LMFDB - Embedded newform 4020.2.a.g.1.3

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:	6.6.195727752.1
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{6} - 15x^{4} - 25x^{3} - 3x^{2} + 9x + 3 $ x^6 - 15x^4 - 25x^3 - 3x^2 + 9x + 3
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			1.3
Root			$0.600664$ 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)

$f(q)$	$=$	$q+1.00000 q^{3} -1.00000 q^{5} +1.05225 q^{7} +1.00000 q^{9} +O(q^{10})$	$q+1.00000 q^{3} -1.00000 q^{5} +1.05225 q^{7} +1.00000 q^{9} +0.701065 q^{11} -5.99448 q^{13} -1.00000 q^{15} -4.59105 q^{17} +2.17442 q^{19} +1.05225 q^{21} +3.01807 q^{23} +1.00000 q^{25} +1.00000 q^{27} +6.98088 q^{29} +6.84462 q^{31} +0.701065 q^{33} -1.05225 q^{35} -4.03312 q^{37} -5.99448 q^{39} +2.54395 q^{41} +3.22796 q^{43} -1.00000 q^{45} +9.92920 q^{47} -5.89278 q^{49} -4.59105 q^{51} -11.6473 q^{53} -0.701065 q^{55} +2.17442 q^{57} +8.38895 q^{59} +12.2090 q^{61} +1.05225 q^{63} +5.99448 q^{65} +1.00000 q^{67} +3.01807 q^{69} +8.35370 q^{71} +10.1748 q^{73} +1.00000 q^{75} +0.737693 q^{77} +16.3593 q^{79} +1.00000 q^{81} +4.35118 q^{83} +4.59105 q^{85} +6.98088 q^{87} +13.6151 q^{89} -6.30767 q^{91} +6.84462 q^{93} -2.17442 q^{95} -14.0544 q^{97} +0.701065 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 6 q + 6 q^{3} - 6 q^{5} + 3 q^{7} + 6 q^{9}+O(q^{10}) $ 6 * q + 6 * q^3 - 6 * q^5 + 3 * q^7 + 6 * q^9	$ 6 q + 6 q^{3} - 6 q^{5} + 3 q^{7} + 6 q^{9} + 3 q^{11} + 3 q^{13} - 6 q^{15} + 6 q^{17} - 3 q^{19} + 3 q^{21} + 6 q^{23} + 6 q^{25} + 6 q^{27} + 21 q^{29} - 3 q^{31} + 3 q^{33} - 3 q^{35} + 3 q^{39} + 9 q^{41} - 3 q^{43} - 6 q^{45} + 21 q^{47} + 3 q^{49} + 6 q^{51} + 15 q^{53} - 3 q^{55} - 3 q^{57} + 15 q^{59} + 15 q^{61} + 3 q^{63} - 3 q^{65} + 6 q^{67} + 6 q^{69} + 18 q^{71} - 6 q^{73} + 6 q^{75} + 33 q^{77} + 9 q^{79} + 6 q^{81} + 24 q^{83} - 6 q^{85} + 21 q^{87} + 24 q^{89} - 21 q^{91} - 3 q^{93} + 3 q^{95} + 9 q^{97} + 3 q^{99}+O(q^{100}) $ 6 * q + 6 * q^3 - 6 * q^5 + 3 * q^7 + 6 * q^9 + 3 * q^11 + 3 * q^13 - 6 * q^15 + 6 * q^17 - 3 * q^19 + 3 * q^21 + 6 * q^23 + 6 * q^25 + 6 * q^27 + 21 * q^29 - 3 * q^31 + 3 * q^33 - 3 * q^35 + 3 * q^39 + 9 * q^41 - 3 * q^43 - 6 * q^45 + 21 * q^47 + 3 * q^49 + 6 * q^51 + 15 * q^53 - 3 * q^55 - 3 * q^57 + 15 * q^59 + 15 * q^61 + 3 * q^63 - 3 * q^65 + 6 * q^67 + 6 * q^69 + 18 * q^71 - 6 * q^73 + 6 * q^75 + 33 * q^77 + 9 * q^79 + 6 * q^81 + 24 * q^83 - 6 * q^85 + 21 * q^87 + 24 * q^89 - 21 * q^91 - 3 * q^93 + 3 * q^95 + 9 * q^97 + 3 * 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
$7$	1.05225	0.397712	0.198856	−	0.980029i	$-0.436277\pi$
$7$	1.05225	0.397712	0.198856	+	0.980029i	$0.436277\pi$
$8$	0	0
$9$	1.00000	0.333333
$10$	0	0
$11$	0.701065	0.211379	0.105690	−	0.994399i	$-0.466295\pi$
$11$	0.701065	0.211379	0.105690	+	0.994399i	$0.466295\pi$
$12$	0	0
$13$	−5.99448	−1.66257	−0.831284	−	0.555847i	$-0.812394\pi$
$13$	−5.99448	−1.66257	−0.831284	+	0.555847i	$0.812394\pi$
$14$	0	0
$15$	−1.00000	−0.258199
$16$	0	0
$17$	−4.59105	−1.11349	−0.556746	−	0.830683i	$-0.687950\pi$
$17$	−4.59105	−1.11349	−0.556746	+	0.830683i	$0.687950\pi$
$18$	0	0
$19$	2.17442	0.498845	0.249423	−	0.968395i	$-0.419759\pi$
$19$	2.17442	0.498845	0.249423	+	0.968395i	$0.419759\pi$
$20$	0	0
$21$	1.05225	0.229619
$22$	0	0
$23$	3.01807	0.629311	0.314656	−	0.949206i	$-0.398111\pi$
$23$	3.01807	0.629311	0.314656	+	0.949206i	$0.398111\pi$
$24$	0	0
$25$	1.00000	0.200000
$26$	0	0
$27$	1.00000	0.192450
$28$	0	0
$29$	6.98088	1.29632	0.648158	−	0.761506i	$-0.275540\pi$
$29$	6.98088	1.29632	0.648158	+	0.761506i	$0.275540\pi$
$30$	0	0
$31$	6.84462	1.22933	0.614665	−	0.788788i	$-0.289291\pi$
$31$	6.84462	1.22933	0.614665	+	0.788788i	$0.289291\pi$
$32$	0	0
$33$	0.701065	0.122040
$34$	0	0
$35$	−1.05225	−0.177862
$36$	0	0
$37$	−4.03312	−0.663042	−0.331521	−	0.943448i	$-0.607562\pi$
$37$	−4.03312	−0.663042	−0.331521	+	0.943448i	$0.607562\pi$
$38$	0	0
$39$	−5.99448	−0.959885
$40$	0	0
$41$	2.54395	0.397298	0.198649	−	0.980071i	$-0.436345\pi$
$41$	2.54395	0.397298	0.198649	+	0.980071i	$0.436345\pi$
$42$	0	0
$43$	3.22796	0.492259	0.246130	−	0.969237i	$-0.420841\pi$
$43$	3.22796	0.492259	0.246130	+	0.969237i	$0.420841\pi$
$44$	0	0
$45$	−1.00000	−0.149071
$46$	0	0
$47$	9.92920	1.44832	0.724161	−	0.689631i	$-0.242227\pi$
$47$	9.92920	1.44832	0.724161	+	0.689631i	$0.242227\pi$
$48$	0	0
$49$	−5.89278	−0.841825
$50$	0	0
$51$	−4.59105	−0.642875
$52$	0	0
$53$	−11.6473	−1.59988	−0.799938	−	0.600082i	$-0.795135\pi$
$53$	−11.6473	−1.59988	−0.799938	+	0.600082i	$0.795135\pi$
$54$	0	0
$55$	−0.701065	−0.0945316
$56$	0	0
$57$	2.17442	0.288008
$58$	0	0
$59$	8.38895	1.09215	0.546074	−	0.837737i	$-0.316122\pi$
$59$	8.38895	1.09215	0.546074	+	0.837737i	$0.316122\pi$
$60$	0	0
$61$	12.2090	1.56320	0.781602	−	0.623778i	$-0.214403\pi$
$61$	12.2090	1.56320	0.781602	+	0.623778i	$0.214403\pi$
$62$	0	0
$63$	1.05225	0.132571
$64$	0	0
$65$	5.99448	0.743523
$66$	0	0
$67$	1.00000	0.122169
$67$	1.00000	0.122169
$68$	0	0
$69$	3.01807	0.363333
$70$	0	0
$71$	8.35370	0.991402	0.495701	−	0.868493i	$-0.334911\pi$
$71$	8.35370	0.991402	0.495701	+	0.868493i	$0.334911\pi$
$72$	0	0
$73$	10.1748	1.19087	0.595437	−	0.803402i	$-0.296979\pi$
$73$	10.1748	1.19087	0.595437	+	0.803402i	$0.296979\pi$
$74$	0	0
$75$	1.00000	0.115470
$76$	0	0
$77$	0.737693	0.0840679
$78$	0	0
$79$	16.3593	1.84057	0.920283	−	0.391254i	$-0.127958\pi$
$79$	16.3593	1.84057	0.920283	+	0.391254i	$0.127958\pi$
$80$	0	0
$81$	1.00000	0.111111
$82$	0	0
$83$	4.35118	0.477604	0.238802	−	0.971068i	$-0.423245\pi$
$83$	4.35118	0.477604	0.238802	+	0.971068i	$0.423245\pi$
$84$	0	0
$85$	4.59105	0.497969
$86$	0	0
$87$	6.98088	0.748429
$88$	0	0
$89$	13.6151	1.44319	0.721597	−	0.692314i	$-0.243409\pi$
$89$	13.6151	1.44319	0.721597	+	0.692314i	$0.243409\pi$
$90$	0	0
$91$	−6.30767	−0.661223
$92$	0	0
$93$	6.84462	0.709754
$94$	0	0
$95$	−2.17442	−0.223090
$96$	0	0
$97$	−14.0544	−1.42701	−0.713504	−	0.700651i	$-0.752893\pi$
$97$	−14.0544	−1.42701	−0.713504	+	0.700651i	$0.752893\pi$
$98$	0	0
$99$	0.701065	0.0704597
$100$	0	0
$101$	2.99906	0.298417	0.149209	−	0.988806i	$-0.452327\pi$
$101$	2.99906	0.298417	0.149209	+	0.988806i	$0.452327\pi$
$102$	0	0
$103$	−7.15546	−0.705049	−0.352524	−	0.935803i	$-0.614677\pi$
$103$	−7.15546	−0.705049	−0.352524	+	0.935803i	$0.614677\pi$
$104$	0	0
$105$	−1.05225	−0.102689
$106$	0	0
$107$	5.14052	0.496953	0.248476	−	0.968638i	$-0.420070\pi$
$107$	5.14052	0.496953	0.248476	+	0.968638i	$0.420070\pi$
$108$	0	0
$109$	−15.5560	−1.48999	−0.744997	−	0.667068i	$-0.767549\pi$
$109$	−15.5560	−1.48999	−0.744997	+	0.667068i	$0.767549\pi$
$110$	0	0
$111$	−4.03312	−0.382807
$112$	0	0
$113$	−1.16568	−0.109658	−0.0548291	−	0.998496i	$-0.517461\pi$
$113$	−1.16568	−0.109658	−0.0548291	+	0.998496i	$0.517461\pi$
$114$	0	0
$115$	−3.01807	−0.281437
$116$	0	0
$117$	−5.99448	−0.554190
$118$	0	0
$119$	−4.83092	−0.442849
$120$	0	0
$121$	−10.5085	−0.955319
$122$	0	0
$123$	2.54395	0.229380
$124$	0	0
$125$	−1.00000	−0.0894427
$126$	0	0
$127$	−0.376517	−0.0334105	−0.0167052	−	0.999860i	$-0.505318\pi$
$127$	−0.376517	−0.0334105	−0.0167052	+	0.999860i	$0.505318\pi$
$128$	0	0
$129$	3.22796	0.284206
$130$	0	0
$131$	0.856165	0.0748035	0.0374018	−	0.999300i	$-0.488092\pi$
$131$	0.856165	0.0748035	0.0374018	+	0.999300i	$0.488092\pi$
$132$	0	0
$133$	2.28802	0.198397
$134$	0	0
$135$	−1.00000	−0.0860663
$136$	0	0
$137$	1.88639	0.161166	0.0805828	−	0.996748i	$-0.474322\pi$
$137$	1.88639	0.161166	0.0805828	+	0.996748i	$0.474322\pi$
$138$	0	0
$139$	−3.81989	−0.323999	−0.161999	−	0.986791i	$-0.551794\pi$
$139$	−3.81989	−0.323999	−0.161999	+	0.986791i	$0.551794\pi$
$140$	0	0
$141$	9.92920	0.836189
$142$	0	0
$143$	−4.20252	−0.351432
$144$	0	0
$145$	−6.98088	−0.579730
$146$	0	0
$147$	−5.89278	−0.486028
$148$	0	0
$149$	6.25620	0.512528	0.256264	−	0.966607i	$-0.417508\pi$
$149$	6.25620	0.512528	0.256264	+	0.966607i	$0.417508\pi$
$150$	0	0
$151$	7.35058	0.598181	0.299091	−	0.954225i	$-0.403317\pi$
$151$	7.35058	0.598181	0.299091	+	0.954225i	$0.403317\pi$
$152$	0	0
$153$	−4.59105	−0.371164
$154$	0	0
$155$	−6.84462	−0.549773
$156$	0	0
$157$	3.37535	0.269382	0.134691	−	0.990888i	$-0.456996\pi$
$157$	3.37535	0.269382	0.134691	+	0.990888i	$0.456996\pi$
$158$	0	0
$159$	−11.6473	−0.923689
$160$	0	0
$161$	3.17575	0.250285
$162$	0	0
$163$	19.0125	1.48917	0.744586	−	0.667527i	$-0.232647\pi$
$163$	19.0125	1.48917	0.744586	+	0.667527i	$0.232647\pi$
$164$	0	0
$165$	−0.701065	−0.0545778
$166$	0	0
$167$	2.82489	0.218597	0.109298	−	0.994009i	$-0.465140\pi$
$167$	2.82489	0.218597	0.109298	+	0.994009i	$0.465140\pi$
$168$	0	0
$169$	22.9338	1.76413
$170$	0	0
$171$	2.17442	0.166282
$172$	0	0
$173$	−19.4931	−1.48203	−0.741016	−	0.671488i	$-0.765656\pi$
$173$	−19.4931	−1.48203	−0.741016	+	0.671488i	$0.765656\pi$
$174$	0	0
$175$	1.05225	0.0795424
$176$	0	0
$177$	8.38895	0.630552
$178$	0	0
$179$	−21.9424	−1.64005	−0.820027	−	0.572325i	$-0.806042\pi$
$179$	−21.9424	−1.64005	−0.820027	+	0.572325i	$0.806042\pi$
$180$	0	0
$181$	−1.07084	−0.0795950	−0.0397975	−	0.999208i	$-0.512671\pi$
$181$	−1.07084	−0.0795950	−0.0397975	+	0.999208i	$0.512671\pi$
$182$	0	0
$183$	12.2090	0.902516
$184$	0	0
$185$	4.03312	0.296521
$186$	0	0
$187$	−3.21862	−0.235369
$188$	0	0
$189$	1.05225	0.0765397
$190$	0	0
$191$	24.9992	1.80888	0.904438	−	0.426606i	$-0.140291\pi$
$191$	24.9992	1.80888	0.904438	+	0.426606i	$0.140291\pi$
$192$	0	0
$193$	−19.5047	−1.40398	−0.701989	−	0.712188i	$-0.747705\pi$
$193$	−19.5047	−1.40398	−0.701989	+	0.712188i	$0.747705\pi$
$194$	0	0
$195$	5.99448	0.429273
$196$	0	0
$197$	22.4851	1.60199	0.800997	−	0.598668i	$-0.204303\pi$
$197$	22.4851	1.60199	0.800997	+	0.598668i	$0.204303\pi$
$198$	0	0
$199$	−16.4697	−1.16751	−0.583754	−	0.811930i	$-0.698417\pi$
$199$	−16.4697	−1.16751	−0.583754	+	0.811930i	$0.698417\pi$
$200$	0	0
$201$	1.00000	0.0705346
$202$	0	0
$203$	7.34561	0.515560
$204$	0	0
$205$	−2.54395	−0.177677
$206$	0	0
$207$	3.01807	0.209770
$208$	0	0
$209$	1.52441	0.105445
$210$	0	0
$211$	−17.6067	−1.21209	−0.606046	−	0.795429i	$-0.707245\pi$
$211$	−17.6067	−1.21209	−0.606046	+	0.795429i	$0.707245\pi$
$212$	0	0
$213$	8.35370	0.572386
$214$	0	0
$215$	−3.22796	−0.220145
$216$	0	0
$217$	7.20223	0.488919
$218$	0	0
$219$	10.1748	0.687551
$220$	0	0
$221$	27.5209	1.85126
$222$	0	0
$223$	23.9158	1.60152	0.800760	−	0.598985i	$-0.204429\pi$
$223$	23.9158	1.60152	0.800760	+	0.598985i	$0.204429\pi$
$224$	0	0
$225$	1.00000	0.0666667
$226$	0	0
$227$	22.3977	1.48659	0.743293	−	0.668965i	$-0.233263\pi$
$227$	22.3977	1.48659	0.743293	+	0.668965i	$0.233263\pi$
$228$	0	0
$229$	−8.56987	−0.566313	−0.283156	−	0.959074i	$-0.591382\pi$
$229$	−8.56987	−0.566313	−0.283156	+	0.959074i	$0.591382\pi$
$230$	0	0
$231$	0.737693	0.0485367
$232$	0	0
$233$	−12.0758	−0.791109	−0.395555	−	0.918442i	$-0.629448\pi$
$233$	−12.0758	−0.791109	−0.395555	+	0.918442i	$0.629448\pi$
$234$	0	0
$235$	−9.92920	−0.647709
$236$	0	0
$237$	16.3593	1.06265
$238$	0	0
$239$	−6.04175	−0.390808	−0.195404	−	0.980723i	$-0.562602\pi$
$239$	−6.04175	−0.390808	−0.195404	+	0.980723i	$0.562602\pi$
$240$	0	0
$241$	−9.20840	−0.593165	−0.296583	−	0.955007i	$-0.595847\pi$
$241$	−9.20840	−0.593165	−0.296583	+	0.955007i	$0.595847\pi$
$242$	0	0
$243$	1.00000	0.0641500
$244$	0	0
$245$	5.89278	0.376476
$246$	0	0
$247$	−13.0345	−0.829364
$248$	0	0
$249$	4.35118	0.275745
$250$	0	0
$251$	−3.79879	−0.239778	−0.119889	−	0.992787i	$-0.538254\pi$
$251$	−3.79879	−0.239778	−0.119889	+	0.992787i	$0.538254\pi$
$252$	0	0
$253$	2.11586	0.133023
$254$	0	0
$255$	4.59105	0.287503
$256$	0	0
$257$	21.5307	1.34305	0.671523	−	0.740984i	$-0.265640\pi$
$257$	21.5307	1.34305	0.671523	+	0.740984i	$0.265640\pi$
$258$	0	0
$259$	−4.24384	−0.263699
$260$	0	0
$261$	6.98088	0.432105
$262$	0	0
$263$	8.97456	0.553395	0.276697	−	0.960957i	$-0.410760\pi$
$263$	8.97456	0.553395	0.276697	+	0.960957i	$0.410760\pi$
$264$	0	0
$265$	11.6473	0.715487
$266$	0	0
$267$	13.6151	0.833228
$268$	0	0
$269$	13.9213	0.848794	0.424397	−	0.905476i	$-0.360486\pi$
$269$	13.9213	0.848794	0.424397	+	0.905476i	$0.360486\pi$
$270$	0	0
$271$	−3.92352	−0.238337	−0.119168	−	0.992874i	$-0.538023\pi$
$271$	−3.92352	−0.238337	−0.119168	+	0.992874i	$0.538023\pi$
$272$	0	0
$273$	−6.30767	−0.381757
$274$	0	0
$275$	0.701065	0.0422758
$276$	0	0
$277$	1.34132	0.0805921	0.0402961	−	0.999188i	$-0.487170\pi$
$277$	1.34132	0.0805921	0.0402961	+	0.999188i	$0.487170\pi$
$278$	0	0
$279$	6.84462	0.409777
$280$	0	0
$281$	−6.49774	−0.387623	−0.193811	−	0.981039i	$-0.562085\pi$
$281$	−6.49774	−0.387623	−0.193811	+	0.981039i	$0.562085\pi$
$282$	0	0
$283$	−20.5479	−1.22145	−0.610724	−	0.791843i	$-0.709122\pi$
$283$	−20.5479	−1.22145	−0.610724	+	0.791843i	$0.709122\pi$
$284$	0	0
$285$	−2.17442	−0.128801
$286$	0	0
$287$	2.67686	0.158010
$288$	0	0
$289$	4.07773	0.239866
$290$	0	0
$291$	−14.0544	−0.823884
$292$	0	0
$293$	−1.90946	−0.111552	−0.0557759	−	0.998443i	$-0.517763\pi$
$293$	−1.90946	−0.111552	−0.0557759	+	0.998443i	$0.517763\pi$
$294$	0	0
$295$	−8.38895	−0.488423
$296$	0	0
$297$	0.701065	0.0406799
$298$	0	0
$299$	−18.0918	−1.04627
$300$	0	0
$301$	3.39661	0.195777
$302$	0	0
$303$	2.99906	0.172291
$304$	0	0
$305$	−12.2090	−0.699086
$306$	0	0
$307$	10.2510	0.585054	0.292527	−	0.956257i	$-0.405504\pi$
$307$	10.2510	0.585054	0.292527	+	0.956257i	$0.405504\pi$
$308$	0	0
$309$	−7.15546	−0.407060
$310$	0	0
$311$	2.75247	0.156078	0.0780391	−	0.996950i	$-0.475134\pi$
$311$	2.75247	0.156078	0.0780391	+	0.996950i	$0.475134\pi$
$312$	0	0
$313$	32.6612	1.84612	0.923060	−	0.384656i	$-0.125680\pi$
$313$	32.6612	1.84612	0.923060	+	0.384656i	$0.125680\pi$
$314$	0	0
$315$	−1.05225	−0.0592874
$316$	0	0
$317$	23.6411	1.32782	0.663908	−	0.747814i	$-0.268897\pi$
$317$	23.6411	1.32782	0.663908	+	0.747814i	$0.268897\pi$
$318$	0	0
$319$	4.89405	0.274014
$320$	0	0
$321$	5.14052	0.286916
$322$	0	0
$323$	−9.98285	−0.555461
$324$	0	0
$325$	−5.99448	−0.332514
$326$	0	0
$327$	−15.5560	−0.860248
$328$	0	0
$329$	10.4480	0.576015
$330$	0	0
$331$	−20.2301	−1.11195	−0.555973	−	0.831200i	$-0.687654\pi$
$331$	−20.2301	−1.11195	−0.555973	+	0.831200i	$0.687654\pi$
$332$	0	0
$333$	−4.03312	−0.221014
$334$	0	0
$335$	−1.00000	−0.0546358
$336$	0	0
$337$	11.5122	0.627107	0.313553	−	0.949571i	$-0.398481\pi$
$337$	11.5122	0.627107	0.313553	+	0.949571i	$0.398481\pi$
$338$	0	0
$339$	−1.16568	−0.0633112
$340$	0	0
$341$	4.79852	0.259855
$342$	0	0
$343$	−13.5664	−0.732516
$344$	0	0
$345$	−3.01807	−0.162487
$346$	0	0
$347$	20.8166	1.11749	0.558747	−	0.829338i	$-0.311282\pi$
$347$	20.8166	1.11749	0.558747	+	0.829338i	$0.311282\pi$
$348$	0	0
$349$	2.65493	0.142115	0.0710576	−	0.997472i	$-0.477363\pi$
$349$	2.65493	0.142115	0.0710576	+	0.997472i	$0.477363\pi$
$350$	0	0
$351$	−5.99448	−0.319962
$352$	0	0
$353$	−25.6474	−1.36508	−0.682538	−	0.730850i	$-0.739124\pi$
$353$	−25.6474	−1.36508	−0.682538	+	0.730850i	$0.739124\pi$
$354$	0	0
$355$	−8.35370	−0.443368
$356$	0	0
$357$	−4.83092	−0.255679
$358$	0	0
$359$	32.4461	1.71244	0.856219	−	0.516613i	$-0.172808\pi$
$359$	32.4461	1.71244	0.856219	+	0.516613i	$0.172808\pi$
$360$	0	0
$361$	−14.2719	−0.751153
$362$	0	0
$363$	−10.5085	−0.551554
$364$	0	0
$365$	−10.1748	−0.532575
$366$	0	0
$367$	−14.2220	−0.742382	−0.371191	−	0.928556i	$-0.621051\pi$
$367$	−14.2220	−0.742382	−0.371191	+	0.928556i	$0.621051\pi$
$368$	0	0
$369$	2.54395	0.132433
$370$	0	0
$371$	−12.2558	−0.636290
$372$	0	0
$373$	−2.94378	−0.152423	−0.0762115	−	0.997092i	$-0.524282\pi$
$373$	−2.94378	−0.152423	−0.0762115	+	0.997092i	$0.524282\pi$
$374$	0	0
$375$	−1.00000	−0.0516398
$376$	0	0
$377$	−41.8467	−2.15522
$378$	0	0
$379$	17.7547	0.911998	0.455999	−	0.889980i	$-0.349282\pi$
$379$	17.7547	0.911998	0.455999	+	0.889980i	$0.349282\pi$
$380$	0	0
$381$	−0.376517	−0.0192896
$382$	0	0
$383$	−4.88908	−0.249820	−0.124910	−	0.992168i	$-0.539864\pi$
$383$	−4.88908	−0.249820	−0.124910	+	0.992168i	$0.539864\pi$
$384$	0	0
$385$	−0.737693	−0.0375963
$386$	0	0
$387$	3.22796	0.164086
$388$	0	0
$389$	7.49001	0.379758	0.189879	−	0.981807i	$-0.439190\pi$
$389$	7.49001	0.379758	0.189879	+	0.981807i	$0.439190\pi$
$390$	0	0
$391$	−13.8561	−0.700734
$392$	0	0
$393$	0.856165	0.0431878
$394$	0	0
$395$	−16.3593	−0.823126
$396$	0	0
$397$	−1.94331	−0.0975317	−0.0487658	−	0.998810i	$-0.515529\pi$
$397$	−1.94331	−0.0975317	−0.0487658	+	0.998810i	$0.515529\pi$
$398$	0	0
$399$	2.28802	0.114544
$400$	0	0
$401$	2.86436	0.143039	0.0715196	−	0.997439i	$-0.477215\pi$
$401$	2.86436	0.143039	0.0715196	+	0.997439i	$0.477215\pi$
$402$	0	0
$403$	−41.0299	−2.04385
$404$	0	0
$405$	−1.00000	−0.0496904
$406$	0	0
$407$	−2.82748	−0.140153
$408$	0	0
$409$	34.2575	1.69392	0.846962	−	0.531654i	$-0.178429\pi$
$409$	34.2575	1.69392	0.846962	+	0.531654i	$0.178429\pi$
$410$	0	0
$411$	1.88639	0.0930490
$412$	0	0
$413$	8.82724	0.434360
$414$	0	0
$415$	−4.35118	−0.213591
$416$	0	0
$417$	−3.81989	−0.187061
$418$	0	0
$419$	9.72207	0.474954	0.237477	−	0.971393i	$-0.423680\pi$
$419$	9.72207	0.474954	0.237477	+	0.971393i	$0.423680\pi$
$420$	0	0
$421$	2.46200	0.119990	0.0599952	−	0.998199i	$-0.480891\pi$
$421$	2.46200	0.119990	0.0599952	+	0.998199i	$0.480891\pi$
$422$	0	0
$423$	9.92920	0.482774
$424$	0	0
$425$	−4.59105	−0.222699
$426$	0	0
$427$	12.8469	0.621704
$428$	0	0
$429$	−4.20252	−0.202899
$430$	0	0
$431$	29.1145	1.40240	0.701199	−	0.712966i	$-0.252648\pi$
$431$	29.1145	1.40240	0.701199	+	0.712966i	$0.252648\pi$
$432$	0	0
$433$	4.49871	0.216194	0.108097	−	0.994140i	$-0.465524\pi$
$433$	4.49871	0.216194	0.108097	+	0.994140i	$0.465524\pi$
$434$	0	0
$435$	−6.98088	−0.334707
$436$	0	0
$437$	6.56254	0.313929
$438$	0	0
$439$	−3.14037	−0.149882	−0.0749409	−	0.997188i	$-0.523877\pi$
$439$	−3.14037	−0.149882	−0.0749409	+	0.997188i	$0.523877\pi$
$440$	0	0
$441$	−5.89278	−0.280608
$442$	0	0
$443$	−14.5043	−0.689122	−0.344561	−	0.938764i	$-0.611972\pi$
$443$	−14.5043	−0.689122	−0.344561	+	0.938764i	$0.611972\pi$
$444$	0	0
$445$	−13.6151	−0.645416
$446$	0	0
$447$	6.25620	0.295908
$448$	0	0
$449$	−39.2442	−1.85205	−0.926024	−	0.377464i	$-0.876796\pi$
$449$	−39.2442	−1.85205	−0.926024	+	0.377464i	$0.876796\pi$
$450$	0	0
$451$	1.78347	0.0839804
$452$	0	0
$453$	7.35058	0.345360
$454$	0	0
$455$	6.30767	0.295708
$456$	0	0
$457$	−5.48153	−0.256415	−0.128208	−	0.991747i	$-0.540922\pi$
$457$	−5.48153	−0.256415	−0.128208	+	0.991747i	$0.540922\pi$
$458$	0	0
$459$	−4.59105	−0.214292
$460$	0	0
$461$	−38.5336	−1.79469	−0.897345	−	0.441329i	$-0.854507\pi$
$461$	−38.5336	−1.79469	−0.897345	+	0.441329i	$0.854507\pi$
$462$	0	0
$463$	−10.0617	−0.467609	−0.233804	−	0.972284i	$-0.575118\pi$
$463$	−10.0617	−0.467609	−0.233804	+	0.972284i	$0.575118\pi$
$464$	0	0
$465$	−6.84462	−0.317412
$466$	0	0
$467$	11.0876	0.513073	0.256536	−	0.966535i	$-0.417419\pi$
$467$	11.0876	0.513073	0.256536	+	0.966535i	$0.417419\pi$
$468$	0	0
$469$	1.05225	0.0485882
$470$	0	0
$471$	3.37535	0.155528
$472$	0	0
$473$	2.26301	0.104053
$474$	0	0
$475$	2.17442	0.0997690
$476$	0	0
$477$	−11.6473	−0.533292
$478$	0	0
$479$	17.8342	0.814866	0.407433	−	0.913235i	$-0.366424\pi$
$479$	17.8342	0.814866	0.407433	+	0.913235i	$0.366424\pi$
$480$	0	0
$481$	24.1765	1.10235
$482$	0	0
$483$	3.17575	0.144502
$484$	0	0
$485$	14.0544	0.638178
$486$	0	0
$487$	30.8623	1.39850	0.699252	−	0.714876i	$-0.253517\pi$
$487$	30.8623	1.39850	0.699252	+	0.714876i	$0.253517\pi$
$488$	0	0
$489$	19.0125	0.859773
$490$	0	0
$491$	−17.2122	−0.776777	−0.388388	−	0.921496i	$-0.626968\pi$
$491$	−17.2122	−0.776777	−0.388388	+	0.921496i	$0.626968\pi$
$492$	0	0
$493$	−32.0495	−1.44344
$494$	0	0
$495$	−0.701065	−0.0315105
$496$	0	0
$497$	8.79015	0.394292
$498$	0	0
$499$	−18.7319	−0.838556	−0.419278	−	0.907858i	$-0.637717\pi$
$499$	−18.7319	−0.838556	−0.419278	+	0.907858i	$0.637717\pi$
$500$	0	0
$501$	2.82489	0.126207
$502$	0	0
$503$	−19.7610	−0.881100	−0.440550	−	0.897728i	$-0.645217\pi$
$503$	−19.7610	−0.881100	−0.440550	+	0.897728i	$0.645217\pi$
$504$	0	0
$505$	−2.99906	−0.133456
$506$	0	0
$507$	22.9338	1.01852
$508$	0	0
$509$	9.50407	0.421260	0.210630	−	0.977566i	$-0.432448\pi$
$509$	9.50407	0.421260	0.210630	+	0.977566i	$0.432448\pi$
$510$	0	0
$511$	10.7064	0.473625
$512$	0	0
$513$	2.17442	0.0960028
$514$	0	0
$515$	7.15546	0.315307
$516$	0	0
$517$	6.96101	0.306145
$518$	0	0
$519$	−19.4931	−0.855651
$520$	0	0
$521$	18.9299	0.829334	0.414667	−	0.909973i	$-0.363898\pi$
$521$	18.9299	0.829334	0.414667	+	0.909973i	$0.363898\pi$
$522$	0	0
$523$	−41.6861	−1.82280	−0.911402	−	0.411517i	$-0.864999\pi$
$523$	−41.6861	−1.82280	−0.911402	+	0.411517i	$0.864999\pi$
$524$	0	0
$525$	1.05225	0.0459238
$526$	0	0
$527$	−31.4240	−1.36885
$528$	0	0
$529$	−13.8913	−0.603967
$530$	0	0
$531$	8.38895	0.364049
$532$	0	0
$533$	−15.2496	−0.660534
$534$	0	0
$535$	−5.14052	−0.222244
$536$	0	0
$537$	−21.9424	−0.946886
$538$	0	0
$539$	−4.13122	−0.177944
$540$	0	0
$541$	−33.1818	−1.42660	−0.713298	−	0.700861i	$-0.752799\pi$
$541$	−33.1818	−1.42660	−0.713298	+	0.700861i	$0.752799\pi$
$542$	0	0
$543$	−1.07084	−0.0459542
$544$	0	0
$545$	15.5560	0.666345
$546$	0	0
$547$	−33.3590	−1.42633	−0.713163	−	0.700998i	$-0.752738\pi$
$547$	−33.3590	−1.42633	−0.713163	+	0.700998i	$0.752738\pi$
$548$	0	0
$549$	12.2090	0.521068
$550$	0	0
$551$	15.1793	0.646661
$552$	0	0
$553$	17.2140	0.732015
$554$	0	0
$555$	4.03312	0.171197
$556$	0	0
$557$	−4.98290	−0.211132	−0.105566	−	0.994412i	$-0.533665\pi$
$557$	−4.98290	−0.211132	−0.105566	+	0.994412i	$0.533665\pi$
$558$	0	0
$559$	−19.3499	−0.818415
$560$	0	0
$561$	−3.21862	−0.135890
$562$	0	0
$563$	12.5785	0.530123	0.265061	−	0.964232i	$-0.414608\pi$
$563$	12.5785	0.530123	0.265061	+	0.964232i	$0.414608\pi$
$564$	0	0
$565$	1.16568	0.0490406
$566$	0	0
$567$	1.05225	0.0441902
$568$	0	0
$569$	−21.2509	−0.890883	−0.445442	−	0.895311i	$-0.646953\pi$
$569$	−21.2509	−0.890883	−0.445442	+	0.895311i	$0.646953\pi$
$570$	0	0
$571$	18.6003	0.778399	0.389200	−	0.921153i	$-0.372752\pi$
$571$	18.6003	0.778399	0.389200	+	0.921153i	$0.372752\pi$
$572$	0	0
$573$	24.9992	1.04435
$574$	0	0
$575$	3.01807	0.125862
$576$	0	0
$577$	−7.96187	−0.331457	−0.165729	−	0.986171i	$-0.552998\pi$
$577$	−7.96187	−0.331457	−0.165729	+	0.986171i	$0.552998\pi$
$578$	0	0
$579$	−19.5047	−0.810587
$580$	0	0
$581$	4.57852	0.189949
$582$	0	0
$583$	−8.16550	−0.338180
$584$	0	0
$585$	5.99448	0.247841
$586$	0	0
$587$	−10.1545	−0.419120	−0.209560	−	0.977796i	$-0.567203\pi$
$587$	−10.1545	−0.419120	−0.209560	+	0.977796i	$0.567203\pi$
$588$	0	0
$589$	14.8831	0.613246
$590$	0	0
$591$	22.4851	0.924912
$592$	0	0
$593$	−24.6956	−1.01412	−0.507062	−	0.861909i	$-0.669269\pi$
$593$	−24.6956	−1.01412	−0.507062	+	0.861909i	$0.669269\pi$
$594$	0	0
$595$	4.83092	0.198048
$596$	0	0
$597$	−16.4697	−0.674062
$598$	0	0
$599$	5.74358	0.234676	0.117338	−	0.993092i	$-0.462564\pi$
$599$	5.74358	0.234676	0.117338	+	0.993092i	$0.462564\pi$
$600$	0	0
$601$	30.4870	1.24359	0.621795	−	0.783180i	$-0.286404\pi$
$601$	30.4870	1.24359	0.621795	+	0.783180i	$0.286404\pi$
$602$	0	0
$603$	1.00000	0.0407231
$604$	0	0
$605$	10.5085	0.427232
$606$	0	0
$607$	44.0741	1.78891	0.894457	−	0.447154i	$-0.147562\pi$
$607$	44.0741	1.78891	0.894457	+	0.447154i	$0.147562\pi$
$608$	0	0
$609$	7.34561	0.297659
$610$	0	0
$611$	−59.5203	−2.40793
$612$	0	0
$613$	−20.1001	−0.811836	−0.405918	−	0.913910i	$-0.633048\pi$
$613$	−20.1001	−0.811836	−0.405918	+	0.913910i	$0.633048\pi$
$614$	0	0
$615$	−2.54395	−0.102582
$616$	0	0
$617$	−9.94553	−0.400392	−0.200196	−	0.979756i	$-0.564158\pi$
$617$	−9.94553	−0.400392	−0.200196	+	0.979756i	$0.564158\pi$
$618$	0	0
$619$	−26.3051	−1.05729	−0.528645	−	0.848843i	$-0.677300\pi$
$619$	−26.3051	−1.05729	−0.528645	+	0.848843i	$0.677300\pi$
$620$	0	0
$621$	3.01807	0.121111
$622$	0	0
$623$	14.3264	0.573975
$624$	0	0
$625$	1.00000	0.0400000
$626$	0	0
$627$	1.52441	0.0608789
$628$	0	0
$629$	18.5163	0.738292
$630$	0	0
$631$	42.6345	1.69725	0.848627	−	0.528992i	$-0.177430\pi$
$631$	42.6345	1.69725	0.848627	+	0.528992i	$0.177430\pi$
$632$	0	0
$633$	−17.6067	−0.699802
$634$	0	0
$635$	0.376517	0.0149416
$636$	0	0
$637$	35.3241	1.39959
$638$	0	0
$639$	8.35370	0.330467
$640$	0	0
$641$	−38.7658	−1.53116	−0.765579	−	0.643342i	$-0.777547\pi$
$641$	−38.7658	−1.53116	−0.765579	+	0.643342i	$0.777547\pi$
$642$	0	0
$643$	−7.78541	−0.307027	−0.153513	−	0.988147i	$-0.549059\pi$
$643$	−7.78541	−0.307027	−0.153513	+	0.988147i	$0.549059\pi$
$644$	0	0
$645$	−3.22796	−0.127101
$646$	0	0
$647$	12.7399	0.500859	0.250429	−	0.968135i	$-0.419428\pi$
$647$	12.7399	0.500859	0.250429	+	0.968135i	$0.419428\pi$
$648$	0	0
$649$	5.88120	0.230857
$650$	0	0
$651$	7.20223	0.282278
$652$	0	0
$653$	6.33923	0.248073	0.124037	−	0.992278i	$-0.460416\pi$
$653$	6.33923	0.248073	0.124037	+	0.992278i	$0.460416\pi$
$654$	0	0
$655$	−0.856165	−0.0334531
$656$	0	0
$657$	10.1748	0.396958
$658$	0	0
$659$	6.34066	0.246997	0.123499	−	0.992345i	$-0.460589\pi$
$659$	6.34066	0.246997	0.123499	+	0.992345i	$0.460589\pi$
$660$	0	0
$661$	13.0564	0.507835	0.253917	−	0.967226i	$-0.418281\pi$
$661$	13.0564	0.507835	0.253917	+	0.967226i	$0.418281\pi$
$662$	0	0
$663$	27.5209	1.06882
$664$	0	0
$665$	−2.28802	−0.0887257
$666$	0	0
$667$	21.0688	0.815786
$668$	0	0
$669$	23.9158	0.924638
$670$	0	0
$671$	8.55931	0.330428
$672$	0	0
$673$	25.0449	0.965409	0.482705	−	0.875783i	$-0.339654\pi$
$673$	25.0449	0.965409	0.482705	+	0.875783i	$0.339654\pi$
$674$	0	0
$675$	1.00000	0.0384900
$676$	0	0
$677$	28.0090	1.07647	0.538237	−	0.842793i	$-0.319090\pi$
$677$	28.0090	1.07647	0.538237	+	0.842793i	$0.319090\pi$
$678$	0	0
$679$	−14.7887	−0.567538
$680$	0	0
$681$	22.3977	0.858281
$682$	0	0
$683$	−28.3931	−1.08643	−0.543215	−	0.839593i	$-0.682793\pi$
$683$	−28.3931	−1.08643	−0.543215	+	0.839593i	$0.682793\pi$
$684$	0	0
$685$	−1.88639	−0.0720754
$686$	0	0
$687$	−8.56987	−0.326961
$688$	0	0
$689$	69.8193	2.65991
$690$	0	0
$691$	14.5749	0.554455	0.277228	−	0.960804i	$-0.410584\pi$
$691$	14.5749	0.554455	0.277228	+	0.960804i	$0.410584\pi$
$692$	0	0
$693$	0.737693	0.0280226
$694$	0	0
$695$	3.81989	0.144897
$696$	0	0
$697$	−11.6794	−0.442388
$698$	0	0
$699$	−12.0758	−0.456747
$700$	0	0
$701$	44.9304	1.69700	0.848499	−	0.529197i	$-0.177507\pi$
$701$	44.9304	1.69700	0.848499	+	0.529197i	$0.177507\pi$
$702$	0	0
$703$	−8.76969	−0.330755
$704$	0	0
$705$	−9.92920	−0.373955
$706$	0	0
$707$	3.15575	0.118684
$708$	0	0
$709$	−36.8906	−1.38546	−0.692728	−	0.721199i	$-0.743592\pi$
$709$	−36.8906	−1.38546	−0.692728	+	0.721199i	$0.743592\pi$
$710$	0	0
$711$	16.3593	0.613522
$712$	0	0
$713$	20.6576	0.773631
$714$	0	0
$715$	4.20252	0.157165
$716$	0	0
$717$	−6.04175	−0.225633
$718$	0	0
$719$	10.2172	0.381036	0.190518	−	0.981684i	$-0.438983\pi$
$719$	10.2172	0.381036	0.190518	+	0.981684i	$0.438983\pi$
$720$	0	0
$721$	−7.52931	−0.280406
$722$	0	0
$723$	−9.20840	−0.342464
$724$	0	0
$725$	6.98088	0.259263
$726$	0	0
$727$	−39.2194	−1.45457	−0.727283	−	0.686338i	$-0.759217\pi$
$727$	−39.2194	−1.45457	−0.727283	+	0.686338i	$0.759217\pi$
$728$	0	0
$729$	1.00000	0.0370370
$730$	0	0
$731$	−14.8197	−0.548127
$732$	0	0
$733$	−13.1375	−0.485246	−0.242623	−	0.970121i	$-0.578008\pi$
$733$	−13.1375	−0.485246	−0.242623	+	0.970121i	$0.578008\pi$
$734$	0	0
$735$	5.89278	0.217358
$736$	0	0
$737$	0.701065	0.0258241
$738$	0	0
$739$	−47.6975	−1.75458	−0.877291	−	0.479958i	$-0.840652\pi$
$739$	−47.6975	−1.75458	−0.877291	+	0.479958i	$0.840652\pi$
$740$	0	0
$741$	−13.0345	−0.478834
$742$	0	0
$743$	−25.6081	−0.939468	−0.469734	−	0.882808i	$-0.655650\pi$
$743$	−25.6081	−0.939468	−0.469734	+	0.882808i	$0.655650\pi$
$744$	0	0
$745$	−6.25620	−0.229209
$746$	0	0
$747$	4.35118	0.159201
$748$	0	0
$749$	5.40909	0.197644
$750$	0	0
$751$	−46.0106	−1.67895	−0.839476	−	0.543397i	$-0.817138\pi$
$751$	−46.0106	−1.67895	−0.839476	+	0.543397i	$0.817138\pi$
$752$	0	0
$753$	−3.79879	−0.138436
$754$	0	0
$755$	−7.35058	−0.267515
$756$	0	0
$757$	−33.8946	−1.23192	−0.615960	−	0.787778i	$-0.711232\pi$
$757$	−33.8946	−1.23192	−0.615960	+	0.787778i	$0.711232\pi$
$758$	0	0
$759$	2.11586	0.0768010
$760$	0	0
$761$	−23.9931	−0.869751	−0.434875	−	0.900491i	$-0.643208\pi$
$761$	−23.9931	−0.869751	−0.434875	+	0.900491i	$0.643208\pi$
$762$	0	0
$763$	−16.3687	−0.592588
$764$	0	0
$765$	4.59105	0.165990
$766$	0	0
$767$	−50.2873	−1.81577
$768$	0	0
$769$	35.5496	1.28195	0.640976	−	0.767561i	$-0.278530\pi$
$769$	35.5496	1.28195	0.640976	+	0.767561i	$0.278530\pi$
$770$	0	0
$771$	21.5307	0.775408
$772$	0	0
$773$	−34.3238	−1.23454	−0.617271	−	0.786750i	$-0.711762\pi$
$773$	−34.3238	−1.23454	−0.617271	+	0.786750i	$0.711762\pi$
$774$	0	0
$775$	6.84462	0.245866
$776$	0	0
$777$	−4.24384	−0.152247
$778$	0	0
$779$	5.53160	0.198190
$780$	0	0
$781$	5.85648	0.209562
$782$	0	0
$783$	6.98088	0.249476
$784$	0	0
$785$	−3.37535	−0.120471
$786$	0	0
$787$	18.2507	0.650566	0.325283	−	0.945617i	$-0.394540\pi$
$787$	18.2507	0.650566	0.325283	+	0.945617i	$0.394540\pi$
$788$	0	0
$789$	8.97456	0.319503
$790$	0	0
$791$	−1.22659	−0.0436124
$792$	0	0
$793$	−73.1866	−2.59893
$794$	0	0
$795$	11.6473	0.413086
$796$	0	0
$797$	54.7995	1.94110	0.970549	−	0.240906i	$-0.0774444\pi$
$797$	54.7995	1.94110	0.970549	+	0.240906i	$0.0774444\pi$
$798$	0	0
$799$	−45.5854	−1.61270
$800$	0	0
$801$	13.6151	0.481064
$802$	0	0
$803$	7.13322	0.251726
$804$	0	0
$805$	−3.17575	−0.111931
$806$	0	0
$807$	13.9213	0.490052
$808$	0	0
$809$	−55.9639	−1.96759	−0.983793	−	0.179310i	$-0.942613\pi$
$809$	−55.9639	−1.96759	−0.983793	+	0.179310i	$0.942613\pi$
$810$	0	0
$811$	20.9992	0.737381	0.368690	−	0.929552i	$-0.379806\pi$
$811$	20.9992	0.737381	0.368690	+	0.929552i	$0.379806\pi$
$812$	0	0
$813$	−3.92352	−0.137604
$814$	0	0
$815$	−19.0125	−0.665978
$816$	0	0
$817$	7.01893	0.245561
$818$	0	0
$819$	−6.30767	−0.220408
$820$	0	0
$821$	2.45781	0.0857782	0.0428891	−	0.999080i	$-0.486344\pi$
$821$	2.45781	0.0857782	0.0428891	+	0.999080i	$0.486344\pi$
$822$	0	0
$823$	−42.9093	−1.49573	−0.747863	−	0.663853i	$-0.768920\pi$
$823$	−42.9093	−1.49573	−0.747863	+	0.663853i	$0.768920\pi$
$824$	0	0
$825$	0.701065	0.0244079
$826$	0	0
$827$	−50.2389	−1.74698	−0.873489	−	0.486843i	$-0.838148\pi$
$827$	−50.2389	−1.74698	−0.873489	+	0.486843i	$0.838148\pi$
$828$	0	0
$829$	43.6739	1.51686	0.758429	−	0.651756i	$-0.225967\pi$
$829$	43.6739	1.51686	0.758429	+	0.651756i	$0.225967\pi$
$830$	0	0
$831$	1.34132	0.0465299
$832$	0	0
$833$	27.0540	0.937366
$834$	0	0
$835$	−2.82489	−0.0977594
$836$	0	0
$837$	6.84462	0.236585
$838$	0	0
$839$	3.96589	0.136918	0.0684588	−	0.997654i	$-0.478192\pi$
$839$	3.96589	0.136918	0.0684588	+	0.997654i	$0.478192\pi$
$840$	0	0
$841$	19.7327	0.680436
$842$	0	0
$843$	−6.49774	−0.223794
$844$	0	0
$845$	−22.9338	−0.788945
$846$	0	0
$847$	−11.0575	−0.379942
$848$	0	0
$849$	−20.5479	−0.705203
$850$	0	0
$851$	−12.1723	−0.417259
$852$	0	0
$853$	−3.49810	−0.119773	−0.0598863	−	0.998205i	$-0.519074\pi$
$853$	−3.49810	−0.119773	−0.0598863	+	0.998205i	$0.519074\pi$
$854$	0	0
$855$	−2.17442	−0.0743634
$856$	0	0
$857$	−5.20269	−0.177721	−0.0888603	−	0.996044i	$-0.528322\pi$
$857$	−5.20269	−0.177721	−0.0888603	+	0.996044i	$0.528322\pi$
$858$	0	0
$859$	−52.3091	−1.78476	−0.892381	−	0.451283i	$-0.850967\pi$
$859$	−52.3091	−1.78476	−0.892381	+	0.451283i	$0.850967\pi$
$860$	0	0
$861$	2.67686	0.0912271
$862$	0	0
$863$	−27.7945	−0.946137	−0.473069	−	0.881026i	$-0.656854\pi$
$863$	−27.7945	−0.946137	−0.473069	+	0.881026i	$0.656854\pi$
$864$	0	0
$865$	19.4931	0.662785
$866$	0	0
$867$	4.07773	0.138487
$868$	0	0
$869$	11.4689	0.389057
$870$	0	0
$871$	−5.99448	−0.203115
$872$	0	0
$873$	−14.0544	−0.475670
$874$	0	0
$875$	−1.05225	−0.0355724
$876$	0	0
$877$	0.0140642	0.000474915 0	0.000237457	−	1.00000i	$-0.499924\pi$
$877$	0.0140642	0.000474915 0	0.000237457		1.00000i	$0.499924\pi$
$878$	0	0
$879$	−1.90946	−0.0644045
$880$	0	0
$881$	13.6980	0.461499	0.230749	−	0.973013i	$-0.425882\pi$
$881$	13.6980	0.461499	0.230749	+	0.973013i	$0.425882\pi$
$882$	0	0
$883$	−45.4390	−1.52914	−0.764572	−	0.644538i	$-0.777050\pi$
$883$	−45.4390	−1.52914	−0.764572	+	0.644538i	$0.777050\pi$
$884$	0	0
$885$	−8.38895	−0.281991
$886$	0	0
$887$	13.8503	0.465046	0.232523	−	0.972591i	$-0.425302\pi$
$887$	13.8503	0.465046	0.232523	+	0.972591i	$0.425302\pi$
$888$	0	0
$889$	−0.396189	−0.0132877
$890$	0	0
$891$	0.701065	0.0234866
$892$	0	0
$893$	21.5902	0.722488
$894$	0	0
$895$	21.9424	0.733455
$896$	0	0
$897$	−18.0918	−0.604066
$898$	0	0
$899$	47.7815	1.59360
$900$	0	0
$901$	53.4732	1.78145
$902$	0	0
$903$	3.39661	0.113032
$904$	0	0
$905$	1.07084	0.0355960
$906$	0	0
$907$	8.51587	0.282765	0.141382	−	0.989955i	$-0.454845\pi$
$907$	8.51587	0.282765	0.141382	+	0.989955i	$0.454845\pi$
$908$	0	0
$909$	2.99906	0.0994724
$910$	0	0
$911$	−14.3277	−0.474697	−0.237348	−	0.971425i	$-0.576278\pi$
$911$	−14.3277	−0.474697	−0.237348	+	0.971425i	$0.576278\pi$
$912$	0	0
$913$	3.05046	0.100956
$914$	0	0
$915$	−12.2090	−0.403617
$916$	0	0
$917$	0.900897	0.0297502
$918$	0	0
$919$	−3.51960	−0.116101	−0.0580504	−	0.998314i	$-0.518488\pi$
$919$	−3.51960	−0.116101	−0.0580504	+	0.998314i	$0.518488\pi$
$920$	0	0
$921$	10.2510	0.337781
$922$	0	0
$923$	−50.0760	−1.64827
$924$	0	0
$925$	−4.03312	−0.132608
$926$	0	0
$927$	−7.15546	−0.235016
$928$	0	0
$929$	39.7428	1.30392	0.651960	−	0.758253i	$-0.273947\pi$
$929$	39.7428	1.30392	0.651960	+	0.758253i	$0.273947\pi$
$930$	0	0
$931$	−12.8133	−0.419940
$932$	0	0
$933$	2.75247	0.0901118
$934$	0	0
$935$	3.21862	0.105260
$936$	0	0
$937$	−27.8825	−0.910883	−0.455441	−	0.890266i	$-0.650519\pi$
$937$	−27.8825	−0.910883	−0.455441	+	0.890266i	$0.650519\pi$
$938$	0	0
$939$	32.6612	1.06586
$940$	0	0
$941$	42.0590	1.37108	0.685542	−	0.728033i	$-0.259565\pi$
$941$	42.0590	1.37108	0.685542	+	0.728033i	$0.259565\pi$
$942$	0	0
$943$	7.67781	0.250024
$944$	0	0
$945$	−1.05225	−0.0342296
$946$	0	0
$947$	35.9187	1.16720	0.583601	−	0.812040i	$-0.301643\pi$
$947$	35.9187	1.16720	0.583601	+	0.812040i	$0.301643\pi$
$948$	0	0
$949$	−60.9928	−1.97991
$950$	0	0
$951$	23.6411	0.766615
$952$	0	0
$953$	−12.1247	−0.392758	−0.196379	−	0.980528i	$-0.562918\pi$
$953$	−12.1247	−0.392758	−0.196379	+	0.980528i	$0.562918\pi$
$954$	0	0
$955$	−24.9992	−0.808954
$956$	0	0
$957$	4.89405	0.158202
$958$	0	0
$959$	1.98495	0.0640974
$960$	0	0
$961$	15.8489	0.511253
$962$	0	0
$963$	5.14052	0.165651
$964$	0	0
$965$	19.5047	0.627878
$966$	0	0
$967$	58.6586	1.88633	0.943166	−	0.332321i	$-0.107832\pi$
$967$	58.6586	1.88633	0.943166	+	0.332321i	$0.107832\pi$
$968$	0	0
$969$	−9.98285	−0.320695
$970$	0	0
$971$	54.6859	1.75495	0.877477	−	0.479618i	$-0.159225\pi$
$971$	54.6859	1.75495	0.877477	+	0.479618i	$0.159225\pi$
$972$	0	0
$973$	−4.01947	−0.128858
$974$	0	0
$975$	−5.99448	−0.191977
$976$	0	0
$977$	−18.2295	−0.583214	−0.291607	−	0.956538i	$-0.594190\pi$
$977$	−18.2295	−0.583214	−0.291607	+	0.956538i	$0.594190\pi$
$978$	0	0
$979$	9.54504	0.305061
$980$	0	0
$981$	−15.5560	−0.496665
$982$	0	0
$983$	19.3585	0.617442	0.308721	−	0.951153i	$-0.400099\pi$
$983$	19.3585	0.617442	0.308721	+	0.951153i	$0.400099\pi$
$984$	0	0
$985$	−22.4851	−0.716434
$986$	0	0
$987$	10.4480	0.332562
$988$	0	0
$989$	9.74221	0.309784
$990$	0	0
$991$	−4.35716	−0.138410	−0.0692049	−	0.997602i	$-0.522046\pi$
$991$	−4.35716	−0.138410	−0.0692049	+	0.997602i	$0.522046\pi$
$992$	0	0
$993$	−20.2301	−0.641983
$994$	0	0
$995$	16.4697	0.522126
$996$	0	0
$997$	−32.4933	−1.02907	−0.514537	−	0.857468i	$-0.672036\pi$
$997$	−32.4933	−1.02907	−0.514537	+	0.857468i	$0.672036\pi$
$998$	0	0
$999$	−4.03312	−0.127602

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	4020.2.a.g.1.3	✓	6

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
4020.2.a.g.1.3	✓	6	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:	\( N \)	\(=\)	\( 4020 = 2^{2} \cdot 3 \cdot 5 \cdot 67 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	4020.a (trivial)

\(f(q)\)	\(=\)	\(q+1.00000 q^{3} -1.00000 q^{5} +1.05225 q^{7} +1.00000 q^{9} +O(q^{10})\)	\(q+1.00000 q^{3} -1.00000 q^{5} +1.05225 q^{7} +1.00000 q^{9} +0.701065 q^{11} -5.99448 q^{13} -1.00000 q^{15} -4.59105 q^{17} +2.17442 q^{19} +1.05225 q^{21} +3.01807 q^{23} +1.00000 q^{25} +1.00000 q^{27} +6.98088 q^{29} +6.84462 q^{31} +0.701065 q^{33} -1.05225 q^{35} -4.03312 q^{37} -5.99448 q^{39} +2.54395 q^{41} +3.22796 q^{43} -1.00000 q^{45} +9.92920 q^{47} -5.89278 q^{49} -4.59105 q^{51} -11.6473 q^{53} -0.701065 q^{55} +2.17442 q^{57} +8.38895 q^{59} +12.2090 q^{61} +1.05225 q^{63} +5.99448 q^{65} +1.00000 q^{67} +3.01807 q^{69} +8.35370 q^{71} +10.1748 q^{73} +1.00000 q^{75} +0.737693 q^{77} +16.3593 q^{79} +1.00000 q^{81} +4.35118 q^{83} +4.59105 q^{85} +6.98088 q^{87} +13.6151 q^{89} -6.30767 q^{91} +6.84462 q^{93} -2.17442 q^{95} -14.0544 q^{97} +0.701065 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 6 q + 6 q^{3} - 6 q^{5} + 3 q^{7} + 6 q^{9}+O(q^{10}) \) 6 * q + 6 * q^3 - 6 * q^5 + 3 * q^7 + 6 * q^9	\( 6 q + 6 q^{3} - 6 q^{5} + 3 q^{7} + 6 q^{9} + 3 q^{11} + 3 q^{13} - 6 q^{15} + 6 q^{17} - 3 q^{19} + 3 q^{21} + 6 q^{23} + 6 q^{25} + 6 q^{27} + 21 q^{29} - 3 q^{31} + 3 q^{33} - 3 q^{35} + 3 q^{39} + 9 q^{41} - 3 q^{43} - 6 q^{45} + 21 q^{47} + 3 q^{49} + 6 q^{51} + 15 q^{53} - 3 q^{55} - 3 q^{57} + 15 q^{59} + 15 q^{61} + 3 q^{63} - 3 q^{65} + 6 q^{67} + 6 q^{69} + 18 q^{71} - 6 q^{73} + 6 q^{75} + 33 q^{77} + 9 q^{79} + 6 q^{81} + 24 q^{83} - 6 q^{85} + 21 q^{87} + 24 q^{89} - 21 q^{91} - 3 q^{93} + 3 q^{95} + 9 q^{97} + 3 q^{99}+O(q^{100}) \) 6 * q + 6 * q^3 - 6 * q^5 + 3 * q^7 + 6 * q^9 + 3 * q^11 + 3 * q^13 - 6 * q^15 + 6 * q^17 - 3 * q^19 + 3 * q^21 + 6 * q^23 + 6 * q^25 + 6 * q^27 + 21 * q^29 - 3 * q^31 + 3 * q^33 - 3 * q^35 + 3 * q^39 + 9 * q^41 - 3 * q^43 - 6 * q^45 + 21 * q^47 + 3 * q^49 + 6 * q^51 + 15 * q^53 - 3 * q^55 - 3 * q^57 + 15 * q^59 + 15 * q^61 + 3 * q^63 - 3 * q^65 + 6 * q^67 + 6 * q^69 + 18 * q^71 - 6 * q^73 + 6 * q^75 + 33 * q^77 + 9 * q^79 + 6 * q^81 + 24 * q^83 - 6 * q^85 + 21 * q^87 + 24 * q^89 - 21 * q^91 - 3 * q^93 + 3 * q^95 + 9 * q^97 + 3 * q^99

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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