LMFDB - Embedded newform 8004.2.a.h.1.13

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [8004,2,Mod(1,8004)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(8004, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([0, 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("8004.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 8004 = 2^{2} \cdot 3 \cdot 23 \cdot 29 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	8004.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:	$63.9122617778$
Analytic rank:	$0$
Dimension:	$13$
Coefficient field:	$\mathbb{Q}[x]/(x^{13} - \cdots)$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{13} - 5 x^{12} - 27 x^{11} + 158 x^{10} + 180 x^{9} - 1652 x^{8} + 65 x^{7} + 7388 x^{6} + \cdots - 5832 $ x^13 - 5x^12 - 27x^11 + 158x^10 + 180x^9 - 1652x^8 + 65x^7 + 7388x^6 - 3259x^5 - 15445x^4 + 7832x^3 + 15102x^2 - 5184x - 5832
Coefficient ring:	$\Z[a_1, \ldots, a_{13}]$
Coefficient ring index:	$ 1 $
Twist minimal:	yes
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.13
Root			$4.44328$ of defining polynomial
Character	$\chi$	$=$	8004.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-1.00000 q^{3} +4.44328 q^{5} -0.0380915 q^{7} +1.00000 q^{9} +O(q^{10})$	$q-1.00000 q^{3} +4.44328 q^{5} -0.0380915 q^{7} +1.00000 q^{9} +4.85545 q^{11} +1.07938 q^{13} -4.44328 q^{15} +2.50758 q^{17} -0.902722 q^{19} +0.0380915 q^{21} -1.00000 q^{23} +14.7428 q^{25} -1.00000 q^{27} -1.00000 q^{29} -3.84286 q^{31} -4.85545 q^{33} -0.169251 q^{35} +8.10565 q^{37} -1.07938 q^{39} +1.14166 q^{41} -7.38920 q^{43} +4.44328 q^{45} +5.54274 q^{47} -6.99855 q^{49} -2.50758 q^{51} +6.54574 q^{53} +21.5742 q^{55} +0.902722 q^{57} -1.34003 q^{59} +6.90876 q^{61} -0.0380915 q^{63} +4.79601 q^{65} +9.06196 q^{67} +1.00000 q^{69} -1.52478 q^{71} -1.88442 q^{73} -14.7428 q^{75} -0.184952 q^{77} -7.21090 q^{79} +1.00000 q^{81} +0.623316 q^{83} +11.1419 q^{85} +1.00000 q^{87} +0.312716 q^{89} -0.0411153 q^{91} +3.84286 q^{93} -4.01105 q^{95} +15.0370 q^{97} +4.85545 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 13 q - 13 q^{3} + 5 q^{5} - 8 q^{7} + 13 q^{9}+O(q^{10}) $ 13 * q - 13 * q^3 + 5 * q^5 - 8 * q^7 + 13 * q^9	$ 13 q - 13 q^{3} + 5 q^{5} - 8 q^{7} + 13 q^{9} + q^{11} + q^{13} - 5 q^{15} - 2 q^{17} - 10 q^{19} + 8 q^{21} - 13 q^{23} + 14 q^{25} - 13 q^{27} - 13 q^{29} - 26 q^{31} - q^{33} + 19 q^{35} + 15 q^{37} - q^{39} + 21 q^{41} - 6 q^{43} + 5 q^{45} + 16 q^{47} + 19 q^{49} + 2 q^{51} + 7 q^{53} + 15 q^{55} + 10 q^{57} - 11 q^{59} + 19 q^{61} - 8 q^{63} + 6 q^{65} - 13 q^{67} + 13 q^{69} + 9 q^{73} - 14 q^{75} + 10 q^{77} - 25 q^{79} + 13 q^{81} + 3 q^{83} + 14 q^{85} + 13 q^{87} + 23 q^{89} + 19 q^{91} + 26 q^{93} + 7 q^{95} + 25 q^{97} + q^{99}+O(q^{100}) $ 13 * q - 13 * q^3 + 5 * q^5 - 8 * q^7 + 13 * q^9 + q^11 + q^13 - 5 * q^15 - 2 * q^17 - 10 * q^19 + 8 * q^21 - 13 * q^23 + 14 * q^25 - 13 * q^27 - 13 * q^29 - 26 * q^31 - q^33 + 19 * q^35 + 15 * q^37 - q^39 + 21 * q^41 - 6 * q^43 + 5 * q^45 + 16 * q^47 + 19 * q^49 + 2 * q^51 + 7 * q^53 + 15 * q^55 + 10 * q^57 - 11 * q^59 + 19 * q^61 - 8 * q^63 + 6 * q^65 - 13 * q^67 + 13 * q^69 + 9 * q^73 - 14 * q^75 + 10 * q^77 - 25 * q^79 + 13 * q^81 + 3 * q^83 + 14 * q^85 + 13 * q^87 + 23 * q^89 + 19 * q^91 + 26 * q^93 + 7 * q^95 + 25 * q^97 + 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
$3$	−1.00000	−0.577350
$3$	−1.00000	−0.577350
$4$	0	0
$5$	4.44328	1.98710	0.993548	−	0.113408i	$-0.0361768\pi$
$5$	4.44328	1.98710	0.993548	+	0.113408i	$0.0361768\pi$
$6$	0	0
$7$	−0.0380915	−0.0143972	−0.00719862	−	0.999974i	$-0.502291\pi$
$7$	−0.0380915	−0.0143972	−0.00719862	+	0.999974i	$0.502291\pi$
$8$	0	0
$9$	1.00000	0.333333
$10$	0	0
$11$	4.85545	1.46397	0.731987	−	0.681318i	$-0.238593\pi$
$11$	4.85545	1.46397	0.731987	+	0.681318i	$0.238593\pi$
$12$	0	0
$13$	1.07938	0.299367	0.149684	−	0.988734i	$-0.452175\pi$
$13$	1.07938	0.299367	0.149684	+	0.988734i	$0.452175\pi$
$14$	0	0
$15$	−4.44328	−1.14725
$16$	0	0
$17$	2.50758	0.608178	0.304089	−	0.952644i	$-0.401648\pi$
$17$	2.50758	0.608178	0.304089	+	0.952644i	$0.401648\pi$
$18$	0	0
$19$	−0.902722	−0.207099	−0.103549	−	0.994624i	$-0.533020\pi$
$19$	−0.902722	−0.207099	−0.103549	+	0.994624i	$0.533020\pi$
$20$	0	0
$21$	0.0380915	0.00831225
$22$	0	0
$23$	−1.00000	−0.208514
$23$	−1.00000	−0.208514
$24$	0	0
$25$	14.7428	2.94855
$26$	0	0
$27$	−1.00000	−0.192450
$28$	0	0
$29$	−1.00000	−0.185695
$29$	−1.00000	−0.185695
$30$	0	0
$31$	−3.84286	−0.690199	−0.345099	−	0.938566i	$-0.612155\pi$
$31$	−3.84286	−0.690199	−0.345099	+	0.938566i	$0.612155\pi$
$32$	0	0
$33$	−4.85545	−0.845226
$34$	0	0
$35$	−0.169251	−0.0286087
$36$	0	0
$37$	8.10565	1.33256	0.666281	−	0.745701i	$-0.267885\pi$
$37$	8.10565	1.33256	0.666281	+	0.745701i	$0.267885\pi$
$38$	0	0
$39$	−1.07938	−0.172840
$40$	0	0
$41$	1.14166	0.178297	0.0891485	−	0.996018i	$-0.471585\pi$
$41$	1.14166	0.178297	0.0891485	+	0.996018i	$0.471585\pi$
$42$	0	0
$43$	−7.38920	−1.12684	−0.563421	−	0.826170i	$-0.690515\pi$
$43$	−7.38920	−1.12684	−0.563421	+	0.826170i	$0.690515\pi$
$44$	0	0
$45$	4.44328	0.662366
$46$	0	0
$47$	5.54274	0.808491	0.404246	−	0.914650i	$-0.367534\pi$
$47$	5.54274	0.808491	0.404246	+	0.914650i	$0.367534\pi$
$48$	0	0
$49$	−6.99855	−0.999793
$50$	0	0
$51$	−2.50758	−0.351131
$52$	0	0
$53$	6.54574	0.899126	0.449563	−	0.893249i	$-0.351580\pi$
$53$	6.54574	0.899126	0.449563	+	0.893249i	$0.351580\pi$
$54$	0	0
$55$	21.5742	2.90906
$56$	0	0
$57$	0.902722	0.119568
$58$	0	0
$59$	−1.34003	−0.174457	−0.0872284	−	0.996188i	$-0.527801\pi$
$59$	−1.34003	−0.174457	−0.0872284	+	0.996188i	$0.527801\pi$
$60$	0	0
$61$	6.90876	0.884576	0.442288	−	0.896873i	$-0.354167\pi$
$61$	6.90876	0.884576	0.442288	+	0.896873i	$0.354167\pi$
$62$	0	0
$63$	−0.0380915	−0.00479908
$64$	0	0
$65$	4.79601	0.594871
$66$	0	0
$67$	9.06196	1.10710	0.553548	−	0.832818i	$-0.313274\pi$
$67$	9.06196	1.10710	0.553548	+	0.832818i	$0.313274\pi$
$68$	0	0
$69$	1.00000	0.120386
$70$	0	0
$71$	−1.52478	−0.180958	−0.0904788	−	0.995898i	$-0.528840\pi$
$71$	−1.52478	−0.180958	−0.0904788	+	0.995898i	$0.528840\pi$
$72$	0	0
$73$	−1.88442	−0.220554	−0.110277	−	0.993901i	$-0.535174\pi$
$73$	−1.88442	−0.220554	−0.110277	+	0.993901i	$0.535174\pi$
$74$	0	0
$75$	−14.7428	−1.70235
$76$	0	0
$77$	−0.184952	−0.0210772
$78$	0	0
$79$	−7.21090	−0.811289	−0.405645	−	0.914031i	$-0.632953\pi$
$79$	−7.21090	−0.811289	−0.405645	+	0.914031i	$0.632953\pi$
$80$	0	0
$81$	1.00000	0.111111
$82$	0	0
$83$	0.623316	0.0684178	0.0342089	−	0.999415i	$-0.489109\pi$
$83$	0.623316	0.0684178	0.0342089	+	0.999415i	$0.489109\pi$
$84$	0	0
$85$	11.1419	1.20851
$86$	0	0
$87$	1.00000	0.107211
$88$	0	0
$89$	0.312716	0.0331478	0.0165739	−	0.999863i	$-0.494724\pi$
$89$	0.312716	0.0331478	0.0165739	+	0.999863i	$0.494724\pi$
$90$	0	0
$91$	−0.0411153	−0.00431006
$92$	0	0
$93$	3.84286	0.398486
$94$	0	0
$95$	−4.01105	−0.411525
$96$	0	0
$97$	15.0370	1.52678	0.763389	−	0.645940i	$-0.223534\pi$
$97$	15.0370	1.52678	0.763389	+	0.645940i	$0.223534\pi$
$98$	0	0
$99$	4.85545	0.487991
$100$	0	0
$101$	−4.42163	−0.439969	−0.219984	−	0.975503i	$-0.570601\pi$
$101$	−4.42163	−0.439969	−0.219984	+	0.975503i	$0.570601\pi$
$102$	0	0
$103$	13.6366	1.34365	0.671827	−	0.740708i	$-0.265510\pi$
$103$	13.6366	1.34365	0.671827	+	0.740708i	$0.265510\pi$
$104$	0	0
$105$	0.169251	0.0165172
$106$	0	0
$107$	−19.4261	−1.87799	−0.938997	−	0.343925i	$-0.888243\pi$
$107$	−19.4261	−1.87799	−0.938997	+	0.343925i	$0.888243\pi$
$108$	0	0
$109$	−2.13710	−0.204697	−0.102349	−	0.994749i	$-0.532636\pi$
$109$	−2.13710	−0.204697	−0.102349	+	0.994749i	$0.532636\pi$
$110$	0	0
$111$	−8.10565	−0.769355
$112$	0	0
$113$	−16.9680	−1.59622	−0.798108	−	0.602514i	$-0.794166\pi$
$113$	−16.9680	−1.59622	−0.798108	+	0.602514i	$0.794166\pi$
$114$	0	0
$115$	−4.44328	−0.414338
$116$	0	0
$117$	1.07938	0.0997890
$118$	0	0
$119$	−0.0955175	−0.00875608
$120$	0	0
$121$	12.5754	1.14322
$122$	0	0
$123$	−1.14166	−0.102940
$124$	0	0
$125$	43.2899	3.87197
$126$	0	0
$127$	0.727332	0.0645402	0.0322701	−	0.999479i	$-0.489726\pi$
$127$	0.727332	0.0645402	0.0322701	+	0.999479i	$0.489726\pi$
$128$	0	0
$129$	7.38920	0.650583
$130$	0	0
$131$	15.9776	1.39597	0.697983	−	0.716114i	$-0.254081\pi$
$131$	15.9776	1.39597	0.697983	+	0.716114i	$0.254081\pi$
$132$	0	0
$133$	0.0343861	0.00298165
$134$	0	0
$135$	−4.44328	−0.382417
$136$	0	0
$137$	−17.7332	−1.51505	−0.757526	−	0.652805i	$-0.773592\pi$
$137$	−17.7332	−1.51505	−0.757526	+	0.652805i	$0.773592\pi$
$138$	0	0
$139$	−22.1154	−1.87581	−0.937903	−	0.346898i	$-0.887235\pi$
$139$	−22.1154	−1.87581	−0.937903	+	0.346898i	$0.887235\pi$
$140$	0	0
$141$	−5.54274	−0.466783
$142$	0	0
$143$	5.24090	0.438266
$144$	0	0
$145$	−4.44328	−0.368995
$146$	0	0
$147$	6.99855	0.577231
$148$	0	0
$149$	−21.0592	−1.72524	−0.862619	−	0.505854i	$-0.831177\pi$
$149$	−21.0592	−1.72524	−0.862619	+	0.505854i	$0.831177\pi$
$150$	0	0
$151$	−11.3382	−0.922692	−0.461346	−	0.887220i	$-0.652633\pi$
$151$	−11.3382	−0.922692	−0.461346	+	0.887220i	$0.652633\pi$
$152$	0	0
$153$	2.50758	0.202726
$154$	0	0
$155$	−17.0749	−1.37149
$156$	0	0
$157$	8.70278	0.694558	0.347279	−	0.937762i	$-0.387106\pi$
$157$	8.70278	0.694558	0.347279	+	0.937762i	$0.387106\pi$
$158$	0	0
$159$	−6.54574	−0.519111
$160$	0	0
$161$	0.0380915	0.00300203
$162$	0	0
$163$	−0.323786	−0.0253609	−0.0126804	−	0.999920i	$-0.504036\pi$
$163$	−0.323786	−0.0253609	−0.0126804	+	0.999920i	$0.504036\pi$
$164$	0	0
$165$	−21.5742	−1.67955
$166$	0	0
$167$	18.5123	1.43253	0.716264	−	0.697829i	$-0.245851\pi$
$167$	18.5123	1.43253	0.716264	+	0.697829i	$0.245851\pi$
$168$	0	0
$169$	−11.8349	−0.910379
$170$	0	0
$171$	−0.902722	−0.0690329
$172$	0	0
$173$	14.2097	1.08034	0.540171	−	0.841556i	$-0.318360\pi$
$173$	14.2097	1.08034	0.540171	+	0.841556i	$0.318360\pi$
$174$	0	0
$175$	−0.561574	−0.0424510
$176$	0	0
$177$	1.34003	0.100723
$178$	0	0
$179$	−3.96497	−0.296356	−0.148178	−	0.988961i	$-0.547341\pi$
$179$	−3.96497	−0.296356	−0.148178	+	0.988961i	$0.547341\pi$
$180$	0	0
$181$	20.2225	1.50313	0.751563	−	0.659661i	$-0.229300\pi$
$181$	20.2225	1.50313	0.751563	+	0.659661i	$0.229300\pi$
$182$	0	0
$183$	−6.90876	−0.510710
$184$	0	0
$185$	36.0157	2.64793
$186$	0	0
$187$	12.1754	0.890356
$188$	0	0
$189$	0.0380915	0.00277075
$190$	0	0
$191$	22.2408	1.60928	0.804642	−	0.593760i	$-0.202357\pi$
$191$	22.2408	1.60928	0.804642	+	0.593760i	$0.202357\pi$
$192$	0	0
$193$	25.9587	1.86855	0.934275	−	0.356552i	$-0.116048\pi$
$193$	25.9587	1.86855	0.934275	+	0.356552i	$0.116048\pi$
$194$	0	0
$195$	−4.79601	−0.343449
$196$	0	0
$197$	−27.3361	−1.94761	−0.973807	−	0.227376i	$-0.926985\pi$
$197$	−27.3361	−1.94761	−0.973807	+	0.227376i	$0.926985\pi$
$198$	0	0
$199$	−2.99825	−0.212540	−0.106270	−	0.994337i	$-0.533891\pi$
$199$	−2.99825	−0.212540	−0.106270	+	0.994337i	$0.533891\pi$
$200$	0	0
$201$	−9.06196	−0.639182
$202$	0	0
$203$	0.0380915	0.00267350
$204$	0	0
$205$	5.07271	0.354293
$206$	0	0
$207$	−1.00000	−0.0695048
$208$	0	0
$209$	−4.38313	−0.303187
$210$	0	0
$211$	25.9074	1.78354	0.891768	−	0.452493i	$-0.149465\pi$
$211$	25.9074	1.78354	0.891768	+	0.452493i	$0.149465\pi$
$212$	0	0
$213$	1.52478	0.104476
$214$	0	0
$215$	−32.8323	−2.23915
$216$	0	0
$217$	0.146380	0.00993695
$218$	0	0
$219$	1.88442	0.127337
$220$	0	0
$221$	2.70664	0.182068
$222$	0	0
$223$	3.85760	0.258324	0.129162	−	0.991624i	$-0.458771\pi$
$223$	3.85760	0.258324	0.129162	+	0.991624i	$0.458771\pi$
$224$	0	0
$225$	14.7428	0.982851
$226$	0	0
$227$	−19.6433	−1.30377	−0.651887	−	0.758316i	$-0.726022\pi$
$227$	−19.6433	−1.30377	−0.651887	+	0.758316i	$0.726022\pi$
$228$	0	0
$229$	18.6467	1.23221	0.616105	−	0.787664i	$-0.288710\pi$
$229$	18.6467	1.23221	0.616105	+	0.787664i	$0.288710\pi$
$230$	0	0
$231$	0.184952	0.0121689
$232$	0	0
$233$	5.49998	0.360316	0.180158	−	0.983638i	$-0.442339\pi$
$233$	5.49998	0.360316	0.180158	+	0.983638i	$0.442339\pi$
$234$	0	0
$235$	24.6280	1.60655
$236$	0	0
$237$	7.21090	0.468398
$238$	0	0
$239$	−0.694600	−0.0449299	−0.0224650	−	0.999748i	$-0.507151\pi$
$239$	−0.694600	−0.0449299	−0.0224650	+	0.999748i	$0.507151\pi$
$240$	0	0
$241$	−3.78910	−0.244077	−0.122039	−	0.992525i	$-0.538943\pi$
$241$	−3.78910	−0.244077	−0.122039	+	0.992525i	$0.538943\pi$
$242$	0	0
$243$	−1.00000	−0.0641500
$244$	0	0
$245$	−31.0965	−1.98669
$246$	0	0
$247$	−0.974383	−0.0619985
$248$	0	0
$249$	−0.623316	−0.0395010
$250$	0	0
$251$	−9.19858	−0.580609	−0.290305	−	0.956934i	$-0.593757\pi$
$251$	−9.19858	−0.580609	−0.290305	+	0.956934i	$0.593757\pi$
$252$	0	0
$253$	−4.85545	−0.305260
$254$	0	0
$255$	−11.1419	−0.697732
$256$	0	0
$257$	−19.8904	−1.24073	−0.620364	−	0.784314i	$-0.713015\pi$
$257$	−19.8904	−1.24073	−0.620364	+	0.784314i	$0.713015\pi$
$258$	0	0
$259$	−0.308757	−0.0191852
$260$	0	0
$261$	−1.00000	−0.0618984
$262$	0	0
$263$	−21.6921	−1.33759	−0.668796	−	0.743446i	$-0.733190\pi$
$263$	−21.6921	−1.33759	−0.668796	+	0.743446i	$0.733190\pi$
$264$	0	0
$265$	29.0846	1.78665
$266$	0	0
$267$	−0.312716	−0.0191379
$268$	0	0
$269$	−27.0524	−1.64941	−0.824706	−	0.565562i	$-0.808659\pi$
$269$	−27.0524	−1.64941	−0.824706	+	0.565562i	$0.808659\pi$
$270$	0	0
$271$	−26.6065	−1.61623	−0.808116	−	0.589024i	$-0.799512\pi$
$271$	−26.6065	−1.61623	−0.808116	+	0.589024i	$0.799512\pi$
$272$	0	0
$273$	0.0411153	0.00248841
$274$	0	0
$275$	71.5828	4.31661
$276$	0	0
$277$	−24.7853	−1.48920	−0.744602	−	0.667509i	$-0.767361\pi$
$277$	−24.7853	−1.48920	−0.744602	+	0.667509i	$0.767361\pi$
$278$	0	0
$279$	−3.84286	−0.230066
$280$	0	0
$281$	13.1785	0.786166	0.393083	−	0.919503i	$-0.371409\pi$
$281$	13.1785	0.786166	0.393083	+	0.919503i	$0.371409\pi$
$282$	0	0
$283$	−7.33033	−0.435743	−0.217871	−	0.975977i	$-0.569911\pi$
$283$	−7.33033	−0.435743	−0.217871	+	0.975977i	$0.569911\pi$
$284$	0	0
$285$	4.01105	0.237594
$286$	0	0
$287$	−0.0434875	−0.00256698
$288$	0	0
$289$	−10.7120	−0.630120
$290$	0	0
$291$	−15.0370	−0.881485
$292$	0	0
$293$	−9.30179	−0.543417	−0.271708	−	0.962380i	$-0.587589\pi$
$293$	−9.30179	−0.543417	−0.271708	+	0.962380i	$0.587589\pi$
$294$	0	0
$295$	−5.95412	−0.346662
$296$	0	0
$297$	−4.85545	−0.281742
$298$	0	0
$299$	−1.07938	−0.0624223
$300$	0	0
$301$	0.281466	0.0162234
$302$	0	0
$303$	4.42163	0.254016
$304$	0	0
$305$	30.6976	1.75774
$306$	0	0
$307$	−12.3008	−0.702043	−0.351022	−	0.936367i	$-0.614166\pi$
$307$	−12.3008	−0.702043	−0.351022	+	0.936367i	$0.614166\pi$
$308$	0	0
$309$	−13.6366	−0.775758
$310$	0	0
$311$	−6.01472	−0.341064	−0.170532	−	0.985352i	$-0.554549\pi$
$311$	−6.01472	−0.341064	−0.170532	+	0.985352i	$0.554549\pi$
$312$	0	0
$313$	−28.3976	−1.60513	−0.802565	−	0.596565i	$-0.796532\pi$
$313$	−28.3976	−1.60513	−0.802565	+	0.596565i	$0.796532\pi$
$314$	0	0
$315$	−0.169251	−0.00953624
$316$	0	0
$317$	−25.4321	−1.42841	−0.714205	−	0.699937i	$-0.753212\pi$
$317$	−25.4321	−1.42841	−0.714205	+	0.699937i	$0.753212\pi$
$318$	0	0
$319$	−4.85545	−0.271853
$320$	0	0
$321$	19.4261	1.08426
$322$	0	0
$323$	−2.26365	−0.125953
$324$	0	0
$325$	15.9131	0.882700
$326$	0	0
$327$	2.13710	0.118182
$328$	0	0
$329$	−0.211131	−0.0116400
$330$	0	0
$331$	22.5950	1.24193	0.620966	−	0.783837i	$-0.286740\pi$
$331$	22.5950	1.24193	0.620966	+	0.783837i	$0.286740\pi$
$332$	0	0
$333$	8.10565	0.444187
$334$	0	0
$335$	40.2649	2.19991
$336$	0	0
$337$	2.58913	0.141039	0.0705193	−	0.997510i	$-0.477534\pi$
$337$	2.58913	0.141039	0.0705193	+	0.997510i	$0.477534\pi$
$338$	0	0
$339$	16.9680	0.921576
$340$	0	0
$341$	−18.6588	−1.01043
$342$	0	0
$343$	0.533226	0.0287915
$344$	0	0
$345$	4.44328	0.239218
$346$	0	0
$347$	−8.98410	−0.482292	−0.241146	−	0.970489i	$-0.577523\pi$
$347$	−8.98410	−0.482292	−0.241146	+	0.970489i	$0.577523\pi$
$348$	0	0
$349$	−16.9174	−0.905569	−0.452784	−	0.891620i	$-0.649569\pi$
$349$	−16.9174	−0.905569	−0.452784	+	0.891620i	$0.649569\pi$
$350$	0	0
$351$	−1.07938	−0.0576132
$352$	0	0
$353$	33.5417	1.78524	0.892621	−	0.450807i	$-0.148864\pi$
$353$	33.5417	1.78524	0.892621	+	0.450807i	$0.148864\pi$
$354$	0	0
$355$	−6.77501	−0.359580
$356$	0	0
$357$	0.0955175	0.00505532
$358$	0	0
$359$	31.9548	1.68651	0.843254	−	0.537515i	$-0.180637\pi$
$359$	31.9548	1.68651	0.843254	+	0.537515i	$0.180637\pi$
$360$	0	0
$361$	−18.1851	−0.957110
$362$	0	0
$363$	−12.5754	−0.660039
$364$	0	0
$365$	−8.37300	−0.438263
$366$	0	0
$367$	−22.9499	−1.19798	−0.598988	−	0.800758i	$-0.704430\pi$
$367$	−22.9499	−1.19798	−0.598988	+	0.800758i	$0.704430\pi$
$368$	0	0
$369$	1.14166	0.0594323
$370$	0	0
$371$	−0.249337	−0.0129449
$372$	0	0
$373$	6.38936	0.330829	0.165414	−	0.986224i	$-0.447104\pi$
$373$	6.38936	0.330829	0.165414	+	0.986224i	$0.447104\pi$
$374$	0	0
$375$	−43.2899	−2.23548
$376$	0	0
$377$	−1.07938	−0.0555911
$378$	0	0
$379$	−7.26759	−0.373311	−0.186655	−	0.982425i	$-0.559765\pi$
$379$	−7.26759	−0.373311	−0.186655	+	0.982425i	$0.559765\pi$
$380$	0	0
$381$	−0.727332	−0.0372623
$382$	0	0
$383$	−23.0105	−1.17578	−0.587891	−	0.808940i	$-0.700042\pi$
$383$	−23.0105	−1.17578	−0.587891	+	0.808940i	$0.700042\pi$
$384$	0	0
$385$	−0.821792	−0.0418824
$386$	0	0
$387$	−7.38920	−0.375614
$388$	0	0
$389$	26.6436	1.35088	0.675442	−	0.737413i	$-0.263953\pi$
$389$	26.6436	1.35088	0.675442	+	0.737413i	$0.263953\pi$
$390$	0	0
$391$	−2.50758	−0.126814
$392$	0	0
$393$	−15.9776	−0.805961
$394$	0	0
$395$	−32.0401	−1.61211
$396$	0	0
$397$	13.6362	0.684379	0.342190	−	0.939631i	$-0.388831\pi$
$397$	13.6362	0.684379	0.342190	+	0.939631i	$0.388831\pi$
$398$	0	0
$399$	−0.0343861	−0.00172146
$400$	0	0
$401$	17.5176	0.874786	0.437393	−	0.899270i	$-0.355902\pi$
$401$	17.5176	0.874786	0.437393	+	0.899270i	$0.355902\pi$
$402$	0	0
$403$	−4.14792	−0.206623
$404$	0	0
$405$	4.44328	0.220789
$406$	0	0
$407$	39.3566	1.95084
$408$	0	0
$409$	11.5172	0.569489	0.284745	−	0.958603i	$-0.408091\pi$
$409$	11.5172	0.569489	0.284745	+	0.958603i	$0.408091\pi$
$410$	0	0
$411$	17.7332	0.874716
$412$	0	0
$413$	0.0510437	0.00251169
$414$	0	0
$415$	2.76957	0.135953
$416$	0	0
$417$	22.1154	1.08300
$418$	0	0
$419$	0.674944	0.0329732	0.0164866	−	0.999864i	$-0.494752\pi$
$419$	0.674944	0.0329732	0.0164866	+	0.999864i	$0.494752\pi$
$420$	0	0
$421$	32.5358	1.58570	0.792850	−	0.609417i	$-0.208596\pi$
$421$	32.5358	1.58570	0.792850	+	0.609417i	$0.208596\pi$
$422$	0	0
$423$	5.54274	0.269497
$424$	0	0
$425$	36.9687	1.79324
$426$	0	0
$427$	−0.263165	−0.0127355
$428$	0	0
$429$	−5.24090	−0.253033
$430$	0	0
$431$	26.1726	1.26069	0.630346	−	0.776314i	$-0.282913\pi$
$431$	26.1726	1.26069	0.630346	+	0.776314i	$0.282913\pi$
$432$	0	0
$433$	5.16462	0.248196	0.124098	−	0.992270i	$-0.460396\pi$
$433$	5.16462	0.248196	0.124098	+	0.992270i	$0.460396\pi$
$434$	0	0
$435$	4.44328	0.213039
$436$	0	0
$437$	0.902722	0.0431831
$438$	0	0
$439$	24.7763	1.18251	0.591255	−	0.806485i	$-0.298633\pi$
$439$	24.7763	1.18251	0.591255	+	0.806485i	$0.298633\pi$
$440$	0	0
$441$	−6.99855	−0.333264
$442$	0	0
$443$	10.8832	0.517076	0.258538	−	0.966001i	$-0.416759\pi$
$443$	10.8832	0.517076	0.258538	+	0.966001i	$0.416759\pi$
$444$	0	0
$445$	1.38949	0.0658679
$446$	0	0
$447$	21.0592	0.996066
$448$	0	0
$449$	24.9141	1.17577	0.587884	−	0.808945i	$-0.299961\pi$
$449$	24.9141	1.17577	0.587884	+	0.808945i	$0.299961\pi$
$450$	0	0
$451$	5.54326	0.261022
$452$	0	0
$453$	11.3382	0.532716
$454$	0	0
$455$	−0.182687	−0.00856450
$456$	0	0
$457$	10.6151	0.496555	0.248277	−	0.968689i	$-0.420136\pi$
$457$	10.6151	0.496555	0.248277	+	0.968689i	$0.420136\pi$
$458$	0	0
$459$	−2.50758	−0.117044
$460$	0	0
$461$	21.6837	1.00991	0.504956	−	0.863145i	$-0.331509\pi$
$461$	21.6837	1.00991	0.504956	+	0.863145i	$0.331509\pi$
$462$	0	0
$463$	−13.7058	−0.636963	−0.318482	−	0.947929i	$-0.603173\pi$
$463$	−13.7058	−0.636963	−0.318482	+	0.947929i	$0.603173\pi$
$464$	0	0
$465$	17.0749	0.791831
$466$	0	0
$467$	−14.3011	−0.661774	−0.330887	−	0.943670i	$-0.607348\pi$
$467$	−14.3011	−0.661774	−0.330887	+	0.943670i	$0.607348\pi$
$468$	0	0
$469$	−0.345184	−0.0159391
$470$	0	0
$471$	−8.70278	−0.401003
$472$	0	0
$473$	−35.8779	−1.64967
$474$	0	0
$475$	−13.3086	−0.610642
$476$	0	0
$477$	6.54574	0.299709
$478$	0	0
$479$	−19.0239	−0.869225	−0.434612	−	0.900618i	$-0.643115\pi$
$479$	−19.0239	−0.869225	−0.434612	+	0.900618i	$0.643115\pi$
$480$	0	0
$481$	8.74911	0.398925
$482$	0	0
$483$	−0.0380915	−0.00173322
$484$	0	0
$485$	66.8137	3.03385
$486$	0	0
$487$	−4.09936	−0.185760	−0.0928799	−	0.995677i	$-0.529607\pi$
$487$	−4.09936	−0.185760	−0.0928799	+	0.995677i	$0.529607\pi$
$488$	0	0
$489$	0.323786	0.0146421
$490$	0	0
$491$	−24.3170	−1.09741	−0.548705	−	0.836016i	$-0.684879\pi$
$491$	−24.3170	−1.09741	−0.548705	+	0.836016i	$0.684879\pi$
$492$	0	0
$493$	−2.50758	−0.112936
$494$	0	0
$495$	21.5742	0.969686
$496$	0	0
$497$	0.0580810	0.00260529
$498$	0	0
$499$	−1.24506	−0.0557364	−0.0278682	−	0.999612i	$-0.508872\pi$
$499$	−1.24506	−0.0557364	−0.0278682	+	0.999612i	$0.508872\pi$
$500$	0	0
$501$	−18.5123	−0.827071
$502$	0	0
$503$	20.9910	0.935941	0.467971	−	0.883744i	$-0.344985\pi$
$503$	20.9910	0.935941	0.467971	+	0.883744i	$0.344985\pi$
$504$	0	0
$505$	−19.6466	−0.874261
$506$	0	0
$507$	11.8349	0.525608
$508$	0	0
$509$	28.2027	1.25006	0.625032	−	0.780599i	$-0.285086\pi$
$509$	28.2027	1.25006	0.625032	+	0.780599i	$0.285086\pi$
$510$	0	0
$511$	0.0717803	0.00317538
$512$	0	0
$513$	0.902722	0.0398562
$514$	0	0
$515$	60.5912	2.66997
$516$	0	0
$517$	26.9125	1.18361
$518$	0	0
$519$	−14.2097	−0.623735
$520$	0	0
$521$	16.4924	0.722545	0.361272	−	0.932460i	$-0.382342\pi$
$521$	16.4924	0.722545	0.361272	+	0.932460i	$0.382342\pi$
$522$	0	0
$523$	−10.5020	−0.459219	−0.229610	−	0.973283i	$-0.573745\pi$
$523$	−10.5020	−0.459219	−0.229610	+	0.973283i	$0.573745\pi$
$524$	0	0
$525$	0.561574	0.0245091
$526$	0	0
$527$	−9.63629	−0.419763
$528$	0	0
$529$	1.00000	0.0434783
$530$	0	0
$531$	−1.34003	−0.0581522
$532$	0	0
$533$	1.23229	0.0533762
$534$	0	0
$535$	−86.3158	−3.73176
$536$	0	0
$537$	3.96497	0.171101
$538$	0	0
$539$	−33.9811	−1.46367
$540$	0	0
$541$	−23.9427	−1.02938	−0.514688	−	0.857378i	$-0.672092\pi$
$541$	−23.9427	−1.02938	−0.514688	+	0.857378i	$0.672092\pi$
$542$	0	0
$543$	−20.2225	−0.867830
$544$	0	0
$545$	−9.49576	−0.406754
$546$	0	0
$547$	9.76306	0.417438	0.208719	−	0.977976i	$-0.433071\pi$
$547$	9.76306	0.417438	0.208719	+	0.977976i	$0.433071\pi$
$548$	0	0
$549$	6.90876	0.294859
$550$	0	0
$551$	0.902722	0.0384573
$552$	0	0
$553$	0.274674	0.0116803
$554$	0	0
$555$	−36.0157	−1.52878
$556$	0	0
$557$	−7.66512	−0.324782	−0.162391	−	0.986727i	$-0.551921\pi$
$557$	−7.66512	−0.324782	−0.162391	+	0.986727i	$0.551921\pi$
$558$	0	0
$559$	−7.97578	−0.337340
$560$	0	0
$561$	−12.1754	−0.514048
$562$	0	0
$563$	12.3779	0.521666	0.260833	−	0.965384i	$-0.416003\pi$
$563$	12.3779	0.521666	0.260833	+	0.965384i	$0.416003\pi$
$564$	0	0
$565$	−75.3937	−3.17184
$566$	0	0
$567$	−0.0380915	−0.00159969
$568$	0	0
$569$	9.49239	0.397942	0.198971	−	0.980005i	$-0.436240\pi$
$569$	9.49239	0.397942	0.198971	+	0.980005i	$0.436240\pi$
$570$	0	0
$571$	6.67084	0.279166	0.139583	−	0.990210i	$-0.455424\pi$
$571$	6.67084	0.279166	0.139583	+	0.990210i	$0.455424\pi$
$572$	0	0
$573$	−22.2408	−0.929121
$574$	0	0
$575$	−14.7428	−0.614816
$576$	0	0
$577$	32.2792	1.34380	0.671901	−	0.740641i	$-0.265478\pi$
$577$	32.2792	1.34380	0.671901	+	0.740641i	$0.265478\pi$
$578$	0	0
$579$	−25.9587	−1.07881
$580$	0	0
$581$	−0.0237430	−0.000985027 0
$582$	0	0
$583$	31.7825	1.31630
$584$	0	0
$585$	4.79601	0.198290
$586$	0	0
$587$	20.5207	0.846982	0.423491	−	0.905900i	$-0.360805\pi$
$587$	20.5207	0.846982	0.423491	+	0.905900i	$0.360805\pi$
$588$	0	0
$589$	3.46904	0.142939
$590$	0	0
$591$	27.3361	1.12446
$592$	0	0
$593$	11.0505	0.453788	0.226894	−	0.973919i	$-0.427143\pi$
$593$	11.0505	0.453788	0.226894	+	0.973919i	$0.427143\pi$
$594$	0	0
$595$	−0.424412	−0.0173992
$596$	0	0
$597$	2.99825	0.122710
$598$	0	0
$599$	−8.83890	−0.361148	−0.180574	−	0.983561i	$-0.557795\pi$
$599$	−8.83890	−0.361148	−0.180574	+	0.983561i	$0.557795\pi$
$600$	0	0
$601$	14.9211	0.608643	0.304321	−	0.952569i	$-0.401570\pi$
$601$	14.9211	0.608643	0.304321	+	0.952569i	$0.401570\pi$
$602$	0	0
$603$	9.06196	0.369032
$604$	0	0
$605$	55.8762	2.27169
$606$	0	0
$607$	−22.7251	−0.922384	−0.461192	−	0.887300i	$-0.652578\pi$
$607$	−22.7251	−0.922384	−0.461192	+	0.887300i	$0.652578\pi$
$608$	0	0
$609$	−0.0380915	−0.00154355
$610$	0	0
$611$	5.98274	0.242036
$612$	0	0
$613$	−1.08590	−0.0438590	−0.0219295	−	0.999760i	$-0.506981\pi$
$613$	−1.08590	−0.0438590	−0.0219295	+	0.999760i	$0.506981\pi$
$614$	0	0
$615$	−5.07271	−0.204551
$616$	0	0
$617$	36.6189	1.47422	0.737110	−	0.675773i	$-0.236190\pi$
$617$	36.6189	1.47422	0.737110	+	0.675773i	$0.236190\pi$
$618$	0	0
$619$	−15.8133	−0.635592	−0.317796	−	0.948159i	$-0.602943\pi$
$619$	−15.8133	−0.635592	−0.317796	+	0.948159i	$0.602943\pi$
$620$	0	0
$621$	1.00000	0.0401286
$622$	0	0
$623$	−0.0119118	−0.000477237 0
$624$	0	0
$625$	118.635	4.74542
$626$	0	0
$627$	4.38313	0.175045
$628$	0	0
$629$	20.3256	0.810434
$630$	0	0
$631$	30.9400	1.23170	0.615851	−	0.787863i	$-0.288812\pi$
$631$	30.9400	1.23170	0.615851	+	0.787863i	$0.288812\pi$
$632$	0	0
$633$	−25.9074	−1.02973
$634$	0	0
$635$	3.23174	0.128248
$636$	0	0
$637$	−7.55412	−0.299305
$638$	0	0
$639$	−1.52478	−0.0603192
$640$	0	0
$641$	32.6517	1.28967	0.644833	−	0.764324i	$-0.276927\pi$
$641$	32.6517	1.28967	0.644833	+	0.764324i	$0.276927\pi$
$642$	0	0
$643$	6.53638	0.257770	0.128885	−	0.991660i	$-0.458860\pi$
$643$	6.53638	0.257770	0.128885	+	0.991660i	$0.458860\pi$
$644$	0	0
$645$	32.8323	1.29277
$646$	0	0
$647$	−20.1089	−0.790564	−0.395282	−	0.918560i	$-0.629353\pi$
$647$	−20.1089	−0.790564	−0.395282	+	0.918560i	$0.629353\pi$
$648$	0	0
$649$	−6.50644	−0.255400
$650$	0	0
$651$	−0.146380	−0.00573710
$652$	0	0
$653$	20.4505	0.800290	0.400145	−	0.916452i	$-0.368960\pi$
$653$	20.4505	0.800290	0.400145	+	0.916452i	$0.368960\pi$
$654$	0	0
$655$	70.9929	2.77392
$656$	0	0
$657$	−1.88442	−0.0735182
$658$	0	0
$659$	−7.55071	−0.294134	−0.147067	−	0.989127i	$-0.546983\pi$
$659$	−7.55071	−0.294134	−0.147067	+	0.989127i	$0.546983\pi$
$660$	0	0
$661$	−32.7321	−1.27313	−0.636566	−	0.771222i	$-0.719646\pi$
$661$	−32.7321	−1.27313	−0.636566	+	0.771222i	$0.719646\pi$
$662$	0	0
$663$	−2.70664	−0.105117
$664$	0	0
$665$	0.152787	0.00592483
$666$	0	0
$667$	1.00000	0.0387202
$668$	0	0
$669$	−3.85760	−0.149143
$670$	0	0
$671$	33.5452	1.29500
$672$	0	0
$673$	−35.7617	−1.37851	−0.689256	−	0.724518i	$-0.742062\pi$
$673$	−35.7617	−1.37851	−0.689256	+	0.724518i	$0.742062\pi$
$674$	0	0
$675$	−14.7428	−0.567450
$676$	0	0
$677$	34.1426	1.31221	0.656104	−	0.754670i	$-0.272203\pi$
$677$	34.1426	1.31221	0.656104	+	0.754670i	$0.272203\pi$
$678$	0	0
$679$	−0.572783	−0.0219814
$680$	0	0
$681$	19.6433	0.752734
$682$	0	0
$683$	16.6013	0.635232	0.317616	−	0.948219i	$-0.397118\pi$
$683$	16.6013	0.635232	0.317616	+	0.948219i	$0.397118\pi$
$684$	0	0
$685$	−78.7938	−3.01056
$686$	0	0
$687$	−18.6467	−0.711417
$688$	0	0
$689$	7.06536	0.269169
$690$	0	0
$691$	−20.8919	−0.794764	−0.397382	−	0.917653i	$-0.630081\pi$
$691$	−20.8919	−0.794764	−0.397382	+	0.917653i	$0.630081\pi$
$692$	0	0
$693$	−0.184952	−0.00702573
$694$	0	0
$695$	−98.2651	−3.72741
$696$	0	0
$697$	2.86280	0.108436
$698$	0	0
$699$	−5.49998	−0.208028
$700$	0	0
$701$	13.4160	0.506714	0.253357	−	0.967373i	$-0.418465\pi$
$701$	13.4160	0.506714	0.253357	+	0.967373i	$0.418465\pi$
$702$	0	0
$703$	−7.31715	−0.275972
$704$	0	0
$705$	−24.6280	−0.927542
$706$	0	0
$707$	0.168427	0.00633434
$708$	0	0
$709$	32.4131	1.21730	0.608649	−	0.793440i	$-0.291712\pi$
$709$	32.4131	1.21730	0.608649	+	0.793440i	$0.291712\pi$
$710$	0	0
$711$	−7.21090	−0.270430
$712$	0	0
$713$	3.84286	0.143916
$714$	0	0
$715$	23.2868	0.870876
$716$	0	0
$717$	0.694600	0.0259403
$718$	0	0
$719$	33.2255	1.23910	0.619551	−	0.784956i	$-0.287315\pi$
$719$	33.2255	1.23910	0.619551	+	0.784956i	$0.287315\pi$
$720$	0	0
$721$	−0.519438	−0.0193449
$722$	0	0
$723$	3.78910	0.140918
$724$	0	0
$725$	−14.7428	−0.547533
$726$	0	0
$727$	−39.1337	−1.45139	−0.725695	−	0.688017i	$-0.758482\pi$
$727$	−39.1337	−1.45139	−0.725695	+	0.688017i	$0.758482\pi$
$728$	0	0
$729$	1.00000	0.0370370
$730$	0	0
$731$	−18.5290	−0.685321
$732$	0	0
$733$	−34.6764	−1.28080	−0.640401	−	0.768041i	$-0.721232\pi$
$733$	−34.6764	−1.28080	−0.640401	+	0.768041i	$0.721232\pi$
$734$	0	0
$735$	31.0965	1.14701
$736$	0	0
$737$	44.0000	1.62076
$738$	0	0
$739$	2.59678	0.0955242	0.0477621	−	0.998859i	$-0.484791\pi$
$739$	2.59678	0.0955242	0.0477621	+	0.998859i	$0.484791\pi$
$740$	0	0
$741$	0.974383	0.0357949
$742$	0	0
$743$	−23.0516	−0.845679	−0.422840	−	0.906204i	$-0.638967\pi$
$743$	−23.0516	−0.845679	−0.422840	+	0.906204i	$0.638967\pi$
$744$	0	0
$745$	−93.5720	−3.42821
$746$	0	0
$747$	0.623316	0.0228059
$748$	0	0
$749$	0.739970	0.0270379
$750$	0	0
$751$	14.0205	0.511616	0.255808	−	0.966728i	$-0.417659\pi$
$751$	14.0205	0.511616	0.255808	+	0.966728i	$0.417659\pi$
$752$	0	0
$753$	9.19858	0.335215
$754$	0	0
$755$	−50.3790	−1.83348
$756$	0	0
$757$	−22.1090	−0.803566	−0.401783	−	0.915735i	$-0.631609\pi$
$757$	−22.1090	−0.803566	−0.401783	+	0.915735i	$0.631609\pi$
$758$	0	0
$759$	4.85545	0.176242
$760$	0	0
$761$	12.5651	0.455486	0.227743	−	0.973721i	$-0.426865\pi$
$761$	12.5651	0.455486	0.227743	+	0.973721i	$0.426865\pi$
$762$	0	0
$763$	0.0814055	0.00294708
$764$	0	0
$765$	11.1419	0.402836
$766$	0	0
$767$	−1.44640	−0.0522266
$768$	0	0
$769$	43.5993	1.57223	0.786115	−	0.618080i	$-0.212089\pi$
$769$	43.5993	1.57223	0.786115	+	0.618080i	$0.212089\pi$
$770$	0	0
$771$	19.8904	0.716335
$772$	0	0
$773$	−24.6734	−0.887440	−0.443720	−	0.896165i	$-0.646342\pi$
$773$	−24.6734	−0.887440	−0.443720	+	0.896165i	$0.646342\pi$
$774$	0	0
$775$	−56.6544	−2.03509
$776$	0	0
$777$	0.308757	0.0110766
$778$	0	0
$779$	−1.03060	−0.0369251
$780$	0	0
$781$	−7.40348	−0.264917
$782$	0	0
$783$	1.00000	0.0357371
$784$	0	0
$785$	38.6689	1.38015
$786$	0	0
$787$	−36.4761	−1.30023	−0.650116	−	0.759835i	$-0.725280\pi$
$787$	−36.4761	−1.30023	−0.650116	+	0.759835i	$0.725280\pi$
$788$	0	0
$789$	21.6921	0.772259
$790$	0	0
$791$	0.646337	0.0229811
$792$	0	0
$793$	7.45720	0.264813
$794$	0	0
$795$	−29.0846	−1.03152
$796$	0	0
$797$	7.89436	0.279633	0.139816	−	0.990177i	$-0.455349\pi$
$797$	7.89436	0.279633	0.139816	+	0.990177i	$0.455349\pi$
$798$	0	0
$799$	13.8989	0.491706
$800$	0	0
$801$	0.312716	0.0110493
$802$	0	0
$803$	−9.14971	−0.322886
$804$	0	0
$805$	0.169251	0.00596533
$806$	0	0
$807$	27.0524	0.952288
$808$	0	0
$809$	38.3859	1.34958	0.674788	−	0.738012i	$-0.264235\pi$
$809$	38.3859	1.34958	0.674788	+	0.738012i	$0.264235\pi$
$810$	0	0
$811$	2.03119	0.0713248	0.0356624	−	0.999364i	$-0.488646\pi$
$811$	2.03119	0.0713248	0.0356624	+	0.999364i	$0.488646\pi$
$812$	0	0
$813$	26.6065	0.933131
$814$	0	0
$815$	−1.43867	−0.0503945
$816$	0	0
$817$	6.67040	0.233368
$818$	0	0
$819$	−0.0411153	−0.00143669
$820$	0	0
$821$	−6.74870	−0.235531	−0.117766	−	0.993041i	$-0.537573\pi$
$821$	−6.74870	−0.235531	−0.117766	+	0.993041i	$0.537573\pi$
$822$	0	0
$823$	−15.4158	−0.537359	−0.268680	−	0.963230i	$-0.586587\pi$
$823$	−15.4158	−0.537359	−0.268680	+	0.963230i	$0.586587\pi$
$824$	0	0
$825$	−71.5828	−2.49219
$826$	0	0
$827$	−28.7489	−0.999696	−0.499848	−	0.866113i	$-0.666611\pi$
$827$	−28.7489	−0.999696	−0.499848	+	0.866113i	$0.666611\pi$
$828$	0	0
$829$	−43.0251	−1.49432	−0.747162	−	0.664642i	$-0.768584\pi$
$829$	−43.0251	−1.49432	−0.747162	+	0.664642i	$0.768584\pi$
$830$	0	0
$831$	24.7853	0.859792
$832$	0	0
$833$	−17.5494	−0.608052
$834$	0	0
$835$	82.2556	2.84657
$836$	0	0
$837$	3.84286	0.132829
$838$	0	0
$839$	−1.97343	−0.0681304	−0.0340652	−	0.999420i	$-0.510845\pi$
$839$	−1.97343	−0.0681304	−0.0340652	+	0.999420i	$0.510845\pi$
$840$	0	0
$841$	1.00000	0.0344828
$842$	0	0
$843$	−13.1785	−0.453893
$844$	0	0
$845$	−52.5860	−1.80901
$846$	0	0
$847$	−0.479017	−0.0164592
$848$	0	0
$849$	7.33033	0.251576
$850$	0	0
$851$	−8.10565	−0.277858
$852$	0	0
$853$	45.3055	1.55123	0.775615	−	0.631206i	$-0.217440\pi$
$853$	45.3055	1.55123	0.775615	+	0.631206i	$0.217440\pi$
$854$	0	0
$855$	−4.01105	−0.137175
$856$	0	0
$857$	−52.3587	−1.78854	−0.894270	−	0.447528i	$-0.852304\pi$
$857$	−52.3587	−1.78854	−0.894270	+	0.447528i	$0.852304\pi$
$858$	0	0
$859$	14.1733	0.483585	0.241793	−	0.970328i	$-0.422265\pi$
$859$	14.1733	0.483585	0.241793	+	0.970328i	$0.422265\pi$
$860$	0	0
$861$	0.0434875	0.00148205
$862$	0	0
$863$	45.1548	1.53709	0.768544	−	0.639797i	$-0.220981\pi$
$863$	45.1548	1.53709	0.768544	+	0.639797i	$0.220981\pi$
$864$	0	0
$865$	63.1376	2.14674
$866$	0	0
$867$	10.7120	0.363800
$868$	0	0
$869$	−35.0122	−1.18771
$870$	0	0
$871$	9.78133	0.331428
$872$	0	0
$873$	15.0370	0.508926
$874$	0	0
$875$	−1.64898	−0.0557456
$876$	0	0
$877$	13.2774	0.448346	0.224173	−	0.974549i	$-0.428032\pi$
$877$	13.2774	0.448346	0.224173	+	0.974549i	$0.428032\pi$
$878$	0	0
$879$	9.30179	0.313742
$880$	0	0
$881$	−10.0272	−0.337826	−0.168913	−	0.985631i	$-0.554026\pi$
$881$	−10.0272	−0.337826	−0.168913	+	0.985631i	$0.554026\pi$
$882$	0	0
$883$	−48.6677	−1.63780	−0.818898	−	0.573938i	$-0.805415\pi$
$883$	−48.6677	−1.63780	−0.818898	+	0.573938i	$0.805415\pi$
$884$	0	0
$885$	5.95412	0.200146
$886$	0	0
$887$	51.0774	1.71501	0.857505	−	0.514476i	$-0.172013\pi$
$887$	51.0774	1.71501	0.857505	+	0.514476i	$0.172013\pi$
$888$	0	0
$889$	−0.0277052	−0.000929201 0
$890$	0	0
$891$	4.85545	0.162664
$892$	0	0
$893$	−5.00355	−0.167437
$894$	0	0
$895$	−17.6175	−0.588888
$896$	0	0
$897$	1.07938	0.0360396
$898$	0	0
$899$	3.84286	0.128167
$900$	0	0
$901$	16.4140	0.546829
$902$	0	0
$903$	−0.281466	−0.00936660
$904$	0	0
$905$	89.8543	2.98686
$906$	0	0
$907$	7.27147	0.241445	0.120723	−	0.992686i	$-0.461479\pi$
$907$	7.27147	0.241445	0.120723	+	0.992686i	$0.461479\pi$
$908$	0	0
$909$	−4.42163	−0.146656
$910$	0	0
$911$	−41.0667	−1.36060	−0.680301	−	0.732933i	$-0.738151\pi$
$911$	−41.0667	−1.36060	−0.680301	+	0.732933i	$0.738151\pi$
$912$	0	0
$913$	3.02648	0.100162
$914$	0	0
$915$	−30.6976	−1.01483
$916$	0	0
$917$	−0.608610	−0.0200981
$918$	0	0
$919$	23.2477	0.766870	0.383435	−	0.923568i	$-0.374741\pi$
$919$	23.2477	0.766870	0.383435	+	0.923568i	$0.374741\pi$
$920$	0	0
$921$	12.3008	0.405325
$922$	0	0
$923$	−1.64582	−0.0541727
$924$	0	0
$925$	119.500	3.92913
$926$	0	0
$927$	13.6366	0.447884
$928$	0	0
$929$	35.0121	1.14871	0.574354	−	0.818607i	$-0.305253\pi$
$929$	35.0121	1.14871	0.574354	+	0.818607i	$0.305253\pi$
$930$	0	0
$931$	6.31775	0.207056
$932$	0	0
$933$	6.01472	0.196913
$934$	0	0
$935$	54.0989	1.76922
$936$	0	0
$937$	1.10229	0.0360103	0.0180052	−	0.999838i	$-0.494268\pi$
$937$	1.10229	0.0360103	0.0180052	+	0.999838i	$0.494268\pi$
$938$	0	0
$939$	28.3976	0.926722
$940$	0	0
$941$	−32.9432	−1.07392	−0.536958	−	0.843609i	$-0.680427\pi$
$941$	−32.9432	−1.07392	−0.536958	+	0.843609i	$0.680427\pi$
$942$	0	0
$943$	−1.14166	−0.0371775
$944$	0	0
$945$	0.169251	0.00550575
$946$	0	0
$947$	−50.6629	−1.64632	−0.823162	−	0.567807i	$-0.807792\pi$
$947$	−50.6629	−1.64632	−0.823162	+	0.567807i	$0.807792\pi$
$948$	0	0
$949$	−2.03401	−0.0660267
$950$	0	0
$951$	25.4321	0.824693
$952$	0	0
$953$	−36.4680	−1.18131	−0.590657	−	0.806923i	$-0.701131\pi$
$953$	−36.4680	−1.18131	−0.590657	+	0.806923i	$0.701131\pi$
$954$	0	0
$955$	98.8220	3.19781
$956$	0	0
$957$	4.85545	0.156955
$958$	0	0
$959$	0.675486	0.0218126
$960$	0	0
$961$	−16.2324	−0.523626
$962$	0	0
$963$	−19.4261	−0.625998
$964$	0	0
$965$	115.342	3.71299
$966$	0	0
$967$	40.8707	1.31431	0.657156	−	0.753754i	$-0.271759\pi$
$967$	40.8707	1.31431	0.657156	+	0.753754i	$0.271759\pi$
$968$	0	0
$969$	2.26365	0.0727189
$970$	0	0
$971$	39.7723	1.27635	0.638177	−	0.769890i	$-0.279689\pi$
$971$	39.7723	1.27635	0.638177	+	0.769890i	$0.279689\pi$
$972$	0	0
$973$	0.842410	0.0270064
$974$	0	0
$975$	−15.9131	−0.509627
$976$	0	0
$977$	−44.8083	−1.43354	−0.716772	−	0.697308i	$-0.754381\pi$
$977$	−44.8083	−1.43354	−0.716772	+	0.697308i	$0.754381\pi$
$978$	0	0
$979$	1.51838	0.0485276
$980$	0	0
$981$	−2.13710	−0.0682325
$982$	0	0
$983$	−43.8720	−1.39930	−0.699649	−	0.714487i	$-0.746660\pi$
$983$	−43.8720	−1.39930	−0.699649	+	0.714487i	$0.746660\pi$
$984$	0	0
$985$	−121.462	−3.87010
$986$	0	0
$987$	0.211131	0.00672038
$988$	0	0
$989$	7.38920	0.234963
$990$	0	0
$991$	17.5077	0.556150	0.278075	−	0.960559i	$-0.410304\pi$
$991$	17.5077	0.556150	0.278075	+	0.960559i	$0.410304\pi$
$992$	0	0
$993$	−22.5950	−0.717030
$994$	0	0
$995$	−13.3221	−0.422338
$996$	0	0
$997$	17.4726	0.553364	0.276682	−	0.960962i	$-0.410765\pi$
$997$	17.4726	0.553364	0.276682	+	0.960962i	$0.410765\pi$
$998$	0	0
$999$	−8.10565	−0.256452

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	8004.2.a.h.1.13	✓	13

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
8004.2.a.h.1.13	✓	13	1.1	even	1	trivial

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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:	\( N \)	\(=\)	\( 8004 = 2^{2} \cdot 3 \cdot 23 \cdot 29 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	8004.a (trivial)

\(f(q)\)	\(=\)	\(q-1.00000 q^{3} +4.44328 q^{5} -0.0380915 q^{7} +1.00000 q^{9} +O(q^{10})\)	\(q-1.00000 q^{3} +4.44328 q^{5} -0.0380915 q^{7} +1.00000 q^{9} +4.85545 q^{11} +1.07938 q^{13} -4.44328 q^{15} +2.50758 q^{17} -0.902722 q^{19} +0.0380915 q^{21} -1.00000 q^{23} +14.7428 q^{25} -1.00000 q^{27} -1.00000 q^{29} -3.84286 q^{31} -4.85545 q^{33} -0.169251 q^{35} +8.10565 q^{37} -1.07938 q^{39} +1.14166 q^{41} -7.38920 q^{43} +4.44328 q^{45} +5.54274 q^{47} -6.99855 q^{49} -2.50758 q^{51} +6.54574 q^{53} +21.5742 q^{55} +0.902722 q^{57} -1.34003 q^{59} +6.90876 q^{61} -0.0380915 q^{63} +4.79601 q^{65} +9.06196 q^{67} +1.00000 q^{69} -1.52478 q^{71} -1.88442 q^{73} -14.7428 q^{75} -0.184952 q^{77} -7.21090 q^{79} +1.00000 q^{81} +0.623316 q^{83} +11.1419 q^{85} +1.00000 q^{87} +0.312716 q^{89} -0.0411153 q^{91} +3.84286 q^{93} -4.01105 q^{95} +15.0370 q^{97} +4.85545 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 13 q - 13 q^{3} + 5 q^{5} - 8 q^{7} + 13 q^{9}+O(q^{10}) \) 13 * q - 13 * q^3 + 5 * q^5 - 8 * q^7 + 13 * q^9	\( 13 q - 13 q^{3} + 5 q^{5} - 8 q^{7} + 13 q^{9} + q^{11} + q^{13} - 5 q^{15} - 2 q^{17} - 10 q^{19} + 8 q^{21} - 13 q^{23} + 14 q^{25} - 13 q^{27} - 13 q^{29} - 26 q^{31} - q^{33} + 19 q^{35} + 15 q^{37} - q^{39} + 21 q^{41} - 6 q^{43} + 5 q^{45} + 16 q^{47} + 19 q^{49} + 2 q^{51} + 7 q^{53} + 15 q^{55} + 10 q^{57} - 11 q^{59} + 19 q^{61} - 8 q^{63} + 6 q^{65} - 13 q^{67} + 13 q^{69} + 9 q^{73} - 14 q^{75} + 10 q^{77} - 25 q^{79} + 13 q^{81} + 3 q^{83} + 14 q^{85} + 13 q^{87} + 23 q^{89} + 19 q^{91} + 26 q^{93} + 7 q^{95} + 25 q^{97} + q^{99}+O(q^{100}) \) 13 * q - 13 * q^3 + 5 * q^5 - 8 * q^7 + 13 * q^9 + q^11 + q^13 - 5 * q^15 - 2 * q^17 - 10 * q^19 + 8 * q^21 - 13 * q^23 + 14 * q^25 - 13 * q^27 - 13 * q^29 - 26 * q^31 - q^33 + 19 * q^35 + 15 * q^37 - q^39 + 21 * q^41 - 6 * q^43 + 5 * q^45 + 16 * q^47 + 19 * q^49 + 2 * q^51 + 7 * q^53 + 15 * q^55 + 10 * q^57 - 11 * q^59 + 19 * q^61 - 8 * q^63 + 6 * q^65 - 13 * q^67 + 13 * q^69 + 9 * q^73 - 14 * q^75 + 10 * q^77 - 25 * q^79 + 13 * q^81 + 3 * q^83 + 14 * q^85 + 13 * q^87 + 23 * q^89 + 19 * q^91 + 26 * q^93 + 7 * q^95 + 25 * q^97 + q^99

Introduction

L-functions

Modular forms

Varieties

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Database

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