LMFDB - Embedded newform 6024.2.a.r.1.11

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [6024,2,Mod(1,6024)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(6024, 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("6024.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 6024 = 2^{3} \cdot 3 \cdot 251 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	6024.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:	$48.1018821776$
Analytic rank:	$0$
Dimension:	$20$
Coefficient field:	$\mathbb{Q}[x]/(x^{20} - \cdots)$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{20} - 9 x^{19} - 31 x^{18} + 471 x^{17} - 82 x^{16} - 9476 x^{15} + 12881 x^{14} + 94079 x^{13} + \cdots + 24832 $ x^20 - 9x^19 - 31x^18 + 471x^17 - 82x^16 - 9476x^15 + 12881x^14 + 94079x^13 - 181272x^12 - 504914x^11 + 1108012x^10 + 1591114x^9 - 3399127x^8 - 3224089x^7 + 5089722x^6 + 4287384x^5 - 2840216x^4 - 2820864x^3 - 208576x^2 + 204992*x + 24832
Coefficient ring:	$\Z[a_1, \ldots, a_{19}]$
Coefficient ring index:	$ 2^{5} $
Twist minimal:	yes
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.11
Root			$0.273378$ of defining polynomial
Character	$\chi$	$=$	6024.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+1.00000 q^{3} +0.273378 q^{5} -2.92800 q^{7} +1.00000 q^{9} +O(q^{10})$	$q+1.00000 q^{3} +0.273378 q^{5} -2.92800 q^{7} +1.00000 q^{9} -4.02080 q^{11} +1.67900 q^{13} +0.273378 q^{15} +0.0598345 q^{17} +5.15858 q^{19} -2.92800 q^{21} +0.580259 q^{23} -4.92526 q^{25} +1.00000 q^{27} +1.19321 q^{29} +3.68555 q^{31} -4.02080 q^{33} -0.800453 q^{35} +6.70422 q^{37} +1.67900 q^{39} -7.90269 q^{41} +7.84401 q^{43} +0.273378 q^{45} -9.18777 q^{47} +1.57320 q^{49} +0.0598345 q^{51} +7.76336 q^{53} -1.09920 q^{55} +5.15858 q^{57} -1.33290 q^{59} +4.78634 q^{61} -2.92800 q^{63} +0.459003 q^{65} -5.67579 q^{67} +0.580259 q^{69} -5.64423 q^{71} +11.2660 q^{73} -4.92526 q^{75} +11.7729 q^{77} -14.1584 q^{79} +1.00000 q^{81} +14.6206 q^{83} +0.0163575 q^{85} +1.19321 q^{87} +8.55882 q^{89} -4.91612 q^{91} +3.68555 q^{93} +1.41024 q^{95} -0.970420 q^{97} -4.02080 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 20 q + 20 q^{3} + 9 q^{5} + 9 q^{7} + 20 q^{9}+O(q^{10}) $ 20 * q + 20 * q^3 + 9 * q^5 + 9 * q^7 + 20 * q^9	$ 20 q + 20 q^{3} + 9 q^{5} + 9 q^{7} + 20 q^{9} + 4 q^{11} + 21 q^{13} + 9 q^{15} + 10 q^{17} + 8 q^{19} + 9 q^{21} + 9 q^{23} + 43 q^{25} + 20 q^{27} + 18 q^{29} + 27 q^{31} + 4 q^{33} - 7 q^{35} + 33 q^{37} + 21 q^{39} + 14 q^{41} - 6 q^{43} + 9 q^{45} + 21 q^{47} + 47 q^{49} + 10 q^{51} + 23 q^{53} + 24 q^{55} + 8 q^{57} + 10 q^{59} + 28 q^{61} + 9 q^{63} + 2 q^{65} + 15 q^{67} + 9 q^{69} + 33 q^{71} + 50 q^{73} + 43 q^{75} + 20 q^{77} + 17 q^{79} + 20 q^{81} - 19 q^{83} + 41 q^{85} + 18 q^{87} + 21 q^{89} + 30 q^{91} + 27 q^{93} + 27 q^{95} + 47 q^{97} + 4 q^{99}+O(q^{100}) $ 20 * q + 20 * q^3 + 9 * q^5 + 9 * q^7 + 20 * q^9 + 4 * q^11 + 21 * q^13 + 9 * q^15 + 10 * q^17 + 8 * q^19 + 9 * q^21 + 9 * q^23 + 43 * q^25 + 20 * q^27 + 18 * q^29 + 27 * q^31 + 4 * q^33 - 7 * q^35 + 33 * q^37 + 21 * q^39 + 14 * q^41 - 6 * q^43 + 9 * q^45 + 21 * q^47 + 47 * q^49 + 10 * q^51 + 23 * q^53 + 24 * q^55 + 8 * q^57 + 10 * q^59 + 28 * q^61 + 9 * q^63 + 2 * q^65 + 15 * q^67 + 9 * q^69 + 33 * q^71 + 50 * q^73 + 43 * q^75 + 20 * q^77 + 17 * q^79 + 20 * q^81 - 19 * q^83 + 41 * q^85 + 18 * q^87 + 21 * q^89 + 30 * q^91 + 27 * q^93 + 27 * q^95 + 47 * q^97 + 4 * q^99

Display 100 coefficients 10 coefficients

Coefficient data

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

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

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

$2$	0	0
$2$	0	0
$3$	1.00000	0.577350
$3$	1.00000	0.577350
$4$	0	0
$5$	0.273378	0.122259	0.0611293	−	0.998130i	$-0.480530\pi$
$5$	0.273378	0.122259	0.0611293	+	0.998130i	$0.480530\pi$
$6$	0	0
$7$	−2.92800	−1.10668	−0.553340	−	0.832955i	$-0.686647\pi$
$7$	−2.92800	−1.10668	−0.553340	+	0.832955i	$0.686647\pi$
$8$	0	0
$9$	1.00000	0.333333
$10$	0	0
$11$	−4.02080	−1.21232	−0.606158	−	0.795344i	$-0.707290\pi$
$11$	−4.02080	−1.21232	−0.606158	+	0.795344i	$0.707290\pi$
$12$	0	0
$13$	1.67900	0.465671	0.232836	−	0.972516i	$-0.425200\pi$
$13$	1.67900	0.465671	0.232836	+	0.972516i	$0.425200\pi$
$14$	0	0
$15$	0.273378	0.0705860
$16$	0	0
$17$	0.0598345	0.0145120	0.00725600	−	0.999974i	$-0.497690\pi$
$17$	0.0598345	0.0145120	0.00725600	+	0.999974i	$0.497690\pi$
$18$	0	0
$19$	5.15858	1.18346	0.591730	−	0.806137i	$-0.298445\pi$
$19$	5.15858	1.18346	0.591730	+	0.806137i	$0.298445\pi$
$20$	0	0
$21$	−2.92800	−0.638942
$22$	0	0
$23$	0.580259	0.120992	0.0604962	−	0.998168i	$-0.480732\pi$
$23$	0.580259	0.120992	0.0604962	+	0.998168i	$0.480732\pi$
$24$	0	0
$25$	−4.92526	−0.985053
$26$	0	0
$27$	1.00000	0.192450
$28$	0	0
$29$	1.19321	0.221573	0.110787	−	0.993844i	$-0.464663\pi$
$29$	1.19321	0.221573	0.110787	+	0.993844i	$0.464663\pi$
$30$	0	0
$31$	3.68555	0.661944	0.330972	−	0.943641i	$-0.392623\pi$
$31$	3.68555	0.661944	0.330972	+	0.943641i	$0.392623\pi$
$32$	0	0
$33$	−4.02080	−0.699931
$34$	0	0
$35$	−0.800453	−0.135301
$36$	0	0
$37$	6.70422	1.10217	0.551084	−	0.834450i	$-0.314214\pi$
$37$	6.70422	1.10217	0.551084	+	0.834450i	$0.314214\pi$
$38$	0	0
$39$	1.67900	0.268855
$40$	0	0
$41$	−7.90269	−1.23419	−0.617096	−	0.786888i	$-0.711691\pi$
$41$	−7.90269	−1.23419	−0.617096	+	0.786888i	$0.711691\pi$
$42$	0	0
$43$	7.84401	1.19620	0.598100	−	0.801422i	$-0.295923\pi$
$43$	7.84401	1.19620	0.598100	+	0.801422i	$0.295923\pi$
$44$	0	0
$45$	0.273378	0.0407528
$46$	0	0
$47$	−9.18777	−1.34017	−0.670087	−	0.742283i	$-0.733743\pi$
$47$	−9.18777	−1.34017	−0.670087	+	0.742283i	$0.733743\pi$
$48$	0	0
$49$	1.57320	0.224742
$50$	0	0
$51$	0.0598345	0.00837851
$52$	0	0
$53$	7.76336	1.06638	0.533190	−	0.845996i	$-0.320993\pi$
$53$	7.76336	1.06638	0.533190	+	0.845996i	$0.320993\pi$
$54$	0	0
$55$	−1.09920	−0.148216
$56$	0	0
$57$	5.15858	0.683270
$58$	0	0
$59$	−1.33290	−0.173529	−0.0867644	−	0.996229i	$-0.527653\pi$
$59$	−1.33290	−0.173529	−0.0867644	+	0.996229i	$0.527653\pi$
$60$	0	0
$61$	4.78634	0.612827	0.306414	−	0.951898i	$-0.400871\pi$
$61$	4.78634	0.612827	0.306414	+	0.951898i	$0.400871\pi$
$62$	0	0
$63$	−2.92800	−0.368894
$64$	0	0
$65$	0.459003	0.0569323
$66$	0	0
$67$	−5.67579	−0.693408	−0.346704	−	0.937975i	$-0.612699\pi$
$67$	−5.67579	−0.693408	−0.346704	+	0.937975i	$0.612699\pi$
$68$	0	0
$69$	0.580259	0.0698550
$70$	0	0
$71$	−5.64423	−0.669847	−0.334924	−	0.942245i	$-0.608711\pi$
$71$	−5.64423	−0.669847	−0.334924	+	0.942245i	$0.608711\pi$
$72$	0	0
$73$	11.2660	1.31858	0.659292	−	0.751887i	$-0.270856\pi$
$73$	11.2660	1.31858	0.659292	+	0.751887i	$0.270856\pi$
$74$	0	0
$75$	−4.92526	−0.568721
$76$	0	0
$77$	11.7729	1.34165
$78$	0	0
$79$	−14.1584	−1.59294	−0.796470	−	0.604678i	$-0.793302\pi$
$79$	−14.1584	−1.59294	−0.796470	+	0.604678i	$0.793302\pi$
$80$	0	0
$81$	1.00000	0.111111
$82$	0	0
$83$	14.6206	1.60482	0.802408	−	0.596775i	$-0.203552\pi$
$83$	14.6206	1.60482	0.802408	+	0.596775i	$0.203552\pi$
$84$	0	0
$85$	0.0163575	0.00177422
$86$	0	0
$87$	1.19321	0.127925
$88$	0	0
$89$	8.55882	0.907233	0.453616	−	0.891197i	$-0.350134\pi$
$89$	8.55882	0.907233	0.453616	+	0.891197i	$0.350134\pi$
$90$	0	0
$91$	−4.91612	−0.515349
$92$	0	0
$93$	3.68555	0.382173
$94$	0	0
$95$	1.41024	0.144688
$96$	0	0
$97$	−0.970420	−0.0985312	−0.0492656	−	0.998786i	$-0.515688\pi$
$97$	−0.970420	−0.0985312	−0.0492656	+	0.998786i	$0.515688\pi$
$98$	0	0
$99$	−4.02080	−0.404105
$100$	0	0
$101$	6.58911	0.655641	0.327821	−	0.944740i	$-0.393686\pi$
$101$	6.58911	0.655641	0.327821	+	0.944740i	$0.393686\pi$
$102$	0	0
$103$	−1.58821	−0.156491	−0.0782454	−	0.996934i	$-0.524932\pi$
$103$	−1.58821	−0.156491	−0.0782454	+	0.996934i	$0.524932\pi$
$104$	0	0
$105$	−0.800453	−0.0781162
$106$	0	0
$107$	10.8965	1.05341	0.526703	−	0.850049i	$-0.323428\pi$
$107$	10.8965	1.05341	0.526703	+	0.850049i	$0.323428\pi$
$108$	0	0
$109$	2.87093	0.274985	0.137493	−	0.990503i	$-0.456096\pi$
$109$	2.87093	0.274985	0.137493	+	0.990503i	$0.456096\pi$
$110$	0	0
$111$	6.70422	0.636337
$112$	0	0
$113$	19.6627	1.84971	0.924857	−	0.380316i	$-0.124185\pi$
$113$	19.6627	1.84971	0.924857	+	0.380316i	$0.124185\pi$
$114$	0	0
$115$	0.158630	0.0147924
$116$	0	0
$117$	1.67900	0.155224
$118$	0	0
$119$	−0.175196	−0.0160601
$120$	0	0
$121$	5.16681	0.469710
$122$	0	0
$123$	−7.90269	−0.712561
$124$	0	0
$125$	−2.71335	−0.242690
$126$	0	0
$127$	4.85311	0.430644	0.215322	−	0.976543i	$-0.430920\pi$
$127$	4.85311	0.430644	0.215322	+	0.976543i	$0.430920\pi$
$128$	0	0
$129$	7.84401	0.690626
$130$	0	0
$131$	−20.4439	−1.78620	−0.893098	−	0.449863i	$-0.851473\pi$
$131$	−20.4439	−1.78620	−0.893098	+	0.449863i	$0.851473\pi$
$132$	0	0
$133$	−15.1043	−1.30971
$134$	0	0
$135$	0.273378	0.0235287
$136$	0	0
$137$	2.12236	0.181326	0.0906629	−	0.995882i	$-0.471101\pi$
$137$	2.12236	0.181326	0.0906629	+	0.995882i	$0.471101\pi$
$138$	0	0
$139$	20.7577	1.76064	0.880322	−	0.474377i	$-0.157327\pi$
$139$	20.7577	1.76064	0.880322	+	0.474377i	$0.157327\pi$
$140$	0	0
$141$	−9.18777	−0.773750
$142$	0	0
$143$	−6.75093	−0.564541
$144$	0	0
$145$	0.326198	0.0270892
$146$	0	0
$147$	1.57320	0.129755
$148$	0	0
$149$	12.3544	1.01211	0.506057	−	0.862500i	$-0.331102\pi$
$149$	12.3544	1.01211	0.506057	+	0.862500i	$0.331102\pi$
$150$	0	0
$151$	−7.37665	−0.600303	−0.300152	−	0.953892i	$-0.597037\pi$
$151$	−7.37665	−0.600303	−0.300152	+	0.953892i	$0.597037\pi$
$152$	0	0
$153$	0.0598345	0.00483733
$154$	0	0
$155$	1.00755	0.0809283
$156$	0	0
$157$	23.5265	1.87762	0.938808	−	0.344440i	$-0.111931\pi$
$157$	23.5265	1.87762	0.938808	+	0.344440i	$0.111931\pi$
$158$	0	0
$159$	7.76336	0.615674
$160$	0	0
$161$	−1.69900	−0.133900
$162$	0	0
$163$	16.7815	1.31443	0.657213	−	0.753705i	$-0.271735\pi$
$163$	16.7815	1.31443	0.657213	+	0.753705i	$0.271735\pi$
$164$	0	0
$165$	−1.09920	−0.0855726
$166$	0	0
$167$	15.1345	1.17114	0.585571	−	0.810621i	$-0.300870\pi$
$167$	15.1345	1.17114	0.585571	+	0.810621i	$0.300870\pi$
$168$	0	0
$169$	−10.1810	−0.783150
$170$	0	0
$171$	5.15858	0.394486
$172$	0	0
$173$	−6.93938	−0.527592	−0.263796	−	0.964579i	$-0.584975\pi$
$173$	−6.93938	−0.527592	−0.263796	+	0.964579i	$0.584975\pi$
$174$	0	0
$175$	14.4212	1.09014
$176$	0	0
$177$	−1.33290	−0.100187
$178$	0	0
$179$	8.11738	0.606721	0.303361	−	0.952876i	$-0.401891\pi$
$179$	8.11738	0.606721	0.303361	+	0.952876i	$0.401891\pi$
$180$	0	0
$181$	12.6503	0.940287	0.470144	−	0.882590i	$-0.344202\pi$
$181$	12.6503	0.940287	0.470144	+	0.882590i	$0.344202\pi$
$182$	0	0
$183$	4.78634	0.353816
$184$	0	0
$185$	1.83279	0.134749
$186$	0	0
$187$	−0.240582	−0.0175931
$188$	0	0
$189$	−2.92800	−0.212981
$190$	0	0
$191$	22.1414	1.60210	0.801049	−	0.598599i	$-0.204276\pi$
$191$	22.1414	1.60210	0.801049	+	0.598599i	$0.204276\pi$
$192$	0	0
$193$	22.1147	1.59185	0.795924	−	0.605396i	$-0.206985\pi$
$193$	22.1147	1.59185	0.795924	+	0.605396i	$0.206985\pi$
$194$	0	0
$195$	0.459003	0.0328699
$196$	0	0
$197$	−6.07138	−0.432568	−0.216284	−	0.976331i	$-0.569394\pi$
$197$	−6.07138	−0.432568	−0.216284	+	0.976331i	$0.569394\pi$
$198$	0	0
$199$	−3.73415	−0.264707	−0.132354	−	0.991203i	$-0.542253\pi$
$199$	−3.73415	−0.264707	−0.132354	+	0.991203i	$0.542253\pi$
$200$	0	0
$201$	−5.67579	−0.400339
$202$	0	0
$203$	−3.49372	−0.245211
$204$	0	0
$205$	−2.16042	−0.150891
$206$	0	0
$207$	0.580259	0.0403308
$208$	0	0
$209$	−20.7416	−1.43473
$210$	0	0
$211$	−19.2956	−1.32836	−0.664180	−	0.747572i	$-0.731219\pi$
$211$	−19.2956	−1.32836	−0.664180	+	0.747572i	$0.731219\pi$
$212$	0	0
$213$	−5.64423	−0.386737
$214$	0	0
$215$	2.14438	0.146246
$216$	0	0
$217$	−10.7913	−0.732560
$218$	0	0
$219$	11.2660	0.761284
$220$	0	0
$221$	0.100462	0.00675782
$222$	0	0
$223$	22.5610	1.51079	0.755397	−	0.655267i	$-0.227444\pi$
$223$	22.5610	1.51079	0.755397	+	0.655267i	$0.227444\pi$
$224$	0	0
$225$	−4.92526	−0.328351
$226$	0	0
$227$	−27.0520	−1.79550	−0.897751	−	0.440503i	$-0.854800\pi$
$227$	−27.0520	−1.79550	−0.897751	+	0.440503i	$0.854800\pi$
$228$	0	0
$229$	26.1271	1.72653	0.863264	−	0.504753i	$-0.168416\pi$
$229$	26.1271	1.72653	0.863264	+	0.504753i	$0.168416\pi$
$230$	0	0
$231$	11.7729	0.774600
$232$	0	0
$233$	−16.1916	−1.06075	−0.530373	−	0.847764i	$-0.677948\pi$
$233$	−16.1916	−1.06075	−0.530373	+	0.847764i	$0.677948\pi$
$234$	0	0
$235$	−2.51174	−0.163848
$236$	0	0
$237$	−14.1584	−0.919684
$238$	0	0
$239$	21.9328	1.41871	0.709357	−	0.704849i	$-0.248985\pi$
$239$	21.9328	1.41871	0.709357	+	0.704849i	$0.248985\pi$
$240$	0	0
$241$	30.3354	1.95407	0.977036	−	0.213074i	$-0.0683475\pi$
$241$	30.3354	1.95407	0.977036	+	0.213074i	$0.0683475\pi$
$242$	0	0
$243$	1.00000	0.0641500
$244$	0	0
$245$	0.430078	0.0274767
$246$	0	0
$247$	8.66126	0.551103
$248$	0	0
$249$	14.6206	0.926541
$250$	0	0
$251$	−1.00000	−0.0631194
$251$	−1.00000	−0.0631194
$252$	0	0
$253$	−2.33310	−0.146681
$254$	0	0
$255$	0.0163575	0.00102434
$256$	0	0
$257$	−12.4119	−0.774234	−0.387117	−	0.922031i	$-0.626529\pi$
$257$	−12.4119	−0.774234	−0.387117	+	0.922031i	$0.626529\pi$
$258$	0	0
$259$	−19.6300	−1.21975
$260$	0	0
$261$	1.19321	0.0738578
$262$	0	0
$263$	2.34910	0.144852	0.0724259	−	0.997374i	$-0.476926\pi$
$263$	2.34910	0.144852	0.0724259	+	0.997374i	$0.476926\pi$
$264$	0	0
$265$	2.12233	0.130374
$266$	0	0
$267$	8.55882	0.523791
$268$	0	0
$269$	20.5512	1.25303	0.626514	−	0.779410i	$-0.284481\pi$
$269$	20.5512	1.25303	0.626514	+	0.779410i	$0.284481\pi$
$270$	0	0
$271$	−32.5662	−1.97825	−0.989127	−	0.147066i	$-0.953017\pi$
$271$	−32.5662	−1.97825	−0.989127	+	0.147066i	$0.953017\pi$
$272$	0	0
$273$	−4.91612	−0.297537
$274$	0	0
$275$	19.8035	1.19420
$276$	0	0
$277$	14.8348	0.891338	0.445669	−	0.895198i	$-0.352966\pi$
$277$	14.8348	0.891338	0.445669	+	0.895198i	$0.352966\pi$
$278$	0	0
$279$	3.68555	0.220648
$280$	0	0
$281$	5.97420	0.356391	0.178196	−	0.983995i	$-0.442974\pi$
$281$	5.97420	0.356391	0.178196	+	0.983995i	$0.442974\pi$
$282$	0	0
$283$	2.54849	0.151492	0.0757461	−	0.997127i	$-0.475866\pi$
$283$	2.54849	0.151492	0.0757461	+	0.997127i	$0.475866\pi$
$284$	0	0
$285$	1.41024	0.0835357
$286$	0	0
$287$	23.1391	1.36586
$288$	0	0
$289$	−16.9964	−0.999789
$290$	0	0
$291$	−0.970420	−0.0568870
$292$	0	0
$293$	−12.5610	−0.733821	−0.366911	−	0.930256i	$-0.619585\pi$
$293$	−12.5610	−0.733821	−0.366911	+	0.930256i	$0.619585\pi$
$294$	0	0
$295$	−0.364386	−0.0212154
$296$	0	0
$297$	−4.02080	−0.233310
$298$	0	0
$299$	0.974256	0.0563427
$300$	0	0
$301$	−22.9673	−1.32381
$302$	0	0
$303$	6.58911	0.378535
$304$	0	0
$305$	1.30848	0.0749234
$306$	0	0
$307$	−13.3645	−0.762754	−0.381377	−	0.924420i	$-0.624550\pi$
$307$	−13.3645	−0.762754	−0.381377	+	0.924420i	$0.624550\pi$
$308$	0	0
$309$	−1.58821	−0.0903500
$310$	0	0
$311$	−15.3964	−0.873049	−0.436524	−	0.899692i	$-0.643791\pi$
$311$	−15.3964	−0.873049	−0.436524	+	0.899692i	$0.643791\pi$
$312$	0	0
$313$	5.54298	0.313308	0.156654	−	0.987654i	$-0.449929\pi$
$313$	5.54298	0.313308	0.156654	+	0.987654i	$0.449929\pi$
$314$	0	0
$315$	−0.800453	−0.0451004
$316$	0	0
$317$	−10.5891	−0.594743	−0.297371	−	0.954762i	$-0.596110\pi$
$317$	−10.5891	−0.594743	−0.297371	+	0.954762i	$0.596110\pi$
$318$	0	0
$319$	−4.79765	−0.268617
$320$	0	0
$321$	10.8965	0.608185
$322$	0	0
$323$	0.308661	0.0171744
$324$	0	0
$325$	−8.26953	−0.458711
$326$	0	0
$327$	2.87093	0.158763
$328$	0	0
$329$	26.9018	1.48314
$330$	0	0
$331$	−21.6567	−1.19036	−0.595179	−	0.803593i	$-0.702919\pi$
$331$	−21.6567	−1.19036	−0.595179	+	0.803593i	$0.702919\pi$
$332$	0	0
$333$	6.70422	0.367389
$334$	0	0
$335$	−1.55164	−0.0847751
$336$	0	0
$337$	−27.8807	−1.51876	−0.759379	−	0.650648i	$-0.774497\pi$
$337$	−27.8807	−1.51876	−0.759379	+	0.650648i	$0.774497\pi$
$338$	0	0
$339$	19.6627	1.06793
$340$	0	0
$341$	−14.8188	−0.802485
$342$	0	0
$343$	15.8897	0.857963
$344$	0	0
$345$	0.158630	0.00854037
$346$	0	0
$347$	−14.0113	−0.752165	−0.376083	−	0.926586i	$-0.622729\pi$
$347$	−14.0113	−0.752165	−0.376083	+	0.926586i	$0.622729\pi$
$348$	0	0
$349$	20.5013	1.09741	0.548706	−	0.836016i	$-0.315121\pi$
$349$	20.5013	1.09741	0.548706	+	0.836016i	$0.315121\pi$
$350$	0	0
$351$	1.67900	0.0896185
$352$	0	0
$353$	−23.7899	−1.26621	−0.633105	−	0.774066i	$-0.718220\pi$
$353$	−23.7899	−1.26621	−0.633105	+	0.774066i	$0.718220\pi$
$354$	0	0
$355$	−1.54301	−0.0818946
$356$	0	0
$357$	−0.175196	−0.00927233
$358$	0	0
$359$	8.17722	0.431577	0.215789	−	0.976440i	$-0.430768\pi$
$359$	8.17722	0.431577	0.215789	+	0.976440i	$0.430768\pi$
$360$	0	0
$361$	7.61093	0.400575
$362$	0	0
$363$	5.16681	0.271187
$364$	0	0
$365$	3.07988	0.161208
$366$	0	0
$367$	−19.9259	−1.04012	−0.520062	−	0.854129i	$-0.674091\pi$
$367$	−19.9259	−1.04012	−0.520062	+	0.854129i	$0.674091\pi$
$368$	0	0
$369$	−7.90269	−0.411398
$370$	0	0
$371$	−22.7311	−1.18014
$372$	0	0
$373$	−6.07412	−0.314506	−0.157253	−	0.987558i	$-0.550264\pi$
$373$	−6.07412	−0.314506	−0.157253	+	0.987558i	$0.550264\pi$
$374$	0	0
$375$	−2.71335	−0.140117
$376$	0	0
$377$	2.00340	0.103180
$378$	0	0
$379$	34.8482	1.79003	0.895015	−	0.446036i	$-0.147165\pi$
$379$	34.8482	1.79003	0.895015	+	0.446036i	$0.147165\pi$
$380$	0	0
$381$	4.85311	0.248633
$382$	0	0
$383$	14.7478	0.753579	0.376789	−	0.926299i	$-0.377028\pi$
$383$	14.7478	0.753579	0.376789	+	0.926299i	$0.377028\pi$
$384$	0	0
$385$	3.21846	0.164028
$386$	0	0
$387$	7.84401	0.398733
$388$	0	0
$389$	4.51913	0.229129	0.114565	−	0.993416i	$-0.463453\pi$
$389$	4.51913	0.229129	0.114565	+	0.993416i	$0.463453\pi$
$390$	0	0
$391$	0.0347195	0.00175584
$392$	0	0
$393$	−20.4439	−1.03126
$394$	0	0
$395$	−3.87059	−0.194750
$396$	0	0
$397$	16.0913	0.807599	0.403800	−	0.914847i	$-0.367689\pi$
$397$	16.0913	0.807599	0.403800	+	0.914847i	$0.367689\pi$
$398$	0	0
$399$	−15.1043	−0.756162
$400$	0	0
$401$	−10.1475	−0.506741	−0.253370	−	0.967369i	$-0.581539\pi$
$401$	−10.1475	−0.506741	−0.253370	+	0.967369i	$0.581539\pi$
$402$	0	0
$403$	6.18804	0.308248
$404$	0	0
$405$	0.273378	0.0135843
$406$	0	0
$407$	−26.9563	−1.33618
$408$	0	0
$409$	−33.1313	−1.63824	−0.819120	−	0.573622i	$-0.805538\pi$
$409$	−33.1313	−1.63824	−0.819120	+	0.573622i	$0.805538\pi$
$410$	0	0
$411$	2.12236	0.104689
$412$	0	0
$413$	3.90273	0.192041
$414$	0	0
$415$	3.99695	0.196203
$416$	0	0
$417$	20.7577	1.01651
$418$	0	0
$419$	−27.3449	−1.33589	−0.667943	−	0.744212i	$-0.732825\pi$
$419$	−27.3449	−1.33589	−0.667943	+	0.744212i	$0.732825\pi$
$420$	0	0
$421$	10.3373	0.503808	0.251904	−	0.967752i	$-0.418943\pi$
$421$	10.3373	0.503808	0.251904	+	0.967752i	$0.418943\pi$
$422$	0	0
$423$	−9.18777	−0.446725
$424$	0	0
$425$	−0.294701	−0.0142951
$426$	0	0
$427$	−14.0144	−0.678204
$428$	0	0
$429$	−6.75093	−0.325938
$430$	0	0
$431$	35.5041	1.71017	0.855086	−	0.518487i	$-0.173504\pi$
$431$	35.5041	1.71017	0.855086	+	0.518487i	$0.173504\pi$
$432$	0	0
$433$	34.1474	1.64102	0.820510	−	0.571632i	$-0.193689\pi$
$433$	34.1474	1.64102	0.820510	+	0.571632i	$0.193689\pi$
$434$	0	0
$435$	0.326198	0.0156400
$436$	0	0
$437$	2.99331	0.143190
$438$	0	0
$439$	−31.0389	−1.48141	−0.740703	−	0.671833i	$-0.765507\pi$
$439$	−31.0389	−1.48141	−0.740703	+	0.671833i	$0.765507\pi$
$440$	0	0
$441$	1.57320	0.0749141
$442$	0	0
$443$	−13.7063	−0.651206	−0.325603	−	0.945507i	$-0.605567\pi$
$443$	−13.7063	−0.651206	−0.325603	+	0.945507i	$0.605567\pi$
$444$	0	0
$445$	2.33980	0.110917
$446$	0	0
$447$	12.3544	0.584344
$448$	0	0
$449$	−4.02929	−0.190154	−0.0950769	−	0.995470i	$-0.530310\pi$
$449$	−4.02929	−0.190154	−0.0950769	+	0.995470i	$0.530310\pi$
$450$	0	0
$451$	31.7751	1.49623
$452$	0	0
$453$	−7.37665	−0.346585
$454$	0	0
$455$	−1.34396	−0.0630059
$456$	0	0
$457$	−28.6333	−1.33941	−0.669705	−	0.742627i	$-0.733579\pi$
$457$	−28.6333	−1.33941	−0.669705	+	0.742627i	$0.733579\pi$
$458$	0	0
$459$	0.0598345	0.00279284
$460$	0	0
$461$	26.5485	1.23649	0.618244	−	0.785986i	$-0.287844\pi$
$461$	26.5485	1.23649	0.618244	+	0.785986i	$0.287844\pi$
$462$	0	0
$463$	−14.8976	−0.692349	−0.346174	−	0.938170i	$-0.612519\pi$
$463$	−14.8976	−0.692349	−0.346174	+	0.938170i	$0.612519\pi$
$464$	0	0
$465$	1.00755	0.0467240
$466$	0	0
$467$	−33.9886	−1.57281	−0.786403	−	0.617713i	$-0.788059\pi$
$467$	−33.9886	−1.57281	−0.786403	+	0.617713i	$0.788059\pi$
$468$	0	0
$469$	16.6187	0.767382
$470$	0	0
$471$	23.5265	1.08404
$472$	0	0
$473$	−31.5392	−1.45017
$474$	0	0
$475$	−25.4074	−1.16577
$476$	0	0
$477$	7.76336	0.355460
$478$	0	0
$479$	33.4738	1.52946	0.764729	−	0.644352i	$-0.222873\pi$
$479$	33.4738	1.52946	0.764729	+	0.644352i	$0.222873\pi$
$480$	0	0
$481$	11.2564	0.513248
$482$	0	0
$483$	−1.69900	−0.0773072
$484$	0	0
$485$	−0.265292	−0.0120463
$486$	0	0
$487$	−17.7429	−0.804007	−0.402004	−	0.915638i	$-0.631686\pi$
$487$	−17.7429	−0.804007	−0.402004	+	0.915638i	$0.631686\pi$
$488$	0	0
$489$	16.7815	0.758884
$490$	0	0
$491$	−28.3266	−1.27836	−0.639181	−	0.769057i	$-0.720726\pi$
$491$	−28.3266	−1.27836	−0.639181	+	0.769057i	$0.720726\pi$
$492$	0	0
$493$	0.0713951	0.00321547
$494$	0	0
$495$	−1.09920	−0.0494053
$496$	0	0
$497$	16.5263	0.741307
$498$	0	0
$499$	18.9138	0.846699	0.423349	−	0.905967i	$-0.360854\pi$
$499$	18.9138	0.846699	0.423349	+	0.905967i	$0.360854\pi$
$500$	0	0
$501$	15.1345	0.676159
$502$	0	0
$503$	−14.0032	−0.624370	−0.312185	−	0.950021i	$-0.601061\pi$
$503$	−14.0032	−0.624370	−0.312185	+	0.950021i	$0.601061\pi$
$504$	0	0
$505$	1.80132	0.0801578
$506$	0	0
$507$	−10.1810	−0.452152
$508$	0	0
$509$	1.95345	0.0865853	0.0432927	−	0.999062i	$-0.486215\pi$
$509$	1.95345	0.0865853	0.0432927	+	0.999062i	$0.486215\pi$
$510$	0	0
$511$	−32.9868	−1.45925
$512$	0	0
$513$	5.15858	0.227757
$514$	0	0
$515$	−0.434182	−0.0191323
$516$	0	0
$517$	36.9422	1.62471
$518$	0	0
$519$	−6.93938	−0.304605
$520$	0	0
$521$	10.1940	0.446607	0.223304	−	0.974749i	$-0.428316\pi$
$521$	10.1940	0.446607	0.223304	+	0.974749i	$0.428316\pi$
$522$	0	0
$523$	25.8449	1.13012	0.565060	−	0.825050i	$-0.308853\pi$
$523$	25.8449	1.13012	0.565060	+	0.825050i	$0.308853\pi$
$524$	0	0
$525$	14.4212	0.629392
$526$	0	0
$527$	0.220523	0.00960613
$528$	0	0
$529$	−22.6633	−0.985361
$530$	0	0
$531$	−1.33290	−0.0578430
$532$	0	0
$533$	−13.2686	−0.574728
$534$	0	0
$535$	2.97887	0.128788
$536$	0	0
$537$	8.11738	0.350291
$538$	0	0
$539$	−6.32550	−0.272459
$540$	0	0
$541$	33.0564	1.42120	0.710602	−	0.703594i	$-0.248423\pi$
$541$	33.0564	1.42120	0.710602	+	0.703594i	$0.248423\pi$
$542$	0	0
$543$	12.6503	0.542875
$544$	0	0
$545$	0.784851	0.0336193
$546$	0	0
$547$	−10.3058	−0.440642	−0.220321	−	0.975427i	$-0.570711\pi$
$547$	−10.3058	−0.440642	−0.220321	+	0.975427i	$0.570711\pi$
$548$	0	0
$549$	4.78634	0.204276
$550$	0	0
$551$	6.15526	0.262223
$552$	0	0
$553$	41.4557	1.76288
$554$	0	0
$555$	1.83279	0.0777976
$556$	0	0
$557$	−21.6427	−0.917032	−0.458516	−	0.888686i	$-0.651619\pi$
$557$	−21.6427	−0.917032	−0.458516	+	0.888686i	$0.651619\pi$
$558$	0	0
$559$	13.1701	0.557036
$560$	0	0
$561$	−0.240582	−0.0101574
$562$	0	0
$563$	26.4559	1.11498	0.557492	−	0.830183i	$-0.311764\pi$
$563$	26.4559	1.11498	0.557492	+	0.830183i	$0.311764\pi$
$564$	0	0
$565$	5.37536	0.226143
$566$	0	0
$567$	−2.92800	−0.122965
$568$	0	0
$569$	4.59646	0.192694	0.0963468	−	0.995348i	$-0.469284\pi$
$569$	4.59646	0.192694	0.0963468	+	0.995348i	$0.469284\pi$
$570$	0	0
$571$	25.6382	1.07293	0.536464	−	0.843923i	$-0.319760\pi$
$571$	25.6382	1.07293	0.536464	+	0.843923i	$0.319760\pi$
$572$	0	0
$573$	22.1414	0.924971
$574$	0	0
$575$	−2.85793	−0.119184
$576$	0	0
$577$	23.3980	0.974071	0.487036	−	0.873382i	$-0.338078\pi$
$577$	23.3980	0.974071	0.487036	+	0.873382i	$0.338078\pi$
$578$	0	0
$579$	22.1147	0.919054
$580$	0	0
$581$	−42.8091	−1.77602
$582$	0	0
$583$	−31.2149	−1.29279
$584$	0	0
$585$	0.459003	0.0189774
$586$	0	0
$587$	−33.5332	−1.38406	−0.692032	−	0.721866i	$-0.743284\pi$
$587$	−33.5332	−1.38406	−0.692032	+	0.721866i	$0.743284\pi$
$588$	0	0
$589$	19.0122	0.783383
$590$	0	0
$591$	−6.07138	−0.249743
$592$	0	0
$593$	−0.284537	−0.0116845	−0.00584227	−	0.999983i	$-0.501860\pi$
$593$	−0.284537	−0.0116845	−0.00584227	+	0.999983i	$0.501860\pi$
$594$	0	0
$595$	−0.0478947	−0.00196349
$596$	0	0
$597$	−3.73415	−0.152829
$598$	0	0
$599$	5.11606	0.209036	0.104518	−	0.994523i	$-0.466670\pi$
$599$	5.11606	0.209036	0.104518	+	0.994523i	$0.466670\pi$
$600$	0	0
$601$	12.7038	0.518197	0.259099	−	0.965851i	$-0.416574\pi$
$601$	12.7038	0.518197	0.259099	+	0.965851i	$0.416574\pi$
$602$	0	0
$603$	−5.67579	−0.231136
$604$	0	0
$605$	1.41250	0.0574261
$606$	0	0
$607$	44.5992	1.81022	0.905112	−	0.425174i	$-0.139787\pi$
$607$	44.5992	1.81022	0.905112	+	0.425174i	$0.139787\pi$
$608$	0	0
$609$	−3.49372	−0.141573
$610$	0	0
$611$	−15.4263	−0.624081
$612$	0	0
$613$	9.31182	0.376101	0.188051	−	0.982159i	$-0.439783\pi$
$613$	9.31182	0.376101	0.188051	+	0.982159i	$0.439783\pi$
$614$	0	0
$615$	−2.16042	−0.0871167
$616$	0	0
$617$	−9.17213	−0.369256	−0.184628	−	0.982808i	$-0.559108\pi$
$617$	−9.17213	−0.369256	−0.184628	+	0.982808i	$0.559108\pi$
$618$	0	0
$619$	−39.2479	−1.57751	−0.788753	−	0.614710i	$-0.789273\pi$
$619$	−39.2479	−1.57751	−0.788753	+	0.614710i	$0.789273\pi$
$620$	0	0
$621$	0.580259	0.0232850
$622$	0	0
$623$	−25.0602	−1.00402
$624$	0	0
$625$	23.8845	0.955382
$626$	0	0
$627$	−20.7416	−0.828340
$628$	0	0
$629$	0.401144	0.0159947
$630$	0	0
$631$	−27.3279	−1.08791	−0.543954	−	0.839115i	$-0.683073\pi$
$631$	−27.3279	−1.08791	−0.543954	+	0.839115i	$0.683073\pi$
$632$	0	0
$633$	−19.2956	−0.766929
$634$	0	0
$635$	1.32674	0.0526499
$636$	0	0
$637$	2.64140	0.104656
$638$	0	0
$639$	−5.64423	−0.223282
$640$	0	0
$641$	−20.9654	−0.828083	−0.414042	−	0.910258i	$-0.635883\pi$
$641$	−20.9654	−0.828083	−0.414042	+	0.910258i	$0.635883\pi$
$642$	0	0
$643$	−33.2335	−1.31060	−0.655301	−	0.755368i	$-0.727458\pi$
$643$	−33.2335	−1.31060	−0.655301	+	0.755368i	$0.727458\pi$
$644$	0	0
$645$	2.14438	0.0844350
$646$	0	0
$647$	−34.8469	−1.36997	−0.684986	−	0.728556i	$-0.740192\pi$
$647$	−34.8469	−1.36997	−0.684986	+	0.728556i	$0.740192\pi$
$648$	0	0
$649$	5.35932	0.210372
$650$	0	0
$651$	−10.7913	−0.422944
$652$	0	0
$653$	3.03152	0.118633	0.0593163	−	0.998239i	$-0.481108\pi$
$653$	3.03152	0.118633	0.0593163	+	0.998239i	$0.481108\pi$
$654$	0	0
$655$	−5.58893	−0.218378
$656$	0	0
$657$	11.2660	0.439528
$658$	0	0
$659$	−33.6696	−1.31158	−0.655791	−	0.754942i	$-0.727665\pi$
$659$	−33.6696	−1.31158	−0.655791	+	0.754942i	$0.727665\pi$
$660$	0	0
$661$	−19.5782	−0.761504	−0.380752	−	0.924677i	$-0.624335\pi$
$661$	−19.5782	−0.761504	−0.380752	+	0.924677i	$0.624335\pi$
$662$	0	0
$663$	0.100462	0.00390163
$664$	0	0
$665$	−4.12920	−0.160123
$666$	0	0
$667$	0.692371	0.0268087
$668$	0	0
$669$	22.5610	0.872257
$670$	0	0
$671$	−19.2449	−0.742941
$672$	0	0
$673$	35.8700	1.38269	0.691344	−	0.722525i	$-0.257019\pi$
$673$	35.8700	1.38269	0.691344	+	0.722525i	$0.257019\pi$
$674$	0	0
$675$	−4.92526	−0.189574
$676$	0	0
$677$	−41.0685	−1.57839	−0.789196	−	0.614142i	$-0.789502\pi$
$677$	−41.0685	−1.57839	−0.789196	+	0.614142i	$0.789502\pi$
$678$	0	0
$679$	2.84139	0.109043
$680$	0	0
$681$	−27.0520	−1.03663
$682$	0	0
$683$	50.5296	1.93346	0.966730	−	0.255800i	$-0.0823387\pi$
$683$	50.5296	1.93346	0.966730	+	0.255800i	$0.0823387\pi$
$684$	0	0
$685$	0.580209	0.0221686
$686$	0	0
$687$	26.1271	0.996811
$688$	0	0
$689$	13.0347	0.496582
$690$	0	0
$691$	18.3109	0.696581	0.348291	−	0.937387i	$-0.386762\pi$
$691$	18.3109	0.696581	0.348291	+	0.937387i	$0.386762\pi$
$692$	0	0
$693$	11.7729	0.447216
$694$	0	0
$695$	5.67470	0.215254
$696$	0	0
$697$	−0.472853	−0.0179106
$698$	0	0
$699$	−16.1916	−0.612422
$700$	0	0
$701$	−15.4600	−0.583918	−0.291959	−	0.956431i	$-0.594307\pi$
$701$	−15.4600	−0.583918	−0.291959	+	0.956431i	$0.594307\pi$
$702$	0	0
$703$	34.5843	1.30437
$704$	0	0
$705$	−2.51174	−0.0945975
$706$	0	0
$707$	−19.2929	−0.725586
$708$	0	0
$709$	35.9051	1.34844	0.674222	−	0.738529i	$-0.264479\pi$
$709$	35.9051	1.34844	0.674222	+	0.738529i	$0.264479\pi$
$710$	0	0
$711$	−14.1584	−0.530980
$712$	0	0
$713$	2.13857	0.0800901
$714$	0	0
$715$	−1.84556	−0.0690199
$716$	0	0
$717$	21.9328	0.819095
$718$	0	0
$719$	−4.17950	−0.155869	−0.0779346	−	0.996958i	$-0.524833\pi$
$719$	−4.17950	−0.155869	−0.0779346	+	0.996958i	$0.524833\pi$
$720$	0	0
$721$	4.65028	0.173185
$722$	0	0
$723$	30.3354	1.12818
$724$	0	0
$725$	−5.87687	−0.218262
$726$	0	0
$727$	−30.9772	−1.14888	−0.574439	−	0.818547i	$-0.694780\pi$
$727$	−30.9772	−1.14888	−0.574439	+	0.818547i	$0.694780\pi$
$728$	0	0
$729$	1.00000	0.0370370
$730$	0	0
$731$	0.469342	0.0173593
$732$	0	0
$733$	−1.02884	−0.0380011	−0.0190005	−	0.999819i	$-0.506048\pi$
$733$	−1.02884	−0.0380011	−0.0190005	+	0.999819i	$0.506048\pi$
$734$	0	0
$735$	0.430078	0.0158637
$736$	0	0
$737$	22.8212	0.840630
$738$	0	0
$739$	26.2727	0.966456	0.483228	−	0.875495i	$-0.339464\pi$
$739$	26.2727	0.966456	0.483228	+	0.875495i	$0.339464\pi$
$740$	0	0
$741$	8.66126	0.318179
$742$	0	0
$743$	−12.6569	−0.464338	−0.232169	−	0.972675i	$-0.574582\pi$
$743$	−12.6569	−0.464338	−0.232169	+	0.972675i	$0.574582\pi$
$744$	0	0
$745$	3.37743	0.123740
$746$	0	0
$747$	14.6206	0.534939
$748$	0	0
$749$	−31.9050	−1.16578
$750$	0	0
$751$	26.1446	0.954030	0.477015	−	0.878895i	$-0.341719\pi$
$751$	26.1446	0.954030	0.477015	+	0.878895i	$0.341719\pi$
$752$	0	0
$753$	−1.00000	−0.0364420
$754$	0	0
$755$	−2.01662	−0.0733922
$756$	0	0
$757$	−28.2286	−1.02599	−0.512994	−	0.858392i	$-0.671464\pi$
$757$	−28.2286	−1.02599	−0.512994	+	0.858392i	$0.671464\pi$
$758$	0	0
$759$	−2.33310	−0.0846863
$760$	0	0
$761$	−32.8095	−1.18934	−0.594671	−	0.803969i	$-0.702718\pi$
$761$	−32.8095	−1.18934	−0.594671	+	0.803969i	$0.702718\pi$
$762$	0	0
$763$	−8.40610	−0.304321
$764$	0	0
$765$	0.0163575	0.000591405 0
$766$	0	0
$767$	−2.23794	−0.0808074
$768$	0	0
$769$	0.625482	0.0225555	0.0112777	−	0.999936i	$-0.496410\pi$
$769$	0.625482	0.0225555	0.0112777	+	0.999936i	$0.496410\pi$
$770$	0	0
$771$	−12.4119	−0.447004
$772$	0	0
$773$	45.5342	1.63775	0.818875	−	0.573971i	$-0.194598\pi$
$773$	45.5342	1.63775	0.818875	+	0.573971i	$0.194598\pi$
$774$	0	0
$775$	−18.1523	−0.652050
$776$	0	0
$777$	−19.6300	−0.704222
$778$	0	0
$779$	−40.7666	−1.46062
$780$	0	0
$781$	22.6943	0.812067
$782$	0	0
$783$	1.19321	0.0426418
$784$	0	0
$785$	6.43163	0.229555
$786$	0	0
$787$	8.35613	0.297864	0.148932	−	0.988847i	$-0.452416\pi$
$787$	8.35613	0.297864	0.148932	+	0.988847i	$0.452416\pi$
$788$	0	0
$789$	2.34910	0.0836302
$790$	0	0
$791$	−57.5725	−2.04704
$792$	0	0
$793$	8.03626	0.285376
$794$	0	0
$795$	2.12233	0.0752715
$796$	0	0
$797$	−24.6794	−0.874190	−0.437095	−	0.899415i	$-0.643993\pi$
$797$	−24.6794	−0.874190	−0.437095	+	0.899415i	$0.643993\pi$
$798$	0	0
$799$	−0.549746	−0.0194486
$800$	0	0
$801$	8.55882	0.302411
$802$	0	0
$803$	−45.2982	−1.59854
$804$	0	0
$805$	−0.464470	−0.0163704
$806$	0	0
$807$	20.5512	0.723436
$808$	0	0
$809$	1.03261	0.0363048	0.0181524	−	0.999835i	$-0.494222\pi$
$809$	1.03261	0.0363048	0.0181524	+	0.999835i	$0.494222\pi$
$810$	0	0
$811$	8.18696	0.287483	0.143742	−	0.989615i	$-0.454087\pi$
$811$	8.18696	0.287483	0.143742	+	0.989615i	$0.454087\pi$
$812$	0	0
$813$	−32.5662	−1.14215
$814$	0	0
$815$	4.58769	0.160700
$816$	0	0
$817$	40.4639	1.41565
$818$	0	0
$819$	−4.91612	−0.171783
$820$	0	0
$821$	47.7323	1.66587	0.832934	−	0.553372i	$-0.186659\pi$
$821$	47.7323	1.66587	0.832934	+	0.553372i	$0.186659\pi$
$822$	0	0
$823$	20.7223	0.722333	0.361167	−	0.932501i	$-0.382379\pi$
$823$	20.7223	0.722333	0.361167	+	0.932501i	$0.382379\pi$
$824$	0	0
$825$	19.8035	0.689469
$826$	0	0
$827$	10.0235	0.348552	0.174276	−	0.984697i	$-0.444241\pi$
$827$	10.0235	0.348552	0.174276	+	0.984697i	$0.444241\pi$
$828$	0	0
$829$	21.5564	0.748685	0.374342	−	0.927291i	$-0.377868\pi$
$829$	21.5564	0.748685	0.374342	+	0.927291i	$0.377868\pi$
$830$	0	0
$831$	14.8348	0.514614
$832$	0	0
$833$	0.0941313	0.00326146
$834$	0	0
$835$	4.13744	0.143182
$836$	0	0
$837$	3.68555	0.127391
$838$	0	0
$839$	38.9167	1.34355	0.671777	−	0.740753i	$-0.265531\pi$
$839$	38.9167	1.34355	0.671777	+	0.740753i	$0.265531\pi$
$840$	0	0
$841$	−27.5763	−0.950905
$842$	0	0
$843$	5.97420	0.205763
$844$	0	0
$845$	−2.78325	−0.0957468
$846$	0	0
$847$	−15.1284	−0.519819
$848$	0	0
$849$	2.54849	0.0874641
$850$	0	0
$851$	3.89019	0.133354
$852$	0	0
$853$	−39.6941	−1.35910	−0.679550	−	0.733630i	$-0.737825\pi$
$853$	−39.6941	−1.35910	−0.679550	+	0.733630i	$0.737825\pi$
$854$	0	0
$855$	1.41024	0.0482293
$856$	0	0
$857$	−31.0574	−1.06090	−0.530449	−	0.847717i	$-0.677977\pi$
$857$	−31.0574	−1.06090	−0.530449	+	0.847717i	$0.677977\pi$
$858$	0	0
$859$	−9.85286	−0.336175	−0.168088	−	0.985772i	$-0.553759\pi$
$859$	−9.85286	−0.336175	−0.168088	+	0.985772i	$0.553759\pi$
$860$	0	0
$861$	23.1391	0.788578
$862$	0	0
$863$	−38.8130	−1.32121	−0.660605	−	0.750733i	$-0.729700\pi$
$863$	−38.8130	−1.32121	−0.660605	+	0.750733i	$0.729700\pi$
$864$	0	0
$865$	−1.89708	−0.0645026
$866$	0	0
$867$	−16.9964	−0.577229
$868$	0	0
$869$	56.9279	1.93115
$870$	0	0
$871$	−9.52966	−0.322900
$872$	0	0
$873$	−0.970420	−0.0328437
$874$	0	0
$875$	7.94470	0.268580
$876$	0	0
$877$	25.6729	0.866912	0.433456	−	0.901175i	$-0.357294\pi$
$877$	25.6729	0.866912	0.433456	+	0.901175i	$0.357294\pi$
$878$	0	0
$879$	−12.5610	−0.423672
$880$	0	0
$881$	−48.1156	−1.62106	−0.810528	−	0.585700i	$-0.800820\pi$
$881$	−48.1156	−1.62106	−0.810528	+	0.585700i	$0.800820\pi$
$882$	0	0
$883$	13.1614	0.442917	0.221459	−	0.975170i	$-0.428918\pi$
$883$	13.1614	0.442917	0.221459	+	0.975170i	$0.428918\pi$
$884$	0	0
$885$	−0.364386	−0.0122487
$886$	0	0
$887$	−8.87248	−0.297909	−0.148954	−	0.988844i	$-0.547591\pi$
$887$	−8.87248	−0.297909	−0.148954	+	0.988844i	$0.547591\pi$
$888$	0	0
$889$	−14.2099	−0.476586
$890$	0	0
$891$	−4.02080	−0.134702
$892$	0	0
$893$	−47.3958	−1.58604
$894$	0	0
$895$	2.21912	0.0741769
$896$	0	0
$897$	0.974256	0.0325295
$898$	0	0
$899$	4.39763	0.146669
$900$	0	0
$901$	0.464517	0.0154753
$902$	0	0
$903$	−22.9673	−0.764303
$904$	0	0
$905$	3.45831	0.114958
$906$	0	0
$907$	3.64725	0.121105	0.0605525	−	0.998165i	$-0.480714\pi$
$907$	3.64725	0.121105	0.0605525	+	0.998165i	$0.480714\pi$
$908$	0	0
$909$	6.58911	0.218547
$910$	0	0
$911$	46.8159	1.55108	0.775540	−	0.631298i	$-0.217477\pi$
$911$	46.8159	1.55108	0.775540	+	0.631298i	$0.217477\pi$
$912$	0	0
$913$	−58.7864	−1.94555
$914$	0	0
$915$	1.30848	0.0432570
$916$	0	0
$917$	59.8599	1.97675
$918$	0	0
$919$	24.2301	0.799277	0.399639	−	0.916673i	$-0.369136\pi$
$919$	24.2301	0.799277	0.399639	+	0.916673i	$0.369136\pi$
$920$	0	0
$921$	−13.3645	−0.440376
$922$	0	0
$923$	−9.47668	−0.311929
$924$	0	0
$925$	−33.0201	−1.08569
$926$	0	0
$927$	−1.58821	−0.0521636
$928$	0	0
$929$	−32.9292	−1.08037	−0.540186	−	0.841545i	$-0.681646\pi$
$929$	−32.9292	−1.08037	−0.540186	+	0.841545i	$0.681646\pi$
$930$	0	0
$931$	8.11545	0.265973
$932$	0	0
$933$	−15.3964	−0.504055
$934$	0	0
$935$	−0.0657700	−0.00215091
$936$	0	0
$937$	41.5772	1.35827	0.679135	−	0.734014i	$-0.262355\pi$
$937$	41.5772	1.35827	0.679135	+	0.734014i	$0.262355\pi$
$938$	0	0
$939$	5.54298	0.180888
$940$	0	0
$941$	45.9889	1.49919	0.749597	−	0.661894i	$-0.230247\pi$
$941$	45.9889	1.49919	0.749597	+	0.661894i	$0.230247\pi$
$942$	0	0
$943$	−4.58561	−0.149328
$944$	0	0
$945$	−0.800453	−0.0260387
$946$	0	0
$947$	49.2229	1.59953	0.799765	−	0.600313i	$-0.204957\pi$
$947$	49.2229	1.59953	0.799765	+	0.600313i	$0.204957\pi$
$948$	0	0
$949$	18.9156	0.614026
$950$	0	0
$951$	−10.5891	−0.343375
$952$	0	0
$953$	−7.82623	−0.253517	−0.126758	−	0.991934i	$-0.540457\pi$
$953$	−7.82623	−0.253517	−0.126758	+	0.991934i	$0.540457\pi$
$954$	0	0
$955$	6.05299	0.195870
$956$	0	0
$957$	−4.79765	−0.155086
$958$	0	0
$959$	−6.21429	−0.200670
$960$	0	0
$961$	−17.4167	−0.561831
$962$	0	0
$963$	10.8965	0.351136
$964$	0	0
$965$	6.04567	0.194617
$966$	0	0
$967$	−31.6763	−1.01864	−0.509321	−	0.860577i	$-0.670103\pi$
$967$	−31.6763	−1.01864	−0.509321	+	0.860577i	$0.670103\pi$
$968$	0	0
$969$	0.308661	0.00991562
$970$	0	0
$971$	−3.48585	−0.111866	−0.0559332	−	0.998435i	$-0.517813\pi$
$971$	−3.48585	−0.111866	−0.0559332	+	0.998435i	$0.517813\pi$
$972$	0	0
$973$	−60.7785	−1.94847
$974$	0	0
$975$	−8.26953	−0.264837
$976$	0	0
$977$	−23.5684	−0.754021	−0.377010	−	0.926209i	$-0.623048\pi$
$977$	−23.5684	−0.754021	−0.377010	+	0.926209i	$0.623048\pi$
$978$	0	0
$979$	−34.4133	−1.09985
$980$	0	0
$981$	2.87093	0.0916618
$982$	0	0
$983$	46.7920	1.49243	0.746216	−	0.665704i	$-0.231868\pi$
$983$	46.7920	1.49243	0.746216	+	0.665704i	$0.231868\pi$
$984$	0	0
$985$	−1.65978	−0.0528851
$986$	0	0
$987$	26.9018	0.856294
$988$	0	0
$989$	4.55156	0.144731
$990$	0	0
$991$	12.1691	0.386563	0.193281	−	0.981143i	$-0.438087\pi$
$991$	12.1691	0.386563	0.193281	+	0.981143i	$0.438087\pi$
$992$	0	0
$993$	−21.6567	−0.687254
$994$	0	0
$995$	−1.02084	−0.0323627
$996$	0	0
$997$	43.1452	1.36642	0.683212	−	0.730220i	$-0.260583\pi$
$997$	43.1452	1.36642	0.683212	+	0.730220i	$0.260583\pi$
$998$	0	0
$999$	6.70422	0.212112

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	6024.2.a.r.1.11	✓	20

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
6024.2.a.r.1.11	✓	20	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:	\( N \)	\(=\)	\( 6024 = 2^{3} \cdot 3 \cdot 251 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	6024.a (trivial)

\(f(q)\)	\(=\)	\(q+1.00000 q^{3} +0.273378 q^{5} -2.92800 q^{7} +1.00000 q^{9} +O(q^{10})\)	\(q+1.00000 q^{3} +0.273378 q^{5} -2.92800 q^{7} +1.00000 q^{9} -4.02080 q^{11} +1.67900 q^{13} +0.273378 q^{15} +0.0598345 q^{17} +5.15858 q^{19} -2.92800 q^{21} +0.580259 q^{23} -4.92526 q^{25} +1.00000 q^{27} +1.19321 q^{29} +3.68555 q^{31} -4.02080 q^{33} -0.800453 q^{35} +6.70422 q^{37} +1.67900 q^{39} -7.90269 q^{41} +7.84401 q^{43} +0.273378 q^{45} -9.18777 q^{47} +1.57320 q^{49} +0.0598345 q^{51} +7.76336 q^{53} -1.09920 q^{55} +5.15858 q^{57} -1.33290 q^{59} +4.78634 q^{61} -2.92800 q^{63} +0.459003 q^{65} -5.67579 q^{67} +0.580259 q^{69} -5.64423 q^{71} +11.2660 q^{73} -4.92526 q^{75} +11.7729 q^{77} -14.1584 q^{79} +1.00000 q^{81} +14.6206 q^{83} +0.0163575 q^{85} +1.19321 q^{87} +8.55882 q^{89} -4.91612 q^{91} +3.68555 q^{93} +1.41024 q^{95} -0.970420 q^{97} -4.02080 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 20 q + 20 q^{3} + 9 q^{5} + 9 q^{7} + 20 q^{9}+O(q^{10}) \) 20 * q + 20 * q^3 + 9 * q^5 + 9 * q^7 + 20 * q^9	\( 20 q + 20 q^{3} + 9 q^{5} + 9 q^{7} + 20 q^{9} + 4 q^{11} + 21 q^{13} + 9 q^{15} + 10 q^{17} + 8 q^{19} + 9 q^{21} + 9 q^{23} + 43 q^{25} + 20 q^{27} + 18 q^{29} + 27 q^{31} + 4 q^{33} - 7 q^{35} + 33 q^{37} + 21 q^{39} + 14 q^{41} - 6 q^{43} + 9 q^{45} + 21 q^{47} + 47 q^{49} + 10 q^{51} + 23 q^{53} + 24 q^{55} + 8 q^{57} + 10 q^{59} + 28 q^{61} + 9 q^{63} + 2 q^{65} + 15 q^{67} + 9 q^{69} + 33 q^{71} + 50 q^{73} + 43 q^{75} + 20 q^{77} + 17 q^{79} + 20 q^{81} - 19 q^{83} + 41 q^{85} + 18 q^{87} + 21 q^{89} + 30 q^{91} + 27 q^{93} + 27 q^{95} + 47 q^{97} + 4 q^{99}+O(q^{100}) \) 20 * q + 20 * q^3 + 9 * q^5 + 9 * q^7 + 20 * q^9 + 4 * q^11 + 21 * q^13 + 9 * q^15 + 10 * q^17 + 8 * q^19 + 9 * q^21 + 9 * q^23 + 43 * q^25 + 20 * q^27 + 18 * q^29 + 27 * q^31 + 4 * q^33 - 7 * q^35 + 33 * q^37 + 21 * q^39 + 14 * q^41 - 6 * q^43 + 9 * q^45 + 21 * q^47 + 47 * q^49 + 10 * q^51 + 23 * q^53 + 24 * q^55 + 8 * q^57 + 10 * q^59 + 28 * q^61 + 9 * q^63 + 2 * q^65 + 15 * q^67 + 9 * q^69 + 33 * q^71 + 50 * q^73 + 43 * q^75 + 20 * q^77 + 17 * q^79 + 20 * q^81 - 19 * q^83 + 41 * q^85 + 18 * q^87 + 21 * q^89 + 30 * q^91 + 27 * q^93 + 27 * q^95 + 47 * q^97 + 4 * q^99

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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