LMFDB - Embedded newform 4020.2.g.b.1609.12

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [4020,2,Mod(1609,4020)]

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, 1, 0]))

N = Newforms(chi, 2, names="a")

//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code

chi := DirichletCharacter("4020.1609");

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.g (of order $2$, degree $1$, minimal)

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:	no
Analytic conductor:	$32.0998616126$
Analytic rank:	$0$
Dimension:	$24$
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label			1609.12
Character	$\chi$	$=$	4020.1609
Dual form			4020.2.g.b.1609.24

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-1.00000i q^{3} +(2.23014 - 0.162710i) q^{5} -0.770377i q^{7} -1.00000 q^{9} +O(q^{10})$	$q-1.00000i q^{3} +(2.23014 - 0.162710i) q^{5} -0.770377i q^{7} -1.00000 q^{9} -0.920084 q^{11} -3.41528i q^{13} +(-0.162710 - 2.23014i) q^{15} -2.39068i q^{17} -5.29643 q^{19} -0.770377 q^{21} -3.81242i q^{23} +(4.94705 - 0.725733i) q^{25} +1.00000i q^{27} -3.65375 q^{29} -10.5963 q^{31} +0.920084i q^{33} +(-0.125348 - 1.71805i) q^{35} -0.493290i q^{37} -3.41528 q^{39} -6.61351 q^{41} +11.1833i q^{43} +(-2.23014 + 0.162710i) q^{45} -4.33651i q^{47} +6.40652 q^{49} -2.39068 q^{51} +10.0784i q^{53} +(-2.05192 + 0.149707i) q^{55} +5.29643i q^{57} -2.20987 q^{59} +1.58317 q^{61} +0.770377i q^{63} +(-0.555701 - 7.61655i) q^{65} +1.00000i q^{67} -3.81242 q^{69} -3.39043 q^{71} -11.1127i q^{73} +(-0.725733 - 4.94705i) q^{75} +0.708811i q^{77} +0.553989 q^{79} +1.00000 q^{81} -7.19859i q^{83} +(-0.388987 - 5.33155i) q^{85} +3.65375i q^{87} -2.49567 q^{89} -2.63105 q^{91} +10.5963i q^{93} +(-11.8118 + 0.861783i) q^{95} -9.99897i q^{97} +0.920084 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 24 q - 4 q^{5} - 24 q^{9}+O(q^{10}) $ 24 * q - 4 * q^5 - 24 * q^9	$ 24 q - 4 q^{5} - 24 q^{9} - 8 q^{11} - 2 q^{15} + 16 q^{19} - 20 q^{21} + 10 q^{25} + 36 q^{29} - 2 q^{35} + 4 q^{39} - 24 q^{41} + 4 q^{45} - 4 q^{51} - 4 q^{55} + 24 q^{59} - 4 q^{61} - 20 q^{65} - 4 q^{69} + 20 q^{71} - 12 q^{75} - 28 q^{79} + 24 q^{81} - 16 q^{85} + 48 q^{89} - 20 q^{91} - 4 q^{95} + 8 q^{99}+O(q^{100}) $ 24 * q - 4 * q^5 - 24 * q^9 - 8 * q^11 - 2 * q^15 + 16 * q^19 - 20 * q^21 + 10 * q^25 + 36 * q^29 - 2 * q^35 + 4 * q^39 - 24 * q^41 + 4 * q^45 - 4 * q^51 - 4 * q^55 + 24 * q^59 - 4 * q^61 - 20 * q^65 - 4 * q^69 + 20 * q^71 - 12 * q^75 - 28 * q^79 + 24 * q^81 - 16 * q^85 + 48 * q^89 - 20 * q^91 - 4 * q^95 + 8 * q^99

Display 100 coefficients 10 coefficients

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/4020\mathbb{Z}\right)^\times$.

$n$	$1141$	$2011$	$2681$	$3217$
$\chi(n)$	$1$	$1$	$1$	$-1$

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
$3$		−	1.00000i		−	0.577350i
$3$		−	1.00000i		−	0.577350i
$4$		0			0
$5$	2.23014	−	0.162710i	0.997349	−	0.0727662i
$5$	2.23014	−	0.162710i	0.997349	−	0.0727662i
$6$		0			0
$7$		−	0.770377i		−	0.291175i	−0.989345	−	0.145588i	$-0.953493\pi$
$7$		−	0.770377i		−	0.291175i	0.989345	−	0.145588i	$-0.0465072\pi$
$8$		0			0
$9$	−1.00000			−0.333333
$10$		0			0
$11$	−0.920084			−0.277416			−0.138708	−	0.990333i	$-0.544295\pi$
$11$	−0.920084			−0.277416			−0.138708	+	0.990333i	$0.544295\pi$
$12$		0			0
$13$		−	3.41528i		−	0.947228i	−0.880732	−	0.473614i	$-0.842949\pi$
$13$		−	3.41528i		−	0.947228i	0.880732	−	0.473614i	$-0.157051\pi$
$14$		0			0
$15$	−0.162710	−	2.23014i	−0.0420116	−	0.575820i
$16$		0			0
$17$		−	2.39068i		−	0.579824i	−0.957053	−	0.289912i	$-0.906374\pi$
$17$		−	2.39068i		−	0.579824i	0.957053	−	0.289912i	$-0.0936261\pi$
$18$		0			0
$19$	−5.29643			−1.21509			−0.607543	−	0.794287i	$-0.707845\pi$
$19$	−5.29643			−1.21509			−0.607543	+	0.794287i	$0.707845\pi$
$20$		0			0
$21$	−0.770377			−0.168110
$22$		0			0
$23$		−	3.81242i		−	0.794945i	−0.917614	−	0.397473i	$-0.869887\pi$
$23$		−	3.81242i		−	0.794945i	0.917614	−	0.397473i	$-0.130113\pi$
$24$		0			0
$25$	4.94705	−	0.725733i	0.989410	−	0.145147i
$26$		0			0
$27$			1.00000i			0.192450i
$28$		0			0
$29$	−3.65375			−0.678485			−0.339243	−	0.940699i	$-0.610171\pi$
$29$	−3.65375			−0.678485			−0.339243	+	0.940699i	$0.610171\pi$
$30$		0			0
$31$	−10.5963			−1.90314			−0.951572	−	0.307427i	$-0.900532\pi$
$31$	−10.5963			−1.90314			−0.951572	+	0.307427i	$0.900532\pi$
$32$		0			0
$33$			0.920084i			0.160166i
$34$		0			0
$35$	−0.125348	−	1.71805i	−0.0211877	−	0.290403i
$36$		0			0
$37$		−	0.493290i		−	0.0810964i	−0.999178	−	0.0405482i	$-0.987090\pi$
$37$		−	0.493290i		−	0.0810964i	0.999178	−	0.0405482i	$-0.0129104\pi$
$38$		0			0
$39$	−3.41528			−0.546882
$40$		0			0
$41$	−6.61351			−1.03286			−0.516429	−	0.856330i	$-0.672739\pi$
$41$	−6.61351			−1.03286			−0.516429	+	0.856330i	$0.672739\pi$
$42$		0			0
$43$			11.1833i			1.70544i	0.522368	+	0.852720i	$0.325049\pi$
$43$			11.1833i			1.70544i	−0.522368	+	0.852720i	$0.674951\pi$
$44$		0			0
$45$	−2.23014	+	0.162710i	−0.332450	+	0.0242554i
$46$		0			0
$47$		−	4.33651i		−	0.632546i	−0.948668	−	0.316273i	$-0.897569\pi$
$47$		−	4.33651i		−	0.632546i	0.948668	−	0.316273i	$-0.102431\pi$
$48$		0			0
$49$	6.40652			0.915217
$50$		0			0
$51$	−2.39068			−0.334762
$52$		0			0
$53$			10.0784i			1.38438i	0.721715	+	0.692190i	$0.243354\pi$
$53$			10.0784i			1.38438i	−0.721715	+	0.692190i	$0.756646\pi$
$54$		0			0
$55$	−2.05192	+	0.149707i	−0.276680	+	0.0201865i
$56$		0			0
$57$			5.29643i			0.701530i
$58$		0			0
$59$	−2.20987			−0.287701			−0.143851	−	0.989599i	$-0.545948\pi$
$59$	−2.20987			−0.287701			−0.143851	+	0.989599i	$0.545948\pi$
$60$		0			0
$61$	1.58317			0.202705			0.101352	−	0.994851i	$-0.467683\pi$
$61$	1.58317			0.202705			0.101352	+	0.994851i	$0.467683\pi$
$62$		0			0
$63$			0.770377i			0.0970583i
$64$		0			0
$65$	−0.555701	−	7.61655i	−0.0689262	−	0.944717i
$66$		0			0
$67$			1.00000i			0.122169i
$67$			1.00000i			0.122169i
$68$		0			0
$69$	−3.81242			−0.458962
$70$		0			0
$71$	−3.39043			−0.402371			−0.201185	−	0.979553i	$-0.564479\pi$
$71$	−3.39043			−0.402371			−0.201185	+	0.979553i	$0.564479\pi$
$72$		0			0
$73$		−	11.1127i		−	1.30065i	−0.759658	−	0.650323i	$-0.774634\pi$
$73$		−	11.1127i		−	1.30065i	0.759658	−	0.650323i	$-0.225366\pi$
$74$		0			0
$75$	−0.725733	−	4.94705i	−0.0838004	−	0.571236i
$76$		0			0
$77$			0.708811i			0.0807765i
$78$		0			0
$79$	0.553989			0.0623286			0.0311643	−	0.999514i	$-0.490078\pi$
$79$	0.553989			0.0623286			0.0311643	+	0.999514i	$0.490078\pi$
$80$		0			0
$81$	1.00000			0.111111
$82$		0			0
$83$		−	7.19859i		−	0.790148i	−0.918649	−	0.395074i	$-0.870719\pi$
$83$		−	7.19859i		−	0.790148i	0.918649	−	0.395074i	$-0.129281\pi$
$84$		0			0
$85$	−0.388987	−	5.33155i	−0.0421916	−	0.578287i
$86$		0			0
$87$			3.65375i			0.391724i
$88$		0			0
$89$	−2.49567			−0.264541			−0.132270	−	0.991214i	$-0.542227\pi$
$89$	−2.49567			−0.264541			−0.132270	+	0.991214i	$0.542227\pi$
$90$		0			0
$91$	−2.63105			−0.275809
$92$		0			0
$93$			10.5963i			1.09878i
$94$		0			0
$95$	−11.8118	+	0.861783i	−1.21186	+	0.0884171i
$96$		0			0
$97$		−	9.99897i		−	1.01524i	−0.861581	−	0.507621i	$-0.830525\pi$
$97$		−	9.99897i		−	1.01524i	0.861581	−	0.507621i	$-0.169475\pi$
$98$		0			0
$99$	0.920084			0.0924719
$100$		0			0
$101$	−11.5549			−1.14976			−0.574880	−	0.818238i	$-0.694951\pi$
$101$	−11.5549			−1.14976			−0.574880	+	0.818238i	$0.694951\pi$
$102$		0			0
$103$			4.43399i			0.436894i	0.975849	+	0.218447i	$0.0700990\pi$
$103$			4.43399i			0.436894i	−0.975849	+	0.218447i	$0.929901\pi$
$104$		0			0
$105$	−1.71805	+	0.125348i	−0.167664	+	0.0122327i
$106$		0			0
$107$		−	1.01113i		−	0.0977497i	−0.998805	−	0.0488748i	$-0.984436\pi$
$107$		−	1.01113i		−	0.0977497i	0.998805	−	0.0488748i	$-0.0155635\pi$
$108$		0			0
$109$	−1.99516			−0.191102			−0.0955511	−	0.995425i	$-0.530461\pi$
$109$	−1.99516			−0.191102			−0.0955511	+	0.995425i	$0.530461\pi$
$110$		0			0
$111$	−0.493290			−0.0468211
$112$		0			0
$113$			4.51783i			0.425002i	0.977161	+	0.212501i	$0.0681608\pi$
$113$			4.51783i			0.425002i	−0.977161	+	0.212501i	$0.931839\pi$
$114$		0			0
$115$	−0.620320	−	8.50224i	−0.0578451	−	0.792838i
$116$		0			0
$117$			3.41528i			0.315743i
$118$		0			0
$119$	−1.84172			−0.168830
$120$		0			0
$121$	−10.1534			−0.923041
$122$		0			0
$123$			6.61351i			0.596321i
$124$		0			0
$125$	10.9145	−	2.42342i	0.976226	−	0.216757i
$126$		0			0
$127$		−	0.688258i		−	0.0610730i	−0.999534	−	0.0305365i	$-0.990278\pi$
$127$		−	0.688258i		−	0.0610730i	0.999534	−	0.0305365i	$-0.00972158\pi$
$128$		0			0
$129$	11.1833			0.984637
$130$		0			0
$131$	6.91821			0.604447			0.302223	−	0.953237i	$-0.402271\pi$
$131$	6.91821			0.604447			0.302223	+	0.953237i	$0.402271\pi$
$132$		0			0
$133$			4.08025i			0.353802i
$134$		0			0
$135$	0.162710	+	2.23014i	0.0140039	+	0.191940i
$136$		0			0
$137$		−	10.3617i		−	0.885260i	−0.896704	−	0.442630i	$-0.854046\pi$
$137$		−	10.3617i		−	0.885260i	0.896704	−	0.442630i	$-0.145954\pi$
$138$		0			0
$139$	3.23127			0.274073			0.137036	−	0.990566i	$-0.456242\pi$
$139$	3.23127			0.274073			0.137036	+	0.990566i	$0.456242\pi$
$140$		0			0
$141$	−4.33651			−0.365200
$142$		0			0
$143$			3.14234i			0.262776i
$144$		0			0
$145$	−8.14839	+	0.594503i	−0.676687	+	0.0493708i
$146$		0			0
$147$		−	6.40652i		−	0.528401i
$148$		0			0
$149$	14.0650			1.15225			0.576123	−	0.817363i	$-0.304565\pi$
$149$	14.0650			1.15225			0.576123	+	0.817363i	$0.304565\pi$
$150$		0			0
$151$	−14.5915			−1.18744			−0.593719	−	0.804673i	$-0.702341\pi$
$151$	−14.5915			−1.18744			−0.593719	+	0.804673i	$0.702341\pi$
$152$		0			0
$153$			2.39068i			0.193275i
$154$		0			0
$155$	−23.6311	+	1.72412i	−1.89810	+	0.138484i
$156$		0			0
$157$		−	14.4240i		−	1.15116i	−0.817747	−	0.575579i	$-0.804777\pi$
$157$		−	14.4240i		−	1.15116i	0.817747	−	0.575579i	$-0.195223\pi$
$158$		0			0
$159$	10.0784			0.799273
$160$		0			0
$161$	−2.93700			−0.231468
$162$		0			0
$163$			0.857554i			0.0671688i	0.999436	+	0.0335844i	$0.0106923\pi$
$163$			0.857554i			0.0671688i	−0.999436	+	0.0335844i	$0.989308\pi$
$164$		0			0
$165$	0.149707	+	2.05192i	0.0116547	+	0.159741i
$166$		0			0
$167$			3.69432i			0.285875i	0.989732	+	0.142937i	$0.0456548\pi$
$167$			3.69432i			0.285875i	−0.989732	+	0.142937i	$0.954345\pi$
$168$		0			0
$169$	1.33586			0.102759
$170$		0			0
$171$	5.29643			0.405028
$172$		0			0
$173$		−	11.0991i		−	0.843848i	−0.906631	−	0.421924i	$-0.861355\pi$
$173$		−	11.0991i		−	0.843848i	0.906631	−	0.421924i	$-0.138645\pi$
$174$		0			0
$175$	−0.559088	−	3.81109i	−0.0422631	−	0.288092i
$176$		0			0
$177$			2.20987i			0.166104i
$178$		0			0
$179$	−0.860578			−0.0643226			−0.0321613	−	0.999483i	$-0.510239\pi$
$179$	−0.860578			−0.0643226			−0.0321613	+	0.999483i	$0.510239\pi$
$180$		0			0
$181$	24.5344			1.82363			0.911814	−	0.410603i	$-0.134682\pi$
$181$	24.5344			1.82363			0.911814	+	0.410603i	$0.134682\pi$
$182$		0			0
$183$		−	1.58317i		−	0.117032i
$184$		0			0
$185$	−0.0802633	−	1.10011i	−0.00590108	−	0.0808815i
$186$		0			0
$187$			2.19962i			0.160852i
$188$		0			0
$189$	0.770377			0.0560367
$190$		0			0
$191$	13.2055			0.955513			0.477757	−	0.878492i	$-0.341450\pi$
$191$	13.2055			0.955513			0.477757	+	0.878492i	$0.341450\pi$
$192$		0			0
$193$		−	7.08757i		−	0.510175i	−0.966918	−	0.255087i	$-0.917896\pi$
$193$		−	7.08757i		−	0.510175i	0.966918	−	0.255087i	$-0.0821042\pi$
$194$		0			0
$195$	−7.61655	+	0.555701i	−0.545433	+	0.0397945i
$196$		0			0
$197$			22.9197i			1.63296i	0.577373	+	0.816480i	$0.304078\pi$
$197$			22.9197i			1.63296i	−0.577373	+	0.816480i	$0.695922\pi$
$198$		0			0
$199$	1.39240			0.0987045			0.0493523	−	0.998781i	$-0.484284\pi$
$199$	1.39240			0.0987045			0.0493523	+	0.998781i	$0.484284\pi$
$200$		0			0
$201$	1.00000			0.0705346
$202$		0			0
$203$			2.81477i			0.197558i
$204$		0			0
$205$	−14.7491	+	1.07609i	−1.03012	+	0.0751571i
$206$		0			0
$207$			3.81242i			0.264982i
$208$		0			0
$209$	4.87316			0.337084
$210$		0			0
$211$	−2.94784			−0.202938			−0.101469	−	0.994839i	$-0.532354\pi$
$211$	−2.94784			−0.202938			−0.101469	+	0.994839i	$0.532354\pi$
$212$		0			0
$213$			3.39043i			0.232309i
$214$		0			0
$215$	1.81964	+	24.9404i	0.124098	+	1.70092i
$216$		0			0
$217$			8.16311i			0.554148i
$218$		0			0
$219$	−11.1127			−0.750928
$220$		0			0
$221$	−8.16483			−0.549226
$222$		0			0
$223$		−	5.22670i		−	0.350006i	−0.984568	−	0.175003i	$-0.944007\pi$
$223$		−	5.22670i		−	0.350006i	0.984568	−	0.175003i	$-0.0559934\pi$
$224$		0			0
$225$	−4.94705	+	0.725733i	−0.329803	+	0.0483822i
$226$		0			0
$227$			8.70449i			0.577737i	0.957369	+	0.288869i	$0.0932791\pi$
$227$			8.70449i			0.577737i	−0.957369	+	0.288869i	$0.906721\pi$
$228$		0			0
$229$	−18.6048			−1.22944			−0.614720	−	0.788745i	$-0.710731\pi$
$229$	−18.6048			−1.22944			−0.614720	+	0.788745i	$0.710731\pi$
$230$		0			0
$231$	0.708811			0.0466363
$232$		0			0
$233$		−	20.8013i		−	1.36274i	−0.731939	−	0.681370i	$-0.761385\pi$
$233$		−	20.8013i		−	1.36274i	0.731939	−	0.681370i	$-0.238615\pi$
$234$		0			0
$235$	−0.705595	−	9.67103i	−0.0460279	−	0.630869i
$236$		0			0
$237$		−	0.553989i		−	0.0359854i
$238$		0			0
$239$	−24.0320			−1.55450			−0.777252	−	0.629189i	$-0.783387\pi$
$239$	−24.0320			−1.55450			−0.777252	+	0.629189i	$0.783387\pi$
$240$		0			0
$241$	−6.08350			−0.391872			−0.195936	−	0.980617i	$-0.562775\pi$
$241$	−6.08350			−0.391872			−0.195936	+	0.980617i	$0.562775\pi$
$242$		0			0
$243$		−	1.00000i		−	0.0641500i
$244$		0			0
$245$	14.2874	−	1.04241i	0.912791	−	0.0665968i
$246$		0			0
$247$			18.0888i			1.15096i
$248$		0			0
$249$	−7.19859			−0.456192
$250$		0			0
$251$	−6.63120			−0.418558			−0.209279	−	0.977856i	$-0.567112\pi$
$251$	−6.63120			−0.418558			−0.209279	+	0.977856i	$0.567112\pi$
$252$		0			0
$253$			3.50775i			0.220530i
$254$		0			0
$255$	−5.33155	+	0.388987i	−0.333874	+	0.0243593i
$256$		0			0
$257$		−	8.49636i		−	0.529988i	−0.964250	−	0.264994i	$-0.914630\pi$
$257$		−	8.49636i		−	0.529988i	0.964250	−	0.264994i	$-0.0853700\pi$
$258$		0			0
$259$	−0.380019			−0.0236133
$260$		0			0
$261$	3.65375			0.226162
$262$		0			0
$263$			3.10113i			0.191224i	0.995419	+	0.0956120i	$0.0304808\pi$
$263$			3.10113i			0.191224i	−0.995419	+	0.0956120i	$0.969519\pi$
$264$		0			0
$265$	1.63986	+	22.4763i	0.100736	+	1.38071i
$266$		0			0
$267$			2.49567i			0.152733i
$268$		0			0
$269$	14.1933			0.865382			0.432691	−	0.901542i	$-0.357564\pi$
$269$	14.1933			0.865382			0.432691	+	0.901542i	$0.357564\pi$
$270$		0			0
$271$	−11.1541			−0.677565			−0.338783	−	0.940865i	$-0.610015\pi$
$271$	−11.1541			−0.677565			−0.338783	+	0.940865i	$0.610015\pi$
$272$		0			0
$273$			2.63105i			0.159239i
$274$		0			0
$275$	−4.55170	+	0.667735i	−0.274478	+	0.0402659i
$276$		0			0
$277$		−	14.8095i		−	0.889815i	−0.895576	−	0.444908i	$-0.853236\pi$
$277$		−	14.8095i		−	0.889815i	0.895576	−	0.444908i	$-0.146764\pi$
$278$		0			0
$279$	10.5963			0.634381
$280$		0			0
$281$	−1.23761			−0.0738295			−0.0369148	−	0.999318i	$-0.511753\pi$
$281$	−1.23761			−0.0738295			−0.0369148	+	0.999318i	$0.511753\pi$
$282$		0			0
$283$		−	12.8258i		−	0.762415i	−0.924489	−	0.381208i	$-0.875508\pi$
$283$		−	12.8258i		−	0.762415i	0.924489	−	0.381208i	$-0.124492\pi$
$284$		0			0
$285$	0.861783	+	11.8118i	0.0510476	+	0.699670i
$286$		0			0
$287$			5.09490i			0.300742i
$288$		0			0
$289$	11.2847			0.663804
$290$		0			0
$291$	−9.99897			−0.586150
$292$		0			0
$293$		−	11.4127i		−	0.666736i	−0.942797	−	0.333368i	$-0.891815\pi$
$293$		−	11.4127i		−	0.666736i	0.942797	−	0.333368i	$-0.108185\pi$
$294$		0			0
$295$	−4.92833	+	0.359569i	−0.286938	+	0.0209349i
$296$		0			0
$297$		−	0.920084i		−	0.0533887i
$298$		0			0
$299$	−13.0205			−0.752995
$300$		0			0
$301$	8.61537			0.496582
$302$		0			0
$303$			11.5549i			0.663814i
$304$		0			0
$305$	3.53070	−	0.257598i	0.202167	−	0.0147500i
$306$		0			0
$307$		−	28.7174i		−	1.63899i	−0.573088	−	0.819494i	$-0.694255\pi$
$307$		−	28.7174i		−	1.63899i	0.573088	−	0.819494i	$-0.305745\pi$
$308$		0			0
$309$	4.43399			0.252241
$310$		0			0
$311$	−7.78964			−0.441710			−0.220855	−	0.975307i	$-0.570885\pi$
$311$	−7.78964			−0.441710			−0.220855	+	0.975307i	$0.570885\pi$
$312$		0			0
$313$			16.8376i			0.951719i	0.879521	+	0.475859i	$0.157863\pi$
$313$			16.8376i			0.951719i	−0.879521	+	0.475859i	$0.842137\pi$
$314$		0			0
$315$	0.125348	+	1.71805i	0.00706256	+	0.0968011i
$316$		0			0
$317$			10.7798i			0.605453i	0.953077	+	0.302727i	$0.0978969\pi$
$317$			10.7798i			0.605453i	−0.953077	+	0.302727i	$0.902103\pi$
$318$		0			0
$319$	3.36176			0.188222
$320$		0			0
$321$	−1.01113			−0.0564358
$322$		0			0
$323$			12.6621i			0.704536i
$324$		0			0
$325$	−2.47858	−	16.8956i	−0.137487	−	0.937197i
$326$		0			0
$327$			1.99516i			0.110333i
$328$		0			0
$329$	−3.34075			−0.184181
$330$		0			0
$331$	−18.1376			−0.996935			−0.498467	−	0.866908i	$-0.666104\pi$
$331$	−18.1376			−0.996935			−0.498467	+	0.866908i	$0.666104\pi$
$332$		0			0
$333$			0.493290i			0.0270321i
$334$		0			0
$335$	0.162710	+	2.23014i	0.00888980	+	0.121846i
$336$		0			0
$337$			9.47247i			0.515998i	0.966145	+	0.257999i	$0.0830631\pi$
$337$			9.47247i			0.515998i	−0.966145	+	0.257999i	$0.916937\pi$
$338$		0			0
$339$	4.51783			0.245375
$340$		0			0
$341$	9.74944			0.527962
$342$		0			0
$343$		−	10.3281i		−	0.557663i
$344$		0			0
$345$	−8.50224	+	0.620320i	−0.457745	+	0.0333969i
$346$		0			0
$347$			23.4014i			1.25625i	0.778112	+	0.628126i	$0.216178\pi$
$347$			23.4014i			1.25625i	−0.778112	+	0.628126i	$0.783822\pi$
$348$		0			0
$349$	5.88313			0.314917			0.157458	−	0.987526i	$-0.449670\pi$
$349$	5.88313			0.314917			0.157458	+	0.987526i	$0.449670\pi$
$350$		0			0
$351$	3.41528			0.182294
$352$		0			0
$353$			23.8146i			1.26752i	0.773529	+	0.633761i	$0.218490\pi$
$353$			23.8146i			1.26752i	−0.773529	+	0.633761i	$0.781510\pi$
$354$		0			0
$355$	−7.56114	+	0.551658i	−0.401304	+	0.0292790i
$356$		0			0
$357$			1.84172i			0.0974743i
$358$		0			0
$359$	−4.36825			−0.230547			−0.115274	−	0.993334i	$-0.536774\pi$
$359$	−4.36825			−0.230547			−0.115274	+	0.993334i	$0.536774\pi$
$360$		0			0
$361$	9.05220			0.476432
$362$		0			0
$363$			10.1534i			0.532918i
$364$		0			0
$365$	−1.80815	−	24.7829i	−0.0946430	−	1.29720i
$366$		0			0
$367$		−	27.3999i		−	1.43026i	−0.698989	−	0.715132i	$-0.746366\pi$
$367$		−	27.3999i		−	1.43026i	0.698989	−	0.715132i	$-0.253634\pi$
$368$		0			0
$369$	6.61351			0.344286
$370$		0			0
$371$	7.76420			0.403097
$372$		0			0
$373$		−	19.2508i		−	0.996770i	−0.866956	−	0.498385i	$-0.833927\pi$
$373$		−	19.2508i		−	0.996770i	0.866956	−	0.498385i	$-0.166073\pi$
$374$		0			0
$375$	−2.42342	−	10.9145i	−0.125145	−	0.563624i
$376$		0			0
$377$			12.4786i			0.642680i
$378$		0			0
$379$	−18.5602			−0.953373			−0.476687	−	0.879073i	$-0.658162\pi$
$379$	−18.5602			−0.953373			−0.476687	+	0.879073i	$0.658162\pi$
$380$		0			0
$381$	−0.688258			−0.0352605
$382$		0			0
$383$			17.1732i			0.877510i	0.898607	+	0.438755i	$0.144580\pi$
$383$			17.1732i			0.877510i	−0.898607	+	0.438755i	$0.855420\pi$
$384$		0			0
$385$	0.115331	+	1.58075i	0.00587780	+	0.0805624i
$386$		0			0
$387$		−	11.1833i		−	0.568480i
$388$		0			0
$389$	3.86752			0.196091			0.0980455	−	0.995182i	$-0.468741\pi$
$389$	3.86752			0.196091			0.0980455	+	0.995182i	$0.468741\pi$
$390$		0			0
$391$	−9.11428			−0.460929
$392$		0			0
$393$		−	6.91821i		−	0.348978i
$394$		0			0
$395$	1.23547	−	0.0901395i	0.0621634	−	0.00453541i
$396$		0			0
$397$		−	37.9554i		−	1.90493i	−0.304653	−	0.952463i	$-0.598541\pi$
$397$		−	37.9554i		−	1.90493i	0.304653	−	0.952463i	$-0.401459\pi$
$398$		0			0
$399$	4.08025			0.204268
$400$		0			0
$401$	−25.7245			−1.28462			−0.642310	−	0.766445i	$-0.722024\pi$
$401$	−25.7245			−1.28462			−0.642310	+	0.766445i	$0.722024\pi$
$402$		0			0
$403$			36.1892i			1.80271i
$404$		0			0
$405$	2.23014	−	0.162710i	0.110817	−	0.00808513i
$406$		0			0
$407$			0.453868i			0.0224974i
$408$		0			0
$409$	20.0320			0.990519			0.495260	−	0.868745i	$-0.335073\pi$
$409$	20.0320			0.990519			0.495260	+	0.868745i	$0.335073\pi$
$410$		0			0
$411$	−10.3617			−0.511105
$412$		0			0
$413$			1.70244i			0.0837714i
$414$		0			0
$415$	−1.17128	−	16.0539i	−0.0574961	−	0.788054i
$416$		0			0
$417$		−	3.23127i		−	0.158236i
$418$		0			0
$419$	−10.5254			−0.514200			−0.257100	−	0.966385i	$-0.582767\pi$
$419$	−10.5254			−0.514200			−0.257100	+	0.966385i	$0.582767\pi$
$420$		0			0
$421$	−23.2559			−1.13342			−0.566712	−	0.823916i	$-0.691785\pi$
$421$	−23.2559			−1.13342			−0.566712	+	0.823916i	$0.691785\pi$
$422$		0			0
$423$			4.33651i			0.210849i
$424$		0			0
$425$	−1.73499	−	11.8268i	−0.0841595	−	0.573684i
$426$		0			0
$427$		−	1.21964i		−	0.0590225i
$428$		0			0
$429$	3.14234			0.151714
$430$		0			0
$431$	−16.6575			−0.802361			−0.401181	−	0.915999i	$-0.631400\pi$
$431$	−16.6575			−0.802361			−0.401181	+	0.915999i	$0.631400\pi$
$432$		0			0
$433$		−	5.94947i		−	0.285913i	−0.989729	−	0.142957i	$-0.954339\pi$
$433$		−	5.94947i		−	0.285913i	0.989729	−	0.142957i	$-0.0456609\pi$
$434$		0			0
$435$	0.594503	+	8.14839i	0.0285042	+	0.390685i
$436$		0			0
$437$			20.1922i			0.965926i
$438$		0			0
$439$	5.98390			0.285596			0.142798	−	0.989752i	$-0.454390\pi$
$439$	5.98390			0.285596			0.142798	+	0.989752i	$0.454390\pi$
$440$		0			0
$441$	−6.40652			−0.305072
$442$		0			0
$443$		−	7.31690i		−	0.347636i	−0.984778	−	0.173818i	$-0.944389\pi$
$443$		−	7.31690i		−	0.347636i	0.984778	−	0.173818i	$-0.0556105\pi$
$444$		0			0
$445$	−5.56570	+	0.406071i	−0.263839	+	0.0192496i
$446$		0			0
$447$		−	14.0650i		−	0.665250i
$448$		0			0
$449$	13.8235			0.652373			0.326186	−	0.945305i	$-0.394236\pi$
$449$	13.8235			0.652373			0.326186	+	0.945305i	$0.394236\pi$
$450$		0			0
$451$	6.08499			0.286531
$452$		0			0
$453$			14.5915i			0.685567i
$454$		0			0
$455$	−5.86762	+	0.428099i	−0.275078	+	0.0200696i
$456$		0			0
$457$		−	12.6567i		−	0.592058i	−0.955179	−	0.296029i	$-0.904338\pi$
$457$		−	12.6567i		−	0.592058i	0.955179	−	0.296029i	$-0.0956625\pi$
$458$		0			0
$459$	2.39068			0.111587
$460$		0			0
$461$	19.3010			0.898939			0.449470	−	0.893296i	$-0.351613\pi$
$461$	19.3010			0.898939			0.449470	+	0.893296i	$0.351613\pi$
$462$		0			0
$463$		−	1.43571i		−	0.0667232i	−0.999443	−	0.0333616i	$-0.989379\pi$
$463$		−	1.43571i		−	0.0667232i	0.999443	−	0.0333616i	$-0.0106213\pi$
$464$		0			0
$465$	1.72412	+	23.6311i	0.0799540	+	1.09587i
$466$		0			0
$467$		−	6.23124i		−	0.288347i	−0.989552	−	0.144174i	$-0.953948\pi$
$467$		−	6.23124i		−	0.288347i	0.989552	−	0.144174i	$-0.0460524\pi$
$468$		0			0
$469$	0.770377			0.0355727
$470$		0			0
$471$	−14.4240			−0.664621
$472$		0			0
$473$		−	10.2896i		−	0.473116i
$474$		0			0
$475$	−26.2017	+	3.84379i	−1.20222	+	0.176365i
$476$		0			0
$477$		−	10.0784i		−	0.461460i
$478$		0			0
$479$	−8.90008			−0.406655			−0.203328	−	0.979111i	$-0.565176\pi$
$479$	−8.90008			−0.406655			−0.203328	+	0.979111i	$0.565176\pi$
$480$		0			0
$481$	−1.68472			−0.0768168
$482$		0			0
$483$			2.93700i			0.133638i
$484$		0			0
$485$	−1.62693	−	22.2991i	−0.0738752	−	1.01255i
$486$		0			0
$487$		−	19.7924i		−	0.896877i	−0.893814	−	0.448439i	$-0.851980\pi$
$487$		−	19.7924i		−	0.896877i	0.893814	−	0.448439i	$-0.148020\pi$
$488$		0			0
$489$	0.857554			0.0387799
$490$		0			0
$491$	−11.0795			−0.500009			−0.250005	−	0.968245i	$-0.580432\pi$
$491$	−11.0795			−0.500009			−0.250005	+	0.968245i	$0.580432\pi$
$492$		0			0
$493$			8.73495i			0.393402i
$494$		0			0
$495$	2.05192	−	0.149707i	0.0922267	−	0.00672883i
$496$		0			0
$497$			2.61191i			0.117160i
$498$		0			0
$499$	32.9441			1.47478			0.737390	−	0.675467i	$-0.236058\pi$
$499$	32.9441			1.47478			0.737390	+	0.675467i	$0.236058\pi$
$500$		0			0
$501$	3.69432			0.165050
$502$		0			0
$503$			33.5278i			1.49493i	0.664302	+	0.747464i	$0.268729\pi$
$503$			33.5278i			1.49493i	−0.664302	+	0.747464i	$0.731271\pi$
$504$		0			0
$505$	−25.7691	+	1.88011i	−1.14671	+	0.0836636i
$506$		0			0
$507$		−	1.33586i		−	0.0593277i
$508$		0			0
$509$	23.0984			1.02382			0.511910	−	0.859039i	$-0.328938\pi$
$509$	23.0984			1.02382			0.511910	+	0.859039i	$0.328938\pi$
$510$		0			0
$511$	−8.56098			−0.378716
$512$		0			0
$513$		−	5.29643i		−	0.233843i
$514$		0			0
$515$	0.721455	+	9.88842i	0.0317911	+	0.435736i
$516$		0			0
$517$			3.98996i			0.175478i
$518$		0			0
$519$	−11.0991			−0.487196
$520$		0			0
$521$	−42.5281			−1.86319			−0.931595	−	0.363499i	$-0.881582\pi$
$521$	−42.5281			−1.86319			−0.931595	+	0.363499i	$0.881582\pi$
$522$		0			0
$523$		−	33.9691i		−	1.48537i	−0.669643	−	0.742683i	$-0.733553\pi$
$523$		−	33.9691i		−	1.48537i	0.669643	−	0.742683i	$-0.266447\pi$
$524$		0			0
$525$	−3.81109	+	0.559088i	−0.166330	+	0.0244006i
$526$		0			0
$527$			25.3322i			1.10349i
$528$		0			0
$529$	8.46542			0.368062
$530$		0			0
$531$	2.20987			0.0959004
$532$		0			0
$533$			22.5870i			0.978352i
$534$		0			0
$535$	−0.164521	−	2.25496i	−0.00711287	−	0.0974906i
$536$		0			0
$537$			0.860578i			0.0371367i
$538$		0			0
$539$	−5.89453			−0.253896
$540$		0			0
$541$	22.9830			0.988118			0.494059	−	0.869429i	$-0.335513\pi$
$541$	22.9830			0.988118			0.494059	+	0.869429i	$0.335513\pi$
$542$		0			0
$543$		−	24.5344i		−	1.05287i
$544$		0			0
$545$	−4.44950	+	0.324633i	−0.190596	+	0.0139058i
$546$		0			0
$547$		−	18.8646i		−	0.806594i	−0.915069	−	0.403297i	$-0.867864\pi$
$547$		−	18.8646i		−	0.806594i	0.915069	−	0.403297i	$-0.132136\pi$
$548$		0			0
$549$	−1.58317			−0.0675682
$550$		0			0
$551$	19.3519			0.824417
$552$		0			0
$553$		−	0.426780i		−	0.0181485i
$554$		0			0
$555$	−1.10011	+	0.0802633i	−0.0466969	+	0.00340699i
$556$		0			0
$557$			18.9726i			0.803896i	0.915663	+	0.401948i	$0.131667\pi$
$557$			18.9726i			0.803896i	−0.915663	+	0.401948i	$0.868333\pi$
$558$		0			0
$559$	38.1942			1.61544
$560$		0			0
$561$	2.19962			0.0928682
$562$		0			0
$563$		−	13.7915i		−	0.581242i	−0.956838	−	0.290621i	$-0.906138\pi$
$563$		−	13.7915i		−	0.581242i	0.956838	−	0.290621i	$-0.0938618\pi$
$564$		0			0
$565$	0.735097	+	10.0754i	0.0309257	+	0.423875i
$566$		0			0
$567$		−	0.770377i		−	0.0323528i
$568$		0			0
$569$	15.7947			0.662148			0.331074	−	0.943605i	$-0.392589\pi$
$569$	15.7947			0.662148			0.331074	+	0.943605i	$0.392589\pi$
$570$		0			0
$571$	−1.75377			−0.0733931			−0.0366966	−	0.999326i	$-0.511684\pi$
$571$	−1.75377			−0.0733931			−0.0366966	+	0.999326i	$0.511684\pi$
$572$		0			0
$573$		−	13.2055i		−	0.551666i
$574$		0			0
$575$	−2.76680	−	18.8603i	−0.115384	−	0.786527i
$576$		0			0
$577$		−	2.81408i		−	0.117151i	−0.998283	−	0.0585757i	$-0.981344\pi$
$577$		−	2.81408i		−	0.117151i	0.998283	−	0.0585757i	$-0.0186559\pi$
$578$		0			0
$579$	−7.08757			−0.294549
$580$		0			0
$581$	−5.54563			−0.230071
$582$		0			0
$583$		−	9.27301i		−	0.384049i
$584$		0			0
$585$	0.555701	+	7.61655i	0.0229754	+	0.314906i
$586$		0			0
$587$			20.0108i			0.825935i	0.910746	+	0.412968i	$0.135508\pi$
$587$			20.0108i			0.825935i	−0.910746	+	0.412968i	$0.864492\pi$
$588$		0			0
$589$	56.1223			2.31248
$590$		0			0
$591$	22.9197			0.942790
$592$		0			0
$593$			13.6992i			0.562560i	0.959626	+	0.281280i	$0.0907589\pi$
$593$			13.6992i			0.562560i	−0.959626	+	0.281280i	$0.909241\pi$
$594$		0			0
$595$	−4.10730	+	0.299667i	−0.168383	+	0.0122851i
$596$		0			0
$597$		−	1.39240i		−	0.0569871i
$598$		0			0
$599$	36.7109			1.49996			0.749982	−	0.661458i	$-0.230062\pi$
$599$	36.7109			1.49996			0.749982	+	0.661458i	$0.230062\pi$
$600$		0			0
$601$	8.93963			0.364655			0.182327	−	0.983238i	$-0.441637\pi$
$601$	8.93963			0.364655			0.182327	+	0.983238i	$0.441637\pi$
$602$		0			0
$603$		−	1.00000i		−	0.0407231i
$604$		0			0
$605$	−22.6436	+	1.65207i	−0.920594	+	0.0671661i
$606$		0			0
$607$			17.3553i			0.704430i	0.935919	+	0.352215i	$0.114571\pi$
$607$			17.3553i			0.704430i	−0.935919	+	0.352215i	$0.885429\pi$
$608$		0			0
$609$	2.81477			0.114060
$610$		0			0
$611$	−14.8104			−0.599165
$612$		0			0
$613$		−	1.60633i		−	0.0648791i	−0.999474	−	0.0324395i	$-0.989672\pi$
$613$		−	1.60633i		−	0.0648791i	0.999474	−	0.0324395i	$-0.0103276\pi$
$614$		0			0
$615$	1.07609	+	14.7491i	0.0433920	+	0.594740i
$616$		0			0
$617$			9.21039i			0.370796i	0.982663	+	0.185398i	$0.0593575\pi$
$617$			9.21039i			0.370796i	−0.982663	+	0.185398i	$0.940643\pi$
$618$		0			0
$619$	11.0130			0.442652			0.221326	−	0.975200i	$-0.428962\pi$
$619$	11.0130			0.442652			0.221326	+	0.975200i	$0.428962\pi$
$620$		0			0
$621$	3.81242			0.152987
$622$		0			0
$623$			1.92261i			0.0770276i
$624$		0			0
$625$	23.9466	−	7.18047i	0.957865	−	0.287219i
$626$		0			0
$627$		−	4.87316i		−	0.194615i
$628$		0			0
$629$	−1.17930			−0.0470217
$630$		0			0
$631$	38.4029			1.52879			0.764397	−	0.644746i	$-0.223037\pi$
$631$	38.4029			1.52879			0.764397	+	0.644746i	$0.223037\pi$
$632$		0			0
$633$			2.94784i			0.117166i
$634$		0			0
$635$	−0.111987	−	1.53491i	−0.00444405	−	0.0609111i
$636$		0			0
$637$		−	21.8801i		−	0.866919i
$638$		0			0
$639$	3.39043			0.134124
$640$		0			0
$641$	38.7806			1.53174			0.765871	−	0.642995i	$-0.222308\pi$
$641$	38.7806			1.53174			0.765871	+	0.642995i	$0.222308\pi$
$642$		0			0
$643$		−	28.4153i		−	1.12059i	−0.828293	−	0.560295i	$-0.810688\pi$
$643$		−	28.4153i		−	1.12059i	0.828293	−	0.560295i	$-0.189312\pi$
$644$		0			0
$645$	24.9404	−	1.81964i	0.982026	−	0.0716482i
$646$		0			0
$647$		−	11.0097i		−	0.432838i	−0.976301	−	0.216419i	$-0.930562\pi$
$647$		−	11.0097i		−	0.432838i	0.976301	−	0.216419i	$-0.0694377\pi$
$648$		0			0
$649$	2.03327			0.0798128
$650$		0			0
$651$	8.16311			0.319937
$652$		0			0
$653$			31.2495i			1.22289i	0.791288	+	0.611443i	$0.209411\pi$
$653$			31.2495i			1.22289i	−0.791288	+	0.611443i	$0.790589\pi$
$654$		0			0
$655$	15.4286	−	1.12566i	0.602844	−	0.0439833i
$656$		0			0
$657$			11.1127i			0.433549i
$658$		0			0
$659$	19.3967			0.755588			0.377794	−	0.925890i	$-0.376683\pi$
$659$	19.3967			0.755588			0.377794	+	0.925890i	$0.376683\pi$
$660$		0			0
$661$	−42.5439			−1.65477			−0.827384	−	0.561637i	$-0.810172\pi$
$661$	−42.5439			−1.65477			−0.827384	+	0.561637i	$0.810172\pi$
$662$		0			0
$663$			8.16483i			0.317096i
$664$		0			0
$665$	0.663898	+	9.09953i	0.0257448	+	0.352865i
$666$		0			0
$667$			13.9297i			0.539359i
$668$		0			0
$669$	−5.22670			−0.202076
$670$		0			0
$671$	−1.45665			−0.0562334
$672$		0			0
$673$		−	14.6493i		−	0.564688i	−0.959313	−	0.282344i	$-0.908888\pi$
$673$		−	14.6493i		−	0.564688i	0.959313	−	0.282344i	$-0.0911120\pi$
$674$		0			0
$675$	0.725733	+	4.94705i	0.0279335	+	0.190412i
$676$		0			0
$677$			23.5892i			0.906608i	0.891356	+	0.453304i	$0.149755\pi$
$677$			23.5892i			0.906608i	−0.891356	+	0.453304i	$0.850245\pi$
$678$		0			0
$679$	−7.70297			−0.295613
$680$		0			0
$681$	8.70449			0.333557
$682$		0			0
$683$		−	43.6476i		−	1.67013i	−0.550151	−	0.835065i	$-0.685430\pi$
$683$		−	43.6476i		−	1.67013i	0.550151	−	0.835065i	$-0.314570\pi$
$684$		0			0
$685$	−1.68595	−	23.1081i	−0.0644170	−	0.882913i
$686$		0			0
$687$			18.6048i			0.709818i
$688$		0			0
$689$	34.4207			1.31132
$690$		0			0
$691$	39.4751			1.50170			0.750852	−	0.660470i	$-0.229643\pi$
$691$	39.4751			1.50170			0.750852	+	0.660470i	$0.229643\pi$
$692$		0			0
$693$		−	0.708811i		−	0.0269255i
$694$		0			0
$695$	7.20619	−	0.525761i	0.273346	−	0.0199432i
$696$		0			0
$697$			15.8108i			0.598876i
$698$		0			0
$699$	−20.8013			−0.786778
$700$		0			0
$701$	19.1075			0.721681			0.360841	−	0.932627i	$-0.382490\pi$
$701$	19.1075			0.721681			0.360841	+	0.932627i	$0.382490\pi$
$702$		0			0
$703$			2.61268i			0.0985391i
$704$		0			0
$705$	−9.67103	+	0.705595i	−0.364232	+	0.0265742i
$706$		0			0
$707$			8.90166i			0.334781i
$708$		0			0
$709$	13.6406			0.512285			0.256143	−	0.966639i	$-0.417548\pi$
$709$	13.6406			0.512285			0.256143	+	0.966639i	$0.417548\pi$
$710$		0			0
$711$	−0.553989			−0.0207762
$712$		0			0
$713$			40.3974i			1.51290i
$714$		0			0
$715$	0.511291	+	7.00787i	0.0191212	+	0.262079i
$716$		0			0
$717$			24.0320i			0.897493i
$718$		0			0
$719$	27.0560			1.00902			0.504509	−	0.863406i	$-0.331674\pi$
$719$	27.0560			1.00902			0.504509	+	0.863406i	$0.331674\pi$
$720$		0			0
$721$	3.41584			0.127213
$722$		0			0
$723$			6.08350i			0.226248i
$724$		0			0
$725$	−18.0753	+	2.65165i	−0.671300	+	0.0984798i
$726$		0			0
$727$		−	39.3246i		−	1.45847i	−0.684265	−	0.729233i	$-0.739877\pi$
$727$		−	39.3246i		−	1.45847i	0.684265	−	0.729233i	$-0.260123\pi$
$728$		0			0
$729$	−1.00000			−0.0370370
$730$		0			0
$731$	26.7357			0.988856
$732$		0			0
$733$		−	31.2476i		−	1.15416i	−0.816689	−	0.577078i	$-0.804193\pi$
$733$		−	31.2476i		−	1.15416i	0.816689	−	0.577078i	$-0.195807\pi$
$734$		0			0
$735$	−1.04241	−	14.2874i	−0.0384497	−	0.527000i
$736$		0			0
$737$		−	0.920084i		−	0.0338917i
$738$		0			0
$739$	14.7439			0.542362			0.271181	−	0.962528i	$-0.412586\pi$
$739$	14.7439			0.542362			0.271181	+	0.962528i	$0.412586\pi$
$740$		0			0
$741$	18.0888			0.664509
$742$		0			0
$743$		−	9.83755i		−	0.360905i	−0.983584	−	0.180452i	$-0.942244\pi$
$743$		−	9.83755i		−	0.360905i	0.983584	−	0.180452i	$-0.0577562\pi$
$744$		0			0
$745$	31.3668	−	2.28851i	1.14919	−	0.0838446i
$746$		0			0
$747$			7.19859i			0.263383i
$748$		0			0
$749$	−0.778952			−0.0284623
$750$		0			0
$751$	10.1400			0.370015			0.185007	−	0.982737i	$-0.440769\pi$
$751$	10.1400			0.370015			0.185007	+	0.982737i	$0.440769\pi$
$752$		0			0
$753$			6.63120i			0.241654i
$754$		0			0
$755$	−32.5410	+	2.37418i	−1.18429	+	0.0864052i
$756$		0			0
$757$		−	1.45440i		−	0.0528610i	−0.999651	−	0.0264305i	$-0.991586\pi$
$757$		−	1.45440i		−	0.0528610i	0.999651	−	0.0264305i	$-0.00841407\pi$
$758$		0			0
$759$	3.50775			0.127323
$760$		0			0
$761$	−24.4868			−0.887644			−0.443822	−	0.896115i	$-0.646378\pi$
$761$	−24.4868			−0.887644			−0.443822	+	0.896115i	$0.646378\pi$
$762$		0			0
$763$			1.53703i			0.0556442i
$764$		0			0
$765$	0.388987	+	5.33155i	0.0140639	+	0.192762i
$766$		0			0
$767$			7.54734i			0.272519i
$768$		0			0
$769$	34.8963			1.25839			0.629197	−	0.777246i	$-0.283384\pi$
$769$	34.8963			1.25839			0.629197	+	0.777246i	$0.283384\pi$
$770$		0			0
$771$	−8.49636			−0.305989
$772$		0			0
$773$			35.2400i			1.26749i	0.773540	+	0.633747i	$0.218484\pi$
$773$			35.2400i			1.26749i	−0.773540	+	0.633747i	$0.781516\pi$
$774$		0			0
$775$	−52.4202	+	7.69005i	−1.88299	+	0.276235i
$776$		0			0
$777$			0.380019i			0.0136331i
$778$		0			0
$779$	35.0280			1.25501
$780$		0			0
$781$	3.11948			0.111624
$782$		0			0
$783$		−	3.65375i		−	0.130575i
$784$		0			0
$785$	−2.34692	−	32.1674i	−0.0837653	−	1.14811i
$786$		0			0
$787$		−	19.0710i		−	0.679809i	−0.940460	−	0.339905i	$-0.889605\pi$
$787$		−	19.0710i		−	0.679809i	0.940460	−	0.339905i	$-0.110395\pi$
$788$		0			0
$789$	3.10113			0.110403
$790$		0			0
$791$	3.48043			0.123750
$792$		0			0
$793$		−	5.40698i		−	0.192008i
$794$		0			0
$795$	22.4763	−	1.63986i	0.797154	−	0.0581600i
$796$		0			0
$797$		−	46.2621i		−	1.63869i	−0.573303	−	0.819344i	$-0.694338\pi$
$797$		−	46.2621i		−	1.63869i	0.573303	−	0.819344i	$-0.305662\pi$
$798$		0			0
$799$	−10.3672			−0.366765
$800$		0			0
$801$	2.49567			0.0881802
$802$		0			0
$803$			10.2246i			0.360819i
$804$		0			0
$805$	−6.54993	+	0.477880i	−0.230855	+	0.0168431i
$806$		0			0
$807$		−	14.1933i		−	0.499629i
$808$		0			0
$809$	41.2160			1.44908			0.724539	−	0.689233i	$-0.242052\pi$
$809$	41.2160			1.44908			0.724539	+	0.689233i	$0.242052\pi$
$810$		0			0
$811$	−28.0508			−0.984997			−0.492499	−	0.870313i	$-0.663916\pi$
$811$	−28.0508			−0.984997			−0.492499	+	0.870313i	$0.663916\pi$
$812$		0			0
$813$			11.1541i			0.391192i
$814$		0			0
$815$	0.139533	+	1.91247i	0.00488762	+	0.0669907i
$816$		0			0
$817$		−	59.2317i		−	2.07226i
$818$		0			0
$819$	2.63105			0.0919364
$820$		0			0
$821$	−36.4935			−1.27363			−0.636816	−	0.771015i	$-0.719749\pi$
$821$	−36.4935			−1.27363			−0.636816	+	0.771015i	$0.719749\pi$
$822$		0			0
$823$			34.7589i			1.21162i	0.795609	+	0.605810i	$0.207151\pi$
$823$			34.7589i			1.21162i	−0.795609	+	0.605810i	$0.792849\pi$
$824$		0			0
$825$	0.667735	+	4.55170i	0.0232475	+	0.158470i
$826$		0			0
$827$		−	16.9400i		−	0.589062i	−0.955642	−	0.294531i	$-0.904837\pi$
$827$		−	16.9400i		−	0.589062i	0.955642	−	0.294531i	$-0.0951634\pi$
$828$		0			0
$829$	−15.0663			−0.523274			−0.261637	−	0.965166i	$-0.584262\pi$
$829$	−15.0663			−0.523274			−0.261637	+	0.965166i	$0.584262\pi$
$830$		0			0
$831$	−14.8095			−0.513735
$832$		0			0
$833$		−	15.3159i		−	0.530665i
$834$		0			0
$835$	0.601103	+	8.23884i	0.0208020	+	0.285117i
$836$		0			0
$837$		−	10.5963i		−	0.366260i
$838$		0			0
$839$	−20.9303			−0.722593			−0.361297	−	0.932451i	$-0.617666\pi$
$839$	−20.9303			−0.722593			−0.361297	+	0.932451i	$0.617666\pi$
$840$		0			0
$841$	−15.6501			−0.539658
$842$		0			0
$843$			1.23761i			0.0426255i
$844$		0			0
$845$	2.97916	−	0.217358i	0.102486	−	0.00747735i
$846$		0			0
$847$			7.82198i			0.268766i
$848$		0			0
$849$	−12.8258			−0.440181
$850$		0			0
$851$	−1.88063			−0.0644672
$852$		0			0
$853$		−	47.5216i		−	1.62711i	−0.581490	−	0.813553i	$-0.697530\pi$
$853$		−	47.5216i		−	1.62711i	0.581490	−	0.813553i	$-0.302470\pi$
$854$		0			0
$855$	11.8118	−	0.861783i	0.403955	−	0.0294724i
$856$		0			0
$857$			38.6811i			1.32132i	0.750685	+	0.660660i	$0.229723\pi$
$857$			38.6811i			1.32132i	−0.750685	+	0.660660i	$0.770277\pi$
$858$		0			0
$859$	−24.4503			−0.834233			−0.417117	−	0.908853i	$-0.636959\pi$
$859$	−24.4503			−0.834233			−0.417117	+	0.908853i	$0.636959\pi$
$860$		0			0
$861$	5.09490			0.173634
$862$		0			0
$863$		−	26.8185i		−	0.912911i	−0.889746	−	0.456456i	$-0.849119\pi$
$863$		−	26.8185i		−	0.912911i	0.889746	−	0.456456i	$-0.150881\pi$
$864$		0			0
$865$	−1.80593	−	24.7525i	−0.0614036	−	0.841611i
$866$		0			0
$867$		−	11.2847i		−	0.383247i
$868$		0			0
$869$	−0.509716			−0.0172909
$870$		0			0
$871$	3.41528			0.115722
$872$		0			0
$873$			9.99897i			0.338414i
$874$		0			0
$875$	−1.86695	−	8.40830i	−0.0631143	−	0.284253i
$876$		0			0
$877$		−	20.8878i		−	0.705332i	−0.935749	−	0.352666i	$-0.885275\pi$
$877$		−	20.8878i		−	0.705332i	0.935749	−	0.352666i	$-0.114725\pi$
$878$		0			0
$879$	−11.4127			−0.384940
$880$		0			0
$881$	17.3955			0.586068			0.293034	−	0.956102i	$-0.405335\pi$
$881$	17.3955			0.586068			0.293034	+	0.956102i	$0.405335\pi$
$882$		0			0
$883$		−	23.6254i		−	0.795058i	−0.917590	−	0.397529i	$-0.869868\pi$
$883$		−	23.6254i		−	0.795058i	0.917590	−	0.397529i	$-0.130132\pi$
$884$		0			0
$885$	0.359569	+	4.92833i	0.0120868	+	0.165664i
$886$		0			0
$887$		−	42.5427i		−	1.42844i	−0.699919	−	0.714222i	$-0.746781\pi$
$887$		−	42.5427i		−	1.42844i	0.699919	−	0.714222i	$-0.253219\pi$
$888$		0			0
$889$	−0.530218			−0.0177829
$890$		0			0
$891$	−0.920084			−0.0308240
$892$		0			0
$893$			22.9681i			0.768597i
$894$		0			0
$895$	−1.91921	+	0.140025i	−0.0641521	+	0.00468051i
$896$		0			0
$897$			13.0205i			0.434742i
$898$		0			0
$899$	38.7161			1.29125
$900$		0			0
$901$	24.0943			0.802698
$902$		0			0
$903$		−	8.61537i		−	0.286702i
$904$		0			0
$905$	54.7152	−	3.99200i	1.81879	−	0.132698i
$906$		0			0
$907$			20.8465i			0.692196i	0.938198	+	0.346098i	$0.112494\pi$
$907$			20.8465i			0.692196i	−0.938198	+	0.346098i	$0.887506\pi$
$908$		0			0
$909$	11.5549			0.383253
$910$		0			0
$911$	−9.12389			−0.302288			−0.151144	−	0.988512i	$-0.548296\pi$
$911$	−9.12389			−0.302288			−0.151144	+	0.988512i	$0.548296\pi$
$912$		0			0
$913$			6.62331i			0.219199i
$914$		0			0
$915$	−0.257598	−	3.53070i	−0.00851594	−	0.116721i
$916$		0			0
$917$		−	5.32963i		−	0.176000i
$918$		0			0
$919$	16.1010			0.531123			0.265561	−	0.964094i	$-0.414443\pi$
$919$	16.1010			0.531123			0.265561	+	0.964094i	$0.414443\pi$
$920$		0			0
$921$	−28.7174			−0.946270
$922$		0			0
$923$			11.5793i			0.381137i
$924$		0			0
$925$	−0.357997	−	2.44033i	−0.0117709	−	0.0802376i
$926$		0			0
$927$		−	4.43399i		−	0.145631i
$928$		0			0
$929$	44.0193			1.44423			0.722113	−	0.691776i	$-0.243171\pi$
$929$	44.0193			1.44423			0.722113	+	0.691776i	$0.243171\pi$
$930$		0			0
$931$	−33.9317			−1.11207
$932$		0			0
$933$			7.78964i			0.255021i
$934$		0			0
$935$	0.357901	+	4.90547i	0.0117046	+	0.160426i
$936$		0			0
$937$		−	36.2421i		−	1.18398i	−0.805947	−	0.591988i	$-0.798343\pi$
$937$		−	36.2421i		−	1.18398i	0.805947	−	0.591988i	$-0.201657\pi$
$938$		0			0
$939$	16.8376			0.549475
$940$		0			0
$941$	12.2154			0.398211			0.199106	−	0.979978i	$-0.436196\pi$
$941$	12.2154			0.398211			0.199106	+	0.979978i	$0.436196\pi$
$942$		0			0
$943$			25.2135i			0.821065i
$944$		0			0
$945$	1.71805	−	0.125348i	0.0558881	−	0.00407757i
$946$		0			0
$947$			47.8298i			1.55426i	0.629340	+	0.777130i	$0.283325\pi$
$947$			47.8298i			1.55426i	−0.629340	+	0.777130i	$0.716675\pi$
$948$		0			0
$949$	−37.9531			−1.23201
$950$		0			0
$951$	10.7798			0.349559
$952$		0			0
$953$		−	42.2855i		−	1.36976i	−0.728655	−	0.684881i	$-0.759854\pi$
$953$		−	42.2855i		−	1.36976i	0.728655	−	0.684881i	$-0.240146\pi$
$954$		0			0
$955$	29.4500	−	2.14866i	0.952980	−	0.0695290i
$956$		0			0
$957$		−	3.36176i		−	0.108670i
$958$		0			0
$959$	−7.98242			−0.257766
$960$		0			0
$961$	81.2806			2.62196
$962$		0			0
$963$			1.01113i			0.0325832i
$964$		0			0
$965$	−1.15322	−	15.8063i	−0.0371234	−	0.508822i
$966$		0			0
$967$			43.4509i			1.39729i	0.715470	+	0.698643i	$0.246213\pi$
$967$			43.4509i			1.39729i	−0.715470	+	0.698643i	$0.753787\pi$
$968$		0			0
$969$	12.6621			0.406764
$970$		0			0
$971$	42.1223			1.35177			0.675885	−	0.737007i	$-0.263762\pi$
$971$	42.1223			1.35177			0.675885	+	0.737007i	$0.263762\pi$
$972$		0			0
$973$		−	2.48930i		−	0.0798032i
$974$		0			0
$975$	−16.8956	+	2.47858i	−0.541091	+	0.0793781i
$976$		0			0
$977$			32.8448i			1.05080i	0.850856	+	0.525399i	$0.176084\pi$
$977$			32.8448i			1.05080i	−0.850856	+	0.525399i	$0.823916\pi$
$978$		0			0
$979$	2.29623			0.0733877
$980$		0			0
$981$	1.99516			0.0637007
$982$		0			0
$983$			49.1940i			1.56905i	0.620100	+	0.784523i	$0.287092\pi$
$983$			49.1940i			1.56905i	−0.620100	+	0.784523i	$0.712908\pi$
$984$		0			0
$985$	3.72927	+	51.1141i	0.118824	+	1.62863i
$986$		0			0
$987$			3.34075i			0.106337i
$988$		0			0
$989$	42.6356			1.35573
$990$		0			0
$991$	18.4892			0.587328			0.293664	−	0.955909i	$-0.405125\pi$
$991$	18.4892			0.587328			0.293664	+	0.955909i	$0.405125\pi$
$992$		0			0
$993$			18.1376i			0.575581i
$994$		0			0
$995$	3.10524	−	0.226557i	0.0984429	−	0.00718235i
$996$		0			0
$997$			48.8050i			1.54567i	0.634608	+	0.772835i	$0.281162\pi$
$997$			48.8050i			1.54567i	−0.634608	+	0.772835i	$0.718838\pi$
$998$		0			0
$999$	0.493290			0.0156070

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	4020.2.g.b.1609.12	✓	24
5.4	even	2	inner	4020.2.g.b.1609.24	yes	24

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
4020.2.g.b.1609.12	✓	24	1.1	even	1	trivial
4020.2.g.b.1609.24	yes	24	5.4	even	2	inner

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.g (of order \(2\), degree \(1\), minimal)

\(f(q)\)	\(=\)	\(q-1.00000i q^{3} +(2.23014 - 0.162710i) q^{5} -0.770377i q^{7} -1.00000 q^{9} +O(q^{10})\)	\(q-1.00000i q^{3} +(2.23014 - 0.162710i) q^{5} -0.770377i q^{7} -1.00000 q^{9} -0.920084 q^{11} -3.41528i q^{13} +(-0.162710 - 2.23014i) q^{15} -2.39068i q^{17} -5.29643 q^{19} -0.770377 q^{21} -3.81242i q^{23} +(4.94705 - 0.725733i) q^{25} +1.00000i q^{27} -3.65375 q^{29} -10.5963 q^{31} +0.920084i q^{33} +(-0.125348 - 1.71805i) q^{35} -0.493290i q^{37} -3.41528 q^{39} -6.61351 q^{41} +11.1833i q^{43} +(-2.23014 + 0.162710i) q^{45} -4.33651i q^{47} +6.40652 q^{49} -2.39068 q^{51} +10.0784i q^{53} +(-2.05192 + 0.149707i) q^{55} +5.29643i q^{57} -2.20987 q^{59} +1.58317 q^{61} +0.770377i q^{63} +(-0.555701 - 7.61655i) q^{65} +1.00000i q^{67} -3.81242 q^{69} -3.39043 q^{71} -11.1127i q^{73} +(-0.725733 - 4.94705i) q^{75} +0.708811i q^{77} +0.553989 q^{79} +1.00000 q^{81} -7.19859i q^{83} +(-0.388987 - 5.33155i) q^{85} +3.65375i q^{87} -2.49567 q^{89} -2.63105 q^{91} +10.5963i q^{93} +(-11.8118 + 0.861783i) q^{95} -9.99897i q^{97} +0.920084 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 24 q - 4 q^{5} - 24 q^{9}+O(q^{10}) \) 24 * q - 4 * q^5 - 24 * q^9	\( 24 q - 4 q^{5} - 24 q^{9} - 8 q^{11} - 2 q^{15} + 16 q^{19} - 20 q^{21} + 10 q^{25} + 36 q^{29} - 2 q^{35} + 4 q^{39} - 24 q^{41} + 4 q^{45} - 4 q^{51} - 4 q^{55} + 24 q^{59} - 4 q^{61} - 20 q^{65} - 4 q^{69} + 20 q^{71} - 12 q^{75} - 28 q^{79} + 24 q^{81} - 16 q^{85} + 48 q^{89} - 20 q^{91} - 4 q^{95} + 8 q^{99}+O(q^{100}) \) 24 * q - 4 * q^5 - 24 * q^9 - 8 * q^11 - 2 * q^15 + 16 * q^19 - 20 * q^21 + 10 * q^25 + 36 * q^29 - 2 * q^35 + 4 * q^39 - 24 * q^41 + 4 * q^45 - 4 * q^51 - 4 * q^55 + 24 * q^59 - 4 * q^61 - 20 * q^65 - 4 * q^69 + 20 * q^71 - 12 * q^75 - 28 * q^79 + 24 * q^81 - 16 * q^85 + 48 * q^89 - 20 * q^91 - 4 * q^95 + 8 * q^99

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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