LMFDB - Embedded newform 4020.2.a.h.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:	$7$
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			1.3
Root			$-0.941196$ 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)

$f(q)$	$=$	$q-1.00000 q^{3} -1.00000 q^{5} -0.941196 q^{7} +1.00000 q^{9} +O(q^{10})$	$q-1.00000 q^{3} -1.00000 q^{5} -0.941196 q^{7} +1.00000 q^{9} +1.81084 q^{11} -5.81690 q^{13} +1.00000 q^{15} +2.76904 q^{17} +0.102297 q^{19} +0.941196 q^{21} +0.798415 q^{23} +1.00000 q^{25} -1.00000 q^{27} -3.73961 q^{29} +8.68655 q^{31} -1.81084 q^{33} +0.941196 q^{35} -4.68081 q^{37} +5.81690 q^{39} -2.78310 q^{41} -9.62943 q^{43} -1.00000 q^{45} -6.94863 q^{47} -6.11415 q^{49} -2.76904 q^{51} +4.87739 q^{53} -1.81084 q^{55} -0.102297 q^{57} -2.95819 q^{59} +8.87739 q^{61} -0.941196 q^{63} +5.81690 q^{65} -1.00000 q^{67} -0.798415 q^{69} +14.9311 q^{71} +11.4688 q^{73} -1.00000 q^{75} -1.70436 q^{77} +2.15978 q^{79} +1.00000 q^{81} +10.6244 q^{83} -2.76904 q^{85} +3.73961 q^{87} -2.16552 q^{89} +5.47485 q^{91} -8.68655 q^{93} -0.102297 q^{95} +11.7531 q^{97} +1.81084 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 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
$7$	−0.941196	−0.355739	−0.177869	−	0.984054i	$-0.556920\pi$
$7$	−0.941196	−0.355739	−0.177869	+	0.984054i	$0.556920\pi$
$8$	0	0
$9$	1.00000	0.333333
$10$	0	0
$11$	1.81084	0.545990	0.272995	−	0.962015i	$-0.411986\pi$
$11$	1.81084	0.545990	0.272995	+	0.962015i	$0.411986\pi$
$12$	0	0
$13$	−5.81690	−1.61332	−0.806660	−	0.591016i	$-0.798727\pi$
$13$	−5.81690	−1.61332	−0.806660	+	0.591016i	$0.798727\pi$
$14$	0	0
$15$	1.00000	0.258199
$16$	0	0
$17$	2.76904	0.671590	0.335795	−	0.941935i	$-0.390995\pi$
$17$	2.76904	0.671590	0.335795	+	0.941935i	$0.390995\pi$
$18$	0	0
$19$	0.102297	0.0234685	0.0117342	−	0.999931i	$-0.496265\pi$
$19$	0.102297	0.0234685	0.0117342	+	0.999931i	$0.496265\pi$
$20$	0	0
$21$	0.941196	0.205386
$22$	0	0
$23$	0.798415	0.166481	0.0832405	−	0.996529i	$-0.473473\pi$
$23$	0.798415	0.166481	0.0832405	+	0.996529i	$0.473473\pi$
$24$	0	0
$25$	1.00000	0.200000
$26$	0	0
$27$	−1.00000	−0.192450
$28$	0	0
$29$	−3.73961	−0.694428	−0.347214	−	0.937786i	$-0.612872\pi$
$29$	−3.73961	−0.694428	−0.347214	+	0.937786i	$0.612872\pi$
$30$	0	0
$31$	8.68655	1.56015	0.780076	−	0.625685i	$-0.215181\pi$
$31$	8.68655	1.56015	0.780076	+	0.625685i	$0.215181\pi$
$32$	0	0
$33$	−1.81084	−0.315228
$34$	0	0
$35$	0.941196	0.159091
$36$	0	0
$37$	−4.68081	−0.769520	−0.384760	−	0.923017i	$-0.625716\pi$
$37$	−4.68081	−0.769520	−0.384760	+	0.923017i	$0.625716\pi$
$38$	0	0
$39$	5.81690	0.931450
$40$	0	0
$41$	−2.78310	−0.434648	−0.217324	−	0.976100i	$-0.569733\pi$
$41$	−2.78310	−0.434648	−0.217324	+	0.976100i	$0.569733\pi$
$42$	0	0
$43$	−9.62943	−1.46847	−0.734237	−	0.678893i	$-0.762460\pi$
$43$	−9.62943	−1.46847	−0.734237	+	0.678893i	$0.762460\pi$
$44$	0	0
$45$	−1.00000	−0.149071
$46$	0	0
$47$	−6.94863	−1.01356	−0.506781	−	0.862075i	$-0.669165\pi$
$47$	−6.94863	−1.01356	−0.506781	+	0.862075i	$0.669165\pi$
$48$	0	0
$49$	−6.11415	−0.873450
$50$	0	0
$51$	−2.76904	−0.387743
$52$	0	0
$53$	4.87739	0.669961	0.334981	−	0.942225i	$-0.391270\pi$
$53$	4.87739	0.669961	0.334981	+	0.942225i	$0.391270\pi$
$54$	0	0
$55$	−1.81084	−0.244174
$56$	0	0
$57$	−0.102297	−0.0135495
$58$	0	0
$59$	−2.95819	−0.385124	−0.192562	−	0.981285i	$-0.561680\pi$
$59$	−2.95819	−0.385124	−0.192562	+	0.981285i	$0.561680\pi$
$60$	0	0
$61$	8.87739	1.13663	0.568317	−	0.822810i	$-0.307595\pi$
$61$	8.87739	1.13663	0.568317	+	0.822810i	$0.307595\pi$
$62$	0	0
$63$	−0.941196	−0.118580
$64$	0	0
$65$	5.81690	0.721498
$66$	0	0
$67$	−1.00000	−0.122169
$67$	−1.00000	−0.122169
$68$	0	0
$69$	−0.798415	−0.0961179
$70$	0	0
$71$	14.9311	1.77199	0.885995	−	0.463695i	$-0.153476\pi$
$71$	14.9311	1.77199	0.885995	+	0.463695i	$0.153476\pi$
$72$	0	0
$73$	11.4688	1.34232	0.671162	−	0.741310i	$-0.265795\pi$
$73$	11.4688	1.34232	0.671162	+	0.741310i	$0.265795\pi$
$74$	0	0
$75$	−1.00000	−0.115470
$76$	0	0
$77$	−1.70436	−0.194230
$78$	0	0
$79$	2.15978	0.242994	0.121497	−	0.992592i	$-0.461231\pi$
$79$	2.15978	0.242994	0.121497	+	0.992592i	$0.461231\pi$
$80$	0	0
$81$	1.00000	0.111111
$82$	0	0
$83$	10.6244	1.16618	0.583092	−	0.812406i	$-0.301843\pi$
$83$	10.6244	1.16618	0.583092	+	0.812406i	$0.301843\pi$
$84$	0	0
$85$	−2.76904	−0.300344
$86$	0	0
$87$	3.73961	0.400928
$88$	0	0
$89$	−2.16552	−0.229545	−0.114773	−	0.993392i	$-0.536614\pi$
$89$	−2.16552	−0.229545	−0.114773	+	0.993392i	$0.536614\pi$
$90$	0	0
$91$	5.47485	0.573920
$92$	0	0
$93$	−8.68655	−0.900754
$94$	0	0
$95$	−0.102297	−0.0104954
$96$	0	0
$97$	11.7531	1.19335	0.596673	−	0.802484i	$-0.296489\pi$
$97$	11.7531	1.19335	0.596673	+	0.802484i	$0.296489\pi$
$98$	0	0
$99$	1.81084	0.181997
$100$	0	0
$101$	5.78810	0.575938	0.287969	−	0.957640i	$-0.407020\pi$
$101$	5.78810	0.575938	0.287969	+	0.957640i	$0.407020\pi$
$102$	0	0
$103$	−7.62943	−0.751750	−0.375875	−	0.926670i	$-0.622658\pi$
$103$	−7.62943	−0.751750	−0.375875	+	0.926670i	$0.622658\pi$
$104$	0	0
$105$	−0.941196	−0.0918513
$106$	0	0
$107$	−9.09348	−0.879100	−0.439550	−	0.898218i	$-0.644862\pi$
$107$	−9.09348	−0.879100	−0.439550	+	0.898218i	$0.644862\pi$
$108$	0	0
$109$	14.8230	1.41979	0.709893	−	0.704309i	$-0.248743\pi$
$109$	14.8230	1.41979	0.709893	+	0.704309i	$0.248743\pi$
$110$	0	0
$111$	4.68081	0.444282
$112$	0	0
$113$	4.88234	0.459292	0.229646	−	0.973274i	$-0.426243\pi$
$113$	4.88234	0.459292	0.229646	+	0.973274i	$0.426243\pi$
$114$	0	0
$115$	−0.798415	−0.0744526
$116$	0	0
$117$	−5.81690	−0.537773
$118$	0	0
$119$	−2.60620	−0.238910
$120$	0	0
$121$	−7.72084	−0.701895
$122$	0	0
$123$	2.78310	0.250944
$124$	0	0
$125$	−1.00000	−0.0894427
$126$	0	0
$127$	−10.2563	−0.910100	−0.455050	−	0.890466i	$-0.650379\pi$
$127$	−10.2563	−0.910100	−0.455050	+	0.890466i	$0.650379\pi$
$128$	0	0
$129$	9.62943	0.847824
$130$	0	0
$131$	−11.7147	−1.02352	−0.511761	−	0.859128i	$-0.671006\pi$
$131$	−11.7147	−1.02352	−0.511761	+	0.859128i	$0.671006\pi$
$132$	0	0
$133$	−0.0962812	−0.00834864
$134$	0	0
$135$	1.00000	0.0860663
$136$	0	0
$137$	8.05097	0.687841	0.343920	−	0.938999i	$-0.388245\pi$
$137$	8.05097	0.687841	0.343920	+	0.938999i	$0.388245\pi$
$138$	0	0
$139$	−0.965623	−0.0819030	−0.0409515	−	0.999161i	$-0.513039\pi$
$139$	−0.965623	−0.0819030	−0.0409515	+	0.999161i	$0.513039\pi$
$140$	0	0
$141$	6.94863	0.585180
$142$	0	0
$143$	−10.5335	−0.880857
$144$	0	0
$145$	3.73961	0.310558
$146$	0	0
$147$	6.11415	0.504287
$148$	0	0
$149$	−7.85371	−0.643401	−0.321701	−	0.946841i	$-0.604254\pi$
$149$	−7.85371	−0.643401	−0.321701	+	0.946841i	$0.604254\pi$
$150$	0	0
$151$	17.4328	1.41866	0.709332	−	0.704874i	$-0.248997\pi$
$151$	17.4328	1.41866	0.709332	+	0.704874i	$0.248997\pi$
$152$	0	0
$153$	2.76904	0.223863
$154$	0	0
$155$	−8.68655	−0.697721
$156$	0	0
$157$	6.67505	0.532727	0.266364	−	0.963873i	$-0.414178\pi$
$157$	6.67505	0.532727	0.266364	+	0.963873i	$0.414178\pi$
$158$	0	0
$159$	−4.87739	−0.386802
$160$	0	0
$161$	−0.751465	−0.0592237
$162$	0	0
$163$	−0.291452	−0.0228283	−0.0114141	−	0.999935i	$-0.503633\pi$
$163$	−0.291452	−0.0228283	−0.0114141	+	0.999935i	$0.503633\pi$
$164$	0	0
$165$	1.81084	0.140974
$166$	0	0
$167$	20.2311	1.56553	0.782766	−	0.622317i	$-0.213808\pi$
$167$	20.2311	1.56553	0.782766	+	0.622317i	$0.213808\pi$
$168$	0	0
$169$	20.8364	1.60280
$170$	0	0
$171$	0.102297	0.00782282
$172$	0	0
$173$	1.23588	0.0939625	0.0469813	−	0.998896i	$-0.485040\pi$
$173$	1.23588	0.0939625	0.0469813	+	0.998896i	$0.485040\pi$
$174$	0	0
$175$	−0.941196	−0.0711477
$176$	0	0
$177$	2.95819	0.222351
$178$	0	0
$179$	−19.6872	−1.47149	−0.735744	−	0.677260i	$-0.763167\pi$
$179$	−19.6872	−1.47149	−0.735744	+	0.677260i	$0.763167\pi$
$180$	0	0
$181$	19.9778	1.48494	0.742470	−	0.669880i	$-0.233654\pi$
$181$	19.9778	1.48494	0.742470	+	0.669880i	$0.233654\pi$
$182$	0	0
$183$	−8.87739	−0.656236
$184$	0	0
$185$	4.68081	0.344140
$186$	0	0
$187$	5.01430	0.366682
$188$	0	0
$189$	0.941196	0.0684619
$190$	0	0
$191$	3.91403	0.283209	0.141605	−	0.989923i	$-0.454774\pi$
$191$	3.91403	0.283209	0.141605	+	0.989923i	$0.454774\pi$
$192$	0	0
$193$	14.5065	1.04420	0.522101	−	0.852884i	$-0.325148\pi$
$193$	14.5065	1.04420	0.522101	+	0.852884i	$0.325148\pi$
$194$	0	0
$195$	−5.81690	−0.416557
$196$	0	0
$197$	11.6832	0.832396	0.416198	−	0.909274i	$-0.363362\pi$
$197$	11.6832	0.832396	0.416198	+	0.909274i	$0.363362\pi$
$198$	0	0
$199$	19.1334	1.35633	0.678165	−	0.734910i	$-0.262775\pi$
$199$	19.1334	1.35633	0.678165	+	0.734910i	$0.262775\pi$
$200$	0	0
$201$	1.00000	0.0705346
$202$	0	0
$203$	3.51971	0.247035
$204$	0	0
$205$	2.78310	0.194380
$206$	0	0
$207$	0.798415	0.0554937
$208$	0	0
$209$	0.185243	0.0128136
$210$	0	0
$211$	10.7975	0.743332	0.371666	−	0.928366i	$-0.378787\pi$
$211$	10.7975	0.743332	0.371666	+	0.928366i	$0.378787\pi$
$212$	0	0
$213$	−14.9311	−1.02306
$214$	0	0
$215$	9.62943	0.656722
$216$	0	0
$217$	−8.17575	−0.555006
$218$	0	0
$219$	−11.4688	−0.774992
$220$	0	0
$221$	−16.1072	−1.08349
$222$	0	0
$223$	9.82164	0.657706	0.328853	−	0.944381i	$-0.393338\pi$
$223$	9.82164	0.657706	0.328853	+	0.944381i	$0.393338\pi$
$224$	0	0
$225$	1.00000	0.0666667
$226$	0	0
$227$	−27.5414	−1.82799	−0.913994	−	0.405727i	$-0.867018\pi$
$227$	−27.5414	−1.82799	−0.913994	+	0.405727i	$0.867018\pi$
$228$	0	0
$229$	23.8866	1.57847	0.789236	−	0.614089i	$-0.210477\pi$
$229$	23.8866	1.57847	0.789236	+	0.614089i	$0.210477\pi$
$230$	0	0
$231$	1.70436	0.112139
$232$	0	0
$233$	−5.48369	−0.359249	−0.179624	−	0.983735i	$-0.557488\pi$
$233$	−5.48369	−0.359249	−0.179624	+	0.983735i	$0.557488\pi$
$234$	0	0
$235$	6.94863	0.453278
$236$	0	0
$237$	−2.15978	−0.140293
$238$	0	0
$239$	27.5312	1.78085	0.890424	−	0.455132i	$-0.150408\pi$
$239$	27.5312	1.78085	0.890424	+	0.455132i	$0.150408\pi$
$240$	0	0
$241$	10.3142	0.664396	0.332198	−	0.943210i	$-0.392210\pi$
$241$	10.3142	0.664396	0.332198	+	0.943210i	$0.392210\pi$
$242$	0	0
$243$	−1.00000	−0.0641500
$244$	0	0
$245$	6.11415	0.390619
$246$	0	0
$247$	−0.595050	−0.0378621
$248$	0	0
$249$	−10.6244	−0.673296
$250$	0	0
$251$	−22.6458	−1.42939	−0.714695	−	0.699436i	$-0.753434\pi$
$251$	−22.6458	−1.42939	−0.714695	+	0.699436i	$0.753434\pi$
$252$	0	0
$253$	1.44581	0.0908970
$254$	0	0
$255$	2.76904	0.173404
$256$	0	0
$257$	−17.5121	−1.09237	−0.546187	−	0.837664i	$-0.683921\pi$
$257$	−17.5121	−1.09237	−0.546187	+	0.837664i	$0.683921\pi$
$258$	0	0
$259$	4.40555	0.273748
$260$	0	0
$261$	−3.73961	−0.231476
$262$	0	0
$263$	−5.36273	−0.330680	−0.165340	−	0.986237i	$-0.552872\pi$
$263$	−5.36273	−0.330680	−0.165340	+	0.986237i	$0.552872\pi$
$264$	0	0
$265$	−4.87739	−0.299616
$266$	0	0
$267$	2.16552	0.132528
$268$	0	0
$269$	24.0226	1.46468	0.732342	−	0.680938i	$-0.238427\pi$
$269$	24.0226	1.46468	0.732342	+	0.680938i	$0.238427\pi$
$270$	0	0
$271$	8.21233	0.498864	0.249432	−	0.968392i	$-0.419756\pi$
$271$	8.21233	0.498864	0.249432	+	0.968392i	$0.419756\pi$
$272$	0	0
$273$	−5.47485	−0.331353
$274$	0	0
$275$	1.81084	0.109198
$276$	0	0
$277$	15.9896	0.960719	0.480360	−	0.877072i	$-0.340506\pi$
$277$	15.9896	0.960719	0.480360	+	0.877072i	$0.340506\pi$
$278$	0	0
$279$	8.68655	0.520050
$280$	0	0
$281$	26.8023	1.59889	0.799447	−	0.600737i	$-0.205126\pi$
$281$	26.8023	1.59889	0.799447	+	0.600737i	$0.205126\pi$
$282$	0	0
$283$	21.7371	1.29213	0.646067	−	0.763280i	$-0.276412\pi$
$283$	21.7371	1.29213	0.646067	+	0.763280i	$0.276412\pi$
$284$	0	0
$285$	0.102297	0.00605953
$286$	0	0
$287$	2.61944	0.154621
$288$	0	0
$289$	−9.33244	−0.548967
$290$	0	0
$291$	−11.7531	−0.688979
$292$	0	0
$293$	−6.15091	−0.359340	−0.179670	−	0.983727i	$-0.557503\pi$
$293$	−6.15091	−0.359340	−0.179670	+	0.983727i	$0.557503\pi$
$294$	0	0
$295$	2.95819	0.172233
$296$	0	0
$297$	−1.81084	−0.105076
$298$	0	0
$299$	−4.64430	−0.268587
$300$	0	0
$301$	9.06318	0.522393
$302$	0	0
$303$	−5.78810	−0.332518
$304$	0	0
$305$	−8.87739	−0.508318
$306$	0	0
$307$	10.3923	0.593121	0.296560	−	0.955014i	$-0.404160\pi$
$307$	10.3923	0.593121	0.296560	+	0.955014i	$0.404160\pi$
$308$	0	0
$309$	7.62943	0.434023
$310$	0	0
$311$	6.24296	0.354006	0.177003	−	0.984210i	$-0.443360\pi$
$311$	6.24296	0.354006	0.177003	+	0.984210i	$0.443360\pi$
$312$	0	0
$313$	−21.8511	−1.23510	−0.617549	−	0.786532i	$-0.711874\pi$
$313$	−21.8511	−1.23510	−0.617549	+	0.786532i	$0.711874\pi$
$314$	0	0
$315$	0.941196	0.0530304
$316$	0	0
$317$	27.7157	1.55667	0.778334	−	0.627850i	$-0.216065\pi$
$317$	27.7157	1.55667	0.778334	+	0.627850i	$0.216065\pi$
$318$	0	0
$319$	−6.77185	−0.379151
$320$	0	0
$321$	9.09348	0.507549
$322$	0	0
$323$	0.283263	0.0157612
$324$	0	0
$325$	−5.81690	−0.322664
$326$	0	0
$327$	−14.8230	−0.819714
$328$	0	0
$329$	6.54002	0.360563
$330$	0	0
$331$	23.1435	1.27208	0.636041	−	0.771655i	$-0.280571\pi$
$331$	23.1435	1.27208	0.636041	+	0.771655i	$0.280571\pi$
$332$	0	0
$333$	−4.68081	−0.256507
$334$	0	0
$335$	1.00000	0.0546358
$336$	0	0
$337$	8.09048	0.440716	0.220358	−	0.975419i	$-0.429277\pi$
$337$	8.09048	0.440716	0.220358	+	0.975419i	$0.429277\pi$
$338$	0	0
$339$	−4.88234	−0.265173
$340$	0	0
$341$	15.7300	0.851827
$342$	0	0
$343$	12.3430	0.666458
$344$	0	0
$345$	0.798415	0.0429852
$346$	0	0
$347$	−7.52740	−0.404092	−0.202046	−	0.979376i	$-0.564759\pi$
$347$	−7.52740	−0.404092	−0.202046	+	0.979376i	$0.564759\pi$
$348$	0	0
$349$	4.55072	0.243595	0.121797	−	0.992555i	$-0.461134\pi$
$349$	4.55072	0.243595	0.121797	+	0.992555i	$0.461134\pi$
$350$	0	0
$351$	5.81690	0.310483
$352$	0	0
$353$	−10.9498	−0.582797	−0.291399	−	0.956602i	$-0.594121\pi$
$353$	−10.9498	−0.582797	−0.291399	+	0.956602i	$0.594121\pi$
$354$	0	0
$355$	−14.9311	−0.792458
$356$	0	0
$357$	2.60620	0.137935
$358$	0	0
$359$	−26.7849	−1.41365	−0.706826	−	0.707387i	$-0.749874\pi$
$359$	−26.7849	−1.41365	−0.706826	+	0.707387i	$0.749874\pi$
$360$	0	0
$361$	−18.9895	−0.999449
$362$	0	0
$363$	7.72084	0.405239
$364$	0	0
$365$	−11.4688	−0.600306
$366$	0	0
$367$	28.7202	1.49918	0.749590	−	0.661902i	$-0.230251\pi$
$367$	28.7202	1.49918	0.749590	+	0.661902i	$0.230251\pi$
$368$	0	0
$369$	−2.78310	−0.144883
$370$	0	0
$371$	−4.59058	−0.238331
$372$	0	0
$373$	0.628372	0.0325359	0.0162679	−	0.999868i	$-0.494822\pi$
$373$	0.628372	0.0325359	0.0162679	+	0.999868i	$0.494822\pi$
$374$	0	0
$375$	1.00000	0.0516398
$376$	0	0
$377$	21.7530	1.12033
$378$	0	0
$379$	−18.8584	−0.968690	−0.484345	−	0.874877i	$-0.660942\pi$
$379$	−18.8584	−0.968690	−0.484345	+	0.874877i	$0.660942\pi$
$380$	0	0
$381$	10.2563	0.525447
$382$	0	0
$383$	−3.01894	−0.154261	−0.0771304	−	0.997021i	$-0.524576\pi$
$383$	−3.01894	−0.154261	−0.0771304	+	0.997021i	$0.524576\pi$
$384$	0	0
$385$	1.70436	0.0868622
$386$	0	0
$387$	−9.62943	−0.489492
$388$	0	0
$389$	23.1003	1.17123	0.585617	−	0.810588i	$-0.300852\pi$
$389$	23.1003	1.17123	0.585617	+	0.810588i	$0.300852\pi$
$390$	0	0
$391$	2.21084	0.111807
$392$	0	0
$393$	11.7147	0.590930
$394$	0	0
$395$	−2.15978	−0.108670
$396$	0	0
$397$	28.2397	1.41731	0.708654	−	0.705556i	$-0.249303\pi$
$397$	28.2397	1.41731	0.708654	+	0.705556i	$0.249303\pi$
$398$	0	0
$399$	0.0962812	0.00482009
$400$	0	0
$401$	7.41434	0.370254	0.185127	−	0.982715i	$-0.440730\pi$
$401$	7.41434	0.370254	0.185127	+	0.982715i	$0.440730\pi$
$402$	0	0
$403$	−50.5289	−2.51702
$404$	0	0
$405$	−1.00000	−0.0496904
$406$	0	0
$407$	−8.47621	−0.420150
$408$	0	0
$409$	3.36007	0.166145	0.0830724	−	0.996544i	$-0.473527\pi$
$409$	3.36007	0.166145	0.0830724	+	0.996544i	$0.473527\pi$
$410$	0	0
$411$	−8.05097	−0.397125
$412$	0	0
$413$	2.78424	0.137003
$414$	0	0
$415$	−10.6244	−0.521533
$416$	0	0
$417$	0.965623	0.0472867
$418$	0	0
$419$	−23.3075	−1.13865	−0.569323	−	0.822114i	$-0.692795\pi$
$419$	−23.3075	−1.13865	−0.569323	+	0.822114i	$0.692795\pi$
$420$	0	0
$421$	−33.0104	−1.60883	−0.804414	−	0.594068i	$-0.797521\pi$
$421$	−33.0104	−1.60883	−0.804414	+	0.594068i	$0.797521\pi$
$422$	0	0
$423$	−6.94863	−0.337854
$424$	0	0
$425$	2.76904	0.134318
$426$	0	0
$427$	−8.35536	−0.404344
$428$	0	0
$429$	10.5335	0.508563
$430$	0	0
$431$	−4.27027	−0.205692	−0.102846	−	0.994697i	$-0.532795\pi$
$431$	−4.27027	−0.205692	−0.102846	+	0.994697i	$0.532795\pi$
$432$	0	0
$433$	7.62850	0.366602	0.183301	−	0.983057i	$-0.441322\pi$
$433$	7.62850	0.366602	0.183301	+	0.983057i	$0.441322\pi$
$434$	0	0
$435$	−3.73961	−0.179301
$436$	0	0
$437$	0.0816752	0.00390706
$438$	0	0
$439$	−31.1016	−1.48440	−0.742199	−	0.670179i	$-0.766217\pi$
$439$	−31.1016	−1.48440	−0.742199	+	0.670179i	$0.766217\pi$
$440$	0	0
$441$	−6.11415	−0.291150
$442$	0	0
$443$	−30.1325	−1.43164	−0.715818	−	0.698287i	$-0.753946\pi$
$443$	−30.1325	−1.43164	−0.715818	+	0.698287i	$0.753946\pi$
$444$	0	0
$445$	2.16552	0.102656
$446$	0	0
$447$	7.85371	0.371468
$448$	0	0
$449$	33.8214	1.59613	0.798064	−	0.602572i	$-0.205857\pi$
$449$	33.8214	1.59613	0.798064	+	0.602572i	$0.205857\pi$
$450$	0	0
$451$	−5.03977	−0.237313
$452$	0	0
$453$	−17.4328	−0.819067
$454$	0	0
$455$	−5.47485	−0.256665
$456$	0	0
$457$	−11.4106	−0.533765	−0.266883	−	0.963729i	$-0.585994\pi$
$457$	−11.4106	−0.533765	−0.266883	+	0.963729i	$0.585994\pi$
$458$	0	0
$459$	−2.76904	−0.129248
$460$	0	0
$461$	6.10092	0.284148	0.142074	−	0.989856i	$-0.454623\pi$
$461$	6.10092	0.284148	0.142074	+	0.989856i	$0.454623\pi$
$462$	0	0
$463$	−0.871037	−0.0404805	−0.0202403	−	0.999795i	$-0.506443\pi$
$463$	−0.871037	−0.0404805	−0.0202403	+	0.999795i	$0.506443\pi$
$464$	0	0
$465$	8.68655	0.402829
$466$	0	0
$467$	29.6693	1.37293	0.686465	−	0.727162i	$-0.259161\pi$
$467$	29.6693	1.37293	0.686465	+	0.727162i	$0.259161\pi$
$468$	0	0
$469$	0.941196	0.0434604
$470$	0	0
$471$	−6.67505	−0.307570
$472$	0	0
$473$	−17.4374	−0.801773
$474$	0	0
$475$	0.102297	0.00469369
$476$	0	0
$477$	4.87739	0.223320
$478$	0	0
$479$	−8.79736	−0.401961	−0.200981	−	0.979595i	$-0.564413\pi$
$479$	−8.79736	−0.401961	−0.200981	+	0.979595i	$0.564413\pi$
$480$	0	0
$481$	27.2278	1.24148
$482$	0	0
$483$	0.751465	0.0341928
$484$	0	0
$485$	−11.7531	−0.533681
$486$	0	0
$487$	−12.2361	−0.554470	−0.277235	−	0.960802i	$-0.589418\pi$
$487$	−12.2361	−0.554470	−0.277235	+	0.960802i	$0.589418\pi$
$488$	0	0
$489$	0.291452	0.0131799
$490$	0	0
$491$	−20.7736	−0.937499	−0.468749	−	0.883331i	$-0.655295\pi$
$491$	−20.7736	−0.937499	−0.468749	+	0.883331i	$0.655295\pi$
$492$	0	0
$493$	−10.3551	−0.466371
$494$	0	0
$495$	−1.81084	−0.0813914
$496$	0	0
$497$	−14.0530	−0.630365
$498$	0	0
$499$	−10.0196	−0.448539	−0.224269	−	0.974527i	$-0.572000\pi$
$499$	−10.0196	−0.448539	−0.224269	+	0.974527i	$0.572000\pi$
$500$	0	0
$501$	−20.2311	−0.903860
$502$	0	0
$503$	13.7834	0.614573	0.307287	−	0.951617i	$-0.400579\pi$
$503$	13.7834	0.614573	0.307287	+	0.951617i	$0.400579\pi$
$504$	0	0
$505$	−5.78810	−0.257567
$506$	0	0
$507$	−20.8364	−0.925376
$508$	0	0
$509$	−20.4766	−0.907610	−0.453805	−	0.891101i	$-0.649934\pi$
$509$	−20.4766	−0.907610	−0.453805	+	0.891101i	$0.649934\pi$
$510$	0	0
$511$	−10.7944	−0.477517
$512$	0	0
$513$	−0.102297	−0.00451651
$514$	0	0
$515$	7.62943	0.336193
$516$	0	0
$517$	−12.5829	−0.553395
$518$	0	0
$519$	−1.23588	−0.0542493
$520$	0	0
$521$	27.9204	1.22322	0.611608	−	0.791161i	$-0.290523\pi$
$521$	27.9204	1.22322	0.611608	+	0.791161i	$0.290523\pi$
$522$	0	0
$523$	12.9483	0.566188	0.283094	−	0.959092i	$-0.408639\pi$
$523$	12.9483	0.566188	0.283094	+	0.959092i	$0.408639\pi$
$524$	0	0
$525$	0.941196	0.0410771
$526$	0	0
$527$	24.0534	1.04778
$528$	0	0
$529$	−22.3625	−0.972284
$530$	0	0
$531$	−2.95819	−0.128375
$532$	0	0
$533$	16.1890	0.701225
$534$	0	0
$535$	9.09348	0.393145
$536$	0	0
$537$	19.6872	0.849564
$538$	0	0
$539$	−11.0718	−0.476895
$540$	0	0
$541$	30.6492	1.31771	0.658856	−	0.752269i	$-0.271040\pi$
$541$	30.6492	1.31771	0.658856	+	0.752269i	$0.271040\pi$
$542$	0	0
$543$	−19.9778	−0.857330
$544$	0	0
$545$	−14.8230	−0.634948
$546$	0	0
$547$	17.4286	0.745195	0.372597	−	0.927993i	$-0.378467\pi$
$547$	17.4286	0.745195	0.372597	+	0.927993i	$0.378467\pi$
$548$	0	0
$549$	8.87739	0.378878
$550$	0	0
$551$	−0.382550	−0.0162972
$552$	0	0
$553$	−2.03277	−0.0864423
$554$	0	0
$555$	−4.68081	−0.198689
$556$	0	0
$557$	−3.00117	−0.127164	−0.0635818	−	0.997977i	$-0.520252\pi$
$557$	−3.00117	−0.127164	−0.0635818	+	0.997977i	$0.520252\pi$
$558$	0	0
$559$	56.0135	2.36912
$560$	0	0
$561$	−5.01430	−0.211704
$562$	0	0
$563$	−1.08270	−0.0456303	−0.0228152	−	0.999740i	$-0.507263\pi$
$563$	−1.08270	−0.0456303	−0.0228152	+	0.999740i	$0.507263\pi$
$564$	0	0
$565$	−4.88234	−0.205402
$566$	0	0
$567$	−0.941196	−0.0395265
$568$	0	0
$569$	−31.0738	−1.30268	−0.651341	−	0.758785i	$-0.725793\pi$
$569$	−31.0738	−1.30268	−0.651341	+	0.758785i	$0.725793\pi$
$570$	0	0
$571$	35.6855	1.49339	0.746695	−	0.665166i	$-0.231639\pi$
$571$	35.6855	1.49339	0.746695	+	0.665166i	$0.231639\pi$
$572$	0	0
$573$	−3.91403	−0.163511
$574$	0	0
$575$	0.798415	0.0332962
$576$	0	0
$577$	4.07472	0.169633	0.0848165	−	0.996397i	$-0.472970\pi$
$577$	4.07472	0.169633	0.0848165	+	0.996397i	$0.472970\pi$
$578$	0	0
$579$	−14.5065	−0.602870
$580$	0	0
$581$	−9.99967	−0.414856
$582$	0	0
$583$	8.83220	0.365792
$584$	0	0
$585$	5.81690	0.240499
$586$	0	0
$587$	2.57045	0.106094	0.0530470	−	0.998592i	$-0.483107\pi$
$587$	2.57045	0.106094	0.0530470	+	0.998592i	$0.483107\pi$
$588$	0	0
$589$	0.888606	0.0366144
$590$	0	0
$591$	−11.6832	−0.480584
$592$	0	0
$593$	2.56059	0.105151	0.0525755	−	0.998617i	$-0.483257\pi$
$593$	2.56059	0.105151	0.0525755	+	0.998617i	$0.483257\pi$
$594$	0	0
$595$	2.60620	0.106844
$596$	0	0
$597$	−19.1334	−0.783077
$598$	0	0
$599$	−46.8956	−1.91610	−0.958051	−	0.286597i	$-0.907476\pi$
$599$	−46.8956	−1.91610	−0.958051	+	0.286597i	$0.907476\pi$
$600$	0	0
$601$	−26.5245	−1.08196	−0.540979	−	0.841036i	$-0.681946\pi$
$601$	−26.5245	−1.08196	−0.540979	+	0.841036i	$0.681946\pi$
$602$	0	0
$603$	−1.00000	−0.0407231
$604$	0	0
$605$	7.72084	0.313897
$606$	0	0
$607$	−26.9319	−1.09313	−0.546565	−	0.837416i	$-0.684065\pi$
$607$	−26.9319	−1.09313	−0.546565	+	0.837416i	$0.684065\pi$
$608$	0	0
$609$	−3.51971	−0.142626
$610$	0	0
$611$	40.4195	1.63520
$612$	0	0
$613$	10.9701	0.443079	0.221540	−	0.975151i	$-0.428892\pi$
$613$	10.9701	0.443079	0.221540	+	0.975151i	$0.428892\pi$
$614$	0	0
$615$	−2.78310	−0.112226
$616$	0	0
$617$	−47.5664	−1.91495	−0.957476	−	0.288513i	$-0.906839\pi$
$617$	−47.5664	−1.91495	−0.957476	+	0.288513i	$0.906839\pi$
$618$	0	0
$619$	20.5462	0.825821	0.412910	−	0.910772i	$-0.364512\pi$
$619$	20.5462	0.825821	0.412910	+	0.910772i	$0.364512\pi$
$620$	0	0
$621$	−0.798415	−0.0320393
$622$	0	0
$623$	2.03818	0.0816580
$624$	0	0
$625$	1.00000	0.0400000
$626$	0	0
$627$	−0.185243	−0.00739791
$628$	0	0
$629$	−12.9613	−0.516802
$630$	0	0
$631$	14.1800	0.564497	0.282249	−	0.959341i	$-0.408920\pi$
$631$	14.1800	0.564497	0.282249	+	0.959341i	$0.408920\pi$
$632$	0	0
$633$	−10.7975	−0.429163
$634$	0	0
$635$	10.2563	0.407009
$636$	0	0
$637$	35.5654	1.40915
$638$	0	0
$639$	14.9311	0.590663
$640$	0	0
$641$	41.4996	1.63914	0.819568	−	0.572981i	$-0.194213\pi$
$641$	41.4996	1.63914	0.819568	+	0.572981i	$0.194213\pi$
$642$	0	0
$643$	−5.45090	−0.214962	−0.107481	−	0.994207i	$-0.534279\pi$
$643$	−5.45090	−0.214962	−0.107481	+	0.994207i	$0.534279\pi$
$644$	0	0
$645$	−9.62943	−0.379159
$646$	0	0
$647$	−20.1288	−0.791346	−0.395673	−	0.918391i	$-0.629489\pi$
$647$	−20.1288	−0.791346	−0.395673	+	0.918391i	$0.629489\pi$
$648$	0	0
$649$	−5.35683	−0.210274
$650$	0	0
$651$	8.17575	0.320433
$652$	0	0
$653$	−4.27472	−0.167283	−0.0836413	−	0.996496i	$-0.526655\pi$
$653$	−4.27472	−0.167283	−0.0836413	+	0.996496i	$0.526655\pi$
$654$	0	0
$655$	11.7147	0.457733
$656$	0	0
$657$	11.4688	0.447442
$658$	0	0
$659$	1.18878	0.0463084	0.0231542	−	0.999732i	$-0.492629\pi$
$659$	1.18878	0.0463084	0.0231542	+	0.999732i	$0.492629\pi$
$660$	0	0
$661$	3.51736	0.136810	0.0684048	−	0.997658i	$-0.478209\pi$
$661$	3.51736	0.136810	0.0684048	+	0.997658i	$0.478209\pi$
$662$	0	0
$663$	16.1072	0.625553
$664$	0	0
$665$	0.0962812	0.00373363
$666$	0	0
$667$	−2.98576	−0.115609
$668$	0	0
$669$	−9.82164	−0.379727
$670$	0	0
$671$	16.0756	0.620591
$672$	0	0
$673$	16.1888	0.624032	0.312016	−	0.950077i	$-0.398996\pi$
$673$	16.1888	0.624032	0.312016	+	0.950077i	$0.398996\pi$
$674$	0	0
$675$	−1.00000	−0.0384900
$676$	0	0
$677$	−2.40088	−0.0922734	−0.0461367	−	0.998935i	$-0.514691\pi$
$677$	−2.40088	−0.0922734	−0.0461367	+	0.998935i	$0.514691\pi$
$678$	0	0
$679$	−11.0620	−0.424519
$680$	0	0
$681$	27.5414	1.05539
$682$	0	0
$683$	−31.3787	−1.20067	−0.600335	−	0.799748i	$-0.704966\pi$
$683$	−31.3787	−1.20067	−0.600335	+	0.799748i	$0.704966\pi$
$684$	0	0
$685$	−8.05097	−0.307612
$686$	0	0
$687$	−23.8866	−0.911332
$688$	0	0
$689$	−28.3713	−1.08086
$690$	0	0
$691$	−40.0964	−1.52534	−0.762670	−	0.646788i	$-0.776112\pi$
$691$	−40.0964	−1.52534	−0.762670	+	0.646788i	$0.776112\pi$
$692$	0	0
$693$	−1.70436	−0.0647433
$694$	0	0
$695$	0.965623	0.0366282
$696$	0	0
$697$	−7.70651	−0.291905
$698$	0	0
$699$	5.48369	0.207412
$700$	0	0
$701$	3.93126	0.148482	0.0742408	−	0.997240i	$-0.476347\pi$
$701$	3.93126	0.148482	0.0742408	+	0.997240i	$0.476347\pi$
$702$	0	0
$703$	−0.478831	−0.0180595
$704$	0	0
$705$	−6.94863	−0.261700
$706$	0	0
$707$	−5.44774	−0.204883
$708$	0	0
$709$	−17.3278	−0.650761	−0.325380	−	0.945583i	$-0.605492\pi$
$709$	−17.3278	−0.650761	−0.325380	+	0.945583i	$0.605492\pi$
$710$	0	0
$711$	2.15978	0.0809979
$712$	0	0
$713$	6.93548	0.259736
$714$	0	0
$715$	10.5335	0.393931
$716$	0	0
$717$	−27.5312	−1.02817
$718$	0	0
$719$	−9.63188	−0.359209	−0.179604	−	0.983739i	$-0.557482\pi$
$719$	−9.63188	−0.359209	−0.179604	+	0.983739i	$0.557482\pi$
$720$	0	0
$721$	7.18079	0.267427
$722$	0	0
$723$	−10.3142	−0.383589
$724$	0	0
$725$	−3.73961	−0.138886
$726$	0	0
$727$	28.2190	1.04658	0.523292	−	0.852154i	$-0.324704\pi$
$727$	28.2190	1.04658	0.523292	+	0.852154i	$0.324704\pi$
$728$	0	0
$729$	1.00000	0.0370370
$730$	0	0
$731$	−26.6642	−0.986213
$732$	0	0
$733$	−36.5015	−1.34821	−0.674106	−	0.738635i	$-0.735471\pi$
$733$	−36.5015	−1.34821	−0.674106	+	0.738635i	$0.735471\pi$
$734$	0	0
$735$	−6.11415	−0.225524
$736$	0	0
$737$	−1.81084	−0.0667033
$738$	0	0
$739$	33.8237	1.24423	0.622113	−	0.782927i	$-0.286274\pi$
$739$	33.8237	1.24423	0.622113	+	0.782927i	$0.286274\pi$
$740$	0	0
$741$	0.595050	0.0218597
$742$	0	0
$743$	−19.6714	−0.721672	−0.360836	−	0.932629i	$-0.617509\pi$
$743$	−19.6714	−0.721672	−0.360836	+	0.932629i	$0.617509\pi$
$744$	0	0
$745$	7.85371	0.287738
$746$	0	0
$747$	10.6244	0.388728
$748$	0	0
$749$	8.55874	0.312730
$750$	0	0
$751$	−2.46340	−0.0898907	−0.0449453	−	0.998989i	$-0.514311\pi$
$751$	−2.46340	−0.0898907	−0.0449453	+	0.998989i	$0.514311\pi$
$752$	0	0
$753$	22.6458	0.825258
$754$	0	0
$755$	−17.4328	−0.634446
$756$	0	0
$757$	−31.8033	−1.15591	−0.577956	−	0.816068i	$-0.696149\pi$
$757$	−31.8033	−1.15591	−0.577956	+	0.816068i	$0.696149\pi$
$758$	0	0
$759$	−1.44581	−0.0524794
$760$	0	0
$761$	7.63140	0.276638	0.138319	−	0.990388i	$-0.455830\pi$
$761$	7.63140	0.276638	0.138319	+	0.990388i	$0.455830\pi$
$762$	0	0
$763$	−13.9514	−0.505073
$764$	0	0
$765$	−2.76904	−0.100115
$766$	0	0
$767$	17.2075	0.621327
$768$	0	0
$769$	−9.27307	−0.334396	−0.167198	−	0.985923i	$-0.553472\pi$
$769$	−9.27307	−0.334396	−0.167198	+	0.985923i	$0.553472\pi$
$770$	0	0
$771$	17.5121	0.630682
$772$	0	0
$773$	46.4127	1.66935	0.834675	−	0.550743i	$-0.185656\pi$
$773$	46.4127	1.66935	0.834675	+	0.550743i	$0.185656\pi$
$774$	0	0
$775$	8.68655	0.312030
$776$	0	0
$777$	−4.40555	−0.158048
$778$	0	0
$779$	−0.284702	−0.0102005
$780$	0	0
$781$	27.0378	0.967489
$782$	0	0
$783$	3.73961	0.133643
$784$	0	0
$785$	−6.67505	−0.238243
$786$	0	0
$787$	42.2700	1.50676	0.753382	−	0.657583i	$-0.228421\pi$
$787$	42.2700	1.50676	0.753382	+	0.657583i	$0.228421\pi$
$788$	0	0
$789$	5.36273	0.190918
$790$	0	0
$791$	−4.59524	−0.163388
$792$	0	0
$793$	−51.6389	−1.83375
$794$	0	0
$795$	4.87739	0.172983
$796$	0	0
$797$	−51.8120	−1.83528	−0.917638	−	0.397416i	$-0.869907\pi$
$797$	−51.8120	−1.83528	−0.917638	+	0.397416i	$0.869907\pi$
$798$	0	0
$799$	−19.2410	−0.680698
$800$	0	0
$801$	−2.16552	−0.0765150
$802$	0	0
$803$	20.7683	0.732896
$804$	0	0
$805$	0.751465	0.0264857
$806$	0	0
$807$	−24.0226	−0.845635
$808$	0	0
$809$	29.9758	1.05389	0.526947	−	0.849898i	$-0.323337\pi$
$809$	29.9758	1.05389	0.526947	+	0.849898i	$0.323337\pi$
$810$	0	0
$811$	15.9578	0.560354	0.280177	−	0.959948i	$-0.409607\pi$
$811$	15.9578	0.560354	0.280177	+	0.959948i	$0.409607\pi$
$812$	0	0
$813$	−8.21233	−0.288019
$814$	0	0
$815$	0.291452	0.0102091
$816$	0	0
$817$	−0.985059	−0.0344629
$818$	0	0
$819$	5.47485	0.191307
$820$	0	0
$821$	−19.7625	−0.689715	−0.344858	−	0.938655i	$-0.612073\pi$
$821$	−19.7625	−0.689715	−0.344858	+	0.938655i	$0.612073\pi$
$822$	0	0
$823$	−51.9048	−1.80929	−0.904645	−	0.426167i	$-0.859864\pi$
$823$	−51.9048	−1.80929	−0.904645	+	0.426167i	$0.859864\pi$
$824$	0	0
$825$	−1.81084	−0.0630455
$826$	0	0
$827$	−16.4388	−0.571632	−0.285816	−	0.958284i	$-0.592265\pi$
$827$	−16.4388	−0.571632	−0.285816	+	0.958284i	$0.592265\pi$
$828$	0	0
$829$	30.6938	1.06604	0.533020	−	0.846103i	$-0.321057\pi$
$829$	30.6938	1.06604	0.533020	+	0.846103i	$0.321057\pi$
$830$	0	0
$831$	−15.9896	−0.554671
$832$	0	0
$833$	−16.9303	−0.586600
$834$	0	0
$835$	−20.2311	−0.700127
$836$	0	0
$837$	−8.68655	−0.300251
$838$	0	0
$839$	−43.6387	−1.50658	−0.753288	−	0.657691i	$-0.771533\pi$
$839$	−43.6387	−1.50658	−0.753288	+	0.657691i	$0.771533\pi$
$840$	0	0
$841$	−15.0153	−0.517769
$842$	0	0
$843$	−26.8023	−0.923122
$844$	0	0
$845$	−20.8364	−0.716793
$846$	0	0
$847$	7.26682	0.249691
$848$	0	0
$849$	−21.7371	−0.746014
$850$	0	0
$851$	−3.73723	−0.128110
$852$	0	0
$853$	−10.0127	−0.342830	−0.171415	−	0.985199i	$-0.554834\pi$
$853$	−10.0127	−0.342830	−0.171415	+	0.985199i	$0.554834\pi$
$854$	0	0
$855$	−0.102297	−0.00349847
$856$	0	0
$857$	10.4875	0.358246	0.179123	−	0.983827i	$-0.442674\pi$
$857$	10.4875	0.358246	0.179123	+	0.983827i	$0.442674\pi$
$858$	0	0
$859$	43.5539	1.48604	0.743020	−	0.669269i	$-0.233393\pi$
$859$	43.5539	1.48604	0.743020	+	0.669269i	$0.233393\pi$
$860$	0	0
$861$	−2.61944	−0.0892704
$862$	0	0
$863$	−47.5596	−1.61895	−0.809474	−	0.587156i	$-0.800247\pi$
$863$	−47.5596	−1.61895	−0.809474	+	0.587156i	$0.800247\pi$
$864$	0	0
$865$	−1.23588	−0.0420213
$866$	0	0
$867$	9.33244	0.316946
$868$	0	0
$869$	3.91102	0.132672
$870$	0	0
$871$	5.81690	0.197098
$872$	0	0
$873$	11.7531	0.397782
$874$	0	0
$875$	0.941196	0.0318182
$876$	0	0
$877$	−50.0010	−1.68841	−0.844207	−	0.536018i	$-0.819928\pi$
$877$	−50.0010	−1.68841	−0.844207	+	0.536018i	$0.819928\pi$
$878$	0	0
$879$	6.15091	0.207465
$880$	0	0
$881$	56.2751	1.89596	0.947978	−	0.318337i	$-0.103124\pi$
$881$	56.2751	1.89596	0.947978	+	0.318337i	$0.103124\pi$
$882$	0	0
$883$	−42.6409	−1.43498	−0.717489	−	0.696569i	$-0.754709\pi$
$883$	−42.6409	−1.43498	−0.717489	+	0.696569i	$0.754709\pi$
$884$	0	0
$885$	−2.95819	−0.0994385
$886$	0	0
$887$	11.7364	0.394070	0.197035	−	0.980396i	$-0.436869\pi$
$887$	11.7364	0.394070	0.197035	+	0.980396i	$0.436869\pi$
$888$	0	0
$889$	9.65319	0.323758
$890$	0	0
$891$	1.81084	0.0606656
$892$	0	0
$893$	−0.710822	−0.0237867
$894$	0	0
$895$	19.6872	0.658069
$896$	0	0
$897$	4.64430	0.155069
$898$	0	0
$899$	−32.4843	−1.08341
$900$	0	0
$901$	13.5057	0.449939
$902$	0	0
$903$	−9.06318	−0.301604
$904$	0	0
$905$	−19.9778	−0.664085
$906$	0	0
$907$	−57.4869	−1.90882	−0.954410	−	0.298498i	$-0.903514\pi$
$907$	−57.4869	−1.90882	−0.954410	+	0.298498i	$0.903514\pi$
$908$	0	0
$909$	5.78810	0.191979
$910$	0	0
$911$	0.257817	0.00854185	0.00427093	−	0.999991i	$-0.498641\pi$
$911$	0.257817	0.00854185	0.00427093	+	0.999991i	$0.498641\pi$
$912$	0	0
$913$	19.2392	0.636725
$914$	0	0
$915$	8.87739	0.293478
$916$	0	0
$917$	11.0259	0.364106
$918$	0	0
$919$	15.7821	0.520604	0.260302	−	0.965527i	$-0.416178\pi$
$919$	15.7821	0.520604	0.260302	+	0.965527i	$0.416178\pi$
$920$	0	0
$921$	−10.3923	−0.342438
$922$	0	0
$923$	−86.8525	−2.85879
$924$	0	0
$925$	−4.68081	−0.153904
$926$	0	0
$927$	−7.62943	−0.250583
$928$	0	0
$929$	−17.1340	−0.562149	−0.281075	−	0.959686i	$-0.590691\pi$
$929$	−17.1340	−0.562149	−0.281075	+	0.959686i	$0.590691\pi$
$930$	0	0
$931$	−0.625457	−0.0204985
$932$	0	0
$933$	−6.24296	−0.204385
$934$	0	0
$935$	−5.01430	−0.163985
$936$	0	0
$937$	16.2849	0.532005	0.266003	−	0.963972i	$-0.414297\pi$
$937$	16.2849	0.532005	0.266003	+	0.963972i	$0.414297\pi$
$938$	0	0
$939$	21.8511	0.713085
$940$	0	0
$941$	−10.5195	−0.342926	−0.171463	−	0.985191i	$-0.554849\pi$
$941$	−10.5195	−0.342926	−0.171463	+	0.985191i	$0.554849\pi$
$942$	0	0
$943$	−2.22207	−0.0723606
$944$	0	0
$945$	−0.941196	−0.0306171
$946$	0	0
$947$	2.33805	0.0759763	0.0379881	−	0.999278i	$-0.487905\pi$
$947$	2.33805	0.0759763	0.0379881	+	0.999278i	$0.487905\pi$
$948$	0	0
$949$	−66.7131	−2.16560
$950$	0	0
$951$	−27.7157	−0.898743
$952$	0	0
$953$	−4.25384	−0.137795	−0.0688976	−	0.997624i	$-0.521948\pi$
$953$	−4.25384	−0.137795	−0.0688976	+	0.997624i	$0.521948\pi$
$954$	0	0
$955$	−3.91403	−0.126655
$956$	0	0
$957$	6.77185	0.218903
$958$	0	0
$959$	−7.57754	−0.244691
$960$	0	0
$961$	44.4562	1.43407
$962$	0	0
$963$	−9.09348	−0.293033
$964$	0	0
$965$	−14.5065	−0.466981
$966$	0	0
$967$	11.4248	0.367397	0.183699	−	0.982983i	$-0.441193\pi$
$967$	11.4248	0.367397	0.183699	+	0.982983i	$0.441193\pi$
$968$	0	0
$969$	−0.283263	−0.00909973
$970$	0	0
$971$	29.2355	0.938213	0.469107	−	0.883142i	$-0.344576\pi$
$971$	29.2355	0.938213	0.469107	+	0.883142i	$0.344576\pi$
$972$	0	0
$973$	0.908840	0.0291361
$974$	0	0
$975$	5.81690	0.186290
$976$	0	0
$977$	24.2169	0.774766	0.387383	−	0.921919i	$-0.373379\pi$
$977$	24.2169	0.774766	0.387383	+	0.921919i	$0.373379\pi$
$978$	0	0
$979$	−3.92143	−0.125329
$980$	0	0
$981$	14.8230	0.473262
$982$	0	0
$983$	−4.22378	−0.134718	−0.0673588	−	0.997729i	$-0.521457\pi$
$983$	−4.22378	−0.134718	−0.0673588	+	0.997729i	$0.521457\pi$
$984$	0	0
$985$	−11.6832	−0.372259
$986$	0	0
$987$	−6.54002	−0.208171
$988$	0	0
$989$	−7.68828	−0.244473
$990$	0	0
$991$	33.9915	1.07978	0.539888	−	0.841737i	$-0.318467\pi$
$991$	33.9915	1.07978	0.539888	+	0.841737i	$0.318467\pi$
$992$	0	0
$993$	−23.1435	−0.734437
$994$	0	0
$995$	−19.1334	−0.606569
$996$	0	0
$997$	33.8867	1.07320	0.536602	−	0.843836i	$-0.319708\pi$
$997$	33.8867	1.07320	0.536602	+	0.843836i	$0.319708\pi$
$998$	0	0
$999$	4.68081	0.148094

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	4020.2.a.h.1.3	✓	7

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
4020.2.a.h.1.3	✓	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)

\(f(q)\)	\(=\)	\(q-1.00000 q^{3} -1.00000 q^{5} -0.941196 q^{7} +1.00000 q^{9} +O(q^{10})\)	\(q-1.00000 q^{3} -1.00000 q^{5} -0.941196 q^{7} +1.00000 q^{9} +1.81084 q^{11} -5.81690 q^{13} +1.00000 q^{15} +2.76904 q^{17} +0.102297 q^{19} +0.941196 q^{21} +0.798415 q^{23} +1.00000 q^{25} -1.00000 q^{27} -3.73961 q^{29} +8.68655 q^{31} -1.81084 q^{33} +0.941196 q^{35} -4.68081 q^{37} +5.81690 q^{39} -2.78310 q^{41} -9.62943 q^{43} -1.00000 q^{45} -6.94863 q^{47} -6.11415 q^{49} -2.76904 q^{51} +4.87739 q^{53} -1.81084 q^{55} -0.102297 q^{57} -2.95819 q^{59} +8.87739 q^{61} -0.941196 q^{63} +5.81690 q^{65} -1.00000 q^{67} -0.798415 q^{69} +14.9311 q^{71} +11.4688 q^{73} -1.00000 q^{75} -1.70436 q^{77} +2.15978 q^{79} +1.00000 q^{81} +10.6244 q^{83} -2.76904 q^{85} +3.73961 q^{87} -2.16552 q^{89} +5.47485 q^{91} -8.68655 q^{93} -0.102297 q^{95} +11.7531 q^{97} +1.81084 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 7 q - 7 q^{3} - 7 q^{5} + 3 q^{7} + 7 q^{9}+O(q^{10}) \) 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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