553115i	$0.813439\pi$
$368$	0	0
$369$	−26.7248	−1.39124
$370$	0	0
$371$	2.86861	0.148931
$372$	0	0
$373$	16.2376	0.840753	0.420376	−	0.907350i	$-0.361898\pi$
$373$	16.2376	0.840753	0.420376	+	0.907350i	$0.361898\pi$
$374$	0	0
$375$	−27.8711	−1.43926
$376$	0	0
$377$	0.109521	0.00564061
$378$	0	0
$379$	21.8277	1.12121	0.560607	−	0.828082i	$-0.310568\pi$
$379$	21.8277	1.12121	0.560607	+	0.828082i	$0.310568\pi$
$380$	0	0
$381$	3.09033	0.158323
$382$	0	0
$383$	5.38270	0.275043	0.137521	−	0.990499i	$-0.456086\pi$
$383$	5.38270	0.275043	0.137521	+	0.990499i	$0.456086\pi$
$384$	0	0
$385$	−2.67085	−0.136119
$386$	0	0
$387$	−20.4658	−1.04034
$388$	0	0
$389$	−30.3894	−1.54080	−0.770401	−	0.637560i	$-0.779944\pi$
$389$	−30.3894	−1.54080	−0.770401	+	0.637560i	$0.779944\pi$
$390$	0	0
$391$	−39.4253	−1.99382
$392$	0	0
$393$	−33.2766	−1.67858
$394$	0	0
$395$	14.0451	0.706687
$396$	0	0
$397$	13.3900	0.672023	0.336011	−	0.941858i	$-0.390922\pi$
$397$	13.3900	0.672023	0.336011	+	0.941858i	$0.390922\pi$
$398$	0	0
$399$	−2.30577	−0.115433
$400$	0	0
$401$	−34.2511	−1.71042	−0.855209	−	0.518284i	$-0.826571\pi$
$401$	−34.2511	−1.71042	−0.855209	+	0.518284i	$0.826571\pi$
$402$	0	0
$403$	−0.505072	−0.0251594
$404$	0	0
$405$	19.0703	0.947610
$406$	0	0
$407$	11.8873	0.589231
$408$	0	0
$409$	13.3152	0.658394	0.329197	−	0.944261i	$-0.393222\pi$
$409$	13.3152	0.658394	0.329197	+	0.944261i	$0.393222\pi$
$410$	0	0
$411$	−11.0279	−0.543965
$412$	0	0
$413$	−0.109920	−0.00540883
$414$	0	0
$415$	9.33050	0.458016
$416$	0	0
$417$	40.5003	1.98331
$418$	0	0
$419$	3.62652	0.177167	0.0885836	−	0.996069i	$-0.471766\pi$
$419$	3.62652	0.177167	0.0885836	+	0.996069i	$0.471766\pi$
$420$	0	0
$421$	−9.05916	−0.441516	−0.220758	−	0.975329i	$-0.570853\pi$
$421$	−9.05916	−0.441516	−0.220758	+	0.975329i	$0.570853\pi$
$422$	0	0
$423$	11.7380	0.570722
$424$	0	0
$425$	−10.0294	−0.486498
$426$	0	0
$427$	7.70368	0.372807
$428$	0	0
$429$	−0.268428	−0.0129598
$430$	0	0
$431$	1.80929	0.0871506	0.0435753	−	0.999050i	$-0.486125\pi$
$431$	1.80929	0.0871506	0.0435753	+	0.999050i	$0.486125\pi$
$432$	0	0
$433$	6.34504	0.304923	0.152462	−	0.988309i	$-0.451280\pi$
$433$	6.34504	0.304923	0.152462	+	0.988309i	$0.451280\pi$
$434$	0	0
$435$	8.52766	0.408870
$436$	0	0
$437$	−10.6975	−0.511733
$438$	0	0
$439$	30.1384	1.43843	0.719214	−	0.694788i	$-0.244502\pi$
$439$	30.1384	1.43843	0.719214	+	0.694788i	$0.244502\pi$
$440$	0	0
$441$	−14.7453	−0.702158
$442$	0	0
$443$	−11.1274	−0.528679	−0.264339	−	0.964430i	$-0.585154\pi$
$443$	−11.1274	−0.528679	−0.264339	+	0.964430i	$0.585154\pi$
$444$	0	0
$445$	4.81998	0.228489
$446$	0	0
$447$	−8.01042	−0.378880
$448$	0	0
$449$	35.6616	1.68297	0.841487	−	0.540277i	$-0.181680\pi$
$449$	35.6616	1.68297	0.841487	+	0.540277i	$0.181680\pi$
$450$	0	0
$451$	−26.4738	−1.24660
$452$	0	0
$453$	6.99527	0.328666
$454$	0	0
$455$	−0.0629036	−0.00294897
$456$	0	0
$457$	−31.2226	−1.46053	−0.730266	−	0.683163i	$-0.760604\pi$
$457$	−31.2226	−1.46053	−0.730266	+	0.683163i	$0.760604\pi$
$458$	0	0
$459$	9.48114	0.442542
$460$	0	0
$461$	10.3296	0.481097	0.240549	−	0.970637i	$-0.422673\pi$
$461$	10.3296	0.481097	0.240549	+	0.970637i	$0.422673\pi$
$462$	0	0
$463$	−24.1578	−1.12271	−0.561355	−	0.827575i	$-0.689720\pi$
$463$	−24.1578	−1.12271	−0.561355	+	0.827575i	$0.689720\pi$
$464$	0	0
$465$	−39.3266	−1.82373
$466$	0	0
$467$	35.6761	1.65089	0.825447	−	0.564479i	$-0.190923\pi$
$467$	35.6761	1.65089	0.825447	+	0.564479i	$0.190923\pi$
$468$	0	0
$469$	−7.17917	−0.331503
$470$	0	0
$471$	6.70960	0.309162
$472$	0	0
$473$	−20.2736	−0.932180
$474$	0	0
$475$	−2.72136	−0.124864
$476$	0	0
$477$	9.62386	0.440646
$478$	0	0
$479$	−4.37843	−0.200055	−0.100028	−	0.994985i	$-0.531893\pi$
$479$	−4.37843	−0.200055	−0.100028	+	0.994985i	$0.531893\pi$
$480$	0	0
$481$	0.279968	0.0127654
$482$	0	0
$483$	10.9709	0.499195
$484$	0	0
$485$	−29.0426	−1.31876
$486$	0	0
$487$	−32.4930	−1.47240	−0.736198	−	0.676766i	$-0.763381\pi$
$487$	−32.4930	−1.47240	−0.736198	+	0.676766i	$0.763381\pi$
$488$	0	0
$489$	−25.4593	−1.15131
$490$	0	0
$491$	12.9926	0.586348	0.293174	−	0.956059i	$-0.405289\pi$
$491$	12.9926	0.586348	0.293174	+	0.956059i	$0.405289\pi$
$492$	0	0
$493$	11.5227	0.518958
$494$	0	0
$495$	−8.96042	−0.402741
$496$	0	0
$497$	−5.53306	−0.248192
$498$	0	0
$499$	2.12523	0.0951383	0.0475691	−	0.998868i	$-0.484853\pi$
$499$	2.12523	0.0951383	0.0475691	+	0.998868i	$0.484853\pi$
$500$	0	0
$501$	−15.5649	−0.695388
$502$	0	0
$503$	28.1675	1.25593	0.627964	−	0.778243i	$-0.283889\pi$
$503$	28.1675	1.25593	0.627964	+	0.778243i	$0.283889\pi$
$504$	0	0
$505$	−13.0544	−0.580914
$506$	0	0
$507$	29.7841	1.32276
$508$	0	0
$509$	−3.65358	−0.161942	−0.0809710	−	0.996716i	$-0.525802\pi$
$509$	−3.65358	−0.161942	−0.0809710	+	0.996716i	$0.525802\pi$
$510$	0	0
$511$	−8.63205	−0.381859
$512$	0	0
$513$	2.57259	0.113582
$514$	0	0
$515$	15.0533	0.663330
$516$	0	0
$517$	11.6278	0.511388
$518$	0	0
$519$	−45.3050	−1.98867
$520$	0	0
$521$	−30.7443	−1.34693	−0.673465	−	0.739219i	$-0.735195\pi$
$521$	−30.7443	−1.34693	−0.673465	+	0.739219i	$0.735195\pi$
$522$	0	0
$523$	25.0146	1.09381	0.546906	−	0.837194i	$-0.315806\pi$
$523$	25.0146	1.09381	0.546906	+	0.837194i	$0.315806\pi$
$524$	0	0
$525$	2.79091	0.121805
$526$	0	0
$527$	−53.1388	−2.31476
$528$	0	0
$529$	27.8993	1.21301
$530$	0	0
$531$	−0.368771	−0.0160033
$532$	0	0
$533$	−0.623507	−0.0270071
$534$	0	0
$535$	23.9567	1.03574
$536$	0	0
$537$	40.2342	1.73623
$538$	0	0
$539$	−14.6068	−0.629160
$540$	0	0
$541$	19.3510	0.831964	0.415982	−	0.909373i	$-0.363438\pi$
$541$	19.3510	0.831964	0.415982	+	0.909373i	$0.363438\pi$
$542$	0	0
$543$	7.36856	0.316215
$544$	0	0
$545$	19.2805	0.825886
$546$	0	0
$547$	−2.14537	−0.0917294	−0.0458647	−	0.998948i	$-0.514604\pi$
$547$	−2.14537	−0.0917294	−0.0458647	+	0.998948i	$0.514604\pi$
$548$	0	0
$549$	25.8450	1.10304
$550$	0	0
$551$	3.12655	0.133195
$552$	0	0
$553$	−5.28105	−0.224573
$554$	0	0
$555$	21.7993	0.925327
$556$	0	0
$557$	42.7554	1.81160	0.905802	−	0.423702i	$-0.139270\pi$
$557$	42.7554	1.81160	0.905802	+	0.423702i	$0.139270\pi$
$558$	0	0
$559$	−0.477481	−0.0201953
$560$	0	0
$561$	−28.2415	−1.19236
$562$	0	0
$563$	17.4443	0.735191	0.367596	−	0.929986i	$-0.380181\pi$
$563$	17.4443	0.735191	0.367596	+	0.929986i	$0.380181\pi$
$564$	0	0
$565$	−11.8166	−0.497126
$566$	0	0
$567$	−7.17054	−0.301134
$568$	0	0
$569$	−36.4900	−1.52974	−0.764871	−	0.644184i	$-0.777197\pi$
$569$	−36.4900	−1.52974	−0.764871	+	0.644184i	$0.777197\pi$
$570$	0	0
$571$	−41.9707	−1.75642	−0.878210	−	0.478275i	$-0.841262\pi$
$571$	−41.9707	−1.75642	−0.878210	+	0.478275i	$0.841262\pi$
$572$	0	0
$573$	27.8172	1.16208
$574$	0	0
$575$	12.9483	0.539981
$576$	0	0
$577$	12.3140	0.512639	0.256320	−	0.966592i	$-0.417490\pi$
$577$	12.3140	0.512639	0.256320	+	0.966592i	$0.417490\pi$
$578$	0	0
$579$	−26.4680	−1.09997
$580$	0	0
$581$	−3.50832	−0.145550
$582$	0	0
$583$	9.53347	0.394836
$584$	0	0
$585$	−0.211035	−0.00872522
$586$	0	0
$587$	13.8231	0.570540	0.285270	−	0.958447i	$-0.407917\pi$
$587$	13.8231	0.570540	0.285270	+	0.958447i	$0.407917\pi$
$588$	0	0
$589$	−14.4185	−0.594105
$590$	0	0
$591$	1.27512	0.0524515
$592$	0	0
$593$	15.5625	0.639075	0.319538	−	0.947574i	$-0.396472\pi$
$593$	15.5625	0.639075	0.319538	+	0.947574i	$0.396472\pi$
$594$	0	0
$595$	−6.61811	−0.271316
$596$	0	0
$597$	−36.4014	−1.48981
$598$	0	0
$599$	6.86070	0.280321	0.140160	−	0.990129i	$-0.455238\pi$
$599$	6.86070	0.280321	0.140160	+	0.990129i	$0.455238\pi$
$600$	0	0
$601$	3.73347	0.152292	0.0761458	−	0.997097i	$-0.475739\pi$
$601$	3.73347	0.152292	0.0761458	+	0.997097i	$0.475739\pi$
$602$	0	0
$603$	−24.0853	−0.980830
$604$	0	0
$605$	10.7552	0.437262
$606$	0	0
$607$	40.4537	1.64197	0.820983	−	0.570953i	$-0.193426\pi$
$607$	40.4537	1.64197	0.820983	+	0.570953i	$0.193426\pi$
$608$	0	0
$609$	−3.20645	−0.129932
$610$	0	0
$611$	0.273856	0.0110790
$612$	0	0
$613$	−18.2453	−0.736921	−0.368460	−	0.929643i	$-0.620115\pi$
$613$	−18.2453	−0.736921	−0.368460	+	0.929643i	$0.620115\pi$
$614$	0	0
$615$	−48.5484	−1.95766
$616$	0	0
$617$	20.5886	0.828864	0.414432	−	0.910080i	$-0.363980\pi$
$617$	20.5886	0.828864	0.414432	+	0.910080i	$0.363980\pi$
$618$	0	0
$619$	−11.8400	−0.475890	−0.237945	−	0.971279i	$-0.576474\pi$
$619$	−11.8400	−0.475890	−0.237945	+	0.971279i	$0.576474\pi$
$620$	0	0
$621$	−12.2404	−0.491192
$622$	0	0
$623$	−1.81234	−0.0726098
$624$	0	0
$625$	−12.6315	−0.505259
$626$	0	0
$627$	−7.66296	−0.306029
$628$	0	0
$629$	29.4555	1.17447
$630$	0	0
$631$	−24.3224	−0.968258	−0.484129	−	0.874997i	$-0.660863\pi$
$631$	−24.3224	−0.968258	−0.484129	+	0.874997i	$0.660863\pi$
$632$	0	0
$633$	54.6677	2.17284
$634$	0	0
$635$	2.40676	0.0955093
$636$	0	0
$637$	−0.344018	−0.0136305
$638$	0	0
$639$	−18.5628	−0.734333
$640$	0	0
$641$	14.8821	0.587808	0.293904	−	0.955835i	$-0.405045\pi$
$641$	14.8821	0.587808	0.293904	+	0.955835i	$0.405045\pi$
$642$	0	0
$643$	−9.38851	−0.370247	−0.185123	−	0.982715i	$-0.559268\pi$
$643$	−9.38851	−0.370247	−0.185123	+	0.982715i	$0.559268\pi$
$644$	0	0
$645$	−37.1783	−1.46389
$646$	0	0
$647$	−24.9638	−0.981430	−0.490715	−	0.871320i	$-0.663264\pi$
$647$	−24.9638	−0.981430	−0.490715	+	0.871320i	$0.663264\pi$
$648$	0	0
$649$	−0.365307	−0.0143396
$650$	0	0
$651$	14.7870	0.579550
$652$	0	0
$653$	34.0819	1.33373	0.666865	−	0.745179i	$-0.267636\pi$
$653$	34.0819	1.33373	0.666865	+	0.745179i	$0.267636\pi$
$654$	0	0
$655$	−25.9158	−1.01262
$656$	0	0
$657$	−28.9596	−1.12982
$658$	0	0
$659$	19.7009	0.767440	0.383720	−	0.923450i	$-0.374643\pi$
$659$	19.7009	0.767440	0.383720	+	0.923450i	$0.374643\pi$
$660$	0	0
$661$	17.9804	0.699358	0.349679	−	0.936870i	$-0.386291\pi$
$661$	17.9804	0.699358	0.349679	+	0.936870i	$0.386291\pi$
$662$	0	0
$663$	−0.665140	−0.0258319
$664$	0	0
$665$	−1.79574	−0.0696359
$666$	0	0
$667$	−14.8762	−0.576009
$668$	0	0
$669$	−14.3683	−0.555512
$670$	0	0
$671$	25.6022	0.988364
$672$	0	0
$673$	−10.6371	−0.410029	−0.205015	−	0.978759i	$-0.565724\pi$
$673$	−10.6371	−0.410029	−0.205015	+	0.978759i	$0.565724\pi$
$674$	0	0
$675$	−3.11386	−0.119852
$676$	0	0
$677$	−11.0551	−0.424883	−0.212441	−	0.977174i	$-0.568141\pi$
$677$	−11.0551	−0.424883	−0.212441	+	0.977174i	$0.568141\pi$
$678$	0	0
$679$	10.9202	0.419078
$680$	0	0
$681$	−13.0414	−0.499748
$682$	0	0
$683$	−26.2786	−1.00552	−0.502762	−	0.864425i	$-0.667683\pi$
$683$	−26.2786	−1.00552	−0.502762	+	0.864425i	$0.667683\pi$
$684$	0	0
$685$	−8.58853	−0.328151
$686$	0	0
$687$	50.2163	1.91587
$688$	0	0
$689$	0.224531	0.00855396
$690$	0	0
$691$	−29.0629	−1.10561	−0.552803	−	0.833312i	$-0.686442\pi$
$691$	−29.0629	−1.10561	−0.552803	+	0.833312i	$0.686442\pi$
$692$	0	0
$693$	3.36917	0.127984
$694$	0	0
$695$	31.5417	1.19644
$696$	0	0
$697$	−65.5994	−2.48476
$698$	0	0
$699$	−26.6487	−1.00795
$700$	0	0
$701$	−29.7473	−1.12354	−0.561771	−	0.827293i	$-0.689880\pi$
$701$	−29.7473	−1.12354	−0.561771	+	0.827293i	$0.689880\pi$
$702$	0	0
$703$	7.99238	0.301438
$704$	0	0
$705$	21.3233	0.803083
$706$	0	0
$707$	4.90854	0.184605
$708$	0	0
$709$	30.8047	1.15689	0.578447	−	0.815720i	$-0.303659\pi$
$709$	30.8047	1.15689	0.578447	+	0.815720i	$0.303659\pi$
$710$	0	0
$711$	−17.7173	−0.664452
$712$	0	0
$713$	68.6038	2.56923
$714$	0	0
$715$	−0.209053	−0.00781812
$716$	0	0
$717$	−39.8668	−1.48885
$718$	0	0
$719$	29.2904	1.09235	0.546175	−	0.837671i	$-0.316083\pi$
$719$	29.2904	1.09235	0.546175	+	0.837671i	$0.316083\pi$
$720$	0	0
$721$	−5.66015	−0.210795
$722$	0	0
$723$	5.40647	0.201069
$724$	0	0
$725$	−3.78437	−0.140548
$726$	0	0
$727$	−37.9357	−1.40696	−0.703479	−	0.710716i	$-0.748371\pi$
$727$	−37.9357	−1.40696	−0.703479	+	0.710716i	$0.748371\pi$
$728$	0	0
$729$	−12.2614	−0.454127
$730$	0	0
$731$	−50.2359	−1.85804
$732$	0	0
$733$	15.8326	0.584792	0.292396	−	0.956297i	$-0.405547\pi$
$733$	15.8326	0.584792	0.292396	+	0.956297i	$0.405547\pi$
$734$	0	0
$735$	−26.7864	−0.988032
$736$	0	0
$737$	−23.8591	−0.878861
$738$	0	0
$739$	9.06045	0.333294	0.166647	−	0.986017i	$-0.446706\pi$
$739$	9.06045	0.333294	0.166647	+	0.986017i	$0.446706\pi$
$740$	0	0
$741$	−0.180477	−0.00663000
$742$	0	0
$743$	−8.42135	−0.308949	−0.154475	−	0.987997i	$-0.549369\pi$
$743$	−8.42135	−0.308949	−0.154475	+	0.987997i	$0.549369\pi$
$744$	0	0
$745$	−6.23853	−0.228562
$746$	0	0
$747$	−11.7700	−0.430643
$748$	0	0
$749$	−9.00787	−0.329140
$750$	0	0
$751$	−19.2016	−0.700677	−0.350338	−	0.936623i	$-0.613933\pi$
$751$	−19.2016	−0.700677	−0.350338	+	0.936623i	$0.613933\pi$
$752$	0	0
$753$	−2.29157	−0.0835095
$754$	0	0
$755$	5.44793	0.198270
$756$	0	0
$757$	−9.32193	−0.338811	−0.169406	−	0.985546i	$-0.554185\pi$
$757$	−9.32193	−0.338811	−0.169406	+	0.985546i	$0.554185\pi$
$758$	0	0
$759$	36.4606	1.32344
$760$	0	0
$761$	51.3651	1.86198	0.930992	−	0.365041i	$-0.118945\pi$
$761$	51.3651	1.86198	0.930992	+	0.365041i	$0.118945\pi$
$762$	0	0
$763$	−7.24958	−0.262452
$764$	0	0
$765$	−22.2030	−0.802753
$766$	0	0
$767$	−0.00860367	−0.000310661 0
$768$	0	0
$769$	43.2738	1.56049	0.780246	−	0.625473i	$-0.215094\pi$
$769$	43.2738	1.56049	0.780246	+	0.625473i	$0.215094\pi$
$770$	0	0
$771$	41.9013	1.50904
$772$	0	0
$773$	−6.59292	−0.237131	−0.118565	−	0.992946i	$-0.537830\pi$
$773$	−6.59292	−0.237131	−0.118565	+	0.992946i	$0.537830\pi$
$774$	0	0
$775$	17.4522	0.626901
$776$	0	0
$777$	−8.19665	−0.294053
$778$	0	0
$779$	−17.7996	−0.637736
$780$	0	0
$781$	−18.3885	−0.657991
$782$	0	0
$783$	3.57749	0.127849
$784$	0	0
$785$	5.22545	0.186504
$786$	0	0
$787$	−51.1075	−1.82179	−0.910893	−	0.412642i	$-0.864606\pi$
$787$	−51.1075	−1.82179	−0.910893	+	0.412642i	$0.864606\pi$
$788$	0	0
$789$	3.71389	0.132218
$790$	0	0
$791$	4.44309	0.157978
$792$	0	0
$793$	0.602981	0.0214125
$794$	0	0
$795$	17.4828	0.620049
$796$	0	0
$797$	37.6794	1.33467	0.667337	−	0.744756i	$-0.267434\pi$
$797$	37.6794	1.33467	0.667337	+	0.744756i	$0.267434\pi$
$798$	0	0
$799$	28.8125	1.01931
$800$	0	0
$801$	−6.08020	−0.214833
$802$	0	0
$803$	−28.6876	−1.01236
$804$	0	0
$805$	8.54419	0.301143
$806$	0	0
$807$	61.7338	2.17313
$808$	0	0
$809$	−38.0342	−1.33721	−0.668605	−	0.743618i	$-0.733108\pi$
$809$	−38.0342	−1.33721	−0.668605	+	0.743618i	$0.733108\pi$
$810$	0	0
$811$	13.6164	0.478138	0.239069	−	0.971003i	$-0.423158\pi$
$811$	13.6164	0.478138	0.239069	+	0.971003i	$0.423158\pi$
$812$	0	0
$813$	−3.90559	−0.136975
$814$	0	0
$815$	−19.8277	−0.694535
$816$	0	0
$817$	−13.6309	−0.476884
$818$	0	0
$819$	0.0793503	0.00277272
$820$	0	0
$821$	30.5818	1.06731	0.533656	−	0.845702i	$-0.320818\pi$
$821$	30.5818	1.06731	0.533656	+	0.845702i	$0.320818\pi$
$822$	0	0
$823$	−18.4303	−0.642439	−0.321220	−	0.947005i	$-0.604093\pi$
$823$	−18.4303	−0.642439	−0.321220	+	0.947005i	$0.604093\pi$
$824$	0	0
$825$	9.27524	0.322922
$826$	0	0
$827$	−2.06070	−0.0716577	−0.0358288	−	0.999358i	$-0.511407\pi$
$827$	−2.06070	−0.0716577	−0.0358288	+	0.999358i	$0.511407\pi$
$828$	0	0
$829$	53.5915	1.86131	0.930655	−	0.365897i	$-0.119238\pi$
$829$	53.5915	1.86131	0.930655	+	0.365897i	$0.119238\pi$
$830$	0	0
$831$	−1.65009	−0.0572410
$832$	0	0
$833$	−36.1943	−1.25406
$834$	0	0
$835$	−12.1220	−0.419498
$836$	0	0
$837$	−16.4981	−0.570258
$838$	0	0
$839$	−9.37280	−0.323585	−0.161792	−	0.986825i	$-0.551728\pi$
$839$	−9.37280	−0.323585	−0.161792	+	0.986825i	$0.551728\pi$
$840$	0	0
$841$	−24.6522	−0.850075
$842$	0	0
$843$	20.7871	0.715946
$844$	0	0
$845$	23.1959	0.797964
$846$	0	0
$847$	−4.04403	−0.138954
$848$	0	0
$849$	23.0199	0.790040
$850$	0	0
$851$	−38.0280	−1.30358
$852$	0	0
$853$	−8.69000	−0.297540	−0.148770	−	0.988872i	$-0.547531\pi$
$853$	−8.69000	−0.297540	−0.148770	+	0.988872i	$0.547531\pi$
$854$	0	0
$855$	−6.02451	−0.206034
$856$	0	0
$857$	−17.4197	−0.595045	−0.297522	−	0.954715i	$-0.596160\pi$
$857$	−17.4197	−0.595045	−0.297522	+	0.954715i	$0.596160\pi$
$858$	0	0
$859$	38.0839	1.29941	0.649703	−	0.760188i	$-0.274893\pi$
$859$	38.0839	1.29941	0.649703	+	0.760188i	$0.274893\pi$
$860$	0	0
$861$	18.2545	0.622111
$862$	0	0
$863$	1.47779	0.0503046	0.0251523	−	0.999684i	$-0.491993\pi$
$863$	1.47779	0.0503046	0.0251523	+	0.999684i	$0.491993\pi$
$864$	0	0
$865$	−35.2836	−1.19968
$866$	0	0
$867$	−31.0229	−1.05359
$868$	0	0
$869$	−17.5509	−0.595374
$870$	0	0
$871$	−0.561927	−0.0190402
$872$	0	0
$873$	36.6360	1.23994
$874$	0	0
$875$	8.16161	0.275913
$876$	0	0
$877$	16.1183	0.544278	0.272139	−	0.962258i	$-0.412269\pi$
$877$	16.1183	0.544278	0.272139	+	0.962258i	$0.412269\pi$
$878$	0	0
$879$	20.8597	0.703579
$880$	0	0
$881$	−25.7458	−0.867399	−0.433699	−	0.901058i	$-0.642792\pi$
$881$	−25.7458	−0.867399	−0.433699	+	0.901058i	$0.642792\pi$
$882$	0	0
$883$	51.3112	1.72676	0.863380	−	0.504554i	$-0.168343\pi$
$883$	51.3112	1.72676	0.863380	+	0.504554i	$0.168343\pi$
$884$	0	0
$885$	−0.669911	−0.0225188
$886$	0	0
$887$	26.4272	0.887339	0.443670	−	0.896190i	$-0.353676\pi$
$887$	26.4272	0.887339	0.443670	+	0.896190i	$0.353676\pi$
$888$	0	0
$889$	−0.904955	−0.0303512
$890$	0	0
$891$	−23.8304	−0.798349
$892$	0	0
$893$	7.81789	0.261616
$894$	0	0
$895$	31.3345	1.04740
$896$	0	0
$897$	0.858716	0.0286717
$898$	0	0
$899$	−20.0507	−0.668727
$900$	0	0
$901$	23.6230	0.786996
$902$	0	0
$903$	13.9793	0.465200
$904$	0	0
$905$	5.73865	0.190759
$906$	0	0
$907$	41.6133	1.38175	0.690873	−	0.722976i	$-0.257226\pi$
$907$	41.6133	1.38175	0.690873	+	0.722976i	$0.257226\pi$
$908$	0	0
$909$	16.4676	0.546196
$910$	0	0
$911$	48.6569	1.61208	0.806038	−	0.591864i	$-0.201608\pi$
$911$	48.6569	1.61208	0.806038	+	0.591864i	$0.201608\pi$
$912$	0	0
$913$	−11.6595	−0.385873
$914$	0	0
$915$	46.9502	1.55212
$916$	0	0
$917$	9.74451	0.321792
$918$	0	0
$919$	32.6597	1.07734	0.538672	−	0.842515i	$-0.318926\pi$
$919$	32.6597	1.07734	0.538672	+	0.842515i	$0.318926\pi$
$920$	0	0
$921$	20.8889	0.688313
$922$	0	0
$923$	−0.433083	−0.0142551
$924$	0	0
$925$	−9.67397	−0.318078
$926$	0	0
$927$	−18.9892	−0.623686
$928$	0	0
$929$	−29.0898	−0.954406	−0.477203	−	0.878793i	$-0.658349\pi$
$929$	−29.0898	−0.954406	−0.477203	+	0.878793i	$0.658349\pi$
$930$	0	0
$931$	−9.82086	−0.321866
$932$	0	0
$933$	−67.6904	−2.21609
$934$	0	0
$935$	−21.9945	−0.719297
$936$	0	0
$937$	39.3014	1.28392	0.641960	−	0.766738i	$-0.278122\pi$
$937$	39.3014	1.28392	0.641960	+	0.766738i	$0.278122\pi$
$938$	0	0
$939$	2.35013	0.0766934
$940$	0	0
$941$	17.7356	0.578163	0.289082	−	0.957304i	$-0.406650\pi$
$941$	17.7356	0.578163	0.289082	+	0.957304i	$0.406650\pi$
$942$	0	0
$943$	84.6909	2.75792
$944$	0	0
$945$	−2.05474	−0.0668407
$946$	0	0
$947$	41.3091	1.34237	0.671183	−	0.741292i	$-0.265787\pi$
$947$	41.3091	1.34237	0.671183	+	0.741292i	$0.265787\pi$
$948$	0	0
$949$	−0.675646	−0.0219324
$950$	0	0
$951$	7.46717	0.242140
$952$	0	0
$953$	19.6289	0.635842	0.317921	−	0.948117i	$-0.397015\pi$
$953$	19.6289	0.635842	0.317921	+	0.948117i	$0.397015\pi$
$954$	0	0
$955$	21.6641	0.701033
$956$	0	0
$957$	−10.6562	−0.344468
$958$	0	0
$959$	3.22934	0.104281
$960$	0	0
$961$	61.4667	1.98280
$962$	0	0
$963$	−30.2204	−0.973839
$964$	0	0
$965$	−20.6134	−0.663567
$966$	0	0
$967$	44.3193	1.42521	0.712606	−	0.701565i	$-0.247515\pi$
$967$	44.3193	1.42521	0.712606	+	0.701565i	$0.247515\pi$
$968$	0	0
$969$	−18.9881	−0.609985
$970$	0	0
$971$	−4.22052	−0.135443	−0.0677214	−	0.997704i	$-0.521573\pi$
$971$	−4.22052	−0.135443	−0.0677214	+	0.997704i	$0.521573\pi$
$972$	0	0
$973$	−11.8599	−0.380210
$974$	0	0
$975$	0.218449	0.00699598
$976$	0	0
$977$	−29.3344	−0.938492	−0.469246	−	0.883068i	$-0.655474\pi$
$977$	−29.3344	−0.938492	−0.469246	+	0.883068i	$0.655474\pi$
$978$	0	0
$979$	−6.02309	−0.192499
$980$	0	0
$981$	−24.3215	−0.776527
$982$	0	0
$983$	55.0110	1.75458	0.877290	−	0.479961i	$-0.159349\pi$
$983$	55.0110	1.75458	0.877290	+	0.479961i	$0.159349\pi$
$984$	0	0
$985$	0.993066	0.0316417
$986$	0	0
$987$	−8.01770	−0.255206
$988$	0	0
$989$	64.8562	2.06231
$990$	0	0
$991$	−8.39630	−0.266717	−0.133359	−	0.991068i	$-0.542576\pi$
$991$	−8.39630	−0.266717	−0.133359	+	0.991068i	$0.542576\pi$
$992$	0	0
$993$	−2.79867	−0.0888131
$994$	0	0
$995$	−28.3495	−0.898738
$996$	0	0
$997$	15.7423	0.498564	0.249282	−	0.968431i	$-0.419805\pi$
$997$	15.7423	0.498564	0.249282	+	0.968431i	$0.419805\pi$
$998$	0	0
$999$	9.14512	0.289339

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	1004.2.a.a.1.2	✓	7
3.2	odd	2		9036.2.a.i.1.6		7
4.3	odd	2		4016.2.a.g.1.6		7

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
1004.2.a.a.1.2	✓	7	1.1	even	1	trivial
4016.2.a.g.1.6		7	4.3	odd	2
9036.2.a.i.1.6		7	3.2	odd	2

Overview	Random
Universe	Knowledge

Classical	Maass
Hilbert	Bianchi

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

Level:	\( N \)	\(=\)	\( 1004 = 2^{2} \cdot 251 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	1004.a (trivial)

\(f(q)\)	\(=\)	\(q-2.29157 q^{3} -1.78468 q^{5} +0.671050 q^{7} +2.25130 q^{9} +O(q^{10})\)	\(q-2.29157 q^{3} -1.78468 q^{5} +0.671050 q^{7} +2.25130 q^{9} +2.23015 q^{11} +0.0525243 q^{13} +4.08972 q^{15} +5.52610 q^{17} +1.49944 q^{19} -1.53776 q^{21} -7.13437 q^{23} -1.81492 q^{25} +1.71570 q^{27} +2.08515 q^{29} -9.61596 q^{31} -5.11055 q^{33} -1.19761 q^{35} +5.33025 q^{37} -0.120363 q^{39} -11.8708 q^{41} -9.09066 q^{43} -4.01785 q^{45} +5.21388 q^{47} -6.54969 q^{49} -12.6635 q^{51} +4.27480 q^{53} -3.98011 q^{55} -3.43607 q^{57} -0.163804 q^{59} +11.4800 q^{61} +1.51073 q^{63} -0.0937391 q^{65} -10.6984 q^{67} +16.3489 q^{69} -8.24538 q^{71} -12.8635 q^{73} +4.15901 q^{75} +1.49654 q^{77} -7.86983 q^{79} -10.6855 q^{81} -5.22811 q^{83} -9.86233 q^{85} -4.77826 q^{87} -2.70075 q^{89} +0.0352465 q^{91} +22.0357 q^{93} -2.67602 q^{95} +16.2733 q^{97} +5.02074 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 7 q - 3 q^{3} - 2 q^{5} - 6 q^{7}+O(q^{10}) \) 7 * q - 3 * q^3 - 2 * q^5 - 6 * q^7	\( 7 q - 3 q^{3} - 2 q^{5} - 6 q^{7} - 5 q^{11} - q^{13} - 6 q^{15} - 8 q^{17} - 15 q^{19} - 3 q^{21} - 5 q^{23} - 9 q^{25} - 9 q^{27} - 21 q^{31} - 7 q^{35} - q^{37} - 23 q^{39} - 10 q^{41} - 23 q^{43} - 4 q^{45} - 10 q^{47} - 13 q^{49} - 20 q^{51} - q^{53} - 23 q^{55} - 6 q^{57} - 4 q^{59} + 3 q^{61} - 4 q^{63} + 4 q^{65} - 28 q^{67} + 18 q^{69} - 18 q^{71} - 7 q^{73} + 11 q^{75} + 6 q^{77} - 30 q^{79} - 5 q^{81} + 13 q^{83} + q^{85} - 7 q^{87} - 18 q^{91} + 36 q^{93} - 2 q^{97} - 10 q^{99}+O(q^{100}) \) 7 * q - 3 * q^3 - 2 * q^5 - 6 * q^7 - 5 * q^11 - q^13 - 6 * q^15 - 8 * q^17 - 15 * q^19 - 3 * q^21 - 5 * q^23 - 9 * q^25 - 9 * q^27 - 21 * q^31 - 7 * q^35 - q^37 - 23 * q^39 - 10 * q^41 - 23 * q^43 - 4 * q^45 - 10 * q^47 - 13 * q^49 - 20 * q^51 - q^53 - 23 * q^55 - 6 * q^57 - 4 * q^59 + 3 * q^61 - 4 * q^63 + 4 * q^65 - 28 * q^67 + 18 * q^69 - 18 * q^71 - 7 * q^73 + 11 * q^75 + 6 * q^77 - 30 * q^79 - 5 * q^81 + 13 * q^83 + q^85 - 7 * q^87 - 18 * q^91 + 36 * q^93 - 2 * q^97 - 10 * q^99

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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