LMFDB - Embedded newform 3920.2.k.a.2351.2

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [3920,2,Mod(2351,3920)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(3920, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([1, 0, 0, 1]))

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

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

chi := DirichletCharacter("3920.2351");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 3920 = 2^{4} \cdot 5 \cdot 7^{2} $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	3920.k (of order $2$, degree $1$, minimal)

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:	no
Analytic conductor:	$31.3013575923$
Analytic rank:	$0$
Dimension:	$8$
Coefficient field:	$\Q(\zeta_{16})$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{8} + 1 $ x^8 + 1
Coefficient ring:	$\Z[a_1, \ldots, a_{5}]$
Coefficient ring index:	$ 2^{3} $
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label			2351.2
Root			$-0.382683 + 0.923880i$ of defining polynomial
Character	$\chi$	$=$	3920.2351
Dual form			3920.2.k.a.2351.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-3.17958 q^{3} +1.00000i q^{5} +7.10973 q^{9} +O(q^{10})$	$q-3.17958 q^{3} +1.00000i q^{5} +7.10973 q^{9} +1.11652i q^{11} +3.17958i q^{13} -3.17958i q^{15} +6.37849i q^{17} +8.30864 q^{19} -6.55967i q^{23} -1.00000 q^{25} -13.0672 q^{27} +7.40743 q^{29} +1.53912 q^{31} -3.55007i q^{33} -0.0147960 q^{37} -10.1097i q^{39} +1.89828i q^{41} +6.07560i q^{43} +7.10973i q^{45} +1.68751 q^{47} -20.2809i q^{51} +9.65980 q^{53} -1.11652 q^{55} -26.4180 q^{57} -5.50461 q^{59} +8.26998i q^{61} -3.17958 q^{65} -4.38303i q^{67} +20.8570i q^{69} -14.8213i q^{71} -11.2149i q^{73} +3.17958 q^{75} -6.29769i q^{79} +20.2191 q^{81} -7.10013 q^{83} -6.37849 q^{85} -23.5525 q^{87} -11.0319i q^{89} -4.89374 q^{93} +8.30864i q^{95} -15.3276i q^{97} +7.93816i q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 8 q - 8 q^{3} + 16 q^{9}+O(q^{10}) $ 8 * q - 8 * q^3 + 16 * q^9	$ 8 q - 8 q^{3} + 16 q^{9} + 16 q^{19} - 8 q^{25} - 8 q^{27} + 8 q^{29} + 48 q^{31} + 24 q^{47} + 32 q^{53} - 8 q^{55} - 32 q^{57} + 16 q^{59} - 8 q^{65} + 8 q^{75} - 8 q^{81} - 48 q^{83} - 24 q^{85} - 24 q^{87} - 32 q^{93}+O(q^{100}) $ 8 * q - 8 * q^3 + 16 * q^9 + 16 * q^19 - 8 * q^25 - 8 * q^27 + 8 * q^29 + 48 * q^31 + 24 * q^47 + 32 * q^53 - 8 * q^55 - 32 * q^57 + 16 * q^59 - 8 * q^65 + 8 * q^75 - 8 * q^81 - 48 * q^83 - 24 * q^85 - 24 * q^87 - 32 * q^93

Display 100 coefficients 10 coefficients

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/3920\mathbb{Z}\right)^\times$.

$n$	$981$	$1471$	$3041$	$3137$
$\chi(n)$	$1$	$-1$	$-1$	$1$

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
$3$	−3.17958	−1.83573	−0.917866	−	0.396891i	$-0.870089\pi$
$3$	−3.17958	−1.83573	−0.917866	+	0.396891i	$0.870089\pi$
$4$	0	0
$5$	1.00000i	0.447214i
$5$	1.00000i	0.447214i
$6$	0	0
$7$	0	0
$7$	0	0
$8$	0	0
$9$	7.10973	2.36991
$10$	0	0
$11$	1.11652i	0.336643i	0.985732	+	0.168322i	$0.0538348\pi$
$11$	1.11652i	0.336643i	−0.985732	+	0.168322i	$0.946165\pi$
$12$	0	0
$13$	3.17958i	0.881857i	0.897542	+	0.440928i	$0.145351\pi$
$13$	3.17958i	0.881857i	−0.897542	+	0.440928i	$0.854649\pi$
$14$	0	0
$15$	− 3.17958i	− 0.820964i
$16$	0	0
$17$	6.37849i	1.54701i	0.633789	+	0.773506i	$0.281499\pi$
$17$	6.37849i	1.54701i	−0.633789	+	0.773506i	$0.718501\pi$
$18$	0	0
$19$	8.30864	1.90613	0.953067	−	0.302760i	$-0.0979081\pi$
$19$	8.30864	1.90613	0.953067	+	0.302760i	$0.0979081\pi$
$20$	0	0
$21$	0	0
$22$	0	0
$23$	− 6.55967i	− 1.36778i	−0.729583	−	0.683892i	$-0.760286\pi$
$23$	− 6.55967i	− 1.36778i	0.729583	−	0.683892i	$-0.239714\pi$
$24$	0	0
$25$	−1.00000	−0.200000
$26$	0	0
$27$	−13.0672	−2.51479
$28$	0	0
$29$	7.40743	1.37552	0.687762	−	0.725936i	$-0.258593\pi$
$29$	7.40743	1.37552	0.687762	+	0.725936i	$0.258593\pi$
$30$	0	0
$31$	1.53912	0.276433	0.138217	−	0.990402i	$-0.455863\pi$
$31$	1.53912	0.276433	0.138217	+	0.990402i	$0.455863\pi$
$32$	0	0
$33$	− 3.55007i	− 0.617987i
$34$	0	0
$35$	0	0
$36$	0	0
$37$	−0.0147960	−0.00243245	−0.00121623	−	0.999999i	$-0.500387\pi$
$37$	−0.0147960	−0.00243245	−0.00121623	+	0.999999i	$0.500387\pi$
$38$	0	0
$39$	− 10.1097i	− 1.61885i
$40$	0	0
$41$	1.89828i	0.296461i	0.988953	+	0.148230i	$0.0473577\pi$
$41$	1.89828i	0.296461i	−0.988953	+	0.148230i	$0.952642\pi$
$42$	0	0
$43$	6.07560i	0.926521i	0.886222	+	0.463260i	$0.153321\pi$
$43$	6.07560i	0.926521i	−0.886222	+	0.463260i	$0.846679\pi$
$44$	0	0
$45$	7.10973i	1.05986i
$46$	0	0
$47$	1.68751	0.246149	0.123074	−	0.992397i	$-0.460725\pi$
$47$	1.68751	0.246149	0.123074	+	0.992397i	$0.460725\pi$
$48$	0	0
$49$	0	0
$50$	0	0
$51$	− 20.2809i	− 2.83990i
$52$	0	0
$53$	9.65980	1.32688	0.663438	−	0.748232i	$-0.269097\pi$
$53$	9.65980	1.32688	0.663438	+	0.748232i	$0.269097\pi$
$54$	0	0
$55$	−1.11652	−0.150552
$56$	0	0
$57$	−26.4180	−3.49915
$58$	0	0
$59$	−5.50461	−0.716640	−0.358320	−	0.933599i	$-0.616650\pi$
$59$	−5.50461	−0.716640	−0.358320	+	0.933599i	$0.616650\pi$
$60$	0	0
$61$	8.26998i	1.05886i	0.848353	+	0.529431i	$0.177595\pi$
$61$	8.26998i	1.05886i	−0.848353	+	0.529431i	$0.822405\pi$
$62$	0	0
$63$	0	0
$64$	0	0
$65$	−3.17958	−0.394378
$66$	0	0
$67$	− 4.38303i	− 0.535472i	−0.963492	−	0.267736i	$-0.913724\pi$
$67$	− 4.38303i	− 0.535472i	0.963492	−	0.267736i	$-0.0862755\pi$
$68$	0	0
$69$	20.8570i	2.51089i
$70$	0	0
$71$	− 14.8213i	− 1.75896i	−0.475935	−	0.879480i	$-0.657890\pi$
$71$	− 14.8213i	− 1.75896i	0.475935	−	0.879480i	$-0.342110\pi$
$72$	0	0
$73$	− 11.2149i	− 1.31261i	−0.754497	−	0.656304i	$-0.772119\pi$
$73$	− 11.2149i	− 1.31261i	0.754497	−	0.656304i	$-0.227881\pi$
$74$	0	0
$75$	3.17958	0.367146
$76$	0	0
$77$	0	0
$78$	0	0
$79$	− 6.29769i	− 0.708546i	−0.935142	−	0.354273i	$-0.884728\pi$
$79$	− 6.29769i	− 0.708546i	0.935142	−	0.354273i	$-0.115272\pi$
$80$	0	0
$81$	20.2191	2.24657
$82$	0	0
$83$	−7.10013	−0.779341	−0.389670	−	0.920954i	$-0.627411\pi$
$83$	−7.10013	−0.779341	−0.389670	+	0.920954i	$0.627411\pi$
$84$	0	0
$85$	−6.37849	−0.691845
$86$	0	0
$87$	−23.5525	−2.52509
$88$	0	0
$89$	− 11.0319i	− 1.16938i	−0.811258	−	0.584688i	$-0.801217\pi$
$89$	− 11.0319i	− 1.16938i	0.811258	−	0.584688i	$-0.198783\pi$
$90$	0	0
$91$	0	0
$92$	0	0
$93$	−4.89374	−0.507457
$94$	0	0
$95$	8.30864i	0.852449i
$96$	0	0
$97$	− 15.3276i	− 1.55628i	−0.628089	−	0.778141i	$-0.716163\pi$
$97$	− 15.3276i	− 1.55628i	0.628089	−	0.778141i	$-0.283837\pi$
$98$	0	0
$99$	7.93816i	0.797815i
$100$	0	0
$101$	− 8.13028i	− 0.808993i	−0.914539	−	0.404497i	$-0.867447\pi$
$101$	− 8.13028i	− 0.808993i	0.914539	−	0.404497i	$-0.132553\pi$
$102$	0	0
$103$	12.5433	1.23593	0.617963	−	0.786207i	$-0.287958\pi$
$103$	12.5433	1.23593	0.617963	+	0.786207i	$0.287958\pi$
$104$	0	0
$105$	0	0
$106$	0	0
$107$	13.6997i	1.32440i	0.749328	+	0.662199i	$0.230377\pi$
$107$	13.6997i	1.32440i	−0.749328	+	0.662199i	$0.769623\pi$
$108$	0	0
$109$	8.55582	0.819499	0.409749	−	0.912198i	$-0.365616\pi$
$109$	8.55582	0.819499	0.409749	+	0.912198i	$0.365616\pi$
$110$	0	0
$111$	0.0470452	0.00446533
$112$	0	0
$113$	−6.36210	−0.598496	−0.299248	−	0.954175i	$-0.596736\pi$
$113$	−6.36210	−0.598496	−0.299248	+	0.954175i	$0.596736\pi$
$114$	0	0
$115$	6.55967	0.611692
$116$	0	0
$117$	22.6060i	2.08992i
$118$	0	0
$119$	0	0
$120$	0	0
$121$	9.75338	0.886671
$122$	0	0
$123$	− 6.03572i	− 0.544223i
$124$	0	0
$125$	− 1.00000i	− 0.0894427i
$126$	0	0
$127$	9.84060i	0.873212i	0.899653	+	0.436606i	$0.143820\pi$
$127$	9.84060i	0.873212i	−0.899653	+	0.436606i	$0.856180\pi$
$128$	0	0
$129$	− 19.3179i	− 1.70084i
$130$	0	0
$131$	13.4122	1.17183	0.585917	−	0.810371i	$-0.300735\pi$
$131$	13.4122	1.17183	0.585917	+	0.810371i	$0.300735\pi$
$132$	0	0
$133$	0	0
$134$	0	0
$135$	− 13.0672i	− 1.12465i
$136$	0	0
$137$	8.90085	0.760451	0.380225	−	0.924894i	$-0.375846\pi$
$137$	8.90085	0.760451	0.380225	+	0.924894i	$0.375846\pi$
$138$	0	0
$139$	−13.0058	−1.10313	−0.551567	−	0.834131i	$-0.685970\pi$
$139$	−13.0058	−1.10313	−0.551567	+	0.834131i	$0.685970\pi$
$140$	0	0
$141$	−5.36558	−0.451863
$142$	0	0
$143$	−3.55007	−0.296871
$144$	0	0
$145$	7.40743i	0.615153i
$146$	0	0
$147$	0	0
$148$	0	0
$149$	12.6656	1.03760	0.518801	−	0.854895i	$-0.326378\pi$
$149$	12.6656	1.03760	0.518801	+	0.854895i	$0.326378\pi$
$150$	0	0
$151$	21.4229i	1.74337i	0.490065	+	0.871686i	$0.336973\pi$
$151$	21.4229i	1.74337i	−0.490065	+	0.871686i	$0.663027\pi$
$152$	0	0
$153$	45.3494i	3.66628i
$154$	0	0
$155$	1.53912i	0.123625i
$156$	0	0
$157$	13.6569i	1.08994i	0.838457	+	0.544968i	$0.183458\pi$
$157$	13.6569i	1.08994i	−0.838457	+	0.544968i	$0.816542\pi$
$158$	0	0
$159$	−30.7141	−2.43579
$160$	0	0
$161$	0	0
$162$	0	0
$163$	15.8364i	1.24041i	0.784442	+	0.620203i	$0.212950\pi$
$163$	15.8364i	1.24041i	−0.784442	+	0.620203i	$0.787050\pi$
$164$	0	0
$165$	3.55007	0.276372
$166$	0	0
$167$	13.5606	1.04935	0.524677	−	0.851301i	$-0.324186\pi$
$167$	13.5606	1.04935	0.524677	+	0.851301i	$0.324186\pi$
$168$	0	0
$169$	2.89027	0.222328
$170$	0	0
$171$	59.0722	4.51737
$172$	0	0
$173$	− 18.0299i	− 1.37079i	−0.728172	−	0.685394i	$-0.759630\pi$
$173$	− 18.0299i	− 1.37079i	0.728172	−	0.685394i	$-0.240370\pi$
$174$	0	0
$175$	0	0
$176$	0	0
$177$	17.5024	1.31556
$178$	0	0
$179$	8.38234i	0.626525i	0.949667	+	0.313263i	$0.101422\pi$
$179$	8.38234i	0.626525i	−0.949667	+	0.313263i	$0.898578\pi$
$180$	0	0
$181$	16.3153i	1.21271i	0.795195	+	0.606353i	$0.207368\pi$
$181$	16.3153i	1.21271i	−0.795195	+	0.606353i	$0.792632\pi$
$182$	0	0
$183$	− 26.2951i	− 1.94379i
$184$	0	0
$185$	− 0.0147960i	− 0.00108783i
$186$	0	0
$187$	−7.12172	−0.520791
$188$	0	0
$189$	0	0
$190$	0	0
$191$	− 5.53943i	− 0.400819i	−0.979712	−	0.200410i	$-0.935773\pi$
$191$	− 5.53943i	− 0.400819i	0.979712	−	0.200410i	$-0.0642273\pi$
$192$	0	0
$193$	−10.7511	−0.773881	−0.386941	−	0.922105i	$-0.626468\pi$
$193$	−10.7511	−0.773881	−0.386941	+	0.922105i	$0.626468\pi$
$194$	0	0
$195$	10.1097	0.723973
$196$	0	0
$197$	−6.91564	−0.492719	−0.246360	−	0.969179i	$-0.579234\pi$
$197$	−6.91564	−0.492719	−0.246360	+	0.969179i	$0.579234\pi$
$198$	0	0
$199$	−25.9411	−1.83892	−0.919458	−	0.393188i	$-0.871372\pi$
$199$	−25.9411	−1.83892	−0.919458	+	0.393188i	$0.871372\pi$
$200$	0	0
$201$	13.9362i	0.982983i
$202$	0	0
$203$	0	0
$204$	0	0
$205$	−1.89828	−0.132581
$206$	0	0
$207$	− 46.6375i	− 3.24153i
$208$	0	0
$209$	9.27677i	0.641688i
$210$	0	0
$211$	− 1.10936i	− 0.0763714i	−0.999271	−	0.0381857i	$-0.987842\pi$
$211$	− 1.10936i	− 0.0763714i	0.999271	−	0.0381857i	$-0.0121578\pi$
$212$	0	0
$213$	47.1254i	3.22898i
$214$	0	0
$215$	−6.07560	−0.414353
$216$	0	0
$217$	0	0
$218$	0	0
$219$	35.6588i	2.40960i
$220$	0	0
$221$	−20.2809	−1.36424
$222$	0	0
$223$	22.0299	1.47523	0.737616	−	0.675221i	$-0.235952\pi$
$223$	22.0299	1.47523	0.737616	+	0.675221i	$0.235952\pi$
$224$	0	0
$225$	−7.10973	−0.473982
$226$	0	0
$227$	−15.6215	−1.03684	−0.518418	−	0.855127i	$-0.673479\pi$
$227$	−15.6215	−1.03684	−0.518418	+	0.855127i	$0.673479\pi$
$228$	0	0
$229$	16.6626i	1.10110i	0.834803	+	0.550548i	$0.185581\pi$
$229$	16.6626i	1.10110i	−0.834803	+	0.550548i	$0.814419\pi$
$230$	0	0
$231$	0	0
$232$	0	0
$233$	−3.18343	−0.208553	−0.104277	−	0.994548i	$-0.533253\pi$
$233$	−3.18343	−0.208553	−0.104277	+	0.994548i	$0.533253\pi$
$234$	0	0
$235$	1.68751i	0.110081i
$236$	0	0
$237$	20.0240i	1.30070i
$238$	0	0
$239$	4.39745i	0.284447i	0.989835	+	0.142224i	$0.0454252\pi$
$239$	4.39745i	0.284447i	−0.989835	+	0.142224i	$0.954575\pi$
$240$	0	0
$241$	8.93882i	0.575800i	0.957660	+	0.287900i	$0.0929571\pi$
$241$	8.93882i	0.575800i	−0.957660	+	0.287900i	$0.907043\pi$
$242$	0	0
$243$	−25.0866	−1.60930
$244$	0	0
$245$	0	0
$246$	0	0
$247$	26.4180i	1.68094i
$248$	0	0
$249$	22.5754	1.43066
$250$	0	0
$251$	23.2075	1.46484	0.732422	−	0.680851i	$-0.238390\pi$
$251$	23.2075	1.46484	0.732422	+	0.680851i	$0.238390\pi$
$252$	0	0
$253$	7.32400	0.460456
$254$	0	0
$255$	20.2809	1.27004
$256$	0	0
$257$	2.88311i	0.179843i	0.995949	+	0.0899216i	$0.0286617\pi$
$257$	2.88311i	0.179843i	−0.995949	+	0.0899216i	$0.971338\pi$
$258$	0	0
$259$	0	0
$260$	0	0
$261$	52.6648	3.25987
$262$	0	0
$263$	9.83840i	0.606662i	0.952885	+	0.303331i	$0.0980988\pi$
$263$	9.83840i	0.606662i	−0.952885	+	0.303331i	$0.901901\pi$
$264$	0	0
$265$	9.65980i	0.593397i
$266$	0	0
$267$	35.0767i	2.14666i
$268$	0	0
$269$	10.5114i	0.640891i	0.947267	+	0.320446i	$0.103833\pi$
$269$	10.5114i	0.640891i	−0.947267	+	0.320446i	$0.896167\pi$
$270$	0	0
$271$	−7.05696	−0.428680	−0.214340	−	0.976759i	$-0.568760\pi$
$271$	−7.05696	−0.428680	−0.214340	+	0.976759i	$0.568760\pi$
$272$	0	0
$273$	0	0
$274$	0	0
$275$	− 1.11652i	− 0.0673287i
$276$	0	0
$277$	5.10454	0.306702	0.153351	−	0.988172i	$-0.450994\pi$
$277$	5.10454	0.306702	0.153351	+	0.988172i	$0.450994\pi$
$278$	0	0
$279$	10.9427	0.655122
$280$	0	0
$281$	−3.78207	−0.225620	−0.112810	−	0.993617i	$-0.535985\pi$
$281$	−3.78207	−0.225620	−0.112810	+	0.993617i	$0.535985\pi$
$282$	0	0
$283$	3.67300	0.218337	0.109169	−	0.994023i	$-0.465181\pi$
$283$	3.67300	0.218337	0.109169	+	0.994023i	$0.465181\pi$
$284$	0	0
$285$	− 26.4180i	− 1.56487i
$286$	0	0
$287$	0	0
$288$	0	0
$289$	−23.6852	−1.39325
$290$	0	0
$291$	48.7353i	2.85692i
$292$	0	0
$293$	0.0626865i	0.00366219i	0.999998	+	0.00183109i	$0.000582855\pi$
$293$	0.0626865i	0.00366219i	−0.999998	+	0.00183109i	$0.999417\pi$
$294$	0	0
$295$	− 5.50461i	− 0.320491i
$296$	0	0
$297$	− 14.5898i	− 0.846587i
$298$	0	0
$299$	20.8570	1.20619
$300$	0	0
$301$	0	0
$302$	0	0
$303$	25.8509i	1.48509i
$304$	0	0
$305$	−8.26998	−0.473538
$306$	0	0
$307$	12.4492	0.710513	0.355256	−	0.934769i	$-0.384394\pi$
$307$	12.4492	0.710513	0.355256	+	0.934769i	$0.384394\pi$
$308$	0	0
$309$	−39.8824	−2.26883
$310$	0	0
$311$	22.8636	1.29648	0.648239	−	0.761437i	$-0.275506\pi$
$311$	22.8636	1.29648	0.648239	+	0.761437i	$0.275506\pi$
$312$	0	0
$313$	23.7227i	1.34089i	0.741960	+	0.670444i	$0.233896\pi$
$313$	23.7227i	1.34089i	−0.741960	+	0.670444i	$0.766104\pi$
$314$	0	0
$315$	0	0
$316$	0	0
$317$	10.9200	0.613331	0.306665	−	0.951817i	$-0.400787\pi$
$317$	10.9200	0.613331	0.306665	+	0.951817i	$0.400787\pi$
$318$	0	0
$319$	8.27054i	0.463061i
$320$	0	0
$321$	− 43.5592i	− 2.43124i
$322$	0	0
$323$	52.9966i	2.94881i
$324$	0	0
$325$	− 3.17958i	− 0.176371i
$326$	0	0
$327$	−27.2039	−1.50438
$328$	0	0
$329$	0	0
$330$	0	0
$331$	− 26.3443i	− 1.44801i	−0.689793	−	0.724007i	$-0.742298\pi$
$331$	− 26.3443i	− 1.44801i	0.689793	−	0.724007i	$-0.257702\pi$
$332$	0	0
$333$	−0.105196	−0.00576469
$334$	0	0
$335$	4.38303	0.239470
$336$	0	0
$337$	33.5329	1.82666	0.913328	−	0.407226i	$-0.133504\pi$
$337$	33.5329	1.82666	0.913328	+	0.407226i	$0.133504\pi$
$338$	0	0
$339$	20.2288	1.09868
$340$	0	0
$341$	1.71845i	0.0930594i
$342$	0	0
$343$	0	0
$344$	0	0
$345$	−20.8570	−1.12290
$346$	0	0
$347$	− 0.0165201i	0 0.000886843i	−1.00000		0.000443422i	$-0.999859\pi$
$347$	− 0.0165201i	0 0.000886843i	1.00000		0.000443422i	$-0.000141146\pi$
$348$	0	0
$349$	− 25.1959i	− 1.34871i	−0.738409	−	0.674353i	$-0.764423\pi$
$349$	− 25.1959i	− 1.34871i	0.738409	−	0.674353i	$-0.235577\pi$
$350$	0	0
$351$	− 41.5483i	− 2.21768i
$352$	0	0
$353$	− 9.33755i	− 0.496987i	−0.968634	−	0.248494i	$-0.920064\pi$
$353$	− 9.33755i	− 0.496987i	0.968634	−	0.248494i	$-0.0799355\pi$
$354$	0	0
$355$	14.8213	0.786631
$356$	0	0
$357$	0	0
$358$	0	0
$359$	30.0867i	1.58792i	0.607973	+	0.793958i	$0.291983\pi$
$359$	30.0867i	1.58792i	−0.607973	+	0.793958i	$0.708017\pi$
$360$	0	0
$361$	50.0336	2.63335
$362$	0	0
$363$	−31.0117	−1.62769
$364$	0	0
$365$	11.2149	0.587016
$366$	0	0
$367$	−25.6626	−1.33958	−0.669789	−	0.742552i	$-0.733615\pi$
$367$	−25.6626	−1.33958	−0.669789	+	0.742552i	$0.733615\pi$
$368$	0	0
$369$	13.4962i	0.702586i
$370$	0	0
$371$	0	0
$372$	0	0
$373$	−16.1540	−0.836422	−0.418211	−	0.908350i	$-0.637343\pi$
$373$	−16.1540	−0.836422	−0.418211	+	0.908350i	$0.637343\pi$
$374$	0	0
$375$	3.17958i	0.164193i
$376$	0	0
$377$	23.5525i	1.21302i
$378$	0	0
$379$	− 18.9373i	− 0.972742i	−0.873753	−	0.486371i	$-0.838320\pi$
$379$	− 18.9373i	− 0.972742i	0.873753	−	0.486371i	$-0.161680\pi$
$380$	0	0
$381$	− 31.2890i	− 1.60298i
$382$	0	0
$383$	−16.2163	−0.828613	−0.414307	−	0.910137i	$-0.635976\pi$
$383$	−16.2163	−0.828613	−0.414307	+	0.910137i	$0.635976\pi$
$384$	0	0
$385$	0	0
$386$	0	0
$387$	43.1959i	2.19577i
$388$	0	0
$389$	−1.35863	−0.0688854	−0.0344427	−	0.999407i	$-0.510966\pi$
$389$	−1.35863	−0.0688854	−0.0344427	+	0.999407i	$0.510966\pi$
$390$	0	0
$391$	41.8408	2.11598
$392$	0	0
$393$	−42.6453	−2.15117
$394$	0	0
$395$	6.29769	0.316871
$396$	0	0
$397$	− 9.20241i	− 0.461856i	−0.972971	−	0.230928i	$-0.925824\pi$
$397$	− 9.20241i	− 0.461856i	0.972971	−	0.230928i	$-0.0741762\pi$
$398$	0	0
$399$	0	0
$400$	0	0
$401$	1.17248	0.0585506	0.0292753	−	0.999571i	$-0.490680\pi$
$401$	1.17248	0.0585506	0.0292753	+	0.999571i	$0.490680\pi$
$402$	0	0
$403$	4.89374i	0.243775i
$404$	0	0
$405$	20.2191i	1.00469i
$406$	0	0
$407$	− 0.0165201i	0 0.000818869i
$408$	0	0
$409$	4.43476i	0.219285i	0.993971	+	0.109642i	$0.0349706\pi$
$409$	4.43476i	0.219285i	−0.993971	+	0.109642i	$0.965029\pi$
$410$	0	0
$411$	−28.3010	−1.39598
$412$	0	0
$413$	0	0
$414$	0	0
$415$	− 7.10013i	− 0.348532i
$416$	0	0
$417$	41.3528	2.02506
$418$	0	0
$419$	9.06910	0.443055	0.221527	−	0.975154i	$-0.428896\pi$
$419$	9.06910	0.443055	0.221527	+	0.975154i	$0.428896\pi$
$420$	0	0
$421$	−16.0388	−0.781684	−0.390842	−	0.920458i	$-0.627816\pi$
$421$	−16.0388	−0.781684	−0.390842	+	0.920458i	$0.627816\pi$
$422$	0	0
$423$	11.9977	0.583350
$424$	0	0
$425$	− 6.37849i	− 0.309402i
$426$	0	0
$427$	0	0
$428$	0	0
$429$	11.2877	0.544976
$430$	0	0
$431$	1.94442i	0.0936594i	0.998903	+	0.0468297i	$0.0149118\pi$
$431$	1.94442i	0.0936594i	−0.998903	+	0.0468297i	$0.985088\pi$
$432$	0	0
$433$	14.4424i	0.694056i	0.937855	+	0.347028i	$0.112809\pi$
$433$	14.4424i	0.694056i	−0.937855	+	0.347028i	$0.887191\pi$
$434$	0	0
$435$	− 23.5525i	− 1.12926i
$436$	0	0
$437$	− 54.5019i	− 2.60718i
$438$	0	0
$439$	−8.52644	−0.406945	−0.203472	−	0.979081i	$-0.565223\pi$
$439$	−8.52644	−0.406945	−0.203472	+	0.979081i	$0.565223\pi$
$440$	0	0
$441$	0	0
$442$	0	0
$443$	− 40.5619i	− 1.92715i	−0.267434	−	0.963576i	$-0.586176\pi$
$443$	− 40.5619i	− 1.92715i	0.267434	−	0.963576i	$-0.413824\pi$
$444$	0	0
$445$	11.0319	0.522961
$446$	0	0
$447$	−40.2711	−1.90476
$448$	0	0
$449$	−12.3518	−0.582920	−0.291460	−	0.956583i	$-0.594141\pi$
$449$	−12.3518	−0.582920	−0.291460	+	0.956583i	$0.594141\pi$
$450$	0	0
$451$	−2.11946	−0.0998016
$452$	0	0
$453$	− 68.1159i	− 3.20036i
$454$	0	0
$455$	0	0
$456$	0	0
$457$	−19.3459	−0.904963	−0.452482	−	0.891774i	$-0.649461\pi$
$457$	−19.3459	−0.904963	−0.452482	+	0.891774i	$0.649461\pi$
$458$	0	0
$459$	− 83.3492i	− 3.89041i
$460$	0	0
$461$	− 22.9259i	− 1.06777i	−0.845558	−	0.533884i	$-0.820732\pi$
$461$	− 22.9259i	− 1.06777i	0.845558	−	0.533884i	$-0.179268\pi$
$462$	0	0
$463$	− 14.9082i	− 0.692842i	−0.938079	−	0.346421i	$-0.887397\pi$
$463$	− 14.9082i	− 0.692842i	0.938079	−	0.346421i	$-0.112603\pi$
$464$	0	0
$465$	− 4.89374i	− 0.226942i
$466$	0	0
$467$	19.5697	0.905578	0.452789	−	0.891618i	$-0.350429\pi$
$467$	19.5697	0.905578	0.452789	+	0.891618i	$0.350429\pi$
$468$	0	0
$469$	0	0
$470$	0	0
$471$	− 43.4231i	− 2.00083i
$472$	0	0
$473$	−6.78353	−0.311907
$474$	0	0
$475$	−8.30864	−0.381227
$476$	0	0
$477$	68.6786	3.14458
$478$	0	0
$479$	−11.0563	−0.505174	−0.252587	−	0.967574i	$-0.581281\pi$
$479$	−11.0563	−0.505174	−0.252587	+	0.967574i	$0.581281\pi$
$480$	0	0
$481$	− 0.0470452i	− 0.00214507i
$482$	0	0
$483$	0	0
$484$	0	0
$485$	15.3276	0.695991
$486$	0	0
$487$	34.3900i	1.55836i	0.626801	+	0.779179i	$0.284364\pi$
$487$	34.3900i	1.55836i	−0.626801	+	0.779179i	$0.715636\pi$
$488$	0	0
$489$	− 50.3532i	− 2.27705i
$490$	0	0
$491$	− 5.06878i	− 0.228751i	−0.993438	−	0.114376i	$-0.963513\pi$
$491$	− 5.06878i	− 0.228751i	0.993438	−	0.114376i	$-0.0364867\pi$
$492$	0	0
$493$	47.2482i	2.12795i
$494$	0	0
$495$	−7.93816	−0.356794
$496$	0	0
$497$	0	0
$498$	0	0
$499$	− 0.566697i	− 0.0253688i	−0.999920	−	0.0126844i	$-0.995962\pi$
$499$	− 0.566697i	− 0.0253688i	0.999920	−	0.0126844i	$-0.00403768\pi$
$500$	0	0
$501$	−43.1172	−1.92633
$502$	0	0
$503$	19.6230	0.874947	0.437473	−	0.899231i	$-0.355873\pi$
$503$	19.6230	0.874947	0.437473	+	0.899231i	$0.355873\pi$
$504$	0	0
$505$	8.13028	0.361793
$506$	0	0
$507$	−9.18984	−0.408135
$508$	0	0
$509$	27.5105i	1.21938i	0.792639	+	0.609691i	$0.208706\pi$
$509$	27.5105i	1.21938i	−0.792639	+	0.609691i	$0.791294\pi$
$510$	0	0
$511$	0	0
$512$	0	0
$513$	−108.571	−4.79352
$514$	0	0
$515$	12.5433i	0.552723i
$516$	0	0
$517$	1.88414i	0.0828644i
$518$	0	0
$519$	57.3275i	2.51640i
$520$	0	0
$521$	− 21.3092i	− 0.933573i	−0.884370	−	0.466787i	$-0.845412\pi$
$521$	− 21.3092i	− 0.933573i	0.884370	−	0.466787i	$-0.154588\pi$
$522$	0	0
$523$	4.23893	0.185355	0.0926777	−	0.995696i	$-0.470457\pi$
$523$	4.23893	0.185355	0.0926777	+	0.995696i	$0.470457\pi$
$524$	0	0
$525$	0	0
$526$	0	0
$527$	9.81723i	0.427645i
$528$	0	0
$529$	−20.0292	−0.870836
$530$	0	0
$531$	−39.1363	−1.69837
$532$	0	0
$533$	−6.03572	−0.261436
$534$	0	0
$535$	−13.6997	−0.592289
$536$	0	0
$537$	− 26.6523i	− 1.15013i
$538$	0	0
$539$	0	0
$540$	0	0
$541$	2.88719	0.124130	0.0620651	−	0.998072i	$-0.480231\pi$
$541$	2.88719	0.124130	0.0620651	+	0.998072i	$0.480231\pi$
$542$	0	0
$543$	− 51.8758i	− 2.22620i
$544$	0	0
$545$	8.55582i	0.366491i
$546$	0	0
$547$	15.3656i	0.656986i	0.944506	+	0.328493i	$0.106541\pi$
$547$	15.3656i	0.656986i	−0.944506	+	0.328493i	$0.893459\pi$
$548$	0	0
$549$	58.7973i	2.50941i
$550$	0	0
$551$	61.5457	2.62193
$552$	0	0
$553$	0	0
$554$	0	0
$555$	0.0470452i	0.00199696i
$556$	0	0
$557$	−37.1405	−1.57369	−0.786846	−	0.617149i	$-0.788288\pi$
$557$	−37.1405	−1.57369	−0.786846	+	0.617149i	$0.788288\pi$
$558$	0	0
$559$	−19.3179	−0.817059
$560$	0	0
$561$	22.6441	0.956033
$562$	0	0
$563$	−30.1869	−1.27222	−0.636112	−	0.771596i	$-0.719459\pi$
$563$	−30.1869	−1.27222	−0.636112	+	0.771596i	$0.719459\pi$
$564$	0	0
$565$	− 6.36210i	− 0.267656i
$566$	0	0
$567$	0	0
$568$	0	0
$569$	15.3337	0.642822	0.321411	−	0.946940i	$-0.395843\pi$
$569$	15.3337	0.642822	0.321411	+	0.946940i	$0.395843\pi$
$570$	0	0
$571$	6.27223i	0.262485i	0.991350	+	0.131242i	$0.0418966\pi$
$571$	6.27223i	0.262485i	−0.991350	+	0.131242i	$0.958103\pi$
$572$	0	0
$573$	17.6131i	0.735797i
$574$	0	0
$575$	6.55967i	0.273557i
$576$	0	0
$577$	− 22.4999i	− 0.936684i	−0.883547	−	0.468342i	$-0.844852\pi$
$577$	− 22.4999i	− 0.936684i	0.883547	−	0.468342i	$-0.155148\pi$
$578$	0	0
$579$	34.1840	1.42064
$580$	0	0
$581$	0	0
$582$	0	0
$583$	10.7854i	0.446684i
$584$	0	0
$585$	−22.6060	−0.934642
$586$	0	0
$587$	−37.8846	−1.56367	−0.781833	−	0.623488i	$-0.785715\pi$
$587$	−37.8846	−1.56367	−0.781833	+	0.623488i	$0.785715\pi$
$588$	0	0
$589$	12.7880	0.526919
$590$	0	0
$591$	21.9888	0.904500
$592$	0	0
$593$	− 8.76181i	− 0.359804i	−0.983685	−	0.179902i	$-0.942422\pi$
$593$	− 8.76181i	− 0.359804i	0.983685	−	0.179902i	$-0.0575781\pi$
$594$	0	0
$595$	0	0
$596$	0	0
$597$	82.4818	3.37576
$598$	0	0
$599$	44.2183i	1.80671i	0.428891	+	0.903356i	$0.358904\pi$
$599$	44.2183i	1.80671i	−0.428891	+	0.903356i	$0.641096\pi$
$600$	0	0
$601$	− 5.76205i	− 0.235039i	−0.993071	−	0.117519i	$-0.962506\pi$
$601$	− 5.76205i	− 0.235039i	0.993071	−	0.117519i	$-0.0374943\pi$
$602$	0	0
$603$	− 31.1622i	− 1.26902i
$604$	0	0
$605$	9.75338i	0.396531i
$606$	0	0
$607$	35.1798	1.42790	0.713951	−	0.700195i	$-0.246904\pi$
$607$	35.1798	1.42790	0.713951	+	0.700195i	$0.246904\pi$
$608$	0	0
$609$	0	0
$610$	0	0
$611$	5.36558i	0.217068i
$612$	0	0
$613$	−13.8850	−0.560809	−0.280405	−	0.959882i	$-0.590469\pi$
$613$	−13.8850	−0.560809	−0.280405	+	0.959882i	$0.590469\pi$
$614$	0	0
$615$	6.03572	0.243384
$616$	0	0
$617$	−10.4778	−0.421822	−0.210911	−	0.977505i	$-0.567643\pi$
$617$	−10.4778	−0.421822	−0.210911	+	0.977505i	$0.567643\pi$
$618$	0	0
$619$	−22.8120	−0.916890	−0.458445	−	0.888723i	$-0.651593\pi$
$619$	−22.8120	−0.916890	−0.458445	+	0.888723i	$0.651593\pi$
$620$	0	0
$621$	85.7166i	3.43969i
$622$	0	0
$623$	0	0
$624$	0	0
$625$	1.00000	0.0400000
$626$	0	0
$627$	− 29.4962i	− 1.17797i
$628$	0	0
$629$	− 0.0943763i	− 0.00376303i
$630$	0	0
$631$	16.9568i	0.675041i	0.941318	+	0.337520i	$0.109588\pi$
$631$	16.9568i	0.675041i	−0.941318	+	0.337520i	$0.890412\pi$
$632$	0	0
$633$	3.52729i	0.140197i
$634$	0	0
$635$	−9.84060	−0.390512
$636$	0	0
$637$	0	0
$638$	0	0
$639$	− 105.375i	− 4.16858i
$640$	0	0
$641$	23.6793	0.935275	0.467638	−	0.883920i	$-0.345105\pi$
$641$	23.6793	0.935275	0.467638	+	0.883920i	$0.345105\pi$
$642$	0	0
$643$	24.4775	0.965297	0.482648	−	0.875814i	$-0.339675\pi$
$643$	24.4775	0.965297	0.482648	+	0.875814i	$0.339675\pi$
$644$	0	0
$645$	19.3179	0.760640
$646$	0	0
$647$	−21.2674	−0.836106	−0.418053	−	0.908423i	$-0.637287\pi$
$647$	−21.2674	−0.836106	−0.418053	+	0.908423i	$0.637287\pi$
$648$	0	0
$649$	− 6.14601i	− 0.241252i
$650$	0	0
$651$	0	0
$652$	0	0
$653$	−9.17303	−0.358969	−0.179484	−	0.983761i	$-0.557443\pi$
$653$	−9.17303	−0.358969	−0.179484	+	0.983761i	$0.557443\pi$
$654$	0	0
$655$	13.4122i	0.524060i
$656$	0	0
$657$	− 79.7351i	− 3.11076i
$658$	0	0
$659$	− 14.3847i	− 0.560347i	−0.959949	−	0.280173i	$-0.909608\pi$
$659$	− 14.3847i	− 0.560347i	0.959949	−	0.280173i	$-0.0903920\pi$
$660$	0	0
$661$	28.7253i	1.11728i	0.829409	+	0.558642i	$0.188677\pi$
$661$	28.7253i	1.11728i	−0.829409	+	0.558642i	$0.811323\pi$
$662$	0	0
$663$	64.4849	2.50438
$664$	0	0
$665$	0	0
$666$	0	0
$667$	− 48.5902i	− 1.88142i
$668$	0	0
$669$	−70.0459	−2.70813
$670$	0	0
$671$	−9.23360	−0.356459
$672$	0	0
$673$	4.78755	0.184547	0.0922733	−	0.995734i	$-0.470587\pi$
$673$	4.78755	0.184547	0.0922733	+	0.995734i	$0.470587\pi$
$674$	0	0
$675$	13.0672	0.502958
$676$	0	0
$677$	3.88459i	0.149297i	0.997210	+	0.0746484i	$0.0237835\pi$
$677$	3.88459i	0.149297i	−0.997210	+	0.0746484i	$0.976217\pi$
$678$	0	0
$679$	0	0
$680$	0	0
$681$	49.6698	1.90335
$682$	0	0
$683$	24.0909i	0.921812i	0.887449	+	0.460906i	$0.152475\pi$
$683$	24.0909i	0.921812i	−0.887449	+	0.460906i	$0.847525\pi$
$684$	0	0
$685$	8.90085i	0.340084i
$686$	0	0
$687$	− 52.9801i	− 2.02132i
$688$	0	0
$689$	30.7141i	1.17011i
$690$	0	0
$691$	11.0555	0.420571	0.210286	−	0.977640i	$-0.432561\pi$
$691$	11.0555	0.420571	0.210286	+	0.977640i	$0.432561\pi$
$692$	0	0
$693$	0	0
$694$	0	0
$695$	− 13.0058i	− 0.493336i
$696$	0	0
$697$	−12.1081	−0.458628
$698$	0	0
$699$	10.1220	0.382848
$700$	0	0
$701$	26.9827	1.01912	0.509561	−	0.860434i	$-0.329808\pi$
$701$	26.9827	1.01912	0.509561	+	0.860434i	$0.329808\pi$
$702$	0	0
$703$	−0.122935	−0.00463658
$704$	0	0
$705$	− 5.36558i	− 0.202079i
$706$	0	0
$707$	0	0
$708$	0	0
$709$	−22.8531	−0.858268	−0.429134	−	0.903241i	$-0.641181\pi$
$709$	−22.8531	−0.858268	−0.429134	+	0.903241i	$0.641181\pi$
$710$	0	0
$711$	− 44.7749i	− 1.67919i
$712$	0	0
$713$	− 10.0961i	− 0.378101i
$714$	0	0
$715$	− 3.55007i	− 0.132765i
$716$	0	0
$717$	− 13.9820i	− 0.522169i
$718$	0	0
$719$	−46.3877	−1.72997	−0.864985	−	0.501798i	$-0.832672\pi$
$719$	−46.3877	−1.72997	−0.864985	+	0.501798i	$0.832672\pi$
$720$	0	0
$721$	0	0
$722$	0	0
$723$	− 28.4217i	− 1.05701i
$724$	0	0
$725$	−7.40743	−0.275105
$726$	0	0
$727$	−21.7751	−0.807594	−0.403797	−	0.914849i	$-0.632310\pi$
$727$	−21.7751	−0.807594	−0.403797	+	0.914849i	$0.632310\pi$
$728$	0	0
$729$	19.1074	0.707683
$730$	0	0
$731$	−38.7532	−1.43334
$732$	0	0
$733$	15.0485i	0.555829i	0.960606	+	0.277915i	$0.0896432\pi$
$733$	15.0485i	0.555829i	−0.960606	+	0.277915i	$0.910357\pi$
$734$	0	0
$735$	0	0
$736$	0	0
$737$	4.89374	0.180263
$738$	0	0
$739$	26.0801i	0.959373i	0.877440	+	0.479687i	$0.159250\pi$
$739$	26.0801i	0.959373i	−0.877440	+	0.479687i	$0.840750\pi$
$740$	0	0
$741$	− 83.9982i	− 3.08575i
$742$	0	0
$743$	− 3.12359i	− 0.114594i	−0.998357	−	0.0572968i	$-0.981752\pi$
$743$	− 3.12359i	− 0.114594i	0.998357	−	0.0572968i	$-0.0182481\pi$
$744$	0	0
$745$	12.6656i	0.464030i
$746$	0	0
$747$	−50.4800	−1.84697
$748$	0	0
$749$	0	0
$750$	0	0
$751$	10.3196i	0.376567i	0.982115	+	0.188284i	$0.0602924\pi$
$751$	10.3196i	0.376567i	−0.982115	+	0.188284i	$0.939708\pi$
$752$	0	0
$753$	−73.7900	−2.68906
$754$	0	0
$755$	−21.4229	−0.779660
$756$	0	0
$757$	27.3952	0.995695	0.497848	−	0.867264i	$-0.334124\pi$
$757$	27.3952	0.995695	0.497848	+	0.867264i	$0.334124\pi$
$758$	0	0
$759$	−23.2872	−0.845273
$760$	0	0
$761$	− 41.9586i	− 1.52100i	−0.649340	−	0.760499i	$-0.724955\pi$
$761$	− 41.9586i	− 1.52100i	0.649340	−	0.760499i	$-0.275045\pi$
$762$	0	0
$763$	0	0
$764$	0	0
$765$	−45.3494	−1.63961
$766$	0	0
$767$	− 17.5024i	− 0.631974i
$768$	0	0
$769$	36.0529i	1.30010i	0.759891	+	0.650050i	$0.225252\pi$
$769$	36.0529i	1.30010i	−0.759891	+	0.650050i	$0.774748\pi$
$770$	0	0
$771$	− 9.16707i	− 0.330144i
$772$	0	0
$773$	− 4.35566i	− 0.156662i	−0.996927	−	0.0783311i	$-0.975041\pi$
$773$	− 4.35566i	− 0.156662i	0.996927	−	0.0783311i	$-0.0249591\pi$
$774$	0	0
$775$	−1.53912	−0.0552866
$776$	0	0
$777$	0	0
$778$	0	0
$779$	15.7721i	0.565094i
$780$	0	0
$781$	16.5482	0.592143
$782$	0	0
$783$	−96.7945	−3.45915
$784$	0	0
$785$	−13.6569	−0.487434
$786$	0	0
$787$	−9.91530	−0.353442	−0.176721	−	0.984261i	$-0.556549\pi$
$787$	−9.91530	−0.353442	−0.176721	+	0.984261i	$0.556549\pi$
$788$	0	0
$789$	− 31.2820i	− 1.11367i
$790$	0	0
$791$	0	0
$792$	0	0
$793$	−26.2951	−0.933765
$794$	0	0
$795$	− 30.7141i	− 1.08932i
$796$	0	0
$797$	13.0901i	0.463674i	0.972755	+	0.231837i	$0.0744735\pi$
$797$	13.0901i	0.463674i	−0.972755	+	0.231837i	$0.925526\pi$
$798$	0	0
$799$	10.7638i	0.380795i
$800$	0	0
$801$	− 78.4337i	− 2.77132i
$802$	0	0
$803$	12.5217	0.441881
$804$	0	0
$805$	0	0
$806$	0	0
$807$	− 33.4218i	− 1.17650i
$808$	0	0
$809$	−0.143890	−0.00505892	−0.00252946	−	0.999997i	$-0.500805\pi$
$809$	−0.143890	−0.00505892	−0.00252946	+	0.999997i	$0.500805\pi$
$810$	0	0
$811$	−18.0191	−0.632738	−0.316369	−	0.948636i	$-0.602464\pi$
$811$	−18.0191	−0.632738	−0.316369	+	0.948636i	$0.602464\pi$
$812$	0	0
$813$	22.4382	0.786941
$814$	0	0
$815$	−15.8364	−0.554726
$816$	0	0
$817$	50.4800i	1.76607i
$818$	0	0
$819$	0	0
$820$	0	0
$821$	−15.9591	−0.556977	−0.278488	−	0.960440i	$-0.589833\pi$
$821$	−15.9591	−0.556977	−0.278488	+	0.960440i	$0.589833\pi$
$822$	0	0
$823$	− 23.5834i	− 0.822067i	−0.911620	−	0.411034i	$-0.865168\pi$
$823$	− 23.5834i	− 0.822067i	0.911620	−	0.411034i	$-0.134832\pi$
$824$	0	0
$825$	3.55007i	0.123597i
$826$	0	0
$827$	− 25.1930i	− 0.876045i	−0.898964	−	0.438023i	$-0.855679\pi$
$827$	− 25.1930i	− 0.876045i	0.898964	−	0.438023i	$-0.144321\pi$
$828$	0	0
$829$	− 44.1665i	− 1.53396i	−0.641668	−	0.766982i	$-0.721757\pi$
$829$	− 44.1665i	− 1.53396i	0.641668	−	0.766982i	$-0.278243\pi$
$830$	0	0
$831$	−16.2303	−0.563022
$832$	0	0
$833$	0	0
$834$	0	0
$835$	13.5606i	0.469285i
$836$	0	0
$837$	−20.1120	−0.695171
$838$	0	0
$839$	−45.9867	−1.58764	−0.793819	−	0.608154i	$-0.791910\pi$
$839$	−45.9867	−1.58764	−0.793819	+	0.608154i	$0.791910\pi$
$840$	0	0
$841$	25.8699	0.892067
$842$	0	0
$843$	12.0254	0.414177
$844$	0	0
$845$	2.89027i	0.0994283i
$846$	0	0
$847$	0	0
$848$	0	0
$849$	−11.6786	−0.400809
$850$	0	0
$851$	0.0970570i	0.00332707i
$852$	0	0
$853$	8.91726i	0.305321i	0.988279	+	0.152661i	$0.0487842\pi$
$853$	8.91726i	0.305321i	−0.988279	+	0.152661i	$0.951216\pi$
$854$	0	0
$855$	59.0722i	2.02023i
$856$	0	0
$857$	10.0059i	0.341795i	0.985289	+	0.170897i	$0.0546666\pi$
$857$	10.0059i	0.341795i	−0.985289	+	0.170897i	$0.945333\pi$
$858$	0	0
$859$	23.1802	0.790900	0.395450	−	0.918488i	$-0.370589\pi$
$859$	23.1802	0.790900	0.395450	+	0.918488i	$0.370589\pi$
$860$	0	0
$861$	0	0
$862$	0	0
$863$	28.7185i	0.977589i	0.872399	+	0.488794i	$0.162563\pi$
$863$	28.7185i	0.977589i	−0.872399	+	0.488794i	$0.837437\pi$
$864$	0	0
$865$	18.0299	0.613035
$866$	0	0
$867$	75.3089	2.55762
$868$	0	0
$869$	7.03150	0.238527
$870$	0	0
$871$	13.9362	0.472210
$872$	0	0
$873$	− 108.975i	− 3.68825i
$874$	0	0
$875$	0	0
$876$	0	0
$877$	−47.5055	−1.60415	−0.802074	−	0.597225i	$-0.796270\pi$
$877$	−47.5055	−1.60415	−0.802074	+	0.597225i	$0.796270\pi$
$878$	0	0
$879$	− 0.199317i	− 0.00672279i
$880$	0	0
$881$	− 42.6584i	− 1.43720i	−0.695425	−	0.718599i	$-0.744784\pi$
$881$	− 42.6584i	− 1.43720i	0.695425	−	0.718599i	$-0.255216\pi$
$882$	0	0
$883$	26.6051i	0.895333i	0.894201	+	0.447666i	$0.147745\pi$
$883$	26.6051i	0.895333i	−0.894201	+	0.447666i	$0.852255\pi$
$884$	0	0
$885$	17.5024i	0.588335i
$886$	0	0
$887$	26.8550	0.901703	0.450852	−	0.892599i	$-0.351120\pi$
$887$	26.8550	0.901703	0.450852	+	0.892599i	$0.351120\pi$
$888$	0	0
$889$	0	0
$890$	0	0
$891$	22.5750i	0.756292i
$892$	0	0
$893$	14.0209	0.469192
$894$	0	0
$895$	−8.38234	−0.280191
$896$	0	0
$897$	−66.3165	−2.21424
$898$	0	0
$899$	11.4009	0.380241
$900$	0	0
$901$	61.6149i	2.05269i
$902$	0	0
$903$	0	0
$904$	0	0
$905$	−16.3153	−0.542339
$906$	0	0
$907$	− 16.7715i	− 0.556889i	−0.960452	−	0.278445i	$-0.910181\pi$
$907$	− 16.7715i	− 0.556889i	0.960452	−	0.278445i	$-0.0898189\pi$
$908$	0	0
$909$	− 57.8041i	− 1.91724i
$910$	0	0
$911$	25.2628i	0.836993i	0.908218	+	0.418497i	$0.137443\pi$
$911$	25.2628i	0.836993i	−0.908218	+	0.418497i	$0.862557\pi$
$912$	0	0
$913$	− 7.92744i	− 0.262360i
$914$	0	0
$915$	26.2951	0.869288
$916$	0	0
$917$	0	0
$918$	0	0
$919$	33.3235i	1.09924i	0.835415	+	0.549620i	$0.185227\pi$
$919$	33.3235i	1.09924i	−0.835415	+	0.549620i	$0.814773\pi$
$920$	0	0
$921$	−39.5832	−1.30431
$922$	0	0
$923$	47.1254	1.55115
$924$	0	0
$925$	0.0147960	0.000486490 0
$926$	0	0
$927$	89.1793	2.92903
$928$	0	0
$929$	− 47.3363i	− 1.55305i	−0.630084	−	0.776527i	$-0.716980\pi$
$929$	− 47.3363i	− 1.55305i	0.630084	−	0.776527i	$-0.283020\pi$
$930$	0	0
$931$	0	0
$932$	0	0
$933$	−72.6968	−2.37999
$934$	0	0
$935$	− 7.12172i	− 0.232905i
$936$	0	0
$937$	− 47.0302i	− 1.53641i	−0.640203	−	0.768206i	$-0.721150\pi$
$937$	− 47.0302i	− 1.53641i	0.640203	−	0.768206i	$-0.278850\pi$
$938$	0	0
$939$	− 75.4283i	− 2.46151i
$940$	0	0
$941$	3.77390i	0.123026i	0.998106	+	0.0615129i	$0.0195925\pi$
$941$	3.77390i	0.123026i	−0.998106	+	0.0615129i	$0.980407\pi$
$942$	0	0
$943$	12.4521	0.405495
$944$	0	0
$945$	0	0
$946$	0	0
$947$	− 42.9655i	− 1.39619i	−0.716005	−	0.698095i	$-0.754031\pi$
$947$	− 42.9655i	− 1.39619i	0.716005	−	0.698095i	$-0.245969\pi$
$948$	0	0
$949$	35.6588	1.15753
$950$	0	0
$951$	−34.7212	−1.12591
$952$	0	0
$953$	7.30904	0.236763	0.118382	−	0.992968i	$-0.462229\pi$
$953$	7.30904	0.236763	0.118382	+	0.992968i	$0.462229\pi$
$954$	0	0
$955$	5.53943	0.179252
$956$	0	0
$957$	− 26.2968i	− 0.850056i
$958$	0	0
$959$	0	0
$960$	0	0
$961$	−28.6311	−0.923585
$962$	0	0
$963$	97.4011i	3.13871i
$964$	0	0
$965$	− 10.7511i	− 0.346090i
$966$	0	0
$967$	48.4508i	1.55807i	0.626979	+	0.779036i	$0.284291\pi$
$967$	48.4508i	1.55807i	−0.626979	+	0.779036i	$0.715709\pi$
$968$	0	0
$969$	− 168.507i	− 5.41323i
$970$	0	0
$971$	−32.0887	−1.02978	−0.514888	−	0.857257i	$-0.672166\pi$
$971$	−32.0887	−1.02978	−0.514888	+	0.857257i	$0.672166\pi$
$972$	0	0
$973$	0	0
$974$	0	0
$975$	10.1097i	0.323771i
$976$	0	0
$977$	−18.2557	−0.584051	−0.292026	−	0.956410i	$-0.594329\pi$
$977$	−18.2557	−0.584051	−0.292026	+	0.956410i	$0.594329\pi$
$978$	0	0
$979$	12.3173	0.393663
$980$	0	0
$981$	60.8296	1.94214
$982$	0	0
$983$	−6.47484	−0.206515	−0.103258	−	0.994655i	$-0.532927\pi$
$983$	−6.47484	−0.206515	−0.103258	+	0.994655i	$0.532927\pi$
$984$	0	0
$985$	− 6.91564i	− 0.220351i
$986$	0	0
$987$	0	0
$988$	0	0
$989$	39.8539	1.26728
$990$	0	0
$991$	− 13.1020i	− 0.416200i	−0.978108	−	0.208100i	$-0.933272\pi$
$991$	− 13.1020i	− 0.416200i	0.978108	−	0.208100i	$-0.0667280\pi$
$992$	0	0
$993$	83.7638i	2.65817i
$994$	0	0
$995$	− 25.9411i	− 0.822388i
$996$	0	0
$997$	32.8675i	1.04093i	0.853884	+	0.520463i	$0.174240\pi$
$997$	32.8675i	1.04093i	−0.853884	+	0.520463i	$0.825760\pi$
$998$	0	0
$999$	0.193343	0.00611710

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	3920.2.k.a.2351.2	yes	8
4.3	odd	2		3920.2.k.c.2351.8	yes	8
7.6	odd	2		3920.2.k.c.2351.7	yes	8
28.27	even	2	inner	3920.2.k.a.2351.1	✓	8

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
3920.2.k.a.2351.1	✓	8	28.27	even	2	inner
3920.2.k.a.2351.2	yes	8	1.1	even	1	trivial
3920.2.k.c.2351.7	yes	8	7.6	odd	2
3920.2.k.c.2351.8	yes	8	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:	\( N \)	\(=\)	\( 3920 = 2^{4} \cdot 5 \cdot 7^{2} \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	3920.k (of order \(2\), degree \(1\), minimal)

\(f(q)\)	\(=\)	\(q-3.17958 q^{3} +1.00000i q^{5} +7.10973 q^{9} +O(q^{10})\)	\(q-3.17958 q^{3} +1.00000i q^{5} +7.10973 q^{9} +1.11652i q^{11} +3.17958i q^{13} -3.17958i q^{15} +6.37849i q^{17} +8.30864 q^{19} -6.55967i q^{23} -1.00000 q^{25} -13.0672 q^{27} +7.40743 q^{29} +1.53912 q^{31} -3.55007i q^{33} -0.0147960 q^{37} -10.1097i q^{39} +1.89828i q^{41} +6.07560i q^{43} +7.10973i q^{45} +1.68751 q^{47} -20.2809i q^{51} +9.65980 q^{53} -1.11652 q^{55} -26.4180 q^{57} -5.50461 q^{59} +8.26998i q^{61} -3.17958 q^{65} -4.38303i q^{67} +20.8570i q^{69} -14.8213i q^{71} -11.2149i q^{73} +3.17958 q^{75} -6.29769i q^{79} +20.2191 q^{81} -7.10013 q^{83} -6.37849 q^{85} -23.5525 q^{87} -11.0319i q^{89} -4.89374 q^{93} +8.30864i q^{95} -15.3276i q^{97} +7.93816i q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 8 q - 8 q^{3} + 16 q^{9}+O(q^{10}) \) 8 * q - 8 * q^3 + 16 * q^9	\( 8 q - 8 q^{3} + 16 q^{9} + 16 q^{19} - 8 q^{25} - 8 q^{27} + 8 q^{29} + 48 q^{31} + 24 q^{47} + 32 q^{53} - 8 q^{55} - 32 q^{57} + 16 q^{59} - 8 q^{65} + 8 q^{75} - 8 q^{81} - 48 q^{83} - 24 q^{85} - 24 q^{87} - 32 q^{93}+O(q^{100}) \) 8 * q - 8 * q^3 + 16 * q^9 + 16 * q^19 - 8 * q^25 - 8 * q^27 + 8 * q^29 + 48 * q^31 + 24 * q^47 + 32 * q^53 - 8 * q^55 - 32 * q^57 + 16 * q^59 - 8 * q^65 + 8 * q^75 - 8 * q^81 - 48 * q^83 - 24 * q^85 - 24 * q^87 - 32 * q^93

Introduction

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Database

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