LMFDB - Embedded newform 4004.2.a.i.1.8

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [4004,2,Mod(1,4004)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(4004, 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("4004.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 4004 = 2^{2} \cdot 7 \cdot 11 \cdot 13 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	4004.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:	$31.9721009693$
Analytic rank:	$0$
Dimension:	$9$
Coefficient field:	$\mathbb{Q}[x]/(x^{9} - \cdots)$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{9} - 4x^{8} - 12x^{7} + 60x^{6} + 15x^{5} - 233x^{4} + 74x^{3} + 271x^{2} - 67x - 87 $ x^9 - 4x^8 - 12x^7 + 60x^6 + 15x^5 - 233x^4 + 74x^3 + 271x^2 - 67x - 87
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			1.8
Root			$2.42810$ of defining polynomial
Character	$\chi$	$=$	4004.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+2.42810 q^{3} +0.790216 q^{5} +1.00000 q^{7} +2.89567 q^{9} +O(q^{10})$	$q+2.42810 q^{3} +0.790216 q^{5} +1.00000 q^{7} +2.89567 q^{9} -1.00000 q^{11} +1.00000 q^{13} +1.91872 q^{15} -3.47084 q^{17} +4.92376 q^{19} +2.42810 q^{21} +8.09164 q^{23} -4.37556 q^{25} -0.253320 q^{27} +7.06355 q^{29} +6.22639 q^{31} -2.42810 q^{33} +0.790216 q^{35} -10.5207 q^{37} +2.42810 q^{39} +2.33249 q^{41} +4.71705 q^{43} +2.28821 q^{45} +0.863114 q^{47} +1.00000 q^{49} -8.42754 q^{51} +2.79269 q^{53} -0.790216 q^{55} +11.9554 q^{57} +4.89334 q^{59} -9.20404 q^{61} +2.89567 q^{63} +0.790216 q^{65} +13.1558 q^{67} +19.6473 q^{69} -10.3701 q^{71} +2.37136 q^{73} -10.6243 q^{75} -1.00000 q^{77} +6.21658 q^{79} -9.30210 q^{81} +3.55905 q^{83} -2.74271 q^{85} +17.1510 q^{87} +8.47351 q^{89} +1.00000 q^{91} +15.1183 q^{93} +3.89084 q^{95} -15.3661 q^{97} -2.89567 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 9 q + 4 q^{3} + 4 q^{5} + 9 q^{7} + 13 q^{9}+O(q^{10}) $ 9 * q + 4 * q^3 + 4 * q^5 + 9 * q^7 + 13 * q^9	$ 9 q + 4 q^{3} + 4 q^{5} + 9 q^{7} + 13 q^{9} - 9 q^{11} + 9 q^{13} - 5 q^{17} + 17 q^{19} + 4 q^{21} - 8 q^{23} + 35 q^{25} + 4 q^{27} - 3 q^{29} + 9 q^{31} - 4 q^{33} + 4 q^{35} + 7 q^{37} + 4 q^{39} + 14 q^{41} + 21 q^{43} + 43 q^{45} + q^{47} + 9 q^{49} + 25 q^{51} - 8 q^{53} - 4 q^{55} + 8 q^{57} - 4 q^{59} + 30 q^{61} + 13 q^{63} + 4 q^{65} + 15 q^{67} + 9 q^{69} - q^{71} + 18 q^{73} - 32 q^{75} - 9 q^{77} + 13 q^{79} + 13 q^{81} + 2 q^{83} + 45 q^{85} + 21 q^{87} + 9 q^{91} - 2 q^{93} + 11 q^{95} + 29 q^{97} - 13 q^{99}+O(q^{100}) $ 9 * q + 4 * q^3 + 4 * q^5 + 9 * q^7 + 13 * q^9 - 9 * q^11 + 9 * q^13 - 5 * q^17 + 17 * q^19 + 4 * q^21 - 8 * q^23 + 35 * q^25 + 4 * q^27 - 3 * q^29 + 9 * q^31 - 4 * q^33 + 4 * q^35 + 7 * q^37 + 4 * q^39 + 14 * q^41 + 21 * q^43 + 43 * q^45 + q^47 + 9 * q^49 + 25 * q^51 - 8 * q^53 - 4 * q^55 + 8 * q^57 - 4 * q^59 + 30 * q^61 + 13 * q^63 + 4 * q^65 + 15 * q^67 + 9 * q^69 - q^71 + 18 * q^73 - 32 * q^75 - 9 * q^77 + 13 * q^79 + 13 * q^81 + 2 * q^83 + 45 * q^85 + 21 * q^87 + 9 * q^91 - 2 * q^93 + 11 * q^95 + 29 * q^97 - 13 * 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
$3$	2.42810	1.40186	0.700932	−	0.713228i	$-0.252768\pi$
$3$	2.42810	1.40186	0.700932	+	0.713228i	$0.252768\pi$
$4$	0	0
$5$	0.790216	0.353395	0.176698	−	0.984265i	$-0.443459\pi$
$5$	0.790216	0.353395	0.176698	+	0.984265i	$0.443459\pi$
$6$	0	0
$7$	1.00000	0.377964
$7$	1.00000	0.377964
$8$	0	0
$9$	2.89567	0.965224
$10$	0	0
$11$	−1.00000	−0.301511
$11$	−1.00000	−0.301511
$12$	0	0
$13$	1.00000	0.277350
$13$	1.00000	0.277350
$14$	0	0
$15$	1.91872	0.495412
$16$	0	0
$17$	−3.47084	−0.841802	−0.420901	−	0.907107i	$-0.638286\pi$
$17$	−3.47084	−0.841802	−0.420901	+	0.907107i	$0.638286\pi$
$18$	0	0
$19$	4.92376	1.12959	0.564794	−	0.825232i	$-0.308956\pi$
$19$	4.92376	1.12959	0.564794	+	0.825232i	$0.308956\pi$
$20$	0	0
$21$	2.42810	0.529855
$22$	0	0
$23$	8.09164	1.68722	0.843612	−	0.536953i	$-0.180425\pi$
$23$	8.09164	1.68722	0.843612	+	0.536953i	$0.180425\pi$
$24$	0	0
$25$	−4.37556	−0.875112
$26$	0	0
$27$	−0.253320	−0.0487514
$28$	0	0
$29$	7.06355	1.31167	0.655834	−	0.754905i	$-0.272317\pi$
$29$	7.06355	1.31167	0.655834	+	0.754905i	$0.272317\pi$
$30$	0	0
$31$	6.22639	1.11829	0.559146	−	0.829069i	$-0.311129\pi$
$31$	6.22639	1.11829	0.559146	+	0.829069i	$0.311129\pi$
$32$	0	0
$33$	−2.42810	−0.422678
$34$	0	0
$35$	0.790216	0.133571
$36$	0	0
$37$	−10.5207	−1.72959	−0.864794	−	0.502126i	$-0.832551\pi$
$37$	−10.5207	−1.72959	−0.864794	+	0.502126i	$0.832551\pi$
$38$	0	0
$39$	2.42810	0.388807
$40$	0	0
$41$	2.33249	0.364273	0.182137	−	0.983273i	$-0.441699\pi$
$41$	2.33249	0.364273	0.182137	+	0.983273i	$0.441699\pi$
$42$	0	0
$43$	4.71705	0.719344	0.359672	−	0.933079i	$-0.382889\pi$
$43$	4.71705	0.719344	0.359672	+	0.933079i	$0.382889\pi$
$44$	0	0
$45$	2.28821	0.341106
$46$	0	0
$47$	0.863114	0.125898	0.0629491	−	0.998017i	$-0.479949\pi$
$47$	0.863114	0.125898	0.0629491	+	0.998017i	$0.479949\pi$
$48$	0	0
$49$	1.00000	0.142857
$50$	0	0
$51$	−8.42754	−1.18009
$52$	0	0
$53$	2.79269	0.383606	0.191803	−	0.981433i	$-0.438567\pi$
$53$	2.79269	0.383606	0.191803	+	0.981433i	$0.438567\pi$
$54$	0	0
$55$	−0.790216	−0.106553
$56$	0	0
$57$	11.9554	1.58353
$58$	0	0
$59$	4.89334	0.637059	0.318529	−	0.947913i	$-0.396811\pi$
$59$	4.89334	0.637059	0.318529	+	0.947913i	$0.396811\pi$
$60$	0	0
$61$	−9.20404	−1.17846	−0.589229	−	0.807966i	$-0.700568\pi$
$61$	−9.20404	−1.17846	−0.589229	+	0.807966i	$0.700568\pi$
$62$	0	0
$63$	2.89567	0.364820
$64$	0	0
$65$	0.790216	0.0980142
$66$	0	0
$67$	13.1558	1.60724	0.803620	−	0.595142i	$-0.202904\pi$
$67$	13.1558	1.60724	0.803620	+	0.595142i	$0.202904\pi$
$68$	0	0
$69$	19.6473	2.36526
$70$	0	0
$71$	−10.3701	−1.23071	−0.615353	−	0.788252i	$-0.710987\pi$
$71$	−10.3701	−1.23071	−0.615353	+	0.788252i	$0.710987\pi$
$72$	0	0
$73$	2.37136	0.277547	0.138774	−	0.990324i	$-0.455684\pi$
$73$	2.37136	0.277547	0.138774	+	0.990324i	$0.455684\pi$
$74$	0	0
$75$	−10.6243	−1.22679
$76$	0	0
$77$	−1.00000	−0.113961
$78$	0	0
$79$	6.21658	0.699420	0.349710	−	0.936858i	$-0.386280\pi$
$79$	6.21658	0.699420	0.349710	+	0.936858i	$0.386280\pi$
$80$	0	0
$81$	−9.30210	−1.03357
$82$	0	0
$83$	3.55905	0.390657	0.195328	−	0.980738i	$-0.437423\pi$
$83$	3.55905	0.390657	0.195328	+	0.980738i	$0.437423\pi$
$84$	0	0
$85$	−2.74271	−0.297489
$86$	0	0
$87$	17.1510	1.83878
$88$	0	0
$89$	8.47351	0.898190	0.449095	−	0.893484i	$-0.351746\pi$
$89$	8.47351	0.898190	0.449095	+	0.893484i	$0.351746\pi$
$90$	0	0
$91$	1.00000	0.104828
$92$	0	0
$93$	15.1183	1.56769
$94$	0	0
$95$	3.89084	0.399191
$96$	0	0
$97$	−15.3661	−1.56019	−0.780096	−	0.625660i	$-0.784830\pi$
$97$	−15.3661	−1.56019	−0.780096	+	0.625660i	$0.784830\pi$
$98$	0	0
$99$	−2.89567	−0.291026
$100$	0	0
$101$	13.7602	1.36919	0.684596	−	0.728923i	$-0.259979\pi$
$101$	13.7602	1.36919	0.684596	+	0.728923i	$0.259979\pi$
$102$	0	0
$103$	12.8132	1.26253	0.631263	−	0.775569i	$-0.282537\pi$
$103$	12.8132	1.26253	0.631263	+	0.775569i	$0.282537\pi$
$104$	0	0
$105$	1.91872	0.187248
$106$	0	0
$107$	−15.3358	−1.48256	−0.741282	−	0.671194i	$-0.765782\pi$
$107$	−15.3358	−1.48256	−0.741282	+	0.671194i	$0.765782\pi$
$108$	0	0
$109$	10.6432	1.01944	0.509718	−	0.860342i	$-0.329750\pi$
$109$	10.6432	1.01944	0.509718	+	0.860342i	$0.329750\pi$
$110$	0	0
$111$	−25.5453	−2.42465
$112$	0	0
$113$	6.27757	0.590544	0.295272	−	0.955413i	$-0.404590\pi$
$113$	6.27757	0.590544	0.295272	+	0.955413i	$0.404590\pi$
$114$	0	0
$115$	6.39415	0.596257
$116$	0	0
$117$	2.89567	0.267705
$118$	0	0
$119$	−3.47084	−0.318171
$120$	0	0
$121$	1.00000	0.0909091
$122$	0	0
$123$	5.66351	0.510661
$124$	0	0
$125$	−7.40872	−0.662656
$126$	0	0
$127$	0.862272	0.0765142	0.0382571	−	0.999268i	$-0.487819\pi$
$127$	0.862272	0.0765142	0.0382571	+	0.999268i	$0.487819\pi$
$128$	0	0
$129$	11.4535	1.00842
$130$	0	0
$131$	−14.5654	−1.27259	−0.636293	−	0.771447i	$-0.719533\pi$
$131$	−14.5654	−1.27259	−0.636293	+	0.771447i	$0.719533\pi$
$132$	0	0
$133$	4.92376	0.426944
$134$	0	0
$135$	−0.200177	−0.0172285
$136$	0	0
$137$	−7.07989	−0.604876	−0.302438	−	0.953169i	$-0.597800\pi$
$137$	−7.07989	−0.604876	−0.302438	+	0.953169i	$0.597800\pi$
$138$	0	0
$139$	8.07256	0.684705	0.342353	−	0.939572i	$-0.388776\pi$
$139$	8.07256	0.684705	0.342353	+	0.939572i	$0.388776\pi$
$140$	0	0
$141$	2.09573	0.176492
$142$	0	0
$143$	−1.00000	−0.0836242
$144$	0	0
$145$	5.58173	0.463538
$146$	0	0
$147$	2.42810	0.200266
$148$	0	0
$149$	−3.04280	−0.249276	−0.124638	−	0.992202i	$-0.539777\pi$
$149$	−3.04280	−0.249276	−0.124638	+	0.992202i	$0.539777\pi$
$150$	0	0
$151$	1.09682	0.0892580	0.0446290	−	0.999004i	$-0.485789\pi$
$151$	1.09682	0.0892580	0.0446290	+	0.999004i	$0.485789\pi$
$152$	0	0
$153$	−10.0504	−0.812527
$154$	0	0
$155$	4.92019	0.395199
$156$	0	0
$157$	−20.4296	−1.63046	−0.815230	−	0.579137i	$-0.803390\pi$
$157$	−20.4296	−1.63046	−0.815230	+	0.579137i	$0.803390\pi$
$158$	0	0
$159$	6.78094	0.537764
$160$	0	0
$161$	8.09164	0.637711
$162$	0	0
$163$	−8.23465	−0.644988	−0.322494	−	0.946572i	$-0.604521\pi$
$163$	−8.23465	−0.644988	−0.322494	+	0.946572i	$0.604521\pi$
$164$	0	0
$165$	−1.91872	−0.149372
$166$	0	0
$167$	−14.2101	−1.09961	−0.549806	−	0.835292i	$-0.685299\pi$
$167$	−14.2101	−1.09961	−0.549806	+	0.835292i	$0.685299\pi$
$168$	0	0
$169$	1.00000	0.0769231
$170$	0	0
$171$	14.2576	1.09031
$172$	0	0
$173$	−2.73039	−0.207588	−0.103794	−	0.994599i	$-0.533098\pi$
$173$	−2.73039	−0.207588	−0.103794	+	0.994599i	$0.533098\pi$
$174$	0	0
$175$	−4.37556	−0.330761
$176$	0	0
$177$	11.8815	0.893070
$178$	0	0
$179$	−17.7507	−1.32675	−0.663374	−	0.748288i	$-0.730876\pi$
$179$	−17.7507	−1.32675	−0.663374	+	0.748288i	$0.730876\pi$
$180$	0	0
$181$	9.97183	0.741200	0.370600	−	0.928793i	$-0.379152\pi$
$181$	9.97183	0.741200	0.370600	+	0.928793i	$0.379152\pi$
$182$	0	0
$183$	−22.3483	−1.65204
$184$	0	0
$185$	−8.31361	−0.611229
$186$	0	0
$187$	3.47084	0.253813
$188$	0	0
$189$	−0.253320	−0.0184263
$190$	0	0
$191$	−1.39143	−0.100680	−0.0503402	−	0.998732i	$-0.516031\pi$
$191$	−1.39143	−0.100680	−0.0503402	+	0.998732i	$0.516031\pi$
$192$	0	0
$193$	12.1355	0.873531	0.436765	−	0.899576i	$-0.356124\pi$
$193$	12.1355	0.873531	0.436765	+	0.899576i	$0.356124\pi$
$194$	0	0
$195$	1.91872	0.137403
$196$	0	0
$197$	5.14499	0.366566	0.183283	−	0.983060i	$-0.441328\pi$
$197$	5.14499	0.366566	0.183283	+	0.983060i	$0.441328\pi$
$198$	0	0
$199$	−5.36868	−0.380576	−0.190288	−	0.981728i	$-0.560942\pi$
$199$	−5.36868	−0.380576	−0.190288	+	0.981728i	$0.560942\pi$
$200$	0	0
$201$	31.9437	2.25313
$202$	0	0
$203$	7.06355	0.495764
$204$	0	0
$205$	1.84317	0.128732
$206$	0	0
$207$	23.4307	1.62855
$208$	0	0
$209$	−4.92376	−0.340584
$210$	0	0
$211$	16.8662	1.16112	0.580558	−	0.814219i	$-0.302834\pi$
$211$	16.8662	1.16112	0.580558	+	0.814219i	$0.302834\pi$
$212$	0	0
$213$	−25.1797	−1.72528
$214$	0	0
$215$	3.72749	0.254213
$216$	0	0
$217$	6.22639	0.422675
$218$	0	0
$219$	5.75791	0.389083
$220$	0	0
$221$	−3.47084	−0.233474
$222$	0	0
$223$	−8.27646	−0.554233	−0.277117	−	0.960836i	$-0.589379\pi$
$223$	−8.27646	−0.554233	−0.277117	+	0.960836i	$0.589379\pi$
$224$	0	0
$225$	−12.6702	−0.844679
$226$	0	0
$227$	−0.0707133	−0.00469340	−0.00234670	−	0.999997i	$-0.500747\pi$
$227$	−0.0707133	−0.00469340	−0.00234670	+	0.999997i	$0.500747\pi$
$228$	0	0
$229$	3.46076	0.228693	0.114347	−	0.993441i	$-0.463523\pi$
$229$	3.46076	0.228693	0.114347	+	0.993441i	$0.463523\pi$
$230$	0	0
$231$	−2.42810	−0.159757
$232$	0	0
$233$	−20.9721	−1.37393	−0.686963	−	0.726692i	$-0.741057\pi$
$233$	−20.9721	−1.37393	−0.686963	+	0.726692i	$0.741057\pi$
$234$	0	0
$235$	0.682047	0.0444918
$236$	0	0
$237$	15.0945	0.980493
$238$	0	0
$239$	25.5361	1.65179	0.825897	−	0.563821i	$-0.190669\pi$
$239$	25.5361	1.65179	0.825897	+	0.563821i	$0.190669\pi$
$240$	0	0
$241$	3.46062	0.222918	0.111459	−	0.993769i	$-0.464448\pi$
$241$	3.46062	0.222918	0.111459	+	0.993769i	$0.464448\pi$
$242$	0	0
$243$	−21.8265	−1.40017
$244$	0	0
$245$	0.790216	0.0504851
$246$	0	0
$247$	4.92376	0.313291
$248$	0	0
$249$	8.64174	0.547648
$250$	0	0
$251$	4.84422	0.305764	0.152882	−	0.988244i	$-0.451145\pi$
$251$	4.84422	0.305764	0.152882	+	0.988244i	$0.451145\pi$
$252$	0	0
$253$	−8.09164	−0.508717
$254$	0	0
$255$	−6.65958	−0.417039
$256$	0	0
$257$	−13.1830	−0.822334	−0.411167	−	0.911560i	$-0.634879\pi$
$257$	−13.1830	−0.822334	−0.411167	+	0.911560i	$0.634879\pi$
$258$	0	0
$259$	−10.5207	−0.653723
$260$	0	0
$261$	20.4537	1.26605
$262$	0	0
$263$	−19.1117	−1.17848	−0.589238	−	0.807959i	$-0.700572\pi$
$263$	−19.1117	−1.17848	−0.589238	+	0.807959i	$0.700572\pi$
$264$	0	0
$265$	2.20683	0.135565
$266$	0	0
$267$	20.5745	1.25914
$268$	0	0
$269$	−24.2623	−1.47930	−0.739649	−	0.672992i	$-0.765009\pi$
$269$	−24.2623	−1.47930	−0.739649	+	0.672992i	$0.765009\pi$
$270$	0	0
$271$	5.01278	0.304505	0.152252	−	0.988342i	$-0.451347\pi$
$271$	5.01278	0.304505	0.152252	+	0.988342i	$0.451347\pi$
$272$	0	0
$273$	2.42810	0.146955
$274$	0	0
$275$	4.37556	0.263856
$276$	0	0
$277$	−21.2259	−1.27534	−0.637670	−	0.770309i	$-0.720102\pi$
$277$	−21.2259	−1.27534	−0.637670	+	0.770309i	$0.720102\pi$
$278$	0	0
$279$	18.0296	1.07940
$280$	0	0
$281$	6.57025	0.391948	0.195974	−	0.980609i	$-0.437213\pi$
$281$	6.57025	0.391948	0.195974	+	0.980609i	$0.437213\pi$
$282$	0	0
$283$	12.6328	0.750942	0.375471	−	0.926834i	$-0.377481\pi$
$283$	12.6328	0.750942	0.375471	+	0.926834i	$0.377481\pi$
$284$	0	0
$285$	9.44734	0.559612
$286$	0	0
$287$	2.33249	0.137682
$288$	0	0
$289$	−4.95328	−0.291370
$290$	0	0
$291$	−37.3105	−2.18718
$292$	0	0
$293$	7.17422	0.419123	0.209561	−	0.977796i	$-0.432796\pi$
$293$	7.17422	0.419123	0.209561	+	0.977796i	$0.432796\pi$
$294$	0	0
$295$	3.86680	0.225134
$296$	0	0
$297$	0.253320	0.0146991
$298$	0	0
$299$	8.09164	0.467952
$300$	0	0
$301$	4.71705	0.271886
$302$	0	0
$303$	33.4112	1.91942
$304$	0	0
$305$	−7.27318	−0.416461
$306$	0	0
$307$	1.64077	0.0936434	0.0468217	−	0.998903i	$-0.485091\pi$
$307$	1.64077	0.0936434	0.0468217	+	0.998903i	$0.485091\pi$
$308$	0	0
$309$	31.1118	1.76989
$310$	0	0
$311$	4.78095	0.271103	0.135551	−	0.990770i	$-0.456719\pi$
$311$	4.78095	0.271103	0.135551	+	0.990770i	$0.456719\pi$
$312$	0	0
$313$	−20.5822	−1.16337	−0.581687	−	0.813413i	$-0.697607\pi$
$313$	−20.5822	−1.16337	−0.581687	+	0.813413i	$0.697607\pi$
$314$	0	0
$315$	2.28821	0.128926
$316$	0	0
$317$	−14.9773	−0.841209	−0.420604	−	0.907244i	$-0.638182\pi$
$317$	−14.9773	−0.841209	−0.420604	+	0.907244i	$0.638182\pi$
$318$	0	0
$319$	−7.06355	−0.395483
$320$	0	0
$321$	−37.2368	−2.07835
$322$	0	0
$323$	−17.0896	−0.950890
$324$	0	0
$325$	−4.37556	−0.242712
$326$	0	0
$327$	25.8428	1.42911
$328$	0	0
$329$	0.863114	0.0475850
$330$	0	0
$331$	23.1949	1.27490	0.637452	−	0.770490i	$-0.279988\pi$
$331$	23.1949	1.27490	0.637452	+	0.770490i	$0.279988\pi$
$332$	0	0
$333$	−30.4644	−1.66944
$334$	0	0
$335$	10.3959	0.567991
$336$	0	0
$337$	18.0303	0.982172	0.491086	−	0.871111i	$-0.336600\pi$
$337$	18.0303	0.982172	0.491086	+	0.871111i	$0.336600\pi$
$338$	0	0
$339$	15.2426	0.827863
$340$	0	0
$341$	−6.22639	−0.337178
$342$	0	0
$343$	1.00000	0.0539949
$344$	0	0
$345$	15.5256	0.835872
$346$	0	0
$347$	−31.6550	−1.69933	−0.849665	−	0.527323i	$-0.823196\pi$
$347$	−31.6550	−1.69933	−0.849665	+	0.527323i	$0.823196\pi$
$348$	0	0
$349$	3.69089	0.197569	0.0987843	−	0.995109i	$-0.468505\pi$
$349$	3.69089	0.197569	0.0987843	+	0.995109i	$0.468505\pi$
$350$	0	0
$351$	−0.253320	−0.0135212
$352$	0	0
$353$	−27.3966	−1.45817	−0.729086	−	0.684422i	$-0.760055\pi$
$353$	−27.3966	−1.45817	−0.729086	+	0.684422i	$0.760055\pi$
$354$	0	0
$355$	−8.19463	−0.434926
$356$	0	0
$357$	−8.42754	−0.446033
$358$	0	0
$359$	8.73001	0.460752	0.230376	−	0.973102i	$-0.426004\pi$
$359$	8.73001	0.460752	0.230376	+	0.973102i	$0.426004\pi$
$360$	0	0
$361$	5.24343	0.275970
$362$	0	0
$363$	2.42810	0.127442
$364$	0	0
$365$	1.87389	0.0980838
$366$	0	0
$367$	−21.1385	−1.10342	−0.551711	−	0.834036i	$-0.686025\pi$
$367$	−21.1385	−1.10342	−0.551711	+	0.834036i	$0.686025\pi$
$368$	0	0
$369$	6.75411	0.351605
$370$	0	0
$371$	2.79269	0.144989
$372$	0	0
$373$	28.8529	1.49395	0.746973	−	0.664854i	$-0.231506\pi$
$373$	28.8529	1.49395	0.746973	+	0.664854i	$0.231506\pi$
$374$	0	0
$375$	−17.9891	−0.928954
$376$	0	0
$377$	7.06355	0.363791
$378$	0	0
$379$	−32.5685	−1.67293	−0.836466	−	0.548019i	$-0.815382\pi$
$379$	−32.5685	−1.67293	−0.836466	+	0.548019i	$0.815382\pi$
$380$	0	0
$381$	2.09368	0.107263
$382$	0	0
$383$	−4.39843	−0.224749	−0.112375	−	0.993666i	$-0.535846\pi$
$383$	−4.39843	−0.224749	−0.112375	+	0.993666i	$0.535846\pi$
$384$	0	0
$385$	−0.790216	−0.0402731
$386$	0	0
$387$	13.6590	0.694328
$388$	0	0
$389$	−25.1125	−1.27325	−0.636627	−	0.771172i	$-0.719671\pi$
$389$	−25.1125	−1.27325	−0.636627	+	0.771172i	$0.719671\pi$
$390$	0	0
$391$	−28.0848	−1.42031
$392$	0	0
$393$	−35.3663	−1.78399
$394$	0	0
$395$	4.91245	0.247172
$396$	0	0
$397$	−26.4405	−1.32701	−0.663506	−	0.748171i	$-0.730932\pi$
$397$	−26.4405	−1.32701	−0.663506	+	0.748171i	$0.730932\pi$
$398$	0	0
$399$	11.9554	0.598518
$400$	0	0
$401$	4.46910	0.223176	0.111588	−	0.993755i	$-0.464406\pi$
$401$	4.46910	0.223176	0.111588	+	0.993755i	$0.464406\pi$
$402$	0	0
$403$	6.22639	0.310159
$404$	0	0
$405$	−7.35067	−0.365258
$406$	0	0
$407$	10.5207	0.521491
$408$	0	0
$409$	31.2043	1.54295	0.771476	−	0.636258i	$-0.219519\pi$
$409$	31.2043	1.54295	0.771476	+	0.636258i	$0.219519\pi$
$410$	0	0
$411$	−17.1907	−0.847954
$412$	0	0
$413$	4.89334	0.240786
$414$	0	0
$415$	2.81242	0.138056
$416$	0	0
$417$	19.6010	0.959864
$418$	0	0
$419$	−13.2765	−0.648599	−0.324299	−	0.945955i	$-0.605129\pi$
$419$	−13.2765	−0.648599	−0.324299	+	0.945955i	$0.605129\pi$
$420$	0	0
$421$	−4.14781	−0.202152	−0.101076	−	0.994879i	$-0.532229\pi$
$421$	−4.14781	−0.202152	−0.101076	+	0.994879i	$0.532229\pi$
$422$	0	0
$423$	2.49929	0.121520
$424$	0	0
$425$	15.1869	0.736671
$426$	0	0
$427$	−9.20404	−0.445415
$428$	0	0
$429$	−2.42810	−0.117230
$430$	0	0
$431$	−34.4336	−1.65861	−0.829304	−	0.558798i	$-0.811263\pi$
$431$	−34.4336	−1.65861	−0.829304	+	0.558798i	$0.811263\pi$
$432$	0	0
$433$	−16.4496	−0.790519	−0.395260	−	0.918569i	$-0.629345\pi$
$433$	−16.4496	−0.790519	−0.395260	+	0.918569i	$0.629345\pi$
$434$	0	0
$435$	13.5530	0.649817
$436$	0	0
$437$	39.8413	1.90587
$438$	0	0
$439$	4.05232	0.193407	0.0967034	−	0.995313i	$-0.469170\pi$
$439$	4.05232	0.193407	0.0967034	+	0.995313i	$0.469170\pi$
$440$	0	0
$441$	2.89567	0.137889
$442$	0	0
$443$	26.2221	1.24585	0.622925	−	0.782282i	$-0.285944\pi$
$443$	26.2221	1.24585	0.622925	+	0.782282i	$0.285944\pi$
$444$	0	0
$445$	6.69590	0.317416
$446$	0	0
$447$	−7.38823	−0.349451
$448$	0	0
$449$	37.1904	1.75512	0.877562	−	0.479463i	$-0.159168\pi$
$449$	37.1904	1.75512	0.877562	+	0.479463i	$0.159168\pi$
$450$	0	0
$451$	−2.33249	−0.109832
$452$	0	0
$453$	2.66319	0.125128
$454$	0	0
$455$	0.790216	0.0370459
$456$	0	0
$457$	37.4522	1.75194	0.875970	−	0.482366i	$-0.160223\pi$
$457$	37.4522	1.75194	0.875970	+	0.482366i	$0.160223\pi$
$458$	0	0
$459$	0.879231	0.0410390
$460$	0	0
$461$	−21.3440	−0.994088	−0.497044	−	0.867725i	$-0.665581\pi$
$461$	−21.3440	−0.994088	−0.497044	+	0.867725i	$0.665581\pi$
$462$	0	0
$463$	2.67202	0.124179	0.0620896	−	0.998071i	$-0.480224\pi$
$463$	2.67202	0.124179	0.0620896	+	0.998071i	$0.480224\pi$
$464$	0	0
$465$	11.9467	0.554016
$466$	0	0
$467$	1.41868	0.0656485	0.0328243	−	0.999461i	$-0.489550\pi$
$467$	1.41868	0.0656485	0.0328243	+	0.999461i	$0.489550\pi$
$468$	0	0
$469$	13.1558	0.607480
$470$	0	0
$471$	−49.6052	−2.28568
$472$	0	0
$473$	−4.71705	−0.216890
$474$	0	0
$475$	−21.5442	−0.988516
$476$	0	0
$477$	8.08672	0.370266
$478$	0	0
$479$	2.04505	0.0934409	0.0467205	−	0.998908i	$-0.485123\pi$
$479$	2.04505	0.0934409	0.0467205	+	0.998908i	$0.485123\pi$
$480$	0	0
$481$	−10.5207	−0.479702
$482$	0	0
$483$	19.6473	0.893984
$484$	0	0
$485$	−12.1425	−0.551365
$486$	0	0
$487$	1.75809	0.0796668	0.0398334	−	0.999206i	$-0.487317\pi$
$487$	1.75809	0.0796668	0.0398334	+	0.999206i	$0.487317\pi$
$488$	0	0
$489$	−19.9946	−0.904185
$490$	0	0
$491$	32.9814	1.48843	0.744215	−	0.667941i	$-0.232824\pi$
$491$	32.9814	1.48843	0.744215	+	0.667941i	$0.232824\pi$
$492$	0	0
$493$	−24.5164	−1.10417
$494$	0	0
$495$	−2.28821	−0.102847
$496$	0	0
$497$	−10.3701	−0.465163
$498$	0	0
$499$	13.5678	0.607379	0.303690	−	0.952771i	$-0.401781\pi$
$499$	13.5678	0.607379	0.303690	+	0.952771i	$0.401781\pi$
$500$	0	0
$501$	−34.5036	−1.54151
$502$	0	0
$503$	−19.4411	−0.866837	−0.433419	−	0.901193i	$-0.642693\pi$
$503$	−19.4411	−0.866837	−0.433419	+	0.901193i	$0.642693\pi$
$504$	0	0
$505$	10.8735	0.483866
$506$	0	0
$507$	2.42810	0.107836
$508$	0	0
$509$	−2.30495	−0.102165	−0.0510826	−	0.998694i	$-0.516267\pi$
$509$	−2.30495	−0.102165	−0.0510826	+	0.998694i	$0.516267\pi$
$510$	0	0
$511$	2.37136	0.104903
$512$	0	0
$513$	−1.24728	−0.0550690
$514$	0	0
$515$	10.1252	0.446171
$516$	0	0
$517$	−0.863114	−0.0379597
$518$	0	0
$519$	−6.62966	−0.291010
$520$	0	0
$521$	−10.1841	−0.446175	−0.223087	−	0.974798i	$-0.571614\pi$
$521$	−10.1841	−0.446175	−0.223087	+	0.974798i	$0.571614\pi$
$522$	0	0
$523$	19.7515	0.863674	0.431837	−	0.901952i	$-0.357866\pi$
$523$	19.7515	0.863674	0.431837	+	0.901952i	$0.357866\pi$
$524$	0	0
$525$	−10.6243	−0.463682
$526$	0	0
$527$	−21.6108	−0.941381
$528$	0	0
$529$	42.4747	1.84673
$530$	0	0
$531$	14.1695	0.614905
$532$	0	0
$533$	2.33249	0.101031
$534$	0	0
$535$	−12.1186	−0.523931
$536$	0	0
$537$	−43.1004	−1.85992
$538$	0	0
$539$	−1.00000	−0.0430730
$540$	0	0
$541$	−35.8720	−1.54226	−0.771130	−	0.636678i	$-0.780308\pi$
$541$	−35.8720	−1.54226	−0.771130	+	0.636678i	$0.780308\pi$
$542$	0	0
$543$	24.2126	1.03906
$544$	0	0
$545$	8.41044	0.360264
$546$	0	0
$547$	10.0234	0.428570	0.214285	−	0.976771i	$-0.431258\pi$
$547$	10.0234	0.428570	0.214285	+	0.976771i	$0.431258\pi$
$548$	0	0
$549$	−26.6519	−1.13747
$550$	0	0
$551$	34.7792	1.48165
$552$	0	0
$553$	6.21658	0.264356
$554$	0	0
$555$	−20.1863	−0.856860
$556$	0	0
$557$	−2.23158	−0.0945550	−0.0472775	−	0.998882i	$-0.515055\pi$
$557$	−2.23158	−0.0945550	−0.0472775	+	0.998882i	$0.515055\pi$
$558$	0	0
$559$	4.71705	0.199510
$560$	0	0
$561$	8.42754	0.355811
$562$	0	0
$563$	24.3349	1.02559	0.512796	−	0.858510i	$-0.328610\pi$
$563$	24.3349	1.02559	0.512796	+	0.858510i	$0.328610\pi$
$564$	0	0
$565$	4.96064	0.208696
$566$	0	0
$567$	−9.30210	−0.390651
$568$	0	0
$569$	−18.1808	−0.762180	−0.381090	−	0.924538i	$-0.624451\pi$
$569$	−18.1808	−0.762180	−0.381090	+	0.924538i	$0.624451\pi$
$570$	0	0
$571$	−42.0366	−1.75918	−0.879588	−	0.475736i	$-0.842182\pi$
$571$	−42.0366	−1.75918	−0.879588	+	0.475736i	$0.842182\pi$
$572$	0	0
$573$	−3.37853	−0.141140
$574$	0	0
$575$	−35.4055	−1.47651
$576$	0	0
$577$	−31.9270	−1.32914	−0.664568	−	0.747227i	$-0.731385\pi$
$577$	−31.9270	−1.32914	−0.664568	+	0.747227i	$0.731385\pi$
$578$	0	0
$579$	29.4662	1.22457
$580$	0	0
$581$	3.55905	0.147654
$582$	0	0
$583$	−2.79269	−0.115662
$584$	0	0
$585$	2.28821	0.0946057
$586$	0	0
$587$	26.0965	1.07712	0.538558	−	0.842588i	$-0.318969\pi$
$587$	26.0965	1.07712	0.538558	+	0.842588i	$0.318969\pi$
$588$	0	0
$589$	30.6573	1.26321
$590$	0	0
$591$	12.4926	0.513875
$592$	0	0
$593$	32.8505	1.34901	0.674504	−	0.738271i	$-0.264358\pi$
$593$	32.8505	1.34901	0.674504	+	0.738271i	$0.264358\pi$
$594$	0	0
$595$	−2.74271	−0.112440
$596$	0	0
$597$	−13.0357	−0.533515
$598$	0	0
$599$	−32.3914	−1.32348	−0.661739	−	0.749734i	$-0.730181\pi$
$599$	−32.3914	−1.32348	−0.661739	+	0.749734i	$0.730181\pi$
$600$	0	0
$601$	−8.74440	−0.356691	−0.178346	−	0.983968i	$-0.557075\pi$
$601$	−8.74440	−0.356691	−0.178346	+	0.983968i	$0.557075\pi$
$602$	0	0
$603$	38.0950	1.55135
$604$	0	0
$605$	0.790216	0.0321269
$606$	0	0
$607$	−34.0477	−1.38195	−0.690976	−	0.722878i	$-0.742819\pi$
$607$	−34.0477	−1.38195	−0.690976	+	0.722878i	$0.742819\pi$
$608$	0	0
$609$	17.1510	0.694994
$610$	0	0
$611$	0.863114	0.0349179
$612$	0	0
$613$	0.883395	0.0356800	0.0178400	−	0.999841i	$-0.494321\pi$
$613$	0.883395	0.0356800	0.0178400	+	0.999841i	$0.494321\pi$
$614$	0	0
$615$	4.47540	0.180465
$616$	0	0
$617$	42.6116	1.71548	0.857739	−	0.514085i	$-0.171868\pi$
$617$	42.6116	1.71548	0.857739	+	0.514085i	$0.171868\pi$
$618$	0	0
$619$	−5.70252	−0.229204	−0.114602	−	0.993412i	$-0.536559\pi$
$619$	−5.70252	−0.229204	−0.114602	+	0.993412i	$0.536559\pi$
$620$	0	0
$621$	−2.04977	−0.0822545
$622$	0	0
$623$	8.47351	0.339484
$624$	0	0
$625$	16.0233	0.640932
$626$	0	0
$627$	−11.9554	−0.477452
$628$	0	0
$629$	36.5156	1.45597
$630$	0	0
$631$	−30.7724	−1.22503	−0.612515	−	0.790459i	$-0.709842\pi$
$631$	−30.7724	−1.22503	−0.612515	+	0.790459i	$0.709842\pi$
$632$	0	0
$633$	40.9528	1.62773
$634$	0	0
$635$	0.681381	0.0270398
$636$	0	0
$637$	1.00000	0.0396214
$638$	0	0
$639$	−30.0284	−1.18791
$640$	0	0
$641$	36.8093	1.45388	0.726940	−	0.686701i	$-0.240942\pi$
$641$	36.8093	1.45388	0.726940	+	0.686701i	$0.240942\pi$
$642$	0	0
$643$	22.8486	0.901061	0.450531	−	0.892761i	$-0.351235\pi$
$643$	22.8486	0.901061	0.450531	+	0.892761i	$0.351235\pi$
$644$	0	0
$645$	9.05072	0.356372
$646$	0	0
$647$	13.6860	0.538052	0.269026	−	0.963133i	$-0.413298\pi$
$647$	13.6860	0.538052	0.269026	+	0.963133i	$0.413298\pi$
$648$	0	0
$649$	−4.89334	−0.192081
$650$	0	0
$651$	15.1183	0.592533
$652$	0	0
$653$	15.6707	0.613243	0.306621	−	0.951832i	$-0.400801\pi$
$653$	15.6707	0.613243	0.306621	+	0.951832i	$0.400801\pi$
$654$	0	0
$655$	−11.5098	−0.449726
$656$	0	0
$657$	6.86669	0.267895
$658$	0	0
$659$	−41.1067	−1.60129	−0.800646	−	0.599138i	$-0.795510\pi$
$659$	−41.1067	−1.60129	−0.800646	+	0.599138i	$0.795510\pi$
$660$	0	0
$661$	14.9682	0.582194	0.291097	−	0.956694i	$-0.405980\pi$
$661$	14.9682	0.582194	0.291097	+	0.956694i	$0.405980\pi$
$662$	0	0
$663$	−8.42754	−0.327299
$664$	0	0
$665$	3.89084	0.150880
$666$	0	0
$667$	57.1557	2.21308
$668$	0	0
$669$	−20.0961	−0.776960
$670$	0	0
$671$	9.20404	0.355318
$672$	0	0
$673$	5.70464	0.219898	0.109949	−	0.993937i	$-0.464931\pi$
$673$	5.70464	0.219898	0.109949	+	0.993937i	$0.464931\pi$
$674$	0	0
$675$	1.10841	0.0426629
$676$	0	0
$677$	−16.8491	−0.647563	−0.323781	−	0.946132i	$-0.604954\pi$
$677$	−16.8491	−0.647563	−0.323781	+	0.946132i	$0.604954\pi$
$678$	0	0
$679$	−15.3661	−0.589697
$680$	0	0
$681$	−0.171699	−0.00657952
$682$	0	0
$683$	−30.1879	−1.15511	−0.577554	−	0.816352i	$-0.695993\pi$
$683$	−30.1879	−1.15511	−0.577554	+	0.816352i	$0.695993\pi$
$684$	0	0
$685$	−5.59464	−0.213760
$686$	0	0
$687$	8.40307	0.320597
$688$	0	0
$689$	2.79269	0.106393
$690$	0	0
$691$	20.5629	0.782248	0.391124	−	0.920338i	$-0.372086\pi$
$691$	20.5629	0.782248	0.391124	+	0.920338i	$0.372086\pi$
$692$	0	0
$693$	−2.89567	−0.109997
$694$	0	0
$695$	6.37906	0.241972
$696$	0	0
$697$	−8.09568	−0.306646
$698$	0	0
$699$	−50.9223	−1.92606
$700$	0	0
$701$	−16.1746	−0.610906	−0.305453	−	0.952207i	$-0.598808\pi$
$701$	−16.1746	−0.610906	−0.305453	+	0.952207i	$0.598808\pi$
$702$	0	0
$703$	−51.8013	−1.95372
$704$	0	0
$705$	1.65608	0.0623715
$706$	0	0
$707$	13.7602	0.517506
$708$	0	0
$709$	−39.7731	−1.49371	−0.746855	−	0.664987i	$-0.768437\pi$
$709$	−39.7731	−1.49371	−0.746855	+	0.664987i	$0.768437\pi$
$710$	0	0
$711$	18.0012	0.675097
$712$	0	0
$713$	50.3817	1.88681
$714$	0	0
$715$	−0.790216	−0.0295524
$716$	0	0
$717$	62.0043	2.31559
$718$	0	0
$719$	−45.0871	−1.68147	−0.840733	−	0.541450i	$-0.817876\pi$
$719$	−45.0871	−1.68147	−0.840733	+	0.541450i	$0.817876\pi$
$720$	0	0
$721$	12.8132	0.477190
$722$	0	0
$723$	8.40272	0.312501
$724$	0	0
$725$	−30.9070	−1.14786
$726$	0	0
$727$	−31.8441	−1.18103	−0.590516	−	0.807026i	$-0.701076\pi$
$727$	−31.8441	−1.18103	−0.590516	+	0.807026i	$0.701076\pi$
$728$	0	0
$729$	−25.0906	−0.929281
$730$	0	0
$731$	−16.3721	−0.605545
$732$	0	0
$733$	21.5324	0.795316	0.397658	−	0.917534i	$-0.369823\pi$
$733$	21.5324	0.795316	0.397658	+	0.917534i	$0.369823\pi$
$734$	0	0
$735$	1.91872	0.0707732
$736$	0	0
$737$	−13.1558	−0.484601
$738$	0	0
$739$	−10.8476	−0.399036	−0.199518	−	0.979894i	$-0.563938\pi$
$739$	−10.8476	−0.399036	−0.199518	+	0.979894i	$0.563938\pi$
$740$	0	0
$741$	11.9554	0.439192
$742$	0	0
$743$	−12.8169	−0.470205	−0.235103	−	0.971971i	$-0.575543\pi$
$743$	−12.8169	−0.470205	−0.235103	+	0.971971i	$0.575543\pi$
$744$	0	0
$745$	−2.40447	−0.0880930
$746$	0	0
$747$	10.3058	0.377071
$748$	0	0
$749$	−15.3358	−0.560356
$750$	0	0
$751$	45.3121	1.65346	0.826730	−	0.562599i	$-0.190198\pi$
$751$	45.3121	1.65346	0.826730	+	0.562599i	$0.190198\pi$
$752$	0	0
$753$	11.7622	0.428640
$754$	0	0
$755$	0.866726	0.0315434
$756$	0	0
$757$	28.7389	1.04453	0.522267	−	0.852782i	$-0.325086\pi$
$757$	28.7389	1.04453	0.522267	+	0.852782i	$0.325086\pi$
$758$	0	0
$759$	−19.6473	−0.713153
$760$	0	0
$761$	−1.45412	−0.0527117	−0.0263559	−	0.999653i	$-0.508390\pi$
$761$	−1.45412	−0.0527117	−0.0263559	+	0.999653i	$0.508390\pi$
$762$	0	0
$763$	10.6432	0.385310
$764$	0	0
$765$	−7.94199	−0.287143
$766$	0	0
$767$	4.89334	0.176688
$768$	0	0
$769$	23.2461	0.838277	0.419139	−	0.907922i	$-0.362332\pi$
$769$	23.2461	0.838277	0.419139	+	0.907922i	$0.362332\pi$
$770$	0	0
$771$	−32.0097	−1.15280
$772$	0	0
$773$	24.9907	0.898853	0.449426	−	0.893317i	$-0.351628\pi$
$773$	24.9907	0.898853	0.449426	+	0.893317i	$0.351628\pi$
$774$	0	0
$775$	−27.2439	−0.978631
$776$	0	0
$777$	−25.5453	−0.916431
$778$	0	0
$779$	11.4846	0.411479
$780$	0	0
$781$	10.3701	0.371072
$782$	0	0
$783$	−1.78934	−0.0639456
$784$	0	0
$785$	−16.1438	−0.576197
$786$	0	0
$787$	30.8670	1.10029	0.550146	−	0.835069i	$-0.314572\pi$
$787$	30.8670	1.10029	0.550146	+	0.835069i	$0.314572\pi$
$788$	0	0
$789$	−46.4051	−1.65206
$790$	0	0
$791$	6.27757	0.223205
$792$	0	0
$793$	−9.20404	−0.326845
$794$	0	0
$795$	5.35841	0.190043
$796$	0	0
$797$	−24.5225	−0.868631	−0.434316	−	0.900761i	$-0.643010\pi$
$797$	−24.5225	−0.868631	−0.434316	+	0.900761i	$0.643010\pi$
$798$	0	0
$799$	−2.99573	−0.105981
$800$	0	0
$801$	24.5365	0.866955
$802$	0	0
$803$	−2.37136	−0.0836836
$804$	0	0
$805$	6.39415	0.225364
$806$	0	0
$807$	−58.9113	−2.07378
$808$	0	0
$809$	−22.3723	−0.786567	−0.393284	−	0.919417i	$-0.628661\pi$
$809$	−22.3723	−0.786567	−0.393284	+	0.919417i	$0.628661\pi$
$810$	0	0
$811$	13.1022	0.460079	0.230039	−	0.973181i	$-0.426114\pi$
$811$	13.1022	0.460079	0.230039	+	0.973181i	$0.426114\pi$
$812$	0	0
$813$	12.1715	0.426874
$814$	0	0
$815$	−6.50715	−0.227936
$816$	0	0
$817$	23.2256	0.812563
$818$	0	0
$819$	2.89567	0.101183
$820$	0	0
$821$	6.71331	0.234296	0.117148	−	0.993114i	$-0.462625\pi$
$821$	6.71331	0.234296	0.117148	+	0.993114i	$0.462625\pi$
$822$	0	0
$823$	−33.1535	−1.15566	−0.577830	−	0.816157i	$-0.696100\pi$
$823$	−33.1535	−1.15566	−0.577830	+	0.816157i	$0.696100\pi$
$824$	0	0
$825$	10.6243	0.369890
$826$	0	0
$827$	−28.6093	−0.994844	−0.497422	−	0.867509i	$-0.665720\pi$
$827$	−28.6093	−0.994844	−0.497422	+	0.867509i	$0.665720\pi$
$828$	0	0
$829$	7.32042	0.254249	0.127124	−	0.991887i	$-0.459425\pi$
$829$	7.32042	0.254249	0.127124	+	0.991887i	$0.459425\pi$
$830$	0	0
$831$	−51.5386	−1.78786
$832$	0	0
$833$	−3.47084	−0.120257
$834$	0	0
$835$	−11.2291	−0.388598
$836$	0	0
$837$	−1.57727	−0.0545183
$838$	0	0
$839$	18.7370	0.646872	0.323436	−	0.946250i	$-0.395162\pi$
$839$	18.7370	0.646872	0.323436	+	0.946250i	$0.395162\pi$
$840$	0	0
$841$	20.8938	0.720475
$842$	0	0
$843$	15.9532	0.549458
$844$	0	0
$845$	0.790216	0.0271843
$846$	0	0
$847$	1.00000	0.0343604
$848$	0	0
$849$	30.6737	1.05272
$850$	0	0
$851$	−85.1295	−2.91820
$852$	0	0
$853$	−23.3290	−0.798769	−0.399385	−	0.916783i	$-0.630776\pi$
$853$	−23.3290	−0.798769	−0.399385	+	0.916783i	$0.630776\pi$
$854$	0	0
$855$	11.2666	0.385309
$856$	0	0
$857$	−20.6420	−0.705116	−0.352558	−	0.935790i	$-0.614688\pi$
$857$	−20.6420	−0.705116	−0.352558	+	0.935790i	$0.614688\pi$
$858$	0	0
$859$	9.67336	0.330051	0.165025	−	0.986289i	$-0.447229\pi$
$859$	9.67336	0.330051	0.165025	+	0.986289i	$0.447229\pi$
$860$	0	0
$861$	5.66351	0.193012
$862$	0	0
$863$	41.9379	1.42758	0.713791	−	0.700359i	$-0.246977\pi$
$863$	41.9379	1.42758	0.713791	+	0.700359i	$0.246977\pi$
$864$	0	0
$865$	−2.15760	−0.0733605
$866$	0	0
$867$	−12.0271	−0.408461
$868$	0	0
$869$	−6.21658	−0.210883
$870$	0	0
$871$	13.1558	0.445768
$872$	0	0
$873$	−44.4952	−1.50593
$874$	0	0
$875$	−7.40872	−0.250460
$876$	0	0
$877$	8.62255	0.291163	0.145581	−	0.989346i	$-0.453495\pi$
$877$	8.62255	0.291163	0.145581	+	0.989346i	$0.453495\pi$
$878$	0	0
$879$	17.4197	0.587553
$880$	0	0
$881$	22.3895	0.754321	0.377160	−	0.926148i	$-0.376901\pi$
$881$	22.3895	0.754321	0.377160	+	0.926148i	$0.376901\pi$
$882$	0	0
$883$	−30.8408	−1.03787	−0.518937	−	0.854812i	$-0.673672\pi$
$883$	−30.8408	−1.03787	−0.518937	+	0.854812i	$0.673672\pi$
$884$	0	0
$885$	9.38897	0.315607
$886$	0	0
$887$	−30.7469	−1.03238	−0.516189	−	0.856474i	$-0.672650\pi$
$887$	−30.7469	−1.03238	−0.516189	+	0.856474i	$0.672650\pi$
$888$	0	0
$889$	0.862272	0.0289197
$890$	0	0
$891$	9.30210	0.311632
$892$	0	0
$893$	4.24977	0.142213
$894$	0	0
$895$	−14.0269	−0.468867
$896$	0	0
$897$	19.6473	0.656005
$898$	0	0
$899$	43.9804	1.46683
$900$	0	0
$901$	−9.69299	−0.322920
$902$	0	0
$903$	11.4535	0.381148
$904$	0	0
$905$	7.87990	0.261937
$906$	0	0
$907$	14.7892	0.491066	0.245533	−	0.969388i	$-0.421037\pi$
$907$	14.7892	0.491066	0.245533	+	0.969388i	$0.421037\pi$
$908$	0	0
$909$	39.8451	1.32158
$910$	0	0
$911$	−25.2715	−0.837281	−0.418641	−	0.908152i	$-0.637493\pi$
$911$	−25.2715	−0.837281	−0.418641	+	0.908152i	$0.637493\pi$
$912$	0	0
$913$	−3.55905	−0.117787
$914$	0	0
$915$	−17.6600	−0.583822
$916$	0	0
$917$	−14.5654	−0.480992
$918$	0	0
$919$	−7.69677	−0.253893	−0.126947	−	0.991910i	$-0.540518\pi$
$919$	−7.69677	−0.253893	−0.126947	+	0.991910i	$0.540518\pi$
$920$	0	0
$921$	3.98394	0.131275
$922$	0	0
$923$	−10.3701	−0.341336
$924$	0	0
$925$	46.0338	1.51358
$926$	0	0
$927$	37.1029	1.21862
$928$	0	0
$929$	−5.05070	−0.165708	−0.0828540	−	0.996562i	$-0.526404\pi$
$929$	−5.05070	−0.165708	−0.0828540	+	0.996562i	$0.526404\pi$
$930$	0	0
$931$	4.92376	0.161370
$932$	0	0
$933$	11.6086	0.380050
$934$	0	0
$935$	2.74271	0.0896963
$936$	0	0
$937$	36.9778	1.20801	0.604006	−	0.796980i	$-0.293570\pi$
$937$	36.9778	1.20801	0.604006	+	0.796980i	$0.293570\pi$
$938$	0	0
$939$	−49.9756	−1.63089
$940$	0	0
$941$	−35.2610	−1.14948	−0.574738	−	0.818338i	$-0.694896\pi$
$941$	−35.2610	−1.14948	−0.574738	+	0.818338i	$0.694896\pi$
$942$	0	0
$943$	18.8736	0.614610
$944$	0	0
$945$	−0.200177	−0.00651176
$946$	0	0
$947$	−39.2327	−1.27489	−0.637445	−	0.770496i	$-0.720009\pi$
$947$	−39.2327	−1.27489	−0.637445	+	0.770496i	$0.720009\pi$
$948$	0	0
$949$	2.37136	0.0769777
$950$	0	0
$951$	−36.3664	−1.17926
$952$	0	0
$953$	10.8083	0.350116	0.175058	−	0.984558i	$-0.443989\pi$
$953$	10.8083	0.350116	0.175058	+	0.984558i	$0.443989\pi$
$954$	0	0
$955$	−1.09953	−0.0355800
$956$	0	0
$957$	−17.1510	−0.554414
$958$	0	0
$959$	−7.07989	−0.228622
$960$	0	0
$961$	7.76793	0.250578
$962$	0	0
$963$	−44.4073	−1.43101
$964$	0	0
$965$	9.58965	0.308702
$966$	0	0
$967$	0.779739	0.0250747	0.0125374	−	0.999921i	$-0.496009\pi$
$967$	0.779739	0.0250747	0.0125374	+	0.999921i	$0.496009\pi$
$968$	0	0
$969$	−41.4952	−1.33302
$970$	0	0
$971$	22.2074	0.712671	0.356335	−	0.934358i	$-0.384026\pi$
$971$	22.2074	0.712671	0.356335	+	0.934358i	$0.384026\pi$
$972$	0	0
$973$	8.07256	0.258794
$974$	0	0
$975$	−10.6243	−0.340250
$976$	0	0
$977$	3.39342	0.108565	0.0542825	−	0.998526i	$-0.482713\pi$
$977$	3.39342	0.108565	0.0542825	+	0.998526i	$0.482713\pi$
$978$	0	0
$979$	−8.47351	−0.270815
$980$	0	0
$981$	30.8193	0.983984
$982$	0	0
$983$	−30.7605	−0.981108	−0.490554	−	0.871411i	$-0.663205\pi$
$983$	−30.7605	−0.981108	−0.490554	+	0.871411i	$0.663205\pi$
$984$	0	0
$985$	4.06566	0.129543
$986$	0	0
$987$	2.09573	0.0667077
$988$	0	0
$989$	38.1687	1.21369
$990$	0	0
$991$	−16.8376	−0.534863	−0.267431	−	0.963577i	$-0.586175\pi$
$991$	−16.8376	−0.534863	−0.267431	+	0.963577i	$0.586175\pi$
$992$	0	0
$993$	56.3194	1.78724
$994$	0	0
$995$	−4.24242	−0.134494
$996$	0	0
$997$	−12.0688	−0.382222	−0.191111	−	0.981568i	$-0.561209\pi$
$997$	−12.0688	−0.382222	−0.191111	+	0.981568i	$0.561209\pi$
$998$	0	0
$999$	2.66509	0.0843198

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	4004.2.a.i.1.8	✓	9

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
4004.2.a.i.1.8	✓	9	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:	\( N \)	\(=\)	\( 4004 = 2^{2} \cdot 7 \cdot 11 \cdot 13 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	4004.a (trivial)

\(f(q)\)	\(=\)	\(q+2.42810 q^{3} +0.790216 q^{5} +1.00000 q^{7} +2.89567 q^{9} +O(q^{10})\)	\(q+2.42810 q^{3} +0.790216 q^{5} +1.00000 q^{7} +2.89567 q^{9} -1.00000 q^{11} +1.00000 q^{13} +1.91872 q^{15} -3.47084 q^{17} +4.92376 q^{19} +2.42810 q^{21} +8.09164 q^{23} -4.37556 q^{25} -0.253320 q^{27} +7.06355 q^{29} +6.22639 q^{31} -2.42810 q^{33} +0.790216 q^{35} -10.5207 q^{37} +2.42810 q^{39} +2.33249 q^{41} +4.71705 q^{43} +2.28821 q^{45} +0.863114 q^{47} +1.00000 q^{49} -8.42754 q^{51} +2.79269 q^{53} -0.790216 q^{55} +11.9554 q^{57} +4.89334 q^{59} -9.20404 q^{61} +2.89567 q^{63} +0.790216 q^{65} +13.1558 q^{67} +19.6473 q^{69} -10.3701 q^{71} +2.37136 q^{73} -10.6243 q^{75} -1.00000 q^{77} +6.21658 q^{79} -9.30210 q^{81} +3.55905 q^{83} -2.74271 q^{85} +17.1510 q^{87} +8.47351 q^{89} +1.00000 q^{91} +15.1183 q^{93} +3.89084 q^{95} -15.3661 q^{97} -2.89567 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 9 q + 4 q^{3} + 4 q^{5} + 9 q^{7} + 13 q^{9}+O(q^{10}) \) 9 * q + 4 * q^3 + 4 * q^5 + 9 * q^7 + 13 * q^9	\( 9 q + 4 q^{3} + 4 q^{5} + 9 q^{7} + 13 q^{9} - 9 q^{11} + 9 q^{13} - 5 q^{17} + 17 q^{19} + 4 q^{21} - 8 q^{23} + 35 q^{25} + 4 q^{27} - 3 q^{29} + 9 q^{31} - 4 q^{33} + 4 q^{35} + 7 q^{37} + 4 q^{39} + 14 q^{41} + 21 q^{43} + 43 q^{45} + q^{47} + 9 q^{49} + 25 q^{51} - 8 q^{53} - 4 q^{55} + 8 q^{57} - 4 q^{59} + 30 q^{61} + 13 q^{63} + 4 q^{65} + 15 q^{67} + 9 q^{69} - q^{71} + 18 q^{73} - 32 q^{75} - 9 q^{77} + 13 q^{79} + 13 q^{81} + 2 q^{83} + 45 q^{85} + 21 q^{87} + 9 q^{91} - 2 q^{93} + 11 q^{95} + 29 q^{97} - 13 q^{99}+O(q^{100}) \) 9 * q + 4 * q^3 + 4 * q^5 + 9 * q^7 + 13 * q^9 - 9 * q^11 + 9 * q^13 - 5 * q^17 + 17 * q^19 + 4 * q^21 - 8 * q^23 + 35 * q^25 + 4 * q^27 - 3 * q^29 + 9 * q^31 - 4 * q^33 + 4 * q^35 + 7 * q^37 + 4 * q^39 + 14 * q^41 + 21 * q^43 + 43 * q^45 + q^47 + 9 * q^49 + 25 * q^51 - 8 * q^53 - 4 * q^55 + 8 * q^57 - 4 * q^59 + 30 * q^61 + 13 * q^63 + 4 * q^65 + 15 * q^67 + 9 * q^69 - q^71 + 18 * q^73 - 32 * q^75 - 9 * q^77 + 13 * q^79 + 13 * q^81 + 2 * q^83 + 45 * q^85 + 21 * q^87 + 9 * q^91 - 2 * q^93 + 11 * q^95 + 29 * q^97 - 13 * q^99

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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