LMFDB - Embedded newform 4008.2.a.g.1.3

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [4008,2,Mod(1,4008)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(4008, 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("4008.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 4008 = 2^{3} \cdot 3 \cdot 167 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	4008.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:	$32.0040411301$
Analytic rank:	$0$
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} - 2x^{6} - 7x^{5} + 11x^{4} + 12x^{3} - 14x^{2} - 6x + 3 $ x^7 - 2x^6 - 7x^5 + 11x^4 + 12x^3 - 14x^2 - 6x + 3
Coefficient ring:	$\Z[a_1, \ldots, a_{7}]$
Coefficient ring index:	$ 2 $
Twist minimal:	yes
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.3
Root			$-0.674271$ of defining polynomial
Character	$\chi$	$=$	4008.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-1.00000 q^{3} -1.68509 q^{5} -2.23045 q^{7} +1.00000 q^{9} +O(q^{10})$	$q-1.00000 q^{3} -1.68509 q^{5} -2.23045 q^{7} +1.00000 q^{9} -3.03364 q^{11} -4.55392 q^{13} +1.68509 q^{15} -2.98814 q^{17} -2.14913 q^{19} +2.23045 q^{21} +1.24231 q^{23} -2.16046 q^{25} -1.00000 q^{27} -6.48215 q^{29} +0.905758 q^{31} +3.03364 q^{33} +3.75852 q^{35} +5.19406 q^{37} +4.55392 q^{39} +0.662614 q^{41} -10.7062 q^{43} -1.68509 q^{45} +6.58538 q^{47} -2.02508 q^{49} +2.98814 q^{51} -8.56261 q^{53} +5.11196 q^{55} +2.14913 q^{57} -0.576526 q^{59} -9.76943 q^{61} -2.23045 q^{63} +7.67378 q^{65} +4.66641 q^{67} -1.24231 q^{69} +2.31288 q^{71} +13.5937 q^{73} +2.16046 q^{75} +6.76638 q^{77} +6.77552 q^{79} +1.00000 q^{81} +11.3095 q^{83} +5.03530 q^{85} +6.48215 q^{87} -1.55582 q^{89} +10.1573 q^{91} -0.905758 q^{93} +3.62148 q^{95} -12.8855 q^{97} -3.03364 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 7 q - 7 q^{3} - 3 q^{5} + 8 q^{7} + 7 q^{9}+O(q^{10}) $ 7 * q - 7 * q^3 - 3 * q^5 + 8 * q^7 + 7 * q^9	$ 7 q - 7 q^{3} - 3 q^{5} + 8 q^{7} + 7 q^{9} + q^{11} - 2 q^{13} + 3 q^{15} + 11 q^{17} + 2 q^{19} - 8 q^{21} + 17 q^{23} + 4 q^{25} - 7 q^{27} - 7 q^{29} + 10 q^{31} - q^{33} + 10 q^{35} - 21 q^{37} + 2 q^{39} + 8 q^{41} - 12 q^{43} - 3 q^{45} + 25 q^{47} - 7 q^{49} - 11 q^{51} - 7 q^{53} + 15 q^{55} - 2 q^{57} + 3 q^{59} - 14 q^{61} + 8 q^{63} + 4 q^{65} + 4 q^{67} - 17 q^{69} + 27 q^{71} - 12 q^{73} - 4 q^{75} + 16 q^{77} + 8 q^{79} + 7 q^{81} + 15 q^{83} - 3 q^{85} + 7 q^{87} + 14 q^{89} - 3 q^{91} - 10 q^{93} + 37 q^{95} + 3 q^{97} + q^{99}+O(q^{100}) $ 7 * q - 7 * q^3 - 3 * q^5 + 8 * q^7 + 7 * q^9 + q^11 - 2 * q^13 + 3 * q^15 + 11 * q^17 + 2 * q^19 - 8 * q^21 + 17 * q^23 + 4 * q^25 - 7 * q^27 - 7 * q^29 + 10 * q^31 - q^33 + 10 * q^35 - 21 * q^37 + 2 * q^39 + 8 * q^41 - 12 * q^43 - 3 * q^45 + 25 * q^47 - 7 * q^49 - 11 * q^51 - 7 * q^53 + 15 * q^55 - 2 * q^57 + 3 * q^59 - 14 * q^61 + 8 * q^63 + 4 * q^65 + 4 * q^67 - 17 * q^69 + 27 * q^71 - 12 * q^73 - 4 * q^75 + 16 * q^77 + 8 * q^79 + 7 * q^81 + 15 * q^83 - 3 * q^85 + 7 * q^87 + 14 * q^89 - 3 * q^91 - 10 * q^93 + 37 * q^95 + 3 * 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
$5$	−1.68509	−0.753597	−0.376799	−	0.926295i	$-0.622975\pi$
$5$	−1.68509	−0.753597	−0.376799	+	0.926295i	$0.622975\pi$
$6$	0	0
$7$	−2.23045	−0.843032	−0.421516	−	0.906821i	$-0.638502\pi$
$7$	−2.23045	−0.843032	−0.421516	+	0.906821i	$0.638502\pi$
$8$	0	0
$9$	1.00000	0.333333
$10$	0	0
$11$	−3.03364	−0.914676	−0.457338	−	0.889293i	$-0.651197\pi$
$11$	−3.03364	−0.914676	−0.457338	+	0.889293i	$0.651197\pi$
$12$	0	0
$13$	−4.55392	−1.26303	−0.631514	−	0.775364i	$-0.717566\pi$
$13$	−4.55392	−1.26303	−0.631514	+	0.775364i	$0.717566\pi$
$14$	0	0
$15$	1.68509	0.435089
$16$	0	0
$17$	−2.98814	−0.724731	−0.362366	−	0.932036i	$-0.618031\pi$
$17$	−2.98814	−0.724731	−0.362366	+	0.932036i	$0.618031\pi$
$18$	0	0
$19$	−2.14913	−0.493043	−0.246522	−	0.969137i	$-0.579288\pi$
$19$	−2.14913	−0.493043	−0.246522	+	0.969137i	$0.579288\pi$
$20$	0	0
$21$	2.23045	0.486725
$22$	0	0
$23$	1.24231	0.259039	0.129520	−	0.991577i	$-0.458656\pi$
$23$	1.24231	0.259039	0.129520	+	0.991577i	$0.458656\pi$
$24$	0	0
$25$	−2.16046	−0.432091
$26$	0	0
$27$	−1.00000	−0.192450
$28$	0	0
$29$	−6.48215	−1.20371	−0.601853	−	0.798607i	$-0.705571\pi$
$29$	−6.48215	−1.20371	−0.601853	+	0.798607i	$0.705571\pi$
$30$	0	0
$31$	0.905758	0.162679	0.0813394	−	0.996686i	$-0.474080\pi$
$31$	0.905758	0.162679	0.0813394	+	0.996686i	$0.474080\pi$
$32$	0	0
$33$	3.03364	0.528088
$34$	0	0
$35$	3.75852	0.635306
$36$	0	0
$37$	5.19406	0.853897	0.426949	−	0.904276i	$-0.359588\pi$
$37$	5.19406	0.853897	0.426949	+	0.904276i	$0.359588\pi$
$38$	0	0
$39$	4.55392	0.729210
$40$	0	0
$41$	0.662614	0.103483	0.0517415	−	0.998661i	$-0.483523\pi$
$41$	0.662614	0.103483	0.0517415	+	0.998661i	$0.483523\pi$
$42$	0	0
$43$	−10.7062	−1.63268	−0.816338	−	0.577574i	$-0.804000\pi$
$43$	−10.7062	−1.63268	−0.816338	+	0.577574i	$0.804000\pi$
$44$	0	0
$45$	−1.68509	−0.251199
$46$	0	0
$47$	6.58538	0.960576	0.480288	−	0.877111i	$-0.340532\pi$
$47$	6.58538	0.960576	0.480288	+	0.877111i	$0.340532\pi$
$48$	0	0
$49$	−2.02508	−0.289297
$50$	0	0
$51$	2.98814	0.418424
$52$	0	0
$53$	−8.56261	−1.17616	−0.588082	−	0.808801i	$-0.700117\pi$
$53$	−8.56261	−1.17616	−0.588082	+	0.808801i	$0.700117\pi$
$54$	0	0
$55$	5.11196	0.689297
$56$	0	0
$57$	2.14913	0.284659
$58$	0	0
$59$	−0.576526	−0.0750573	−0.0375286	−	0.999296i	$-0.511949\pi$
$59$	−0.576526	−0.0750573	−0.0375286	+	0.999296i	$0.511949\pi$
$60$	0	0
$61$	−9.76943	−1.25085	−0.625424	−	0.780285i	$-0.715074\pi$
$61$	−9.76943	−1.25085	−0.625424	+	0.780285i	$0.715074\pi$
$62$	0	0
$63$	−2.23045	−0.281011
$64$	0	0
$65$	7.67378	0.951815
$66$	0	0
$67$	4.66641	0.570093	0.285046	−	0.958514i	$-0.407991\pi$
$67$	4.66641	0.570093	0.285046	+	0.958514i	$0.407991\pi$
$68$	0	0
$69$	−1.24231	−0.149557
$70$	0	0
$71$	2.31288	0.274489	0.137244	−	0.990537i	$-0.456175\pi$
$71$	2.31288	0.274489	0.137244	+	0.990537i	$0.456175\pi$
$72$	0	0
$73$	13.5937	1.59102	0.795511	−	0.605939i	$-0.207203\pi$
$73$	13.5937	1.59102	0.795511	+	0.605939i	$0.207203\pi$
$74$	0	0
$75$	2.16046	0.249468
$76$	0	0
$77$	6.76638	0.771101
$78$	0	0
$79$	6.77552	0.762305	0.381153	−	0.924512i	$-0.375527\pi$
$79$	6.77552	0.762305	0.381153	+	0.924512i	$0.375527\pi$
$80$	0	0
$81$	1.00000	0.111111
$82$	0	0
$83$	11.3095	1.24138	0.620692	−	0.784054i	$-0.286852\pi$
$83$	11.3095	1.24138	0.620692	+	0.784054i	$0.286852\pi$
$84$	0	0
$85$	5.03530	0.546155
$86$	0	0
$87$	6.48215	0.694960
$88$	0	0
$89$	−1.55582	−0.164916	−0.0824582	−	0.996595i	$-0.526277\pi$
$89$	−1.55582	−0.164916	−0.0824582	+	0.996595i	$0.526277\pi$
$90$	0	0
$91$	10.1573	1.06477
$92$	0	0
$93$	−0.905758	−0.0939227
$94$	0	0
$95$	3.62148	0.371556
$96$	0	0
$97$	−12.8855	−1.30833	−0.654164	−	0.756352i	$-0.726980\pi$
$97$	−12.8855	−1.30833	−0.654164	+	0.756352i	$0.726980\pi$
$98$	0	0
$99$	−3.03364	−0.304892
$100$	0	0
$101$	3.21883	0.320285	0.160143	−	0.987094i	$-0.448805\pi$
$101$	3.21883	0.320285	0.160143	+	0.987094i	$0.448805\pi$
$102$	0	0
$103$	−12.7950	−1.26073	−0.630365	−	0.776299i	$-0.717095\pi$
$103$	−12.7950	−1.26073	−0.630365	+	0.776299i	$0.717095\pi$
$104$	0	0
$105$	−3.75852	−0.366794
$106$	0	0
$107$	−5.79714	−0.560431	−0.280216	−	0.959937i	$-0.590406\pi$
$107$	−5.79714	−0.560431	−0.280216	+	0.959937i	$0.590406\pi$
$108$	0	0
$109$	−11.0690	−1.06021	−0.530107	−	0.847931i	$-0.677848\pi$
$109$	−11.0690	−1.06021	−0.530107	+	0.847931i	$0.677848\pi$
$110$	0	0
$111$	−5.19406	−0.492998
$112$	0	0
$113$	−12.1323	−1.14131	−0.570654	−	0.821191i	$-0.693310\pi$
$113$	−12.1323	−1.14131	−0.570654	+	0.821191i	$0.693310\pi$
$114$	0	0
$115$	−2.09341	−0.195211
$116$	0	0
$117$	−4.55392	−0.421010
$118$	0	0
$119$	6.66491	0.610972
$120$	0	0
$121$	−1.79705	−0.163368
$122$	0	0
$123$	−0.662614	−0.0597459
$124$	0	0
$125$	12.0660	1.07922
$126$	0	0
$127$	16.7289	1.48445	0.742226	−	0.670150i	$-0.233770\pi$
$127$	16.7289	1.48445	0.742226	+	0.670150i	$0.233770\pi$
$128$	0	0
$129$	10.7062	0.942626
$130$	0	0
$131$	9.52434	0.832145	0.416073	−	0.909331i	$-0.363406\pi$
$131$	9.52434	0.832145	0.416073	+	0.909331i	$0.363406\pi$
$132$	0	0
$133$	4.79352	0.415651
$134$	0	0
$135$	1.68509	0.145030
$136$	0	0
$137$	11.1055	0.948810	0.474405	−	0.880307i	$-0.342663\pi$
$137$	11.1055	0.948810	0.474405	+	0.880307i	$0.342663\pi$
$138$	0	0
$139$	−3.18986	−0.270561	−0.135280	−	0.990807i	$-0.543193\pi$
$139$	−3.18986	−0.270561	−0.135280	+	0.990807i	$0.543193\pi$
$140$	0	0
$141$	−6.58538	−0.554589
$142$	0	0
$143$	13.8149	1.15526
$144$	0	0
$145$	10.9230	0.907109
$146$	0	0
$147$	2.02508	0.167026
$148$	0	0
$149$	8.87024	0.726678	0.363339	−	0.931657i	$-0.381637\pi$
$149$	8.87024	0.726678	0.363339	+	0.931657i	$0.381637\pi$
$150$	0	0
$151$	−1.14248	−0.0929738	−0.0464869	−	0.998919i	$-0.514803\pi$
$151$	−1.14248	−0.0929738	−0.0464869	+	0.998919i	$0.514803\pi$
$152$	0	0
$153$	−2.98814	−0.241577
$154$	0	0
$155$	−1.52629	−0.122594
$156$	0	0
$157$	9.53736	0.761164	0.380582	−	0.924747i	$-0.375724\pi$
$157$	9.53736	0.761164	0.380582	+	0.924747i	$0.375724\pi$
$158$	0	0
$159$	8.56261	0.679059
$160$	0	0
$161$	−2.77091	−0.218379
$162$	0	0
$163$	20.9850	1.64367	0.821836	−	0.569724i	$-0.192950\pi$
$163$	20.9850	1.64367	0.821836	+	0.569724i	$0.192950\pi$
$164$	0	0
$165$	−5.11196	−0.397966
$166$	0	0
$167$	−1.00000	−0.0773823
$167$	−1.00000	−0.0773823
$168$	0	0
$169$	7.73814	0.595242
$170$	0	0
$171$	−2.14913	−0.164348
$172$	0	0
$173$	−9.53795	−0.725157	−0.362578	−	0.931953i	$-0.618103\pi$
$173$	−9.53795	−0.725157	−0.362578	+	0.931953i	$0.618103\pi$
$174$	0	0
$175$	4.81880	0.364267
$176$	0	0
$177$	0.576526	0.0433343
$178$	0	0
$179$	−2.12282	−0.158667	−0.0793336	−	0.996848i	$-0.525279\pi$
$179$	−2.12282	−0.158667	−0.0793336	+	0.996848i	$0.525279\pi$
$180$	0	0
$181$	−8.06276	−0.599300	−0.299650	−	0.954049i	$-0.596870\pi$
$181$	−8.06276	−0.599300	−0.299650	+	0.954049i	$0.596870\pi$
$182$	0	0
$183$	9.76943	0.722177
$184$	0	0
$185$	−8.75247	−0.643495
$186$	0	0
$187$	9.06494	0.662894
$188$	0	0
$189$	2.23045	0.162242
$190$	0	0
$191$	−2.87699	−0.208172	−0.104086	−	0.994568i	$-0.533192\pi$
$191$	−2.87699	−0.208172	−0.104086	+	0.994568i	$0.533192\pi$
$192$	0	0
$193$	−26.0937	−1.87826	−0.939132	−	0.343557i	$-0.888368\pi$
$193$	−26.0937	−1.87826	−0.939132	+	0.343557i	$0.888368\pi$
$194$	0	0
$195$	−7.67378	−0.549531
$196$	0	0
$197$	24.0013	1.71003	0.855013	−	0.518607i	$-0.173549\pi$
$197$	24.0013	1.71003	0.855013	+	0.518607i	$0.173549\pi$
$198$	0	0
$199$	−17.2330	−1.22162	−0.610809	−	0.791778i	$-0.709156\pi$
$199$	−17.2330	−1.22162	−0.610809	+	0.791778i	$0.709156\pi$
$200$	0	0
$201$	−4.66641	−0.329143
$202$	0	0
$203$	14.4581	1.01476
$204$	0	0
$205$	−1.11657	−0.0779845
$206$	0	0
$207$	1.24231	0.0863465
$208$	0	0
$209$	6.51967	0.450975
$210$	0	0
$211$	−28.7011	−1.97587	−0.987934	−	0.154876i	$-0.950502\pi$
$211$	−28.7011	−1.97587	−0.987934	+	0.154876i	$0.950502\pi$
$212$	0	0
$213$	−2.31288	−0.158476
$214$	0	0
$215$	18.0409	1.23038
$216$	0	0
$217$	−2.02025	−0.137143
$218$	0	0
$219$	−13.5937	−0.918577
$220$	0	0
$221$	13.6078	0.915356
$222$	0	0
$223$	13.2997	0.890614	0.445307	−	0.895378i	$-0.353095\pi$
$223$	13.2997	0.890614	0.445307	+	0.895378i	$0.353095\pi$
$224$	0	0
$225$	−2.16046	−0.144030
$226$	0	0
$227$	5.09741	0.338327	0.169164	−	0.985588i	$-0.445893\pi$
$227$	5.09741	0.338327	0.169164	+	0.985588i	$0.445893\pi$
$228$	0	0
$229$	26.5563	1.75489	0.877444	−	0.479679i	$-0.159247\pi$
$229$	26.5563	1.75489	0.877444	+	0.479679i	$0.159247\pi$
$230$	0	0
$231$	−6.76638	−0.445195
$232$	0	0
$233$	28.9849	1.89886	0.949432	−	0.313972i	$-0.101660\pi$
$233$	28.9849	1.89886	0.949432	+	0.313972i	$0.101660\pi$
$234$	0	0
$235$	−11.0970	−0.723887
$236$	0	0
$237$	−6.77552	−0.440117
$238$	0	0
$239$	15.1035	0.976965	0.488483	−	0.872574i	$-0.337551\pi$
$239$	15.1035	0.976965	0.488483	+	0.872574i	$0.337551\pi$
$240$	0	0
$241$	19.7378	1.27142	0.635711	−	0.771927i	$-0.280707\pi$
$241$	19.7378	1.27142	0.635711	+	0.771927i	$0.280707\pi$
$242$	0	0
$243$	−1.00000	−0.0641500
$244$	0	0
$245$	3.41245	0.218013
$246$	0	0
$247$	9.78694	0.622728
$248$	0	0
$249$	−11.3095	−0.716714
$250$	0	0
$251$	11.0371	0.696655	0.348328	−	0.937373i	$-0.386750\pi$
$251$	11.0371	0.696655	0.348328	+	0.937373i	$0.386750\pi$
$252$	0	0
$253$	−3.76872	−0.236937
$254$	0	0
$255$	−5.03530	−0.315323
$256$	0	0
$257$	−21.9098	−1.36669	−0.683347	−	0.730094i	$-0.739476\pi$
$257$	−21.9098	−1.36669	−0.683347	+	0.730094i	$0.739476\pi$
$258$	0	0
$259$	−11.5851	−0.719863
$260$	0	0
$261$	−6.48215	−0.401235
$262$	0	0
$263$	20.7484	1.27940	0.639702	−	0.768623i	$-0.279058\pi$
$263$	20.7484	1.27940	0.639702	+	0.768623i	$0.279058\pi$
$264$	0	0
$265$	14.4288	0.886354
$266$	0	0
$267$	1.55582	0.0952145
$268$	0	0
$269$	−9.59849	−0.585230	−0.292615	−	0.956230i	$-0.594525\pi$
$269$	−9.59849	−0.585230	−0.292615	+	0.956230i	$0.594525\pi$
$270$	0	0
$271$	8.78291	0.533524	0.266762	−	0.963762i	$-0.414046\pi$
$271$	8.78291	0.533524	0.266762	+	0.963762i	$0.414046\pi$
$272$	0	0
$273$	−10.1573	−0.614747
$274$	0	0
$275$	6.55404	0.395224
$276$	0	0
$277$	−10.9897	−0.660304	−0.330152	−	0.943928i	$-0.607100\pi$
$277$	−10.9897	−0.660304	−0.330152	+	0.943928i	$0.607100\pi$
$278$	0	0
$279$	0.905758	0.0542263
$280$	0	0
$281$	1.74882	0.104326	0.0521629	−	0.998639i	$-0.483388\pi$
$281$	1.74882	0.104326	0.0521629	+	0.998639i	$0.483388\pi$
$282$	0	0
$283$	5.33783	0.317301	0.158650	−	0.987335i	$-0.449286\pi$
$283$	5.33783	0.317301	0.158650	+	0.987335i	$0.449286\pi$
$284$	0	0
$285$	−3.62148	−0.214518
$286$	0	0
$287$	−1.47793	−0.0872395
$288$	0	0
$289$	−8.07100	−0.474765
$290$	0	0
$291$	12.8855	0.755364
$292$	0	0
$293$	−23.7683	−1.38856	−0.694280	−	0.719705i	$-0.744277\pi$
$293$	−23.7683	−1.38856	−0.694280	+	0.719705i	$0.744277\pi$
$294$	0	0
$295$	0.971500	0.0565629
$296$	0	0
$297$	3.03364	0.176029
$298$	0	0
$299$	−5.65737	−0.327174
$300$	0	0
$301$	23.8796	1.37640
$302$	0	0
$303$	−3.21883	−0.184917
$304$	0	0
$305$	16.4624	0.942635
$306$	0	0
$307$	−25.4316	−1.45146	−0.725728	−	0.687981i	$-0.758497\pi$
$307$	−25.4316	−1.45146	−0.725728	+	0.687981i	$0.758497\pi$
$308$	0	0
$309$	12.7950	0.727883
$310$	0	0
$311$	34.0914	1.93315	0.966573	−	0.256392i	$-0.0825337\pi$
$311$	34.0914	1.93315	0.966573	+	0.256392i	$0.0825337\pi$
$312$	0	0
$313$	5.72030	0.323330	0.161665	−	0.986846i	$-0.448314\pi$
$313$	5.72030	0.323330	0.161665	+	0.986846i	$0.448314\pi$
$314$	0	0
$315$	3.75852	0.211769
$316$	0	0
$317$	1.84667	0.103719	0.0518596	−	0.998654i	$-0.483485\pi$
$317$	1.84667	0.103719	0.0518596	+	0.998654i	$0.483485\pi$
$318$	0	0
$319$	19.6645	1.10100
$320$	0	0
$321$	5.79714	0.323565
$322$	0	0
$323$	6.42190	0.357324
$324$	0	0
$325$	9.83854	0.545744
$326$	0	0
$327$	11.0690	0.612115
$328$	0	0
$329$	−14.6884	−0.809796
$330$	0	0
$331$	−28.9536	−1.59143	−0.795716	−	0.605670i	$-0.792905\pi$
$331$	−28.9536	−1.59143	−0.795716	+	0.605670i	$0.792905\pi$
$332$	0	0
$333$	5.19406	0.284632
$334$	0	0
$335$	−7.86334	−0.429620
$336$	0	0
$337$	2.43852	0.132835	0.0664173	−	0.997792i	$-0.478843\pi$
$337$	2.43852	0.132835	0.0664173	+	0.997792i	$0.478843\pi$
$338$	0	0
$339$	12.1323	0.658934
$340$	0	0
$341$	−2.74774	−0.148798
$342$	0	0
$343$	20.1300	1.08692
$344$	0	0
$345$	2.09341	0.112705
$346$	0	0
$347$	4.29284	0.230452	0.115226	−	0.993339i	$-0.463241\pi$
$347$	4.29284	0.230452	0.115226	+	0.993339i	$0.463241\pi$
$348$	0	0
$349$	−22.7374	−1.21710	−0.608551	−	0.793515i	$-0.708249\pi$
$349$	−22.7374	−1.21710	−0.608551	+	0.793515i	$0.708249\pi$
$350$	0	0
$351$	4.55392	0.243070
$352$	0	0
$353$	−6.60400	−0.351496	−0.175748	−	0.984435i	$-0.556234\pi$
$353$	−6.60400	−0.351496	−0.175748	+	0.984435i	$0.556234\pi$
$354$	0	0
$355$	−3.89743	−0.206854
$356$	0	0
$357$	−6.66491	−0.352745
$358$	0	0
$359$	34.3313	1.81194	0.905969	−	0.423343i	$-0.139144\pi$
$359$	34.3313	1.81194	0.905969	+	0.423343i	$0.139144\pi$
$360$	0	0
$361$	−14.3813	−0.756908
$362$	0	0
$363$	1.79705	0.0943206
$364$	0	0
$365$	−22.9067	−1.19899
$366$	0	0
$367$	12.3087	0.642510	0.321255	−	0.946993i	$-0.395895\pi$
$367$	12.3087	0.642510	0.321255	+	0.946993i	$0.395895\pi$
$368$	0	0
$369$	0.662614	0.0344943
$370$	0	0
$371$	19.0985	0.991544
$372$	0	0
$373$	10.6805	0.553013	0.276507	−	0.961012i	$-0.410823\pi$
$373$	10.6805	0.553013	0.276507	+	0.961012i	$0.410823\pi$
$374$	0	0
$375$	−12.0660	−0.623088
$376$	0	0
$377$	29.5192	1.52031
$378$	0	0
$379$	−24.4104	−1.25388	−0.626938	−	0.779069i	$-0.715692\pi$
$379$	−24.4104	−1.25388	−0.626938	+	0.779069i	$0.715692\pi$
$380$	0	0
$381$	−16.7289	−0.857049
$382$	0	0
$383$	−5.92490	−0.302748	−0.151374	−	0.988477i	$-0.548370\pi$
$383$	−5.92490	−0.302748	−0.151374	+	0.988477i	$0.548370\pi$
$384$	0	0
$385$	−11.4020	−0.581099
$386$	0	0
$387$	−10.7062	−0.544226
$388$	0	0
$389$	−21.7011	−1.10029	−0.550146	−	0.835069i	$-0.685428\pi$
$389$	−21.7011	−1.10029	−0.550146	+	0.835069i	$0.685428\pi$
$390$	0	0
$391$	−3.71220	−0.187734
$392$	0	0
$393$	−9.52434	−0.480439
$394$	0	0
$395$	−11.4174	−0.574471
$396$	0	0
$397$	20.8145	1.04465	0.522326	−	0.852746i	$-0.325064\pi$
$397$	20.8145	1.04465	0.522326	+	0.852746i	$0.325064\pi$
$398$	0	0
$399$	−4.79352	−0.239976
$400$	0	0
$401$	−13.4584	−0.672080	−0.336040	−	0.941848i	$-0.609088\pi$
$401$	−13.4584	−0.672080	−0.336040	+	0.941848i	$0.609088\pi$
$402$	0	0
$403$	−4.12474	−0.205468
$404$	0	0
$405$	−1.68509	−0.0837330
$406$	0	0
$407$	−15.7569	−0.781039
$408$	0	0
$409$	−0.684734	−0.0338579	−0.0169290	−	0.999857i	$-0.505389\pi$
$409$	−0.684734	−0.0338579	−0.0169290	+	0.999857i	$0.505389\pi$
$410$	0	0
$411$	−11.1055	−0.547796
$412$	0	0
$413$	1.28591	0.0632757
$414$	0	0
$415$	−19.0577	−0.935504
$416$	0	0
$417$	3.18986	0.156208
$418$	0	0
$419$	23.5683	1.15139	0.575694	−	0.817665i	$-0.304732\pi$
$419$	23.5683	1.15139	0.575694	+	0.817665i	$0.304732\pi$
$420$	0	0
$421$	−8.78562	−0.428185	−0.214092	−	0.976813i	$-0.568679\pi$
$421$	−8.78562	−0.428185	−0.214092	+	0.976813i	$0.568679\pi$
$422$	0	0
$423$	6.58538	0.320192
$424$	0	0
$425$	6.45575	0.313150
$426$	0	0
$427$	21.7903	1.05450
$428$	0	0
$429$	−13.8149	−0.666991
$430$	0	0
$431$	−15.9236	−0.767014	−0.383507	−	0.923538i	$-0.625284\pi$
$431$	−15.9236	−0.767014	−0.383507	+	0.923538i	$0.625284\pi$
$432$	0	0
$433$	33.9313	1.63063	0.815316	−	0.579016i	$-0.196563\pi$
$433$	33.9313	1.63063	0.815316	+	0.579016i	$0.196563\pi$
$434$	0	0
$435$	−10.9230	−0.523720
$436$	0	0
$437$	−2.66988	−0.127718
$438$	0	0
$439$	6.67326	0.318497	0.159249	−	0.987239i	$-0.449093\pi$
$439$	6.67326	0.318497	0.159249	+	0.987239i	$0.449093\pi$
$440$	0	0
$441$	−2.02508	−0.0964324
$442$	0	0
$443$	−2.91953	−0.138711	−0.0693555	−	0.997592i	$-0.522094\pi$
$443$	−2.91953	−0.138711	−0.0693555	+	0.997592i	$0.522094\pi$
$444$	0	0
$445$	2.62170	0.124281
$446$	0	0
$447$	−8.87024	−0.419548
$448$	0	0
$449$	−33.6453	−1.58782	−0.793910	−	0.608035i	$-0.791958\pi$
$449$	−33.6453	−1.58782	−0.793910	+	0.608035i	$0.791958\pi$
$450$	0	0
$451$	−2.01013	−0.0946534
$452$	0	0
$453$	1.14248	0.0536784
$454$	0	0
$455$	−17.1160	−0.802410
$456$	0	0
$457$	−11.8562	−0.554608	−0.277304	−	0.960782i	$-0.589441\pi$
$457$	−11.8562	−0.554608	−0.277304	+	0.960782i	$0.589441\pi$
$458$	0	0
$459$	2.98814	0.139475
$460$	0	0
$461$	−30.1143	−1.40256	−0.701282	−	0.712884i	$-0.747389\pi$
$461$	−30.1143	−1.40256	−0.701282	+	0.712884i	$0.747389\pi$
$462$	0	0
$463$	−30.5085	−1.41785	−0.708924	−	0.705284i	$-0.750819\pi$
$463$	−30.5085	−1.41785	−0.708924	+	0.705284i	$0.750819\pi$
$464$	0	0
$465$	1.52629	0.0707799
$466$	0	0
$467$	2.58448	0.119596	0.0597978	−	0.998211i	$-0.480954\pi$
$467$	2.58448	0.119596	0.0597978	+	0.998211i	$0.480954\pi$
$468$	0	0
$469$	−10.4082	−0.480606
$470$	0	0
$471$	−9.53736	−0.439458
$472$	0	0
$473$	32.4787	1.49337
$474$	0	0
$475$	4.64309	0.213040
$476$	0	0
$477$	−8.56261	−0.392055
$478$	0	0
$479$	25.6801	1.17336	0.586678	−	0.809820i	$-0.300436\pi$
$479$	25.6801	1.17336	0.586678	+	0.809820i	$0.300436\pi$
$480$	0	0
$481$	−23.6533	−1.07850
$482$	0	0
$483$	2.77091	0.126081
$484$	0	0
$485$	21.7134	0.985953
$486$	0	0
$487$	1.86501	0.0845119	0.0422559	−	0.999107i	$-0.486546\pi$
$487$	1.86501	0.0845119	0.0422559	+	0.999107i	$0.486546\pi$
$488$	0	0
$489$	−20.9850	−0.948974
$490$	0	0
$491$	26.7924	1.20913	0.604563	−	0.796557i	$-0.293348\pi$
$491$	26.7924	1.20913	0.604563	+	0.796557i	$0.293348\pi$
$492$	0	0
$493$	19.3696	0.872363
$494$	0	0
$495$	5.11196	0.229766
$496$	0	0
$497$	−5.15878	−0.231403
$498$	0	0
$499$	−20.9253	−0.936746	−0.468373	−	0.883531i	$-0.655160\pi$
$499$	−20.9253	−0.936746	−0.468373	+	0.883531i	$0.655160\pi$
$500$	0	0
$501$	1.00000	0.0446767
$502$	0	0
$503$	−4.03993	−0.180132	−0.0900658	−	0.995936i	$-0.528708\pi$
$503$	−4.03993	−0.180132	−0.0900658	+	0.995936i	$0.528708\pi$
$504$	0	0
$505$	−5.42403	−0.241366
$506$	0	0
$507$	−7.73814	−0.343663
$508$	0	0
$509$	−6.65081	−0.294792	−0.147396	−	0.989078i	$-0.547089\pi$
$509$	−6.65081	−0.294792	−0.147396	+	0.989078i	$0.547089\pi$
$510$	0	0
$511$	−30.3201	−1.34128
$512$	0	0
$513$	2.14913	0.0948862
$514$	0	0
$515$	21.5608	0.950082
$516$	0	0
$517$	−19.9776	−0.878616
$518$	0	0
$519$	9.53795	0.418669
$520$	0	0
$521$	−23.7666	−1.04124	−0.520618	−	0.853790i	$-0.674298\pi$
$521$	−23.7666	−1.04124	−0.520618	+	0.853790i	$0.674298\pi$
$522$	0	0
$523$	−6.42494	−0.280943	−0.140471	−	0.990085i	$-0.544862\pi$
$523$	−6.42494	−0.280943	−0.140471	+	0.990085i	$0.544862\pi$
$524$	0	0
$525$	−4.81880	−0.210310
$526$	0	0
$527$	−2.70653	−0.117898
$528$	0	0
$529$	−21.4567	−0.932899
$530$	0	0
$531$	−0.576526	−0.0250191
$532$	0	0
$533$	−3.01749	−0.130702
$534$	0	0
$535$	9.76874	0.422339
$536$	0	0
$537$	2.12282	0.0916066
$538$	0	0
$539$	6.14336	0.264613
$540$	0	0
$541$	6.56151	0.282101	0.141051	−	0.990002i	$-0.454952\pi$
$541$	6.56151	0.282101	0.141051	+	0.990002i	$0.454952\pi$
$542$	0	0
$543$	8.06276	0.346006
$544$	0	0
$545$	18.6522	0.798975
$546$	0	0
$547$	12.7097	0.543429	0.271714	−	0.962378i	$-0.412409\pi$
$547$	12.7097	0.543429	0.271714	+	0.962378i	$0.412409\pi$
$548$	0	0
$549$	−9.76943	−0.416949
$550$	0	0
$551$	13.9310	0.593479
$552$	0	0
$553$	−15.1125	−0.642648
$554$	0	0
$555$	8.75247	0.371522
$556$	0	0
$557$	35.7402	1.51436	0.757180	−	0.653207i	$-0.226577\pi$
$557$	35.7402	1.51436	0.757180	+	0.653207i	$0.226577\pi$
$558$	0	0
$559$	48.7550	2.06212
$560$	0	0
$561$	−9.06494	−0.382722
$562$	0	0
$563$	41.4699	1.74775	0.873873	−	0.486154i	$-0.161601\pi$
$563$	41.4699	1.74775	0.873873	+	0.486154i	$0.161601\pi$
$564$	0	0
$565$	20.4440	0.860086
$566$	0	0
$567$	−2.23045	−0.0936702
$568$	0	0
$569$	−17.6005	−0.737852	−0.368926	−	0.929459i	$-0.620274\pi$
$569$	−17.6005	−0.737852	−0.368926	+	0.929459i	$0.620274\pi$
$570$	0	0
$571$	3.72137	0.155734	0.0778672	−	0.996964i	$-0.475189\pi$
$571$	3.72137	0.155734	0.0778672	+	0.996964i	$0.475189\pi$
$572$	0	0
$573$	2.87699	0.120188
$574$	0	0
$575$	−2.68396	−0.111929
$576$	0	0
$577$	−24.7882	−1.03194	−0.515972	−	0.856605i	$-0.672569\pi$
$577$	−24.7882	−1.03194	−0.515972	+	0.856605i	$0.672569\pi$
$578$	0	0
$579$	26.0937	1.08442
$580$	0	0
$581$	−25.2254	−1.04653
$582$	0	0
$583$	25.9758	1.07581
$584$	0	0
$585$	7.67378	0.317272
$586$	0	0
$587$	−10.7714	−0.444582	−0.222291	−	0.974980i	$-0.571353\pi$
$587$	−10.7714	−0.444582	−0.222291	+	0.974980i	$0.571353\pi$
$588$	0	0
$589$	−1.94659	−0.0802077
$590$	0	0
$591$	−24.0013	−0.987283
$592$	0	0
$593$	3.45137	0.141731	0.0708653	−	0.997486i	$-0.477424\pi$
$593$	3.45137	0.141731	0.0708653	+	0.997486i	$0.477424\pi$
$594$	0	0
$595$	−11.2310	−0.460426
$596$	0	0
$597$	17.2330	0.705301
$598$	0	0
$599$	18.8839	0.771574	0.385787	−	0.922588i	$-0.373930\pi$
$599$	18.8839	0.771574	0.385787	+	0.922588i	$0.373930\pi$
$600$	0	0
$601$	−37.9861	−1.54949	−0.774743	−	0.632276i	$-0.782121\pi$
$601$	−37.9861	−1.54949	−0.774743	+	0.632276i	$0.782121\pi$
$602$	0	0
$603$	4.66641	0.190031
$604$	0	0
$605$	3.02820	0.123114
$606$	0	0
$607$	30.1849	1.22517	0.612583	−	0.790406i	$-0.290130\pi$
$607$	30.1849	1.22517	0.612583	+	0.790406i	$0.290130\pi$
$608$	0	0
$609$	−14.4581	−0.585873
$610$	0	0
$611$	−29.9892	−1.21324
$612$	0	0
$613$	−5.96309	−0.240847	−0.120423	−	0.992723i	$-0.538425\pi$
$613$	−5.96309	−0.240847	−0.120423	+	0.992723i	$0.538425\pi$
$614$	0	0
$615$	1.11657	0.0450244
$616$	0	0
$617$	−16.0868	−0.647630	−0.323815	−	0.946120i	$-0.604966\pi$
$617$	−16.0868	−0.647630	−0.323815	+	0.946120i	$0.604966\pi$
$618$	0	0
$619$	−17.6051	−0.707610	−0.353805	−	0.935319i	$-0.615112\pi$
$619$	−17.6051	−0.707610	−0.353805	+	0.935319i	$0.615112\pi$
$620$	0	0
$621$	−1.24231	−0.0498522
$622$	0	0
$623$	3.47018	0.139030
$624$	0	0
$625$	−9.53014	−0.381206
$626$	0	0
$627$	−6.51967	−0.260370
$628$	0	0
$629$	−15.5206	−0.618846
$630$	0	0
$631$	−20.7751	−0.827044	−0.413522	−	0.910494i	$-0.635702\pi$
$631$	−20.7751	−0.827044	−0.413522	+	0.910494i	$0.635702\pi$
$632$	0	0
$633$	28.7011	1.14077
$634$	0	0
$635$	−28.1898	−1.11868
$636$	0	0
$637$	9.22204	0.365391
$638$	0	0
$639$	2.31288	0.0914962
$640$	0	0
$641$	35.2600	1.39269	0.696343	−	0.717709i	$-0.254809\pi$
$641$	35.2600	1.39269	0.696343	+	0.717709i	$0.254809\pi$
$642$	0	0
$643$	−7.93781	−0.313036	−0.156518	−	0.987675i	$-0.550027\pi$
$643$	−7.93781	−0.313036	−0.156518	+	0.987675i	$0.550027\pi$
$644$	0	0
$645$	−18.0409	−0.710360
$646$	0	0
$647$	−13.5098	−0.531124	−0.265562	−	0.964094i	$-0.585557\pi$
$647$	−13.5098	−0.531124	−0.265562	+	0.964094i	$0.585557\pi$
$648$	0	0
$649$	1.74897	0.0686531
$650$	0	0
$651$	2.02025	0.0791798
$652$	0	0
$653$	−18.8369	−0.737146	−0.368573	−	0.929599i	$-0.620154\pi$
$653$	−18.8369	−0.737146	−0.368573	+	0.929599i	$0.620154\pi$
$654$	0	0
$655$	−16.0494	−0.627102
$656$	0	0
$657$	13.5937	0.530341
$658$	0	0
$659$	−4.71786	−0.183782	−0.0918909	−	0.995769i	$-0.529291\pi$
$659$	−4.71786	−0.183782	−0.0918909	+	0.995769i	$0.529291\pi$
$660$	0	0
$661$	20.3849	0.792882	0.396441	−	0.918060i	$-0.370245\pi$
$661$	20.3849	0.792882	0.396441	+	0.918060i	$0.370245\pi$
$662$	0	0
$663$	−13.6078	−0.528481
$664$	0	0
$665$	−8.07754	−0.313234
$666$	0	0
$667$	−8.05284	−0.311807
$668$	0	0
$669$	−13.2997	−0.514196
$670$	0	0
$671$	29.6369	1.14412
$672$	0	0
$673$	−20.2303	−0.779822	−0.389911	−	0.920853i	$-0.627494\pi$
$673$	−20.2303	−0.779822	−0.389911	+	0.920853i	$0.627494\pi$
$674$	0	0
$675$	2.16046	0.0831560
$676$	0	0
$677$	−33.1478	−1.27397	−0.636987	−	0.770875i	$-0.719819\pi$
$677$	−33.1478	−1.27397	−0.636987	+	0.770875i	$0.719819\pi$
$678$	0	0
$679$	28.7406	1.10296
$680$	0	0
$681$	−5.09741	−0.195333
$682$	0	0
$683$	−16.5351	−0.632698	−0.316349	−	0.948643i	$-0.602457\pi$
$683$	−16.5351	−0.632698	−0.316349	+	0.948643i	$0.602457\pi$
$684$	0	0
$685$	−18.7139	−0.715020
$686$	0	0
$687$	−26.5563	−1.01319
$688$	0	0
$689$	38.9934	1.48553
$690$	0	0
$691$	−32.9967	−1.25525	−0.627627	−	0.778514i	$-0.715974\pi$
$691$	−32.9967	−1.25525	−0.627627	+	0.778514i	$0.715974\pi$
$692$	0	0
$693$	6.76638	0.257034
$694$	0	0
$695$	5.37522	0.203894
$696$	0	0
$697$	−1.97999	−0.0749973
$698$	0	0
$699$	−28.9849	−1.09631
$700$	0	0
$701$	36.1210	1.36427	0.682136	−	0.731225i	$-0.261051\pi$
$701$	36.1210	1.36427	0.682136	+	0.731225i	$0.261051\pi$
$702$	0	0
$703$	−11.1627	−0.421008
$704$	0	0
$705$	11.0970	0.417936
$706$	0	0
$707$	−7.17944	−0.270011
$708$	0	0
$709$	−5.15320	−0.193533	−0.0967663	−	0.995307i	$-0.530850\pi$
$709$	−5.15320	−0.193533	−0.0967663	+	0.995307i	$0.530850\pi$
$710$	0	0
$711$	6.77552	0.254102
$712$	0	0
$713$	1.12523	0.0421402
$714$	0	0
$715$	−23.2794	−0.870602
$716$	0	0
$717$	−15.1035	−0.564051
$718$	0	0
$719$	−38.8596	−1.44922	−0.724609	−	0.689160i	$-0.757980\pi$
$719$	−38.8596	−1.44922	−0.724609	+	0.689160i	$0.757980\pi$
$720$	0	0
$721$	28.5387	1.06284
$722$	0	0
$723$	−19.7378	−0.734055
$724$	0	0
$725$	14.0044	0.520111
$726$	0	0
$727$	7.08653	0.262825	0.131412	−	0.991328i	$-0.458049\pi$
$727$	7.08653	0.262825	0.131412	+	0.991328i	$0.458049\pi$
$728$	0	0
$729$	1.00000	0.0370370
$730$	0	0
$731$	31.9916	1.18325
$732$	0	0
$733$	−9.95448	−0.367677	−0.183839	−	0.982956i	$-0.558852\pi$
$733$	−9.95448	−0.367677	−0.183839	+	0.982956i	$0.558852\pi$
$734$	0	0
$735$	−3.41245	−0.125870
$736$	0	0
$737$	−14.1562	−0.521450
$738$	0	0
$739$	−41.9772	−1.54416	−0.772078	−	0.635528i	$-0.780782\pi$
$739$	−41.9772	−1.54416	−0.772078	+	0.635528i	$0.780782\pi$
$740$	0	0
$741$	−9.78694	−0.359532
$742$	0	0
$743$	−8.11387	−0.297669	−0.148834	−	0.988862i	$-0.547552\pi$
$743$	−8.11387	−0.297669	−0.148834	+	0.988862i	$0.547552\pi$
$744$	0	0
$745$	−14.9472	−0.547623
$746$	0	0
$747$	11.3095	0.413795
$748$	0	0
$749$	12.9303	0.472461
$750$	0	0
$751$	−35.3736	−1.29080	−0.645400	−	0.763845i	$-0.723309\pi$
$751$	−35.3736	−1.29080	−0.645400	+	0.763845i	$0.723309\pi$
$752$	0	0
$753$	−11.0371	−0.402214
$754$	0	0
$755$	1.92519	0.0700648
$756$	0	0
$757$	7.15401	0.260017	0.130008	−	0.991513i	$-0.458500\pi$
$757$	7.15401	0.260017	0.130008	+	0.991513i	$0.458500\pi$
$758$	0	0
$759$	3.76872	0.136796
$760$	0	0
$761$	2.69995	0.0978730	0.0489365	−	0.998802i	$-0.484417\pi$
$761$	2.69995	0.0978730	0.0489365	+	0.998802i	$0.484417\pi$
$762$	0	0
$763$	24.6888	0.893795
$764$	0	0
$765$	5.03530	0.182052
$766$	0	0
$767$	2.62545	0.0947995
$768$	0	0
$769$	39.3940	1.42058	0.710291	−	0.703908i	$-0.248563\pi$
$769$	39.3940	1.42058	0.710291	+	0.703908i	$0.248563\pi$
$770$	0	0
$771$	21.9098	0.789061
$772$	0	0
$773$	15.7509	0.566522	0.283261	−	0.959043i	$-0.408584\pi$
$773$	15.7509	0.566522	0.283261	+	0.959043i	$0.408584\pi$
$774$	0	0
$775$	−1.95685	−0.0702921
$776$	0	0
$777$	11.5851	0.415613
$778$	0	0
$779$	−1.42404	−0.0510216
$780$	0	0
$781$	−7.01645	−0.251068
$782$	0	0
$783$	6.48215	0.231653
$784$	0	0
$785$	−16.0713	−0.573611
$786$	0	0
$787$	−26.1812	−0.933261	−0.466630	−	0.884452i	$-0.654532\pi$
$787$	−26.1812	−0.933261	−0.466630	+	0.884452i	$0.654532\pi$
$788$	0	0
$789$	−20.7484	−0.738664
$790$	0	0
$791$	27.0604	0.962158
$792$	0	0
$793$	44.4892	1.57986
$794$	0	0
$795$	−14.4288	−0.511737
$796$	0	0
$797$	14.3201	0.507242	0.253621	−	0.967304i	$-0.418378\pi$
$797$	14.3201	0.507242	0.253621	+	0.967304i	$0.418378\pi$
$798$	0	0
$799$	−19.6780	−0.696159
$800$	0	0
$801$	−1.55582	−0.0549721
$802$	0	0
$803$	−41.2383	−1.45527
$804$	0	0
$805$	4.66925	0.164569
$806$	0	0
$807$	9.59849	0.337883
$808$	0	0
$809$	−30.4050	−1.06898	−0.534492	−	0.845174i	$-0.679497\pi$
$809$	−30.4050	−1.06898	−0.534492	+	0.845174i	$0.679497\pi$
$810$	0	0
$811$	18.4602	0.648224	0.324112	−	0.946019i	$-0.394935\pi$
$811$	18.4602	0.648224	0.324112	+	0.946019i	$0.394935\pi$
$812$	0	0
$813$	−8.78291	−0.308030
$814$	0	0
$815$	−35.3617	−1.23867
$816$	0	0
$817$	23.0089	0.804980
$818$	0	0
$819$	10.1573	0.354925
$820$	0	0
$821$	19.7093	0.687859	0.343930	−	0.938995i	$-0.388242\pi$
$821$	19.7093	0.687859	0.343930	+	0.938995i	$0.388242\pi$
$822$	0	0
$823$	2.37531	0.0827982	0.0413991	−	0.999143i	$-0.486818\pi$
$823$	2.37531	0.0827982	0.0413991	+	0.999143i	$0.486818\pi$
$824$	0	0
$825$	−6.55404	−0.228182
$826$	0	0
$827$	1.85683	0.0645683	0.0322841	−	0.999479i	$-0.489722\pi$
$827$	1.85683	0.0645683	0.0322841	+	0.999479i	$0.489722\pi$
$828$	0	0
$829$	−2.41323	−0.0838148	−0.0419074	−	0.999121i	$-0.513343\pi$
$829$	−2.41323	−0.0838148	−0.0419074	+	0.999121i	$0.513343\pi$
$830$	0	0
$831$	10.9897	0.381227
$832$	0	0
$833$	6.05123	0.209663
$834$	0	0
$835$	1.68509	0.0583151
$836$	0	0
$837$	−0.905758	−0.0313076
$838$	0	0
$839$	−9.76856	−0.337248	−0.168624	−	0.985680i	$-0.553932\pi$
$839$	−9.76856	−0.337248	−0.168624	+	0.985680i	$0.553932\pi$
$840$	0	0
$841$	13.0183	0.448907
$842$	0	0
$843$	−1.74882	−0.0602326
$844$	0	0
$845$	−13.0395	−0.448572
$846$	0	0
$847$	4.00823	0.137725
$848$	0	0
$849$	−5.33783	−0.183194
$850$	0	0
$851$	6.45263	0.221193
$852$	0	0
$853$	39.4709	1.35146	0.675729	−	0.737150i	$-0.263829\pi$
$853$	39.4709	1.35146	0.675729	+	0.737150i	$0.263829\pi$
$854$	0	0
$855$	3.62148	0.123852
$856$	0	0
$857$	−7.76665	−0.265304	−0.132652	−	0.991163i	$-0.542349\pi$
$857$	−7.76665	−0.265304	−0.132652	+	0.991163i	$0.542349\pi$
$858$	0	0
$859$	12.0800	0.412163	0.206081	−	0.978535i	$-0.433929\pi$
$859$	12.0800	0.412163	0.206081	+	0.978535i	$0.433929\pi$
$860$	0	0
$861$	1.47793	0.0503677
$862$	0	0
$863$	−23.3890	−0.796170	−0.398085	−	0.917349i	$-0.630325\pi$
$863$	−23.3890	−0.796170	−0.398085	+	0.917349i	$0.630325\pi$
$864$	0	0
$865$	16.0723	0.546476
$866$	0	0
$867$	8.07100	0.274106
$868$	0	0
$869$	−20.5545	−0.697262
$870$	0	0
$871$	−21.2504	−0.720043
$872$	0	0
$873$	−12.8855	−0.436110
$874$	0	0
$875$	−26.9127	−0.909817
$876$	0	0
$877$	43.6990	1.47561	0.737805	−	0.675014i	$-0.235862\pi$
$877$	43.6990	1.47561	0.737805	+	0.675014i	$0.235862\pi$
$878$	0	0
$879$	23.7683	0.801685
$880$	0	0
$881$	36.3074	1.22323	0.611614	−	0.791156i	$-0.290520\pi$
$881$	36.3074	1.22323	0.611614	+	0.791156i	$0.290520\pi$
$882$	0	0
$883$	−28.2760	−0.951563	−0.475782	−	0.879563i	$-0.657835\pi$
$883$	−28.2760	−0.951563	−0.475782	+	0.879563i	$0.657835\pi$
$884$	0	0
$885$	−0.971500	−0.0326566
$886$	0	0
$887$	19.2679	0.646952	0.323476	−	0.946236i	$-0.395149\pi$
$887$	19.2679	0.646952	0.323476	+	0.946236i	$0.395149\pi$
$888$	0	0
$889$	−37.3131	−1.25144
$890$	0	0
$891$	−3.03364	−0.101631
$892$	0	0
$893$	−14.1528	−0.473605
$894$	0	0
$895$	3.57716	0.119571
$896$	0	0
$897$	5.65737	0.188894
$898$	0	0
$899$	−5.87126	−0.195817
$900$	0	0
$901$	25.5863	0.852403
$902$	0	0
$903$	−23.8796	−0.794664
$904$	0	0
$905$	13.5865	0.451631
$906$	0	0
$907$	−28.4693	−0.945306	−0.472653	−	0.881249i	$-0.656704\pi$
$907$	−28.4693	−0.945306	−0.472653	+	0.881249i	$0.656704\pi$
$908$	0	0
$909$	3.21883	0.106762
$910$	0	0
$911$	−15.3135	−0.507360	−0.253680	−	0.967288i	$-0.581641\pi$
$911$	−15.3135	−0.507360	−0.253680	+	0.967288i	$0.581641\pi$
$912$	0	0
$913$	−34.3091	−1.13546
$914$	0	0
$915$	−16.4624	−0.544231
$916$	0	0
$917$	−21.2436	−0.701525
$918$	0	0
$919$	15.0291	0.495765	0.247883	−	0.968790i	$-0.420265\pi$
$919$	15.0291	0.495765	0.247883	+	0.968790i	$0.420265\pi$
$920$	0	0
$921$	25.4316	0.837999
$922$	0	0
$923$	−10.5327	−0.346687
$924$	0	0
$925$	−11.2215	−0.368962
$926$	0	0
$927$	−12.7950	−0.420243
$928$	0	0
$929$	−15.5114	−0.508912	−0.254456	−	0.967084i	$-0.581896\pi$
$929$	−15.5114	−0.508912	−0.254456	+	0.967084i	$0.581896\pi$
$930$	0	0
$931$	4.35215	0.142636
$932$	0	0
$933$	−34.0914	−1.11610
$934$	0	0
$935$	−15.2753	−0.499555
$936$	0	0
$937$	29.5850	0.966500	0.483250	−	0.875482i	$-0.339456\pi$
$937$	29.5850	0.966500	0.483250	+	0.875482i	$0.339456\pi$
$938$	0	0
$939$	−5.72030	−0.186675
$940$	0	0
$941$	−14.1685	−0.461881	−0.230940	−	0.972968i	$-0.574180\pi$
$941$	−14.1685	−0.461881	−0.230940	+	0.972968i	$0.574180\pi$
$942$	0	0
$943$	0.823172	0.0268062
$944$	0	0
$945$	−3.75852	−0.122265
$946$	0	0
$947$	−27.9383	−0.907874	−0.453937	−	0.891034i	$-0.649981\pi$
$947$	−27.9383	−0.907874	−0.453937	+	0.891034i	$0.649981\pi$
$948$	0	0
$949$	−61.9045	−2.00951
$950$	0	0
$951$	−1.84667	−0.0598823
$952$	0	0
$953$	14.8870	0.482238	0.241119	−	0.970496i	$-0.422486\pi$
$953$	14.8870	0.482238	0.241119	+	0.970496i	$0.422486\pi$
$954$	0	0
$955$	4.84800	0.156877
$956$	0	0
$957$	−19.6645	−0.635663
$958$	0	0
$959$	−24.7704	−0.799877
$960$	0	0
$961$	−30.1796	−0.973536
$962$	0	0
$963$	−5.79714	−0.186810
$964$	0	0
$965$	43.9703	1.41545
$966$	0	0
$967$	−21.6077	−0.694857	−0.347429	−	0.937706i	$-0.612945\pi$
$967$	−21.6077	−0.694857	−0.347429	+	0.937706i	$0.612945\pi$
$968$	0	0
$969$	−6.42190	−0.206301
$970$	0	0
$971$	0.916978	0.0294272	0.0147136	−	0.999892i	$-0.495316\pi$
$971$	0.916978	0.0294272	0.0147136	+	0.999892i	$0.495316\pi$
$972$	0	0
$973$	7.11484	0.228091
$974$	0	0
$975$	−9.83854	−0.315085
$976$	0	0
$977$	26.3437	0.842810	0.421405	−	0.906873i	$-0.361537\pi$
$977$	26.3437	0.842810	0.421405	+	0.906873i	$0.361537\pi$
$978$	0	0
$979$	4.71979	0.150845
$980$	0	0
$981$	−11.0690	−0.353405
$982$	0	0
$983$	−1.16941	−0.0372984	−0.0186492	−	0.999826i	$-0.505937\pi$
$983$	−1.16941	−0.0372984	−0.0186492	+	0.999826i	$0.505937\pi$
$984$	0	0
$985$	−40.4445	−1.28867
$986$	0	0
$987$	14.6884	0.467536
$988$	0	0
$989$	−13.3004	−0.422928
$990$	0	0
$991$	−47.2036	−1.49947	−0.749737	−	0.661736i	$-0.769820\pi$
$991$	−47.2036	−1.49947	−0.749737	+	0.661736i	$0.769820\pi$
$992$	0	0
$993$	28.9536	0.918814
$994$	0	0
$995$	29.0393	0.920607
$996$	0	0
$997$	−12.0724	−0.382337	−0.191168	−	0.981557i	$-0.561228\pi$
$997$	−12.0724	−0.382337	−0.191168	+	0.981557i	$0.561228\pi$
$998$	0	0
$999$	−5.19406	−0.164333

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	4008.2.a.g.1.3	✓	7
4.3	odd	2		8016.2.a.w.1.3		7

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
4008.2.a.g.1.3	✓	7	1.1	even	1	trivial
8016.2.a.w.1.3		7	4.3	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:	\( N \)	\(=\)	\( 4008 = 2^{3} \cdot 3 \cdot 167 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	4008.a (trivial)

\(f(q)\)	\(=\)	\(q-1.00000 q^{3} -1.68509 q^{5} -2.23045 q^{7} +1.00000 q^{9} +O(q^{10})\)	\(q-1.00000 q^{3} -1.68509 q^{5} -2.23045 q^{7} +1.00000 q^{9} -3.03364 q^{11} -4.55392 q^{13} +1.68509 q^{15} -2.98814 q^{17} -2.14913 q^{19} +2.23045 q^{21} +1.24231 q^{23} -2.16046 q^{25} -1.00000 q^{27} -6.48215 q^{29} +0.905758 q^{31} +3.03364 q^{33} +3.75852 q^{35} +5.19406 q^{37} +4.55392 q^{39} +0.662614 q^{41} -10.7062 q^{43} -1.68509 q^{45} +6.58538 q^{47} -2.02508 q^{49} +2.98814 q^{51} -8.56261 q^{53} +5.11196 q^{55} +2.14913 q^{57} -0.576526 q^{59} -9.76943 q^{61} -2.23045 q^{63} +7.67378 q^{65} +4.66641 q^{67} -1.24231 q^{69} +2.31288 q^{71} +13.5937 q^{73} +2.16046 q^{75} +6.76638 q^{77} +6.77552 q^{79} +1.00000 q^{81} +11.3095 q^{83} +5.03530 q^{85} +6.48215 q^{87} -1.55582 q^{89} +10.1573 q^{91} -0.905758 q^{93} +3.62148 q^{95} -12.8855 q^{97} -3.03364 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 7 q - 7 q^{3} - 3 q^{5} + 8 q^{7} + 7 q^{9}+O(q^{10}) \) 7 * q - 7 * q^3 - 3 * q^5 + 8 * q^7 + 7 * q^9	\( 7 q - 7 q^{3} - 3 q^{5} + 8 q^{7} + 7 q^{9} + q^{11} - 2 q^{13} + 3 q^{15} + 11 q^{17} + 2 q^{19} - 8 q^{21} + 17 q^{23} + 4 q^{25} - 7 q^{27} - 7 q^{29} + 10 q^{31} - q^{33} + 10 q^{35} - 21 q^{37} + 2 q^{39} + 8 q^{41} - 12 q^{43} - 3 q^{45} + 25 q^{47} - 7 q^{49} - 11 q^{51} - 7 q^{53} + 15 q^{55} - 2 q^{57} + 3 q^{59} - 14 q^{61} + 8 q^{63} + 4 q^{65} + 4 q^{67} - 17 q^{69} + 27 q^{71} - 12 q^{73} - 4 q^{75} + 16 q^{77} + 8 q^{79} + 7 q^{81} + 15 q^{83} - 3 q^{85} + 7 q^{87} + 14 q^{89} - 3 q^{91} - 10 q^{93} + 37 q^{95} + 3 q^{97} + q^{99}+O(q^{100}) \) 7 * q - 7 * q^3 - 3 * q^5 + 8 * q^7 + 7 * q^9 + q^11 - 2 * q^13 + 3 * q^15 + 11 * q^17 + 2 * q^19 - 8 * q^21 + 17 * q^23 + 4 * q^25 - 7 * q^27 - 7 * q^29 + 10 * q^31 - q^33 + 10 * q^35 - 21 * q^37 + 2 * q^39 + 8 * q^41 - 12 * q^43 - 3 * q^45 + 25 * q^47 - 7 * q^49 - 11 * q^51 - 7 * q^53 + 15 * q^55 - 2 * q^57 + 3 * q^59 - 14 * q^61 + 8 * q^63 + 4 * q^65 + 4 * q^67 - 17 * q^69 + 27 * q^71 - 12 * q^73 - 4 * q^75 + 16 * q^77 + 8 * q^79 + 7 * q^81 + 15 * q^83 - 3 * q^85 + 7 * q^87 + 14 * q^89 - 3 * q^91 - 10 * q^93 + 37 * q^95 + 3 * q^97 + q^99

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