LMFDB - Embedded newform 4016.2.a.g.1.7

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [4016,2,Mod(1,4016)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(4016, base_ring=CyclotomicField(2))

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

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

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

chi := DirichletCharacter("4016.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 4016 = 2^{4} \cdot 251 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	4016.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:	$32.0679214517$
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} - 3x^{6} - 6x^{5} + 18x^{4} + 8x^{3} - 17x^{2} - 9x - 1 $ x^7 - 3x^6 - 6x^5 + 18x^4 + 8x^3 - 17x^2 - 9x - 1
Coefficient ring:	$\Z[a_1, \ldots, a_{5}]$
Coefficient ring index:	$ 1 $
Twist minimal:	no (minimal twist has level 1004)
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.7
Root			$2.85375$ of defining polynomial
Character	$\chi$	$=$	4016.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+2.85375 q^{3} +0.454452 q^{5} +2.18653 q^{7} +5.14389 q^{9} +O(q^{10})$	$q+2.85375 q^{3} +0.454452 q^{5} +2.18653 q^{7} +5.14389 q^{9} +1.15877 q^{11} +6.03215 q^{13} +1.29689 q^{15} +0.0535246 q^{17} +2.08924 q^{19} +6.23980 q^{21} -3.22849 q^{23} -4.79347 q^{25} +6.11813 q^{27} -1.62034 q^{29} +2.01992 q^{31} +3.30683 q^{33} +0.993671 q^{35} -3.49384 q^{37} +17.2143 q^{39} +2.85650 q^{41} -2.78023 q^{43} +2.33765 q^{45} +9.03663 q^{47} -2.21910 q^{49} +0.152746 q^{51} -4.09037 q^{53} +0.526603 q^{55} +5.96218 q^{57} +4.76605 q^{59} -14.6219 q^{61} +11.2473 q^{63} +2.74132 q^{65} +4.98740 q^{67} -9.21331 q^{69} +2.04056 q^{71} -0.0937424 q^{73} -13.6794 q^{75} +2.53367 q^{77} +1.92521 q^{79} +2.02793 q^{81} -7.96981 q^{83} +0.0243243 q^{85} -4.62404 q^{87} -16.7462 q^{89} +13.1895 q^{91} +5.76433 q^{93} +0.949460 q^{95} +3.00301 q^{97} +5.96057 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 7 q + 3 q^{3} - 2 q^{5} + 6 q^{7}+O(q^{10}) $ 7 * q + 3 * q^3 - 2 * q^5 + 6 * q^7	$ 7 q + 3 q^{3} - 2 q^{5} + 6 q^{7} + 5 q^{11} - q^{13} + 6 q^{15} - 8 q^{17} + 15 q^{19} - 3 q^{21} + 5 q^{23} - 9 q^{25} + 9 q^{27} + 21 q^{31} + 7 q^{35} - q^{37} + 23 q^{39} - 10 q^{41} + 23 q^{43} - 4 q^{45} + 10 q^{47} - 13 q^{49} + 20 q^{51} - q^{53} + 23 q^{55} - 6 q^{57} + 4 q^{59} + 3 q^{61} + 4 q^{63} + 4 q^{65} + 28 q^{67} + 18 q^{69} + 18 q^{71} - 7 q^{73} - 11 q^{75} + 6 q^{77} + 30 q^{79} - 5 q^{81} - 13 q^{83} + q^{85} + 7 q^{87} + 18 q^{91} + 36 q^{93} - 2 q^{97} + 10 q^{99}+O(q^{100}) $ 7 * q + 3 * q^3 - 2 * q^5 + 6 * q^7 + 5 * q^11 - q^13 + 6 * q^15 - 8 * q^17 + 15 * q^19 - 3 * q^21 + 5 * q^23 - 9 * q^25 + 9 * q^27 + 21 * q^31 + 7 * q^35 - q^37 + 23 * q^39 - 10 * q^41 + 23 * q^43 - 4 * q^45 + 10 * q^47 - 13 * q^49 + 20 * q^51 - q^53 + 23 * q^55 - 6 * q^57 + 4 * q^59 + 3 * q^61 + 4 * q^63 + 4 * q^65 + 28 * q^67 + 18 * q^69 + 18 * q^71 - 7 * q^73 - 11 * q^75 + 6 * q^77 + 30 * q^79 - 5 * q^81 - 13 * q^83 + q^85 + 7 * q^87 + 18 * q^91 + 36 * q^93 - 2 * q^97 + 10 * q^99

Display 100 coefficients 10 coefficients

Coefficient data

For each $n$ we display the coefficients of the $q$-expansion $a_n$, the Satake parameters $\alpha_p$, and the Satake angles $\theta_p = \textrm{Arg}(\alpha_p)$.

Display $a_p$ with $p$ up to: 50 250 1000 (See $a_n$ instead) (See $a_n$ instead) (See $a_n$ instead) Display $a_n$ with $n$ up to: 50 250 1000 (See only $a_p$) (See only $a_p$) (See only $a_p$)

$n$	$a_n$	$a_n / n^{(k-1)/2}$	$ \alpha_n $			$ \theta_n $
$p$	$a_p$	$a_p / p^{(k-1)/2}$	$ \alpha_p$			$ \theta_p $

$2$	0	0
$2$	0	0
$3$	2.85375	1.64761	0.823807	−	0.566871i	$-0.191846\pi$
$3$	2.85375	1.64761	0.823807	+	0.566871i	$0.191846\pi$
$4$	0	0
$5$	0.454452	0.203237	0.101618	−	0.994823i	$-0.467598\pi$
$5$	0.454452	0.203237	0.101618	+	0.994823i	$0.467598\pi$
$6$	0	0
$7$	2.18653	0.826429	0.413215	−	0.910634i	$-0.364406\pi$
$7$	2.18653	0.826429	0.413215	+	0.910634i	$0.364406\pi$
$8$	0	0
$9$	5.14389	1.71463
$10$	0	0
$11$	1.15877	0.349381	0.174691	−	0.984623i	$-0.444107\pi$
$11$	1.15877	0.349381	0.174691	+	0.984623i	$0.444107\pi$
$12$	0	0
$13$	6.03215	1.67302	0.836509	−	0.547953i	$-0.184593\pi$
$13$	6.03215	1.67302	0.836509	+	0.547953i	$0.184593\pi$
$14$	0	0
$15$	1.29689	0.334856
$16$	0	0
$17$	0.0535246	0.0129816	0.00649081	−	0.999979i	$-0.497934\pi$
$17$	0.0535246	0.0129816	0.00649081	+	0.999979i	$0.497934\pi$
$18$	0	0
$19$	2.08924	0.479305	0.239653	−	0.970859i	$-0.422966\pi$
$19$	2.08924	0.479305	0.239653	+	0.970859i	$0.422966\pi$
$20$	0	0
$21$	6.23980	1.36164
$22$	0	0
$23$	−3.22849	−0.673187	−0.336593	−	0.941650i	$-0.609275\pi$
$23$	−3.22849	−0.673187	−0.336593	+	0.941650i	$0.609275\pi$
$24$	0	0
$25$	−4.79347	−0.958695
$26$	0	0
$27$	6.11813	1.17743
$28$	0	0
$29$	−1.62034	−0.300889	−0.150445	−	0.988618i	$-0.548071\pi$
$29$	−1.62034	−0.300889	−0.150445	+	0.988618i	$0.548071\pi$
$30$	0	0
$31$	2.01992	0.362788	0.181394	−	0.983411i	$-0.441939\pi$
$31$	2.01992	0.362788	0.181394	+	0.983411i	$0.441939\pi$
$32$	0	0
$33$	3.30683	0.575645
$34$	0	0
$35$	0.993671	0.167961
$36$	0	0
$37$	−3.49384	−0.574383	−0.287192	−	0.957873i	$-0.592722\pi$
$37$	−3.49384	−0.574383	−0.287192	+	0.957873i	$0.592722\pi$
$38$	0	0
$39$	17.2143	2.75649
$40$	0	0
$41$	2.85650	0.446111	0.223055	−	0.974806i	$-0.428397\pi$
$41$	2.85650	0.446111	0.223055	+	0.974806i	$0.428397\pi$
$42$	0	0
$43$	−2.78023	−0.423980	−0.211990	−	0.977272i	$-0.567995\pi$
$43$	−2.78023	−0.423980	−0.211990	+	0.977272i	$0.567995\pi$
$44$	0	0
$45$	2.33765	0.348476
$46$	0	0
$47$	9.03663	1.31813	0.659064	−	0.752087i	$-0.270953\pi$
$47$	9.03663	1.31813	0.659064	+	0.752087i	$0.270953\pi$
$48$	0	0
$49$	−2.21910	−0.317014
$50$	0	0
$51$	0.152746	0.0213887
$52$	0	0
$53$	−4.09037	−0.561855	−0.280927	−	0.959729i	$-0.590642\pi$
$53$	−4.09037	−0.561855	−0.280927	+	0.959729i	$0.590642\pi$
$54$	0	0
$55$	0.526603	0.0710072
$56$	0	0
$57$	5.96218	0.789710
$58$	0	0
$59$	4.76605	0.620487	0.310244	−	0.950657i	$-0.399589\pi$
$59$	4.76605	0.620487	0.310244	+	0.950657i	$0.399589\pi$
$60$	0	0
$61$	−14.6219	−1.87214	−0.936068	−	0.351819i	$-0.885563\pi$
$61$	−14.6219	−1.87214	−0.936068	+	0.351819i	$0.885563\pi$
$62$	0	0
$63$	11.2473	1.41702
$64$	0	0
$65$	2.74132	0.340019
$66$	0	0
$67$	4.98740	0.609308	0.304654	−	0.952463i	$-0.401459\pi$
$67$	4.98740	0.609308	0.304654	+	0.952463i	$0.401459\pi$
$68$	0	0
$69$	−9.21331	−1.10915
$70$	0	0
$71$	2.04056	0.242170	0.121085	−	0.992642i	$-0.461363\pi$
$71$	2.04056	0.242170	0.121085	+	0.992642i	$0.461363\pi$
$72$	0	0
$73$	−0.0937424	−0.0109717	−0.00548586	−	0.999985i	$-0.501746\pi$
$73$	−0.0937424	−0.0109717	−0.00548586	+	0.999985i	$0.501746\pi$
$74$	0	0
$75$	−13.6794	−1.57956
$76$	0	0
$77$	2.53367	0.288739
$78$	0	0
$79$	1.92521	0.216603	0.108302	−	0.994118i	$-0.465459\pi$
$79$	1.92521	0.216603	0.108302	+	0.994118i	$0.465459\pi$
$80$	0	0
$81$	2.02793	0.225326
$82$	0	0
$83$	−7.96981	−0.874801	−0.437400	−	0.899267i	$-0.644101\pi$
$83$	−7.96981	−0.874801	−0.437400	+	0.899267i	$0.644101\pi$
$84$	0	0
$85$	0.0243243	0.00263834
$86$	0	0
$87$	−4.62404	−0.495749
$88$	0	0
$89$	−16.7462	−1.77510	−0.887549	−	0.460713i	$-0.847594\pi$
$89$	−16.7462	−1.77510	−0.887549	+	0.460713i	$0.847594\pi$
$90$	0	0
$91$	13.1895	1.38263
$92$	0	0
$93$	5.76433	0.597734
$94$	0	0
$95$	0.949460	0.0974125
$96$	0	0
$97$	3.00301	0.304910	0.152455	−	0.988310i	$-0.451282\pi$
$97$	3.00301	0.304910	0.152455	+	0.988310i	$0.451282\pi$
$98$	0	0
$99$	5.96057	0.599059
$100$	0	0
$101$	−3.55655	−0.353890	−0.176945	−	0.984221i	$-0.556621\pi$
$101$	−3.55655	−0.353890	−0.176945	+	0.984221i	$0.556621\pi$
$102$	0	0
$103$	3.10568	0.306011	0.153006	−	0.988225i	$-0.451105\pi$
$103$	3.10568	0.306011	0.153006	+	0.988225i	$0.451105\pi$
$104$	0	0
$105$	2.83569	0.276735
$106$	0	0
$107$	−10.8110	−1.04514	−0.522571	−	0.852596i	$-0.675027\pi$
$107$	−10.8110	−1.04514	−0.522571	+	0.852596i	$0.675027\pi$
$108$	0	0
$109$	7.53872	0.722078	0.361039	−	0.932551i	$-0.382422\pi$
$109$	7.53872	0.722078	0.361039	+	0.932551i	$0.382422\pi$
$110$	0	0
$111$	−9.97054	−0.946362
$112$	0	0
$113$	−3.30774	−0.311166	−0.155583	−	0.987823i	$-0.549726\pi$
$113$	−3.30774	−0.311166	−0.155583	+	0.987823i	$0.549726\pi$
$114$	0	0
$115$	−1.46719	−0.136816
$116$	0	0
$117$	31.0287	2.86861
$118$	0	0
$119$	0.117033	0.0107284
$120$	0	0
$121$	−9.65726	−0.877933
$122$	0	0
$123$	8.15174	0.735018
$124$	0	0
$125$	−4.45066	−0.398079
$126$	0	0
$127$	13.1951	1.17087	0.585437	−	0.810718i	$-0.300923\pi$
$127$	13.1951	1.17087	0.585437	+	0.810718i	$0.300923\pi$
$128$	0	0
$129$	−7.93407	−0.698556
$130$	0	0
$131$	−4.18286	−0.365458	−0.182729	−	0.983163i	$-0.558493\pi$
$131$	−4.18286	−0.365458	−0.182729	+	0.983163i	$0.558493\pi$
$132$	0	0
$133$	4.56819	0.396112
$134$	0	0
$135$	2.78039	0.239298
$136$	0	0
$137$	−12.2951	−1.05044	−0.525219	−	0.850967i	$-0.676017\pi$
$137$	−12.2951	−1.05044	−0.525219	+	0.850967i	$0.676017\pi$
$138$	0	0
$139$	−6.49640	−0.551018	−0.275509	−	0.961299i	$-0.588846\pi$
$139$	−6.49640	−0.551018	−0.275509	+	0.961299i	$0.588846\pi$
$140$	0	0
$141$	25.7883	2.17176
$142$	0	0
$143$	6.98985	0.584521
$144$	0	0
$145$	−0.736365	−0.0611518
$146$	0	0
$147$	−6.33276	−0.522317
$148$	0	0
$149$	11.0170	0.902548	0.451274	−	0.892385i	$-0.350970\pi$
$149$	11.0170	0.902548	0.451274	+	0.892385i	$0.350970\pi$
$150$	0	0
$151$	21.4320	1.74411	0.872055	−	0.489408i	$-0.162787\pi$
$151$	21.4320	1.74411	0.872055	+	0.489408i	$0.162787\pi$
$152$	0	0
$153$	0.275324	0.0222587
$154$	0	0
$155$	0.917954	0.0737318
$156$	0	0
$157$	11.7026	0.933967	0.466984	−	0.884266i	$-0.345341\pi$
$157$	11.7026	0.933967	0.466984	+	0.884266i	$0.345341\pi$
$158$	0	0
$159$	−11.6729	−0.925720
$160$	0	0
$161$	−7.05918	−0.556341
$162$	0	0
$163$	3.32371	0.260333	0.130167	−	0.991492i	$-0.458449\pi$
$163$	3.32371	0.260333	0.130167	+	0.991492i	$0.458449\pi$
$164$	0	0
$165$	1.50279	0.116992
$166$	0	0
$167$	−23.4924	−1.81789	−0.908947	−	0.416913i	$-0.863112\pi$
$167$	−23.4924	−1.81789	−0.908947	+	0.416913i	$0.863112\pi$
$168$	0	0
$169$	23.3869	1.79899
$170$	0	0
$171$	10.7468	0.821831
$172$	0	0
$173$	−4.48627	−0.341085	−0.170543	−	0.985350i	$-0.554552\pi$
$173$	−4.48627	−0.341085	−0.170543	+	0.985350i	$0.554552\pi$
$174$	0	0
$175$	−10.4811	−0.792294
$176$	0	0
$177$	13.6011	1.02232
$178$	0	0
$179$	6.40995	0.479102	0.239551	−	0.970884i	$-0.423000\pi$
$179$	6.40995	0.479102	0.239551	+	0.970884i	$0.423000\pi$
$180$	0	0
$181$	6.25690	0.465072	0.232536	−	0.972588i	$-0.425298\pi$
$181$	6.25690	0.465072	0.232536	+	0.972588i	$0.425298\pi$
$182$	0	0
$183$	−41.7271	−3.08456
$184$	0	0
$185$	−1.58778	−0.116736
$186$	0	0
$187$	0.0620225	0.00453553
$188$	0	0
$189$	13.3774	0.973066
$190$	0	0
$191$	11.8680	0.858738	0.429369	−	0.903129i	$-0.358736\pi$
$191$	11.8680	0.858738	0.429369	+	0.903129i	$0.358736\pi$
$192$	0	0
$193$	7.71261	0.555166	0.277583	−	0.960702i	$-0.410467\pi$
$193$	7.71261	0.555166	0.277583	+	0.960702i	$0.410467\pi$
$194$	0	0
$195$	7.82305	0.560220
$196$	0	0
$197$	11.2830	0.803881	0.401941	−	0.915666i	$-0.368336\pi$
$197$	11.2830	0.803881	0.401941	+	0.915666i	$0.368336\pi$
$198$	0	0
$199$	4.78352	0.339095	0.169547	−	0.985522i	$-0.445769\pi$
$199$	4.78352	0.339095	0.169547	+	0.985522i	$0.445769\pi$
$200$	0	0
$201$	14.2328	1.00390
$202$	0	0
$203$	−3.54291	−0.248664
$204$	0	0
$205$	1.29814	0.0906661
$206$	0	0
$207$	−16.6070	−1.15427
$208$	0	0
$209$	2.42094	0.167460
$210$	0	0
$211$	−12.7879	−0.880358	−0.440179	−	0.897910i	$-0.645085\pi$
$211$	−12.7879	−0.880358	−0.440179	+	0.897910i	$0.645085\pi$
$212$	0	0
$213$	5.82324	0.399002
$214$	0	0
$215$	−1.26348	−0.0861685
$216$	0	0
$217$	4.41660	0.299818
$218$	0	0
$219$	−0.267517	−0.0180772
$220$	0	0
$221$	0.322868	0.0217185
$222$	0	0
$223$	22.5770	1.51187	0.755933	−	0.654649i	$-0.227183\pi$
$223$	22.5770	1.51187	0.755933	+	0.654649i	$0.227183\pi$
$224$	0	0
$225$	−24.6571	−1.64381
$226$	0	0
$227$	−23.7505	−1.57637	−0.788186	−	0.615437i	$-0.788980\pi$
$227$	−23.7505	−1.57637	−0.788186	+	0.615437i	$0.788980\pi$
$228$	0	0
$229$	−6.30407	−0.416585	−0.208292	−	0.978067i	$-0.566791\pi$
$229$	−6.30407	−0.416585	−0.208292	+	0.978067i	$0.566791\pi$
$230$	0	0
$231$	7.23047	0.475730
$232$	0	0
$233$	−16.4538	−1.07793	−0.538963	−	0.842329i	$-0.681184\pi$
$233$	−16.4538	−1.07793	−0.538963	+	0.842329i	$0.681184\pi$
$234$	0	0
$235$	4.10671	0.267892
$236$	0	0
$237$	5.49407	0.356878
$238$	0	0
$239$	1.59054	0.102884	0.0514418	−	0.998676i	$-0.483618\pi$
$239$	1.59054	0.102884	0.0514418	+	0.998676i	$0.483618\pi$
$240$	0	0
$241$	3.38944	0.218333	0.109167	−	0.994023i	$-0.465182\pi$
$241$	3.38944	0.218333	0.109167	+	0.994023i	$0.465182\pi$
$242$	0	0
$243$	−12.5672	−0.806184
$244$	0	0
$245$	−1.00847	−0.0644291
$246$	0	0
$247$	12.6026	0.801886
$248$	0	0
$249$	−22.7439	−1.44133
$250$	0	0
$251$	−1.00000	−0.0631194
$251$	−1.00000	−0.0631194
$252$	0	0
$253$	−3.74107	−0.235199
$254$	0	0
$255$	0.0694156	0.00434697
$256$	0	0
$257$	26.8858	1.67709	0.838546	−	0.544831i	$-0.183406\pi$
$257$	26.8858	1.67709	0.838546	+	0.544831i	$0.183406\pi$
$258$	0	0
$259$	−7.63937	−0.474687
$260$	0	0
$261$	−8.33484	−0.515913
$262$	0	0
$263$	−4.39336	−0.270906	−0.135453	−	0.990784i	$-0.543249\pi$
$263$	−4.39336	−0.270906	−0.135453	+	0.990784i	$0.543249\pi$
$264$	0	0
$265$	−1.85887	−0.114190
$266$	0	0
$267$	−47.7896	−2.92468
$268$	0	0
$269$	22.6845	1.38310	0.691549	−	0.722329i	$-0.256928\pi$
$269$	22.6845	1.38310	0.691549	+	0.722329i	$0.256928\pi$
$270$	0	0
$271$	15.0589	0.914763	0.457382	−	0.889271i	$-0.348787\pi$
$271$	15.0589	0.914763	0.457382	+	0.889271i	$0.348787\pi$
$272$	0	0
$273$	37.6394	2.27804
$274$	0	0
$275$	−5.55452	−0.334950
$276$	0	0
$277$	−10.7918	−0.648419	−0.324210	−	0.945985i	$-0.605098\pi$
$277$	−10.7918	−0.648419	−0.324210	+	0.945985i	$0.605098\pi$
$278$	0	0
$279$	10.3902	0.622046
$280$	0	0
$281$	−21.9382	−1.30872	−0.654362	−	0.756182i	$-0.727063\pi$
$281$	−21.9382	−1.30872	−0.654362	+	0.756182i	$0.727063\pi$
$282$	0	0
$283$	12.1997	0.725198	0.362599	−	0.931945i	$-0.381889\pi$
$283$	12.1997	0.725198	0.362599	+	0.931945i	$0.381889\pi$
$284$	0	0
$285$	2.70952	0.160498
$286$	0	0
$287$	6.24582	0.368679
$288$	0	0
$289$	−16.9971	−0.999831
$290$	0	0
$291$	8.56985	0.502374
$292$	0	0
$293$	29.4675	1.72151	0.860756	−	0.509017i	$-0.169991\pi$
$293$	29.4675	1.72151	0.860756	+	0.509017i	$0.169991\pi$
$294$	0	0
$295$	2.16594	0.126106
$296$	0	0
$297$	7.08948	0.411373
$298$	0	0
$299$	−19.4747	−1.12625
$300$	0	0
$301$	−6.07904	−0.350390
$302$	0	0
$303$	−10.1495	−0.583074
$304$	0	0
$305$	−6.64492	−0.380487
$306$	0	0
$307$	21.3436	1.21814	0.609072	−	0.793115i	$-0.291542\pi$
$307$	21.3436	1.21814	0.609072	+	0.793115i	$0.291542\pi$
$308$	0	0
$309$	8.86282	0.504188
$310$	0	0
$311$	30.9771	1.75655	0.878275	−	0.478155i	$-0.158694\pi$
$311$	30.9771	1.75655	0.878275	+	0.478155i	$0.158694\pi$
$312$	0	0
$313$	−16.2372	−0.917783	−0.458892	−	0.888492i	$-0.651753\pi$
$313$	−16.2372	−0.917783	−0.458892	+	0.888492i	$0.651753\pi$
$314$	0	0
$315$	5.11133	0.287991
$316$	0	0
$317$	−27.5305	−1.54627	−0.773133	−	0.634244i	$-0.781312\pi$
$317$	−27.5305	−1.54627	−0.773133	+	0.634244i	$0.781312\pi$
$318$	0	0
$319$	−1.87759	−0.105125
$320$	0	0
$321$	−30.8520	−1.72199
$322$	0	0
$323$	0.111826	0.00622216
$324$	0	0
$325$	−28.9150	−1.60391
$326$	0	0
$327$	21.5136	1.18971
$328$	0	0
$329$	19.7588	1.08934
$330$	0	0
$331$	31.9237	1.75469	0.877344	−	0.479862i	$-0.159313\pi$
$331$	31.9237	1.75469	0.877344	+	0.479862i	$0.159313\pi$
$332$	0	0
$333$	−17.9719	−0.984855
$334$	0	0
$335$	2.26653	0.123834
$336$	0	0
$337$	−18.7610	−1.02198	−0.510990	−	0.859587i	$-0.670721\pi$
$337$	−18.7610	−1.02198	−0.510990	+	0.859587i	$0.670721\pi$
$338$	0	0
$339$	−9.43947	−0.512682
$340$	0	0
$341$	2.34061	0.126751
$342$	0	0
$343$	−20.1578	−1.08842
$344$	0	0
$345$	−4.18700	−0.225421
$346$	0	0
$347$	4.43093	0.237865	0.118932	−	0.992902i	$-0.462053\pi$
$347$	4.43093	0.237865	0.118932	+	0.992902i	$0.462053\pi$
$348$	0	0
$349$	−13.3203	−0.713018	−0.356509	−	0.934292i	$-0.616033\pi$
$349$	−13.3203	−0.713018	−0.356509	+	0.934292i	$0.616033\pi$
$350$	0	0
$351$	36.9055	1.96987
$352$	0	0
$353$	21.3691	1.13736	0.568681	−	0.822558i	$-0.307454\pi$
$353$	21.3691	1.13736	0.568681	+	0.822558i	$0.307454\pi$
$354$	0	0
$355$	0.927334	0.0492178
$356$	0	0
$357$	0.333983	0.0176762
$358$	0	0
$359$	6.72114	0.354728	0.177364	−	0.984145i	$-0.443243\pi$
$359$	6.72114	0.354728	0.177364	+	0.984145i	$0.443243\pi$
$360$	0	0
$361$	−14.6351	−0.770267
$362$	0	0
$363$	−27.5594	−1.44649
$364$	0	0
$365$	−0.0426014	−0.00222986
$366$	0	0
$367$	−4.60106	−0.240174	−0.120087	−	0.992763i	$-0.538317\pi$
$367$	−4.60106	−0.240174	−0.120087	+	0.992763i	$0.538317\pi$
$368$	0	0
$369$	14.6935	0.764914
$370$	0	0
$371$	−8.94369	−0.464333
$372$	0	0
$373$	−25.0550	−1.29730	−0.648650	−	0.761087i	$-0.724666\pi$
$373$	−25.0550	−1.29730	−0.648650	+	0.761087i	$0.724666\pi$
$374$	0	0
$375$	−12.7011	−0.655881
$376$	0	0
$377$	−9.77412	−0.503393
$378$	0	0
$379$	10.3050	0.529333	0.264666	−	0.964340i	$-0.414738\pi$
$379$	10.3050	0.529333	0.264666	+	0.964340i	$0.414738\pi$
$380$	0	0
$381$	37.6554	1.92915
$382$	0	0
$383$	−9.92110	−0.506944	−0.253472	−	0.967343i	$-0.581573\pi$
$383$	−9.92110	−0.506944	−0.253472	+	0.967343i	$0.581573\pi$
$384$	0	0
$385$	1.15143	0.0586824
$386$	0	0
$387$	−14.3012	−0.726970
$388$	0	0
$389$	−33.3777	−1.69232	−0.846158	−	0.532932i	$-0.821090\pi$
$389$	−33.3777	−1.69232	−0.846158	+	0.532932i	$0.821090\pi$
$390$	0	0
$391$	−0.172804	−0.00873905
$392$	0	0
$393$	−11.9368	−0.602134
$394$	0	0
$395$	0.874915	0.0440218
$396$	0	0
$397$	−2.43725	−0.122322	−0.0611611	−	0.998128i	$-0.519480\pi$
$397$	−2.43725	−0.122322	−0.0611611	+	0.998128i	$0.519480\pi$
$398$	0	0
$399$	13.0365	0.652639
$400$	0	0
$401$	11.0399	0.551306	0.275653	−	0.961257i	$-0.411106\pi$
$401$	11.0399	0.551306	0.275653	+	0.961257i	$0.411106\pi$
$402$	0	0
$403$	12.1844	0.606950
$404$	0	0
$405$	0.921597	0.0457945
$406$	0	0
$407$	−4.04854	−0.200679
$408$	0	0
$409$	−7.10281	−0.351211	−0.175606	−	0.984461i	$-0.556188\pi$
$409$	−7.10281	−0.351211	−0.175606	+	0.984461i	$0.556188\pi$
$410$	0	0
$411$	−35.0871	−1.73072
$412$	0	0
$413$	10.4211	0.512789
$414$	0	0
$415$	−3.62189	−0.177792
$416$	0	0
$417$	−18.5391	−0.907864
$418$	0	0
$419$	−22.4900	−1.09871	−0.549354	−	0.835590i	$-0.685126\pi$
$419$	−22.4900	−1.09871	−0.549354	+	0.835590i	$0.685126\pi$
$420$	0	0
$421$	−23.3162	−1.13636	−0.568180	−	0.822904i	$-0.692352\pi$
$421$	−23.3162	−1.13636	−0.568180	+	0.822904i	$0.692352\pi$
$422$	0	0
$423$	46.4834	2.26010
$424$	0	0
$425$	−0.256569	−0.0124454
$426$	0	0
$427$	−31.9711	−1.54719
$428$	0	0
$429$	19.9473	0.963065
$430$	0	0
$431$	−5.63794	−0.271570	−0.135785	−	0.990738i	$-0.543356\pi$
$431$	−5.63794	−0.271570	−0.135785	+	0.990738i	$0.543356\pi$
$432$	0	0
$433$	26.5522	1.27602	0.638009	−	0.770029i	$-0.279758\pi$
$433$	26.5522	1.27602	0.638009	+	0.770029i	$0.279758\pi$
$434$	0	0
$435$	−2.10140	−0.100755
$436$	0	0
$437$	−6.74510	−0.322662
$438$	0	0
$439$	−37.8832	−1.80807	−0.904034	−	0.427462i	$-0.859408\pi$
$439$	−37.8832	−1.80807	−0.904034	+	0.427462i	$0.859408\pi$
$440$	0	0
$441$	−11.4148	−0.543562
$442$	0	0
$443$	−37.3614	−1.77509	−0.887547	−	0.460717i	$-0.847592\pi$
$443$	−37.3614	−1.77509	−0.887547	+	0.460717i	$0.847592\pi$
$444$	0	0
$445$	−7.61036	−0.360766
$446$	0	0
$447$	31.4398	1.48705
$448$	0	0
$449$	3.53079	0.166628	0.0833141	−	0.996523i	$-0.473450\pi$
$449$	3.53079	0.166628	0.0833141	+	0.996523i	$0.473450\pi$
$450$	0	0
$451$	3.31002	0.155863
$452$	0	0
$453$	61.1615	2.87362
$454$	0	0
$455$	5.99397	0.281002
$456$	0	0
$457$	19.8957	0.930681	0.465341	−	0.885132i	$-0.345932\pi$
$457$	19.8957	0.930681	0.465341	+	0.885132i	$0.345932\pi$
$458$	0	0
$459$	0.327470	0.0152850
$460$	0	0
$461$	25.9530	1.20875	0.604375	−	0.796700i	$-0.293423\pi$
$461$	25.9530	1.20875	0.604375	+	0.796700i	$0.293423\pi$
$462$	0	0
$463$	23.5757	1.09566	0.547828	−	0.836591i	$-0.315455\pi$
$463$	23.5757	1.09566	0.547828	+	0.836591i	$0.315455\pi$
$464$	0	0
$465$	2.61961	0.121482
$466$	0	0
$467$	−3.93839	−0.182247	−0.0911234	−	0.995840i	$-0.529046\pi$
$467$	−3.93839	−0.182247	−0.0911234	+	0.995840i	$0.529046\pi$
$468$	0	0
$469$	10.9051	0.503550
$470$	0	0
$471$	33.3962	1.53882
$472$	0	0
$473$	−3.22163	−0.148131
$474$	0	0
$475$	−10.0147	−0.459507
$476$	0	0
$477$	−21.0404	−0.963373
$478$	0	0
$479$	−17.7826	−0.812506	−0.406253	−	0.913761i	$-0.633165\pi$
$479$	−17.7826	−0.812506	−0.406253	+	0.913761i	$0.633165\pi$
$480$	0	0
$481$	−21.0754	−0.960954
$482$	0	0
$483$	−20.1451	−0.916636
$484$	0	0
$485$	1.36472	0.0619689
$486$	0	0
$487$	−38.4530	−1.74247	−0.871237	−	0.490863i	$-0.836681\pi$
$487$	−38.4530	−1.74247	−0.871237	+	0.490863i	$0.836681\pi$
$488$	0	0
$489$	9.48504	0.428928
$490$	0	0
$491$	−4.93938	−0.222911	−0.111456	−	0.993769i	$-0.535551\pi$
$491$	−4.93938	−0.222911	−0.111456	+	0.993769i	$0.535551\pi$
$492$	0	0
$493$	−0.0867279	−0.00390603
$494$	0	0
$495$	2.70879	0.121751
$496$	0	0
$497$	4.46173	0.200136
$498$	0	0
$499$	9.78960	0.438243	0.219121	−	0.975698i	$-0.429681\pi$
$499$	9.78960	0.438243	0.219121	+	0.975698i	$0.429681\pi$
$500$	0	0
$501$	−67.0413	−2.99519
$502$	0	0
$503$	−16.9870	−0.757412	−0.378706	−	0.925517i	$-0.623631\pi$
$503$	−16.9870	−0.757412	−0.378706	+	0.925517i	$0.623631\pi$
$504$	0	0
$505$	−1.61628	−0.0719235
$506$	0	0
$507$	66.7403	2.96404
$508$	0	0
$509$	20.2558	0.897822	0.448911	−	0.893577i	$-0.351812\pi$
$509$	20.2558	0.897822	0.448911	+	0.893577i	$0.351812\pi$
$510$	0	0
$511$	−0.204970	−0.00906735
$512$	0	0
$513$	12.7823	0.564350
$514$	0	0
$515$	1.41138	0.0621928
$516$	0	0
$517$	10.4713	0.460529
$518$	0	0
$519$	−12.8027	−0.561976
$520$	0	0
$521$	−14.9138	−0.653384	−0.326692	−	0.945131i	$-0.605934\pi$
$521$	−14.9138	−0.653384	−0.326692	+	0.945131i	$0.605934\pi$
$522$	0	0
$523$	30.4438	1.33121	0.665607	−	0.746302i	$-0.268173\pi$
$523$	30.4438	1.33121	0.665607	+	0.746302i	$0.268173\pi$
$524$	0	0
$525$	−29.9103	−1.30539
$526$	0	0
$527$	0.108115	0.00470957
$528$	0	0
$529$	−12.5768	−0.546820
$530$	0	0
$531$	24.5160	1.06391
$532$	0	0
$533$	17.2308	0.746351
$534$	0	0
$535$	−4.91309	−0.212412
$536$	0	0
$537$	18.2924	0.789375
$538$	0	0
$539$	−2.57142	−0.110759
$540$	0	0
$541$	2.65322	0.114071	0.0570353	−	0.998372i	$-0.481835\pi$
$541$	2.65322	0.114071	0.0570353	+	0.998372i	$0.481835\pi$
$542$	0	0
$543$	17.8556	0.766259
$544$	0	0
$545$	3.42598	0.146753
$546$	0	0
$547$	−18.8391	−0.805500	−0.402750	−	0.915310i	$-0.631946\pi$
$547$	−18.8391	−0.805500	−0.402750	+	0.915310i	$0.631946\pi$
$548$	0	0
$549$	−75.2132	−3.21002
$550$	0	0
$551$	−3.38528	−0.144218
$552$	0	0
$553$	4.20952	0.179007
$554$	0	0
$555$	−4.53113	−0.192336
$556$	0	0
$557$	−3.30321	−0.139962	−0.0699808	−	0.997548i	$-0.522294\pi$
$557$	−3.30321	−0.139962	−0.0699808	+	0.997548i	$0.522294\pi$
$558$	0	0
$559$	−16.7707	−0.709327
$560$	0	0
$561$	0.176997	0.00747280
$562$	0	0
$563$	−8.19333	−0.345308	−0.172654	−	0.984983i	$-0.555234\pi$
$563$	−8.19333	−0.345308	−0.172654	+	0.984983i	$0.555234\pi$
$564$	0	0
$565$	−1.50321	−0.0632405
$566$	0	0
$567$	4.43413	0.186216
$568$	0	0
$569$	−26.3058	−1.10280	−0.551398	−	0.834242i	$-0.685906\pi$
$569$	−26.3058	−1.10280	−0.551398	+	0.834242i	$0.685906\pi$
$570$	0	0
$571$	29.5821	1.23797	0.618987	−	0.785401i	$-0.287543\pi$
$571$	29.5821	1.23797	0.618987	+	0.785401i	$0.287543\pi$
$572$	0	0
$573$	33.8683	1.41487
$574$	0	0
$575$	15.4757	0.645381
$576$	0	0
$577$	22.7949	0.948964	0.474482	−	0.880265i	$-0.342635\pi$
$577$	22.7949	0.948964	0.474482	+	0.880265i	$0.342635\pi$
$578$	0	0
$579$	22.0099	0.914699
$580$	0	0
$581$	−17.4262	−0.722961
$582$	0	0
$583$	−4.73978	−0.196302
$584$	0	0
$585$	14.1011	0.583007
$586$	0	0
$587$	21.3149	0.879761	0.439881	−	0.898056i	$-0.355021\pi$
$587$	21.3149	0.879761	0.439881	+	0.898056i	$0.355021\pi$
$588$	0	0
$589$	4.22009	0.173886
$590$	0	0
$591$	32.1989	1.32449
$592$	0	0
$593$	−4.84036	−0.198770	−0.0993850	−	0.995049i	$-0.531688\pi$
$593$	−4.84036	−0.198770	−0.0993850	+	0.995049i	$0.531688\pi$
$594$	0	0
$595$	0.0531858	0.00218040
$596$	0	0
$597$	13.6510	0.558697
$598$	0	0
$599$	−43.7104	−1.78596	−0.892978	−	0.450100i	$-0.851388\pi$
$599$	−43.7104	−1.78596	−0.892978	+	0.450100i	$0.851388\pi$
$600$	0	0
$601$	−16.0174	−0.653365	−0.326682	−	0.945134i	$-0.605931\pi$
$601$	−16.0174	−0.653365	−0.326682	+	0.945134i	$0.605931\pi$
$602$	0	0
$603$	25.6546	1.04474
$604$	0	0
$605$	−4.38876	−0.178428
$606$	0	0
$607$	3.55146	0.144149	0.0720747	−	0.997399i	$-0.477038\pi$
$607$	3.55146	0.144149	0.0720747	+	0.997399i	$0.477038\pi$
$608$	0	0
$609$	−10.1106	−0.409702
$610$	0	0
$611$	54.5103	2.20525
$612$	0	0
$613$	20.6687	0.834801	0.417400	−	0.908723i	$-0.362941\pi$
$613$	20.6687	0.834801	0.417400	+	0.908723i	$0.362941\pi$
$614$	0	0
$615$	3.70457	0.149383
$616$	0	0
$617$	−34.0094	−1.36917	−0.684583	−	0.728935i	$-0.740016\pi$
$617$	−34.0094	−1.36917	−0.684583	+	0.728935i	$0.740016\pi$
$618$	0	0
$619$	−1.90390	−0.0765243	−0.0382621	−	0.999268i	$-0.512182\pi$
$619$	−1.90390	−0.0765243	−0.0382621	+	0.999268i	$0.512182\pi$
$620$	0	0
$621$	−19.7523	−0.792633
$622$	0	0
$623$	−36.6161	−1.46699
$624$	0	0
$625$	21.9448	0.877790
$626$	0	0
$627$	6.90877	0.275910
$628$	0	0
$629$	−0.187006	−0.00745642
$630$	0	0
$631$	−11.4055	−0.454048	−0.227024	−	0.973889i	$-0.572900\pi$
$631$	−11.4055	−0.454048	−0.227024	+	0.973889i	$0.572900\pi$
$632$	0	0
$633$	−36.4936	−1.45049
$634$	0	0
$635$	5.99652	0.237965
$636$	0	0
$637$	−13.3860	−0.530371
$638$	0	0
$639$	10.4964	0.415231
$640$	0	0
$641$	1.81401	0.0716491	0.0358245	−	0.999358i	$-0.488594\pi$
$641$	1.81401	0.0716491	0.0358245	+	0.999358i	$0.488594\pi$
$642$	0	0
$643$	10.4605	0.412523	0.206261	−	0.978497i	$-0.433870\pi$
$643$	10.4605	0.412523	0.206261	+	0.978497i	$0.433870\pi$
$644$	0	0
$645$	−3.60565	−0.141972
$646$	0	0
$647$	22.7209	0.893249	0.446624	−	0.894722i	$-0.352626\pi$
$647$	22.7209	0.893249	0.446624	+	0.894722i	$0.352626\pi$
$648$	0	0
$649$	5.52274	0.216787
$650$	0	0
$651$	12.6039	0.493985
$652$	0	0
$653$	12.8539	0.503013	0.251507	−	0.967856i	$-0.419074\pi$
$653$	12.8539	0.503013	0.251507	+	0.967856i	$0.419074\pi$
$654$	0	0
$655$	−1.90091	−0.0742746
$656$	0	0
$657$	−0.482201	−0.0188124
$658$	0	0
$659$	−33.7873	−1.31617	−0.658083	−	0.752946i	$-0.728632\pi$
$659$	−33.7873	−1.31617	−0.658083	+	0.752946i	$0.728632\pi$
$660$	0	0
$661$	27.6933	1.07714	0.538572	−	0.842580i	$-0.318964\pi$
$661$	27.6933	1.07714	0.538572	+	0.842580i	$0.318964\pi$
$662$	0	0
$663$	0.921386	0.0357836
$664$	0	0
$665$	2.07602	0.0805046
$666$	0	0
$667$	5.23124	0.202555
$668$	0	0
$669$	64.4291	2.49097
$670$	0	0
$671$	−16.9433	−0.654089
$672$	0	0
$673$	17.1280	0.660237	0.330118	−	0.943940i	$-0.392911\pi$
$673$	17.1280	0.660237	0.330118	+	0.943940i	$0.392911\pi$
$674$	0	0
$675$	−29.3271	−1.12880
$676$	0	0
$677$	10.9570	0.421111	0.210555	−	0.977582i	$-0.432473\pi$
$677$	10.9570	0.421111	0.210555	+	0.977582i	$0.432473\pi$
$678$	0	0
$679$	6.56617	0.251986
$680$	0	0
$681$	−67.7779	−2.59725
$682$	0	0
$683$	34.6594	1.32620	0.663102	−	0.748529i	$-0.269240\pi$
$683$	34.6594	1.32620	0.663102	+	0.748529i	$0.269240\pi$
$684$	0	0
$685$	−5.58752	−0.213488
$686$	0	0
$687$	−17.9902	−0.686371
$688$	0	0
$689$	−24.6737	−0.939993
$690$	0	0
$691$	−9.23393	−0.351275	−0.175638	−	0.984455i	$-0.556199\pi$
$691$	−9.23393	−0.351275	−0.175638	+	0.984455i	$0.556199\pi$
$692$	0	0
$693$	13.0329	0.495080
$694$	0	0
$695$	−2.95230	−0.111987
$696$	0	0
$697$	0.152893	0.00579123
$698$	0	0
$699$	−46.9551	−1.77601
$700$	0	0
$701$	13.2605	0.500842	0.250421	−	0.968137i	$-0.419431\pi$
$701$	13.2605	0.500842	0.250421	+	0.968137i	$0.419431\pi$
$702$	0	0
$703$	−7.29947	−0.275305
$704$	0	0
$705$	11.7195	0.441383
$706$	0	0
$707$	−7.77649	−0.292465
$708$	0	0
$709$	46.0334	1.72882	0.864410	−	0.502787i	$-0.167692\pi$
$709$	46.0334	1.72882	0.864410	+	0.502787i	$0.167692\pi$
$710$	0	0
$711$	9.90307	0.371394
$712$	0	0
$713$	−6.52128	−0.244224
$714$	0	0
$715$	3.17655	0.118796
$716$	0	0
$717$	4.53901	0.169513
$718$	0	0
$719$	1.81518	0.0676949	0.0338474	−	0.999427i	$-0.489224\pi$
$719$	1.81518	0.0676949	0.0338474	+	0.999427i	$0.489224\pi$
$720$	0	0
$721$	6.79064	0.252897
$722$	0	0
$723$	9.67262	0.359729
$724$	0	0
$725$	7.76705	0.288461
$726$	0	0
$727$	34.6521	1.28518	0.642588	−	0.766212i	$-0.277861\pi$
$727$	34.6521	1.28518	0.642588	+	0.766212i	$0.277861\pi$
$728$	0	0
$729$	−41.9473	−1.55361
$730$	0	0
$731$	−0.148810	−0.00550395
$732$	0	0
$733$	26.5655	0.981220	0.490610	−	0.871379i	$-0.336774\pi$
$733$	26.5655	0.981220	0.490610	+	0.871379i	$0.336774\pi$
$734$	0	0
$735$	−2.87793	−0.106154
$736$	0	0
$737$	5.77923	0.212881
$738$	0	0
$739$	−14.3106	−0.526424	−0.263212	−	0.964738i	$-0.584782\pi$
$739$	−14.3106	−0.526424	−0.263212	+	0.964738i	$0.584782\pi$
$740$	0	0
$741$	35.9648	1.32120
$742$	0	0
$743$	9.40007	0.344855	0.172428	−	0.985022i	$-0.444839\pi$
$743$	9.40007	0.344855	0.172428	+	0.985022i	$0.444839\pi$
$744$	0	0
$745$	5.00670	0.183431
$746$	0	0
$747$	−40.9958	−1.49996
$748$	0	0
$749$	−23.6386	−0.863736
$750$	0	0
$751$	−21.4083	−0.781198	−0.390599	−	0.920561i	$-0.627732\pi$
$751$	−21.4083	−0.781198	−0.390599	+	0.920561i	$0.627732\pi$
$752$	0	0
$753$	−2.85375	−0.103996
$754$	0	0
$755$	9.73980	0.354468
$756$	0	0
$757$	−19.4330	−0.706304	−0.353152	−	0.935566i	$-0.614890\pi$
$757$	−19.4330	−0.706304	−0.353152	+	0.935566i	$0.614890\pi$
$758$	0	0
$759$	−10.6761	−0.387517
$760$	0	0
$761$	36.1110	1.30902	0.654511	−	0.756053i	$-0.272875\pi$
$761$	36.1110	1.30902	0.654511	+	0.756053i	$0.272875\pi$
$762$	0	0
$763$	16.4836	0.596747
$764$	0	0
$765$	0.125122	0.00452378
$766$	0	0
$767$	28.7496	1.03809
$768$	0	0
$769$	32.4205	1.16911	0.584556	−	0.811353i	$-0.301269\pi$
$769$	32.4205	1.16911	0.584556	+	0.811353i	$0.301269\pi$
$770$	0	0
$771$	76.7254	2.76320
$772$	0	0
$773$	−25.6367	−0.922089	−0.461045	−	0.887377i	$-0.652525\pi$
$773$	−25.6367	−0.922089	−0.461045	+	0.887377i	$0.652525\pi$
$774$	0	0
$775$	−9.68241	−0.347803
$776$	0	0
$777$	−21.8008	−0.782101
$778$	0	0
$779$	5.96792	0.213823
$780$	0	0
$781$	2.36453	0.0846095
$782$	0	0
$783$	−9.91343	−0.354277
$784$	0	0
$785$	5.31826	0.189817
$786$	0	0
$787$	−2.24434	−0.0800022	−0.0400011	−	0.999200i	$-0.512736\pi$
$787$	−2.24434	−0.0800022	−0.0400011	+	0.999200i	$0.512736\pi$
$788$	0	0
$789$	−12.5375	−0.446348
$790$	0	0
$791$	−7.23247	−0.257157
$792$	0	0
$793$	−88.2012	−3.13212
$794$	0	0
$795$	−5.30476	−0.188140
$796$	0	0
$797$	−20.7390	−0.734613	−0.367307	−	0.930100i	$-0.619720\pi$
$797$	−20.7390	−0.734613	−0.367307	+	0.930100i	$0.619720\pi$
$798$	0	0
$799$	0.483682	0.0171114
$800$	0	0
$801$	−86.1408	−3.04364
$802$	0	0
$803$	−0.108626	−0.00383331
$804$	0	0
$805$	−3.20806	−0.113069
$806$	0	0
$807$	64.7359	2.27881
$808$	0	0
$809$	−23.7671	−0.835606	−0.417803	−	0.908538i	$-0.637200\pi$
$809$	−23.7671	−0.835606	−0.417803	+	0.908538i	$0.637200\pi$
$810$	0	0
$811$	6.35066	0.223002	0.111501	−	0.993764i	$-0.464434\pi$
$811$	6.35066	0.223002	0.111501	+	0.993764i	$0.464434\pi$
$812$	0	0
$813$	42.9743	1.50718
$814$	0	0
$815$	1.51047	0.0529093
$816$	0	0
$817$	−5.80857	−0.203216
$818$	0	0
$819$	67.8451	2.37070
$820$	0	0
$821$	10.6403	0.371350	0.185675	−	0.982611i	$-0.440553\pi$
$821$	10.6403	0.371350	0.185675	+	0.982611i	$0.440553\pi$
$822$	0	0
$823$	−31.4709	−1.09701	−0.548503	−	0.836149i	$-0.684802\pi$
$823$	−31.4709	−1.09701	−0.548503	+	0.836149i	$0.684802\pi$
$824$	0	0
$825$	−15.8512	−0.551868
$826$	0	0
$827$	−11.2833	−0.392360	−0.196180	−	0.980568i	$-0.562854\pi$
$827$	−11.2833	−0.392360	−0.196180	+	0.980568i	$0.562854\pi$
$828$	0	0
$829$	2.17919	0.0756864	0.0378432	−	0.999284i	$-0.487951\pi$
$829$	2.17919	0.0756864	0.0378432	+	0.999284i	$0.487951\pi$
$830$	0	0
$831$	−30.7972	−1.06834
$832$	0	0
$833$	−0.118776	−0.00411536
$834$	0	0
$835$	−10.6761	−0.369463
$836$	0	0
$837$	12.3581	0.427158
$838$	0	0
$839$	−41.1253	−1.41980	−0.709901	−	0.704301i	$-0.751260\pi$
$839$	−41.1253	−1.41980	−0.709901	+	0.704301i	$0.751260\pi$
$840$	0	0
$841$	−26.3745	−0.909466
$842$	0	0
$843$	−62.6061	−2.15627
$844$	0	0
$845$	10.6282	0.365621
$846$	0	0
$847$	−21.1159	−0.725549
$848$	0	0
$849$	34.8150	1.19485
$850$	0	0
$851$	11.2798	0.386667
$852$	0	0
$853$	20.1056	0.688403	0.344202	−	0.938896i	$-0.388150\pi$
$853$	20.1056	0.688403	0.344202	+	0.938896i	$0.388150\pi$
$854$	0	0
$855$	4.88392	0.167026
$856$	0	0
$857$	−4.75122	−0.162299	−0.0811493	−	0.996702i	$-0.525859\pi$
$857$	−4.75122	−0.162299	−0.0811493	+	0.996702i	$0.525859\pi$
$858$	0	0
$859$	−27.2044	−0.928203	−0.464102	−	0.885782i	$-0.653623\pi$
$859$	−27.2044	−0.928203	−0.464102	+	0.885782i	$0.653623\pi$
$860$	0	0
$861$	17.8240	0.607440
$862$	0	0
$863$	−43.0961	−1.46701	−0.733505	−	0.679684i	$-0.762117\pi$
$863$	−43.0961	−1.46701	−0.733505	+	0.679684i	$0.762117\pi$
$864$	0	0
$865$	−2.03879	−0.0693211
$866$	0	0
$867$	−48.5056	−1.64734
$868$	0	0
$869$	2.23087	0.0756771
$870$	0	0
$871$	30.0848	1.01938
$872$	0	0
$873$	15.4472	0.522808
$874$	0	0
$875$	−9.73149	−0.328984
$876$	0	0
$877$	−37.2416	−1.25756	−0.628780	−	0.777583i	$-0.716445\pi$
$877$	−37.2416	−1.25756	−0.628780	+	0.777583i	$0.716445\pi$
$878$	0	0
$879$	84.0930	2.83639
$880$	0	0
$881$	23.0353	0.776079	0.388039	−	0.921643i	$-0.373152\pi$
$881$	23.0353	0.776079	0.388039	+	0.921643i	$0.373152\pi$
$882$	0	0
$883$	−13.7702	−0.463403	−0.231701	−	0.972787i	$-0.574429\pi$
$883$	−13.7702	−0.463403	−0.231701	+	0.972787i	$0.574429\pi$
$884$	0	0
$885$	6.18105	0.207774
$886$	0	0
$887$	26.3219	0.883804	0.441902	−	0.897063i	$-0.354304\pi$
$887$	26.3219	0.883804	0.441902	+	0.897063i	$0.354304\pi$
$888$	0	0
$889$	28.8514	0.967644
$890$	0	0
$891$	2.34990	0.0787246
$892$	0	0
$893$	18.8797	0.631785
$894$	0	0
$895$	2.91301	0.0973712
$896$	0	0
$897$	−55.5761	−1.85563
$898$	0	0
$899$	−3.27295	−0.109159
$900$	0	0
$901$	−0.218935	−0.00729378
$902$	0	0
$903$	−17.3481	−0.577307
$904$	0	0
$905$	2.84346	0.0945198
$906$	0	0
$907$	32.6153	1.08297	0.541486	−	0.840710i	$-0.317862\pi$
$907$	32.6153	1.08297	0.541486	+	0.840710i	$0.317862\pi$
$908$	0	0
$909$	−18.2945	−0.606790
$910$	0	0
$911$	−47.5936	−1.57685	−0.788423	−	0.615133i	$-0.789102\pi$
$911$	−47.5936	−1.57685	−0.788423	+	0.615133i	$0.789102\pi$
$912$	0	0
$913$	−9.23515	−0.305639
$914$	0	0
$915$	−18.9630	−0.626896
$916$	0	0
$917$	−9.14593	−0.302025
$918$	0	0
$919$	16.4448	0.542463	0.271232	−	0.962514i	$-0.412569\pi$
$919$	16.4448	0.542463	0.271232	+	0.962514i	$0.412569\pi$
$920$	0	0
$921$	60.9093	2.00703
$922$	0	0
$923$	12.3089	0.405154
$924$	0	0
$925$	16.7476	0.550658
$926$	0	0
$927$	15.9753	0.524696
$928$	0	0
$929$	58.6216	1.92331	0.961657	−	0.274256i	$-0.0884314\pi$
$929$	58.6216	1.92331	0.961657	+	0.274256i	$0.0884314\pi$
$930$	0	0
$931$	−4.63624	−0.151947
$932$	0	0
$933$	88.4009	2.89412
$934$	0	0
$935$	0.0281862	0.000921788 0
$936$	0	0
$937$	12.5654	0.410494	0.205247	−	0.978710i	$-0.434200\pi$
$937$	12.5654	0.410494	0.205247	+	0.978710i	$0.434200\pi$
$938$	0	0
$939$	−46.3370	−1.51215
$940$	0	0
$941$	−4.27635	−0.139405	−0.0697025	−	0.997568i	$-0.522205\pi$
$941$	−4.27635	−0.139405	−0.0697025	+	0.997568i	$0.522205\pi$
$942$	0	0
$943$	−9.22219	−0.300316
$944$	0	0
$945$	6.07940	0.197763
$946$	0	0
$947$	−28.9545	−0.940895	−0.470447	−	0.882428i	$-0.655907\pi$
$947$	−28.9545	−0.940895	−0.470447	+	0.882428i	$0.655907\pi$
$948$	0	0
$949$	−0.565469	−0.0183559
$950$	0	0
$951$	−78.5652	−2.54765
$952$	0	0
$953$	−11.0253	−0.357144	−0.178572	−	0.983927i	$-0.557148\pi$
$953$	−11.0253	−0.357144	−0.178572	+	0.983927i	$0.557148\pi$
$954$	0	0
$955$	5.39343	0.174527
$956$	0	0
$957$	−5.35818	−0.173205
$958$	0	0
$959$	−26.8835	−0.868114
$960$	0	0
$961$	−26.9199	−0.868385
$962$	0	0
$963$	−55.6108	−1.79203
$964$	0	0
$965$	3.50501	0.112830
$966$	0	0
$967$	29.3057	0.942409	0.471205	−	0.882024i	$-0.343819\pi$
$967$	29.3057	0.942409	0.471205	+	0.882024i	$0.343819\pi$
$968$	0	0
$969$	0.319123	0.0102517
$970$	0	0
$971$	−1.75260	−0.0562437	−0.0281219	−	0.999605i	$-0.508953\pi$
$971$	−1.75260	−0.0562437	−0.0281219	+	0.999605i	$0.508953\pi$
$972$	0	0
$973$	−14.2046	−0.455377
$974$	0	0
$975$	−82.5161	−2.64263
$976$	0	0
$977$	19.9928	0.639626	0.319813	−	0.947481i	$-0.396380\pi$
$977$	19.9928	0.639626	0.319813	+	0.947481i	$0.396380\pi$
$978$	0	0
$979$	−19.4050	−0.620186
$980$	0	0
$981$	38.7783	1.23810
$982$	0	0
$983$	33.6346	1.07278	0.536389	−	0.843971i	$-0.319788\pi$
$983$	33.6346	1.07278	0.536389	+	0.843971i	$0.319788\pi$
$984$	0	0
$985$	5.12758	0.163378
$986$	0	0
$987$	56.3868	1.79481
$988$	0	0
$989$	8.97593	0.285418
$990$	0	0
$991$	−54.2527	−1.72339	−0.861696	−	0.507424i	$-0.830598\pi$
$991$	−54.2527	−1.72339	−0.861696	+	0.507424i	$0.830598\pi$
$992$	0	0
$993$	91.1024	2.89105
$994$	0	0
$995$	2.17388	0.0689166
$996$	0	0
$997$	15.3787	0.487048	0.243524	−	0.969895i	$-0.421697\pi$
$997$	15.3787	0.487048	0.243524	+	0.969895i	$0.421697\pi$
$998$	0	0
$999$	−21.3757	−0.676298

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	4016.2.a.g.1.7		7
4.3	odd	2		1004.2.a.a.1.1	✓	7
12.11	even	2		9036.2.a.i.1.3		7

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
1004.2.a.a.1.1	✓	7	4.3	odd	2
4016.2.a.g.1.7		7	1.1	even	1	trivial
9036.2.a.i.1.3		7	12.11	even	2

Overview	Random
Universe	Knowledge

Classical	Maass
Hilbert	Bianchi

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

Level:	\( N \)	\(=\)	\( 4016 = 2^{4} \cdot 251 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	4016.a (trivial)

\(f(q)\)	\(=\)	\(q+2.85375 q^{3} +0.454452 q^{5} +2.18653 q^{7} +5.14389 q^{9} +O(q^{10})\)	\(q+2.85375 q^{3} +0.454452 q^{5} +2.18653 q^{7} +5.14389 q^{9} +1.15877 q^{11} +6.03215 q^{13} +1.29689 q^{15} +0.0535246 q^{17} +2.08924 q^{19} +6.23980 q^{21} -3.22849 q^{23} -4.79347 q^{25} +6.11813 q^{27} -1.62034 q^{29} +2.01992 q^{31} +3.30683 q^{33} +0.993671 q^{35} -3.49384 q^{37} +17.2143 q^{39} +2.85650 q^{41} -2.78023 q^{43} +2.33765 q^{45} +9.03663 q^{47} -2.21910 q^{49} +0.152746 q^{51} -4.09037 q^{53} +0.526603 q^{55} +5.96218 q^{57} +4.76605 q^{59} -14.6219 q^{61} +11.2473 q^{63} +2.74132 q^{65} +4.98740 q^{67} -9.21331 q^{69} +2.04056 q^{71} -0.0937424 q^{73} -13.6794 q^{75} +2.53367 q^{77} +1.92521 q^{79} +2.02793 q^{81} -7.96981 q^{83} +0.0243243 q^{85} -4.62404 q^{87} -16.7462 q^{89} +13.1895 q^{91} +5.76433 q^{93} +0.949460 q^{95} +3.00301 q^{97} +5.96057 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 7 q + 3 q^{3} - 2 q^{5} + 6 q^{7}+O(q^{10}) \) 7 * q + 3 * q^3 - 2 * q^5 + 6 * q^7	\( 7 q + 3 q^{3} - 2 q^{5} + 6 q^{7} + 5 q^{11} - q^{13} + 6 q^{15} - 8 q^{17} + 15 q^{19} - 3 q^{21} + 5 q^{23} - 9 q^{25} + 9 q^{27} + 21 q^{31} + 7 q^{35} - q^{37} + 23 q^{39} - 10 q^{41} + 23 q^{43} - 4 q^{45} + 10 q^{47} - 13 q^{49} + 20 q^{51} - q^{53} + 23 q^{55} - 6 q^{57} + 4 q^{59} + 3 q^{61} + 4 q^{63} + 4 q^{65} + 28 q^{67} + 18 q^{69} + 18 q^{71} - 7 q^{73} - 11 q^{75} + 6 q^{77} + 30 q^{79} - 5 q^{81} - 13 q^{83} + q^{85} + 7 q^{87} + 18 q^{91} + 36 q^{93} - 2 q^{97} + 10 q^{99}+O(q^{100}) \) 7 * q + 3 * q^3 - 2 * q^5 + 6 * q^7 + 5 * q^11 - q^13 + 6 * q^15 - 8 * q^17 + 15 * q^19 - 3 * q^21 + 5 * q^23 - 9 * q^25 + 9 * q^27 + 21 * q^31 + 7 * q^35 - q^37 + 23 * q^39 - 10 * q^41 + 23 * q^43 - 4 * q^45 + 10 * q^47 - 13 * q^49 + 20 * q^51 - q^53 + 23 * q^55 - 6 * q^57 + 4 * q^59 + 3 * q^61 + 4 * q^63 + 4 * q^65 + 28 * q^67 + 18 * q^69 + 18 * q^71 - 7 * q^73 - 11 * q^75 + 6 * q^77 + 30 * q^79 - 5 * q^81 - 13 * q^83 + q^85 + 7 * q^87 + 18 * q^91 + 36 * q^93 - 2 * q^97 + 10 * q^99

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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