LMFDB - Embedded newform 6032.2.a.be.1.1

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [6032,2,Mod(1,6032)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(6032, base_ring=CyclotomicField(2))

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

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

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

chi := DirichletCharacter("6032.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 6032 = 2^{4} \cdot 13 \cdot 29 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	6032.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:	$48.1657624992$
Analytic rank:	$0$
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} - 4 x^{12} - 22 x^{11} + 96 x^{10} + 159 x^{9} - 827 x^{8} - 362 x^{7} + 3029 x^{6} - 142 x^{5} + \cdots + 16 $ x^13 - 4x^12 - 22x^11 + 96x^10 + 159x^9 - 827x^8 - 362x^7 + 3029x^6 - 142x^5 - 4316x^4 + 208x^3 + 2228x^2 + 376x + 16
Coefficient ring:	$\Z[a_1, \ldots, a_{7}]$
Coefficient ring index:	$ 2^{2} $
Twist minimal:	no (minimal twist has level 3016)
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.1
Root			$-2.90994$ of defining polynomial
Character	$\chi$	$=$	6032.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-2.90994 q^{3} -1.66472 q^{5} -2.42557 q^{7} +5.46773 q^{9} +O(q^{10})$	$q-2.90994 q^{3} -1.66472 q^{5} -2.42557 q^{7} +5.46773 q^{9} -0.493837 q^{11} -1.00000 q^{13} +4.84422 q^{15} -1.59559 q^{17} -4.35217 q^{19} +7.05825 q^{21} -5.41356 q^{23} -2.22872 q^{25} -7.18092 q^{27} +1.00000 q^{29} -0.312230 q^{31} +1.43703 q^{33} +4.03789 q^{35} +9.46701 q^{37} +2.90994 q^{39} -10.7627 q^{41} -1.06750 q^{43} -9.10222 q^{45} -11.4753 q^{47} -1.11662 q^{49} +4.64306 q^{51} +12.7515 q^{53} +0.822099 q^{55} +12.6645 q^{57} -8.68999 q^{59} -7.26557 q^{61} -13.2623 q^{63} +1.66472 q^{65} +3.92038 q^{67} +15.7531 q^{69} -15.1418 q^{71} -13.5174 q^{73} +6.48542 q^{75} +1.19783 q^{77} +9.87383 q^{79} +4.49285 q^{81} -14.3884 q^{83} +2.65620 q^{85} -2.90994 q^{87} -9.48951 q^{89} +2.42557 q^{91} +0.908571 q^{93} +7.24513 q^{95} -1.40579 q^{97} -2.70016 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 13 q + 4 q^{3} + 5 q^{5} - 6 q^{7} + 21 q^{9}+O(q^{10}) $ 13 * q + 4 * q^3 + 5 * q^5 - 6 * q^7 + 21 * q^9	$ 13 q + 4 q^{3} + 5 q^{5} - 6 q^{7} + 21 q^{9} - 6 q^{11} - 13 q^{13} - 6 q^{15} + 4 q^{17} + 3 q^{19} + 5 q^{21} - 13 q^{23} + 24 q^{25} + 16 q^{27} + 13 q^{29} - 21 q^{31} + 21 q^{33} - 8 q^{35} + 23 q^{37} - 4 q^{39} + 4 q^{41} + 24 q^{43} + 9 q^{45} - 7 q^{47} + 41 q^{49} + q^{51} + 17 q^{53} + 8 q^{55} + 24 q^{57} - 11 q^{59} + 26 q^{61} - 17 q^{63} - 5 q^{65} + 35 q^{67} + 30 q^{69} - 30 q^{71} + 17 q^{73} + 43 q^{75} + 34 q^{77} - 3 q^{79} + 37 q^{81} + 18 q^{83} + 9 q^{85} + 4 q^{87} + 38 q^{89} + 6 q^{91} + 25 q^{93} + 9 q^{95} + 7 q^{97} + 40 q^{99}+O(q^{100}) $ 13 * q + 4 * q^3 + 5 * q^5 - 6 * q^7 + 21 * q^9 - 6 * q^11 - 13 * q^13 - 6 * q^15 + 4 * q^17 + 3 * q^19 + 5 * q^21 - 13 * q^23 + 24 * q^25 + 16 * q^27 + 13 * q^29 - 21 * q^31 + 21 * q^33 - 8 * q^35 + 23 * q^37 - 4 * q^39 + 4 * q^41 + 24 * q^43 + 9 * q^45 - 7 * q^47 + 41 * q^49 + q^51 + 17 * q^53 + 8 * q^55 + 24 * q^57 - 11 * q^59 + 26 * q^61 - 17 * q^63 - 5 * q^65 + 35 * q^67 + 30 * q^69 - 30 * q^71 + 17 * q^73 + 43 * q^75 + 34 * q^77 - 3 * q^79 + 37 * q^81 + 18 * q^83 + 9 * q^85 + 4 * q^87 + 38 * q^89 + 6 * q^91 + 25 * q^93 + 9 * q^95 + 7 * q^97 + 40 * 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
$3$	−2.90994	−1.68005	−0.840026	−	0.542546i	$-0.817460\pi$
$3$	−2.90994	−1.68005	−0.840026	+	0.542546i	$0.817460\pi$
$4$	0	0
$5$	−1.66472	−0.744484	−0.372242	−	0.928136i	$-0.621411\pi$
$5$	−1.66472	−0.744484	−0.372242	+	0.928136i	$0.621411\pi$
$6$	0	0
$7$	−2.42557	−0.916779	−0.458389	−	0.888751i	$-0.651573\pi$
$7$	−2.42557	−0.916779	−0.458389	+	0.888751i	$0.651573\pi$
$8$	0	0
$9$	5.46773	1.82258
$10$	0	0
$11$	−0.493837	−0.148897	−0.0744487	−	0.997225i	$-0.523720\pi$
$11$	−0.493837	−0.148897	−0.0744487	+	0.997225i	$0.523720\pi$
$12$	0	0
$13$	−1.00000	−0.277350
$13$	−1.00000	−0.277350
$14$	0	0
$15$	4.84422	1.25077
$16$	0	0
$17$	−1.59559	−0.386987	−0.193493	−	0.981102i	$-0.561982\pi$
$17$	−1.59559	−0.386987	−0.193493	+	0.981102i	$0.561982\pi$
$18$	0	0
$19$	−4.35217	−0.998456	−0.499228	−	0.866471i	$-0.666383\pi$
$19$	−4.35217	−0.998456	−0.499228	+	0.866471i	$0.666383\pi$
$20$	0	0
$21$	7.05825	1.54024
$22$	0	0
$23$	−5.41356	−1.12880	−0.564402	−	0.825500i	$-0.690893\pi$
$23$	−5.41356	−1.12880	−0.564402	+	0.825500i	$0.690893\pi$
$24$	0	0
$25$	−2.22872	−0.445743
$26$	0	0
$27$	−7.18092	−1.38197
$28$	0	0
$29$	1.00000	0.185695
$29$	1.00000	0.185695
$30$	0	0
$31$	−0.312230	−0.0560782	−0.0280391	−	0.999607i	$-0.508926\pi$
$31$	−0.312230	−0.0560782	−0.0280391	+	0.999607i	$0.508926\pi$
$32$	0	0
$33$	1.43703	0.250155
$34$	0	0
$35$	4.03789	0.682527
$36$	0	0
$37$	9.46701	1.55637	0.778183	−	0.628037i	$-0.216142\pi$
$37$	9.46701	1.55637	0.778183	+	0.628037i	$0.216142\pi$
$38$	0	0
$39$	2.90994	0.465963
$40$	0	0
$41$	−10.7627	−1.68085	−0.840425	−	0.541928i	$-0.817695\pi$
$41$	−10.7627	−1.68085	−0.840425	+	0.541928i	$0.817695\pi$
$42$	0	0
$43$	−1.06750	−0.162793	−0.0813964	−	0.996682i	$-0.525938\pi$
$43$	−1.06750	−0.162793	−0.0813964	+	0.996682i	$0.525938\pi$
$44$	0	0
$45$	−9.10222	−1.35688
$46$	0	0
$47$	−11.4753	−1.67384	−0.836919	−	0.547327i	$-0.815645\pi$
$47$	−11.4753	−1.67384	−0.836919	+	0.547327i	$0.815645\pi$
$48$	0	0
$49$	−1.11662	−0.159517
$50$	0	0
$51$	4.64306	0.650158
$52$	0	0
$53$	12.7515	1.75156	0.875778	−	0.482714i	$-0.160349\pi$
$53$	12.7515	1.75156	0.875778	+	0.482714i	$0.160349\pi$
$54$	0	0
$55$	0.822099	0.110852
$56$	0	0
$57$	12.6645	1.67746
$58$	0	0
$59$	−8.68999	−1.13134	−0.565670	−	0.824632i	$-0.691382\pi$
$59$	−8.68999	−1.13134	−0.565670	+	0.824632i	$0.691382\pi$
$60$	0	0
$61$	−7.26557	−0.930261	−0.465131	−	0.885242i	$-0.653993\pi$
$61$	−7.26557	−0.930261	−0.465131	+	0.885242i	$0.653993\pi$
$62$	0	0
$63$	−13.2623	−1.67090
$64$	0	0
$65$	1.66472	0.206483
$66$	0	0
$67$	3.92038	0.478950	0.239475	−	0.970902i	$-0.423025\pi$
$67$	3.92038	0.478950	0.239475	+	0.970902i	$0.423025\pi$
$68$	0	0
$69$	15.7531	1.89645
$70$	0	0
$71$	−15.1418	−1.79700	−0.898499	−	0.438976i	$-0.855341\pi$
$71$	−15.1418	−1.79700	−0.898499	+	0.438976i	$0.855341\pi$
$72$	0	0
$73$	−13.5174	−1.58210	−0.791048	−	0.611754i	$-0.790464\pi$
$73$	−13.5174	−1.58210	−0.791048	+	0.611754i	$0.790464\pi$
$74$	0	0
$75$	6.48542	0.748872
$76$	0	0
$77$	1.19783	0.136506
$78$	0	0
$79$	9.87383	1.11089	0.555447	−	0.831552i	$-0.312547\pi$
$79$	9.87383	1.11089	0.555447	+	0.831552i	$0.312547\pi$
$80$	0	0
$81$	4.49285	0.499205
$82$	0	0
$83$	−14.3884	−1.57933	−0.789667	−	0.613535i	$-0.789747\pi$
$83$	−14.3884	−1.57933	−0.789667	+	0.613535i	$0.789747\pi$
$84$	0	0
$85$	2.65620	0.288106
$86$	0	0
$87$	−2.90994	−0.311978
$88$	0	0
$89$	−9.48951	−1.00589	−0.502943	−	0.864320i	$-0.667749\pi$
$89$	−9.48951	−1.00589	−0.502943	+	0.864320i	$0.667749\pi$
$90$	0	0
$91$	2.42557	0.254269
$92$	0	0
$93$	0.908571	0.0942144
$94$	0	0
$95$	7.24513	0.743334
$96$	0	0
$97$	−1.40579	−0.142736	−0.0713680	−	0.997450i	$-0.522736\pi$
$97$	−1.40579	−0.142736	−0.0713680	+	0.997450i	$0.522736\pi$
$98$	0	0
$99$	−2.70016	−0.271377
$100$	0	0
$101$	−9.08908	−0.904397	−0.452199	−	0.891917i	$-0.649360\pi$
$101$	−9.08908	−0.904397	−0.452199	+	0.891917i	$0.649360\pi$
$102$	0	0
$103$	1.75705	0.173127	0.0865637	−	0.996246i	$-0.472411\pi$
$103$	1.75705	0.173127	0.0865637	+	0.996246i	$0.472411\pi$
$104$	0	0
$105$	−11.7500	−1.14668
$106$	0	0
$107$	−1.94039	−0.187584	−0.0937922	−	0.995592i	$-0.529899\pi$
$107$	−1.94039	−0.187584	−0.0937922	+	0.995592i	$0.529899\pi$
$108$	0	0
$109$	−18.7642	−1.79728	−0.898642	−	0.438682i	$-0.855446\pi$
$109$	−18.7642	−1.79728	−0.898642	+	0.438682i	$0.855446\pi$
$110$	0	0
$111$	−27.5484	−2.61478
$112$	0	0
$113$	5.73076	0.539105	0.269552	−	0.962986i	$-0.413124\pi$
$113$	5.73076	0.539105	0.269552	+	0.962986i	$0.413124\pi$
$114$	0	0
$115$	9.01204	0.840377
$116$	0	0
$117$	−5.46773	−0.505491
$118$	0	0
$119$	3.87021	0.354781
$120$	0	0
$121$	−10.7561	−0.977830
$122$	0	0
$123$	31.3187	2.82392
$124$	0	0
$125$	12.0338	1.07633
$126$	0	0
$127$	−19.6629	−1.74480	−0.872399	−	0.488794i	$-0.837437\pi$
$127$	−19.6629	−1.74480	−0.872399	+	0.488794i	$0.837437\pi$
$128$	0	0
$129$	3.10637	0.273501
$130$	0	0
$131$	2.19090	0.191420	0.0957098	−	0.995409i	$-0.469488\pi$
$131$	2.19090	0.191420	0.0957098	+	0.995409i	$0.469488\pi$
$132$	0	0
$133$	10.5565	0.915363
$134$	0	0
$135$	11.9542	1.02885
$136$	0	0
$137$	4.89880	0.418533	0.209266	−	0.977859i	$-0.432892\pi$
$137$	4.89880	0.418533	0.209266	+	0.977859i	$0.432892\pi$
$138$	0	0
$139$	10.9444	0.928290	0.464145	−	0.885759i	$-0.346362\pi$
$139$	10.9444	0.928290	0.464145	+	0.885759i	$0.346362\pi$
$140$	0	0
$141$	33.3923	2.81213
$142$	0	0
$143$	0.493837	0.0412967
$144$	0	0
$145$	−1.66472	−0.138247
$146$	0	0
$147$	3.24929	0.267997
$148$	0	0
$149$	−8.00528	−0.655818	−0.327909	−	0.944709i	$-0.606344\pi$
$149$	−8.00528	−0.655818	−0.327909	+	0.944709i	$0.606344\pi$
$150$	0	0
$151$	−21.8255	−1.77614	−0.888068	−	0.459713i	$-0.847952\pi$
$151$	−21.8255	−1.77614	−0.888068	+	0.459713i	$0.847952\pi$
$152$	0	0
$153$	−8.72423	−0.705312
$154$	0	0
$155$	0.519775	0.0417494
$156$	0	0
$157$	5.79605	0.462575	0.231287	−	0.972885i	$-0.425706\pi$
$157$	5.79605	0.462575	0.231287	+	0.972885i	$0.425706\pi$
$158$	0	0
$159$	−37.1061	−2.94271
$160$	0	0
$161$	13.1310	1.03486
$162$	0	0
$163$	−9.28013	−0.726876	−0.363438	−	0.931618i	$-0.618397\pi$
$163$	−9.28013	−0.726876	−0.363438	+	0.931618i	$0.618397\pi$
$164$	0	0
$165$	−2.39225	−0.186237
$166$	0	0
$167$	−1.17680	−0.0910638	−0.0455319	−	0.998963i	$-0.514498\pi$
$167$	−1.17680	−0.0910638	−0.0455319	+	0.998963i	$0.514498\pi$
$168$	0	0
$169$	1.00000	0.0769231
$170$	0	0
$171$	−23.7965	−1.81976
$172$	0	0
$173$	−16.3807	−1.24540	−0.622700	−	0.782461i	$-0.713964\pi$
$173$	−16.3807	−1.24540	−0.622700	+	0.782461i	$0.713964\pi$
$174$	0	0
$175$	5.40590	0.408648
$176$	0	0
$177$	25.2873	1.90071
$178$	0	0
$179$	1.22795	0.0917812	0.0458906	−	0.998946i	$-0.485387\pi$
$179$	1.22795	0.0917812	0.0458906	+	0.998946i	$0.485387\pi$
$180$	0	0
$181$	−19.4613	−1.44655	−0.723274	−	0.690562i	$-0.757363\pi$
$181$	−19.4613	−1.44655	−0.723274	+	0.690562i	$0.757363\pi$
$182$	0	0
$183$	21.1423	1.56289
$184$	0	0
$185$	−15.7599	−1.15869
$186$	0	0
$187$	0.787960	0.0576213
$188$	0	0
$189$	17.4178	1.26696
$190$	0	0
$191$	15.5089	1.12219	0.561093	−	0.827753i	$-0.310381\pi$
$191$	15.5089	1.12219	0.561093	+	0.827753i	$0.310381\pi$
$192$	0	0
$193$	15.0881	1.08607	0.543033	−	0.839711i	$-0.317276\pi$
$193$	15.0881	1.08607	0.543033	+	0.839711i	$0.317276\pi$
$194$	0	0
$195$	−4.84422	−0.346902
$196$	0	0
$197$	15.7115	1.11940	0.559700	−	0.828695i	$-0.310916\pi$
$197$	15.7115	1.11940	0.559700	+	0.828695i	$0.310916\pi$
$198$	0	0
$199$	−10.7261	−0.760352	−0.380176	−	0.924914i	$-0.624137\pi$
$199$	−10.7261	−0.760352	−0.380176	+	0.924914i	$0.624137\pi$
$200$	0	0
$201$	−11.4080	−0.804662
$202$	0	0
$203$	−2.42557	−0.170242
$204$	0	0
$205$	17.9168	1.25137
$206$	0	0
$207$	−29.5998	−2.05733
$208$	0	0
$209$	2.14926	0.148667
$210$	0	0
$211$	21.7330	1.49616	0.748079	−	0.663609i	$-0.230976\pi$
$211$	21.7330	1.49616	0.748079	+	0.663609i	$0.230976\pi$
$212$	0	0
$213$	44.0616	3.01905
$214$	0	0
$215$	1.77709	0.121197
$216$	0	0
$217$	0.757336	0.0514113
$218$	0	0
$219$	39.3349	2.65801
$220$	0	0
$221$	1.59559	0.107331
$222$	0	0
$223$	13.5127	0.904879	0.452439	−	0.891795i	$-0.350554\pi$
$223$	13.5127	0.904879	0.452439	+	0.891795i	$0.350554\pi$
$224$	0	0
$225$	−12.1860	−0.812400
$226$	0	0
$227$	19.8927	1.32033	0.660163	−	0.751123i	$-0.270487\pi$
$227$	19.8927	1.32033	0.660163	+	0.751123i	$0.270487\pi$
$228$	0	0
$229$	−1.39202	−0.0919871	−0.0459935	−	0.998942i	$-0.514645\pi$
$229$	−1.39202	−0.0919871	−0.0459935	+	0.998942i	$0.514645\pi$
$230$	0	0
$231$	−3.48562	−0.229337
$232$	0	0
$233$	13.8534	0.907569	0.453784	−	0.891111i	$-0.350074\pi$
$233$	13.8534	0.907569	0.453784	+	0.891111i	$0.350074\pi$
$234$	0	0
$235$	19.1031	1.24615
$236$	0	0
$237$	−28.7322	−1.86636
$238$	0	0
$239$	−1.07290	−0.0693998	−0.0346999	−	0.999398i	$-0.511048\pi$
$239$	−1.07290	−0.0693998	−0.0346999	+	0.999398i	$0.511048\pi$
$240$	0	0
$241$	−3.27470	−0.210942	−0.105471	−	0.994422i	$-0.533635\pi$
$241$	−3.27470	−0.210942	−0.105471	+	0.994422i	$0.533635\pi$
$242$	0	0
$243$	8.46888	0.543279
$244$	0	0
$245$	1.85885	0.118758
$246$	0	0
$247$	4.35217	0.276922
$248$	0	0
$249$	41.8694	2.65336
$250$	0	0
$251$	11.4270	0.721263	0.360632	−	0.932708i	$-0.382561\pi$
$251$	11.4270	0.721263	0.360632	+	0.932708i	$0.382561\pi$
$252$	0	0
$253$	2.67341	0.168076
$254$	0	0
$255$	−7.72938	−0.484032
$256$	0	0
$257$	26.5369	1.65532	0.827662	−	0.561227i	$-0.189671\pi$
$257$	26.5369	1.65532	0.827662	+	0.561227i	$0.189671\pi$
$258$	0	0
$259$	−22.9629	−1.42684
$260$	0	0
$261$	5.46773	0.338444
$262$	0	0
$263$	−5.21818	−0.321767	−0.160883	−	0.986973i	$-0.551434\pi$
$263$	−5.21818	−0.321767	−0.160883	+	0.986973i	$0.551434\pi$
$264$	0	0
$265$	−21.2277	−1.30401
$266$	0	0
$267$	27.6139	1.68994
$268$	0	0
$269$	−3.95664	−0.241241	−0.120620	−	0.992699i	$-0.538488\pi$
$269$	−3.95664	−0.241241	−0.120620	+	0.992699i	$0.538488\pi$
$270$	0	0
$271$	5.98221	0.363394	0.181697	−	0.983355i	$-0.441841\pi$
$271$	5.98221	0.363394	0.181697	+	0.983355i	$0.441841\pi$
$272$	0	0
$273$	−7.05825	−0.427185
$274$	0	0
$275$	1.10062	0.0663700
$276$	0	0
$277$	−9.94587	−0.597589	−0.298795	−	0.954317i	$-0.596585\pi$
$277$	−9.94587	−0.597589	−0.298795	+	0.954317i	$0.596585\pi$
$278$	0	0
$279$	−1.70719	−0.102207
$280$	0	0
$281$	13.7385	0.819571	0.409785	−	0.912182i	$-0.365604\pi$
$281$	13.7385	0.819571	0.409785	+	0.912182i	$0.365604\pi$
$282$	0	0
$283$	7.55869	0.449317	0.224659	−	0.974438i	$-0.427873\pi$
$283$	7.55869	0.449317	0.224659	+	0.974438i	$0.427873\pi$
$284$	0	0
$285$	−21.0829	−1.24884
$286$	0	0
$287$	26.1056	1.54097
$288$	0	0
$289$	−14.4541	−0.850241
$290$	0	0
$291$	4.09075	0.239804
$292$	0	0
$293$	−32.4773	−1.89734	−0.948672	−	0.316263i	$-0.897572\pi$
$293$	−32.4773	−1.89734	−0.948672	+	0.316263i	$0.897572\pi$
$294$	0	0
$295$	14.4664	0.842265
$296$	0	0
$297$	3.54620	0.205772
$298$	0	0
$299$	5.41356	0.313074
$300$	0	0
$301$	2.58930	0.149245
$302$	0	0
$303$	26.4486	1.51943
$304$	0	0
$305$	12.0951	0.692565
$306$	0	0
$307$	27.4410	1.56614	0.783072	−	0.621932i	$-0.213652\pi$
$307$	27.4410	1.56614	0.783072	+	0.621932i	$0.213652\pi$
$308$	0	0
$309$	−5.11291	−0.290863
$310$	0	0
$311$	−16.3704	−0.928280	−0.464140	−	0.885762i	$-0.653637\pi$
$311$	−16.3704	−0.928280	−0.464140	+	0.885762i	$0.653637\pi$
$312$	0	0
$313$	13.1496	0.743259	0.371629	−	0.928381i	$-0.378799\pi$
$313$	13.1496	0.743259	0.371629	+	0.928381i	$0.378799\pi$
$314$	0	0
$315$	22.0781	1.24396
$316$	0	0
$317$	−15.6639	−0.879770	−0.439885	−	0.898054i	$-0.644981\pi$
$317$	−15.6639	−0.879770	−0.439885	+	0.898054i	$0.644981\pi$
$318$	0	0
$319$	−0.493837	−0.0276495
$320$	0	0
$321$	5.64641	0.315152
$322$	0	0
$323$	6.94426	0.386389
$324$	0	0
$325$	2.22872	0.123627
$326$	0	0
$327$	54.6026	3.01953
$328$	0	0
$329$	27.8340	1.53454
$330$	0	0
$331$	5.30703	0.291701	0.145850	−	0.989307i	$-0.453408\pi$
$331$	5.30703	0.291701	0.145850	+	0.989307i	$0.453408\pi$
$332$	0	0
$333$	51.7630	2.83659
$334$	0	0
$335$	−6.52632	−0.356571
$336$	0	0
$337$	−24.8767	−1.35512	−0.677561	−	0.735467i	$-0.736963\pi$
$337$	−24.8767	−1.35512	−0.677561	+	0.735467i	$0.736963\pi$
$338$	0	0
$339$	−16.6762	−0.905724
$340$	0	0
$341$	0.154191	0.00834990
$342$	0	0
$343$	19.6874	1.06302
$344$	0	0
$345$	−26.2245	−1.41188
$346$	0	0
$347$	9.71444	0.521499	0.260749	−	0.965407i	$-0.416030\pi$
$347$	9.71444	0.521499	0.260749	+	0.965407i	$0.416030\pi$
$348$	0	0
$349$	24.1010	1.29009	0.645047	−	0.764143i	$-0.276838\pi$
$349$	24.1010	1.29009	0.645047	+	0.764143i	$0.276838\pi$
$350$	0	0
$351$	7.18092	0.383289
$352$	0	0
$353$	−9.29935	−0.494955	−0.247477	−	0.968894i	$-0.579602\pi$
$353$	−9.29935	−0.494955	−0.247477	+	0.968894i	$0.579602\pi$
$354$	0	0
$355$	25.2068	1.33784
$356$	0	0
$357$	−11.2621	−0.596051
$358$	0	0
$359$	−2.14408	−0.113160	−0.0565800	−	0.998398i	$-0.518020\pi$
$359$	−2.14408	−0.113160	−0.0565800	+	0.998398i	$0.518020\pi$
$360$	0	0
$361$	−0.0586412	−0.00308638
$362$	0	0
$363$	31.2996	1.64280
$364$	0	0
$365$	22.5027	1.17785
$366$	0	0
$367$	12.4010	0.647329	0.323664	−	0.946172i	$-0.395085\pi$
$367$	12.4010	0.647329	0.323664	+	0.946172i	$0.395085\pi$
$368$	0	0
$369$	−58.8475	−3.06348
$370$	0	0
$371$	−30.9297	−1.60579
$372$	0	0
$373$	−24.4683	−1.26692	−0.633461	−	0.773774i	$-0.718366\pi$
$373$	−24.4683	−1.26692	−0.633461	+	0.773774i	$0.718366\pi$
$374$	0	0
$375$	−35.0175	−1.80830
$376$	0	0
$377$	−1.00000	−0.0515026
$378$	0	0
$379$	−24.9993	−1.28413	−0.642064	−	0.766651i	$-0.721922\pi$
$379$	−24.9993	−1.28413	−0.642064	+	0.766651i	$0.721922\pi$
$380$	0	0
$381$	57.2177	2.93135
$382$	0	0
$383$	8.53623	0.436181	0.218090	−	0.975929i	$-0.430017\pi$
$383$	8.53623	0.436181	0.218090	+	0.975929i	$0.430017\pi$
$384$	0	0
$385$	−1.99406	−0.101627
$386$	0	0
$387$	−5.83682	−0.296702
$388$	0	0
$389$	−24.0394	−1.21884	−0.609422	−	0.792846i	$-0.708598\pi$
$389$	−24.0394	−1.21884	−0.609422	+	0.792846i	$0.708598\pi$
$390$	0	0
$391$	8.63780	0.436832
$392$	0	0
$393$	−6.37537	−0.321595
$394$	0	0
$395$	−16.4371	−0.827043
$396$	0	0
$397$	21.8260	1.09542	0.547708	−	0.836670i	$-0.315501\pi$
$397$	21.8260	1.09542	0.547708	+	0.836670i	$0.315501\pi$
$398$	0	0
$399$	−30.7187	−1.53786
$400$	0	0
$401$	17.5947	0.878638	0.439319	−	0.898331i	$-0.355220\pi$
$401$	17.5947	0.878638	0.439319	+	0.898331i	$0.355220\pi$
$402$	0	0
$403$	0.312230	0.0155533
$404$	0	0
$405$	−7.47932	−0.371650
$406$	0	0
$407$	−4.67516	−0.231739
$408$	0	0
$409$	23.5558	1.16476	0.582379	−	0.812917i	$-0.302122\pi$
$409$	23.5558	1.16476	0.582379	+	0.812917i	$0.302122\pi$
$410$	0	0
$411$	−14.2552	−0.703157
$412$	0	0
$413$	21.0782	1.03719
$414$	0	0
$415$	23.9527	1.17579
$416$	0	0
$417$	−31.8474	−1.55958
$418$	0	0
$419$	−15.9844	−0.780887	−0.390443	−	0.920627i	$-0.627678\pi$
$419$	−15.9844	−0.780887	−0.390443	+	0.920627i	$0.627678\pi$
$420$	0	0
$421$	−19.9877	−0.974141	−0.487070	−	0.873363i	$-0.661935\pi$
$421$	−19.9877	−0.974141	−0.487070	+	0.873363i	$0.661935\pi$
$422$	0	0
$423$	−62.7435	−3.05070
$424$	0	0
$425$	3.55611	0.172497
$426$	0	0
$427$	17.6231	0.852844
$428$	0	0
$429$	−1.43703	−0.0693806
$430$	0	0
$431$	−27.8846	−1.34316	−0.671578	−	0.740934i	$-0.734383\pi$
$431$	−27.8846	−1.34316	−0.671578	+	0.740934i	$0.734383\pi$
$432$	0	0
$433$	−12.6307	−0.606992	−0.303496	−	0.952833i	$-0.598154\pi$
$433$	−12.6307	−0.606992	−0.303496	+	0.952833i	$0.598154\pi$
$434$	0	0
$435$	4.84422	0.232263
$436$	0	0
$437$	23.5607	1.12706
$438$	0	0
$439$	37.7510	1.80176	0.900878	−	0.434072i	$-0.142924\pi$
$439$	37.7510	1.80176	0.900878	+	0.434072i	$0.142924\pi$
$440$	0	0
$441$	−6.10536	−0.290731
$442$	0	0
$443$	21.7748	1.03455	0.517276	−	0.855819i	$-0.326946\pi$
$443$	21.7748	1.03455	0.517276	+	0.855819i	$0.326946\pi$
$444$	0	0
$445$	15.7973	0.748866
$446$	0	0
$447$	23.2949	1.10181
$448$	0	0
$449$	−10.6048	−0.500471	−0.250235	−	0.968185i	$-0.580508\pi$
$449$	−10.6048	−0.500471	−0.250235	+	0.968185i	$0.580508\pi$
$450$	0	0
$451$	5.31501	0.250274
$452$	0	0
$453$	63.5108	2.98400
$454$	0	0
$455$	−4.03789	−0.189299
$456$	0	0
$457$	8.02935	0.375597	0.187799	−	0.982208i	$-0.439865\pi$
$457$	8.02935	0.375597	0.187799	+	0.982208i	$0.439865\pi$
$458$	0	0
$459$	11.4578	0.534804
$460$	0	0
$461$	−1.47394	−0.0686482	−0.0343241	−	0.999411i	$-0.510928\pi$
$461$	−1.47394	−0.0686482	−0.0343241	+	0.999411i	$0.510928\pi$
$462$	0	0
$463$	−16.5973	−0.771340	−0.385670	−	0.922637i	$-0.626030\pi$
$463$	−16.5973	−0.771340	−0.385670	+	0.922637i	$0.626030\pi$
$464$	0	0
$465$	−1.51251	−0.0701411
$466$	0	0
$467$	−27.7901	−1.28597	−0.642986	−	0.765878i	$-0.722305\pi$
$467$	−27.7901	−1.28597	−0.642986	+	0.765878i	$0.722305\pi$
$468$	0	0
$469$	−9.50915	−0.439092
$470$	0	0
$471$	−16.8661	−0.777150
$472$	0	0
$473$	0.527173	0.0242394
$474$	0	0
$475$	9.69974	0.445055
$476$	0	0
$477$	69.7218	3.19234
$478$	0	0
$479$	25.8976	1.18329	0.591646	−	0.806198i	$-0.298478\pi$
$479$	25.8976	1.18329	0.591646	+	0.806198i	$0.298478\pi$
$480$	0	0
$481$	−9.46701	−0.431658
$482$	0	0
$483$	−38.2102	−1.73863
$484$	0	0
$485$	2.34024	0.106265
$486$	0	0
$487$	−10.0280	−0.454412	−0.227206	−	0.973847i	$-0.572959\pi$
$487$	−10.0280	−0.454412	−0.227206	+	0.973847i	$0.572959\pi$
$488$	0	0
$489$	27.0046	1.22119
$490$	0	0
$491$	8.53837	0.385331	0.192666	−	0.981264i	$-0.438287\pi$
$491$	8.53837	0.385331	0.192666	+	0.981264i	$0.438287\pi$
$492$	0	0
$493$	−1.59559	−0.0718616
$494$	0	0
$495$	4.49501	0.202036
$496$	0	0
$497$	36.7274	1.64745
$498$	0	0
$499$	1.74961	0.0783234	0.0391617	−	0.999233i	$-0.487531\pi$
$499$	1.74961	0.0783234	0.0391617	+	0.999233i	$0.487531\pi$
$500$	0	0
$501$	3.42442	0.152992
$502$	0	0
$503$	5.42352	0.241823	0.120911	−	0.992663i	$-0.461418\pi$
$503$	5.42352	0.241823	0.120911	+	0.992663i	$0.461418\pi$
$504$	0	0
$505$	15.1308	0.673310
$506$	0	0
$507$	−2.90994	−0.129235
$508$	0	0
$509$	−43.0892	−1.90990	−0.954949	−	0.296771i	$-0.904090\pi$
$509$	−43.0892	−1.90990	−0.954949	+	0.296771i	$0.904090\pi$
$510$	0	0
$511$	32.7875	1.45043
$512$	0	0
$513$	31.2526	1.37983
$514$	0	0
$515$	−2.92499	−0.128891
$516$	0	0
$517$	5.66690	0.249230
$518$	0	0
$519$	47.6667	2.09234
$520$	0	0
$521$	−15.7286	−0.689084	−0.344542	−	0.938771i	$-0.611966\pi$
$521$	−15.7286	−0.689084	−0.344542	+	0.938771i	$0.611966\pi$
$522$	0	0
$523$	−4.09909	−0.179241	−0.0896204	−	0.995976i	$-0.528565\pi$
$523$	−4.09909	−0.179241	−0.0896204	+	0.995976i	$0.528565\pi$
$524$	0	0
$525$	−15.7308	−0.686550
$526$	0	0
$527$	0.498191	0.0217015
$528$	0	0
$529$	6.30659	0.274200
$530$	0	0
$531$	−47.5145	−2.06195
$532$	0	0
$533$	10.7627	0.466184
$534$	0	0
$535$	3.23020	0.139654
$536$	0	0
$537$	−3.57325	−0.154197
$538$	0	0
$539$	0.551427	0.0237516
$540$	0	0
$541$	25.9696	1.11652	0.558261	−	0.829666i	$-0.311469\pi$
$541$	25.9696	1.11652	0.558261	+	0.829666i	$0.311469\pi$
$542$	0	0
$543$	56.6311	2.43027
$544$	0	0
$545$	31.2371	1.33805
$546$	0	0
$547$	−38.6218	−1.65135	−0.825675	−	0.564146i	$-0.809206\pi$
$547$	−38.6218	−1.65135	−0.825675	+	0.564146i	$0.809206\pi$
$548$	0	0
$549$	−39.7262	−1.69547
$550$	0	0
$551$	−4.35217	−0.185409
$552$	0	0
$553$	−23.9497	−1.01844
$554$	0	0
$555$	45.8603	1.94666
$556$	0	0
$557$	−43.4878	−1.84264	−0.921320	−	0.388806i	$-0.872888\pi$
$557$	−43.4878	−1.84264	−0.921320	+	0.388806i	$0.872888\pi$
$558$	0	0
$559$	1.06750	0.0451506
$560$	0	0
$561$	−2.29291	−0.0968068
$562$	0	0
$563$	−26.9071	−1.13400	−0.566999	−	0.823719i	$-0.691896\pi$
$563$	−26.9071	−1.13400	−0.566999	+	0.823719i	$0.691896\pi$
$564$	0	0
$565$	−9.54010	−0.401355
$566$	0	0
$567$	−10.8977	−0.457661
$568$	0	0
$569$	−11.6296	−0.487537	−0.243768	−	0.969834i	$-0.578384\pi$
$569$	−11.6296	−0.487537	−0.243768	+	0.969834i	$0.578384\pi$
$570$	0	0
$571$	32.5286	1.36128	0.680641	−	0.732617i	$-0.261702\pi$
$571$	32.5286	1.36128	0.680641	+	0.732617i	$0.261702\pi$
$572$	0	0
$573$	−45.1300	−1.88533
$574$	0	0
$575$	12.0653	0.503157
$576$	0	0
$577$	25.7611	1.07245	0.536223	−	0.844076i	$-0.319850\pi$
$577$	25.7611	1.07245	0.536223	+	0.844076i	$0.319850\pi$
$578$	0	0
$579$	−43.9055	−1.82465
$580$	0	0
$581$	34.9001	1.44790
$582$	0	0
$583$	−6.29717	−0.260802
$584$	0	0
$585$	9.10222	0.376330
$586$	0	0
$587$	27.2937	1.12653	0.563266	−	0.826275i	$-0.309544\pi$
$587$	27.2937	1.12653	0.563266	+	0.826275i	$0.309544\pi$
$588$	0	0
$589$	1.35888	0.0559916
$590$	0	0
$591$	−45.7195	−1.88065
$592$	0	0
$593$	−2.83741	−0.116519	−0.0582593	−	0.998301i	$-0.518555\pi$
$593$	−2.83741	−0.116519	−0.0582593	+	0.998301i	$0.518555\pi$
$594$	0	0
$595$	−6.44280	−0.264129
$596$	0	0
$597$	31.2122	1.27743
$598$	0	0
$599$	43.0898	1.76060	0.880302	−	0.474415i	$-0.157340\pi$
$599$	43.0898	1.76060	0.880302	+	0.474415i	$0.157340\pi$
$600$	0	0
$601$	6.93951	0.283069	0.141534	−	0.989933i	$-0.454796\pi$
$601$	6.93951	0.283069	0.141534	+	0.989933i	$0.454796\pi$
$602$	0	0
$603$	21.4356	0.872923
$604$	0	0
$605$	17.9059	0.727979
$606$	0	0
$607$	−5.56958	−0.226062	−0.113031	−	0.993591i	$-0.536056\pi$
$607$	−5.56958	−0.226062	−0.113031	+	0.993591i	$0.536056\pi$
$608$	0	0
$609$	7.05825	0.286015
$610$	0	0
$611$	11.4753	0.464239
$612$	0	0
$613$	1.56680	0.0632826	0.0316413	−	0.999499i	$-0.489927\pi$
$613$	1.56680	0.0632826	0.0316413	+	0.999499i	$0.489927\pi$
$614$	0	0
$615$	−52.1369	−2.10236
$616$	0	0
$617$	−22.8333	−0.919234	−0.459617	−	0.888117i	$-0.652013\pi$
$617$	−22.8333	−0.919234	−0.459617	+	0.888117i	$0.652013\pi$
$618$	0	0
$619$	−19.0043	−0.763845	−0.381923	−	0.924194i	$-0.624738\pi$
$619$	−19.0043	−0.763845	−0.381923	+	0.924194i	$0.624738\pi$
$620$	0	0
$621$	38.8743	1.55997
$622$	0	0
$623$	23.0175	0.922175
$624$	0	0
$625$	−8.88925	−0.355570
$626$	0	0
$627$	−6.25421	−0.249769
$628$	0	0
$629$	−15.1054	−0.602293
$630$	0	0
$631$	−25.2670	−1.00586	−0.502931	−	0.864327i	$-0.667745\pi$
$631$	−25.2670	−1.00586	−0.502931	+	0.864327i	$0.667745\pi$
$632$	0	0
$633$	−63.2415	−2.51362
$634$	0	0
$635$	32.7331	1.29897
$636$	0	0
$637$	1.11662	0.0442420
$638$	0	0
$639$	−82.7910	−3.27516
$640$	0	0
$641$	−9.27225	−0.366232	−0.183116	−	0.983091i	$-0.558618\pi$
$641$	−9.27225	−0.366232	−0.183116	+	0.983091i	$0.558618\pi$
$642$	0	0
$643$	21.0813	0.831364	0.415682	−	0.909510i	$-0.363543\pi$
$643$	21.0813	0.831364	0.415682	+	0.909510i	$0.363543\pi$
$644$	0	0
$645$	−5.17123	−0.203617
$646$	0	0
$647$	−9.75628	−0.383559	−0.191780	−	0.981438i	$-0.561426\pi$
$647$	−9.75628	−0.383559	−0.191780	+	0.981438i	$0.561426\pi$
$648$	0	0
$649$	4.29144	0.168454
$650$	0	0
$651$	−2.20380	−0.0863737
$652$	0	0
$653$	29.2393	1.14422	0.572111	−	0.820176i	$-0.306125\pi$
$653$	29.2393	1.14422	0.572111	+	0.820176i	$0.306125\pi$
$654$	0	0
$655$	−3.64722	−0.142509
$656$	0	0
$657$	−73.9097	−2.88349
$658$	0	0
$659$	−8.37345	−0.326183	−0.163092	−	0.986611i	$-0.552147\pi$
$659$	−8.37345	−0.326183	−0.163092	+	0.986611i	$0.552147\pi$
$660$	0	0
$661$	−9.13306	−0.355235	−0.177617	−	0.984100i	$-0.556839\pi$
$661$	−9.13306	−0.355235	−0.177617	+	0.984100i	$0.556839\pi$
$662$	0	0
$663$	−4.64306	−0.180321
$664$	0	0
$665$	−17.5736	−0.681473
$666$	0	0
$667$	−5.41356	−0.209614
$668$	0	0
$669$	−39.3211	−1.52024
$670$	0	0
$671$	3.58801	0.138513
$672$	0	0
$673$	42.7299	1.64712	0.823559	−	0.567231i	$-0.191985\pi$
$673$	42.7299	1.64712	0.823559	+	0.567231i	$0.191985\pi$
$674$	0	0
$675$	16.0042	0.616003
$676$	0	0
$677$	−16.0897	−0.618377	−0.309189	−	0.951001i	$-0.600057\pi$
$677$	−16.0897	−0.618377	−0.309189	+	0.951001i	$0.600057\pi$
$678$	0	0
$679$	3.40983	0.130857
$680$	0	0
$681$	−57.8865	−2.21822
$682$	0	0
$683$	−4.97216	−0.190255	−0.0951273	−	0.995465i	$-0.530326\pi$
$683$	−4.97216	−0.190255	−0.0951273	+	0.995465i	$0.530326\pi$
$684$	0	0
$685$	−8.15512	−0.311591
$686$	0	0
$687$	4.05068	0.154543
$688$	0	0
$689$	−12.7515	−0.485794
$690$	0	0
$691$	4.57819	0.174163	0.0870813	−	0.996201i	$-0.472246\pi$
$691$	4.57819	0.174163	0.0870813	+	0.996201i	$0.472246\pi$
$692$	0	0
$693$	6.54943	0.248792
$694$	0	0
$695$	−18.2193	−0.691097
$696$	0	0
$697$	17.1728	0.650467
$698$	0	0
$699$	−40.3126	−1.52476
$700$	0	0
$701$	−7.78441	−0.294013	−0.147007	−	0.989136i	$-0.546964\pi$
$701$	−7.78441	−0.294013	−0.147007	+	0.989136i	$0.546964\pi$
$702$	0	0
$703$	−41.2020	−1.55396
$704$	0	0
$705$	−55.5887	−2.09359
$706$	0	0
$707$	22.0462	0.829132
$708$	0	0
$709$	−30.0908	−1.13008	−0.565042	−	0.825062i	$-0.691140\pi$
$709$	−30.0908	−1.13008	−0.565042	+	0.825062i	$0.691140\pi$
$710$	0	0
$711$	53.9874	2.02469
$712$	0	0
$713$	1.69028	0.0633014
$714$	0	0
$715$	−0.822099	−0.0307447
$716$	0	0
$717$	3.12206	0.116595
$718$	0	0
$719$	−43.7600	−1.63197	−0.815986	−	0.578071i	$-0.803806\pi$
$719$	−43.7600	−1.63197	−0.815986	+	0.578071i	$0.803806\pi$
$720$	0	0
$721$	−4.26185	−0.158720
$722$	0	0
$723$	9.52916	0.354393
$724$	0	0
$725$	−2.22872	−0.0827724
$726$	0	0
$727$	11.5712	0.429152	0.214576	−	0.976707i	$-0.431163\pi$
$727$	11.5712	0.429152	0.214576	+	0.976707i	$0.431163\pi$
$728$	0	0
$729$	−38.1224	−1.41194
$730$	0	0
$731$	1.70330	0.0629987
$732$	0	0
$733$	24.5294	0.906013	0.453006	−	0.891507i	$-0.350351\pi$
$733$	24.5294	0.906013	0.453006	+	0.891507i	$0.350351\pi$
$734$	0	0
$735$	−5.40914	−0.199519
$736$	0	0
$737$	−1.93603	−0.0713145
$738$	0	0
$739$	27.6804	1.01824	0.509120	−	0.860696i	$-0.329971\pi$
$739$	27.6804	1.01824	0.509120	+	0.860696i	$0.329971\pi$
$740$	0	0
$741$	−12.6645	−0.465243
$742$	0	0
$743$	−20.9955	−0.770250	−0.385125	−	0.922864i	$-0.625842\pi$
$743$	−20.9955	−0.770250	−0.385125	+	0.922864i	$0.625842\pi$
$744$	0	0
$745$	13.3265	0.488246
$746$	0	0
$747$	−78.6720	−2.87846
$748$	0	0
$749$	4.70655	0.171973
$750$	0	0
$751$	38.0790	1.38952	0.694760	−	0.719241i	$-0.255510\pi$
$751$	38.0790	1.38952	0.694760	+	0.719241i	$0.255510\pi$
$752$	0	0
$753$	−33.2517	−1.21176
$754$	0	0
$755$	36.3333	1.32230
$756$	0	0
$757$	11.6875	0.424790	0.212395	−	0.977184i	$-0.431874\pi$
$757$	11.6875	0.424790	0.212395	+	0.977184i	$0.431874\pi$
$758$	0	0
$759$	−7.77946	−0.282376
$760$	0	0
$761$	−18.5723	−0.673246	−0.336623	−	0.941639i	$-0.609285\pi$
$761$	−18.5723	−0.673246	−0.336623	+	0.941639i	$0.609285\pi$
$762$	0	0
$763$	45.5139	1.64771
$764$	0	0
$765$	14.5234	0.525094
$766$	0	0
$767$	8.68999	0.313777
$768$	0	0
$769$	5.21648	0.188111	0.0940556	−	0.995567i	$-0.470017\pi$
$769$	5.21648	0.188111	0.0940556	+	0.995567i	$0.470017\pi$
$770$	0	0
$771$	−77.2205	−2.78103
$772$	0	0
$773$	−26.2958	−0.945793	−0.472897	−	0.881118i	$-0.656792\pi$
$773$	−26.2958	−0.945793	−0.472897	+	0.881118i	$0.656792\pi$
$774$	0	0
$775$	0.695873	0.0249965
$776$	0	0
$777$	66.8205	2.39717
$778$	0	0
$779$	46.8410	1.67825
$780$	0	0
$781$	7.47756	0.267568
$782$	0	0
$783$	−7.18092	−0.256625
$784$	0	0
$785$	−9.64878	−0.344380
$786$	0	0
$787$	21.2206	0.756434	0.378217	−	0.925717i	$-0.376537\pi$
$787$	21.2206	0.756434	0.378217	+	0.925717i	$0.376537\pi$
$788$	0	0
$789$	15.1846	0.540585
$790$	0	0
$791$	−13.9004	−0.494240
$792$	0	0
$793$	7.26557	0.258008
$794$	0	0
$795$	61.7712	2.19080
$796$	0	0
$797$	21.9901	0.778929	0.389465	−	0.921041i	$-0.372660\pi$
$797$	21.9901	0.778929	0.389465	+	0.921041i	$0.372660\pi$
$798$	0	0
$799$	18.3098	0.647753
$800$	0	0
$801$	−51.8860	−1.83330
$802$	0	0
$803$	6.67541	0.235570
$804$	0	0
$805$	−21.8593	−0.770440
$806$	0	0
$807$	11.5136	0.405297
$808$	0	0
$809$	10.3909	0.365323	0.182662	−	0.983176i	$-0.441529\pi$
$809$	10.3909	0.365323	0.182662	+	0.983176i	$0.441529\pi$
$810$	0	0
$811$	9.54847	0.335292	0.167646	−	0.985847i	$-0.446383\pi$
$811$	9.54847	0.335292	0.167646	+	0.985847i	$0.446383\pi$
$812$	0	0
$813$	−17.4079	−0.610520
$814$	0	0
$815$	15.4488	0.541148
$816$	0	0
$817$	4.64596	0.162541
$818$	0	0
$819$	13.2623	0.463424
$820$	0	0
$821$	37.7911	1.31892	0.659459	−	0.751741i	$-0.270786\pi$
$821$	37.7911	1.31892	0.659459	+	0.751741i	$0.270786\pi$
$822$	0	0
$823$	−1.04619	−0.0364680	−0.0182340	−	0.999834i	$-0.505804\pi$
$823$	−1.04619	−0.0364680	−0.0182340	+	0.999834i	$0.505804\pi$
$824$	0	0
$825$	−3.20274	−0.111505
$826$	0	0
$827$	−39.5426	−1.37503	−0.687516	−	0.726170i	$-0.741299\pi$
$827$	−39.5426	−1.37503	−0.687516	+	0.726170i	$0.741299\pi$
$828$	0	0
$829$	−22.4661	−0.780279	−0.390139	−	0.920756i	$-0.627573\pi$
$829$	−22.4661	−0.780279	−0.390139	+	0.920756i	$0.627573\pi$
$830$	0	0
$831$	28.9418	1.00398
$832$	0	0
$833$	1.78166	0.0617309
$834$	0	0
$835$	1.95905	0.0677956
$836$	0	0
$837$	2.24210	0.0774984
$838$	0	0
$839$	−5.35980	−0.185041	−0.0925205	−	0.995711i	$-0.529492\pi$
$839$	−5.35980	−0.185041	−0.0925205	+	0.995711i	$0.529492\pi$
$840$	0	0
$841$	1.00000	0.0344828
$842$	0	0
$843$	−39.9782	−1.37692
$844$	0	0
$845$	−1.66472	−0.0572680
$846$	0	0
$847$	26.0897	0.896453
$848$	0	0
$849$	−21.9953	−0.754877
$850$	0	0
$851$	−51.2502	−1.75683
$852$	0	0
$853$	47.9909	1.64318	0.821589	−	0.570080i	$-0.193088\pi$
$853$	47.9909	1.64318	0.821589	+	0.570080i	$0.193088\pi$
$854$	0	0
$855$	39.6144	1.35478
$856$	0	0
$857$	−47.4675	−1.62146	−0.810730	−	0.585420i	$-0.800930\pi$
$857$	−47.4675	−1.62146	−0.810730	+	0.585420i	$0.800930\pi$
$858$	0	0
$859$	40.5638	1.38402	0.692009	−	0.721889i	$-0.256726\pi$
$859$	40.5638	1.38402	0.692009	+	0.721889i	$0.256726\pi$
$860$	0	0
$861$	−75.9658	−2.58891
$862$	0	0
$863$	41.3801	1.40860	0.704298	−	0.709904i	$-0.251262\pi$
$863$	41.3801	1.40860	0.704298	+	0.709904i	$0.251262\pi$
$864$	0	0
$865$	27.2692	0.927180
$866$	0	0
$867$	42.0605	1.42845
$868$	0	0
$869$	−4.87606	−0.165409
$870$	0	0
$871$	−3.92038	−0.132837
$872$	0	0
$873$	−7.68645	−0.260147
$874$	0	0
$875$	−29.1887	−0.986759
$876$	0	0
$877$	48.5332	1.63885	0.819424	−	0.573188i	$-0.194293\pi$
$877$	48.5332	1.63885	0.819424	+	0.573188i	$0.194293\pi$
$878$	0	0
$879$	94.5068	3.18764
$880$	0	0
$881$	−2.53610	−0.0854434	−0.0427217	−	0.999087i	$-0.513603\pi$
$881$	−2.53610	−0.0854434	−0.0427217	+	0.999087i	$0.513603\pi$
$882$	0	0
$883$	−3.21381	−0.108153	−0.0540766	−	0.998537i	$-0.517222\pi$
$883$	−3.21381	−0.108153	−0.0540766	+	0.998537i	$0.517222\pi$
$884$	0	0
$885$	−42.0962	−1.41505
$886$	0	0
$887$	19.4914	0.654457	0.327229	−	0.944945i	$-0.393885\pi$
$887$	19.4914	0.654457	0.327229	+	0.944945i	$0.393885\pi$
$888$	0	0
$889$	47.6937	1.59959
$890$	0	0
$891$	−2.21873	−0.0743303
$892$	0	0
$893$	49.9422	1.67125
$894$	0	0
$895$	−2.04419	−0.0683297
$896$	0	0
$897$	−15.7531	−0.525981
$898$	0	0
$899$	−0.312230	−0.0104135
$900$	0	0
$901$	−20.3462	−0.677829
$902$	0	0
$903$	−7.53471	−0.250739
$904$	0	0
$905$	32.3976	1.07693
$906$	0	0
$907$	54.2041	1.79982	0.899908	−	0.436080i	$-0.143633\pi$
$907$	54.2041	1.79982	0.899908	+	0.436080i	$0.143633\pi$
$908$	0	0
$909$	−49.6966	−1.64833
$910$	0	0
$911$	−53.1760	−1.76180	−0.880900	−	0.473303i	$-0.843062\pi$
$911$	−53.1760	−1.76180	−0.880900	+	0.473303i	$0.843062\pi$
$912$	0	0
$913$	7.10553	0.235159
$914$	0	0
$915$	−35.1960	−1.16355
$916$	0	0
$917$	−5.31417	−0.175489
$918$	0	0
$919$	12.8700	0.424544	0.212272	−	0.977211i	$-0.431914\pi$
$919$	12.8700	0.424544	0.212272	+	0.977211i	$0.431914\pi$
$920$	0	0
$921$	−79.8517	−2.63120
$922$	0	0
$923$	15.1418	0.498397
$924$	0	0
$925$	−21.0993	−0.693740
$926$	0	0
$927$	9.60707	0.315538
$928$	0	0
$929$	22.0985	0.725029	0.362515	−	0.931978i	$-0.381918\pi$
$929$	22.0985	0.725029	0.362515	+	0.931978i	$0.381918\pi$
$930$	0	0
$931$	4.85971	0.159271
$932$	0	0
$933$	47.6368	1.55956
$934$	0	0
$935$	−1.31173	−0.0428982
$936$	0	0
$937$	−36.0132	−1.17650	−0.588250	−	0.808679i	$-0.700183\pi$
$937$	−36.0132	−1.17650	−0.588250	+	0.808679i	$0.700183\pi$
$938$	0	0
$939$	−38.2645	−1.24871
$940$	0	0
$941$	8.37682	0.273076	0.136538	−	0.990635i	$-0.456402\pi$
$941$	8.37682	0.273076	0.136538	+	0.990635i	$0.456402\pi$
$942$	0	0
$943$	58.2644	1.89735
$944$	0	0
$945$	−28.9957	−0.943232
$946$	0	0
$947$	37.0792	1.20491	0.602457	−	0.798152i	$-0.294189\pi$
$947$	37.0792	1.20491	0.602457	+	0.798152i	$0.294189\pi$
$948$	0	0
$949$	13.5174	0.438795
$950$	0	0
$951$	45.5808	1.47806
$952$	0	0
$953$	11.1205	0.360227	0.180114	−	0.983646i	$-0.442353\pi$
$953$	11.1205	0.360227	0.180114	+	0.983646i	$0.442353\pi$
$954$	0	0
$955$	−25.8180	−0.835450
$956$	0	0
$957$	1.43703	0.0464527
$958$	0	0
$959$	−11.8824	−0.383702
$960$	0	0
$961$	−30.9025	−0.996855
$962$	0	0
$963$	−10.6095	−0.341887
$964$	0	0
$965$	−25.1175	−0.808560
$966$	0	0
$967$	−3.61047	−0.116105	−0.0580525	−	0.998314i	$-0.518489\pi$
$967$	−3.61047	−0.116105	−0.0580525	+	0.998314i	$0.518489\pi$
$968$	0	0
$969$	−20.2074	−0.649154
$970$	0	0
$971$	−27.5364	−0.883686	−0.441843	−	0.897092i	$-0.645675\pi$
$971$	−27.5364	−0.883686	−0.441843	+	0.897092i	$0.645675\pi$
$972$	0	0
$973$	−26.5463	−0.851037
$974$	0	0
$975$	−6.48542	−0.207700
$976$	0	0
$977$	−41.1615	−1.31687	−0.658436	−	0.752636i	$-0.728782\pi$
$977$	−41.1615	−1.31687	−0.658436	+	0.752636i	$0.728782\pi$
$978$	0	0
$979$	4.68627	0.149774
$980$	0	0
$981$	−102.598	−3.27569
$982$	0	0
$983$	−55.2536	−1.76232	−0.881159	−	0.472820i	$-0.843236\pi$
$983$	−55.2536	−1.76232	−0.881159	+	0.472820i	$0.843236\pi$
$984$	0	0
$985$	−26.1553	−0.833376
$986$	0	0
$987$	−80.9952	−2.57811
$988$	0	0
$989$	5.77899	0.183761
$990$	0	0
$991$	−43.7301	−1.38913	−0.694565	−	0.719430i	$-0.744403\pi$
$991$	−43.7301	−1.38913	−0.694565	+	0.719430i	$0.744403\pi$
$992$	0	0
$993$	−15.4431	−0.490072
$994$	0	0
$995$	17.8559	0.566070
$996$	0	0
$997$	−14.1513	−0.448175	−0.224087	−	0.974569i	$-0.571940\pi$
$997$	−14.1513	−0.448175	−0.224087	+	0.974569i	$0.571940\pi$
$998$	0	0
$999$	−67.9818	−2.15085

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	6032.2.a.be.1.1		13
4.3	odd	2		3016.2.a.k.1.13	✓	13

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
3016.2.a.k.1.13	✓	13	4.3	odd	2
6032.2.a.be.1.1		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:	\( N \)	\(=\)	\( 6032 = 2^{4} \cdot 13 \cdot 29 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	6032.a (trivial)

\(f(q)\)	\(=\)	\(q-2.90994 q^{3} -1.66472 q^{5} -2.42557 q^{7} +5.46773 q^{9} +O(q^{10})\)	\(q-2.90994 q^{3} -1.66472 q^{5} -2.42557 q^{7} +5.46773 q^{9} -0.493837 q^{11} -1.00000 q^{13} +4.84422 q^{15} -1.59559 q^{17} -4.35217 q^{19} +7.05825 q^{21} -5.41356 q^{23} -2.22872 q^{25} -7.18092 q^{27} +1.00000 q^{29} -0.312230 q^{31} +1.43703 q^{33} +4.03789 q^{35} +9.46701 q^{37} +2.90994 q^{39} -10.7627 q^{41} -1.06750 q^{43} -9.10222 q^{45} -11.4753 q^{47} -1.11662 q^{49} +4.64306 q^{51} +12.7515 q^{53} +0.822099 q^{55} +12.6645 q^{57} -8.68999 q^{59} -7.26557 q^{61} -13.2623 q^{63} +1.66472 q^{65} +3.92038 q^{67} +15.7531 q^{69} -15.1418 q^{71} -13.5174 q^{73} +6.48542 q^{75} +1.19783 q^{77} +9.87383 q^{79} +4.49285 q^{81} -14.3884 q^{83} +2.65620 q^{85} -2.90994 q^{87} -9.48951 q^{89} +2.42557 q^{91} +0.908571 q^{93} +7.24513 q^{95} -1.40579 q^{97} -2.70016 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 13 q + 4 q^{3} + 5 q^{5} - 6 q^{7} + 21 q^{9}+O(q^{10}) \) 13 * q + 4 * q^3 + 5 * q^5 - 6 * q^7 + 21 * q^9	\( 13 q + 4 q^{3} + 5 q^{5} - 6 q^{7} + 21 q^{9} - 6 q^{11} - 13 q^{13} - 6 q^{15} + 4 q^{17} + 3 q^{19} + 5 q^{21} - 13 q^{23} + 24 q^{25} + 16 q^{27} + 13 q^{29} - 21 q^{31} + 21 q^{33} - 8 q^{35} + 23 q^{37} - 4 q^{39} + 4 q^{41} + 24 q^{43} + 9 q^{45} - 7 q^{47} + 41 q^{49} + q^{51} + 17 q^{53} + 8 q^{55} + 24 q^{57} - 11 q^{59} + 26 q^{61} - 17 q^{63} - 5 q^{65} + 35 q^{67} + 30 q^{69} - 30 q^{71} + 17 q^{73} + 43 q^{75} + 34 q^{77} - 3 q^{79} + 37 q^{81} + 18 q^{83} + 9 q^{85} + 4 q^{87} + 38 q^{89} + 6 q^{91} + 25 q^{93} + 9 q^{95} + 7 q^{97} + 40 q^{99}+O(q^{100}) \) 13 * q + 4 * q^3 + 5 * q^5 - 6 * q^7 + 21 * q^9 - 6 * q^11 - 13 * q^13 - 6 * q^15 + 4 * q^17 + 3 * q^19 + 5 * q^21 - 13 * q^23 + 24 * q^25 + 16 * q^27 + 13 * q^29 - 21 * q^31 + 21 * q^33 - 8 * q^35 + 23 * q^37 - 4 * q^39 + 4 * q^41 + 24 * q^43 + 9 * q^45 - 7 * q^47 + 41 * q^49 + q^51 + 17 * q^53 + 8 * q^55 + 24 * q^57 - 11 * q^59 + 26 * q^61 - 17 * q^63 - 5 * q^65 + 35 * q^67 + 30 * q^69 - 30 * q^71 + 17 * q^73 + 43 * q^75 + 34 * q^77 - 3 * q^79 + 37 * q^81 + 18 * q^83 + 9 * q^85 + 4 * q^87 + 38 * q^89 + 6 * q^91 + 25 * q^93 + 9 * q^95 + 7 * q^97 + 40 * q^99

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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