LMFDB - Embedded newform 4020.2.a.h.1.2

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:	$7$
Coefficient field:	$\mathbb{Q}[x]/(x^{7} - \cdots)$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{7} - 3x^{6} - 28x^{5} + 90x^{4} + 143x^{3} - 418x^{2} - 256x + 160 $ x^7 - 3x^6 - 28x^5 + 90x^4 + 143x^3 - 418x^2 - 256x + 160
Coefficient ring:	$\Z[a_1, \ldots, a_{17}]$
Coefficient ring index:	$ 1 $
Twist minimal:	yes
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.2
Root			$-2.11010$ 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)

$f(q)$	$=$	$q-1.00000 q^{3} -1.00000 q^{5} -2.11010 q^{7} +1.00000 q^{9} +O(q^{10})$	$q-1.00000 q^{3} -1.00000 q^{5} -2.11010 q^{7} +1.00000 q^{9} -1.80390 q^{11} +2.27174 q^{13} +1.00000 q^{15} -7.04852 q^{17} +7.64308 q^{19} +2.11010 q^{21} -8.99511 q^{23} +1.00000 q^{25} -1.00000 q^{27} +4.88501 q^{29} -4.18575 q^{31} +1.80390 q^{33} +2.11010 q^{35} +2.77490 q^{37} -2.27174 q^{39} -2.86817 q^{41} -0.432915 q^{43} -1.00000 q^{45} -5.20782 q^{47} -2.54746 q^{49} +7.04852 q^{51} -1.87329 q^{53} +1.80390 q^{55} -7.64308 q^{57} +3.24462 q^{59} +2.12671 q^{61} -2.11010 q^{63} -2.27174 q^{65} -1.00000 q^{67} +8.99511 q^{69} +3.27572 q^{71} +9.69100 q^{73} -1.00000 q^{75} +3.80642 q^{77} +5.75049 q^{79} +1.00000 q^{81} -7.66058 q^{83} +7.04852 q^{85} -4.88501 q^{87} -0.339642 q^{89} -4.79361 q^{91} +4.18575 q^{93} -7.64308 q^{95} -4.25514 q^{97} -1.80390 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 7 q - 7 q^{3} - 7 q^{5} + 3 q^{7} + 7 q^{9}+O(q^{10}) $ 7 * q - 7 * q^3 - 7 * q^5 + 3 * q^7 + 7 * q^9	$ 7 q - 7 q^{3} - 7 q^{5} + 3 q^{7} + 7 q^{9} - 5 q^{11} + 5 q^{13} + 7 q^{15} + 3 q^{17} + 2 q^{19} - 3 q^{21} - 3 q^{23} + 7 q^{25} - 7 q^{27} - 8 q^{29} + 7 q^{31} + 5 q^{33} - 3 q^{35} - 5 q^{37} - 5 q^{39} + 7 q^{41} + 3 q^{43} - 7 q^{45} - 6 q^{47} + 16 q^{49} - 3 q^{51} - 9 q^{53} + 5 q^{55} - 2 q^{57} - 22 q^{59} + 19 q^{61} + 3 q^{63} - 5 q^{65} - 7 q^{67} + 3 q^{69} + 23 q^{73} - 7 q^{75} + 9 q^{77} + 25 q^{79} + 7 q^{81} - 20 q^{83} - 3 q^{85} + 8 q^{87} + q^{89} + 23 q^{91} - 7 q^{93} - 2 q^{95} + 3 q^{97} - 5 q^{99}+O(q^{100}) $ 7 * q - 7 * q^3 - 7 * q^5 + 3 * q^7 + 7 * q^9 - 5 * q^11 + 5 * q^13 + 7 * q^15 + 3 * q^17 + 2 * q^19 - 3 * q^21 - 3 * q^23 + 7 * q^25 - 7 * q^27 - 8 * q^29 + 7 * q^31 + 5 * q^33 - 3 * q^35 - 5 * q^37 - 5 * q^39 + 7 * q^41 + 3 * q^43 - 7 * q^45 - 6 * q^47 + 16 * q^49 - 3 * q^51 - 9 * q^53 + 5 * q^55 - 2 * q^57 - 22 * q^59 + 19 * q^61 + 3 * q^63 - 5 * q^65 - 7 * q^67 + 3 * q^69 + 23 * q^73 - 7 * q^75 + 9 * q^77 + 25 * q^79 + 7 * q^81 - 20 * q^83 - 3 * q^85 + 8 * q^87 + q^89 + 23 * q^91 - 7 * q^93 - 2 * q^95 + 3 * q^97 - 5 * 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
$7$	−2.11010	−0.797545	−0.398772	−	0.917050i	$-0.630564\pi$
$7$	−2.11010	−0.797545	−0.398772	+	0.917050i	$0.630564\pi$
$8$	0	0
$9$	1.00000	0.333333
$10$	0	0
$11$	−1.80390	−0.543896	−0.271948	−	0.962312i	$-0.587668\pi$
$11$	−1.80390	−0.543896	−0.271948	+	0.962312i	$0.587668\pi$
$12$	0	0
$13$	2.27174	0.630068	0.315034	−	0.949080i	$-0.397984\pi$
$13$	2.27174	0.630068	0.315034	+	0.949080i	$0.397984\pi$
$14$	0	0
$15$	1.00000	0.258199
$16$	0	0
$17$	−7.04852	−1.70952	−0.854759	−	0.519025i	$-0.826295\pi$
$17$	−7.04852	−1.70952	−0.854759	+	0.519025i	$0.826295\pi$
$18$	0	0
$19$	7.64308	1.75344	0.876721	−	0.480999i	$-0.159726\pi$
$19$	7.64308	1.75344	0.876721	+	0.480999i	$0.159726\pi$
$20$	0	0
$21$	2.11010	0.460463
$22$	0	0
$23$	−8.99511	−1.87561	−0.937805	−	0.347162i	$-0.887145\pi$
$23$	−8.99511	−1.87561	−0.937805	+	0.347162i	$0.887145\pi$
$24$	0	0
$25$	1.00000	0.200000
$26$	0	0
$27$	−1.00000	−0.192450
$28$	0	0
$29$	4.88501	0.907123	0.453561	−	0.891225i	$-0.350153\pi$
$29$	4.88501	0.907123	0.453561	+	0.891225i	$0.350153\pi$
$30$	0	0
$31$	−4.18575	−0.751782	−0.375891	−	0.926664i	$-0.622663\pi$
$31$	−4.18575	−0.751782	−0.375891	+	0.926664i	$0.622663\pi$
$32$	0	0
$33$	1.80390	0.314019
$34$	0	0
$35$	2.11010	0.356673
$36$	0	0
$37$	2.77490	0.456191	0.228096	−	0.973639i	$-0.426750\pi$
$37$	2.77490	0.456191	0.228096	+	0.973639i	$0.426750\pi$
$38$	0	0
$39$	−2.27174	−0.363770
$40$	0	0
$41$	−2.86817	−0.447934	−0.223967	−	0.974597i	$-0.571901\pi$
$41$	−2.86817	−0.447934	−0.223967	+	0.974597i	$0.571901\pi$
$42$	0	0
$43$	−0.432915	−0.0660189	−0.0330095	−	0.999455i	$-0.510509\pi$
$43$	−0.432915	−0.0660189	−0.0330095	+	0.999455i	$0.510509\pi$
$44$	0	0
$45$	−1.00000	−0.149071
$46$	0	0
$47$	−5.20782	−0.759638	−0.379819	−	0.925061i	$-0.624014\pi$
$47$	−5.20782	−0.759638	−0.379819	+	0.925061i	$0.624014\pi$
$48$	0	0
$49$	−2.54746	−0.363923
$50$	0	0
$51$	7.04852	0.986991
$52$	0	0
$53$	−1.87329	−0.257316	−0.128658	−	0.991689i	$-0.541067\pi$
$53$	−1.87329	−0.257316	−0.128658	+	0.991689i	$0.541067\pi$
$54$	0	0
$55$	1.80390	0.243238
$56$	0	0
$57$	−7.64308	−1.01235
$58$	0	0
$59$	3.24462	0.422414	0.211207	−	0.977441i	$-0.432261\pi$
$59$	3.24462	0.422414	0.211207	+	0.977441i	$0.432261\pi$
$60$	0	0
$61$	2.12671	0.272297	0.136149	−	0.990688i	$-0.456528\pi$
$61$	2.12671	0.272297	0.136149	+	0.990688i	$0.456528\pi$
$62$	0	0
$63$	−2.11010	−0.265848
$64$	0	0
$65$	−2.27174	−0.281775
$66$	0	0
$67$	−1.00000	−0.122169
$67$	−1.00000	−0.122169
$68$	0	0
$69$	8.99511	1.08288
$70$	0	0
$71$	3.27572	0.388756	0.194378	−	0.980927i	$-0.437731\pi$
$71$	3.27572	0.388756	0.194378	+	0.980927i	$0.437731\pi$
$72$	0	0
$73$	9.69100	1.13425	0.567123	−	0.823633i	$-0.308056\pi$
$73$	9.69100	1.13425	0.567123	+	0.823633i	$0.308056\pi$
$74$	0	0
$75$	−1.00000	−0.115470
$76$	0	0
$77$	3.80642	0.433781
$78$	0	0
$79$	5.75049	0.646980	0.323490	−	0.946232i	$-0.395144\pi$
$79$	5.75049	0.646980	0.323490	+	0.946232i	$0.395144\pi$
$80$	0	0
$81$	1.00000	0.111111
$82$	0	0
$83$	−7.66058	−0.840858	−0.420429	−	0.907325i	$-0.638120\pi$
$83$	−7.66058	−0.840858	−0.420429	+	0.907325i	$0.638120\pi$
$84$	0	0
$85$	7.04852	0.764520
$86$	0	0
$87$	−4.88501	−0.523728
$88$	0	0
$89$	−0.339642	−0.0360020	−0.0180010	−	0.999838i	$-0.505730\pi$
$89$	−0.339642	−0.0360020	−0.0180010	+	0.999838i	$0.505730\pi$
$90$	0	0
$91$	−4.79361	−0.502507
$92$	0	0
$93$	4.18575	0.434042
$94$	0	0
$95$	−7.64308	−0.784163
$96$	0	0
$97$	−4.25514	−0.432044	−0.216022	−	0.976389i	$-0.569308\pi$
$97$	−4.25514	−0.432044	−0.216022	+	0.976389i	$0.569308\pi$
$98$	0	0
$99$	−1.80390	−0.181299
$100$	0	0
$101$	14.9617	1.48874	0.744371	−	0.667766i	$-0.232749\pi$
$101$	14.9617	1.48874	0.744371	+	0.667766i	$0.232749\pi$
$102$	0	0
$103$	1.56708	0.154409	0.0772047	−	0.997015i	$-0.475400\pi$
$103$	1.56708	0.154409	0.0772047	+	0.997015i	$0.475400\pi$
$104$	0	0
$105$	−2.11010	−0.205925
$106$	0	0
$107$	12.4755	1.20605	0.603027	−	0.797721i	$-0.293961\pi$
$107$	12.4755	1.20605	0.603027	+	0.797721i	$0.293961\pi$
$108$	0	0
$109$	1.04348	0.0999477	0.0499739	−	0.998751i	$-0.484086\pi$
$109$	1.04348	0.0999477	0.0499739	+	0.998751i	$0.484086\pi$
$110$	0	0
$111$	−2.77490	−0.263382
$112$	0	0
$113$	8.43714	0.793699	0.396850	−	0.917884i	$-0.370103\pi$
$113$	8.43714	0.793699	0.396850	+	0.917884i	$0.370103\pi$
$114$	0	0
$115$	8.99511	0.838798
$116$	0	0
$117$	2.27174	0.210023
$118$	0	0
$119$	14.8731	1.36342
$120$	0	0
$121$	−7.74595	−0.704177
$122$	0	0
$123$	2.86817	0.258615
$124$	0	0
$125$	−1.00000	−0.0894427
$126$	0	0
$127$	−1.58278	−0.140449	−0.0702243	−	0.997531i	$-0.522372\pi$
$127$	−1.58278	−0.140449	−0.0702243	+	0.997531i	$0.522372\pi$
$128$	0	0
$129$	0.432915	0.0381161
$130$	0	0
$131$	−17.4307	−1.52293	−0.761465	−	0.648207i	$-0.775519\pi$
$131$	−17.4307	−1.52293	−0.761465	+	0.648207i	$0.775519\pi$
$132$	0	0
$133$	−16.1277	−1.39845
$134$	0	0
$135$	1.00000	0.0860663
$136$	0	0
$137$	12.6340	1.07939	0.539696	−	0.841860i	$-0.318539\pi$
$137$	12.6340	1.07939	0.539696	+	0.841860i	$0.318539\pi$
$138$	0	0
$139$	8.14691	0.691012	0.345506	−	0.938417i	$-0.387707\pi$
$139$	8.14691	0.691012	0.345506	+	0.938417i	$0.387707\pi$
$140$	0	0
$141$	5.20782	0.438577
$142$	0	0
$143$	−4.09799	−0.342691
$144$	0	0
$145$	−4.88501	−0.405678
$146$	0	0
$147$	2.54746	0.210111
$148$	0	0
$149$	3.12062	0.255651	0.127825	−	0.991797i	$-0.459200\pi$
$149$	3.12062	0.255651	0.127825	+	0.991797i	$0.459200\pi$
$150$	0	0
$151$	7.53130	0.612889	0.306444	−	0.951889i	$-0.400861\pi$
$151$	7.53130	0.612889	0.306444	+	0.951889i	$0.400861\pi$
$152$	0	0
$153$	−7.04852	−0.569839
$154$	0	0
$155$	4.18575	0.336207
$156$	0	0
$157$	24.0535	1.91968	0.959841	−	0.280546i	$-0.0905154\pi$
$157$	24.0535	1.91968	0.959841	+	0.280546i	$0.0905154\pi$
$158$	0	0
$159$	1.87329	0.148562
$160$	0	0
$161$	18.9806	1.49588
$162$	0	0
$163$	−11.4470	−0.896596	−0.448298	−	0.893884i	$-0.647970\pi$
$163$	−11.4470	−0.896596	−0.448298	+	0.893884i	$0.647970\pi$
$164$	0	0
$165$	−1.80390	−0.140433
$166$	0	0
$167$	5.53790	0.428536	0.214268	−	0.976775i	$-0.431263\pi$
$167$	5.53790	0.428536	0.214268	+	0.976775i	$0.431263\pi$
$168$	0	0
$169$	−7.83919	−0.603014
$170$	0	0
$171$	7.64308	0.584481
$172$	0	0
$173$	16.6463	1.26559	0.632796	−	0.774319i	$-0.281907\pi$
$173$	16.6463	1.26559	0.632796	+	0.774319i	$0.281907\pi$
$174$	0	0
$175$	−2.11010	−0.159509
$176$	0	0
$177$	−3.24462	−0.243881
$178$	0	0
$179$	21.7551	1.62605	0.813025	−	0.582228i	$-0.197819\pi$
$179$	21.7551	1.62605	0.813025	+	0.582228i	$0.197819\pi$
$180$	0	0
$181$	24.8308	1.84566	0.922828	−	0.385211i	$-0.125871\pi$
$181$	24.8308	1.84566	0.922828	+	0.385211i	$0.125871\pi$
$182$	0	0
$183$	−2.12671	−0.157211
$184$	0	0
$185$	−2.77490	−0.204015
$186$	0	0
$187$	12.7148	0.929800
$188$	0	0
$189$	2.11010	0.153488
$190$	0	0
$191$	26.0690	1.88629	0.943144	−	0.332386i	$-0.107854\pi$
$191$	26.0690	1.88629	0.943144	+	0.332386i	$0.107854\pi$
$192$	0	0
$193$	−2.38338	−0.171559	−0.0857796	−	0.996314i	$-0.527338\pi$
$193$	−2.38338	−0.171559	−0.0857796	+	0.996314i	$0.527338\pi$
$194$	0	0
$195$	2.27174	0.162683
$196$	0	0
$197$	−7.77069	−0.553639	−0.276819	−	0.960922i	$-0.589280\pi$
$197$	−7.77069	−0.553639	−0.276819	+	0.960922i	$0.589280\pi$
$198$	0	0
$199$	−16.3002	−1.15549	−0.577746	−	0.816217i	$-0.696067\pi$
$199$	−16.3002	−1.15549	−0.577746	+	0.816217i	$0.696067\pi$
$200$	0	0
$201$	1.00000	0.0705346
$202$	0	0
$203$	−10.3079	−0.723471
$204$	0	0
$205$	2.86817	0.200322
$206$	0	0
$207$	−8.99511	−0.625203
$208$	0	0
$209$	−13.7873	−0.953690
$210$	0	0
$211$	−17.2249	−1.18581	−0.592907	−	0.805271i	$-0.702020\pi$
$211$	−17.2249	−1.18581	−0.592907	+	0.805271i	$0.702020\pi$
$212$	0	0
$213$	−3.27572	−0.224448
$214$	0	0
$215$	0.432915	0.0295246
$216$	0	0
$217$	8.83236	0.599580
$218$	0	0
$219$	−9.69100	−0.654857
$220$	0	0
$221$	−16.0124	−1.07711
$222$	0	0
$223$	8.31093	0.556541	0.278270	−	0.960503i	$-0.410239\pi$
$223$	8.31093	0.556541	0.278270	+	0.960503i	$0.410239\pi$
$224$	0	0
$225$	1.00000	0.0666667
$226$	0	0
$227$	12.3016	0.816484	0.408242	−	0.912874i	$-0.366142\pi$
$227$	12.3016	0.816484	0.408242	+	0.912874i	$0.366142\pi$
$228$	0	0
$229$	24.2455	1.60219	0.801095	−	0.598538i	$-0.204251\pi$
$229$	24.2455	1.60219	0.801095	+	0.598538i	$0.204251\pi$
$230$	0	0
$231$	−3.80642	−0.250444
$232$	0	0
$233$	21.1803	1.38757	0.693783	−	0.720184i	$-0.255943\pi$
$233$	21.1803	1.38757	0.693783	+	0.720184i	$0.255943\pi$
$234$	0	0
$235$	5.20782	0.339721
$236$	0	0
$237$	−5.75049	−0.373534
$238$	0	0
$239$	−29.1527	−1.88573	−0.942866	−	0.333171i	$-0.891881\pi$
$239$	−29.1527	−1.88573	−0.942866	+	0.333171i	$0.891881\pi$
$240$	0	0
$241$	8.68140	0.559218	0.279609	−	0.960114i	$-0.409795\pi$
$241$	8.68140	0.559218	0.279609	+	0.960114i	$0.409795\pi$
$242$	0	0
$243$	−1.00000	−0.0641500
$244$	0	0
$245$	2.54746	0.162751
$246$	0	0
$247$	17.3631	1.10479
$248$	0	0
$249$	7.66058	0.485470
$250$	0	0
$251$	−16.7064	−1.05450	−0.527250	−	0.849710i	$-0.676777\pi$
$251$	−16.7064	−1.05450	−0.527250	+	0.849710i	$0.676777\pi$
$252$	0	0
$253$	16.2263	1.02014
$254$	0	0
$255$	−7.04852	−0.441396
$256$	0	0
$257$	17.6423	1.10049	0.550247	−	0.835002i	$-0.314533\pi$
$257$	17.6423	1.10049	0.550247	+	0.835002i	$0.314533\pi$
$258$	0	0
$259$	−5.85533	−0.363833
$260$	0	0
$261$	4.88501	0.302374
$262$	0	0
$263$	−20.3112	−1.25244	−0.626220	−	0.779647i	$-0.715399\pi$
$263$	−20.3112	−1.25244	−0.626220	+	0.779647i	$0.715399\pi$
$264$	0	0
$265$	1.87329	0.115075
$266$	0	0
$267$	0.339642	0.0207858
$268$	0	0
$269$	−28.0522	−1.71037	−0.855186	−	0.518321i	$-0.826557\pi$
$269$	−28.0522	−1.71037	−0.855186	+	0.518321i	$0.826557\pi$
$270$	0	0
$271$	21.6778	1.31683	0.658415	−	0.752655i	$-0.271227\pi$
$271$	21.6778	1.31683	0.658415	+	0.752655i	$0.271227\pi$
$272$	0	0
$273$	4.79361	0.290123
$274$	0	0
$275$	−1.80390	−0.108779
$276$	0	0
$277$	−17.1950	−1.03315	−0.516573	−	0.856243i	$-0.672793\pi$
$277$	−17.1950	−1.03315	−0.516573	+	0.856243i	$0.672793\pi$
$278$	0	0
$279$	−4.18575	−0.250594
$280$	0	0
$281$	14.0872	0.840372	0.420186	−	0.907438i	$-0.361965\pi$
$281$	14.0872	0.840372	0.420186	+	0.907438i	$0.361965\pi$
$282$	0	0
$283$	3.67393	0.218393	0.109196	−	0.994020i	$-0.465172\pi$
$283$	3.67393	0.218393	0.109196	+	0.994020i	$0.465172\pi$
$284$	0	0
$285$	7.64308	0.452737
$286$	0	0
$287$	6.05215	0.357247
$288$	0	0
$289$	32.6817	1.92245
$290$	0	0
$291$	4.25514	0.249441
$292$	0	0
$293$	18.1512	1.06040	0.530202	−	0.847872i	$-0.322116\pi$
$293$	18.1512	1.06040	0.530202	+	0.847872i	$0.322116\pi$
$294$	0	0
$295$	−3.24462	−0.188909
$296$	0	0
$297$	1.80390	0.104673
$298$	0	0
$299$	−20.4346	−1.18176
$300$	0	0
$301$	0.913496	0.0526530
$302$	0	0
$303$	−14.9617	−0.859526
$304$	0	0
$305$	−2.12671	−0.121775
$306$	0	0
$307$	−1.71390	−0.0978177	−0.0489089	−	0.998803i	$-0.515574\pi$
$307$	−1.71390	−0.0978177	−0.0489089	+	0.998803i	$0.515574\pi$
$308$	0	0
$309$	−1.56708	−0.0891483
$310$	0	0
$311$	−0.399704	−0.0226651	−0.0113326	−	0.999936i	$-0.503607\pi$
$311$	−0.399704	−0.0226651	−0.0113326	+	0.999936i	$0.503607\pi$
$312$	0	0
$313$	2.57845	0.145743	0.0728713	−	0.997341i	$-0.476784\pi$
$313$	2.57845	0.145743	0.0728713	+	0.997341i	$0.476784\pi$
$314$	0	0
$315$	2.11010	0.118891
$316$	0	0
$317$	24.0051	1.34826	0.674129	−	0.738613i	$-0.264519\pi$
$317$	24.0051	1.34826	0.674129	+	0.738613i	$0.264519\pi$
$318$	0	0
$319$	−8.81206	−0.493381
$320$	0	0
$321$	−12.4755	−0.696316
$322$	0	0
$323$	−53.8724	−2.99754
$324$	0	0
$325$	2.27174	0.126014
$326$	0	0
$327$	−1.04348	−0.0577048
$328$	0	0
$329$	10.9890	0.605845
$330$	0	0
$331$	26.8814	1.47754	0.738768	−	0.673959i	$-0.235408\pi$
$331$	26.8814	1.47754	0.738768	+	0.673959i	$0.235408\pi$
$332$	0	0
$333$	2.77490	0.152064
$334$	0	0
$335$	1.00000	0.0546358
$336$	0	0
$337$	−19.1175	−1.04139	−0.520697	−	0.853741i	$-0.674328\pi$
$337$	−19.1175	−1.04139	−0.520697	+	0.853741i	$0.674328\pi$
$338$	0	0
$339$	−8.43714	−0.458243
$340$	0	0
$341$	7.55066	0.408891
$342$	0	0
$343$	20.1461	1.08779
$344$	0	0
$345$	−8.99511	−0.484280
$346$	0	0
$347$	−7.13367	−0.382955	−0.191478	−	0.981497i	$-0.561328\pi$
$347$	−7.13367	−0.382955	−0.191478	+	0.981497i	$0.561328\pi$
$348$	0	0
$349$	−5.52897	−0.295959	−0.147980	−	0.988990i	$-0.547277\pi$
$349$	−5.52897	−0.295959	−0.147980	+	0.988990i	$0.547277\pi$
$350$	0	0
$351$	−2.27174	−0.121257
$352$	0	0
$353$	−18.5598	−0.987841	−0.493920	−	0.869507i	$-0.664437\pi$
$353$	−18.5598	−0.987841	−0.493920	+	0.869507i	$0.664437\pi$
$354$	0	0
$355$	−3.27572	−0.173857
$356$	0	0
$357$	−14.8731	−0.787169
$358$	0	0
$359$	−6.43394	−0.339570	−0.169785	−	0.985481i	$-0.554307\pi$
$359$	−6.43394	−0.339570	−0.169785	+	0.985481i	$0.554307\pi$
$360$	0	0
$361$	39.4166	2.07456
$362$	0	0
$363$	7.74595	0.406557
$364$	0	0
$365$	−9.69100	−0.507250
$366$	0	0
$367$	13.8930	0.725208	0.362604	−	0.931943i	$-0.381888\pi$
$367$	13.8930	0.725208	0.362604	+	0.931943i	$0.381888\pi$
$368$	0	0
$369$	−2.86817	−0.149311
$370$	0	0
$371$	3.95284	0.205221
$372$	0	0
$373$	4.99426	0.258593	0.129296	−	0.991606i	$-0.458728\pi$
$373$	4.99426	0.258593	0.129296	+	0.991606i	$0.458728\pi$
$374$	0	0
$375$	1.00000	0.0516398
$376$	0	0
$377$	11.0975	0.571549
$378$	0	0
$379$	28.6093	1.46956	0.734782	−	0.678304i	$-0.237285\pi$
$379$	28.6093	1.46956	0.734782	+	0.678304i	$0.237285\pi$
$380$	0	0
$381$	1.58278	0.0810881
$382$	0	0
$383$	−0.313760	−0.0160324	−0.00801619	−	0.999968i	$-0.502552\pi$
$383$	−0.313760	−0.0160324	−0.00801619	+	0.999968i	$0.502552\pi$
$384$	0	0
$385$	−3.80642	−0.193993
$386$	0	0
$387$	−0.432915	−0.0220063
$388$	0	0
$389$	−18.6646	−0.946335	−0.473167	−	0.880973i	$-0.656889\pi$
$389$	−18.6646	−0.946335	−0.473167	+	0.880973i	$0.656889\pi$
$390$	0	0
$391$	63.4023	3.20639
$392$	0	0
$393$	17.4307	0.879264
$394$	0	0
$395$	−5.75049	−0.289338
$396$	0	0
$397$	−3.79176	−0.190303	−0.0951514	−	0.995463i	$-0.530334\pi$
$397$	−3.79176	−0.190303	−0.0951514	+	0.995463i	$0.530334\pi$
$398$	0	0
$399$	16.1277	0.807394
$400$	0	0
$401$	21.5506	1.07618	0.538092	−	0.842886i	$-0.319145\pi$
$401$	21.5506	1.07618	0.538092	+	0.842886i	$0.319145\pi$
$402$	0	0
$403$	−9.50893	−0.473674
$404$	0	0
$405$	−1.00000	−0.0496904
$406$	0	0
$407$	−5.00564	−0.248121
$408$	0	0
$409$	−19.4110	−0.959811	−0.479905	−	0.877320i	$-0.659329\pi$
$409$	−19.4110	−0.959811	−0.479905	+	0.877320i	$0.659329\pi$
$410$	0	0
$411$	−12.6340	−0.623187
$412$	0	0
$413$	−6.84650	−0.336894
$414$	0	0
$415$	7.66058	0.376043
$416$	0	0
$417$	−8.14691	−0.398956
$418$	0	0
$419$	−17.3599	−0.848087	−0.424043	−	0.905642i	$-0.639390\pi$
$419$	−17.3599	−0.848087	−0.424043	+	0.905642i	$0.639390\pi$
$420$	0	0
$421$	−17.4477	−0.850348	−0.425174	−	0.905112i	$-0.639787\pi$
$421$	−17.4477	−0.850348	−0.425174	+	0.905112i	$0.639787\pi$
$422$	0	0
$423$	−5.20782	−0.253213
$424$	0	0
$425$	−7.04852	−0.341904
$426$	0	0
$427$	−4.48758	−0.217169
$428$	0	0
$429$	4.09799	0.197853
$430$	0	0
$431$	−28.4423	−1.37002	−0.685008	−	0.728536i	$-0.740201\pi$
$431$	−28.4423	−1.37002	−0.685008	+	0.728536i	$0.740201\pi$
$432$	0	0
$433$	−23.8137	−1.14441	−0.572206	−	0.820110i	$-0.693912\pi$
$433$	−23.8137	−1.14441	−0.572206	+	0.820110i	$0.693912\pi$
$434$	0	0
$435$	4.88501	0.234218
$436$	0	0
$437$	−68.7503	−3.28877
$438$	0	0
$439$	19.1473	0.913851	0.456925	−	0.889505i	$-0.348951\pi$
$439$	19.1473	0.913851	0.456925	+	0.889505i	$0.348951\pi$
$440$	0	0
$441$	−2.54746	−0.121308
$442$	0	0
$443$	22.3131	1.06013	0.530064	−	0.847958i	$-0.322168\pi$
$443$	22.3131	1.06013	0.530064	+	0.847958i	$0.322168\pi$
$444$	0	0
$445$	0.339642	0.0161006
$446$	0	0
$447$	−3.12062	−0.147600
$448$	0	0
$449$	−9.25453	−0.436748	−0.218374	−	0.975865i	$-0.570075\pi$
$449$	−9.25453	−0.436748	−0.218374	+	0.975865i	$0.570075\pi$
$450$	0	0
$451$	5.17390	0.243629
$452$	0	0
$453$	−7.53130	−0.353852
$454$	0	0
$455$	4.79361	0.224728
$456$	0	0
$457$	2.71830	0.127157	0.0635783	−	0.997977i	$-0.479749\pi$
$457$	2.71830	0.127157	0.0635783	+	0.997977i	$0.479749\pi$
$458$	0	0
$459$	7.04852	0.328997
$460$	0	0
$461$	31.4951	1.46687	0.733437	−	0.679757i	$-0.237915\pi$
$461$	31.4951	1.46687	0.733437	+	0.679757i	$0.237915\pi$
$462$	0	0
$463$	29.0466	1.34991	0.674955	−	0.737859i	$-0.264163\pi$
$463$	29.0466	1.34991	0.674955	+	0.737859i	$0.264163\pi$
$464$	0	0
$465$	−4.18575	−0.194109
$466$	0	0
$467$	−22.9811	−1.06344	−0.531719	−	0.846921i	$-0.678454\pi$
$467$	−22.9811	−1.06344	−0.531719	+	0.846921i	$0.678454\pi$
$468$	0	0
$469$	2.11010	0.0974356
$470$	0	0
$471$	−24.0535	−1.10833
$472$	0	0
$473$	0.780935	0.0359074
$474$	0	0
$475$	7.64308	0.350688
$476$	0	0
$477$	−1.87329	−0.0857721
$478$	0	0
$479$	1.61766	0.0739130	0.0369565	−	0.999317i	$-0.488234\pi$
$479$	1.61766	0.0739130	0.0369565	+	0.999317i	$0.488234\pi$
$480$	0	0
$481$	6.30386	0.287431
$482$	0	0
$483$	−18.9806	−0.863648
$484$	0	0
$485$	4.25514	0.193216
$486$	0	0
$487$	−7.84833	−0.355642	−0.177821	−	0.984063i	$-0.556905\pi$
$487$	−7.84833	−0.355642	−0.177821	+	0.984063i	$0.556905\pi$
$488$	0	0
$489$	11.4470	0.517650
$490$	0	0
$491$	33.0498	1.49152	0.745758	−	0.666217i	$-0.232087\pi$
$491$	33.0498	1.49152	0.745758	+	0.666217i	$0.232087\pi$
$492$	0	0
$493$	−34.4321	−1.55074
$494$	0	0
$495$	1.80390	0.0810792
$496$	0	0
$497$	−6.91211	−0.310050
$498$	0	0
$499$	8.28860	0.371049	0.185524	−	0.982640i	$-0.440602\pi$
$499$	8.28860	0.371049	0.185524	+	0.982640i	$0.440602\pi$
$500$	0	0
$501$	−5.53790	−0.247415
$502$	0	0
$503$	−11.6274	−0.518439	−0.259220	−	0.965818i	$-0.583465\pi$
$503$	−11.6274	−0.518439	−0.259220	+	0.965818i	$0.583465\pi$
$504$	0	0
$505$	−14.9617	−0.665786
$506$	0	0
$507$	7.83919	0.348151
$508$	0	0
$509$	−15.5148	−0.687681	−0.343840	−	0.939028i	$-0.611728\pi$
$509$	−15.5148	−0.687681	−0.343840	+	0.939028i	$0.611728\pi$
$510$	0	0
$511$	−20.4490	−0.904612
$512$	0	0
$513$	−7.64308	−0.337450
$514$	0	0
$515$	−1.56708	−0.0690540
$516$	0	0
$517$	9.39438	0.413164
$518$	0	0
$519$	−16.6463	−0.730689
$520$	0	0
$521$	33.9386	1.48688	0.743439	−	0.668804i	$-0.233194\pi$
$521$	33.9386	1.48688	0.743439	+	0.668804i	$0.233194\pi$
$522$	0	0
$523$	−38.3912	−1.67873	−0.839365	−	0.543569i	$-0.817073\pi$
$523$	−38.3912	−1.67873	−0.839365	+	0.543569i	$0.817073\pi$
$524$	0	0
$525$	2.11010	0.0920925
$526$	0	0
$527$	29.5033	1.28519
$528$	0	0
$529$	57.9120	2.51791
$530$	0	0
$531$	3.24462	0.140805
$532$	0	0
$533$	−6.51575	−0.282229
$534$	0	0
$535$	−12.4755	−0.539364
$536$	0	0
$537$	−21.7551	−0.938801
$538$	0	0
$539$	4.59536	0.197936
$540$	0	0
$541$	7.04217	0.302767	0.151383	−	0.988475i	$-0.451627\pi$
$541$	7.04217	0.302767	0.151383	+	0.988475i	$0.451627\pi$
$542$	0	0
$543$	−24.8308	−1.06559
$544$	0	0
$545$	−1.04348	−0.0446980
$546$	0	0
$547$	−31.1165	−1.33044	−0.665222	−	0.746646i	$-0.731663\pi$
$547$	−31.1165	−1.33044	−0.665222	+	0.746646i	$0.731663\pi$
$548$	0	0
$549$	2.12671	0.0907658
$550$	0	0
$551$	37.3365	1.59059
$552$	0	0
$553$	−12.1341	−0.515996
$554$	0	0
$555$	2.77490	0.117788
$556$	0	0
$557$	20.2758	0.859111	0.429556	−	0.903040i	$-0.358670\pi$
$557$	20.2758	0.859111	0.429556	+	0.903040i	$0.358670\pi$
$558$	0	0
$559$	−0.983472	−0.0415964
$560$	0	0
$561$	−12.7148	−0.536820
$562$	0	0
$563$	−21.6980	−0.914461	−0.457230	−	0.889348i	$-0.651159\pi$
$563$	−21.6980	−0.914461	−0.457230	+	0.889348i	$0.651159\pi$
$564$	0	0
$565$	−8.43714	−0.354953
$566$	0	0
$567$	−2.11010	−0.0886161
$568$	0	0
$569$	14.2583	0.597739	0.298869	−	0.954294i	$-0.403390\pi$
$569$	14.2583	0.597739	0.298869	+	0.954294i	$0.403390\pi$
$570$	0	0
$571$	18.4345	0.771461	0.385730	−	0.922612i	$-0.373950\pi$
$571$	18.4345	0.771461	0.385730	+	0.922612i	$0.373950\pi$
$572$	0	0
$573$	−26.0690	−1.08905
$574$	0	0
$575$	−8.99511	−0.375122
$576$	0	0
$577$	35.7947	1.49015	0.745076	−	0.666980i	$-0.232413\pi$
$577$	35.7947	1.49015	0.745076	+	0.666980i	$0.232413\pi$
$578$	0	0
$579$	2.38338	0.0990497
$580$	0	0
$581$	16.1646	0.670622
$582$	0	0
$583$	3.37923	0.139953
$584$	0	0
$585$	−2.27174	−0.0939250
$586$	0	0
$587$	31.5679	1.30295	0.651473	−	0.758672i	$-0.274151\pi$
$587$	31.5679	1.30295	0.651473	+	0.758672i	$0.274151\pi$
$588$	0	0
$589$	−31.9920	−1.31821
$590$	0	0
$591$	7.77069	0.319643
$592$	0	0
$593$	8.37918	0.344092	0.172046	−	0.985089i	$-0.444962\pi$
$593$	8.37918	0.344092	0.172046	+	0.985089i	$0.444962\pi$
$594$	0	0
$595$	−14.8731	−0.609739
$596$	0	0
$597$	16.3002	0.667123
$598$	0	0
$599$	4.94908	0.202214	0.101107	−	0.994876i	$-0.467762\pi$
$599$	4.94908	0.202214	0.101107	+	0.994876i	$0.467762\pi$
$600$	0	0
$601$	−1.56571	−0.0638665	−0.0319332	−	0.999490i	$-0.510166\pi$
$601$	−1.56571	−0.0638665	−0.0319332	+	0.999490i	$0.510166\pi$
$602$	0	0
$603$	−1.00000	−0.0407231
$604$	0	0
$605$	7.74595	0.314918
$606$	0	0
$607$	20.9545	0.850516	0.425258	−	0.905072i	$-0.360183\pi$
$607$	20.9545	0.850516	0.425258	+	0.905072i	$0.360183\pi$
$608$	0	0
$609$	10.3079	0.417696
$610$	0	0
$611$	−11.8308	−0.478624
$612$	0	0
$613$	−35.6976	−1.44181	−0.720905	−	0.693034i	$-0.756274\pi$
$613$	−35.6976	−1.44181	−0.720905	+	0.693034i	$0.756274\pi$
$614$	0	0
$615$	−2.86817	−0.115656
$616$	0	0
$617$	−22.0088	−0.886041	−0.443020	−	0.896512i	$-0.646093\pi$
$617$	−22.0088	−0.886041	−0.443020	+	0.896512i	$0.646093\pi$
$618$	0	0
$619$	17.7364	0.712888	0.356444	−	0.934317i	$-0.383989\pi$
$619$	17.7364	0.712888	0.356444	+	0.934317i	$0.383989\pi$
$620$	0	0
$621$	8.99511	0.360961
$622$	0	0
$623$	0.716680	0.0287132
$624$	0	0
$625$	1.00000	0.0400000
$626$	0	0
$627$	13.7873	0.550613
$628$	0	0
$629$	−19.5590	−0.779867
$630$	0	0
$631$	0.256552	0.0102132	0.00510659	−	0.999987i	$-0.498375\pi$
$631$	0.256552	0.0102132	0.00510659	+	0.999987i	$0.498375\pi$
$632$	0	0
$633$	17.2249	0.684630
$634$	0	0
$635$	1.58278	0.0628105
$636$	0	0
$637$	−5.78717	−0.229296
$638$	0	0
$639$	3.27572	0.129585
$640$	0	0
$641$	−30.2888	−1.19634	−0.598168	−	0.801371i	$-0.704105\pi$
$641$	−30.2888	−1.19634	−0.598168	+	0.801371i	$0.704105\pi$
$642$	0	0
$643$	−44.9971	−1.77451	−0.887256	−	0.461277i	$-0.847391\pi$
$643$	−44.9971	−1.77451	−0.887256	+	0.461277i	$0.847391\pi$
$644$	0	0
$645$	−0.432915	−0.0170460
$646$	0	0
$647$	−16.9615	−0.666826	−0.333413	−	0.942781i	$-0.608200\pi$
$647$	−16.9615	−0.666826	−0.333413	+	0.942781i	$0.608200\pi$
$648$	0	0
$649$	−5.85298	−0.229749
$650$	0	0
$651$	−8.83236	−0.346168
$652$	0	0
$653$	31.8409	1.24603	0.623016	−	0.782209i	$-0.285907\pi$
$653$	31.8409	1.24603	0.623016	+	0.782209i	$0.285907\pi$
$654$	0	0
$655$	17.4307	0.681075
$656$	0	0
$657$	9.69100	0.378082
$658$	0	0
$659$	−39.5614	−1.54109	−0.770547	−	0.637384i	$-0.780017\pi$
$659$	−39.5614	−1.54109	−0.770547	+	0.637384i	$0.780017\pi$
$660$	0	0
$661$	16.9698	0.660050	0.330025	−	0.943972i	$-0.392943\pi$
$661$	16.9698	0.660050	0.330025	+	0.943972i	$0.392943\pi$
$662$	0	0
$663$	16.0124	0.621871
$664$	0	0
$665$	16.1277	0.625405
$666$	0	0
$667$	−43.9412	−1.70141
$668$	0	0
$669$	−8.31093	−0.321319
$670$	0	0
$671$	−3.83637	−0.148101
$672$	0	0
$673$	−49.5577	−1.91031	−0.955155	−	0.296106i	$-0.904312\pi$
$673$	−49.5577	−1.91031	−0.955155	+	0.296106i	$0.904312\pi$
$674$	0	0
$675$	−1.00000	−0.0384900
$676$	0	0
$677$	22.3309	0.858245	0.429122	−	0.903246i	$-0.358823\pi$
$677$	22.3309	0.858245	0.429122	+	0.903246i	$0.358823\pi$
$678$	0	0
$679$	8.97878	0.344574
$680$	0	0
$681$	−12.3016	−0.471397
$682$	0	0
$683$	−13.1122	−0.501726	−0.250863	−	0.968023i	$-0.580714\pi$
$683$	−13.1122	−0.501726	−0.250863	+	0.968023i	$0.580714\pi$
$684$	0	0
$685$	−12.6340	−0.482719
$686$	0	0
$687$	−24.2455	−0.925024
$688$	0	0
$689$	−4.25563	−0.162127
$690$	0	0
$691$	−1.16174	−0.0441946	−0.0220973	−	0.999756i	$-0.507034\pi$
$691$	−1.16174	−0.0441946	−0.0220973	+	0.999756i	$0.507034\pi$
$692$	0	0
$693$	3.80642	0.144594
$694$	0	0
$695$	−8.14691	−0.309030
$696$	0	0
$697$	20.2164	0.765751
$698$	0	0
$699$	−21.1803	−0.801111
$700$	0	0
$701$	−21.5744	−0.814853	−0.407426	−	0.913238i	$-0.633574\pi$
$701$	−21.5744	−0.814853	−0.407426	+	0.913238i	$0.633574\pi$
$702$	0	0
$703$	21.2088	0.799904
$704$	0	0
$705$	−5.20782	−0.196138
$706$	0	0
$707$	−31.5707	−1.18734
$708$	0	0
$709$	35.2660	1.32444	0.662222	−	0.749308i	$-0.269614\pi$
$709$	35.2660	1.32444	0.662222	+	0.749308i	$0.269614\pi$
$710$	0	0
$711$	5.75049	0.215660
$712$	0	0
$713$	37.6512	1.41005
$714$	0	0
$715$	4.09799	0.153256
$716$	0	0
$717$	29.1527	1.08873
$718$	0	0
$719$	−18.8925	−0.704570	−0.352285	−	0.935893i	$-0.614595\pi$
$719$	−18.8925	−0.704570	−0.352285	+	0.935893i	$0.614595\pi$
$720$	0	0
$721$	−3.30671	−0.123148
$722$	0	0
$723$	−8.68140	−0.322865
$724$	0	0
$725$	4.88501	0.181425
$726$	0	0
$727$	17.7609	0.658715	0.329357	−	0.944205i	$-0.393168\pi$
$727$	17.7609	0.658715	0.329357	+	0.944205i	$0.393168\pi$
$728$	0	0
$729$	1.00000	0.0370370
$730$	0	0
$731$	3.05141	0.112861
$732$	0	0
$733$	−38.4382	−1.41975	−0.709874	−	0.704329i	$-0.751248\pi$
$733$	−38.4382	−1.41975	−0.709874	+	0.704329i	$0.751248\pi$
$734$	0	0
$735$	−2.54746	−0.0939644
$736$	0	0
$737$	1.80390	0.0664475
$738$	0	0
$739$	−19.6825	−0.724032	−0.362016	−	0.932172i	$-0.617911\pi$
$739$	−19.6825	−0.724032	−0.362016	+	0.932172i	$0.617911\pi$
$740$	0	0
$741$	−17.3631	−0.637849
$742$	0	0
$743$	−24.1901	−0.887448	−0.443724	−	0.896163i	$-0.646343\pi$
$743$	−24.1901	−0.887448	−0.443724	+	0.896163i	$0.646343\pi$
$744$	0	0
$745$	−3.12062	−0.114331
$746$	0	0
$747$	−7.66058	−0.280286
$748$	0	0
$749$	−26.3247	−0.961882
$750$	0	0
$751$	−20.8778	−0.761843	−0.380922	−	0.924607i	$-0.624393\pi$
$751$	−20.8778	−0.761843	−0.380922	+	0.924607i	$0.624393\pi$
$752$	0	0
$753$	16.7064	0.608816
$754$	0	0
$755$	−7.53130	−0.274092
$756$	0	0
$757$	16.0811	0.584477	0.292239	−	0.956345i	$-0.405600\pi$
$757$	16.0811	0.584477	0.292239	+	0.956345i	$0.405600\pi$
$758$	0	0
$759$	−16.2263	−0.588976
$760$	0	0
$761$	9.65385	0.349952	0.174976	−	0.984573i	$-0.444015\pi$
$761$	9.65385	0.349952	0.174976	+	0.984573i	$0.444015\pi$
$762$	0	0
$763$	−2.20186	−0.0797128
$764$	0	0
$765$	7.04852	0.254840
$766$	0	0
$767$	7.37095	0.266150
$768$	0	0
$769$	23.6368	0.852364	0.426182	−	0.904637i	$-0.359858\pi$
$769$	23.6368	0.852364	0.426182	+	0.904637i	$0.359858\pi$
$770$	0	0
$771$	−17.6423	−0.635371
$772$	0	0
$773$	−8.67398	−0.311981	−0.155991	−	0.987759i	$-0.549857\pi$
$773$	−8.67398	−0.311981	−0.155991	+	0.987759i	$0.549857\pi$
$774$	0	0
$775$	−4.18575	−0.150356
$776$	0	0
$777$	5.85533	0.210059
$778$	0	0
$779$	−21.9217	−0.785426
$780$	0	0
$781$	−5.90906	−0.211443
$782$	0	0
$783$	−4.88501	−0.174576
$784$	0	0
$785$	−24.0535	−0.858508
$786$	0	0
$787$	−37.7718	−1.34642	−0.673209	−	0.739452i	$-0.735085\pi$
$787$	−37.7718	−1.34642	−0.673209	+	0.739452i	$0.735085\pi$
$788$	0	0
$789$	20.3112	0.723096
$790$	0	0
$791$	−17.8032	−0.633011
$792$	0	0
$793$	4.83134	0.171566
$794$	0	0
$795$	−1.87329	−0.0664387
$796$	0	0
$797$	14.7466	0.522351	0.261176	−	0.965291i	$-0.415890\pi$
$797$	14.7466	0.522351	0.261176	+	0.965291i	$0.415890\pi$
$798$	0	0
$799$	36.7074	1.29862
$800$	0	0
$801$	−0.339642	−0.0120007
$802$	0	0
$803$	−17.4816	−0.616912
$804$	0	0
$805$	−18.9806	−0.668979
$806$	0	0
$807$	28.0522	0.987484
$808$	0	0
$809$	10.9148	0.383743	0.191872	−	0.981420i	$-0.438544\pi$
$809$	10.9148	0.383743	0.191872	+	0.981420i	$0.438544\pi$
$810$	0	0
$811$	−38.9717	−1.36848	−0.684241	−	0.729256i	$-0.739866\pi$
$811$	−38.9717	−1.36848	−0.684241	+	0.729256i	$0.739866\pi$
$812$	0	0
$813$	−21.6778	−0.760272
$814$	0	0
$815$	11.4470	0.400970
$816$	0	0
$817$	−3.30880	−0.115760
$818$	0	0
$819$	−4.79361	−0.167502
$820$	0	0
$821$	−26.6305	−0.929410	−0.464705	−	0.885466i	$-0.653840\pi$
$821$	−26.6305	−0.929410	−0.464705	+	0.885466i	$0.653840\pi$
$822$	0	0
$823$	35.7999	1.24790	0.623952	−	0.781462i	$-0.285526\pi$
$823$	35.7999	1.24790	0.623952	+	0.781462i	$0.285526\pi$
$824$	0	0
$825$	1.80390	0.0628037
$826$	0	0
$827$	−20.8982	−0.726701	−0.363350	−	0.931653i	$-0.618367\pi$
$827$	−20.8982	−0.726701	−0.363350	+	0.931653i	$0.618367\pi$
$828$	0	0
$829$	7.29761	0.253457	0.126728	−	0.991937i	$-0.459552\pi$
$829$	7.29761	0.253457	0.126728	+	0.991937i	$0.459552\pi$
$830$	0	0
$831$	17.1950	0.596487
$832$	0	0
$833$	17.9558	0.622132
$834$	0	0
$835$	−5.53790	−0.191647
$836$	0	0
$837$	4.18575	0.144681
$838$	0	0
$839$	−14.7623	−0.509650	−0.254825	−	0.966987i	$-0.582018\pi$
$839$	−14.7623	−0.509650	−0.254825	+	0.966987i	$0.582018\pi$
$840$	0	0
$841$	−5.13671	−0.177128
$842$	0	0
$843$	−14.0872	−0.485189
$844$	0	0
$845$	7.83919	0.269676
$846$	0	0
$847$	16.3448	0.561613
$848$	0	0
$849$	−3.67393	−0.126089
$850$	0	0
$851$	−24.9605	−0.855637
$852$	0	0
$853$	40.0193	1.37023	0.685117	−	0.728433i	$-0.259751\pi$
$853$	40.0193	1.37023	0.685117	+	0.728433i	$0.259751\pi$
$854$	0	0
$855$	−7.64308	−0.261388
$856$	0	0
$857$	9.77446	0.333889	0.166945	−	0.985966i	$-0.446610\pi$
$857$	9.77446	0.333889	0.166945	+	0.985966i	$0.446610\pi$
$858$	0	0
$859$	−43.5980	−1.48754	−0.743772	−	0.668434i	$-0.766965\pi$
$859$	−43.5980	−1.48754	−0.743772	+	0.668434i	$0.766965\pi$
$860$	0	0
$861$	−6.05215	−0.206257
$862$	0	0
$863$	−1.92636	−0.0655740	−0.0327870	−	0.999462i	$-0.510438\pi$
$863$	−1.92636	−0.0655740	−0.0327870	+	0.999462i	$0.510438\pi$
$864$	0	0
$865$	−16.6463	−0.565990
$866$	0	0
$867$	−32.6817	−1.10993
$868$	0	0
$869$	−10.3733	−0.351890
$870$	0	0
$871$	−2.27174	−0.0769750
$872$	0	0
$873$	−4.25514	−0.144015
$874$	0	0
$875$	2.11010	0.0713346
$876$	0	0
$877$	16.6705	0.562921	0.281461	−	0.959573i	$-0.409181\pi$
$877$	16.6705	0.562921	0.281461	+	0.959573i	$0.409181\pi$
$878$	0	0
$879$	−18.1512	−0.612224
$880$	0	0
$881$	−41.1169	−1.38526	−0.692631	−	0.721292i	$-0.743549\pi$
$881$	−41.1169	−1.38526	−0.692631	+	0.721292i	$0.743549\pi$
$882$	0	0
$883$	13.1796	0.443529	0.221765	−	0.975100i	$-0.428818\pi$
$883$	13.1796	0.443529	0.221765	+	0.975100i	$0.428818\pi$
$884$	0	0
$885$	3.24462	0.109067
$886$	0	0
$887$	−38.9270	−1.30704	−0.653520	−	0.756909i	$-0.726709\pi$
$887$	−38.9270	−1.30704	−0.653520	+	0.756909i	$0.726709\pi$
$888$	0	0
$889$	3.33982	0.112014
$890$	0	0
$891$	−1.80390	−0.0604329
$892$	0	0
$893$	−39.8037	−1.33198
$894$	0	0
$895$	−21.7551	−0.727192
$896$	0	0
$897$	20.4346	0.682290
$898$	0	0
$899$	−20.4474	−0.681959
$900$	0	0
$901$	13.2039	0.439887
$902$	0	0
$903$	−0.913496	−0.0303993
$904$	0	0
$905$	−24.8308	−0.825403
$906$	0	0
$907$	28.6714	0.952018	0.476009	−	0.879440i	$-0.342083\pi$
$907$	28.6714	0.952018	0.476009	+	0.879440i	$0.342083\pi$
$908$	0	0
$909$	14.9617	0.496247
$910$	0	0
$911$	31.1162	1.03093	0.515463	−	0.856912i	$-0.327620\pi$
$911$	31.1162	1.03093	0.515463	+	0.856912i	$0.327620\pi$
$912$	0	0
$913$	13.8189	0.457340
$914$	0	0
$915$	2.12671	0.0703069
$916$	0	0
$917$	36.7806	1.21460
$918$	0	0
$919$	−49.4498	−1.63120	−0.815600	−	0.578617i	$-0.803593\pi$
$919$	−49.4498	−1.63120	−0.815600	+	0.578617i	$0.803593\pi$
$920$	0	0
$921$	1.71390	0.0564751
$922$	0	0
$923$	7.44158	0.244943
$924$	0	0
$925$	2.77490	0.0912382
$926$	0	0
$927$	1.56708	0.0514698
$928$	0	0
$929$	9.25715	0.303717	0.151859	−	0.988402i	$-0.451474\pi$
$929$	9.25715	0.303717	0.151859	+	0.988402i	$0.451474\pi$
$930$	0	0
$931$	−19.4704	−0.638117
$932$	0	0
$933$	0.399704	0.0130857
$934$	0	0
$935$	−12.7148	−0.415819
$936$	0	0
$937$	−8.31480	−0.271633	−0.135816	−	0.990734i	$-0.543366\pi$
$937$	−8.31480	−0.271633	−0.135816	+	0.990734i	$0.543366\pi$
$938$	0	0
$939$	−2.57845	−0.0841446
$940$	0	0
$941$	1.80193	0.0587414	0.0293707	−	0.999569i	$-0.490650\pi$
$941$	1.80193	0.0587414	0.0293707	+	0.999569i	$0.490650\pi$
$942$	0	0
$943$	25.7995	0.840149
$944$	0	0
$945$	−2.11010	−0.0686417
$946$	0	0
$947$	25.0573	0.814254	0.407127	−	0.913372i	$-0.366531\pi$
$947$	25.0573	0.814254	0.407127	+	0.913372i	$0.366531\pi$
$948$	0	0
$949$	22.0155	0.714652
$950$	0	0
$951$	−24.0051	−0.778417
$952$	0	0
$953$	34.5250	1.11837	0.559187	−	0.829041i	$-0.311113\pi$
$953$	34.5250	1.11837	0.559187	+	0.829041i	$0.311113\pi$
$954$	0	0
$955$	−26.0690	−0.843573
$956$	0	0
$957$	8.81206	0.284853
$958$	0	0
$959$	−26.6590	−0.860863
$960$	0	0
$961$	−13.4795	−0.434824
$962$	0	0
$963$	12.4755	0.402018
$964$	0	0
$965$	2.38338	0.0767236
$966$	0	0
$967$	34.0029	1.09346	0.546730	−	0.837309i	$-0.315872\pi$
$967$	34.0029	1.09346	0.546730	+	0.837309i	$0.315872\pi$
$968$	0	0
$969$	53.8724	1.73063
$970$	0	0
$971$	25.7622	0.826747	0.413374	−	0.910561i	$-0.364350\pi$
$971$	25.7622	0.826747	0.413374	+	0.910561i	$0.364350\pi$
$972$	0	0
$973$	−17.1908	−0.551113
$974$	0	0
$975$	−2.27174	−0.0727540
$976$	0	0
$977$	−24.2646	−0.776293	−0.388146	−	0.921598i	$-0.626885\pi$
$977$	−24.2646	−0.776293	−0.388146	+	0.921598i	$0.626885\pi$
$978$	0	0
$979$	0.612680	0.0195813
$980$	0	0
$981$	1.04348	0.0333159
$982$	0	0
$983$	−45.8089	−1.46108	−0.730538	−	0.682872i	$-0.760731\pi$
$983$	−45.8089	−1.46108	−0.730538	+	0.682872i	$0.760731\pi$
$984$	0	0
$985$	7.77069	0.247595
$986$	0	0
$987$	−10.9890	−0.349785
$988$	0	0
$989$	3.89412	0.123826
$990$	0	0
$991$	−37.5874	−1.19400	−0.597001	−	0.802240i	$-0.703641\pi$
$991$	−37.5874	−1.19400	−0.597001	+	0.802240i	$0.703641\pi$
$992$	0	0
$993$	−26.8814	−0.853056
$994$	0	0
$995$	16.3002	0.516752
$996$	0	0
$997$	30.3694	0.961809	0.480905	−	0.876773i	$-0.340308\pi$
$997$	30.3694	0.961809	0.480905	+	0.876773i	$0.340308\pi$
$998$	0	0
$999$	−2.77490	−0.0877940

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	4020.2.a.h.1.2	✓	7

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
4020.2.a.h.1.2	✓	7	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)

\(f(q)\)	\(=\)	\(q-1.00000 q^{3} -1.00000 q^{5} -2.11010 q^{7} +1.00000 q^{9} +O(q^{10})\)	\(q-1.00000 q^{3} -1.00000 q^{5} -2.11010 q^{7} +1.00000 q^{9} -1.80390 q^{11} +2.27174 q^{13} +1.00000 q^{15} -7.04852 q^{17} +7.64308 q^{19} +2.11010 q^{21} -8.99511 q^{23} +1.00000 q^{25} -1.00000 q^{27} +4.88501 q^{29} -4.18575 q^{31} +1.80390 q^{33} +2.11010 q^{35} +2.77490 q^{37} -2.27174 q^{39} -2.86817 q^{41} -0.432915 q^{43} -1.00000 q^{45} -5.20782 q^{47} -2.54746 q^{49} +7.04852 q^{51} -1.87329 q^{53} +1.80390 q^{55} -7.64308 q^{57} +3.24462 q^{59} +2.12671 q^{61} -2.11010 q^{63} -2.27174 q^{65} -1.00000 q^{67} +8.99511 q^{69} +3.27572 q^{71} +9.69100 q^{73} -1.00000 q^{75} +3.80642 q^{77} +5.75049 q^{79} +1.00000 q^{81} -7.66058 q^{83} +7.04852 q^{85} -4.88501 q^{87} -0.339642 q^{89} -4.79361 q^{91} +4.18575 q^{93} -7.64308 q^{95} -4.25514 q^{97} -1.80390 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 7 q - 7 q^{3} - 7 q^{5} + 3 q^{7} + 7 q^{9}+O(q^{10}) \) 7 * q - 7 * q^3 - 7 * q^5 + 3 * q^7 + 7 * q^9	\( 7 q - 7 q^{3} - 7 q^{5} + 3 q^{7} + 7 q^{9} - 5 q^{11} + 5 q^{13} + 7 q^{15} + 3 q^{17} + 2 q^{19} - 3 q^{21} - 3 q^{23} + 7 q^{25} - 7 q^{27} - 8 q^{29} + 7 q^{31} + 5 q^{33} - 3 q^{35} - 5 q^{37} - 5 q^{39} + 7 q^{41} + 3 q^{43} - 7 q^{45} - 6 q^{47} + 16 q^{49} - 3 q^{51} - 9 q^{53} + 5 q^{55} - 2 q^{57} - 22 q^{59} + 19 q^{61} + 3 q^{63} - 5 q^{65} - 7 q^{67} + 3 q^{69} + 23 q^{73} - 7 q^{75} + 9 q^{77} + 25 q^{79} + 7 q^{81} - 20 q^{83} - 3 q^{85} + 8 q^{87} + q^{89} + 23 q^{91} - 7 q^{93} - 2 q^{95} + 3 q^{97} - 5 q^{99}+O(q^{100}) \) 7 * q - 7 * q^3 - 7 * q^5 + 3 * q^7 + 7 * q^9 - 5 * q^11 + 5 * q^13 + 7 * q^15 + 3 * q^17 + 2 * q^19 - 3 * q^21 - 3 * q^23 + 7 * q^25 - 7 * q^27 - 8 * q^29 + 7 * q^31 + 5 * q^33 - 3 * q^35 - 5 * q^37 - 5 * q^39 + 7 * q^41 + 3 * q^43 - 7 * q^45 - 6 * q^47 + 16 * q^49 - 3 * q^51 - 9 * q^53 + 5 * q^55 - 2 * q^57 - 22 * q^59 + 19 * q^61 + 3 * q^63 - 5 * q^65 - 7 * q^67 + 3 * q^69 + 23 * q^73 - 7 * q^75 + 9 * q^77 + 25 * q^79 + 7 * q^81 - 20 * q^83 - 3 * q^85 + 8 * q^87 + q^89 + 23 * q^91 - 7 * q^93 - 2 * q^95 + 3 * q^97 - 5 * q^99

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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