LMFDB - Embedded newform 4020.2.a.f.1.4

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [4020,2,Mod(1,4020)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(4020, 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("4020.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 4020 = 2^{2} \cdot 3 \cdot 5 \cdot 67 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	4020.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:	$32.0998616126$
Analytic rank:	$0$
Dimension:	$6$
Coefficient field:	$\mathbb{Q}[x]/(x^{6} - \cdots)$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{6} - x^{5} - 28x^{4} - 12x^{3} + 209x^{2} + 360x + 144 $ x^6 - x^5 - 28x^4 - 12x^3 + 209x^2 + 360x + 144
Coefficient ring:	$\Z[a_1, \ldots, a_{13}]$
Coefficient ring index:	$ 1 $
Twist minimal:	yes
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.4
Root			$-0.620094$ of defining polynomial
Character	$\chi$	$=$	4020.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-1.00000 q^{3} +1.00000 q^{5} -0.620094 q^{7} +1.00000 q^{9} +O(q^{10})$	$q-1.00000 q^{3} +1.00000 q^{5} -0.620094 q^{7} +1.00000 q^{9} +4.33117 q^{11} +4.44645 q^{13} -1.00000 q^{15} +3.88472 q^{17} +4.73642 q^{19} +0.620094 q^{21} -2.44937 q^{23} +1.00000 q^{25} -1.00000 q^{27} +3.01878 q^{29} -2.50481 q^{31} -4.33117 q^{33} -0.620094 q^{35} +4.44937 q^{37} -4.44645 q^{39} -6.37530 q^{41} +1.02534 q^{43} +1.00000 q^{45} +1.42403 q^{47} -6.61548 q^{49} -3.88472 q^{51} +7.97661 q^{53} +4.33117 q^{55} -4.73642 q^{57} -2.44645 q^{59} -2.73642 q^{61} -0.620094 q^{63} +4.44645 q^{65} +1.00000 q^{67} +2.44937 q^{69} -12.8217 q^{71} -1.71764 q^{73} -1.00000 q^{75} -2.68573 q^{77} -15.6088 q^{79} +1.00000 q^{81} +11.0902 q^{83} +3.88472 q^{85} -3.01878 q^{87} -2.52157 q^{89} -2.75722 q^{91} +2.50481 q^{93} +4.73642 q^{95} +4.82181 q^{97} +4.33117 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 6 q - 6 q^{3} + 6 q^{5} + q^{7} + 6 q^{9}+O(q^{10}) $ 6 * q - 6 * q^3 + 6 * q^5 + q^7 + 6 * q^9	$ 6 q - 6 q^{3} + 6 q^{5} + q^{7} + 6 q^{9} - 7 q^{11} + 7 q^{13} - 6 q^{15} + 10 q^{17} - 3 q^{19} - q^{21} - 4 q^{23} + 6 q^{25} - 6 q^{27} + 9 q^{29} + 3 q^{31} + 7 q^{33} + q^{35} + 16 q^{37} - 7 q^{39} + 7 q^{41} + 3 q^{43} + 6 q^{45} + q^{47} + 15 q^{49} - 10 q^{51} + 7 q^{53} - 7 q^{55} + 3 q^{57} + 5 q^{59} + 15 q^{61} + q^{63} + 7 q^{65} + 6 q^{67} + 4 q^{69} - 12 q^{71} + 12 q^{73} - 6 q^{75} + 9 q^{77} + 9 q^{79} + 6 q^{81} - 2 q^{83} + 10 q^{85} - 9 q^{87} + 10 q^{89} - 5 q^{91} - 3 q^{93} - 3 q^{95} + 37 q^{97} - 7 q^{99}+O(q^{100}) $ 6 * q - 6 * q^3 + 6 * q^5 + q^7 + 6 * q^9 - 7 * q^11 + 7 * q^13 - 6 * q^15 + 10 * q^17 - 3 * q^19 - q^21 - 4 * q^23 + 6 * q^25 - 6 * q^27 + 9 * q^29 + 3 * q^31 + 7 * q^33 + q^35 + 16 * q^37 - 7 * q^39 + 7 * q^41 + 3 * q^43 + 6 * q^45 + q^47 + 15 * q^49 - 10 * q^51 + 7 * q^53 - 7 * q^55 + 3 * q^57 + 5 * q^59 + 15 * q^61 + q^63 + 7 * q^65 + 6 * q^67 + 4 * q^69 - 12 * q^71 + 12 * q^73 - 6 * q^75 + 9 * q^77 + 9 * q^79 + 6 * q^81 - 2 * q^83 + 10 * q^85 - 9 * q^87 + 10 * q^89 - 5 * q^91 - 3 * q^93 - 3 * q^95 + 37 * 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
$5$	1.00000	0.447214
$5$	1.00000	0.447214
$6$	0	0
$7$	−0.620094	−0.234374	−0.117187	−	0.993110i	$-0.537388\pi$
$7$	−0.620094	−0.234374	−0.117187	+	0.993110i	$0.537388\pi$
$8$	0	0
$9$	1.00000	0.333333
$10$	0	0
$11$	4.33117	1.30590	0.652948	−	0.757402i	$-0.273532\pi$
$11$	4.33117	1.30590	0.652948	+	0.757402i	$0.273532\pi$
$12$	0	0
$13$	4.44645	1.23322	0.616612	−	0.787267i	$-0.288505\pi$
$13$	4.44645	1.23322	0.616612	+	0.787267i	$0.288505\pi$
$14$	0	0
$15$	−1.00000	−0.258199
$16$	0	0
$17$	3.88472	0.942183	0.471091	−	0.882084i	$-0.343860\pi$
$17$	3.88472	0.942183	0.471091	+	0.882084i	$0.343860\pi$
$18$	0	0
$19$	4.73642	1.08661	0.543304	−	0.839536i	$-0.317173\pi$
$19$	4.73642	1.08661	0.543304	+	0.839536i	$0.317173\pi$
$20$	0	0
$21$	0.620094	0.135316
$22$	0	0
$23$	−2.44937	−0.510730	−0.255365	−	0.966845i	$-0.582196\pi$
$23$	−2.44937	−0.510730	−0.255365	+	0.966845i	$0.582196\pi$
$24$	0	0
$25$	1.00000	0.200000
$26$	0	0
$27$	−1.00000	−0.192450
$28$	0	0
$29$	3.01878	0.560574	0.280287	−	0.959916i	$-0.409570\pi$
$29$	3.01878	0.560574	0.280287	+	0.959916i	$0.409570\pi$
$30$	0	0
$31$	−2.50481	−0.449878	−0.224939	−	0.974373i	$-0.572218\pi$
$31$	−2.50481	−0.449878	−0.224939	+	0.974373i	$0.572218\pi$
$32$	0	0
$33$	−4.33117	−0.753960
$34$	0	0
$35$	−0.620094	−0.104815
$36$	0	0
$37$	4.44937	0.731473	0.365736	−	0.930719i	$-0.380817\pi$
$37$	4.44937	0.731473	0.365736	+	0.930719i	$0.380817\pi$
$38$	0	0
$39$	−4.44645	−0.712002
$40$	0	0
$41$	−6.37530	−0.995654	−0.497827	−	0.867276i	$-0.665869\pi$
$41$	−6.37530	−0.995654	−0.497827	+	0.867276i	$0.665869\pi$
$42$	0	0
$43$	1.02534	0.156363	0.0781817	−	0.996939i	$-0.475089\pi$
$43$	1.02534	0.156363	0.0781817	+	0.996939i	$0.475089\pi$
$44$	0	0
$45$	1.00000	0.149071
$46$	0	0
$47$	1.42403	0.207716	0.103858	−	0.994592i	$-0.466881\pi$
$47$	1.42403	0.207716	0.103858	+	0.994592i	$0.466881\pi$
$48$	0	0
$49$	−6.61548	−0.945069
$50$	0	0
$51$	−3.88472	−0.543970
$52$	0	0
$53$	7.97661	1.09567	0.547836	−	0.836586i	$-0.315452\pi$
$53$	7.97661	1.09567	0.547836	+	0.836586i	$0.315452\pi$
$54$	0	0
$55$	4.33117	0.584015
$56$	0	0
$57$	−4.73642	−0.627354
$58$	0	0
$59$	−2.44645	−0.318501	−0.159250	−	0.987238i	$-0.550908\pi$
$59$	−2.44645	−0.318501	−0.159250	+	0.987238i	$0.550908\pi$
$60$	0	0
$61$	−2.73642	−0.350362	−0.175181	−	0.984536i	$-0.556051\pi$
$61$	−2.73642	−0.350362	−0.175181	+	0.984536i	$0.556051\pi$
$62$	0	0
$63$	−0.620094	−0.0781245
$64$	0	0
$65$	4.44645	0.551514
$66$	0	0
$67$	1.00000	0.122169
$67$	1.00000	0.122169
$68$	0	0
$69$	2.44937	0.294870
$70$	0	0
$71$	−12.8217	−1.52166	−0.760831	−	0.648951i	$-0.775208\pi$
$71$	−12.8217	−1.52166	−0.760831	+	0.648951i	$0.775208\pi$
$72$	0	0
$73$	−1.71764	−0.201034	−0.100517	−	0.994935i	$-0.532050\pi$
$73$	−1.71764	−0.201034	−0.100517	+	0.994935i	$0.532050\pi$
$74$	0	0
$75$	−1.00000	−0.115470
$76$	0	0
$77$	−2.68573	−0.306068
$78$	0	0
$79$	−15.6088	−1.75613	−0.878066	−	0.478539i	$-0.841167\pi$
$79$	−15.6088	−1.75613	−0.878066	+	0.478539i	$0.841167\pi$
$80$	0	0
$81$	1.00000	0.111111
$82$	0	0
$83$	11.0902	1.21731	0.608654	−	0.793436i	$-0.291710\pi$
$83$	11.0902	1.21731	0.608654	+	0.793436i	$0.291710\pi$
$84$	0	0
$85$	3.88472	0.421357
$86$	0	0
$87$	−3.01878	−0.323647
$88$	0	0
$89$	−2.52157	−0.267286	−0.133643	−	0.991030i	$-0.542668\pi$
$89$	−2.52157	−0.267286	−0.133643	+	0.991030i	$0.542668\pi$
$90$	0	0
$91$	−2.75722	−0.289035
$92$	0	0
$93$	2.50481	0.259737
$94$	0	0
$95$	4.73642	0.485946
$96$	0	0
$97$	4.82181	0.489581	0.244790	−	0.969576i	$-0.421281\pi$
$97$	4.82181	0.489581	0.244790	+	0.969576i	$0.421281\pi$
$98$	0	0
$99$	4.33117	0.435299
$100$	0	0
$101$	8.26365	0.822264	0.411132	−	0.911576i	$-0.365134\pi$
$101$	8.26365	0.822264	0.411132	+	0.911576i	$0.365134\pi$
$102$	0	0
$103$	−1.87341	−0.184592	−0.0922961	−	0.995732i	$-0.529421\pi$
$103$	−1.87341	−0.184592	−0.0922961	+	0.995732i	$0.529421\pi$
$104$	0	0
$105$	0.620094	0.0605150
$106$	0	0
$107$	2.04413	0.197613	0.0988065	−	0.995107i	$-0.468498\pi$
$107$	2.04413	0.197613	0.0988065	+	0.995107i	$0.468498\pi$
$108$	0	0
$109$	17.2346	1.65078	0.825388	−	0.564567i	$-0.190957\pi$
$109$	17.2346	1.65078	0.825388	+	0.564567i	$0.190957\pi$
$110$	0	0
$111$	−4.44937	−0.422316
$112$	0	0
$113$	9.46231	0.890139	0.445070	−	0.895496i	$-0.353179\pi$
$113$	9.46231	0.890139	0.445070	+	0.895496i	$0.353179\pi$
$114$	0	0
$115$	−2.44937	−0.228405
$116$	0	0
$117$	4.44645	0.411074
$118$	0	0
$119$	−2.40889	−0.220823
$120$	0	0
$121$	7.75903	0.705366
$122$	0	0
$123$	6.37530	0.574841
$124$	0	0
$125$	1.00000	0.0894427
$126$	0	0
$127$	9.41488	0.835435	0.417718	−	0.908577i	$-0.362830\pi$
$127$	9.41488	0.835435	0.417718	+	0.908577i	$0.362830\pi$
$128$	0	0
$129$	−1.02534	−0.0902764
$130$	0	0
$131$	−5.88472	−0.514150	−0.257075	−	0.966391i	$-0.582759\pi$
$131$	−5.88472	−0.514150	−0.257075	+	0.966391i	$0.582759\pi$
$132$	0	0
$133$	−2.93703	−0.254672
$134$	0	0
$135$	−1.00000	−0.0860663
$136$	0	0
$137$	−8.54998	−0.730474	−0.365237	−	0.930914i	$-0.619012\pi$
$137$	−8.54998	−0.730474	−0.365237	+	0.930914i	$0.619012\pi$
$138$	0	0
$139$	3.68008	0.312140	0.156070	−	0.987746i	$-0.450117\pi$
$139$	3.68008	0.312140	0.156070	+	0.987746i	$0.450117\pi$
$140$	0	0
$141$	−1.42403	−0.119925
$142$	0	0
$143$	19.2583	1.61046
$144$	0	0
$145$	3.01878	0.250696
$146$	0	0
$147$	6.61548	0.545636
$148$	0	0
$149$	0.828042	0.0678359	0.0339179	−	0.999425i	$-0.489202\pi$
$149$	0.828042	0.0678359	0.0339179	+	0.999425i	$0.489202\pi$
$150$	0	0
$151$	10.2618	0.835095	0.417548	−	0.908655i	$-0.362890\pi$
$151$	10.2618	0.835095	0.417548	+	0.908655i	$0.362890\pi$
$152$	0	0
$153$	3.88472	0.314061
$154$	0	0
$155$	−2.50481	−0.201191
$156$	0	0
$157$	−13.3845	−1.06820	−0.534100	−	0.845421i	$-0.679350\pi$
$157$	−13.3845	−1.06820	−0.534100	+	0.845421i	$0.679350\pi$
$158$	0	0
$159$	−7.97661	−0.632586
$160$	0	0
$161$	1.51884	0.119702
$162$	0	0
$163$	6.40525	0.501698	0.250849	−	0.968026i	$-0.419290\pi$
$163$	6.40525	0.501698	0.250849	+	0.968026i	$0.419290\pi$
$164$	0	0
$165$	−4.33117	−0.337181
$166$	0	0
$167$	−9.90909	−0.766788	−0.383394	−	0.923585i	$-0.625245\pi$
$167$	−9.90909	−0.766788	−0.383394	+	0.923585i	$0.625245\pi$
$168$	0	0
$169$	6.77092	0.520840
$170$	0	0
$171$	4.73642	0.362203
$172$	0	0
$173$	12.9990	0.988298	0.494149	−	0.869377i	$-0.335480\pi$
$173$	12.9990	0.988298	0.494149	+	0.869377i	$0.335480\pi$
$174$	0	0
$175$	−0.620094	−0.0468747
$176$	0	0
$177$	2.44645	0.183886
$178$	0	0
$179$	−8.21114	−0.613729	−0.306865	−	0.951753i	$-0.599280\pi$
$179$	−8.21114	−0.613729	−0.306865	+	0.951753i	$0.599280\pi$
$180$	0	0
$181$	8.35268	0.620850	0.310425	−	0.950598i	$-0.399529\pi$
$181$	8.35268	0.620850	0.310425	+	0.950598i	$0.399529\pi$
$182$	0	0
$183$	2.73642	0.202282
$184$	0	0
$185$	4.44937	0.327124
$186$	0	0
$187$	16.8254	1.23039
$188$	0	0
$189$	0.620094	0.0451052
$190$	0	0
$191$	9.03295	0.653602	0.326801	−	0.945093i	$-0.394029\pi$
$191$	9.03295	0.653602	0.326801	+	0.945093i	$0.394029\pi$
$192$	0	0
$193$	1.97927	0.142471	0.0712354	−	0.997460i	$-0.477306\pi$
$193$	1.97927	0.142471	0.0712354	+	0.997460i	$0.477306\pi$
$194$	0	0
$195$	−4.44645	−0.318417
$196$	0	0
$197$	−8.47011	−0.603470	−0.301735	−	0.953392i	$-0.597566\pi$
$197$	−8.47011	−0.603470	−0.301735	+	0.953392i	$0.597566\pi$
$198$	0	0
$199$	17.4589	1.23763	0.618816	−	0.785536i	$-0.287613\pi$
$199$	17.4589	1.23763	0.618816	+	0.785536i	$0.287613\pi$
$200$	0	0
$201$	−1.00000	−0.0705346
$202$	0	0
$203$	−1.87193	−0.131384
$204$	0	0
$205$	−6.37530	−0.445270
$206$	0	0
$207$	−2.44937	−0.170243
$208$	0	0
$209$	20.5142	1.41900
$210$	0	0
$211$	−19.3228	−1.33024	−0.665120	−	0.746737i	$-0.731620\pi$
$211$	−19.3228	−1.33024	−0.665120	+	0.746737i	$0.731620\pi$
$212$	0	0
$213$	12.8217	0.878531
$214$	0	0
$215$	1.02534	0.0699278
$216$	0	0
$217$	1.55322	0.105439
$218$	0	0
$219$	1.71764	0.116067
$220$	0	0
$221$	17.2732	1.16192
$222$	0	0
$223$	6.68502	0.447662	0.223831	−	0.974628i	$-0.428144\pi$
$223$	6.68502	0.447662	0.223831	+	0.974628i	$0.428144\pi$
$224$	0	0
$225$	1.00000	0.0666667
$226$	0	0
$227$	−20.3201	−1.34869	−0.674347	−	0.738415i	$-0.735575\pi$
$227$	−20.3201	−1.34869	−0.674347	+	0.738415i	$0.735575\pi$
$228$	0	0
$229$	15.6228	1.03238	0.516190	−	0.856474i	$-0.327350\pi$
$229$	15.6228	1.03238	0.516190	+	0.856474i	$0.327350\pi$
$230$	0	0
$231$	2.68573	0.176708
$232$	0	0
$233$	−4.00649	−0.262474	−0.131237	−	0.991351i	$-0.541895\pi$
$233$	−4.00649	−0.262474	−0.131237	+	0.991351i	$0.541895\pi$
$234$	0	0
$235$	1.42403	0.0928936
$236$	0	0
$237$	15.6088	1.01390
$238$	0	0
$239$	−15.1306	−0.978719	−0.489360	−	0.872082i	$-0.662769\pi$
$239$	−15.1306	−0.978719	−0.489360	+	0.872082i	$0.662769\pi$
$240$	0	0
$241$	−5.44807	−0.350941	−0.175470	−	0.984485i	$-0.556145\pi$
$241$	−5.44807	−0.350941	−0.175470	+	0.984485i	$0.556145\pi$
$242$	0	0
$243$	−1.00000	−0.0641500
$244$	0	0
$245$	−6.61548	−0.422648
$246$	0	0
$247$	21.0602	1.34003
$248$	0	0
$249$	−11.0902	−0.702813
$250$	0	0
$251$	−26.7889	−1.69090	−0.845449	−	0.534056i	$-0.820667\pi$
$251$	−26.7889	−1.69090	−0.845449	+	0.534056i	$0.820667\pi$
$252$	0	0
$253$	−10.6087	−0.666960
$254$	0	0
$255$	−3.88472	−0.243271
$256$	0	0
$257$	1.38939	0.0866678	0.0433339	−	0.999061i	$-0.486202\pi$
$257$	1.38939	0.0866678	0.0433339	+	0.999061i	$0.486202\pi$
$258$	0	0
$259$	−2.75903	−0.171438
$260$	0	0
$261$	3.01878	0.186858
$262$	0	0
$263$	1.61886	0.0998229	0.0499115	−	0.998754i	$-0.484106\pi$
$263$	1.61886	0.0998229	0.0499115	+	0.998754i	$0.484106\pi$
$264$	0	0
$265$	7.97661	0.489999
$266$	0	0
$267$	2.52157	0.154318
$268$	0	0
$269$	28.7917	1.75546	0.877731	−	0.479154i	$-0.159056\pi$
$269$	28.7917	1.75546	0.877731	+	0.479154i	$0.159056\pi$
$270$	0	0
$271$	1.43423	0.0871231	0.0435616	−	0.999051i	$-0.486130\pi$
$271$	1.43423	0.0871231	0.0435616	+	0.999051i	$0.486130\pi$
$272$	0	0
$273$	2.75722	0.166874
$274$	0	0
$275$	4.33117	0.261179
$276$	0	0
$277$	13.0538	0.784324	0.392162	−	0.919896i	$-0.371727\pi$
$277$	13.0538	0.784324	0.392162	+	0.919896i	$0.371727\pi$
$278$	0	0
$279$	−2.50481	−0.149959
$280$	0	0
$281$	−16.3311	−0.974232	−0.487116	−	0.873337i	$-0.661951\pi$
$281$	−16.3311	−0.974232	−0.487116	+	0.873337i	$0.661951\pi$
$282$	0	0
$283$	5.03450	0.299270	0.149635	−	0.988741i	$-0.452190\pi$
$283$	5.03450	0.299270	0.149635	+	0.988741i	$0.452190\pi$
$284$	0	0
$285$	−4.73642	−0.280561
$286$	0	0
$287$	3.95328	0.233355
$288$	0	0
$289$	−1.90895	−0.112291
$290$	0	0
$291$	−4.82181	−0.282660
$292$	0	0
$293$	−8.25890	−0.482490	−0.241245	−	0.970464i	$-0.577556\pi$
$293$	−8.25890	−0.482490	−0.241245	+	0.970464i	$0.577556\pi$
$294$	0	0
$295$	−2.44645	−0.142438
$296$	0	0
$297$	−4.33117	−0.251320
$298$	0	0
$299$	−10.8910	−0.629844
$300$	0	0
$301$	−0.635809	−0.0366474
$302$	0	0
$303$	−8.26365	−0.474734
$304$	0	0
$305$	−2.73642	−0.156687
$306$	0	0
$307$	15.8492	0.904564	0.452282	−	0.891875i	$-0.350610\pi$
$307$	15.8492	0.904564	0.452282	+	0.891875i	$0.350610\pi$
$308$	0	0
$309$	1.87341	0.106574
$310$	0	0
$311$	−1.57471	−0.0892939	−0.0446469	−	0.999003i	$-0.514216\pi$
$311$	−1.57471	−0.0892939	−0.0446469	+	0.999003i	$0.514216\pi$
$312$	0	0
$313$	20.5562	1.16191	0.580953	−	0.813937i	$-0.302680\pi$
$313$	20.5562	1.16191	0.580953	+	0.813937i	$0.302680\pi$
$314$	0	0
$315$	−0.620094	−0.0349383
$316$	0	0
$317$	9.86061	0.553827	0.276914	−	0.960895i	$-0.410688\pi$
$317$	9.86061	0.553827	0.276914	+	0.960895i	$0.410688\pi$
$318$	0	0
$319$	13.0749	0.732052
$320$	0	0
$321$	−2.04413	−0.114092
$322$	0	0
$323$	18.3997	1.02378
$324$	0	0
$325$	4.44645	0.246645
$326$	0	0
$327$	−17.2346	−0.953075
$328$	0	0
$329$	−0.883033	−0.0486832
$330$	0	0
$331$	−8.06124	−0.443086	−0.221543	−	0.975151i	$-0.571109\pi$
$331$	−8.06124	−0.443086	−0.221543	+	0.975151i	$0.571109\pi$
$332$	0	0
$333$	4.44937	0.243824
$334$	0	0
$335$	1.00000	0.0546358
$336$	0	0
$337$	4.99986	0.272360	0.136180	−	0.990684i	$-0.456518\pi$
$337$	4.99986	0.272360	0.136180	+	0.990684i	$0.456518\pi$
$338$	0	0
$339$	−9.46231	−0.513922
$340$	0	0
$341$	−10.8488	−0.587494
$342$	0	0
$343$	8.44288	0.455873
$344$	0	0
$345$	2.44937	0.131870
$346$	0	0
$347$	−5.32591	−0.285910	−0.142955	−	0.989729i	$-0.545660\pi$
$347$	−5.32591	−0.285910	−0.142955	+	0.989729i	$0.545660\pi$
$348$	0	0
$349$	20.4675	1.09560	0.547800	−	0.836609i	$-0.315465\pi$
$349$	20.4675	1.09560	0.547800	+	0.836609i	$0.315465\pi$
$350$	0	0
$351$	−4.44645	−0.237334
$352$	0	0
$353$	0.608854	0.0324060	0.0162030	−	0.999869i	$-0.494842\pi$
$353$	0.608854	0.0324060	0.0162030	+	0.999869i	$0.494842\pi$
$354$	0	0
$355$	−12.8217	−0.680508
$356$	0	0
$357$	2.40889	0.127492
$358$	0	0
$359$	−12.4041	−0.654663	−0.327331	−	0.944910i	$-0.606149\pi$
$359$	−12.4041	−0.654663	−0.327331	+	0.944910i	$0.606149\pi$
$360$	0	0
$361$	3.43366	0.180719
$362$	0	0
$363$	−7.75903	−0.407243
$364$	0	0
$365$	−1.71764	−0.0899052
$366$	0	0
$367$	5.51599	0.287932	0.143966	−	0.989583i	$-0.454014\pi$
$367$	5.51599	0.287932	0.143966	+	0.989583i	$0.454014\pi$
$368$	0	0
$369$	−6.37530	−0.331885
$370$	0	0
$371$	−4.94625	−0.256796
$372$	0	0
$373$	4.78501	0.247758	0.123879	−	0.992297i	$-0.460466\pi$
$373$	4.78501	0.247758	0.123879	+	0.992297i	$0.460466\pi$
$374$	0	0
$375$	−1.00000	−0.0516398
$376$	0	0
$377$	13.4229	0.691313
$378$	0	0
$379$	27.7740	1.42665	0.713327	−	0.700832i	$-0.247188\pi$
$379$	27.7740	1.42665	0.713327	+	0.700832i	$0.247188\pi$
$380$	0	0
$381$	−9.41488	−0.482339
$382$	0	0
$383$	1.81433	0.0927078	0.0463539	−	0.998925i	$-0.485240\pi$
$383$	1.81433	0.0927078	0.0463539	+	0.998925i	$0.485240\pi$
$384$	0	0
$385$	−2.68573	−0.136878
$386$	0	0
$387$	1.02534	0.0521211
$388$	0	0
$389$	22.2844	1.12986	0.564932	−	0.825138i	$-0.308903\pi$
$389$	22.2844	1.12986	0.564932	+	0.825138i	$0.308903\pi$
$390$	0	0
$391$	−9.51513	−0.481201
$392$	0	0
$393$	5.88472	0.296845
$394$	0	0
$395$	−15.6088	−0.785366
$396$	0	0
$397$	1.19937	0.0601946	0.0300973	−	0.999547i	$-0.490418\pi$
$397$	1.19937	0.0601946	0.0300973	+	0.999547i	$0.490418\pi$
$398$	0	0
$399$	2.93703	0.147035
$400$	0	0
$401$	−11.1869	−0.558648	−0.279324	−	0.960197i	$-0.590110\pi$
$401$	−11.1869	−0.558648	−0.279324	+	0.960197i	$0.590110\pi$
$402$	0	0
$403$	−11.1375	−0.554800
$404$	0	0
$405$	1.00000	0.0496904
$406$	0	0
$407$	19.2710	0.955228
$408$	0	0
$409$	−8.79875	−0.435070	−0.217535	−	0.976052i	$-0.569802\pi$
$409$	−8.79875	−0.435070	−0.217535	+	0.976052i	$0.569802\pi$
$410$	0	0
$411$	8.54998	0.421740
$412$	0	0
$413$	1.51703	0.0746481
$414$	0	0
$415$	11.0902	0.544397
$416$	0	0
$417$	−3.68008	−0.180214
$418$	0	0
$419$	−6.72218	−0.328400	−0.164200	−	0.986427i	$-0.552504\pi$
$419$	−6.72218	−0.328400	−0.164200	+	0.986427i	$0.552504\pi$
$420$	0	0
$421$	28.6889	1.39821	0.699105	−	0.715019i	$-0.253582\pi$
$421$	28.6889	1.39821	0.699105	+	0.715019i	$0.253582\pi$
$422$	0	0
$423$	1.42403	0.0692388
$424$	0	0
$425$	3.88472	0.188437
$426$	0	0
$427$	1.69684	0.0821157
$428$	0	0
$429$	−19.2583	−0.929801
$430$	0	0
$431$	−15.8708	−0.764470	−0.382235	−	0.924065i	$-0.624846\pi$
$431$	−15.8708	−0.764470	−0.382235	+	0.924065i	$0.624846\pi$
$432$	0	0
$433$	13.7198	0.659331	0.329665	−	0.944098i	$-0.393064\pi$
$433$	13.7198	0.659331	0.329665	+	0.944098i	$0.393064\pi$
$434$	0	0
$435$	−3.01878	−0.144740
$436$	0	0
$437$	−11.6013	−0.554964
$438$	0	0
$439$	−25.0386	−1.19503	−0.597514	−	0.801859i	$-0.703845\pi$
$439$	−25.0386	−1.19503	−0.597514	+	0.801859i	$0.703845\pi$
$440$	0	0
$441$	−6.61548	−0.315023
$442$	0	0
$443$	−6.98046	−0.331651	−0.165826	−	0.986155i	$-0.553029\pi$
$443$	−6.98046	−0.331651	−0.165826	+	0.986155i	$0.553029\pi$
$444$	0	0
$445$	−2.52157	−0.119534
$446$	0	0
$447$	−0.828042	−0.0391650
$448$	0	0
$449$	5.36419	0.253152	0.126576	−	0.991957i	$-0.459601\pi$
$449$	5.36419	0.253152	0.126576	+	0.991957i	$0.459601\pi$
$450$	0	0
$451$	−27.6125	−1.30022
$452$	0	0
$453$	−10.2618	−0.482142
$454$	0	0
$455$	−2.75722	−0.129260
$456$	0	0
$457$	19.2981	0.902729	0.451365	−	0.892340i	$-0.350937\pi$
$457$	19.2981	0.902729	0.451365	+	0.892340i	$0.350937\pi$
$458$	0	0
$459$	−3.88472	−0.181323
$460$	0	0
$461$	−11.9390	−0.556057	−0.278028	−	0.960573i	$-0.589681\pi$
$461$	−11.9390	−0.556057	−0.278028	+	0.960573i	$0.589681\pi$
$462$	0	0
$463$	20.2732	0.942176	0.471088	−	0.882086i	$-0.343861\pi$
$463$	20.2732	0.942176	0.471088	+	0.882086i	$0.343861\pi$
$464$	0	0
$465$	2.50481	0.116158
$466$	0	0
$467$	1.17883	0.0545495	0.0272748	−	0.999628i	$-0.491317\pi$
$467$	1.17883	0.0545495	0.0272748	+	0.999628i	$0.491317\pi$
$468$	0	0
$469$	−0.620094	−0.0286333
$470$	0	0
$471$	13.3845	0.616726
$472$	0	0
$473$	4.44093	0.204194
$474$	0	0
$475$	4.73642	0.217322
$476$	0	0
$477$	7.97661	0.365224
$478$	0	0
$479$	5.79640	0.264844	0.132422	−	0.991193i	$-0.457725\pi$
$479$	5.79640	0.264844	0.132422	+	0.991193i	$0.457725\pi$
$480$	0	0
$481$	19.7839	0.902069
$482$	0	0
$483$	−1.51884	−0.0691097
$484$	0	0
$485$	4.82181	0.218947
$486$	0	0
$487$	1.07637	0.0487748	0.0243874	−	0.999703i	$-0.492236\pi$
$487$	1.07637	0.0487748	0.0243874	+	0.999703i	$0.492236\pi$
$488$	0	0
$489$	−6.40525	−0.289655
$490$	0	0
$491$	−4.16975	−0.188178	−0.0940890	−	0.995564i	$-0.529994\pi$
$491$	−4.16975	−0.188178	−0.0940890	+	0.995564i	$0.529994\pi$
$492$	0	0
$493$	11.7271	0.528163
$494$	0	0
$495$	4.33117	0.194672
$496$	0	0
$497$	7.95069	0.356637
$498$	0	0
$499$	38.0228	1.70213	0.851066	−	0.525058i	$-0.175957\pi$
$499$	38.0228	1.70213	0.851066	+	0.525058i	$0.175957\pi$
$500$	0	0
$501$	9.90909	0.442705
$502$	0	0
$503$	−17.1898	−0.766453	−0.383227	−	0.923654i	$-0.625187\pi$
$503$	−17.1898	−0.766453	−0.383227	+	0.923654i	$0.625187\pi$
$504$	0	0
$505$	8.26365	0.367728
$506$	0	0
$507$	−6.77092	−0.300707
$508$	0	0
$509$	0.0817070	0.00362160	0.00181080	−	0.999998i	$-0.499424\pi$
$509$	0.0817070	0.00362160	0.00181080	+	0.999998i	$0.499424\pi$
$510$	0	0
$511$	1.06510	0.0471171
$512$	0	0
$513$	−4.73642	−0.209118
$514$	0	0
$515$	−1.87341	−0.0825521
$516$	0	0
$517$	6.16772	0.271256
$518$	0	0
$519$	−12.9990	−0.570594
$520$	0	0
$521$	−42.8332	−1.87656	−0.938279	−	0.345878i	$-0.887581\pi$
$521$	−42.8332	−1.87656	−0.938279	+	0.345878i	$0.887581\pi$
$522$	0	0
$523$	3.78411	0.165468	0.0827339	−	0.996572i	$-0.473635\pi$
$523$	3.78411	0.165468	0.0827339	+	0.996572i	$0.473635\pi$
$524$	0	0
$525$	0.620094	0.0270631
$526$	0	0
$527$	−9.73050	−0.423867
$528$	0	0
$529$	−17.0006	−0.739155
$530$	0	0
$531$	−2.44645	−0.106167
$532$	0	0
$533$	−28.3474	−1.22786
$534$	0	0
$535$	2.04413	0.0883753
$536$	0	0
$537$	8.21114	0.354337
$538$	0	0
$539$	−28.6528	−1.23416
$540$	0	0
$541$	−38.5024	−1.65535	−0.827674	−	0.561209i	$-0.810336\pi$
$541$	−38.5024	−1.65535	−0.827674	+	0.561209i	$0.810336\pi$
$542$	0	0
$543$	−8.35268	−0.358448
$544$	0	0
$545$	17.2346	0.738249
$546$	0	0
$547$	4.52664	0.193545	0.0967726	−	0.995307i	$-0.469148\pi$
$547$	4.52664	0.193545	0.0967726	+	0.995307i	$0.469148\pi$
$548$	0	0
$549$	−2.73642	−0.116787
$550$	0	0
$551$	14.2982	0.609125
$552$	0	0
$553$	9.67896	0.411591
$554$	0	0
$555$	−4.44937	−0.188865
$556$	0	0
$557$	8.81363	0.373445	0.186723	−	0.982413i	$-0.440213\pi$
$557$	8.81363	0.373445	0.186723	+	0.982413i	$0.440213\pi$
$558$	0	0
$559$	4.55914	0.192831
$560$	0	0
$561$	−16.8254	−0.710368
$562$	0	0
$563$	22.8420	0.962677	0.481339	−	0.876535i	$-0.340151\pi$
$563$	22.8420	0.962677	0.481339	+	0.876535i	$0.340151\pi$
$564$	0	0
$565$	9.46231	0.398082
$566$	0	0
$567$	−0.620094	−0.0260415
$568$	0	0
$569$	30.9278	1.29656	0.648280	−	0.761402i	$-0.275488\pi$
$569$	30.9278	1.29656	0.648280	+	0.761402i	$0.275488\pi$
$570$	0	0
$571$	−12.4220	−0.519845	−0.259923	−	0.965629i	$-0.583697\pi$
$571$	−12.4220	−0.519845	−0.259923	+	0.965629i	$0.583697\pi$
$572$	0	0
$573$	−9.03295	−0.377357
$574$	0	0
$575$	−2.44937	−0.102146
$576$	0	0
$577$	−16.9825	−0.706992	−0.353496	−	0.935436i	$-0.615007\pi$
$577$	−16.9825	−0.706992	−0.353496	+	0.935436i	$0.615007\pi$
$578$	0	0
$579$	−1.97927	−0.0822556
$580$	0	0
$581$	−6.87697	−0.285305
$582$	0	0
$583$	34.5480	1.43083
$584$	0	0
$585$	4.44645	0.183838
$586$	0	0
$587$	−11.7195	−0.483716	−0.241858	−	0.970312i	$-0.577757\pi$
$587$	−11.7195	−0.483716	−0.241858	+	0.970312i	$0.577757\pi$
$588$	0	0
$589$	−11.8638	−0.488841
$590$	0	0
$591$	8.47011	0.348414
$592$	0	0
$593$	−11.3772	−0.467207	−0.233604	−	0.972332i	$-0.575052\pi$
$593$	−11.3772	−0.467207	−0.233604	+	0.972332i	$0.575052\pi$
$594$	0	0
$595$	−2.40889	−0.0987549
$596$	0	0
$597$	−17.4589	−0.714547
$598$	0	0
$599$	26.0994	1.06639	0.533195	−	0.845992i	$-0.320991\pi$
$599$	26.0994	1.06639	0.533195	+	0.845992i	$0.320991\pi$
$600$	0	0
$601$	17.4410	0.711433	0.355717	−	0.934594i	$-0.384237\pi$
$601$	17.4410	0.711433	0.355717	+	0.934594i	$0.384237\pi$
$602$	0	0
$603$	1.00000	0.0407231
$604$	0	0
$605$	7.75903	0.315449
$606$	0	0
$607$	−18.5781	−0.754060	−0.377030	−	0.926201i	$-0.623055\pi$
$607$	−18.5781	−0.754060	−0.377030	+	0.926201i	$0.623055\pi$
$608$	0	0
$609$	1.87193	0.0758544
$610$	0	0
$611$	6.33188	0.256161
$612$	0	0
$613$	1.97193	0.0796454	0.0398227	−	0.999207i	$-0.487321\pi$
$613$	1.97193	0.0796454	0.0398227	+	0.999207i	$0.487321\pi$
$614$	0	0
$615$	6.37530	0.257077
$616$	0	0
$617$	−3.32592	−0.133896	−0.0669482	−	0.997756i	$-0.521326\pi$
$617$	−3.32592	−0.133896	−0.0669482	+	0.997756i	$0.521326\pi$
$618$	0	0
$619$	−40.6792	−1.63504	−0.817518	−	0.575903i	$-0.804651\pi$
$619$	−40.6792	−1.63504	−0.817518	+	0.575903i	$0.804651\pi$
$620$	0	0
$621$	2.44937	0.0982900
$622$	0	0
$623$	1.56361	0.0626448
$624$	0	0
$625$	1.00000	0.0400000
$626$	0	0
$627$	−20.5142	−0.819259
$628$	0	0
$629$	17.2846	0.689181
$630$	0	0
$631$	2.00760	0.0799215	0.0399607	−	0.999201i	$-0.487277\pi$
$631$	2.00760	0.0799215	0.0399607	+	0.999201i	$0.487277\pi$
$632$	0	0
$633$	19.3228	0.768014
$634$	0	0
$635$	9.41488	0.373618
$636$	0	0
$637$	−29.4154	−1.16548
$638$	0	0
$639$	−12.8217	−0.507220
$640$	0	0
$641$	−17.8109	−0.703489	−0.351745	−	0.936096i	$-0.614411\pi$
$641$	−17.8109	−0.703489	−0.351745	+	0.936096i	$0.614411\pi$
$642$	0	0
$643$	18.2403	0.719328	0.359664	−	0.933082i	$-0.382891\pi$
$643$	18.2403	0.719328	0.359664	+	0.933082i	$0.382891\pi$
$644$	0	0
$645$	−1.02534	−0.0403728
$646$	0	0
$647$	−30.6397	−1.20457	−0.602286	−	0.798281i	$-0.705743\pi$
$647$	−30.6397	−1.20457	−0.602286	+	0.798281i	$0.705743\pi$
$648$	0	0
$649$	−10.5960	−0.415929
$650$	0	0
$651$	−1.55322	−0.0608755
$652$	0	0
$653$	10.2231	0.400060	0.200030	−	0.979790i	$-0.435896\pi$
$653$	10.2231	0.400060	0.200030	+	0.979790i	$0.435896\pi$
$654$	0	0
$655$	−5.88472	−0.229935
$656$	0	0
$657$	−1.71764	−0.0670114
$658$	0	0
$659$	12.3636	0.481616	0.240808	−	0.970573i	$-0.422588\pi$
$659$	12.3636	0.481616	0.240808	+	0.970573i	$0.422588\pi$
$660$	0	0
$661$	30.2602	1.17699	0.588493	−	0.808502i	$-0.299721\pi$
$661$	30.2602	1.17699	0.588493	+	0.808502i	$0.299721\pi$
$662$	0	0
$663$	−17.2732	−0.670836
$664$	0	0
$665$	−2.93703	−0.113893
$666$	0	0
$667$	−7.39413	−0.286302
$668$	0	0
$669$	−6.68502	−0.258458
$670$	0	0
$671$	−11.8519	−0.457537
$672$	0	0
$673$	11.3678	0.438198	0.219099	−	0.975703i	$-0.429688\pi$
$673$	11.3678	0.438198	0.219099	+	0.975703i	$0.429688\pi$
$674$	0	0
$675$	−1.00000	−0.0384900
$676$	0	0
$677$	2.26190	0.0869320	0.0434660	−	0.999055i	$-0.486160\pi$
$677$	2.26190	0.0869320	0.0434660	+	0.999055i	$0.486160\pi$
$678$	0	0
$679$	−2.98998	−0.114745
$680$	0	0
$681$	20.3201	0.778669
$682$	0	0
$683$	−43.6211	−1.66911	−0.834557	−	0.550922i	$-0.814276\pi$
$683$	−43.6211	−1.66911	−0.834557	+	0.550922i	$0.814276\pi$
$684$	0	0
$685$	−8.54998	−0.326678
$686$	0	0
$687$	−15.6228	−0.596045
$688$	0	0
$689$	35.4676	1.35121
$690$	0	0
$691$	−28.7596	−1.09407	−0.547034	−	0.837110i	$-0.684243\pi$
$691$	−28.7596	−1.09407	−0.547034	+	0.837110i	$0.684243\pi$
$692$	0	0
$693$	−2.68573	−0.102023
$694$	0	0
$695$	3.68008	0.139593
$696$	0	0
$697$	−24.7662	−0.938088
$698$	0	0
$699$	4.00649	0.151539
$700$	0	0
$701$	−43.5020	−1.64305	−0.821523	−	0.570175i	$-0.806875\pi$
$701$	−43.5020	−1.64305	−0.821523	+	0.570175i	$0.806875\pi$
$702$	0	0
$703$	21.0741	0.794825
$704$	0	0
$705$	−1.42403	−0.0536321
$706$	0	0
$707$	−5.12424	−0.192717
$708$	0	0
$709$	−22.4164	−0.841867	−0.420933	−	0.907092i	$-0.638297\pi$
$709$	−22.4164	−0.841867	−0.420933	+	0.907092i	$0.638297\pi$
$710$	0	0
$711$	−15.6088	−0.585378
$712$	0	0
$713$	6.13523	0.229766
$714$	0	0
$715$	19.2583	0.720221
$716$	0	0
$717$	15.1306	0.565064
$718$	0	0
$719$	−51.0958	−1.90555	−0.952776	−	0.303675i	$-0.901786\pi$
$719$	−51.0958	−1.90555	−0.952776	+	0.303675i	$0.901786\pi$
$720$	0	0
$721$	1.16169	0.0432635
$722$	0	0
$723$	5.44807	0.202616
$724$	0	0
$725$	3.01878	0.112115
$726$	0	0
$727$	−48.9934	−1.81706	−0.908532	−	0.417816i	$-0.862796\pi$
$727$	−48.9934	−1.81706	−0.908532	+	0.417816i	$0.862796\pi$
$728$	0	0
$729$	1.00000	0.0370370
$730$	0	0
$731$	3.98317	0.147323
$732$	0	0
$733$	5.32375	0.196637	0.0983187	−	0.995155i	$-0.468654\pi$
$733$	5.32375	0.196637	0.0983187	+	0.995155i	$0.468654\pi$
$734$	0	0
$735$	6.61548	0.244016
$736$	0	0
$737$	4.33117	0.159541
$738$	0	0
$739$	−2.03866	−0.0749932	−0.0374966	−	0.999297i	$-0.511938\pi$
$739$	−2.03866	−0.0749932	−0.0374966	+	0.999297i	$0.511938\pi$
$740$	0	0
$741$	−21.0602	−0.773667
$742$	0	0
$743$	−46.4379	−1.70364	−0.851820	−	0.523834i	$-0.824501\pi$
$743$	−46.4379	−1.70364	−0.851820	+	0.523834i	$0.824501\pi$
$744$	0	0
$745$	0.828042	0.0303371
$746$	0	0
$747$	11.0902	0.405769
$748$	0	0
$749$	−1.26755	−0.0463153
$750$	0	0
$751$	16.0845	0.586933	0.293467	−	0.955969i	$-0.405191\pi$
$751$	16.0845	0.586933	0.293467	+	0.955969i	$0.405191\pi$
$752$	0	0
$753$	26.7889	0.976241
$754$	0	0
$755$	10.2618	0.373466
$756$	0	0
$757$	−45.2980	−1.64638	−0.823192	−	0.567763i	$-0.807809\pi$
$757$	−45.2980	−1.64638	−0.823192	+	0.567763i	$0.807809\pi$
$758$	0	0
$759$	10.6087	0.385070
$760$	0	0
$761$	25.4661	0.923147	0.461573	−	0.887102i	$-0.347285\pi$
$761$	25.4661	0.923147	0.461573	+	0.887102i	$0.347285\pi$
$762$	0	0
$763$	−10.6871	−0.386898
$764$	0	0
$765$	3.88472	0.140452
$766$	0	0
$767$	−10.8780	−0.392782
$768$	0	0
$769$	20.2778	0.731236	0.365618	−	0.930765i	$-0.380858\pi$
$769$	20.2778	0.731236	0.365618	+	0.930765i	$0.380858\pi$
$770$	0	0
$771$	−1.38939	−0.0500377
$772$	0	0
$773$	−19.3963	−0.697636	−0.348818	−	0.937190i	$-0.613417\pi$
$773$	−19.3963	−0.697636	−0.348818	+	0.937190i	$0.613417\pi$
$774$	0	0
$775$	−2.50481	−0.0899756
$776$	0	0
$777$	2.75903	0.0989797
$778$	0	0
$779$	−30.1961	−1.08189
$780$	0	0
$781$	−55.5332	−1.98713
$782$	0	0
$783$	−3.01878	−0.107882
$784$	0	0
$785$	−13.3845	−0.477714
$786$	0	0
$787$	14.6651	0.522755	0.261378	−	0.965237i	$-0.415823\pi$
$787$	14.6651	0.522755	0.261378	+	0.965237i	$0.415823\pi$
$788$	0	0
$789$	−1.61886	−0.0576328
$790$	0	0
$791$	−5.86752	−0.208625
$792$	0	0
$793$	−12.1673	−0.432075
$794$	0	0
$795$	−7.97661	−0.282901
$796$	0	0
$797$	−30.2704	−1.07223	−0.536116	−	0.844144i	$-0.680109\pi$
$797$	−30.2704	−1.07223	−0.536116	+	0.844144i	$0.680109\pi$
$798$	0	0
$799$	5.53196	0.195707
$800$	0	0
$801$	−2.52157	−0.0890954
$802$	0	0
$803$	−7.43937	−0.262530
$804$	0	0
$805$	1.51884	0.0535322
$806$	0	0
$807$	−28.7917	−1.01352
$808$	0	0
$809$	44.6949	1.57139	0.785694	−	0.618615i	$-0.212306\pi$
$809$	44.6949	1.57139	0.785694	+	0.618615i	$0.212306\pi$
$810$	0	0
$811$	−5.91908	−0.207847	−0.103923	−	0.994585i	$-0.533140\pi$
$811$	−5.91908	−0.207847	−0.103923	+	0.994585i	$0.533140\pi$
$812$	0	0
$813$	−1.43423	−0.0503006
$814$	0	0
$815$	6.40525	0.224366
$816$	0	0
$817$	4.85645	0.169906
$818$	0	0
$819$	−2.75722	−0.0963450
$820$	0	0
$821$	−10.4691	−0.365373	−0.182686	−	0.983171i	$-0.558479\pi$
$821$	−10.4691	−0.365373	−0.182686	+	0.983171i	$0.558479\pi$
$822$	0	0
$823$	−13.7780	−0.480272	−0.240136	−	0.970739i	$-0.577192\pi$
$823$	−13.7780	−0.480272	−0.240136	+	0.970739i	$0.577192\pi$
$824$	0	0
$825$	−4.33117	−0.150792
$826$	0	0
$827$	−23.0855	−0.802763	−0.401381	−	0.915911i	$-0.631470\pi$
$827$	−23.0855	−0.802763	−0.401381	+	0.915911i	$0.631470\pi$
$828$	0	0
$829$	−16.1809	−0.561987	−0.280993	−	0.959710i	$-0.590664\pi$
$829$	−16.1809	−0.561987	−0.280993	+	0.959710i	$0.590664\pi$
$830$	0	0
$831$	−13.0538	−0.452830
$832$	0	0
$833$	−25.6993	−0.890428
$834$	0	0
$835$	−9.90909	−0.342918
$836$	0	0
$837$	2.50481	0.0865790
$838$	0	0
$839$	−4.99502	−0.172447	−0.0862237	−	0.996276i	$-0.527480\pi$
$839$	−4.99502	−0.172447	−0.0862237	+	0.996276i	$0.527480\pi$
$840$	0	0
$841$	−19.8870	−0.685757
$842$	0	0
$843$	16.3311	0.562473
$844$	0	0
$845$	6.77092	0.232927
$846$	0	0
$847$	−4.81133	−0.165319
$848$	0	0
$849$	−5.03450	−0.172784
$850$	0	0
$851$	−10.8982	−0.373585
$852$	0	0
$853$	−29.5183	−1.01069	−0.505344	−	0.862918i	$-0.668635\pi$
$853$	−29.5183	−1.01069	−0.505344	+	0.862918i	$0.668635\pi$
$854$	0	0
$855$	4.73642	0.161982
$856$	0	0
$857$	−24.9255	−0.851440	−0.425720	−	0.904855i	$-0.639979\pi$
$857$	−24.9255	−0.851440	−0.425720	+	0.904855i	$0.639979\pi$
$858$	0	0
$859$	48.5089	1.65510	0.827550	−	0.561391i	$-0.189734\pi$
$859$	48.5089	1.65510	0.827550	+	0.561391i	$0.189734\pi$
$860$	0	0
$861$	−3.95328	−0.134728
$862$	0	0
$863$	53.3224	1.81511	0.907557	−	0.419929i	$-0.137945\pi$
$863$	53.3224	1.81511	0.907557	+	0.419929i	$0.137945\pi$
$864$	0	0
$865$	12.9990	0.441980
$866$	0	0
$867$	1.90895	0.0648314
$868$	0	0
$869$	−67.6046	−2.29333
$870$	0	0
$871$	4.44645	0.150662
$872$	0	0
$873$	4.82181	0.163194
$874$	0	0
$875$	−0.620094	−0.0209630
$876$	0	0
$877$	−21.3440	−0.720737	−0.360368	−	0.932810i	$-0.617349\pi$
$877$	−21.3440	−0.720737	−0.360368	+	0.932810i	$0.617349\pi$
$878$	0	0
$879$	8.25890	0.278566
$880$	0	0
$881$	−34.8241	−1.17325	−0.586627	−	0.809857i	$-0.699545\pi$
$881$	−34.8241	−1.17325	−0.586627	+	0.809857i	$0.699545\pi$
$882$	0	0
$883$	51.6794	1.73915	0.869576	−	0.493800i	$-0.164392\pi$
$883$	51.6794	1.73915	0.869576	+	0.493800i	$0.164392\pi$
$884$	0	0
$885$	2.44645	0.0822365
$886$	0	0
$887$	−35.2513	−1.18362	−0.591812	−	0.806076i	$-0.701587\pi$
$887$	−35.2513	−1.18362	−0.591812	+	0.806076i	$0.701587\pi$
$888$	0	0
$889$	−5.83811	−0.195804
$890$	0	0
$891$	4.33117	0.145100
$892$	0	0
$893$	6.74481	0.225706
$894$	0	0
$895$	−8.21114	−0.274468
$896$	0	0
$897$	10.8910	0.363641
$898$	0	0
$899$	−7.56149	−0.252190
$900$	0	0
$901$	30.9869	1.03232
$902$	0	0
$903$	0.635809	0.0211584
$904$	0	0
$905$	8.35268	0.277653
$906$	0	0
$907$	22.4579	0.745702	0.372851	−	0.927891i	$-0.378380\pi$
$907$	22.4579	0.745702	0.372851	+	0.927891i	$0.378380\pi$
$908$	0	0
$909$	8.26365	0.274088
$910$	0	0
$911$	10.2411	0.339303	0.169652	−	0.985504i	$-0.445736\pi$
$911$	10.2411	0.339303	0.169652	+	0.985504i	$0.445736\pi$
$912$	0	0
$913$	48.0335	1.58968
$914$	0	0
$915$	2.73642	0.0904632
$916$	0	0
$917$	3.64908	0.120503
$918$	0	0
$919$	−31.2272	−1.03009	−0.515045	−	0.857163i	$-0.672225\pi$
$919$	−31.2272	−1.03009	−0.515045	+	0.857163i	$0.672225\pi$
$920$	0	0
$921$	−15.8492	−0.522250
$922$	0	0
$923$	−57.0112	−1.87655
$924$	0	0
$925$	4.44937	0.146295
$926$	0	0
$927$	−1.87341	−0.0615307
$928$	0	0
$929$	−50.7692	−1.66568	−0.832842	−	0.553511i	$-0.813288\pi$
$929$	−50.7692	−1.66568	−0.832842	+	0.553511i	$0.813288\pi$
$930$	0	0
$931$	−31.3337	−1.02692
$932$	0	0
$933$	1.57471	0.0515538
$934$	0	0
$935$	16.8254	0.550249
$936$	0	0
$937$	33.4738	1.09354	0.546771	−	0.837282i	$-0.315857\pi$
$937$	33.4738	1.09354	0.546771	+	0.837282i	$0.315857\pi$
$938$	0	0
$939$	−20.5562	−0.670827
$940$	0	0
$941$	−48.4513	−1.57947	−0.789734	−	0.613449i	$-0.789782\pi$
$941$	−48.4513	−1.57947	−0.789734	+	0.613449i	$0.789782\pi$
$942$	0	0
$943$	15.6155	0.508510
$944$	0	0
$945$	0.620094	0.0201717
$946$	0	0
$947$	3.65383	0.118733	0.0593667	−	0.998236i	$-0.481092\pi$
$947$	3.65383	0.118733	0.0593667	+	0.998236i	$0.481092\pi$
$948$	0	0
$949$	−7.63738	−0.247920
$950$	0	0
$951$	−9.86061	−0.319752
$952$	0	0
$953$	−16.1415	−0.522873	−0.261437	−	0.965221i	$-0.584196\pi$
$953$	−16.1415	−0.522873	−0.261437	+	0.965221i	$0.584196\pi$
$954$	0	0
$955$	9.03295	0.292300
$956$	0	0
$957$	−13.0749	−0.422650
$958$	0	0
$959$	5.30179	0.171204
$960$	0	0
$961$	−24.7259	−0.797610
$962$	0	0
$963$	2.04413	0.0658710
$964$	0	0
$965$	1.97927	0.0637149
$966$	0	0
$967$	−42.0311	−1.35163	−0.675814	−	0.737072i	$-0.736208\pi$
$967$	−42.0311	−1.35163	−0.675814	+	0.737072i	$0.736208\pi$
$968$	0	0
$969$	−18.3997	−0.591082
$970$	0	0
$971$	3.27515	0.105105	0.0525524	−	0.998618i	$-0.483264\pi$
$971$	3.27515	0.105105	0.0525524	+	0.998618i	$0.483264\pi$
$972$	0	0
$973$	−2.28199	−0.0731574
$974$	0	0
$975$	−4.44645	−0.142400
$976$	0	0
$977$	43.0801	1.37825	0.689127	−	0.724641i	$-0.257994\pi$
$977$	43.0801	1.37825	0.689127	+	0.724641i	$0.257994\pi$
$978$	0	0
$979$	−10.9214	−0.349048
$980$	0	0
$981$	17.2346	0.550258
$982$	0	0
$983$	4.85022	0.154698	0.0773489	−	0.997004i	$-0.475354\pi$
$983$	4.85022	0.154698	0.0773489	+	0.997004i	$0.475354\pi$
$984$	0	0
$985$	−8.47011	−0.269880
$986$	0	0
$987$	0.883033	0.0281073
$988$	0	0
$989$	−2.51145	−0.0798594
$990$	0	0
$991$	32.7009	1.03878	0.519389	−	0.854538i	$-0.326160\pi$
$991$	32.7009	1.03878	0.519389	+	0.854538i	$0.326160\pi$
$992$	0	0
$993$	8.06124	0.255816
$994$	0	0
$995$	17.4589	0.553485
$996$	0	0
$997$	34.6169	1.09633	0.548164	−	0.836371i	$-0.315327\pi$
$997$	34.6169	1.09633	0.548164	+	0.836371i	$0.315327\pi$
$998$	0	0
$999$	−4.44937	−0.140772

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	4020.2.a.f.1.4	✓	6

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
4020.2.a.f.1.4	✓	6	1.1	even	1	trivial

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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:	\( N \)	\(=\)	\( 4020 = 2^{2} \cdot 3 \cdot 5 \cdot 67 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	4020.a (trivial)

\(f(q)\)	\(=\)	\(q-1.00000 q^{3} +1.00000 q^{5} -0.620094 q^{7} +1.00000 q^{9} +O(q^{10})\)	\(q-1.00000 q^{3} +1.00000 q^{5} -0.620094 q^{7} +1.00000 q^{9} +4.33117 q^{11} +4.44645 q^{13} -1.00000 q^{15} +3.88472 q^{17} +4.73642 q^{19} +0.620094 q^{21} -2.44937 q^{23} +1.00000 q^{25} -1.00000 q^{27} +3.01878 q^{29} -2.50481 q^{31} -4.33117 q^{33} -0.620094 q^{35} +4.44937 q^{37} -4.44645 q^{39} -6.37530 q^{41} +1.02534 q^{43} +1.00000 q^{45} +1.42403 q^{47} -6.61548 q^{49} -3.88472 q^{51} +7.97661 q^{53} +4.33117 q^{55} -4.73642 q^{57} -2.44645 q^{59} -2.73642 q^{61} -0.620094 q^{63} +4.44645 q^{65} +1.00000 q^{67} +2.44937 q^{69} -12.8217 q^{71} -1.71764 q^{73} -1.00000 q^{75} -2.68573 q^{77} -15.6088 q^{79} +1.00000 q^{81} +11.0902 q^{83} +3.88472 q^{85} -3.01878 q^{87} -2.52157 q^{89} -2.75722 q^{91} +2.50481 q^{93} +4.73642 q^{95} +4.82181 q^{97} +4.33117 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 6 q - 6 q^{3} + 6 q^{5} + q^{7} + 6 q^{9}+O(q^{10}) \) 6 * q - 6 * q^3 + 6 * q^5 + q^7 + 6 * q^9	\( 6 q - 6 q^{3} + 6 q^{5} + q^{7} + 6 q^{9} - 7 q^{11} + 7 q^{13} - 6 q^{15} + 10 q^{17} - 3 q^{19} - q^{21} - 4 q^{23} + 6 q^{25} - 6 q^{27} + 9 q^{29} + 3 q^{31} + 7 q^{33} + q^{35} + 16 q^{37} - 7 q^{39} + 7 q^{41} + 3 q^{43} + 6 q^{45} + q^{47} + 15 q^{49} - 10 q^{51} + 7 q^{53} - 7 q^{55} + 3 q^{57} + 5 q^{59} + 15 q^{61} + q^{63} + 7 q^{65} + 6 q^{67} + 4 q^{69} - 12 q^{71} + 12 q^{73} - 6 q^{75} + 9 q^{77} + 9 q^{79} + 6 q^{81} - 2 q^{83} + 10 q^{85} - 9 q^{87} + 10 q^{89} - 5 q^{91} - 3 q^{93} - 3 q^{95} + 37 q^{97} - 7 q^{99}+O(q^{100}) \) 6 * q - 6 * q^3 + 6 * q^5 + q^7 + 6 * q^9 - 7 * q^11 + 7 * q^13 - 6 * q^15 + 10 * q^17 - 3 * q^19 - q^21 - 4 * q^23 + 6 * q^25 - 6 * q^27 + 9 * q^29 + 3 * q^31 + 7 * q^33 + q^35 + 16 * q^37 - 7 * q^39 + 7 * q^41 + 3 * q^43 + 6 * q^45 + q^47 + 15 * q^49 - 10 * q^51 + 7 * q^53 - 7 * q^55 + 3 * q^57 + 5 * q^59 + 15 * q^61 + q^63 + 7 * q^65 + 6 * q^67 + 4 * q^69 - 12 * q^71 + 12 * q^73 - 6 * q^75 + 9 * q^77 + 9 * q^79 + 6 * q^81 - 2 * q^83 + 10 * q^85 - 9 * q^87 + 10 * q^89 - 5 * q^91 - 3 * q^93 - 3 * q^95 + 37 * q^97 - 7 * q^99

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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