LMFDB - Embedded newform 4304.2.a.j.1.7

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [4304,2,Mod(1,4304)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(4304, 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("4304.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 4304 = 2^{4} \cdot 269 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	4304.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:	$34.3676130300$
Analytic rank:	$1$
Dimension:	$13$
Coefficient field:	$\mathbb{Q}[x]/(x^{13} - \cdots)$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{13} - 3 x^{12} - 15 x^{11} + 45 x^{10} + 82 x^{9} - 242 x^{8} - 201 x^{7} + 574 x^{6} + 200 x^{5} + \cdots + 4 $ x^13 - 3x^12 - 15x^11 + 45x^10 + 82x^9 - 242x^8 - 201x^7 + 574x^6 + 200x^5 - 595x^4 - 4x^3 + 220x^2 - 67x + 4
Coefficient ring:	$\Z[a_1, \ldots, a_{5}]$
Coefficient ring index:	$ 1 $
Twist minimal:	no (minimal twist has level 2152)
Fricke sign:	$+1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.7
Root			$0.326022$ of defining polynomial
Character	$\chi$	$=$	4304.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+0.326022 q^{3} +1.25336 q^{5} -3.69529 q^{7} -2.89371 q^{9} +5.00996 q^{11} +2.41345 q^{13} +0.408623 q^{15} -5.30028 q^{17} -0.886964 q^{19} -1.20475 q^{21} +7.57500 q^{23} -3.42909 q^{25} -1.92148 q^{27} -6.03262 q^{29} +7.35225 q^{31} +1.63336 q^{33} -4.63153 q^{35} -3.61677 q^{37} +0.786836 q^{39} -2.17395 q^{41} -12.5138 q^{43} -3.62686 q^{45} -4.49061 q^{47} +6.65518 q^{49} -1.72801 q^{51} +0.796691 q^{53} +6.27929 q^{55} -0.289170 q^{57} +4.80584 q^{59} +0.838597 q^{61} +10.6931 q^{63} +3.02491 q^{65} +6.72297 q^{67} +2.46962 q^{69} +2.08679 q^{71} +1.21332 q^{73} -1.11796 q^{75} -18.5133 q^{77} -10.2032 q^{79} +8.05468 q^{81} +3.77744 q^{83} -6.64316 q^{85} -1.96677 q^{87} -16.7296 q^{89} -8.91838 q^{91} +2.39700 q^{93} -1.11169 q^{95} +17.4798 q^{97} -14.4974 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 13 q + 3 q^{3} - 9 q^{5} - 6 q^{7} + 6 q^{11} - 8 q^{13} + q^{15} - 14 q^{17} + 6 q^{19} - 13 q^{21} + 11 q^{23} - 2 q^{25} + 9 q^{27} - 19 q^{29} - 14 q^{31} - 15 q^{33} + 20 q^{35} - 23 q^{37} - 8 q^{39}+ \cdots + 7 q^{99}+O(q^{100}) $ 13 * q + 3 * q^3 - 9 * q^5 - 6 * q^7 + 6 * q^11 - 8 * q^13 + q^15 - 14 * q^17 + 6 * q^19 - 13 * q^21 + 11 * q^23 - 2 * q^25 + 9 * q^27 - 19 * q^29 - 14 * q^31 - 15 * q^33 + 20 * q^35 - 23 * q^37 - 8 * q^39 - 23 * q^41 + 9 * q^43 - 14 * q^45 + 8 * q^47 - 7 * q^49 + 9 * q^51 - 24 * q^53 - 19 * q^55 - 30 * q^57 + 17 * q^59 - 23 * q^61 - 4 * q^63 - 34 * q^65 + 15 * q^67 - 21 * q^69 + 14 * q^71 - 21 * q^73 + 9 * q^75 - 50 * q^77 - 42 * q^79 - 31 * q^81 + 29 * q^83 - 33 * q^85 - 15 * q^87 - 33 * q^89 + 21 * q^91 - 37 * q^93 - 2 * q^95 - 27 * q^97 + 7 * q^99

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
$3$	0.326022	0.188229	0.0941144	−	0.995561i	$-0.469998\pi$
$3$	0.326022	0.188229	0.0941144	+	0.995561i	$0.469998\pi$
$4$	0	0
$5$	1.25336	0.560519	0.280260	−	0.959924i	$-0.409579\pi$
$5$	1.25336	0.560519	0.280260	+	0.959924i	$0.409579\pi$
$6$	0	0
$7$	−3.69529	−1.39669	−0.698344	−	0.715762i	$-0.746080\pi$
$7$	−3.69529	−1.39669	−0.698344	+	0.715762i	$0.746080\pi$
$8$	0	0
$9$	−2.89371	−0.964570
$10$	0	0
$11$	5.00996	1.51056	0.755280	−	0.655402i	$-0.227501\pi$
$11$	5.00996	1.51056	0.755280	+	0.655402i	$0.227501\pi$
$12$	0	0
$13$	2.41345	0.669369	0.334685	−	0.942330i	$-0.391370\pi$
$13$	2.41345	0.669369	0.334685	+	0.942330i	$0.391370\pi$
$14$	0	0
$15$	0.408623	0.105506
$16$	0	0
$17$	−5.30028	−1.28551	−0.642754	−	0.766073i	$-0.722208\pi$
$17$	−5.30028	−1.28551	−0.642754	+	0.766073i	$0.722208\pi$
$18$	0	0
$19$	−0.886964	−0.203484	−0.101742	−	0.994811i	$-0.532442\pi$
$19$	−0.886964	−0.203484	−0.101742	+	0.994811i	$0.532442\pi$
$20$	0	0
$21$	−1.20475	−0.262897
$22$	0	0
$23$	7.57500	1.57950	0.789748	−	0.613431i	$-0.210211\pi$
$23$	7.57500	1.57950	0.789748	+	0.613431i	$0.210211\pi$
$24$	0	0
$25$	−3.42909	−0.685818
$26$	0	0
$27$	−1.92148	−0.369789
$28$	0	0
$29$	−6.03262	−1.12023	−0.560115	−	0.828415i	$-0.689243\pi$
$29$	−6.03262	−1.12023	−0.560115	+	0.828415i	$0.689243\pi$
$30$	0	0
$31$	7.35225	1.32050	0.660252	−	0.751044i	$-0.270450\pi$
$31$	7.35225	1.32050	0.660252	+	0.751044i	$0.270450\pi$
$32$	0	0
$33$	1.63336	0.284331
$34$	0	0
$35$	−4.63153	−0.782871
$36$	0	0
$37$	−3.61677	−0.594593	−0.297296	−	0.954785i	$-0.596085\pi$
$37$	−3.61677	−0.594593	−0.297296	+	0.954785i	$0.596085\pi$
$38$	0	0
$39$	0.786836	0.125995
$40$	0	0
$41$	−2.17395	−0.339513	−0.169757	−	0.985486i	$-0.554298\pi$
$41$	−2.17395	−0.339513	−0.169757	+	0.985486i	$0.554298\pi$
$42$	0	0
$43$	−12.5138	−1.90834	−0.954168	−	0.299273i	$-0.903256\pi$
$43$	−12.5138	−1.90834	−0.954168	+	0.299273i	$0.903256\pi$
$44$	0	0
$45$	−3.62686	−0.540660
$46$	0	0
$47$	−4.49061	−0.655023	−0.327511	−	0.944847i	$-0.606210\pi$
$47$	−4.49061	−0.655023	−0.327511	+	0.944847i	$0.606210\pi$
$48$	0	0
$49$	6.65518	0.950740
$50$	0	0
$51$	−1.72801	−0.241970
$52$	0	0
$53$	0.796691	0.109434	0.0547170	−	0.998502i	$-0.482574\pi$
$53$	0.796691	0.109434	0.0547170	+	0.998502i	$0.482574\pi$
$54$	0	0
$55$	6.27929	0.846699
$56$	0	0
$57$	−0.289170	−0.0383015
$58$	0	0
$59$	4.80584	0.625667	0.312834	−	0.949808i	$-0.398722\pi$
$59$	4.80584	0.625667	0.312834	+	0.949808i	$0.398722\pi$
$60$	0	0
$61$	0.838597	0.107371	0.0536857	−	0.998558i	$-0.482903\pi$
$61$	0.838597	0.107371	0.0536857	+	0.998558i	$0.482903\pi$
$62$	0	0
$63$	10.6931	1.34720
$64$	0	0
$65$	3.02491	0.375195
$66$	0	0
$67$	6.72297	0.821341	0.410671	−	0.911784i	$-0.365295\pi$
$67$	6.72297	0.821341	0.410671	+	0.911784i	$0.365295\pi$
$68$	0	0
$69$	2.46962	0.297307
$70$	0	0
$71$	2.08679	0.247656	0.123828	−	0.992304i	$-0.460483\pi$
$71$	2.08679	0.247656	0.123828	+	0.992304i	$0.460483\pi$
$72$	0	0
$73$	1.21332	0.142009	0.0710043	−	0.997476i	$-0.477380\pi$
$73$	1.21332	0.142009	0.0710043	+	0.997476i	$0.477380\pi$
$74$	0	0
$75$	−1.11796	−0.129091
$76$	0	0
$77$	−18.5133	−2.10978
$78$	0	0
$79$	−10.2032	−1.14795	−0.573975	−	0.818873i	$-0.694599\pi$
$79$	−10.2032	−1.14795	−0.573975	+	0.818873i	$0.694599\pi$
$80$	0	0
$81$	8.05468	0.894965
$82$	0	0
$83$	3.77744	0.414628	0.207314	−	0.978274i	$-0.433528\pi$
$83$	3.77744	0.414628	0.207314	+	0.978274i	$0.433528\pi$
$84$	0	0
$85$	−6.64316	−0.720552
$86$	0	0
$87$	−1.96677	−0.210860
$88$	0	0
$89$	−16.7296	−1.77334	−0.886669	−	0.462405i	$-0.846987\pi$
$89$	−16.7296	−1.77334	−0.886669	+	0.462405i	$0.846987\pi$
$90$	0	0
$91$	−8.91838	−0.934901
$92$	0	0
$93$	2.39700	0.248557
$94$	0	0
$95$	−1.11169	−0.114056
$96$	0	0
$97$	17.4798	1.77481	0.887403	−	0.460995i	$-0.152507\pi$
$97$	17.4798	1.77481	0.887403	+	0.460995i	$0.152507\pi$
$98$	0	0
$99$	−14.4974	−1.45704
$100$	0	0
$101$	−15.3226	−1.52465	−0.762326	−	0.647193i	$-0.775943\pi$
$101$	−15.3226	−1.52465	−0.762326	+	0.647193i	$0.775943\pi$
$102$	0	0
$103$	−18.8960	−1.86188	−0.930938	−	0.365176i	$-0.881009\pi$
$103$	−18.8960	−1.86188	−0.930938	+	0.365176i	$0.881009\pi$
$104$	0	0
$105$	−1.50998	−0.147359
$106$	0	0
$107$	−7.56339	−0.731180	−0.365590	−	0.930776i	$-0.619133\pi$
$107$	−7.56339	−0.731180	−0.365590	+	0.930776i	$0.619133\pi$
$108$	0	0
$109$	−19.1055	−1.82998	−0.914988	−	0.403481i	$-0.867800\pi$
$109$	−19.1055	−1.82998	−0.914988	+	0.403481i	$0.867800\pi$
$110$	0	0
$111$	−1.17915	−0.111919
$112$	0	0
$113$	4.76999	0.448723	0.224362	−	0.974506i	$-0.427970\pi$
$113$	4.76999	0.448723	0.224362	+	0.974506i	$0.427970\pi$
$114$	0	0
$115$	9.49419	0.885338
$116$	0	0
$117$	−6.98381	−0.645654
$118$	0	0
$119$	19.5861	1.79545
$120$	0	0
$121$	14.0997	1.28179
$122$	0	0
$123$	−0.708754	−0.0639062
$124$	0	0
$125$	−10.5647	−0.944934
$126$	0	0
$127$	−0.363345	−0.0322417	−0.0161208	−	0.999870i	$-0.505132\pi$
$127$	−0.363345	−0.0322417	−0.0161208	+	0.999870i	$0.505132\pi$
$128$	0	0
$129$	−4.07977	−0.359204
$130$	0	0
$131$	−22.4233	−1.95913	−0.979566	−	0.201122i	$-0.935541\pi$
$131$	−22.4233	−1.95913	−0.979566	+	0.201122i	$0.935541\pi$
$132$	0	0
$133$	3.27759	0.284203
$134$	0	0
$135$	−2.40830	−0.207274
$136$	0	0
$137$	10.8876	0.930191	0.465096	−	0.885260i	$-0.346020\pi$
$137$	10.8876	0.930191	0.465096	+	0.885260i	$0.346020\pi$
$138$	0	0
$139$	14.2991	1.21283	0.606417	−	0.795147i	$-0.292606\pi$
$139$	14.2991	1.21283	0.606417	+	0.795147i	$0.292606\pi$
$140$	0	0
$141$	−1.46404	−0.123294
$142$	0	0
$143$	12.0913	1.01112
$144$	0	0
$145$	−7.56104	−0.627911
$146$	0	0
$147$	2.16973	0.178957
$148$	0	0
$149$	−14.5630	−1.19305	−0.596525	−	0.802595i	$-0.703452\pi$
$149$	−14.5630	−1.19305	−0.596525	+	0.802595i	$0.703452\pi$
$150$	0	0
$151$	−0.354938	−0.0288845	−0.0144422	−	0.999896i	$-0.504597\pi$
$151$	−0.354938	−0.0288845	−0.0144422	+	0.999896i	$0.504597\pi$
$152$	0	0
$153$	15.3375	1.23996
$154$	0	0
$155$	9.21502	0.740168
$156$	0	0
$157$	−4.96121	−0.395948	−0.197974	−	0.980207i	$-0.563436\pi$
$157$	−4.96121	−0.395948	−0.197974	+	0.980207i	$0.563436\pi$
$158$	0	0
$159$	0.259739	0.0205986
$160$	0	0
$161$	−27.9918	−2.20606
$162$	0	0
$163$	14.4104	1.12871	0.564357	−	0.825531i	$-0.309124\pi$
$163$	14.4104	1.12871	0.564357	+	0.825531i	$0.309124\pi$
$164$	0	0
$165$	2.04718	0.159373
$166$	0	0
$167$	−15.0597	−1.16535	−0.582676	−	0.812704i	$-0.697994\pi$
$167$	−15.0597	−1.16535	−0.582676	+	0.812704i	$0.697994\pi$
$168$	0	0
$169$	−7.17528	−0.551945
$170$	0	0
$171$	2.56662	0.196274
$172$	0	0
$173$	−2.07090	−0.157448	−0.0787240	−	0.996896i	$-0.525085\pi$
$173$	−2.07090	−0.157448	−0.0787240	+	0.996896i	$0.525085\pi$
$174$	0	0
$175$	12.6715	0.957874
$176$	0	0
$177$	1.56681	0.117769
$178$	0	0
$179$	−8.32293	−0.622085	−0.311042	−	0.950396i	$-0.600678\pi$
$179$	−8.32293	−0.622085	−0.311042	+	0.950396i	$0.600678\pi$
$180$	0	0
$181$	−7.53110	−0.559782	−0.279891	−	0.960032i	$-0.590298\pi$
$181$	−7.53110	−0.559782	−0.279891	+	0.960032i	$0.590298\pi$
$182$	0	0
$183$	0.273401	0.0202104
$184$	0	0
$185$	−4.53311	−0.333281
$186$	0	0
$187$	−26.5542	−1.94184
$188$	0	0
$189$	7.10042	0.516480
$190$	0	0
$191$	5.87599	0.425172	0.212586	−	0.977142i	$-0.431811\pi$
$191$	5.87599	0.425172	0.212586	+	0.977142i	$0.431811\pi$
$192$	0	0
$193$	−18.7956	−1.35293	−0.676467	−	0.736473i	$-0.736490\pi$
$193$	−18.7956	−1.35293	−0.676467	+	0.736473i	$0.736490\pi$
$194$	0	0
$195$	0.986189	0.0706224
$196$	0	0
$197$	−6.91975	−0.493012	−0.246506	−	0.969141i	$-0.579282\pi$
$197$	−6.91975	−0.493012	−0.246506	+	0.969141i	$0.579282\pi$
$198$	0	0
$199$	−19.1138	−1.35494	−0.677471	−	0.735550i	$-0.736924\pi$
$199$	−19.1138	−1.35494	−0.677471	+	0.735550i	$0.736924\pi$
$200$	0	0
$201$	2.19183	0.154600
$202$	0	0
$203$	22.2923	1.56461
$204$	0	0
$205$	−2.72473	−0.190304
$206$	0	0
$207$	−21.9198	−1.52353
$208$	0	0
$209$	−4.44366	−0.307374
$210$	0	0
$211$	27.5643	1.89760	0.948801	−	0.315875i	$-0.102298\pi$
$211$	27.5643	1.89760	0.948801	+	0.315875i	$0.102298\pi$
$212$	0	0
$213$	0.680339	0.0466161
$214$	0	0
$215$	−15.6843	−1.06966
$216$	0	0
$217$	−27.1687	−1.84433
$218$	0	0
$219$	0.395569	0.0267301
$220$	0	0
$221$	−12.7919	−0.860479
$222$	0	0
$223$	22.5228	1.50824	0.754121	−	0.656736i	$-0.228063\pi$
$223$	22.5228	1.50824	0.754121	+	0.656736i	$0.228063\pi$
$224$	0	0
$225$	9.92279	0.661519
$226$	0	0
$227$	5.19737	0.344962	0.172481	−	0.985013i	$-0.444822\pi$
$227$	5.19737	0.344962	0.172481	+	0.985013i	$0.444822\pi$
$228$	0	0
$229$	−17.2822	−1.14204	−0.571021	−	0.820935i	$-0.693453\pi$
$229$	−17.2822	−1.14204	−0.571021	+	0.820935i	$0.693453\pi$
$230$	0	0
$231$	−6.03573	−0.397122
$232$	0	0
$233$	9.67253	0.633669	0.316834	−	0.948481i	$-0.397380\pi$
$233$	9.67253	0.633669	0.316834	+	0.948481i	$0.397380\pi$
$234$	0	0
$235$	−5.62835	−0.367153
$236$	0	0
$237$	−3.32647	−0.216077
$238$	0	0
$239$	−24.9782	−1.61571	−0.807853	−	0.589383i	$-0.799371\pi$
$239$	−24.9782	−1.61571	−0.807853	+	0.589383i	$0.799371\pi$
$240$	0	0
$241$	−23.5977	−1.52006	−0.760032	−	0.649886i	$-0.774817\pi$
$241$	−23.5977	−1.52006	−0.760032	+	0.649886i	$0.774817\pi$
$242$	0	0
$243$	8.39044	0.538247
$244$	0	0
$245$	8.34133	0.532908
$246$	0	0
$247$	−2.14064	−0.136206
$248$	0	0
$249$	1.23153	0.0780450
$250$	0	0
$251$	8.55653	0.540083	0.270042	−	0.962849i	$-0.412962\pi$
$251$	8.55653	0.540083	0.270042	+	0.962849i	$0.412962\pi$
$252$	0	0
$253$	37.9505	2.38592
$254$	0	0
$255$	−2.16582	−0.135629
$256$	0	0
$257$	15.7949	0.985261	0.492630	−	0.870239i	$-0.336035\pi$
$257$	15.7949	0.985261	0.492630	+	0.870239i	$0.336035\pi$
$258$	0	0
$259$	13.3650	0.830461
$260$	0	0
$261$	17.4567	1.08054
$262$	0	0
$263$	−26.5167	−1.63509	−0.817545	−	0.575865i	$-0.804665\pi$
$263$	−26.5167	−1.63509	−0.817545	+	0.575865i	$0.804665\pi$
$264$	0	0
$265$	0.998540	0.0613398
$266$	0	0
$267$	−5.45423	−0.333793
$268$	0	0
$269$	−1.00000	−0.0609711
$269$	−1.00000	−0.0609711
$270$	0	0
$271$	17.9669	1.09141	0.545707	−	0.837976i	$-0.316261\pi$
$271$	17.9669	1.09141	0.545707	+	0.837976i	$0.316261\pi$
$272$	0	0
$273$	−2.90759	−0.175975
$274$	0	0
$275$	−17.1796	−1.03597
$276$	0	0
$277$	25.1450	1.51082	0.755408	−	0.655254i	$-0.227438\pi$
$277$	25.1450	1.51082	0.755408	+	0.655254i	$0.227438\pi$
$278$	0	0
$279$	−21.2753	−1.27372
$280$	0	0
$281$	−4.30104	−0.256579	−0.128289	−	0.991737i	$-0.540949\pi$
$281$	−4.30104	−0.256579	−0.128289	+	0.991737i	$0.540949\pi$
$282$	0	0
$283$	11.5434	0.686185	0.343092	−	0.939302i	$-0.388526\pi$
$283$	11.5434	0.686185	0.343092	+	0.939302i	$0.388526\pi$
$284$	0	0
$285$	−0.362434	−0.0214687
$286$	0	0
$287$	8.03336	0.474194
$288$	0	0
$289$	11.0930	0.652529
$290$	0	0
$291$	5.69880	0.334070
$292$	0	0
$293$	−6.42354	−0.375267	−0.187633	−	0.982239i	$-0.560082\pi$
$293$	−6.42354	−0.375267	−0.187633	+	0.982239i	$0.560082\pi$
$294$	0	0
$295$	6.02344	0.350699
$296$	0	0
$297$	−9.62654	−0.558588
$298$	0	0
$299$	18.2818	1.05727
$300$	0	0
$301$	46.2421	2.66535
$302$	0	0
$303$	−4.99549	−0.286983
$304$	0	0
$305$	1.05106	0.0601837
$306$	0	0
$307$	−15.3186	−0.874278	−0.437139	−	0.899394i	$-0.644008\pi$
$307$	−15.3186	−0.874278	−0.437139	+	0.899394i	$0.644008\pi$
$308$	0	0
$309$	−6.16051	−0.350459
$310$	0	0
$311$	12.0076	0.680891	0.340445	−	0.940264i	$-0.389422\pi$
$311$	12.0076	0.680891	0.340445	+	0.940264i	$0.389422\pi$
$312$	0	0
$313$	22.9542	1.29745	0.648723	−	0.761025i	$-0.275303\pi$
$313$	22.9542	1.29745	0.648723	+	0.761025i	$0.275303\pi$
$314$	0	0
$315$	13.4023	0.755134
$316$	0	0
$317$	−0.469514	−0.0263706	−0.0131853	−	0.999913i	$-0.504197\pi$
$317$	−0.469514	−0.0263706	−0.0131853	+	0.999913i	$0.504197\pi$
$318$	0	0
$319$	−30.2232	−1.69218
$320$	0	0
$321$	−2.46583	−0.137629
$322$	0	0
$323$	4.70116	0.261580
$324$	0	0
$325$	−8.27592	−0.459066
$326$	0	0
$327$	−6.22882	−0.344454
$328$	0	0
$329$	16.5941	0.914863
$330$	0	0
$331$	−15.4417	−0.848752	−0.424376	−	0.905486i	$-0.639506\pi$
$331$	−15.4417	−0.848752	−0.424376	+	0.905486i	$0.639506\pi$
$332$	0	0
$333$	10.4659	0.573526
$334$	0	0
$335$	8.42629	0.460378
$336$	0	0
$337$	20.1414	1.09717	0.548587	−	0.836093i	$-0.315166\pi$
$337$	20.1414	1.09717	0.548587	+	0.836093i	$0.315166\pi$
$338$	0	0
$339$	1.55512	0.0844627
$340$	0	0
$341$	36.8345	1.99470
$342$	0	0
$343$	1.27422	0.0688015
$344$	0	0
$345$	3.09532	0.166646
$346$	0	0
$347$	−24.9254	−1.33807	−0.669033	−	0.743232i	$-0.733292\pi$
$347$	−24.9254	−1.33807	−0.669033	+	0.743232i	$0.733292\pi$
$348$	0	0
$349$	−21.9372	−1.17427	−0.587137	−	0.809488i	$-0.699745\pi$
$349$	−21.9372	−1.17427	−0.587137	+	0.809488i	$0.699745\pi$
$350$	0	0
$351$	−4.63738	−0.247525
$352$	0	0
$353$	−4.04498	−0.215292	−0.107646	−	0.994189i	$-0.534331\pi$
$353$	−4.04498	−0.215292	−0.107646	+	0.994189i	$0.534331\pi$
$354$	0	0
$355$	2.61550	0.138816
$356$	0	0
$357$	6.38549	0.337956
$358$	0	0
$359$	−20.3192	−1.07241	−0.536203	−	0.844089i	$-0.680142\pi$
$359$	−20.3192	−1.07241	−0.536203	+	0.844089i	$0.680142\pi$
$360$	0	0
$361$	−18.2133	−0.958594
$362$	0	0
$363$	4.59682	0.241271
$364$	0	0
$365$	1.52073	0.0795985
$366$	0	0
$367$	7.44266	0.388504	0.194252	−	0.980952i	$-0.437772\pi$
$367$	7.44266	0.388504	0.194252	+	0.980952i	$0.437772\pi$
$368$	0	0
$369$	6.29077	0.327484
$370$	0	0
$371$	−2.94401	−0.152845
$372$	0	0
$373$	23.4596	1.21469	0.607346	−	0.794437i	$-0.292234\pi$
$373$	23.4596	1.21469	0.607346	+	0.794437i	$0.292234\pi$
$374$	0	0
$375$	−3.44432	−0.177864
$376$	0	0
$377$	−14.5594	−0.749848
$378$	0	0
$379$	−6.49048	−0.333393	−0.166697	−	0.986008i	$-0.553310\pi$
$379$	−6.49048	−0.333393	−0.166697	+	0.986008i	$0.553310\pi$
$380$	0	0
$381$	−0.118459	−0.00606881
$382$	0	0
$383$	−6.81379	−0.348169	−0.174084	−	0.984731i	$-0.555697\pi$
$383$	−6.81379	−0.348169	−0.174084	+	0.984731i	$0.555697\pi$
$384$	0	0
$385$	−23.2038	−1.18257
$386$	0	0
$387$	36.2113	1.84072
$388$	0	0
$389$	24.2209	1.22805	0.614024	−	0.789287i	$-0.289550\pi$
$389$	24.2209	1.22805	0.614024	+	0.789287i	$0.289550\pi$
$390$	0	0
$391$	−40.1496	−2.03045
$392$	0	0
$393$	−7.31049	−0.368765
$394$	0	0
$395$	−12.7883	−0.643448
$396$	0	0
$397$	−9.76461	−0.490072	−0.245036	−	0.969514i	$-0.578800\pi$
$397$	−9.76461	−0.490072	−0.245036	+	0.969514i	$0.578800\pi$
$398$	0	0
$399$	1.06857	0.0534952
$400$	0	0
$401$	26.5925	1.32796	0.663982	−	0.747748i	$-0.268865\pi$
$401$	26.5925	1.32796	0.663982	+	0.747748i	$0.268865\pi$
$402$	0	0
$403$	17.7443	0.883905
$404$	0	0
$405$	10.0954	0.501645
$406$	0	0
$407$	−18.1199	−0.898168
$408$	0	0
$409$	−13.4495	−0.665035	−0.332518	−	0.943097i	$-0.607898\pi$
$409$	−13.4495	−0.665035	−0.332518	+	0.943097i	$0.607898\pi$
$410$	0	0
$411$	3.54960	0.175089
$412$	0	0
$413$	−17.7590	−0.873862
$414$	0	0
$415$	4.73450	0.232407
$416$	0	0
$417$	4.66182	0.228290
$418$	0	0
$419$	3.54342	0.173108	0.0865538	−	0.996247i	$-0.472415\pi$
$419$	3.54342	0.173108	0.0865538	+	0.996247i	$0.472415\pi$
$420$	0	0
$421$	40.0980	1.95426	0.977129	−	0.212649i	$-0.0682090\pi$
$421$	40.0980	1.95426	0.977129	+	0.212649i	$0.0682090\pi$
$422$	0	0
$423$	12.9945	0.631815
$424$	0	0
$425$	18.1751	0.881624
$426$	0	0
$427$	−3.09886	−0.149964
$428$	0	0
$429$	3.94202	0.190323
$430$	0	0
$431$	−0.431537	−0.0207864	−0.0103932	−	0.999946i	$-0.503308\pi$
$431$	−0.431537	−0.0207864	−0.0103932	+	0.999946i	$0.503308\pi$
$432$	0	0
$433$	−1.01440	−0.0487490	−0.0243745	−	0.999703i	$-0.507759\pi$
$433$	−1.01440	−0.0487490	−0.0243745	+	0.999703i	$0.507759\pi$
$434$	0	0
$435$	−2.46507	−0.118191
$436$	0	0
$437$	−6.71875	−0.321402
$438$	0	0
$439$	12.1382	0.579324	0.289662	−	0.957129i	$-0.406457\pi$
$439$	12.1382	0.579324	0.289662	+	0.957129i	$0.406457\pi$
$440$	0	0
$441$	−19.2582	−0.917055
$442$	0	0
$443$	25.8786	1.22953	0.614764	−	0.788711i	$-0.289251\pi$
$443$	25.8786	1.22953	0.614764	+	0.788711i	$0.289251\pi$
$444$	0	0
$445$	−20.9682	−0.993990
$446$	0	0
$447$	−4.74787	−0.224566
$448$	0	0
$449$	17.2940	0.816153	0.408077	−	0.912948i	$-0.366200\pi$
$449$	17.2940	0.816153	0.408077	+	0.912948i	$0.366200\pi$
$450$	0	0
$451$	−10.8914	−0.512855
$452$	0	0
$453$	−0.115718	−0.00543689
$454$	0	0
$455$	−11.1779	−0.524030
$456$	0	0
$457$	−17.4083	−0.814324	−0.407162	−	0.913356i	$-0.633482\pi$
$457$	−17.4083	−0.814324	−0.407162	+	0.913356i	$0.633482\pi$
$458$	0	0
$459$	10.1844	0.475366
$460$	0	0
$461$	1.89558	0.0882858	0.0441429	−	0.999025i	$-0.485944\pi$
$461$	1.89558	0.0882858	0.0441429	+	0.999025i	$0.485944\pi$
$462$	0	0
$463$	−9.59253	−0.445803	−0.222901	−	0.974841i	$-0.571553\pi$
$463$	−9.59253	−0.445803	−0.222901	+	0.974841i	$0.571553\pi$
$464$	0	0
$465$	3.00430	0.139321
$466$	0	0
$467$	−29.1818	−1.35037	−0.675186	−	0.737648i	$-0.735937\pi$
$467$	−29.1818	−1.35037	−0.675186	+	0.737648i	$0.735937\pi$
$468$	0	0
$469$	−24.8433	−1.14716
$470$	0	0
$471$	−1.61746	−0.0745288
$472$	0	0
$473$	−62.6936	−2.88266
$474$	0	0
$475$	3.04148	0.139553
$476$	0	0
$477$	−2.30539	−0.105557
$478$	0	0
$479$	40.9033	1.86892	0.934460	−	0.356068i	$-0.115883\pi$
$479$	40.9033	1.86892	0.934460	+	0.356068i	$0.115883\pi$
$480$	0	0
$481$	−8.72887	−0.398002
$482$	0	0
$483$	−9.12595	−0.415245
$484$	0	0
$485$	21.9085	0.994813
$486$	0	0
$487$	0.880098	0.0398810	0.0199405	−	0.999801i	$-0.493652\pi$
$487$	0.880098	0.0398810	0.0199405	+	0.999801i	$0.493652\pi$
$488$	0	0
$489$	4.69812	0.212456
$490$	0	0
$491$	23.7235	1.07063	0.535313	−	0.844654i	$-0.320194\pi$
$491$	23.7235	1.07063	0.535313	+	0.844654i	$0.320194\pi$
$492$	0	0
$493$	31.9746	1.44006
$494$	0	0
$495$	−18.1704	−0.816700
$496$	0	0
$497$	−7.71129	−0.345899
$498$	0	0
$499$	−37.2473	−1.66742	−0.833709	−	0.552204i	$-0.813787\pi$
$499$	−37.2473	−1.66742	−0.833709	+	0.552204i	$0.813787\pi$
$500$	0	0
$501$	−4.90978	−0.219353
$502$	0	0
$503$	7.72236	0.344323	0.172161	−	0.985069i	$-0.444925\pi$
$503$	7.72236	0.344323	0.172161	+	0.985069i	$0.444925\pi$
$504$	0	0
$505$	−19.2047	−0.854597
$506$	0	0
$507$	−2.33930	−0.103892
$508$	0	0
$509$	−8.55860	−0.379353	−0.189677	−	0.981847i	$-0.560744\pi$
$509$	−8.55860	−0.379353	−0.189677	+	0.981847i	$0.560744\pi$
$510$	0	0
$511$	−4.48358	−0.198342
$512$	0	0
$513$	1.70428	0.0752459
$514$	0	0
$515$	−23.6835	−1.04362
$516$	0	0
$517$	−22.4978	−0.989452
$518$	0	0
$519$	−0.675160	−0.0296362
$520$	0	0
$521$	−21.7291	−0.951969	−0.475984	−	0.879454i	$-0.657908\pi$
$521$	−21.7291	−0.951969	−0.475984	+	0.879454i	$0.657908\pi$
$522$	0	0
$523$	0.142219	0.00621879	0.00310939	−	0.999995i	$-0.499010\pi$
$523$	0.142219	0.00621879	0.00310939	+	0.999995i	$0.499010\pi$
$524$	0	0
$525$	4.13118	0.180300
$526$	0	0
$527$	−38.9690	−1.69752
$528$	0	0
$529$	34.3806	1.49481
$530$	0	0
$531$	−13.9067	−0.603500
$532$	0	0
$533$	−5.24670	−0.227260
$534$	0	0
$535$	−9.47964	−0.409841
$536$	0	0
$537$	−2.71346	−0.117094
$538$	0	0
$539$	33.3422	1.43615
$540$	0	0
$541$	−36.4213	−1.56587	−0.782936	−	0.622102i	$-0.786279\pi$
$541$	−36.4213	−1.56587	−0.782936	+	0.622102i	$0.786279\pi$
$542$	0	0
$543$	−2.45530	−0.105367
$544$	0	0
$545$	−23.9461	−1.02574
$546$	0	0
$547$	6.36581	0.272182	0.136091	−	0.990696i	$-0.456546\pi$
$547$	6.36581	0.272182	0.136091	+	0.990696i	$0.456546\pi$
$548$	0	0
$549$	−2.42666	−0.103567
$550$	0	0
$551$	5.35072	0.227948
$552$	0	0
$553$	37.7038	1.60333
$554$	0	0
$555$	−1.47789	−0.0627330
$556$	0	0
$557$	9.89776	0.419382	0.209691	−	0.977768i	$-0.432754\pi$
$557$	9.89776	0.419382	0.209691	+	0.977768i	$0.432754\pi$
$558$	0	0
$559$	−30.2014	−1.27738
$560$	0	0
$561$	−8.65726	−0.365510
$562$	0	0
$563$	−13.0240	−0.548897	−0.274449	−	0.961602i	$-0.588495\pi$
$563$	−13.0240	−0.548897	−0.274449	+	0.961602i	$0.588495\pi$
$564$	0	0
$565$	5.97852	0.251518
$566$	0	0
$567$	−29.7644	−1.24999
$568$	0	0
$569$	7.36726	0.308852	0.154426	−	0.988004i	$-0.450647\pi$
$569$	7.36726	0.308852	0.154426	+	0.988004i	$0.450647\pi$
$570$	0	0
$571$	43.5841	1.82394	0.911969	−	0.410258i	$-0.134561\pi$
$571$	43.5841	1.82394	0.911969	+	0.410258i	$0.134561\pi$
$572$	0	0
$573$	1.91570	0.0800296
$574$	0	0
$575$	−25.9753	−1.08325
$576$	0	0
$577$	31.6865	1.31913	0.659563	−	0.751650i	$-0.270742\pi$
$577$	31.6865	1.31913	0.659563	+	0.751650i	$0.270742\pi$
$578$	0	0
$579$	−6.12776	−0.254661
$580$	0	0
$581$	−13.9588	−0.579107
$582$	0	0
$583$	3.99139	0.165307
$584$	0	0
$585$	−8.75323	−0.361901
$586$	0	0
$587$	−12.3294	−0.508887	−0.254443	−	0.967088i	$-0.581892\pi$
$587$	−12.3294	−0.508887	−0.254443	+	0.967088i	$0.581892\pi$
$588$	0	0
$589$	−6.52119	−0.268701
$590$	0	0
$591$	−2.25599	−0.0927990
$592$	0	0
$593$	3.11572	0.127947	0.0639735	−	0.997952i	$-0.479623\pi$
$593$	3.11572	0.127947	0.0639735	+	0.997952i	$0.479623\pi$
$594$	0	0
$595$	24.5484	1.00639
$596$	0	0
$597$	−6.23152	−0.255039
$598$	0	0
$599$	14.7795	0.603875	0.301937	−	0.953328i	$-0.402367\pi$
$599$	14.7795	0.603875	0.301937	+	0.953328i	$0.402367\pi$
$600$	0	0
$601$	30.4492	1.24205	0.621024	−	0.783792i	$-0.286717\pi$
$601$	30.4492	1.24205	0.621024	+	0.783792i	$0.286717\pi$
$602$	0	0
$603$	−19.4543	−0.792241
$604$	0	0
$605$	17.6720	0.718471
$606$	0	0
$607$	8.80913	0.357552	0.178776	−	0.983890i	$-0.442786\pi$
$607$	8.80913	0.357552	0.178776	+	0.983890i	$0.442786\pi$
$608$	0	0
$609$	7.26778	0.294505
$610$	0	0
$611$	−10.8378	−0.438452
$612$	0	0
$613$	−9.85071	−0.397866	−0.198933	−	0.980013i	$-0.563748\pi$
$613$	−9.85071	−0.397866	−0.198933	+	0.980013i	$0.563748\pi$
$614$	0	0
$615$	−0.888323	−0.0358207
$616$	0	0
$617$	−26.3185	−1.05954	−0.529771	−	0.848141i	$-0.677722\pi$
$617$	−26.3185	−1.05954	−0.529771	+	0.848141i	$0.677722\pi$
$618$	0	0
$619$	−40.8536	−1.64205	−0.821023	−	0.570895i	$-0.806596\pi$
$619$	−40.8536	−1.64205	−0.821023	+	0.570895i	$0.806596\pi$
$620$	0	0
$621$	−14.5552	−0.584080
$622$	0	0
$623$	61.8209	2.47680
$624$	0	0
$625$	3.90411	0.156164
$626$	0	0
$627$	−1.44873	−0.0578567
$628$	0	0
$629$	19.1699	0.764353
$630$	0	0
$631$	−34.3064	−1.36572	−0.682858	−	0.730551i	$-0.739263\pi$
$631$	−34.3064	−1.36572	−0.682858	+	0.730551i	$0.739263\pi$
$632$	0	0
$633$	8.98655	0.357183
$634$	0	0
$635$	−0.455402	−0.0180721
$636$	0	0
$637$	16.0619	0.636396
$638$	0	0
$639$	−6.03856	−0.238882
$640$	0	0
$641$	−43.8934	−1.73369	−0.866843	−	0.498581i	$-0.833855\pi$
$641$	−43.8934	−1.73369	−0.866843	+	0.498581i	$0.833855\pi$
$642$	0	0
$643$	−10.1680	−0.400987	−0.200494	−	0.979695i	$-0.564255\pi$
$643$	−10.1680	−0.400987	−0.200494	+	0.979695i	$0.564255\pi$
$644$	0	0
$645$	−5.11342	−0.201341
$646$	0	0
$647$	−27.0736	−1.06437	−0.532186	−	0.846627i	$-0.678629\pi$
$647$	−27.0736	−1.06437	−0.532186	+	0.846627i	$0.678629\pi$
$648$	0	0
$649$	24.0771	0.945108
$650$	0	0
$651$	−8.85760	−0.347157
$652$	0	0
$653$	2.01231	0.0787477	0.0393738	−	0.999225i	$-0.487464\pi$
$653$	2.01231	0.0787477	0.0393738	+	0.999225i	$0.487464\pi$
$654$	0	0
$655$	−28.1044	−1.09813
$656$	0	0
$657$	−3.51100	−0.136977
$658$	0	0
$659$	28.5071	1.11048	0.555240	−	0.831690i	$-0.312626\pi$
$659$	28.5071	1.11048	0.555240	+	0.831690i	$0.312626\pi$
$660$	0	0
$661$	−25.4035	−0.988081	−0.494040	−	0.869439i	$-0.664481\pi$
$661$	−25.4035	−0.988081	−0.494040	+	0.869439i	$0.664481\pi$
$662$	0	0
$663$	−4.17045	−0.161967
$664$	0	0
$665$	4.10800	0.159301
$666$	0	0
$667$	−45.6971	−1.76940
$668$	0	0
$669$	7.34294	0.283894
$670$	0	0
$671$	4.20134	0.162191
$672$	0	0
$673$	−12.6470	−0.487505	−0.243753	−	0.969837i	$-0.578378\pi$
$673$	−12.6470	−0.487505	−0.243753	+	0.969837i	$0.578378\pi$
$674$	0	0
$675$	6.58892	0.253608
$676$	0	0
$677$	24.1188	0.926961	0.463480	−	0.886107i	$-0.346600\pi$
$677$	24.1188	0.926961	0.463480	+	0.886107i	$0.346600\pi$
$678$	0	0
$679$	−64.5930	−2.47885
$680$	0	0
$681$	1.69446	0.0649317
$682$	0	0
$683$	10.9260	0.418074	0.209037	−	0.977908i	$-0.432967\pi$
$683$	10.9260	0.418074	0.209037	+	0.977908i	$0.432967\pi$
$684$	0	0
$685$	13.6461	0.521390
$686$	0	0
$687$	−5.63439	−0.214965
$688$	0	0
$689$	1.92277	0.0732517
$690$	0	0
$691$	32.4216	1.23338	0.616688	−	0.787208i	$-0.288474\pi$
$691$	32.4216	1.23338	0.616688	+	0.787208i	$0.288474\pi$
$692$	0	0
$693$	53.5720	2.03503
$694$	0	0
$695$	17.9219	0.679817
$696$	0	0
$697$	11.5225	0.436447
$698$	0	0
$699$	3.15346	0.119275
$700$	0	0
$701$	10.9234	0.412572	0.206286	−	0.978492i	$-0.433862\pi$
$701$	10.9234	0.412572	0.206286	+	0.978492i	$0.433862\pi$
$702$	0	0
$703$	3.20794	0.120990
$704$	0	0
$705$	−1.83496	−0.0691088
$706$	0	0
$707$	56.6213	2.12946
$708$	0	0
$709$	−23.9585	−0.899779	−0.449890	−	0.893084i	$-0.648537\pi$
$709$	−23.9585	−0.899779	−0.449890	+	0.893084i	$0.648537\pi$
$710$	0	0
$711$	29.5251	1.10728
$712$	0	0
$713$	55.6933	2.08573
$714$	0	0
$715$	15.1547	0.566754
$716$	0	0
$717$	−8.14345	−0.304123
$718$	0	0
$719$	41.3440	1.54187	0.770935	−	0.636914i	$-0.219789\pi$
$719$	41.3440	1.54187	0.770935	+	0.636914i	$0.219789\pi$
$720$	0	0
$721$	69.8262	2.60046
$722$	0	0
$723$	−7.69338	−0.286120
$724$	0	0
$725$	20.6864	0.768274
$726$	0	0
$727$	16.4022	0.608323	0.304161	−	0.952620i	$-0.401624\pi$
$727$	16.4022	0.608323	0.304161	+	0.952620i	$0.401624\pi$
$728$	0	0
$729$	−21.4286	−0.793651
$730$	0	0
$731$	66.3266	2.45318
$732$	0	0
$733$	−5.10118	−0.188416	−0.0942082	−	0.995553i	$-0.530032\pi$
$733$	−5.10118	−0.188416	−0.0942082	+	0.995553i	$0.530032\pi$
$734$	0	0
$735$	2.71946	0.100309
$736$	0	0
$737$	33.6818	1.24069
$738$	0	0
$739$	−10.9087	−0.401284	−0.200642	−	0.979665i	$-0.564303\pi$
$739$	−10.9087	−0.401284	−0.200642	+	0.979665i	$0.564303\pi$
$740$	0	0
$741$	−0.697896	−0.0256378
$742$	0	0
$743$	12.3861	0.454401	0.227200	−	0.973848i	$-0.427043\pi$
$743$	12.3861	0.454401	0.227200	+	0.973848i	$0.427043\pi$
$744$	0	0
$745$	−18.2527	−0.668728
$746$	0	0
$747$	−10.9308	−0.399938
$748$	0	0
$749$	27.9489	1.02123
$750$	0	0
$751$	−9.37979	−0.342273	−0.171137	−	0.985247i	$-0.554744\pi$
$751$	−9.37979	−0.342273	−0.171137	+	0.985247i	$0.554744\pi$
$752$	0	0
$753$	2.78962	0.101659
$754$	0	0
$755$	−0.444865	−0.0161903
$756$	0	0
$757$	7.55577	0.274619	0.137310	−	0.990528i	$-0.456154\pi$
$757$	7.55577	0.274619	0.137310	+	0.990528i	$0.456154\pi$
$758$	0	0
$759$	12.3727	0.449100
$760$	0	0
$761$	−32.6833	−1.18477	−0.592385	−	0.805655i	$-0.701813\pi$
$761$	−32.6833	−1.18477	−0.592385	+	0.805655i	$0.701813\pi$
$762$	0	0
$763$	70.6004	2.55591
$764$	0	0
$765$	19.2234	0.695023
$766$	0	0
$767$	11.5986	0.418802
$768$	0	0
$769$	5.17796	0.186722	0.0933611	−	0.995632i	$-0.470239\pi$
$769$	5.17796	0.186722	0.0933611	+	0.995632i	$0.470239\pi$
$770$	0	0
$771$	5.14949	0.185454
$772$	0	0
$773$	37.8283	1.36059	0.680295	−	0.732939i	$-0.261852\pi$
$773$	37.8283	1.36059	0.680295	+	0.732939i	$0.261852\pi$
$774$	0	0
$775$	−25.2115	−0.905625
$776$	0	0
$777$	4.35728	0.156317
$778$	0	0
$779$	1.92821	0.0690854
$780$	0	0
$781$	10.4547	0.374100
$782$	0	0
$783$	11.5916	0.414248
$784$	0	0
$785$	−6.21818	−0.221937
$786$	0	0
$787$	31.8773	1.13630	0.568152	−	0.822924i	$-0.307659\pi$
$787$	31.8773	1.13630	0.568152	+	0.822924i	$0.307659\pi$
$788$	0	0
$789$	−8.64502	−0.307771
$790$	0	0
$791$	−17.6265	−0.626727
$792$	0	0
$793$	2.02391	0.0718711
$794$	0	0
$795$	0.325546	0.0115459
$796$	0	0
$797$	−30.7449	−1.08904	−0.544520	−	0.838748i	$-0.683288\pi$
$797$	−30.7449	−1.08904	−0.544520	+	0.838748i	$0.683288\pi$
$798$	0	0
$799$	23.8015	0.842036
$800$	0	0
$801$	48.4107	1.71051
$802$	0	0
$803$	6.07870	0.214513
$804$	0	0
$805$	−35.0838	−1.23654
$806$	0	0
$807$	−0.326022	−0.0114765
$808$	0	0
$809$	39.5415	1.39021	0.695103	−	0.718910i	$-0.255359\pi$
$809$	39.5415	1.39021	0.695103	+	0.718910i	$0.255359\pi$
$810$	0	0
$811$	33.9458	1.19200	0.595999	−	0.802985i	$-0.296756\pi$
$811$	33.9458	1.19200	0.595999	+	0.802985i	$0.296756\pi$
$812$	0	0
$813$	5.85762	0.205436
$814$	0	0
$815$	18.0615	0.632666
$816$	0	0
$817$	11.0993	0.388315
$818$	0	0
$819$	25.8072	0.901777
$820$	0	0
$821$	−21.3102	−0.743730	−0.371865	−	0.928287i	$-0.621282\pi$
$821$	−21.3102	−0.743730	−0.371865	+	0.928287i	$0.621282\pi$
$822$	0	0
$823$	37.7609	1.31626	0.658131	−	0.752904i	$-0.271347\pi$
$823$	37.7609	1.31626	0.658131	+	0.752904i	$0.271347\pi$
$824$	0	0
$825$	−5.60093	−0.194999
$826$	0	0
$827$	12.0994	0.420737	0.210369	−	0.977622i	$-0.432534\pi$
$827$	12.0994	0.420737	0.210369	+	0.977622i	$0.432534\pi$
$828$	0	0
$829$	−41.8630	−1.45396	−0.726981	−	0.686658i	$-0.759077\pi$
$829$	−41.8630	−1.45396	−0.726981	+	0.686658i	$0.759077\pi$
$830$	0	0
$831$	8.19782	0.284379
$832$	0	0
$833$	−35.2743	−1.22218
$834$	0	0
$835$	−18.8752	−0.653203
$836$	0	0
$837$	−14.1272	−0.488307
$838$	0	0
$839$	37.8172	1.30560	0.652798	−	0.757532i	$-0.273595\pi$
$839$	37.8172	1.30560	0.652798	+	0.757532i	$0.273595\pi$
$840$	0	0
$841$	7.39253	0.254915
$842$	0	0
$843$	−1.40223	−0.0482955
$844$	0	0
$845$	−8.99320	−0.309376
$846$	0	0
$847$	−52.1026	−1.79027
$848$	0	0
$849$	3.76341	0.129160
$850$	0	0
$851$	−27.3970	−0.939157
$852$	0	0
$853$	−6.32008	−0.216395	−0.108198	−	0.994129i	$-0.534508\pi$
$853$	−6.32008	−0.216395	−0.108198	+	0.994129i	$0.534508\pi$
$854$	0	0
$855$	3.21689	0.110015
$856$	0	0
$857$	19.9518	0.681540	0.340770	−	0.940147i	$-0.389312\pi$
$857$	19.9518	0.681540	0.340770	+	0.940147i	$0.389312\pi$
$858$	0	0
$859$	43.7001	1.49103	0.745514	−	0.666490i	$-0.232204\pi$
$859$	43.7001	1.49103	0.745514	+	0.666490i	$0.232204\pi$
$860$	0	0
$861$	2.61905	0.0892570
$862$	0	0
$863$	30.8136	1.04891	0.524454	−	0.851439i	$-0.324269\pi$
$863$	30.8136	1.04891	0.524454	+	0.851439i	$0.324269\pi$
$864$	0	0
$865$	−2.59559	−0.0882526
$866$	0	0
$867$	3.61656	0.122825
$868$	0	0
$869$	−51.1177	−1.73405
$870$	0	0
$871$	16.2255	0.549781
$872$	0	0
$873$	−50.5815	−1.71192
$874$	0	0
$875$	39.0396	1.31978
$876$	0	0
$877$	2.64592	0.0893462	0.0446731	−	0.999002i	$-0.485775\pi$
$877$	2.64592	0.0893462	0.0446731	+	0.999002i	$0.485775\pi$
$878$	0	0
$879$	−2.09421	−0.0706361
$880$	0	0
$881$	3.50336	0.118031	0.0590157	−	0.998257i	$-0.481204\pi$
$881$	3.50336	0.118031	0.0590157	+	0.998257i	$0.481204\pi$
$882$	0	0
$883$	38.9577	1.31103	0.655516	−	0.755181i	$-0.272451\pi$
$883$	38.9577	1.31103	0.655516	+	0.755181i	$0.272451\pi$
$884$	0	0
$885$	1.96377	0.0660116
$886$	0	0
$887$	37.4317	1.25683	0.628416	−	0.777877i	$-0.283703\pi$
$887$	37.4317	1.25683	0.628416	+	0.777877i	$0.283703\pi$
$888$	0	0
$889$	1.34267	0.0450316
$890$	0	0
$891$	40.3537	1.35190
$892$	0	0
$893$	3.98301	0.133286
$894$	0	0
$895$	−10.4316	−0.348691
$896$	0	0
$897$	5.96028	0.199008
$898$	0	0
$899$	−44.3534	−1.47927
$900$	0	0
$901$	−4.22269	−0.140678
$902$	0	0
$903$	15.0759	0.501696
$904$	0	0
$905$	−9.43917	−0.313769
$906$	0	0
$907$	−58.8154	−1.95293	−0.976467	−	0.215669i	$-0.930807\pi$
$907$	−58.8154	−1.95293	−0.976467	+	0.215669i	$0.930807\pi$
$908$	0	0
$909$	44.3390	1.47063
$910$	0	0
$911$	−46.4802	−1.53996	−0.769978	−	0.638070i	$-0.779733\pi$
$911$	−46.4802	−1.53996	−0.769978	+	0.638070i	$0.779733\pi$
$912$	0	0
$913$	18.9249	0.626322
$914$	0	0
$915$	0.342670	0.0113283
$916$	0	0
$917$	82.8606	2.73630
$918$	0	0
$919$	−24.1146	−0.795468	−0.397734	−	0.917501i	$-0.630203\pi$
$919$	−24.1146	−0.795468	−0.397734	+	0.917501i	$0.630203\pi$
$920$	0	0
$921$	−4.99419	−0.164564
$922$	0	0
$923$	5.03635	0.165774
$924$	0	0
$925$	12.4022	0.407782
$926$	0	0
$927$	54.6795	1.79591
$928$	0	0
$929$	27.5803	0.904881	0.452440	−	0.891795i	$-0.350553\pi$
$929$	27.5803	0.904881	0.452440	+	0.891795i	$0.350553\pi$
$930$	0	0
$931$	−5.90291	−0.193460
$932$	0	0
$933$	3.91475	0.128163
$934$	0	0
$935$	−33.2820	−1.08844
$936$	0	0
$937$	−55.9382	−1.82742	−0.913710	−	0.406366i	$-0.866796\pi$
$937$	−55.9382	−1.82742	−0.913710	+	0.406366i	$0.866796\pi$
$938$	0	0
$939$	7.48356	0.244217
$940$	0	0
$941$	−4.57223	−0.149051	−0.0745253	−	0.997219i	$-0.523744\pi$
$941$	−4.57223	−0.149051	−0.0745253	+	0.997219i	$0.523744\pi$
$942$	0	0
$943$	−16.4676	−0.536260
$944$	0	0
$945$	8.89938	0.289497
$946$	0	0
$947$	43.5393	1.41484	0.707419	−	0.706795i	$-0.249860\pi$
$947$	43.5393	1.41484	0.707419	+	0.706795i	$0.249860\pi$
$948$	0	0
$949$	2.92829	0.0950562
$950$	0	0
$951$	−0.153072	−0.00496370
$952$	0	0
$953$	−15.0276	−0.486791	−0.243395	−	0.969927i	$-0.578261\pi$
$953$	−15.0276	−0.486791	−0.243395	+	0.969927i	$0.578261\pi$
$954$	0	0
$955$	7.36473	0.238317
$956$	0	0
$957$	−9.85343	−0.318516
$958$	0	0
$959$	−40.2329	−1.29919
$960$	0	0
$961$	23.0556	0.743730
$962$	0	0
$963$	21.8862	0.705274
$964$	0	0
$965$	−23.5576	−0.758346
$966$	0	0
$967$	36.2094	1.16441	0.582207	−	0.813040i	$-0.302189\pi$
$967$	36.2094	1.16441	0.582207	+	0.813040i	$0.302189\pi$
$968$	0	0
$969$	1.53268	0.0492368
$970$	0	0
$971$	−16.8140	−0.539588	−0.269794	−	0.962918i	$-0.586956\pi$
$971$	−16.8140	−0.539588	−0.269794	+	0.962918i	$0.586956\pi$
$972$	0	0
$973$	−52.8393	−1.69395
$974$	0	0
$975$	−2.69813	−0.0864094
$976$	0	0
$977$	33.6418	1.07630	0.538148	−	0.842850i	$-0.319124\pi$
$977$	33.6418	1.07630	0.538148	+	0.842850i	$0.319124\pi$
$978$	0	0
$979$	−83.8149	−2.67873
$980$	0	0
$981$	55.2858	1.76514
$982$	0	0
$983$	−37.6528	−1.20094	−0.600469	−	0.799648i	$-0.705020\pi$
$983$	−37.6528	−1.20094	−0.600469	+	0.799648i	$0.705020\pi$
$984$	0	0
$985$	−8.67293	−0.276343
$986$	0	0
$987$	5.41004	0.172204
$988$	0	0
$989$	−94.7919	−3.01421
$990$	0	0
$991$	−28.2958	−0.898845	−0.449422	−	0.893319i	$-0.648370\pi$
$991$	−28.2958	−0.898845	−0.449422	+	0.893319i	$0.648370\pi$
$992$	0	0
$993$	−5.03433	−0.159760
$994$	0	0
$995$	−23.9565	−0.759471
$996$	0	0
$997$	13.7229	0.434610	0.217305	−	0.976104i	$-0.430273\pi$
$997$	13.7229	0.434610	0.217305	+	0.976104i	$0.430273\pi$
$998$	0	0
$999$	6.94954	0.219874

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	4304.2.a.j.1.7		13
4.3	odd	2		2152.2.a.b.1.7	✓	13

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
2152.2.a.b.1.7	✓	13	4.3	odd	2
4304.2.a.j.1.7		13	1.1	even	1	trivial

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:	\( N \)	\(=\)	\( 4304 = 2^{4} \cdot 269 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	4304.a (trivial)

\(f(q)\)	\(=\)	\(q+0.326022 q^{3} +1.25336 q^{5} -3.69529 q^{7} -2.89371 q^{9} +5.00996 q^{11} +2.41345 q^{13} +0.408623 q^{15} -5.30028 q^{17} -0.886964 q^{19} -1.20475 q^{21} +7.57500 q^{23} -3.42909 q^{25} -1.92148 q^{27} -6.03262 q^{29} +7.35225 q^{31} +1.63336 q^{33} -4.63153 q^{35} -3.61677 q^{37} +0.786836 q^{39} -2.17395 q^{41} -12.5138 q^{43} -3.62686 q^{45} -4.49061 q^{47} +6.65518 q^{49} -1.72801 q^{51} +0.796691 q^{53} +6.27929 q^{55} -0.289170 q^{57} +4.80584 q^{59} +0.838597 q^{61} +10.6931 q^{63} +3.02491 q^{65} +6.72297 q^{67} +2.46962 q^{69} +2.08679 q^{71} +1.21332 q^{73} -1.11796 q^{75} -18.5133 q^{77} -10.2032 q^{79} +8.05468 q^{81} +3.77744 q^{83} -6.64316 q^{85} -1.96677 q^{87} -16.7296 q^{89} -8.91838 q^{91} +2.39700 q^{93} -1.11169 q^{95} +17.4798 q^{97} -14.4974 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 13 q + 3 q^{3} - 9 q^{5} - 6 q^{7} + 6 q^{11} - 8 q^{13} + q^{15} - 14 q^{17} + 6 q^{19} - 13 q^{21} + 11 q^{23} - 2 q^{25} + 9 q^{27} - 19 q^{29} - 14 q^{31} - 15 q^{33} + 20 q^{35} - 23 q^{37} - 8 q^{39}+ \cdots + 7 q^{99}+O(q^{100}) \) 13 * q + 3 * q^3 - 9 * q^5 - 6 * q^7 + 6 * q^11 - 8 * q^13 + q^15 - 14 * q^17 + 6 * q^19 - 13 * q^21 + 11 * q^23 - 2 * q^25 + 9 * q^27 - 19 * q^29 - 14 * q^31 - 15 * q^33 + 20 * q^35 - 23 * q^37 - 8 * q^39 - 23 * q^41 + 9 * q^43 - 14 * q^45 + 8 * q^47 - 7 * q^49 + 9 * q^51 - 24 * q^53 - 19 * q^55 - 30 * q^57 + 17 * q^59 - 23 * q^61 - 4 * q^63 - 34 * q^65 + 15 * q^67 - 21 * q^69 + 14 * q^71 - 21 * q^73 + 9 * q^75 - 50 * q^77 - 42 * q^79 - 31 * q^81 + 29 * q^83 - 33 * q^85 - 15 * q^87 - 33 * q^89 + 21 * q^91 - 37 * q^93 - 2 * q^95 - 27 * q^97 + 7 * q^99

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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