LMFDB - Embedded newform 6024.2.a.p.1.13

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:	$14$
Coefficient field:	$\mathbb{Q}[x]/(x^{14} - \cdots)$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{14} - 6 x^{13} - 29 x^{12} + 210 x^{11} + 280 x^{10} - 2796 x^{9} - 863 x^{8} + 17652 x^{7} + \cdots - 3364 $ x^14 - 6x^13 - 29x^12 + 210x^11 + 280x^10 - 2796x^9 - 863x^8 + 17652x^7 - 1300x^6 - 53458x^5 + 10892x^4 + 67298x^3 - 13761x^2 - 22140*x - 3364
Coefficient ring:	$\Z[a_1, \ldots, a_{19}]$
Coefficient ring index:	$ 2 $
Twist minimal:	yes
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.13
Root			$3.63608$ 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)

$f(q)$	$=$	$q-1.00000 q^{3} +3.63608 q^{5} +2.74254 q^{7} +1.00000 q^{9} +O(q^{10})$	$q-1.00000 q^{3} +3.63608 q^{5} +2.74254 q^{7} +1.00000 q^{9} +3.30758 q^{11} +5.31864 q^{13} -3.63608 q^{15} +3.11451 q^{17} -4.91358 q^{19} -2.74254 q^{21} -1.16838 q^{23} +8.22110 q^{25} -1.00000 q^{27} -3.53214 q^{29} +2.72293 q^{31} -3.30758 q^{33} +9.97209 q^{35} +2.05276 q^{37} -5.31864 q^{39} +4.50924 q^{41} +3.85770 q^{43} +3.63608 q^{45} +3.91425 q^{47} +0.521509 q^{49} -3.11451 q^{51} +9.24140 q^{53} +12.0266 q^{55} +4.91358 q^{57} -8.59898 q^{59} +2.00967 q^{61} +2.74254 q^{63} +19.3390 q^{65} +5.49320 q^{67} +1.16838 q^{69} +4.41623 q^{71} -5.91685 q^{73} -8.22110 q^{75} +9.07115 q^{77} -16.0135 q^{79} +1.00000 q^{81} -14.2902 q^{83} +11.3246 q^{85} +3.53214 q^{87} -11.4711 q^{89} +14.5866 q^{91} -2.72293 q^{93} -17.8662 q^{95} +7.25459 q^{97} +3.30758 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 14 q - 14 q^{3} + 6 q^{5} + 3 q^{7} + 14 q^{9}+O(q^{10}) $ 14 * q - 14 * q^3 + 6 * q^5 + 3 * q^7 + 14 * q^9	$ 14 q - 14 q^{3} + 6 q^{5} + 3 q^{7} + 14 q^{9} + 7 q^{11} + 5 q^{13} - 6 q^{15} + 28 q^{17} - q^{19} - 3 q^{21} - 3 q^{23} + 24 q^{25} - 14 q^{27} + 15 q^{29} + 9 q^{31} - 7 q^{33} - 14 q^{35} + 4 q^{37} - 5 q^{39} + 32 q^{41} - 3 q^{43} + 6 q^{45} + 15 q^{47} + 27 q^{49} - 28 q^{51} + 8 q^{53} + q^{57} - 11 q^{59} + 25 q^{61} + 3 q^{63} + 18 q^{65} - 20 q^{67} + 3 q^{69} + 33 q^{71} + 8 q^{73} - 24 q^{75} + 14 q^{77} - 17 q^{79} + 14 q^{81} - 4 q^{83} + 16 q^{85} - 15 q^{87} + 14 q^{89} - 28 q^{91} - 9 q^{93} + 3 q^{95} + 9 q^{97} + 7 q^{99}+O(q^{100}) $ 14 * q - 14 * q^3 + 6 * q^5 + 3 * q^7 + 14 * q^9 + 7 * q^11 + 5 * q^13 - 6 * q^15 + 28 * q^17 - q^19 - 3 * q^21 - 3 * q^23 + 24 * q^25 - 14 * q^27 + 15 * q^29 + 9 * q^31 - 7 * q^33 - 14 * q^35 + 4 * q^37 - 5 * q^39 + 32 * q^41 - 3 * q^43 + 6 * q^45 + 15 * q^47 + 27 * q^49 - 28 * q^51 + 8 * q^53 + q^57 - 11 * q^59 + 25 * q^61 + 3 * q^63 + 18 * q^65 - 20 * q^67 + 3 * q^69 + 33 * q^71 + 8 * q^73 - 24 * q^75 + 14 * q^77 - 17 * q^79 + 14 * q^81 - 4 * q^83 + 16 * q^85 - 15 * q^87 + 14 * q^89 - 28 * q^91 - 9 * q^93 + 3 * q^95 + 9 * q^97 + 7 * 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
$5$	3.63608	1.62611	0.813053	−	0.582190i	$-0.197804\pi$
$5$	3.63608	1.62611	0.813053	+	0.582190i	$0.197804\pi$
$6$	0	0
$7$	2.74254	1.03658	0.518291	−	0.855204i	$-0.326569\pi$
$7$	2.74254	1.03658	0.518291	+	0.855204i	$0.326569\pi$
$8$	0	0
$9$	1.00000	0.333333
$10$	0	0
$11$	3.30758	0.997272	0.498636	−	0.866811i	$-0.333834\pi$
$11$	3.30758	0.997272	0.498636	+	0.866811i	$0.333834\pi$
$12$	0	0
$13$	5.31864	1.47512	0.737562	−	0.675279i	$-0.235977\pi$
$13$	5.31864	1.47512	0.737562	+	0.675279i	$0.235977\pi$
$14$	0	0
$15$	−3.63608	−0.938833
$16$	0	0
$17$	3.11451	0.755380	0.377690	−	0.925932i	$-0.376718\pi$
$17$	3.11451	0.755380	0.377690	+	0.925932i	$0.376718\pi$
$18$	0	0
$19$	−4.91358	−1.12725	−0.563626	−	0.826030i	$-0.690594\pi$
$19$	−4.91358	−1.12725	−0.563626	+	0.826030i	$0.690594\pi$
$20$	0	0
$21$	−2.74254	−0.598471
$22$	0	0
$23$	−1.16838	−0.243625	−0.121812	−	0.992553i	$-0.538871\pi$
$23$	−1.16838	−0.243625	−0.121812	+	0.992553i	$0.538871\pi$
$24$	0	0
$25$	8.22110	1.64422
$26$	0	0
$27$	−1.00000	−0.192450
$28$	0	0
$29$	−3.53214	−0.655902	−0.327951	−	0.944695i	$-0.606358\pi$
$29$	−3.53214	−0.655902	−0.327951	+	0.944695i	$0.606358\pi$
$30$	0	0
$31$	2.72293	0.489052	0.244526	−	0.969643i	$-0.421368\pi$
$31$	2.72293	0.489052	0.244526	+	0.969643i	$0.421368\pi$
$32$	0	0
$33$	−3.30758	−0.575775
$34$	0	0
$35$	9.97209	1.68559
$36$	0	0
$37$	2.05276	0.337472	0.168736	−	0.985661i	$-0.446031\pi$
$37$	2.05276	0.337472	0.168736	+	0.985661i	$0.446031\pi$
$38$	0	0
$39$	−5.31864	−0.851663
$40$	0	0
$41$	4.50924	0.704225	0.352112	−	0.935958i	$-0.385463\pi$
$41$	4.50924	0.704225	0.352112	+	0.935958i	$0.385463\pi$
$42$	0	0
$43$	3.85770	0.588294	0.294147	−	0.955760i	$-0.404965\pi$
$43$	3.85770	0.588294	0.294147	+	0.955760i	$0.404965\pi$
$44$	0	0
$45$	3.63608	0.542035
$46$	0	0
$47$	3.91425	0.570951	0.285476	−	0.958386i	$-0.407848\pi$
$47$	3.91425	0.570951	0.285476	+	0.958386i	$0.407848\pi$
$48$	0	0
$49$	0.521509	0.0745012
$50$	0	0
$51$	−3.11451	−0.436119
$52$	0	0
$53$	9.24140	1.26940	0.634702	−	0.772757i	$-0.281123\pi$
$53$	9.24140	1.26940	0.634702	+	0.772757i	$0.281123\pi$
$54$	0	0
$55$	12.0266	1.62167
$56$	0	0
$57$	4.91358	0.650819
$58$	0	0
$59$	−8.59898	−1.11949	−0.559746	−	0.828664i	$-0.689101\pi$
$59$	−8.59898	−1.11949	−0.559746	+	0.828664i	$0.689101\pi$
$60$	0	0
$61$	2.00967	0.257312	0.128656	−	0.991689i	$-0.458934\pi$
$61$	2.00967	0.257312	0.128656	+	0.991689i	$0.458934\pi$
$62$	0	0
$63$	2.74254	0.345527
$64$	0	0
$65$	19.3390	2.39871
$66$	0	0
$67$	5.49320	0.671101	0.335551	−	0.942022i	$-0.391078\pi$
$67$	5.49320	0.671101	0.335551	+	0.942022i	$0.391078\pi$
$68$	0	0
$69$	1.16838	0.140657
$70$	0	0
$71$	4.41623	0.524110	0.262055	−	0.965053i	$-0.415600\pi$
$71$	4.41623	0.524110	0.262055	+	0.965053i	$0.415600\pi$
$72$	0	0
$73$	−5.91685	−0.692515	−0.346258	−	0.938139i	$-0.612548\pi$
$73$	−5.91685	−0.692515	−0.346258	+	0.938139i	$0.612548\pi$
$74$	0	0
$75$	−8.22110	−0.949291
$76$	0	0
$77$	9.07115	1.03375
$78$	0	0
$79$	−16.0135	−1.80166	−0.900829	−	0.434174i	$-0.857040\pi$
$79$	−16.0135	−1.80166	−0.900829	+	0.434174i	$0.857040\pi$
$80$	0	0
$81$	1.00000	0.111111
$82$	0	0
$83$	−14.2902	−1.56856	−0.784278	−	0.620410i	$-0.786966\pi$
$83$	−14.2902	−1.56856	−0.784278	+	0.620410i	$0.786966\pi$
$84$	0	0
$85$	11.3246	1.22833
$86$	0	0
$87$	3.53214	0.378685
$88$	0	0
$89$	−11.4711	−1.21594	−0.607969	−	0.793960i	$-0.708016\pi$
$89$	−11.4711	−1.21594	−0.607969	+	0.793960i	$0.708016\pi$
$90$	0	0
$91$	14.5866	1.52909
$92$	0	0
$93$	−2.72293	−0.282354
$94$	0	0
$95$	−17.8662	−1.83303
$96$	0	0
$97$	7.25459	0.736592	0.368296	−	0.929708i	$-0.379941\pi$
$97$	7.25459	0.736592	0.368296	+	0.929708i	$0.379941\pi$
$98$	0	0
$99$	3.30758	0.332424
$100$	0	0
$101$	3.67926	0.366100	0.183050	−	0.983104i	$-0.441403\pi$
$101$	3.67926	0.366100	0.183050	+	0.983104i	$0.441403\pi$
$102$	0	0
$103$	−2.99402	−0.295010	−0.147505	−	0.989061i	$-0.547124\pi$
$103$	−2.99402	−0.295010	−0.147505	+	0.989061i	$0.547124\pi$
$104$	0	0
$105$	−9.97209	−0.973177
$106$	0	0
$107$	−3.22631	−0.311900	−0.155950	−	0.987765i	$-0.549844\pi$
$107$	−3.22631	−0.311900	−0.155950	+	0.987765i	$0.549844\pi$
$108$	0	0
$109$	−15.0539	−1.44191	−0.720953	−	0.692984i	$-0.756296\pi$
$109$	−15.0539	−1.44191	−0.720953	+	0.692984i	$0.756296\pi$
$110$	0	0
$111$	−2.05276	−0.194840
$112$	0	0
$113$	2.69850	0.253854	0.126927	−	0.991912i	$-0.459489\pi$
$113$	2.69850	0.253854	0.126927	+	0.991912i	$0.459489\pi$
$114$	0	0
$115$	−4.24834	−0.396159
$116$	0	0
$117$	5.31864	0.491708
$118$	0	0
$119$	8.54166	0.783013
$120$	0	0
$121$	−0.0599350	−0.00544864
$122$	0	0
$123$	−4.50924	−0.406584
$124$	0	0
$125$	11.7122	1.04757
$126$	0	0
$127$	4.28643	0.380359	0.190179	−	0.981749i	$-0.439093\pi$
$127$	4.28643	0.380359	0.190179	+	0.981749i	$0.439093\pi$
$128$	0	0
$129$	−3.85770	−0.339652
$130$	0	0
$131$	−6.36296	−0.555935	−0.277967	−	0.960591i	$-0.589661\pi$
$131$	−6.36296	−0.555935	−0.277967	+	0.960591i	$0.589661\pi$
$132$	0	0
$133$	−13.4757	−1.16849
$134$	0	0
$135$	−3.63608	−0.312944
$136$	0	0
$137$	−1.00976	−0.0862697	−0.0431349	−	0.999069i	$-0.513735\pi$
$137$	−1.00976	−0.0862697	−0.0431349	+	0.999069i	$0.513735\pi$
$138$	0	0
$139$	−3.15450	−0.267561	−0.133781	−	0.991011i	$-0.542712\pi$
$139$	−3.15450	−0.267561	−0.133781	+	0.991011i	$0.542712\pi$
$140$	0	0
$141$	−3.91425	−0.329639
$142$	0	0
$143$	17.5918	1.47110
$144$	0	0
$145$	−12.8432	−1.06657
$146$	0	0
$147$	−0.521509	−0.0430133
$148$	0	0
$149$	−14.7567	−1.20892	−0.604459	−	0.796636i	$-0.706611\pi$
$149$	−14.7567	−1.20892	−0.604459	+	0.796636i	$0.706611\pi$
$150$	0	0
$151$	−4.89674	−0.398491	−0.199246	−	0.979950i	$-0.563849\pi$
$151$	−4.89674	−0.398491	−0.199246	+	0.979950i	$0.563849\pi$
$152$	0	0
$153$	3.11451	0.251793
$154$	0	0
$155$	9.90079	0.795251
$156$	0	0
$157$	6.24805	0.498649	0.249324	−	0.968420i	$-0.419791\pi$
$157$	6.24805	0.498649	0.249324	+	0.968420i	$0.419791\pi$
$158$	0	0
$159$	−9.24140	−0.732890
$160$	0	0
$161$	−3.20433	−0.252537
$162$	0	0
$163$	3.24473	0.254147	0.127074	−	0.991893i	$-0.459442\pi$
$163$	3.24473	0.254147	0.127074	+	0.991893i	$0.459442\pi$
$164$	0	0
$165$	−12.0266	−0.936271
$166$	0	0
$167$	1.78067	0.137792	0.0688961	−	0.997624i	$-0.478052\pi$
$167$	1.78067	0.137792	0.0688961	+	0.997624i	$0.478052\pi$
$168$	0	0
$169$	15.2879	1.17599
$170$	0	0
$171$	−4.91358	−0.375750
$172$	0	0
$173$	2.68221	0.203925	0.101962	−	0.994788i	$-0.467488\pi$
$173$	2.68221	0.203925	0.101962	+	0.994788i	$0.467488\pi$
$174$	0	0
$175$	22.5467	1.70437
$176$	0	0
$177$	8.59898	0.646339
$178$	0	0
$179$	−4.91782	−0.367575	−0.183788	−	0.982966i	$-0.558836\pi$
$179$	−4.91782	−0.367575	−0.183788	+	0.982966i	$0.558836\pi$
$180$	0	0
$181$	9.26513	0.688672	0.344336	−	0.938846i	$-0.388104\pi$
$181$	9.26513	0.688672	0.344336	+	0.938846i	$0.388104\pi$
$182$	0	0
$183$	−2.00967	−0.148559
$184$	0	0
$185$	7.46402	0.548766
$186$	0	0
$187$	10.3015	0.753319
$188$	0	0
$189$	−2.74254	−0.199490
$190$	0	0
$191$	−22.0938	−1.59865	−0.799326	−	0.600898i	$-0.794810\pi$
$191$	−22.0938	−1.59865	−0.799326	+	0.600898i	$0.794810\pi$
$192$	0	0
$193$	−1.29131	−0.0929504	−0.0464752	−	0.998919i	$-0.514799\pi$
$193$	−1.29131	−0.0929504	−0.0464752	+	0.998919i	$0.514799\pi$
$194$	0	0
$195$	−19.3390	−1.38489
$196$	0	0
$197$	−25.3659	−1.80725	−0.903623	−	0.428329i	$-0.859103\pi$
$197$	−25.3659	−1.80725	−0.903623	+	0.428329i	$0.859103\pi$
$198$	0	0
$199$	−3.71745	−0.263523	−0.131762	−	0.991281i	$-0.542063\pi$
$199$	−3.71745	−0.263523	−0.131762	+	0.991281i	$0.542063\pi$
$200$	0	0
$201$	−5.49320	−0.387461
$202$	0	0
$203$	−9.68702	−0.679896
$204$	0	0
$205$	16.3960	1.14514
$206$	0	0
$207$	−1.16838	−0.0812082
$208$	0	0
$209$	−16.2520	−1.12418
$210$	0	0
$211$	−8.18344	−0.563371	−0.281686	−	0.959507i	$-0.590894\pi$
$211$	−8.18344	−0.563371	−0.281686	+	0.959507i	$0.590894\pi$
$212$	0	0
$213$	−4.41623	−0.302595
$214$	0	0
$215$	14.0269	0.956628
$216$	0	0
$217$	7.46773	0.506943
$218$	0	0
$219$	5.91685	0.399824
$220$	0	0
$221$	16.5650	1.11428
$222$	0	0
$223$	−22.6888	−1.51936	−0.759679	−	0.650298i	$-0.774644\pi$
$223$	−22.6888	−1.51936	−0.759679	+	0.650298i	$0.774644\pi$
$224$	0	0
$225$	8.22110	0.548073
$226$	0	0
$227$	21.4427	1.42320	0.711601	−	0.702584i	$-0.247970\pi$
$227$	21.4427	1.42320	0.711601	+	0.702584i	$0.247970\pi$
$228$	0	0
$229$	−3.39774	−0.224529	−0.112265	−	0.993678i	$-0.535810\pi$
$229$	−3.39774	−0.224529	−0.112265	+	0.993678i	$0.535810\pi$
$230$	0	0
$231$	−9.07115	−0.596838
$232$	0	0
$233$	−9.95915	−0.652446	−0.326223	−	0.945293i	$-0.605776\pi$
$233$	−9.95915	−0.652446	−0.326223	+	0.945293i	$0.605776\pi$
$234$	0	0
$235$	14.2325	0.928427
$236$	0	0
$237$	16.0135	1.04019
$238$	0	0
$239$	3.09684	0.200318	0.100159	−	0.994971i	$-0.468065\pi$
$239$	3.09684	0.200318	0.100159	+	0.994971i	$0.468065\pi$
$240$	0	0
$241$	4.17708	0.269069	0.134535	−	0.990909i	$-0.457046\pi$
$241$	4.17708	0.269069	0.134535	+	0.990909i	$0.457046\pi$
$242$	0	0
$243$	−1.00000	−0.0641500
$244$	0	0
$245$	1.89625	0.121147
$246$	0	0
$247$	−26.1335	−1.66284
$248$	0	0
$249$	14.2902	0.905606
$250$	0	0
$251$	1.00000	0.0631194
$251$	1.00000	0.0631194
$252$	0	0
$253$	−3.86452	−0.242960
$254$	0	0
$255$	−11.3246	−0.709175
$256$	0	0
$257$	−5.38217	−0.335731	−0.167865	−	0.985810i	$-0.553687\pi$
$257$	−5.38217	−0.335731	−0.167865	+	0.985810i	$0.553687\pi$
$258$	0	0
$259$	5.62978	0.349818
$260$	0	0
$261$	−3.53214	−0.218634
$262$	0	0
$263$	7.42640	0.457932	0.228966	−	0.973434i	$-0.426466\pi$
$263$	7.42640	0.457932	0.228966	+	0.973434i	$0.426466\pi$
$264$	0	0
$265$	33.6025	2.06418
$266$	0	0
$267$	11.4711	0.702023
$268$	0	0
$269$	20.1284	1.22725	0.613624	−	0.789598i	$-0.289711\pi$
$269$	20.1284	1.22725	0.613624	+	0.789598i	$0.289711\pi$
$270$	0	0
$271$	−9.46169	−0.574757	−0.287378	−	0.957817i	$-0.592784\pi$
$271$	−9.46169	−0.574757	−0.287378	+	0.957817i	$0.592784\pi$
$272$	0	0
$273$	−14.5866	−0.882818
$274$	0	0
$275$	27.1919	1.63973
$276$	0	0
$277$	−6.87311	−0.412965	−0.206483	−	0.978450i	$-0.566202\pi$
$277$	−6.87311	−0.412965	−0.206483	+	0.978450i	$0.566202\pi$
$278$	0	0
$279$	2.72293	0.163017
$280$	0	0
$281$	32.0446	1.91162	0.955810	−	0.293985i	$-0.0949817\pi$
$281$	32.0446	1.91162	0.955810	+	0.293985i	$0.0949817\pi$
$282$	0	0
$283$	4.16536	0.247605	0.123802	−	0.992307i	$-0.460491\pi$
$283$	4.16536	0.247605	0.123802	+	0.992307i	$0.460491\pi$
$284$	0	0
$285$	17.8662	1.05830
$286$	0	0
$287$	12.3668	0.729986
$288$	0	0
$289$	−7.29982	−0.429401
$290$	0	0
$291$	−7.25459	−0.425272
$292$	0	0
$293$	22.0512	1.28824	0.644122	−	0.764923i	$-0.277223\pi$
$293$	22.0512	1.28824	0.644122	+	0.764923i	$0.277223\pi$
$294$	0	0
$295$	−31.2666	−1.82041
$296$	0	0
$297$	−3.30758	−0.191925
$298$	0	0
$299$	−6.21420	−0.359377
$300$	0	0
$301$	10.5799	0.609814
$302$	0	0
$303$	−3.67926	−0.211368
$304$	0	0
$305$	7.30734	0.418417
$306$	0	0
$307$	−12.1942	−0.695961	−0.347980	−	0.937502i	$-0.613132\pi$
$307$	−12.1942	−0.695961	−0.347980	+	0.937502i	$0.613132\pi$
$308$	0	0
$309$	2.99402	0.170324
$310$	0	0
$311$	−3.27647	−0.185792	−0.0928958	−	0.995676i	$-0.529612\pi$
$311$	−3.27647	−0.185792	−0.0928958	+	0.995676i	$0.529612\pi$
$312$	0	0
$313$	4.51307	0.255094	0.127547	−	0.991833i	$-0.459290\pi$
$313$	4.51307	0.255094	0.127547	+	0.991833i	$0.459290\pi$
$314$	0	0
$315$	9.97209	0.561864
$316$	0	0
$317$	−20.6736	−1.16114	−0.580571	−	0.814209i	$-0.697171\pi$
$317$	−20.6736	−1.16114	−0.580571	+	0.814209i	$0.697171\pi$
$318$	0	0
$319$	−11.6828	−0.654113
$320$	0	0
$321$	3.22631	0.180075
$322$	0	0
$323$	−15.3034	−0.851503
$324$	0	0
$325$	43.7250	2.42543
$326$	0	0
$327$	15.0539	0.832485
$328$	0	0
$329$	10.7350	0.591838
$330$	0	0
$331$	−18.4327	−1.01315	−0.506576	−	0.862195i	$-0.669089\pi$
$331$	−18.4327	−1.01315	−0.506576	+	0.862195i	$0.669089\pi$
$332$	0	0
$333$	2.05276	0.112491
$334$	0	0
$335$	19.9737	1.09128
$336$	0	0
$337$	7.11276	0.387457	0.193728	−	0.981055i	$-0.437942\pi$
$337$	7.11276	0.387457	0.193728	+	0.981055i	$0.437942\pi$
$338$	0	0
$339$	−2.69850	−0.146563
$340$	0	0
$341$	9.00629	0.487718
$342$	0	0
$343$	−17.7675	−0.959355
$344$	0	0
$345$	4.24834	0.228723
$346$	0	0
$347$	3.93879	0.211445	0.105723	−	0.994396i	$-0.466284\pi$
$347$	3.93879	0.211445	0.105723	+	0.994396i	$0.466284\pi$
$348$	0	0
$349$	−22.4953	−1.20414	−0.602072	−	0.798442i	$-0.705658\pi$
$349$	−22.4953	−1.20414	−0.602072	+	0.798442i	$0.705658\pi$
$350$	0	0
$351$	−5.31864	−0.283888
$352$	0	0
$353$	23.0497	1.22681	0.613405	−	0.789768i	$-0.289799\pi$
$353$	23.0497	1.22681	0.613405	+	0.789768i	$0.289799\pi$
$354$	0	0
$355$	16.0578	0.852258
$356$	0	0
$357$	−8.54166	−0.452073
$358$	0	0
$359$	21.6846	1.14447	0.572234	−	0.820090i	$-0.306077\pi$
$359$	21.6846	1.14447	0.572234	+	0.820090i	$0.306077\pi$
$360$	0	0
$361$	5.14322	0.270696
$362$	0	0
$363$	0.0599350	0.00314577
$364$	0	0
$365$	−21.5142	−1.12610
$366$	0	0
$367$	−18.3562	−0.958185	−0.479093	−	0.877764i	$-0.659034\pi$
$367$	−18.3562	−0.958185	−0.479093	+	0.877764i	$0.659034\pi$
$368$	0	0
$369$	4.50924	0.234742
$370$	0	0
$371$	25.3449	1.31584
$372$	0	0
$373$	32.5365	1.68468	0.842339	−	0.538948i	$-0.181178\pi$
$373$	32.5365	1.68468	0.842339	+	0.538948i	$0.181178\pi$
$374$	0	0
$375$	−11.7122	−0.604815
$376$	0	0
$377$	−18.7862	−0.967537
$378$	0	0
$379$	37.3419	1.91813	0.959063	−	0.283191i	$-0.0913931\pi$
$379$	37.3419	1.91813	0.959063	+	0.283191i	$0.0913931\pi$
$380$	0	0
$381$	−4.28643	−0.219600
$382$	0	0
$383$	2.45975	0.125687	0.0628437	−	0.998023i	$-0.479983\pi$
$383$	2.45975	0.125687	0.0628437	+	0.998023i	$0.479983\pi$
$384$	0	0
$385$	32.9835	1.68099
$386$	0	0
$387$	3.85770	0.196098
$388$	0	0
$389$	1.47471	0.0747710	0.0373855	−	0.999301i	$-0.488097\pi$
$389$	1.47471	0.0747710	0.0373855	+	0.999301i	$0.488097\pi$
$390$	0	0
$391$	−3.63894	−0.184029
$392$	0	0
$393$	6.36296	0.320969
$394$	0	0
$395$	−58.2264	−2.92969
$396$	0	0
$397$	4.07233	0.204384	0.102192	−	0.994765i	$-0.467414\pi$
$397$	4.07233	0.204384	0.102192	+	0.994765i	$0.467414\pi$
$398$	0	0
$399$	13.4757	0.674627
$400$	0	0
$401$	3.98842	0.199172	0.0995861	−	0.995029i	$-0.468248\pi$
$401$	3.98842	0.199172	0.0995861	+	0.995029i	$0.468248\pi$
$402$	0	0
$403$	14.4823	0.721413
$404$	0	0
$405$	3.63608	0.180678
$406$	0	0
$407$	6.78967	0.336552
$408$	0	0
$409$	−7.81431	−0.386393	−0.193196	−	0.981160i	$-0.561885\pi$
$409$	−7.81431	−0.386393	−0.193196	+	0.981160i	$0.561885\pi$
$410$	0	0
$411$	1.00976	0.0498079
$412$	0	0
$413$	−23.5830	−1.16044
$414$	0	0
$415$	−51.9604	−2.55064
$416$	0	0
$417$	3.15450	0.154477
$418$	0	0
$419$	−34.6777	−1.69412	−0.847059	−	0.531500i	$-0.821629\pi$
$419$	−34.6777	−1.69412	−0.847059	+	0.531500i	$0.821629\pi$
$420$	0	0
$421$	−6.57468	−0.320430	−0.160215	−	0.987082i	$-0.551219\pi$
$421$	−6.57468	−0.320430	−0.160215	+	0.987082i	$0.551219\pi$
$422$	0	0
$423$	3.91425	0.190317
$424$	0	0
$425$	25.6047	1.24201
$426$	0	0
$427$	5.51160	0.266725
$428$	0	0
$429$	−17.5918	−0.849340
$430$	0	0
$431$	−20.4479	−0.984943	−0.492472	−	0.870329i	$-0.663906\pi$
$431$	−20.4479	−0.984943	−0.492472	+	0.870329i	$0.663906\pi$
$432$	0	0
$433$	33.1769	1.59438	0.797191	−	0.603728i	$-0.206319\pi$
$433$	33.1769	1.59438	0.797191	+	0.603728i	$0.206319\pi$
$434$	0	0
$435$	12.8432	0.615782
$436$	0	0
$437$	5.74094	0.274626
$438$	0	0
$439$	−20.1691	−0.962618	−0.481309	−	0.876551i	$-0.659838\pi$
$439$	−20.1691	−0.962618	−0.481309	+	0.876551i	$0.659838\pi$
$440$	0	0
$441$	0.521509	0.0248337
$442$	0	0
$443$	0.223059	0.0105979	0.00529893	−	0.999986i	$-0.498313\pi$
$443$	0.223059	0.0105979	0.00529893	+	0.999986i	$0.498313\pi$
$444$	0	0
$445$	−41.7100	−1.97725
$446$	0	0
$447$	14.7567	0.697969
$448$	0	0
$449$	21.7821	1.02796	0.513980	−	0.857802i	$-0.328170\pi$
$449$	21.7821	1.02796	0.513980	+	0.857802i	$0.328170\pi$
$450$	0	0
$451$	14.9147	0.702304
$452$	0	0
$453$	4.89674	0.230069
$454$	0	0
$455$	53.0379	2.48646
$456$	0	0
$457$	−1.35804	−0.0635265	−0.0317632	−	0.999495i	$-0.510112\pi$
$457$	−1.35804	−0.0635265	−0.0317632	+	0.999495i	$0.510112\pi$
$458$	0	0
$459$	−3.11451	−0.145373
$460$	0	0
$461$	−33.2634	−1.54923	−0.774615	−	0.632433i	$-0.782056\pi$
$461$	−33.2634	−1.54923	−0.774615	+	0.632433i	$0.782056\pi$
$462$	0	0
$463$	32.7502	1.52203	0.761015	−	0.648735i	$-0.224702\pi$
$463$	32.7502	1.52203	0.761015	+	0.648735i	$0.224702\pi$
$464$	0	0
$465$	−9.90079	−0.459138
$466$	0	0
$467$	28.7601	1.33086	0.665430	−	0.746461i	$-0.268248\pi$
$467$	28.7601	1.33086	0.665430	+	0.746461i	$0.268248\pi$
$468$	0	0
$469$	15.0653	0.695651
$470$	0	0
$471$	−6.24805	−0.287895
$472$	0	0
$473$	12.7596	0.586689
$474$	0	0
$475$	−40.3950	−1.85345
$476$	0	0
$477$	9.24140	0.423135
$478$	0	0
$479$	−13.2309	−0.604534	−0.302267	−	0.953223i	$-0.597743\pi$
$479$	−13.2309	−0.604534	−0.302267	+	0.953223i	$0.597743\pi$
$480$	0	0
$481$	10.9179	0.497814
$482$	0	0
$483$	3.20433	0.145802
$484$	0	0
$485$	26.3783	1.19778
$486$	0	0
$487$	−0.865076	−0.0392004	−0.0196002	−	0.999808i	$-0.506239\pi$
$487$	−0.865076	−0.0392004	−0.0196002	+	0.999808i	$0.506239\pi$
$488$	0	0
$489$	−3.24473	−0.146732
$490$	0	0
$491$	38.2292	1.72526	0.862630	−	0.505836i	$-0.168816\pi$
$491$	38.2292	1.72526	0.862630	+	0.505836i	$0.168816\pi$
$492$	0	0
$493$	−11.0009	−0.495455
$494$	0	0
$495$	12.0266	0.540557
$496$	0	0
$497$	12.1117	0.543283
$498$	0	0
$499$	36.8770	1.65084	0.825421	−	0.564518i	$-0.190938\pi$
$499$	36.8770	1.65084	0.825421	+	0.564518i	$0.190938\pi$
$500$	0	0
$501$	−1.78067	−0.0795543
$502$	0	0
$503$	−22.3161	−0.995027	−0.497514	−	0.867456i	$-0.665754\pi$
$503$	−22.3161	−0.995027	−0.497514	+	0.867456i	$0.665754\pi$
$504$	0	0
$505$	13.3781	0.595317
$506$	0	0
$507$	−15.2879	−0.678959
$508$	0	0
$509$	15.9250	0.705865	0.352932	−	0.935649i	$-0.385185\pi$
$509$	15.9250	0.705865	0.352932	+	0.935649i	$0.385185\pi$
$510$	0	0
$511$	−16.2272	−0.717848
$512$	0	0
$513$	4.91358	0.216940
$514$	0	0
$515$	−10.8865	−0.479717
$516$	0	0
$517$	12.9467	0.569394
$518$	0	0
$519$	−2.68221	−0.117736
$520$	0	0
$521$	44.1225	1.93304	0.966521	−	0.256586i	$-0.0825976\pi$
$521$	44.1225	1.93304	0.966521	+	0.256586i	$0.0825976\pi$
$522$	0	0
$523$	2.20274	0.0963191	0.0481595	−	0.998840i	$-0.484664\pi$
$523$	2.20274	0.0963191	0.0481595	+	0.998840i	$0.484664\pi$
$524$	0	0
$525$	−22.5467	−0.984017
$526$	0	0
$527$	8.48059	0.369420
$528$	0	0
$529$	−21.6349	−0.940647
$530$	0	0
$531$	−8.59898	−0.373164
$532$	0	0
$533$	23.9830	1.03882
$534$	0	0
$535$	−11.7311	−0.507182
$536$	0	0
$537$	4.91782	0.212220
$538$	0	0
$539$	1.72493	0.0742980
$540$	0	0
$541$	−37.6663	−1.61940	−0.809700	−	0.586844i	$-0.800370\pi$
$541$	−37.6663	−1.61940	−0.809700	+	0.586844i	$0.800370\pi$
$542$	0	0
$543$	−9.26513	−0.397605
$544$	0	0
$545$	−54.7374	−2.34469
$546$	0	0
$547$	−11.8587	−0.507041	−0.253520	−	0.967330i	$-0.581588\pi$
$547$	−11.8587	−0.507041	−0.253520	+	0.967330i	$0.581588\pi$
$548$	0	0
$549$	2.00967	0.0857707
$550$	0	0
$551$	17.3554	0.739366
$552$	0	0
$553$	−43.9176	−1.86757
$554$	0	0
$555$	−7.46402	−0.316830
$556$	0	0
$557$	−5.56491	−0.235793	−0.117896	−	0.993026i	$-0.537615\pi$
$557$	−5.56491	−0.235793	−0.117896	+	0.993026i	$0.537615\pi$
$558$	0	0
$559$	20.5177	0.867806
$560$	0	0
$561$	−10.3015	−0.434929
$562$	0	0
$563$	−4.20101	−0.177052	−0.0885258	−	0.996074i	$-0.528216\pi$
$563$	−4.20101	−0.177052	−0.0885258	+	0.996074i	$0.528216\pi$
$564$	0	0
$565$	9.81198	0.412793
$566$	0	0
$567$	2.74254	0.115176
$568$	0	0
$569$	5.71502	0.239586	0.119793	−	0.992799i	$-0.461777\pi$
$569$	5.71502	0.239586	0.119793	+	0.992799i	$0.461777\pi$
$570$	0	0
$571$	−7.07869	−0.296234	−0.148117	−	0.988970i	$-0.547321\pi$
$571$	−7.07869	−0.296234	−0.148117	+	0.988970i	$0.547321\pi$
$572$	0	0
$573$	22.0938	0.922982
$574$	0	0
$575$	−9.60539	−0.400572
$576$	0	0
$577$	−40.3212	−1.67860	−0.839298	−	0.543672i	$-0.817033\pi$
$577$	−40.3212	−1.67860	−0.839298	+	0.543672i	$0.817033\pi$
$578$	0	0
$579$	1.29131	0.0536649
$580$	0	0
$581$	−39.1914	−1.62594
$582$	0	0
$583$	30.5666	1.26594
$584$	0	0
$585$	19.3390	0.799569
$586$	0	0
$587$	24.7903	1.02321	0.511603	−	0.859222i	$-0.329052\pi$
$587$	24.7903	1.02321	0.511603	+	0.859222i	$0.329052\pi$
$588$	0	0
$589$	−13.3793	−0.551285
$590$	0	0
$591$	25.3659	1.04341
$592$	0	0
$593$	−11.7195	−0.481260	−0.240630	−	0.970617i	$-0.577354\pi$
$593$	−11.7195	−0.481260	−0.240630	+	0.970617i	$0.577354\pi$
$594$	0	0
$595$	31.0582	1.27326
$596$	0	0
$597$	3.71745	0.152145
$598$	0	0
$599$	39.3646	1.60840	0.804198	−	0.594362i	$-0.202595\pi$
$599$	39.3646	1.60840	0.804198	+	0.594362i	$0.202595\pi$
$600$	0	0
$601$	7.16577	0.292298	0.146149	−	0.989263i	$-0.453312\pi$
$601$	7.16577	0.292298	0.146149	+	0.989263i	$0.453312\pi$
$602$	0	0
$603$	5.49320	0.223700
$604$	0	0
$605$	−0.217929	−0.00886006
$606$	0	0
$607$	18.3317	0.744059	0.372030	−	0.928221i	$-0.378662\pi$
$607$	18.3317	0.744059	0.372030	+	0.928221i	$0.378662\pi$
$608$	0	0
$609$	9.68702	0.392538
$610$	0	0
$611$	20.8184	0.842224
$612$	0	0
$613$	−9.20496	−0.371785	−0.185892	−	0.982570i	$-0.559518\pi$
$613$	−9.20496	−0.371785	−0.185892	+	0.982570i	$0.559518\pi$
$614$	0	0
$615$	−16.3960	−0.661149
$616$	0	0
$617$	25.0317	1.00774	0.503868	−	0.863780i	$-0.331910\pi$
$617$	25.0317	1.00774	0.503868	+	0.863780i	$0.331910\pi$
$618$	0	0
$619$	−10.4450	−0.419820	−0.209910	−	0.977721i	$-0.567317\pi$
$619$	−10.4450	−0.419820	−0.209910	+	0.977721i	$0.567317\pi$
$620$	0	0
$621$	1.16838	0.0468856
$622$	0	0
$623$	−31.4600	−1.26042
$624$	0	0
$625$	1.48100	0.0592398
$626$	0	0
$627$	16.2520	0.649043
$628$	0	0
$629$	6.39336	0.254920
$630$	0	0
$631$	5.26520	0.209604	0.104802	−	0.994493i	$-0.466579\pi$
$631$	5.26520	0.209604	0.104802	+	0.994493i	$0.466579\pi$
$632$	0	0
$633$	8.18344	0.325263
$634$	0	0
$635$	15.5858	0.618504
$636$	0	0
$637$	2.77371	0.109899
$638$	0	0
$639$	4.41623	0.174703
$640$	0	0
$641$	20.0870	0.793390	0.396695	−	0.917951i	$-0.370157\pi$
$641$	20.0870	0.793390	0.396695	+	0.917951i	$0.370157\pi$
$642$	0	0
$643$	6.72845	0.265344	0.132672	−	0.991160i	$-0.457644\pi$
$643$	6.72845	0.265344	0.132672	+	0.991160i	$0.457644\pi$
$644$	0	0
$645$	−14.0269	−0.552309
$646$	0	0
$647$	20.5734	0.808825	0.404413	−	0.914577i	$-0.367476\pi$
$647$	20.5734	0.808825	0.404413	+	0.914577i	$0.367476\pi$
$648$	0	0
$649$	−28.4418	−1.11644
$650$	0	0
$651$	−7.46773	−0.292683
$652$	0	0
$653$	6.29408	0.246306	0.123153	−	0.992388i	$-0.460699\pi$
$653$	6.29408	0.246306	0.123153	+	0.992388i	$0.460699\pi$
$654$	0	0
$655$	−23.1363	−0.904008
$656$	0	0
$657$	−5.91685	−0.230838
$658$	0	0
$659$	−18.5412	−0.722264	−0.361132	−	0.932515i	$-0.617610\pi$
$659$	−18.5412	−0.722264	−0.361132	+	0.932515i	$0.617610\pi$
$660$	0	0
$661$	8.06451	0.313673	0.156836	−	0.987625i	$-0.449870\pi$
$661$	8.06451	0.313673	0.156836	+	0.987625i	$0.449870\pi$
$662$	0	0
$663$	−16.5650	−0.643330
$664$	0	0
$665$	−48.9986	−1.90009
$666$	0	0
$667$	4.12689	0.159794
$668$	0	0
$669$	22.6888	0.877202
$670$	0	0
$671$	6.64715	0.256610
$672$	0	0
$673$	37.6780	1.45238	0.726189	−	0.687495i	$-0.241290\pi$
$673$	37.6780	1.45238	0.726189	+	0.687495i	$0.241290\pi$
$674$	0	0
$675$	−8.22110	−0.316430
$676$	0	0
$677$	0.442982	0.0170252	0.00851259	−	0.999964i	$-0.497290\pi$
$677$	0.442982	0.0170252	0.00851259	+	0.999964i	$0.497290\pi$
$678$	0	0
$679$	19.8960	0.763538
$680$	0	0
$681$	−21.4427	−0.821686
$682$	0	0
$683$	−7.19845	−0.275441	−0.137721	−	0.990471i	$-0.543978\pi$
$683$	−7.19845	−0.275441	−0.137721	+	0.990471i	$0.543978\pi$
$684$	0	0
$685$	−3.67158	−0.140284
$686$	0	0
$687$	3.39774	0.129632
$688$	0	0
$689$	49.1516	1.87253
$690$	0	0
$691$	36.0929	1.37304	0.686520	−	0.727111i	$-0.259137\pi$
$691$	36.0929	1.37304	0.686520	+	0.727111i	$0.259137\pi$
$692$	0	0
$693$	9.07115	0.344585
$694$	0	0
$695$	−11.4700	−0.435083
$696$	0	0
$697$	14.0441	0.531957
$698$	0	0
$699$	9.95915	0.376690
$700$	0	0
$701$	50.6063	1.91137	0.955687	−	0.294385i	$-0.0951147\pi$
$701$	50.6063	1.91137	0.955687	+	0.294385i	$0.0951147\pi$
$702$	0	0
$703$	−10.0864	−0.380416
$704$	0	0
$705$	−14.2325	−0.536028
$706$	0	0
$707$	10.0905	0.379492
$708$	0	0
$709$	2.66171	0.0999626	0.0499813	−	0.998750i	$-0.484084\pi$
$709$	2.66171	0.0999626	0.0499813	+	0.998750i	$0.484084\pi$
$710$	0	0
$711$	−16.0135	−0.600553
$712$	0	0
$713$	−3.18142	−0.119145
$714$	0	0
$715$	63.9652	2.39216
$716$	0	0
$717$	−3.09684	−0.115654
$718$	0	0
$719$	−11.1075	−0.414241	−0.207121	−	0.978315i	$-0.566409\pi$
$719$	−11.1075	−0.414241	−0.207121	+	0.978315i	$0.566409\pi$
$720$	0	0
$721$	−8.21122	−0.305802
$722$	0	0
$723$	−4.17708	−0.155347
$724$	0	0
$725$	−29.0381	−1.07845
$726$	0	0
$727$	17.1394	0.635666	0.317833	−	0.948147i	$-0.397045\pi$
$727$	17.1394	0.635666	0.317833	+	0.948147i	$0.397045\pi$
$728$	0	0
$729$	1.00000	0.0370370
$730$	0	0
$731$	12.0149	0.444385
$732$	0	0
$733$	45.4453	1.67856	0.839280	−	0.543700i	$-0.182977\pi$
$733$	45.4453	1.67856	0.839280	+	0.543700i	$0.182977\pi$
$734$	0	0
$735$	−1.89625	−0.0699442
$736$	0	0
$737$	18.1692	0.669271
$738$	0	0
$739$	−21.1637	−0.778519	−0.389260	−	0.921128i	$-0.627269\pi$
$739$	−21.1637	−0.778519	−0.389260	+	0.921128i	$0.627269\pi$
$740$	0	0
$741$	26.1335	0.960039
$742$	0	0
$743$	34.2143	1.25520	0.627601	−	0.778535i	$-0.284037\pi$
$743$	34.2143	1.25520	0.627601	+	0.778535i	$0.284037\pi$
$744$	0	0
$745$	−53.6567	−1.96583
$746$	0	0
$747$	−14.2902	−0.522852
$748$	0	0
$749$	−8.84829	−0.323309
$750$	0	0
$751$	17.3780	0.634132	0.317066	−	0.948404i	$-0.397302\pi$
$751$	17.3780	0.634132	0.317066	+	0.948404i	$0.397302\pi$
$752$	0	0
$753$	−1.00000	−0.0364420
$754$	0	0
$755$	−17.8050	−0.647989
$756$	0	0
$757$	−15.9503	−0.579723	−0.289862	−	0.957069i	$-0.593609\pi$
$757$	−15.9503	−0.579723	−0.289862	+	0.957069i	$0.593609\pi$
$758$	0	0
$759$	3.86452	0.140273
$760$	0	0
$761$	33.3317	1.20827	0.604137	−	0.796881i	$-0.293518\pi$
$761$	33.3317	1.20827	0.604137	+	0.796881i	$0.293518\pi$
$762$	0	0
$763$	−41.2860	−1.49465
$764$	0	0
$765$	11.3246	0.409443
$766$	0	0
$767$	−45.7349	−1.65139
$768$	0	0
$769$	−33.2961	−1.20069	−0.600343	−	0.799742i	$-0.704969\pi$
$769$	−33.2961	−1.20069	−0.600343	+	0.799742i	$0.704969\pi$
$770$	0	0
$771$	5.38217	0.193834
$772$	0	0
$773$	−47.5339	−1.70968	−0.854838	−	0.518895i	$-0.826343\pi$
$773$	−47.5339	−1.70968	−0.854838	+	0.518895i	$0.826343\pi$
$774$	0	0
$775$	22.3855	0.804110
$776$	0	0
$777$	−5.62978	−0.201967
$778$	0	0
$779$	−22.1565	−0.793839
$780$	0	0
$781$	14.6070	0.522680
$782$	0	0
$783$	3.53214	0.126228
$784$	0	0
$785$	22.7184	0.810855
$786$	0	0
$787$	−17.2334	−0.614305	−0.307153	−	0.951660i	$-0.599376\pi$
$787$	−17.2334	−0.614305	−0.307153	+	0.951660i	$0.599376\pi$
$788$	0	0
$789$	−7.42640	−0.264387
$790$	0	0
$791$	7.40075	0.263140
$792$	0	0
$793$	10.6887	0.379567
$794$	0	0
$795$	−33.6025	−1.19176
$796$	0	0
$797$	−11.3524	−0.402124	−0.201062	−	0.979579i	$-0.564439\pi$
$797$	−11.3524	−0.402124	−0.201062	+	0.979579i	$0.564439\pi$
$798$	0	0
$799$	12.1910	0.431285
$800$	0	0
$801$	−11.4711	−0.405313
$802$	0	0
$803$	−19.5704	−0.690626
$804$	0	0
$805$	−11.6512	−0.410651
$806$	0	0
$807$	−20.1284	−0.708552
$808$	0	0
$809$	20.2528	0.712051	0.356025	−	0.934476i	$-0.384132\pi$
$809$	20.2528	0.712051	0.356025	+	0.934476i	$0.384132\pi$
$810$	0	0
$811$	−5.98950	−0.210320	−0.105160	−	0.994455i	$-0.533535\pi$
$811$	−5.98950	−0.210320	−0.105160	+	0.994455i	$0.533535\pi$
$812$	0	0
$813$	9.46169	0.331836
$814$	0	0
$815$	11.7981	0.413270
$816$	0	0
$817$	−18.9551	−0.663155
$818$	0	0
$819$	14.5866	0.509695
$820$	0	0
$821$	3.42472	0.119524	0.0597619	−	0.998213i	$-0.480966\pi$
$821$	3.42472	0.119524	0.0597619	+	0.998213i	$0.480966\pi$
$822$	0	0
$823$	−17.0342	−0.593775	−0.296888	−	0.954912i	$-0.595949\pi$
$823$	−17.0342	−0.593775	−0.296888	+	0.954912i	$0.595949\pi$
$824$	0	0
$825$	−27.1919	−0.946701
$826$	0	0
$827$	−13.7866	−0.479406	−0.239703	−	0.970846i	$-0.577050\pi$
$827$	−13.7866	−0.479406	−0.239703	+	0.970846i	$0.577050\pi$
$828$	0	0
$829$	−32.3763	−1.12448	−0.562238	−	0.826975i	$-0.690060\pi$
$829$	−32.3763	−1.12448	−0.562238	+	0.826975i	$0.690060\pi$
$830$	0	0
$831$	6.87311	0.238426
$832$	0	0
$833$	1.62424	0.0562768
$834$	0	0
$835$	6.47465	0.224065
$836$	0	0
$837$	−2.72293	−0.0941181
$838$	0	0
$839$	25.2428	0.871477	0.435738	−	0.900073i	$-0.356487\pi$
$839$	25.2428	0.871477	0.435738	+	0.900073i	$0.356487\pi$
$840$	0	0
$841$	−16.5240	−0.569793
$842$	0	0
$843$	−32.0446	−1.10367
$844$	0	0
$845$	55.5880	1.91229
$846$	0	0
$847$	−0.164374	−0.00564796
$848$	0	0
$849$	−4.16536	−0.142955
$850$	0	0
$851$	−2.39841	−0.0822166
$852$	0	0
$853$	−34.2065	−1.17121	−0.585605	−	0.810597i	$-0.699143\pi$
$853$	−34.2065	−1.17121	−0.585605	+	0.810597i	$0.699143\pi$
$854$	0	0
$855$	−17.8662	−0.611010
$856$	0	0
$857$	−34.3971	−1.17498	−0.587491	−	0.809231i	$-0.699884\pi$
$857$	−34.3971	−1.17498	−0.587491	+	0.809231i	$0.699884\pi$
$858$	0	0
$859$	−21.4230	−0.730943	−0.365471	−	0.930823i	$-0.619092\pi$
$859$	−21.4230	−0.730943	−0.365471	+	0.930823i	$0.619092\pi$
$860$	0	0
$861$	−12.3668	−0.421458
$862$	0	0
$863$	27.4581	0.934686	0.467343	−	0.884076i	$-0.345211\pi$
$863$	27.4581	0.934686	0.467343	+	0.884076i	$0.345211\pi$
$864$	0	0
$865$	9.75274	0.331603
$866$	0	0
$867$	7.29982	0.247915
$868$	0	0
$869$	−52.9658	−1.79674
$870$	0	0
$871$	29.2163	0.989958
$872$	0	0
$873$	7.25459	0.245531
$874$	0	0
$875$	32.1211	1.08589
$876$	0	0
$877$	−3.98919	−0.134705	−0.0673526	−	0.997729i	$-0.521455\pi$
$877$	−3.98919	−0.134705	−0.0673526	+	0.997729i	$0.521455\pi$
$878$	0	0
$879$	−22.0512	−0.743768
$880$	0	0
$881$	−1.93637	−0.0652381	−0.0326190	−	0.999468i	$-0.510385\pi$
$881$	−1.93637	−0.0652381	−0.0326190	+	0.999468i	$0.510385\pi$
$882$	0	0
$883$	−16.7206	−0.562692	−0.281346	−	0.959606i	$-0.590781\pi$
$883$	−16.7206	−0.562692	−0.281346	+	0.959606i	$0.590781\pi$
$884$	0	0
$885$	31.2666	1.05102
$886$	0	0
$887$	34.2888	1.15130	0.575652	−	0.817695i	$-0.304748\pi$
$887$	34.2888	1.15130	0.575652	+	0.817695i	$0.304748\pi$
$888$	0	0
$889$	11.7557	0.394273
$890$	0	0
$891$	3.30758	0.110808
$892$	0	0
$893$	−19.2329	−0.643606
$894$	0	0
$895$	−17.8816	−0.597716
$896$	0	0
$897$	6.21420	0.207486
$898$	0	0
$899$	−9.61776	−0.320770
$900$	0	0
$901$	28.7824	0.958882
$902$	0	0
$903$	−10.5799	−0.352077
$904$	0	0
$905$	33.6888	1.11985
$906$	0	0
$907$	−30.6508	−1.01774	−0.508871	−	0.860843i	$-0.669937\pi$
$907$	−30.6508	−1.01774	−0.508871	+	0.860843i	$0.669937\pi$
$908$	0	0
$909$	3.67926	0.122033
$910$	0	0
$911$	−32.3864	−1.07301	−0.536505	−	0.843897i	$-0.680256\pi$
$911$	−32.3864	−1.07301	−0.536505	+	0.843897i	$0.680256\pi$
$912$	0	0
$913$	−47.2660	−1.56428
$914$	0	0
$915$	−7.30734	−0.241573
$916$	0	0
$917$	−17.4507	−0.576272
$918$	0	0
$919$	−9.24292	−0.304896	−0.152448	−	0.988312i	$-0.548716\pi$
$919$	−9.24292	−0.304896	−0.152448	+	0.988312i	$0.548716\pi$
$920$	0	0
$921$	12.1942	0.401813
$922$	0	0
$923$	23.4883	0.773127
$924$	0	0
$925$	16.8760	0.554879
$926$	0	0
$927$	−2.99402	−0.0983367
$928$	0	0
$929$	−2.46370	−0.0808315	−0.0404158	−	0.999183i	$-0.512868\pi$
$929$	−2.46370	−0.0808315	−0.0404158	+	0.999183i	$0.512868\pi$
$930$	0	0
$931$	−2.56247	−0.0839816
$932$	0	0
$933$	3.27647	0.107267
$934$	0	0
$935$	37.4571	1.22498
$936$	0	0
$937$	−22.4657	−0.733922	−0.366961	−	0.930236i	$-0.619602\pi$
$937$	−22.4657	−0.733922	−0.366961	+	0.930236i	$0.619602\pi$
$938$	0	0
$939$	−4.51307	−0.147279
$940$	0	0
$941$	46.5179	1.51644	0.758220	−	0.651999i	$-0.226069\pi$
$941$	46.5179	1.51644	0.758220	+	0.651999i	$0.226069\pi$
$942$	0	0
$943$	−5.26852	−0.171566
$944$	0	0
$945$	−9.97209	−0.324392
$946$	0	0
$947$	18.3859	0.597463	0.298731	−	0.954337i	$-0.403437\pi$
$947$	18.3859	0.597463	0.298731	+	0.954337i	$0.403437\pi$
$948$	0	0
$949$	−31.4696	−1.02155
$950$	0	0
$951$	20.6736	0.670386
$952$	0	0
$953$	−42.1623	−1.36577	−0.682885	−	0.730526i	$-0.739275\pi$
$953$	−42.1623	−1.36577	−0.682885	+	0.730526i	$0.739275\pi$
$954$	0	0
$955$	−80.3349	−2.59958
$956$	0	0
$957$	11.6828	0.377652
$958$	0	0
$959$	−2.76931	−0.0894256
$960$	0	0
$961$	−23.5857	−0.760828
$962$	0	0
$963$	−3.22631	−0.103967
$964$	0	0
$965$	−4.69530	−0.151147
$966$	0	0
$967$	2.31993	0.0746039	0.0373020	−	0.999304i	$-0.488124\pi$
$967$	2.31993	0.0746039	0.0373020	+	0.999304i	$0.488124\pi$
$968$	0	0
$969$	15.3034	0.491616
$970$	0	0
$971$	−0.750000	−0.0240686	−0.0120343	−	0.999928i	$-0.503831\pi$
$971$	−0.750000	−0.0240686	−0.0120343	+	0.999928i	$0.503831\pi$
$972$	0	0
$973$	−8.65134	−0.277349
$974$	0	0
$975$	−43.7250	−1.40032
$976$	0	0
$977$	11.9268	0.381571	0.190786	−	0.981632i	$-0.438896\pi$
$977$	11.9268	0.381571	0.190786	+	0.981632i	$0.438896\pi$
$978$	0	0
$979$	−37.9417	−1.21262
$980$	0	0
$981$	−15.0539	−0.480635
$982$	0	0
$983$	58.1619	1.85508	0.927538	−	0.373730i	$-0.121921\pi$
$983$	58.1619	1.85508	0.927538	+	0.373730i	$0.121921\pi$
$984$	0	0
$985$	−92.2326	−2.93877
$986$	0	0
$987$	−10.7350	−0.341698
$988$	0	0
$989$	−4.50727	−0.143323
$990$	0	0
$991$	−24.4634	−0.777105	−0.388553	−	0.921426i	$-0.627025\pi$
$991$	−24.4634	−0.777105	−0.388553	+	0.921426i	$0.627025\pi$
$992$	0	0
$993$	18.4327	0.584943
$994$	0	0
$995$	−13.5170	−0.428516
$996$	0	0
$997$	0.300217	0.00950796	0.00475398	−	0.999989i	$-0.498487\pi$
$997$	0.300217	0.00950796	0.00475398	+	0.999989i	$0.498487\pi$
$998$	0	0
$999$	−2.05276	−0.0649466

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	6024.2.a.p.1.13	✓	14

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
6024.2.a.p.1.13	✓	14	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)

\(f(q)\)	\(=\)	\(q-1.00000 q^{3} +3.63608 q^{5} +2.74254 q^{7} +1.00000 q^{9} +O(q^{10})\)	\(q-1.00000 q^{3} +3.63608 q^{5} +2.74254 q^{7} +1.00000 q^{9} +3.30758 q^{11} +5.31864 q^{13} -3.63608 q^{15} +3.11451 q^{17} -4.91358 q^{19} -2.74254 q^{21} -1.16838 q^{23} +8.22110 q^{25} -1.00000 q^{27} -3.53214 q^{29} +2.72293 q^{31} -3.30758 q^{33} +9.97209 q^{35} +2.05276 q^{37} -5.31864 q^{39} +4.50924 q^{41} +3.85770 q^{43} +3.63608 q^{45} +3.91425 q^{47} +0.521509 q^{49} -3.11451 q^{51} +9.24140 q^{53} +12.0266 q^{55} +4.91358 q^{57} -8.59898 q^{59} +2.00967 q^{61} +2.74254 q^{63} +19.3390 q^{65} +5.49320 q^{67} +1.16838 q^{69} +4.41623 q^{71} -5.91685 q^{73} -8.22110 q^{75} +9.07115 q^{77} -16.0135 q^{79} +1.00000 q^{81} -14.2902 q^{83} +11.3246 q^{85} +3.53214 q^{87} -11.4711 q^{89} +14.5866 q^{91} -2.72293 q^{93} -17.8662 q^{95} +7.25459 q^{97} +3.30758 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 14 q - 14 q^{3} + 6 q^{5} + 3 q^{7} + 14 q^{9}+O(q^{10}) \) 14 * q - 14 * q^3 + 6 * q^5 + 3 * q^7 + 14 * q^9	\( 14 q - 14 q^{3} + 6 q^{5} + 3 q^{7} + 14 q^{9} + 7 q^{11} + 5 q^{13} - 6 q^{15} + 28 q^{17} - q^{19} - 3 q^{21} - 3 q^{23} + 24 q^{25} - 14 q^{27} + 15 q^{29} + 9 q^{31} - 7 q^{33} - 14 q^{35} + 4 q^{37} - 5 q^{39} + 32 q^{41} - 3 q^{43} + 6 q^{45} + 15 q^{47} + 27 q^{49} - 28 q^{51} + 8 q^{53} + q^{57} - 11 q^{59} + 25 q^{61} + 3 q^{63} + 18 q^{65} - 20 q^{67} + 3 q^{69} + 33 q^{71} + 8 q^{73} - 24 q^{75} + 14 q^{77} - 17 q^{79} + 14 q^{81} - 4 q^{83} + 16 q^{85} - 15 q^{87} + 14 q^{89} - 28 q^{91} - 9 q^{93} + 3 q^{95} + 9 q^{97} + 7 q^{99}+O(q^{100}) \) 14 * q - 14 * q^3 + 6 * q^5 + 3 * q^7 + 14 * q^9 + 7 * q^11 + 5 * q^13 - 6 * q^15 + 28 * q^17 - q^19 - 3 * q^21 - 3 * q^23 + 24 * q^25 - 14 * q^27 + 15 * q^29 + 9 * q^31 - 7 * q^33 - 14 * q^35 + 4 * q^37 - 5 * q^39 + 32 * q^41 - 3 * q^43 + 6 * q^45 + 15 * q^47 + 27 * q^49 - 28 * q^51 + 8 * q^53 + q^57 - 11 * q^59 + 25 * q^61 + 3 * q^63 + 18 * q^65 - 20 * q^67 + 3 * q^69 + 33 * q^71 + 8 * q^73 - 24 * q^75 + 14 * q^77 - 17 * q^79 + 14 * q^81 - 4 * q^83 + 16 * q^85 - 15 * q^87 + 14 * q^89 - 28 * q^91 - 9 * q^93 + 3 * q^95 + 9 * q^97 + 7 * q^99

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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