LMFDB - Embedded newform 1840.3.k.c.321.6

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [1840,3,Mod(321,1840)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(1840, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([0, 0, 0, 1]))

N = Newforms(chi, 3, names="a")

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

chi := DirichletCharacter("1840.321");

S:= CuspForms(chi, 3);

N := Newforms(S);

Level:	$ N $	$=$	$ 1840 = 2^{4} \cdot 5 \cdot 23 $
Weight:	$ k $	$=$	$ 3 $
Character orbit:	$[\chi]$	$=$	1840.k (of order $2$, degree $1$, not 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:	$50.1363686423$
Analytic rank:	$0$
Dimension:	$16$
Coefficient field:	$\mathbb{Q}[x]/(x^{16} - \cdots)$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{16} - 64 x^{14} - 16 x^{13} + 2252 x^{12} + 648 x^{11} - 30106 x^{10} + 12360 x^{9} + \cdots + 1535848276 $ x^16 - 64x^14 - 16x^13 + 2252x^12 + 648x^11 - 30106x^10 + 12360x^9 + 374528x^8 + 196544x^7 + 1261236x^6 - 4237944x^5 + 38013345x^4 - 8913096x^3 + 327235800x^2 + 72566336x + 1535848276
Coefficient ring:	$\Z[a_1, \ldots, a_{7}]$
Coefficient ring index:	$ 2^{6} $
Twist minimal:	no (minimal twist has level 460)
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label			321.6
Root			$2.47022 + 2.23607i$ of defining polynomial
Character	$\chi$	$=$	1840.321
Dual form			1840.3.k.c.321.5

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-2.47022 q^{3} +2.23607i q^{5} -8.25674i q^{7} -2.89803 q^{9} +O(q^{10})$	$q-2.47022 q^{3} +2.23607i q^{5} -8.25674i q^{7} -2.89803 q^{9} -0.596769i q^{11} -0.219868 q^{13} -5.52357i q^{15} +25.0471i q^{17} +1.81284i q^{19} +20.3959i q^{21} +(22.4752 - 4.88514i) q^{23} -5.00000 q^{25} +29.3907 q^{27} -1.88636 q^{29} -18.8533 q^{31} +1.47415i q^{33} +18.4626 q^{35} +19.4877i q^{37} +0.543122 q^{39} +33.6825 q^{41} -19.3697i q^{43} -6.48019i q^{45} -74.0247 q^{47} -19.1738 q^{49} -61.8717i q^{51} +18.1176i q^{53} +1.33441 q^{55} -4.47811i q^{57} +27.5793 q^{59} -37.6457i q^{61} +23.9283i q^{63} -0.491641i q^{65} -15.5802i q^{67} +(-55.5187 + 12.0674i) q^{69} -49.8789 q^{71} -13.8538 q^{73} +12.3511 q^{75} -4.92737 q^{77} -46.8236i q^{79} -46.5192 q^{81} -11.5849i q^{83} -56.0070 q^{85} +4.65972 q^{87} +32.4824i q^{89} +1.81540i q^{91} +46.5718 q^{93} -4.05363 q^{95} -101.474i q^{97} +1.72945i q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 16 q + 64 q^{9}+O(q^{10}) $ 16 * q + 64 * q^9	$ 16 q + 64 q^{9} - 12 q^{13} + 14 q^{23} - 80 q^{25} - 48 q^{27} + 90 q^{29} - 10 q^{31} - 30 q^{35} - 20 q^{39} + 186 q^{41} + 320 q^{47} + 2 q^{49} + 120 q^{55} + 90 q^{59} - 232 q^{69} + 238 q^{71} - 280 q^{73} + 324 q^{77} + 704 q^{81} - 30 q^{85} - 724 q^{87} - 380 q^{93} - 80 q^{95}+O(q^{100}) $ 16 * q + 64 * q^9 - 12 * q^13 + 14 * q^23 - 80 * q^25 - 48 * q^27 + 90 * q^29 - 10 * q^31 - 30 * q^35 - 20 * q^39 + 186 * q^41 + 320 * q^47 + 2 * q^49 + 120 * q^55 + 90 * q^59 - 232 * q^69 + 238 * q^71 - 280 * q^73 + 324 * q^77 + 704 * q^81 - 30 * q^85 - 724 * q^87 - 380 * q^93 - 80 * q^95

Display 100 coefficients 10 coefficients

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/1840\mathbb{Z}\right)^\times$.

$n$	$737$	$1151$	$1201$	$1381$
$\chi(n)$	$1$	$1$	$-1$	$1$

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
$3$	−2.47022			−0.823406			−0.411703	−	0.911318i	$-0.635066\pi$
$3$	−2.47022			−0.823406			−0.411703	+	0.911318i	$0.635066\pi$
$4$		0			0
$5$			2.23607i			0.447214i
$5$			2.23607i			0.447214i
$6$		0			0
$7$		−	8.25674i		−	1.17953i	−0.807573	−	0.589767i	$-0.799219\pi$
$7$		−	8.25674i		−	1.17953i	0.807573	−	0.589767i	$-0.200781\pi$
$8$		0			0
$9$	−2.89803			−0.322003
$10$		0			0
$11$		−	0.596769i		−	0.0542517i	−0.999632	−	0.0271258i	$-0.991365\pi$
$11$		−	0.596769i		−	0.0542517i	0.999632	−	0.0271258i	$-0.00863548\pi$
$12$		0			0
$13$	−0.219868			−0.0169129			−0.00845647	−	0.999964i	$-0.502692\pi$
$13$	−0.219868			−0.0169129			−0.00845647	+	0.999964i	$0.502692\pi$
$14$		0			0
$15$		−	5.52357i		−	0.368238i
$16$		0			0
$17$			25.0471i			1.47336i	0.676243	+	0.736679i	$0.263607\pi$
$17$			25.0471i			1.47336i	−0.676243	+	0.736679i	$0.736393\pi$
$18$		0			0
$19$			1.81284i			0.0954126i	0.998861	+	0.0477063i	$0.0151912\pi$
$19$			1.81284i			0.0954126i	−0.998861	+	0.0477063i	$0.984809\pi$
$20$		0			0
$21$			20.3959i			0.971236i
$22$		0			0
$23$	22.4752	−	4.88514i	0.977183	−	0.212397i
$23$	22.4752	−	4.88514i	0.977183	−	0.212397i
$24$		0			0
$25$	−5.00000			−0.200000
$26$		0			0
$27$	29.3907			1.08854
$28$		0			0
$29$	−1.88636			−0.0650469			−0.0325235	−	0.999471i	$-0.510354\pi$
$29$	−1.88636			−0.0650469			−0.0325235	+	0.999471i	$0.510354\pi$
$30$		0			0
$31$	−18.8533			−0.608171			−0.304086	−	0.952645i	$-0.598351\pi$
$31$	−18.8533			−0.608171			−0.304086	+	0.952645i	$0.598351\pi$
$32$		0			0
$33$			1.47415i			0.0446711i
$34$		0			0
$35$	18.4626			0.527504
$36$		0			0
$37$			19.4877i			0.526696i	0.964701	+	0.263348i	$0.0848267\pi$
$37$			19.4877i			0.526696i	−0.964701	+	0.263348i	$0.915173\pi$
$38$		0			0
$39$	0.543122			0.0139262
$40$		0			0
$41$	33.6825			0.821524			0.410762	−	0.911743i	$-0.365263\pi$
$41$	33.6825			0.821524			0.410762	+	0.911743i	$0.365263\pi$
$42$		0			0
$43$		−	19.3697i		−	0.450458i	−0.974306	−	0.225229i	$-0.927687\pi$
$43$		−	19.3697i		−	0.450458i	0.974306	−	0.225229i	$-0.0723130\pi$
$44$		0			0
$45$		−	6.48019i		−	0.144004i
$46$		0			0
$47$	−74.0247			−1.57499			−0.787497	−	0.616319i	$-0.788623\pi$
$47$	−74.0247			−1.57499			−0.787497	+	0.616319i	$0.788623\pi$
$48$		0			0
$49$	−19.1738			−0.391303
$50$		0			0
$51$		−	61.8717i		−	1.21317i
$52$		0			0
$53$			18.1176i			0.341841i	0.985285	+	0.170920i	$0.0546741\pi$
$53$			18.1176i			0.341841i	−0.985285	+	0.170920i	$0.945326\pi$
$54$		0			0
$55$	1.33441			0.0242621
$56$		0			0
$57$		−	4.47811i		−	0.0785633i
$58$		0			0
$59$	27.5793			0.467445			0.233723	−	0.972303i	$-0.424909\pi$
$59$	27.5793			0.467445			0.233723	+	0.972303i	$0.424909\pi$
$60$		0			0
$61$		−	37.6457i		−	0.617143i	−0.951201	−	0.308571i	$-0.900149\pi$
$61$		−	37.6457i		−	0.617143i	0.951201	−	0.308571i	$-0.0998508\pi$
$62$		0			0
$63$			23.9283i			0.379814i
$64$		0			0
$65$		−	0.491641i		−	0.00756370i
$66$		0			0
$67$		−	15.5802i		−	0.232541i	−0.993218	−	0.116271i	$-0.962906\pi$
$67$		−	15.5802i		−	0.232541i	0.993218	−	0.116271i	$-0.0370940\pi$
$68$		0			0
$69$	−55.5187	+	12.0674i	−0.804618	+	0.174889i
$70$		0			0
$71$	−49.8789			−0.702519			−0.351260	−	0.936278i	$-0.614247\pi$
$71$	−49.8789			−0.702519			−0.351260	+	0.936278i	$0.614247\pi$
$72$		0			0
$73$	−13.8538			−0.189777			−0.0948887	−	0.995488i	$-0.530250\pi$
$73$	−13.8538			−0.189777			−0.0948887	+	0.995488i	$0.530250\pi$
$74$		0			0
$75$	12.3511			0.164681
$76$		0			0
$77$	−4.92737			−0.0639918
$78$		0			0
$79$		−	46.8236i		−	0.592704i	−0.955079	−	0.296352i	$-0.904230\pi$
$79$		−	46.8236i		−	0.592704i	0.955079	−	0.296352i	$-0.0957701\pi$
$80$		0			0
$81$	−46.5192			−0.574311
$82$		0			0
$83$		−	11.5849i		−	0.139577i	−0.997562	−	0.0697884i	$-0.977768\pi$
$83$		−	11.5849i		−	0.139577i	0.997562	−	0.0697884i	$-0.0222324\pi$
$84$		0			0
$85$	−56.0070			−0.658906
$86$		0			0
$87$	4.65972			0.0535600
$88$		0			0
$89$			32.4824i			0.364970i	0.983209	+	0.182485i	$0.0584142\pi$
$89$			32.4824i			0.364970i	−0.983209	+	0.182485i	$0.941586\pi$
$90$		0			0
$91$			1.81540i			0.0199494i
$92$		0			0
$93$	46.5718			0.500772
$94$		0			0
$95$	−4.05363			−0.0426698
$96$		0			0
$97$		−	101.474i		−	1.04612i	−0.852295	−	0.523062i	$-0.824790\pi$
$97$		−	101.474i		−	1.04612i	0.852295	−	0.523062i	$-0.175210\pi$
$98$		0			0
$99$			1.72945i			0.0174692i
$100$		0			0
$101$	122.925			1.21708			0.608539	−	0.793524i	$-0.291756\pi$
$101$	122.925			1.21708			0.608539	+	0.793524i	$0.291756\pi$
$102$		0			0
$103$			26.7742i			0.259944i	0.991518	+	0.129972i	$0.0414887\pi$
$103$			26.7742i			0.259944i	−0.991518	+	0.129972i	$0.958511\pi$
$104$		0			0
$105$	−45.6067			−0.434350
$106$		0			0
$107$		−	100.317i		−	0.937544i	−0.883319	−	0.468772i	$-0.844697\pi$
$107$		−	100.317i		−	0.937544i	0.883319	−	0.468772i	$-0.155303\pi$
$108$		0			0
$109$		−	148.064i		−	1.35838i	−0.733961	−	0.679192i	$-0.762330\pi$
$109$		−	148.064i		−	1.35838i	0.733961	−	0.679192i	$-0.237670\pi$
$110$		0			0
$111$		−	48.1389i		−	0.433684i
$112$		0			0
$113$			72.5714i			0.642225i	0.947041	+	0.321112i	$0.104057\pi$
$113$			72.5714i			0.642225i	−0.947041	+	0.321112i	$0.895943\pi$
$114$		0			0
$115$	10.9235	+	50.2561i	0.0949870	+	0.437010i
$116$		0			0
$117$	0.637185			0.00544602
$118$		0			0
$119$	206.807			1.73788
$120$		0			0
$121$	120.644			0.997057
$122$		0			0
$123$	−83.2031			−0.676448
$124$		0			0
$125$		−	11.1803i		−	0.0894427i
$126$		0			0
$127$	−188.604			−1.48507			−0.742536	−	0.669806i	$-0.766377\pi$
$127$	−188.604			−1.48507			−0.742536	+	0.669806i	$0.766377\pi$
$128$		0			0
$129$			47.8473i			0.370909i
$130$		0			0
$131$	157.588			1.20297			0.601483	−	0.798886i	$-0.294577\pi$
$131$	157.588			1.20297			0.601483	+	0.798886i	$0.294577\pi$
$132$		0			0
$133$	14.9682			0.112543
$134$		0			0
$135$			65.7196i			0.486812i
$136$		0			0
$137$		−	241.444i		−	1.76237i	−0.472774	−	0.881184i	$-0.656747\pi$
$137$		−	241.444i		−	1.76237i	0.472774	−	0.881184i	$-0.343253\pi$
$138$		0			0
$139$	77.6599			0.558704			0.279352	−	0.960189i	$-0.409880\pi$
$139$	77.6599			0.558704			0.279352	+	0.960189i	$0.409880\pi$
$140$		0			0
$141$	182.857			1.29686
$142$		0			0
$143$			0.131211i			0.000917556i
$144$		0			0
$145$		−	4.21803i		−	0.0290899i
$146$		0			0
$147$	47.3635			0.322201
$148$		0			0
$149$		−	111.378i		−	0.747505i	−0.927529	−	0.373752i	$-0.878071\pi$
$149$		−	111.378i		−	0.747505i	0.927529	−	0.373752i	$-0.121929\pi$
$150$		0			0
$151$	−11.7575			−0.0778641			−0.0389321	−	0.999242i	$-0.512396\pi$
$151$	−11.7575			−0.0778641			−0.0389321	+	0.999242i	$0.512396\pi$
$152$		0			0
$153$		−	72.5872i		−	0.474426i
$154$		0			0
$155$		−	42.1573i		−	0.271982i
$156$		0			0
$157$		−	186.009i		−	1.18477i	−0.805654	−	0.592386i	$-0.798186\pi$
$157$		−	186.009i		−	1.18477i	0.805654	−	0.592386i	$-0.201814\pi$
$158$		0			0
$159$		−	44.7543i		−	0.281474i
$160$		0			0
$161$	−40.3353	−	185.572i	−0.250530	−	1.15262i
$162$		0			0
$163$	−188.029			−1.15355			−0.576775	−	0.816903i	$-0.695689\pi$
$163$	−188.029			−1.15355			−0.576775	+	0.816903i	$0.695689\pi$
$164$		0			0
$165$	−3.29629			−0.0199775
$166$		0			0
$167$	−21.0962			−0.126325			−0.0631624	−	0.998003i	$-0.520119\pi$
$167$	−21.0962			−0.126325			−0.0631624	+	0.998003i	$0.520119\pi$
$168$		0			0
$169$	−168.952			−0.999714
$170$		0			0
$171$		−	5.25366i		−	0.0307232i
$172$		0			0
$173$	244.057			1.41073			0.705366	−	0.708843i	$-0.250783\pi$
$173$	244.057			1.41073			0.705366	+	0.708843i	$0.250783\pi$
$174$		0			0
$175$			41.2837i			0.235907i
$176$		0			0
$177$	−68.1268			−0.384897
$178$		0			0
$179$	204.186			1.14070			0.570351	−	0.821401i	$-0.306807\pi$
$179$	204.186			1.14070			0.570351	+	0.821401i	$0.306807\pi$
$180$		0			0
$181$		−	224.135i		−	1.23832i	−0.785267	−	0.619158i	$-0.787474\pi$
$181$		−	224.135i		−	1.23832i	0.785267	−	0.619158i	$-0.212526\pi$
$182$		0			0
$183$			92.9930i			0.508159i
$184$		0			0
$185$	−43.5759			−0.235545
$186$		0			0
$187$	14.9473			0.0799322
$188$		0			0
$189$		−	242.672i		−	1.28398i
$190$		0			0
$191$		−	117.560i		−	0.615497i	−0.951468	−	0.307749i	$-0.900424\pi$
$191$		−	117.560i		−	0.615497i	0.951468	−	0.307749i	$-0.0995756\pi$
$192$		0			0
$193$	204.891			1.06161			0.530806	−	0.847493i	$-0.321889\pi$
$193$	204.891			1.06161			0.530806	+	0.847493i	$0.321889\pi$
$194$		0			0
$195$			1.21446i			0.00622799i
$196$		0			0
$197$	−125.626			−0.637698			−0.318849	−	0.947806i	$-0.603296\pi$
$197$	−125.626			−0.637698			−0.318849	+	0.947806i	$0.603296\pi$
$198$		0			0
$199$			119.866i			0.602344i	0.953570	+	0.301172i	$0.0973778\pi$
$199$			119.866i			0.602344i	−0.953570	+	0.301172i	$0.902622\pi$
$200$		0			0
$201$			38.4866i			0.191476i
$202$		0			0
$203$			15.5752i			0.0767251i
$204$		0			0
$205$			75.3163i			0.367397i
$206$		0			0
$207$	−65.1338	+	14.1573i	−0.314656	+	0.0683926i
$208$		0			0
$209$	1.08185			0.00517629
$210$		0			0
$211$	−60.5329			−0.286886			−0.143443	−	0.989659i	$-0.545817\pi$
$211$	−60.5329			−0.286886			−0.143443	+	0.989659i	$0.545817\pi$
$212$		0			0
$213$	123.212			0.578458
$214$		0			0
$215$	43.3119			0.201451
$216$		0			0
$217$			155.667i			0.717359i
$218$		0			0
$219$	34.2218			0.156264
$220$		0			0
$221$		−	5.50706i		−	0.0249188i
$222$		0			0
$223$	−44.2442			−0.198405			−0.0992024	−	0.995067i	$-0.531629\pi$
$223$	−44.2442			−0.198405			−0.0992024	+	0.995067i	$0.531629\pi$
$224$		0			0
$225$	14.4901			0.0644006
$226$		0			0
$227$		−	436.822i		−	1.92432i	−0.272478	−	0.962162i	$-0.587843\pi$
$227$		−	436.822i		−	1.92432i	0.272478	−	0.962162i	$-0.412157\pi$
$228$		0			0
$229$		−	165.166i		−	0.721247i	−0.932712	−	0.360623i	$-0.882564\pi$
$229$		−	165.166i		−	0.721247i	0.932712	−	0.360623i	$-0.117436\pi$
$230$		0			0
$231$	12.1717			0.0526912
$232$		0			0
$233$	220.867			0.947928			0.473964	−	0.880544i	$-0.342823\pi$
$233$	220.867			0.947928			0.473964	+	0.880544i	$0.342823\pi$
$234$		0			0
$235$		−	165.524i		−	0.704359i
$236$		0			0
$237$			115.664i			0.488035i
$238$		0			0
$239$	150.110			0.628076			0.314038	−	0.949410i	$-0.398318\pi$
$239$	150.110			0.628076			0.314038	+	0.949410i	$0.398318\pi$
$240$		0			0
$241$		−	264.391i		−	1.09706i	−0.836132	−	0.548529i	$-0.815188\pi$
$241$		−	264.391i		−	1.09706i	0.836132	−	0.548529i	$-0.184812\pi$
$242$		0			0
$243$	−149.604			−0.615654
$244$		0			0
$245$		−	42.8740i		−	0.174996i
$246$		0			0
$247$		−	0.398586i		−	0.00161371i
$248$		0			0
$249$			28.6171i			0.114928i
$250$		0			0
$251$		−	418.511i		−	1.66737i	−0.552237	−	0.833687i	$-0.686226\pi$
$251$		−	418.511i		−	1.66737i	0.552237	−	0.833687i	$-0.313774\pi$
$252$		0			0
$253$	−2.91530	−	13.4125i	−0.0115229	−	0.0530138i
$254$		0			0
$255$	138.349			0.542547
$256$		0			0
$257$	−203.107			−0.790300			−0.395150	−	0.918617i	$-0.629307\pi$
$257$	−203.107			−0.790300			−0.395150	+	0.918617i	$0.629307\pi$
$258$		0			0
$259$	160.905			0.621256
$260$		0			0
$261$	5.46673			0.0209453
$262$		0			0
$263$		−	182.612i		−	0.694344i	−0.937801	−	0.347172i	$-0.887142\pi$
$263$		−	182.612i		−	0.694344i	0.937801	−	0.347172i	$-0.112858\pi$
$264$		0			0
$265$	−40.5121			−0.152876
$266$		0			0
$267$		−	80.2385i		−	0.300519i
$268$		0			0
$269$	−92.2556			−0.342958			−0.171479	−	0.985188i	$-0.554855\pi$
$269$	−92.2556			−0.342958			−0.171479	+	0.985188i	$0.554855\pi$
$270$		0			0
$271$	167.824			0.619277			0.309639	−	0.950854i	$-0.399792\pi$
$271$	167.824			0.619277			0.309639	+	0.950854i	$0.399792\pi$
$272$		0			0
$273$		−	4.48442i		−	0.0164265i
$274$		0			0
$275$			2.98384i			0.0108503i
$276$		0			0
$277$	−130.608			−0.471510			−0.235755	−	0.971812i	$-0.575756\pi$
$277$	−130.608			−0.471510			−0.235755	+	0.971812i	$0.575756\pi$
$278$		0			0
$279$	54.6374			0.195833
$280$		0			0
$281$			465.802i			1.65766i	0.559501	+	0.828830i	$0.310993\pi$
$281$			465.802i			1.65766i	−0.559501	+	0.828830i	$0.689007\pi$
$282$		0			0
$283$		−	400.983i		−	1.41690i	−0.705760	−	0.708451i	$-0.749394\pi$
$283$		−	400.983i		−	1.41690i	0.705760	−	0.708451i	$-0.250606\pi$
$284$		0			0
$285$	10.0134			0.0351346
$286$		0			0
$287$		−	278.108i		−	0.969016i
$288$		0			0
$289$	−338.357			−1.17078
$290$		0			0
$291$			250.663i			0.861384i
$292$		0			0
$293$		−	33.3297i		−	0.113753i	−0.998381	−	0.0568767i	$-0.981886\pi$
$293$		−	33.3297i		−	0.113753i	0.998381	−	0.0568767i	$-0.0181142\pi$
$294$		0			0
$295$			61.6691i			0.209048i
$296$		0			0
$297$		−	17.5395i		−	0.0590554i
$298$		0			0
$299$	−4.94159	+	1.07409i	−0.0165271	+	0.00359227i
$300$		0			0
$301$	−159.931			−0.531331
$302$		0			0
$303$	−303.651			−1.00215
$304$		0			0
$305$	84.1783			0.275995
$306$		0			0
$307$	−408.916			−1.33197			−0.665987	−	0.745963i	$-0.731990\pi$
$307$	−408.916			−1.33197			−0.665987	+	0.745963i	$0.731990\pi$
$308$		0			0
$309$		−	66.1381i		−	0.214039i
$310$		0			0
$311$	−78.9088			−0.253726			−0.126863	−	0.991920i	$-0.540491\pi$
$311$	−78.9088			−0.253726			−0.126863	+	0.991920i	$0.540491\pi$
$312$		0			0
$313$			421.949i			1.34808i	0.738695	+	0.674040i	$0.235442\pi$
$313$			421.949i			1.34808i	−0.738695	+	0.674040i	$0.764558\pi$
$314$		0			0
$315$	−53.5053			−0.169858
$316$		0			0
$317$	2.40226			0.00757810			0.00378905	−	0.999993i	$-0.498794\pi$
$317$	2.40226			0.00757810			0.00378905	+	0.999993i	$0.498794\pi$
$318$		0			0
$319$			1.12572i			0.00352891i
$320$		0			0
$321$			247.805i			0.771979i
$322$		0			0
$323$	−45.4064			−0.140577
$324$		0			0
$325$	1.09934			0.00338259
$326$		0			0
$327$			365.750i			1.11850i
$328$		0			0
$329$			611.203i			1.85776i
$330$		0			0
$331$	369.886			1.11748			0.558739	−	0.829343i	$-0.311285\pi$
$331$	369.886			1.11748			0.558739	+	0.829343i	$0.311285\pi$
$332$		0			0
$333$		−	56.4760i		−	0.169598i
$334$		0			0
$335$	34.8385			0.103996
$336$		0			0
$337$			91.2849i			0.270875i	0.990786	+	0.135438i	$0.0432440\pi$
$337$			91.2849i			0.270875i	−0.990786	+	0.135438i	$0.956756\pi$
$338$		0			0
$339$		−	179.267i		−	0.528812i
$340$		0			0
$341$			11.2511i			0.0329943i
$342$		0			0
$343$		−	246.267i		−	0.717980i
$344$		0			0
$345$	−26.9834	−	124.143i	−0.0782128	−	0.359836i
$346$		0			0
$347$	−357.398			−1.02996			−0.514982	−	0.857201i	$-0.672202\pi$
$347$	−357.398			−1.02996			−0.514982	+	0.857201i	$0.672202\pi$
$348$		0			0
$349$	−158.842			−0.455135			−0.227568	−	0.973762i	$-0.573077\pi$
$349$	−158.842			−0.455135			−0.227568	+	0.973762i	$0.573077\pi$
$350$		0			0
$351$	−6.46209			−0.0184105
$352$		0			0
$353$	125.964			0.356837			0.178419	−	0.983955i	$-0.442902\pi$
$353$	125.964			0.356837			0.178419	+	0.983955i	$0.442902\pi$
$354$		0			0
$355$		−	111.533i		−	0.314176i
$356$		0			0
$357$	−510.859			−1.43098
$358$		0			0
$359$		−	409.017i		−	1.13932i	−0.821879	−	0.569662i	$-0.807074\pi$
$359$		−	409.017i		−	1.13932i	0.821879	−	0.569662i	$-0.192926\pi$
$360$		0			0
$361$	357.714			0.990896
$362$		0			0
$363$	−298.017			−0.820982
$364$		0			0
$365$		−	30.9779i		−	0.0848710i
$366$		0			0
$367$			686.483i			1.87053i	0.353955	+	0.935263i	$0.384837\pi$
$367$			686.483i			1.87053i	−0.353955	+	0.935263i	$0.615163\pi$
$368$		0			0
$369$	−97.6128			−0.264533
$370$		0			0
$371$	149.592			0.403213
$372$		0			0
$373$		−	492.015i		−	1.31907i	−0.751672	−	0.659537i	$-0.770752\pi$
$373$		−	492.015i		−	1.31907i	0.751672	−	0.659537i	$-0.229248\pi$
$374$		0			0
$375$			27.6179i			0.0736476i
$376$		0			0
$377$	0.414751			0.00110014
$378$		0			0
$379$			1.63529i			0.00431474i	0.999998	+	0.00215737i	$0.000686713\pi$
$379$			1.63529i			0.00431474i	−0.999998	+	0.00215737i	$0.999313\pi$
$380$		0			0
$381$	465.893			1.22282
$382$		0			0
$383$		−	223.309i		−	0.583051i	−0.956563	−	0.291526i	$-0.905837\pi$
$383$		−	223.309i		−	0.583051i	0.956563	−	0.291526i	$-0.0941628\pi$
$384$		0			0
$385$		−	11.0179i		−	0.0286180i
$386$		0			0
$387$			56.1339i			0.145049i
$388$		0			0
$389$			296.882i			0.763192i	0.924329	+	0.381596i	$0.124625\pi$
$389$			296.882i			0.763192i	−0.924329	+	0.381596i	$0.875375\pi$
$390$		0			0
$391$	122.359	+	562.939i	0.312937	+	1.43974i
$392$		0			0
$393$	−389.278			−0.990529
$394$		0			0
$395$	104.701			0.265065
$396$		0			0
$397$	−27.3096			−0.0687899			−0.0343950	−	0.999408i	$-0.510950\pi$
$397$	−27.3096			−0.0687899			−0.0343950	+	0.999408i	$0.510950\pi$
$398$		0			0
$399$	−36.9746			−0.0926681
$400$		0			0
$401$			622.315i			1.55191i	0.630789	+	0.775954i	$0.282731\pi$
$401$			622.315i			1.55191i	−0.630789	+	0.775954i	$0.717269\pi$
$402$		0			0
$403$	4.14525			0.0102860
$404$		0			0
$405$		−	104.020i		−	0.256840i
$406$		0			0
$407$	11.6297			0.0285741
$408$		0			0
$409$	679.902			1.66235			0.831175	−	0.556010i	$-0.187668\pi$
$409$	679.902			1.66235			0.831175	+	0.556010i	$0.187668\pi$
$410$		0			0
$411$			596.420i			1.45114i
$412$		0			0
$413$		−	227.715i		−	0.551368i
$414$		0			0
$415$	25.9046			0.0624206
$416$		0			0
$417$	−191.837			−0.460040
$418$		0			0
$419$			538.609i			1.28546i	0.766092	+	0.642731i	$0.222199\pi$
$419$			538.609i			1.28546i	−0.766092	+	0.642731i	$0.777801\pi$
$420$		0			0
$421$			298.554i			0.709154i	0.935027	+	0.354577i	$0.115375\pi$
$421$			298.554i			0.709154i	−0.935027	+	0.354577i	$0.884625\pi$
$422$		0			0
$423$	214.526			0.507153
$424$		0			0
$425$		−	125.235i		−	0.294672i
$426$		0			0
$427$	−310.831			−0.727941
$428$		0			0
$429$		−	0.324118i		−	0.000755521i
$430$		0			0
$431$		−	271.961i		−	0.630999i	−0.948926	−	0.315499i	$-0.897828\pi$
$431$		−	271.961i		−	0.630999i	0.948926	−	0.315499i	$-0.102172\pi$
$432$		0			0
$433$			50.1785i			0.115886i	0.998320	+	0.0579428i	$0.0184541\pi$
$433$			50.1785i			0.115886i	−0.998320	+	0.0579428i	$0.981546\pi$
$434$		0			0
$435$			10.4195i			0.0239528i
$436$		0			0
$437$	8.85597	+	40.7440i	0.0202654	+	0.0932356i
$438$		0			0
$439$	572.063			1.30310			0.651552	−	0.758604i	$-0.274118\pi$
$439$	572.063			1.30310			0.651552	+	0.758604i	$0.274118\pi$
$440$		0			0
$441$	55.5663			0.126001
$442$		0			0
$443$	436.287			0.984848			0.492424	−	0.870356i	$-0.336111\pi$
$443$	436.287			0.984848			0.492424	+	0.870356i	$0.336111\pi$
$444$		0			0
$445$	−72.6328			−0.163220
$446$		0			0
$447$			275.128i			0.615499i
$448$		0			0
$449$	−391.151			−0.871161			−0.435581	−	0.900150i	$-0.643457\pi$
$449$	−391.151			−0.871161			−0.435581	+	0.900150i	$0.643457\pi$
$450$		0			0
$451$		−	20.1006i		−	0.0445691i
$452$		0			0
$453$	29.0435			0.0641138
$454$		0			0
$455$	−4.05935			−0.00892165
$456$		0			0
$457$			148.735i			0.325460i	0.986671	+	0.162730i	$0.0520299\pi$
$457$			148.735i			0.325460i	−0.986671	+	0.162730i	$0.947970\pi$
$458$		0			0
$459$			736.152i			1.60382i
$460$		0			0
$461$	463.228			1.00483			0.502416	−	0.864626i	$-0.332444\pi$
$461$	463.228			1.00483			0.502416	+	0.864626i	$0.332444\pi$
$462$		0			0
$463$	710.802			1.53521			0.767605	−	0.640923i	$-0.221448\pi$
$463$	710.802			1.53521			0.767605	+	0.640923i	$0.221448\pi$
$464$		0			0
$465$			104.138i			0.223952i
$466$		0			0
$467$		−	286.551i		−	0.613600i	−0.951774	−	0.306800i	$-0.900742\pi$
$467$		−	286.551i		−	0.613600i	0.951774	−	0.306800i	$-0.0992582\pi$
$468$		0			0
$469$	−128.642			−0.274290
$470$		0			0
$471$			459.483i			0.975549i
$472$		0			0
$473$	−11.5592			−0.0244381
$474$		0			0
$475$		−	9.06420i		−	0.0190825i
$476$		0			0
$477$		−	52.5052i		−	0.110074i
$478$		0			0
$479$		−	15.6682i		−	0.0327102i	−0.999866	−	0.0163551i	$-0.994794\pi$
$479$		−	15.6682i		−	0.0327102i	0.999866	−	0.0163551i	$-0.00520623\pi$
$480$		0			0
$481$		−	4.28474i		−	0.00890798i
$482$		0			0
$483$	99.6371	+	458.403i	0.206288	+	0.949075i
$484$		0			0
$485$	226.903			0.467841
$486$		0			0
$487$	−845.442			−1.73602			−0.868010	−	0.496546i	$-0.834601\pi$
$487$	−845.442			−1.73602			−0.868010	+	0.496546i	$0.834601\pi$
$488$		0			0
$489$	464.471			0.949839
$490$		0			0
$491$	−22.3768			−0.0455739			−0.0227870	−	0.999740i	$-0.507254\pi$
$491$	−22.3768			−0.0455739			−0.0227870	+	0.999740i	$0.507254\pi$
$492$		0			0
$493$		−	47.2478i		−	0.0958374i
$494$		0			0
$495$	−3.86717			−0.00781247
$496$		0			0
$497$			411.837i			0.828646i
$498$		0			0
$499$	−624.639			−1.25178			−0.625891	−	0.779911i	$-0.715264\pi$
$499$	−624.639			−1.25178			−0.625891	+	0.779911i	$0.715264\pi$
$500$		0			0
$501$	52.1123			0.104017
$502$		0			0
$503$			221.993i			0.441338i	0.975349	+	0.220669i	$0.0708240\pi$
$503$			221.993i			0.441338i	−0.975349	+	0.220669i	$0.929176\pi$
$504$		0			0
$505$			274.869i			0.544294i
$506$		0			0
$507$	417.347			0.823170
$508$		0			0
$509$	122.910			0.241473			0.120737	−	0.992685i	$-0.461474\pi$
$509$	122.910			0.241473			0.120737	+	0.992685i	$0.461474\pi$
$510$		0			0
$511$			114.387i			0.223849i
$512$		0			0
$513$			53.2806i			0.103861i
$514$		0			0
$515$	−59.8690			−0.116250
$516$		0			0
$517$			44.1756i			0.0854461i
$518$		0			0
$519$	−602.873			−1.16160
$520$		0			0
$521$			370.245i			0.710643i	0.934744	+	0.355321i	$0.115629\pi$
$521$			370.245i			0.710643i	−0.934744	+	0.355321i	$0.884371\pi$
$522$		0			0
$523$		−	149.485i		−	0.285823i	−0.989735	−	0.142912i	$-0.954354\pi$
$523$		−	149.485i		−	0.285823i	0.989735	−	0.142912i	$-0.0456464\pi$
$524$		0			0
$525$		−	101.980i		−	0.194247i
$526$		0			0
$527$		−	472.221i		−	0.896054i
$528$		0			0
$529$	481.271	−	219.589i	0.909775	−	0.415102i
$530$		0			0
$531$	−79.9255			−0.150519
$532$		0			0
$533$	−7.40571			−0.0138944
$534$		0			0
$535$	224.316			0.419283
$536$		0			0
$537$	−504.383			−0.939261
$538$		0			0
$539$			11.4423i			0.0212288i
$540$		0			0
$541$	222.304			0.410913			0.205456	−	0.978666i	$-0.434132\pi$
$541$	222.304			0.410913			0.205456	+	0.978666i	$0.434132\pi$
$542$		0			0
$543$			553.662i			1.01964i
$544$		0			0
$545$	331.081			0.607488
$546$		0			0
$547$	−289.014			−0.528362			−0.264181	−	0.964473i	$-0.585102\pi$
$547$	−289.014			−0.528362			−0.264181	+	0.964473i	$0.585102\pi$
$548$		0			0
$549$			109.098i			0.198722i
$550$		0			0
$551$		−	3.41967i		−	0.00620630i
$552$		0			0
$553$	−386.610			−0.699115
$554$		0			0
$555$	107.642			0.193949
$556$		0			0
$557$		−	976.361i		−	1.75289i	−0.481500	−	0.876446i	$-0.659908\pi$
$557$		−	976.361i		−	1.75289i	0.481500	−	0.876446i	$-0.340092\pi$
$558$		0			0
$559$			4.25878i			0.00761857i
$560$		0			0
$561$	−36.9231			−0.0658166
$562$		0			0
$563$			203.128i			0.360797i	0.983594	+	0.180398i	$0.0577387\pi$
$563$			203.128i			0.360797i	−0.983594	+	0.180398i	$0.942261\pi$
$564$		0			0
$565$	−162.275			−0.287212
$566$		0			0
$567$			384.097i			0.677420i
$568$		0			0
$569$		−	54.0125i		−	0.0949254i	−0.998873	−	0.0474627i	$-0.984886\pi$
$569$		−	54.0125i		−	0.0949254i	0.998873	−	0.0474627i	$-0.0151135\pi$
$570$		0			0
$571$			782.388i			1.37021i	0.728446	+	0.685103i	$0.240243\pi$
$571$			782.388i			1.37021i	−0.728446	+	0.685103i	$0.759757\pi$
$572$		0			0
$573$			290.399i			0.506804i
$574$		0			0
$575$	−112.376	+	24.4257i	−0.195437	+	0.0424795i
$576$		0			0
$577$	−161.020			−0.279064			−0.139532	−	0.990218i	$-0.544560\pi$
$577$	−161.020			−0.279064			−0.139532	+	0.990218i	$0.544560\pi$
$578$		0			0
$579$	−506.126			−0.874137
$580$		0			0
$581$	−95.6533			−0.164636
$582$		0			0
$583$	10.8120			0.0185454
$584$		0			0
$585$			1.42479i			0.00243554i
$586$		0			0
$587$	340.427			0.579943			0.289972	−	0.957035i	$-0.406354\pi$
$587$	340.427			0.579943			0.289972	+	0.957035i	$0.406354\pi$
$588$		0			0
$589$		−	34.1780i		−	0.0580272i
$590$		0			0
$591$	310.325			0.525084
$592$		0			0
$593$	−823.560			−1.38880			−0.694401	−	0.719588i	$-0.744331\pi$
$593$	−823.560			−1.38880			−0.694401	+	0.719588i	$0.744331\pi$
$594$		0			0
$595$			462.435i			0.777202i
$596$		0			0
$597$		−	296.096i		−	0.495973i
$598$		0			0
$599$	−487.521			−0.813892			−0.406946	−	0.913452i	$-0.633406\pi$
$599$	−487.521			−0.813892			−0.406946	+	0.913452i	$0.633406\pi$
$600$		0			0
$601$	−600.376			−0.998962			−0.499481	−	0.866325i	$-0.666476\pi$
$601$	−600.376			−0.998962			−0.499481	+	0.866325i	$0.666476\pi$
$602$		0			0
$603$			45.1520i			0.0748790i
$604$		0			0
$605$			269.768i			0.445897i
$606$		0			0
$607$	−440.562			−0.725802			−0.362901	−	0.931828i	$-0.618214\pi$
$607$	−440.562			−0.725802			−0.362901	+	0.931828i	$0.618214\pi$
$608$		0			0
$609$		−	38.4741i		−	0.0631759i
$610$		0			0
$611$	16.2757			0.0266378
$612$		0			0
$613$		−	552.134i		−	0.900708i	−0.892850	−	0.450354i	$-0.851298\pi$
$613$		−	552.134i		−	0.900708i	0.892850	−	0.450354i	$-0.148702\pi$
$614$		0			0
$615$		−	186.048i		−	0.302517i
$616$		0			0
$617$			824.577i			1.33643i	0.743968	+	0.668215i	$0.232941\pi$
$617$			824.577i			1.33643i	−0.743968	+	0.668215i	$0.767059\pi$
$618$		0			0
$619$		−	87.5843i		−	0.141493i	−0.997494	−	0.0707466i	$-0.977462\pi$
$619$		−	87.5843i		−	0.141493i	0.997494	−	0.0707466i	$-0.0225382\pi$
$620$		0			0
$621$	660.563	−	143.578i	1.06371	−	0.231204i
$622$		0			0
$623$	268.199			0.430495
$624$		0			0
$625$	25.0000			0.0400000
$626$		0			0
$627$	−2.67239			−0.00426219
$628$		0			0
$629$	−488.111			−0.776011
$630$		0			0
$631$			73.4341i			0.116377i	0.998306	+	0.0581887i	$0.0185325\pi$
$631$			73.4341i			0.116377i	−0.998306	+	0.0581887i	$0.981468\pi$
$632$		0			0
$633$	149.529			0.236223
$634$		0			0
$635$		−	421.732i		−	0.664145i
$636$		0			0
$637$	4.21572			0.00661808
$638$		0			0
$639$	144.550			0.226213
$640$		0			0
$641$		−	1027.28i		−	1.60262i	−0.598252	−	0.801308i	$-0.704138\pi$
$641$		−	1027.28i		−	1.60262i	0.598252	−	0.801308i	$-0.295862\pi$
$642$		0			0
$643$			860.413i			1.33812i	0.743207	+	0.669062i	$0.233304\pi$
$643$			860.413i			1.33812i	−0.743207	+	0.669062i	$0.766696\pi$
$644$		0			0
$645$	−106.990			−0.165876
$646$		0			0
$647$	−349.194			−0.539713			−0.269856	−	0.962901i	$-0.586976\pi$
$647$	−349.194			−0.539713			−0.269856	+	0.962901i	$0.586976\pi$
$648$		0			0
$649$		−	16.4584i		−	0.0253597i
$650$		0			0
$651$		−	384.531i		−	0.590678i
$652$		0			0
$653$	−515.430			−0.789326			−0.394663	−	0.918826i	$-0.629139\pi$
$653$	−515.430			−0.789326			−0.394663	+	0.918826i	$0.629139\pi$
$654$		0			0
$655$			352.379i			0.537983i
$656$		0			0
$657$	40.1486			0.0611089
$658$		0			0
$659$			81.1143i			0.123087i	0.998104	+	0.0615435i	$0.0196023\pi$
$659$			81.1143i			0.123087i	−0.998104	+	0.0615435i	$0.980398\pi$
$660$		0			0
$661$			213.989i			0.323735i	0.986812	+	0.161868i	$0.0517518\pi$
$661$			213.989i			0.323735i	−0.986812	+	0.161868i	$0.948248\pi$
$662$		0			0
$663$			13.6036i			0.0205183i
$664$		0			0
$665$			33.4698i			0.0503305i
$666$		0			0
$667$	−42.3964	+	9.21514i	−0.0635628	+	0.0138158i
$668$		0			0
$669$	109.293			0.163368
$670$		0			0
$671$	−22.4658			−0.0334810
$672$		0			0
$673$	−573.388			−0.851988			−0.425994	−	0.904726i	$-0.640076\pi$
$673$	−573.388			−0.851988			−0.425994	+	0.904726i	$0.640076\pi$
$674$		0			0
$675$	−146.954			−0.217709
$676$		0			0
$677$		−	798.023i		−	1.17876i	−0.807855	−	0.589382i	$-0.799371\pi$
$677$		−	798.023i		−	1.17876i	0.807855	−	0.589382i	$-0.200629\pi$
$678$		0			0
$679$	−837.845			−1.23394
$680$		0			0
$681$			1079.04i			1.58450i
$682$		0			0
$683$	808.667			1.18399			0.591997	−	0.805940i	$-0.298340\pi$
$683$	808.667			1.18399			0.591997	+	0.805940i	$0.298340\pi$
$684$		0			0
$685$	539.886			0.788155
$686$		0			0
$687$			407.995i			0.593879i
$688$		0			0
$689$		−	3.98348i		−	0.00578154i
$690$		0			0
$691$	1127.48			1.63167			0.815834	−	0.578286i	$-0.196278\pi$
$691$	1127.48			1.63167			0.815834	+	0.578286i	$0.196278\pi$
$692$		0			0
$693$	14.2796			0.0206056
$694$		0			0
$695$			173.653i			0.249860i
$696$		0			0
$697$			843.648i			1.21040i
$698$		0			0
$699$	−545.590			−0.780529
$700$		0			0
$701$		−	1055.92i		−	1.50631i	−0.657845	−	0.753153i	$-0.728532\pi$
$701$		−	1055.92i		−	1.50631i	0.657845	−	0.753153i	$-0.271468\pi$
$702$		0			0
$703$	−35.3281			−0.0502534
$704$		0			0
$705$			408.881i			0.579973i
$706$		0			0
$707$		−	1014.96i		−	1.43559i
$708$		0			0
$709$		−	39.7402i		−	0.0560510i	−0.999607	−	0.0280255i	$-0.991078\pi$
$709$		−	39.7402i		−	0.0560510i	0.999607	−	0.0280255i	$-0.00892196\pi$
$710$		0			0
$711$			135.696i			0.190852i
$712$		0			0
$713$	−423.732	+	92.1011i	−0.594295	+	0.129174i
$714$		0			0
$715$	−0.293396			−0.000410344
$716$		0			0
$717$	−370.805			−0.517161
$718$		0			0
$719$	−1158.15			−1.61078			−0.805389	−	0.592746i	$-0.798044\pi$
$719$	−1158.15			−1.61078			−0.805389	+	0.592746i	$0.798044\pi$
$720$		0			0
$721$	221.068			0.306613
$722$		0			0
$723$			653.103i			0.903324i
$724$		0			0
$725$	9.43180			0.0130094
$726$		0			0
$727$		−	180.994i		−	0.248960i	−0.992222	−	0.124480i	$-0.960274\pi$
$727$		−	180.994i		−	0.248960i	0.992222	−	0.124480i	$-0.0397262\pi$
$728$		0			0
$729$	788.227			1.08124
$730$		0			0
$731$	485.154			0.663686
$732$		0			0
$733$		−	334.983i		−	0.457003i	−0.973544	−	0.228501i	$-0.926617\pi$
$733$		−	334.983i		−	0.457003i	0.973544	−	0.228501i	$-0.0733825\pi$
$734$		0			0
$735$			105.908i			0.144093i
$736$		0			0
$737$	−9.29780			−0.0126157
$738$		0			0
$739$	−959.742			−1.29870			−0.649352	−	0.760488i	$-0.724960\pi$
$739$	−959.742			−1.29870			−0.649352	+	0.760488i	$0.724960\pi$
$740$		0			0
$741$			0.984594i			0.00132874i
$742$		0			0
$743$		−	705.966i		−	0.950156i	−0.879944	−	0.475078i	$-0.842420\pi$
$743$		−	705.966i		−	0.950156i	0.879944	−	0.475078i	$-0.157580\pi$
$744$		0			0
$745$	249.049			0.334294
$746$		0			0
$747$			33.5733i			0.0449442i
$748$		0			0
$749$	−828.294			−1.10587
$750$		0			0
$751$		−	1480.59i		−	1.97149i	−0.168241	−	0.985746i	$-0.553809\pi$
$751$		−	1480.59i		−	1.97149i	0.168241	−	0.985746i	$-0.446191\pi$
$752$		0			0
$753$			1033.81i			1.37293i
$754$		0			0
$755$		−	26.2905i		−	0.0348219i
$756$		0			0
$757$			911.552i			1.20416i	0.798434	+	0.602082i	$0.205662\pi$
$757$			911.552i			1.20416i	−0.798434	+	0.602082i	$0.794338\pi$
$758$		0			0
$759$	7.20142	+	33.1318i	0.00948803	+	0.0436519i
$760$		0			0
$761$	1129.29			1.48396			0.741979	−	0.670423i	$-0.233887\pi$
$761$	1129.29			1.48396			0.741979	+	0.670423i	$0.233887\pi$
$762$		0			0
$763$	−1222.53			−1.60226
$764$		0			0
$765$	162.310			0.212170
$766$		0			0
$767$	−6.06381			−0.00790588
$768$		0			0
$769$		−	264.516i		−	0.343974i	−0.985099	−	0.171987i	$-0.944981\pi$
$769$		−	264.516i		−	0.343974i	0.985099	−	0.171987i	$-0.0550187\pi$
$770$		0			0
$771$	501.718			0.650737
$772$		0			0
$773$			1379.04i			1.78401i	0.452027	+	0.892004i	$0.350701\pi$
$773$			1379.04i			1.78401i	−0.452027	+	0.892004i	$0.649299\pi$
$774$		0			0
$775$	94.2666			0.121634
$776$		0			0
$777$	−397.471			−0.511546
$778$		0			0
$779$			61.0609i			0.0783838i
$780$		0			0
$781$			29.7661i			0.0381128i
$782$		0			0
$783$	−55.4415			−0.0708065
$784$		0			0
$785$	415.929			0.529846
$786$		0			0
$787$			1080.69i			1.37318i	0.727045	+	0.686590i	$0.240893\pi$
$787$			1080.69i			1.37318i	−0.727045	+	0.686590i	$0.759107\pi$
$788$		0			0
$789$			451.092i			0.571727i
$790$		0			0
$791$	599.204			0.757527
$792$		0			0
$793$			8.27710i			0.0104377i
$794$		0			0
$795$	100.074			0.125879
$796$		0			0
$797$		−	673.128i		−	0.844577i	−0.906462	−	0.422288i	$-0.861227\pi$
$797$		−	673.128i		−	0.844577i	0.906462	−	0.422288i	$-0.138773\pi$
$798$		0			0
$799$		−	1854.10i		−	2.32053i
$800$		0			0
$801$		−	94.1349i		−	0.117522i
$802$		0			0
$803$			8.26748i			0.0102957i
$804$		0			0
$805$	414.952	−	90.1926i	0.515468	−	0.112040i
$806$		0			0
$807$	227.891			0.282393
$808$		0			0
$809$	−662.758			−0.819231			−0.409616	−	0.912258i	$-0.634337\pi$
$809$	−662.758			−0.819231			−0.409616	+	0.912258i	$0.634337\pi$
$810$		0			0
$811$	1286.90			1.58681			0.793405	−	0.608694i	$-0.208306\pi$
$811$	1286.90			1.58681			0.793405	+	0.608694i	$0.208306\pi$
$812$		0			0
$813$	−414.562			−0.509916
$814$		0			0
$815$		−	420.445i		−	0.515883i
$816$		0			0
$817$	35.1141			0.0429793
$818$		0			0
$819$		−	5.26107i		−	0.00642378i
$820$		0			0
$821$	−90.1269			−0.109777			−0.0548885	−	0.998492i	$-0.517480\pi$
$821$	−90.1269			−0.109777			−0.0548885	+	0.998492i	$0.517480\pi$
$822$		0			0
$823$	1014.08			1.23218			0.616088	−	0.787677i	$-0.288717\pi$
$823$	1014.08			1.23218			0.616088	+	0.787677i	$0.288717\pi$
$824$		0			0
$825$		−	7.37074i		−	0.00893423i
$826$		0			0
$827$			310.832i			0.375855i	0.982183	+	0.187928i	$0.0601770\pi$
$827$			310.832i			0.375855i	−0.982183	+	0.187928i	$0.939823\pi$
$828$		0			0
$829$	804.117			0.969985			0.484992	−	0.874518i	$-0.338822\pi$
$829$	804.117			0.969985			0.484992	+	0.874518i	$0.338822\pi$
$830$		0			0
$831$	322.631			0.388244
$832$		0			0
$833$		−	480.249i		−	0.576529i
$834$		0			0
$835$		−	47.1726i		−	0.0564942i
$836$		0			0
$837$	−554.112			−0.662022
$838$		0			0
$839$		−	1075.82i		−	1.28227i	−0.767430	−	0.641133i	$-0.778465\pi$
$839$		−	1075.82i		−	1.28227i	0.767430	−	0.641133i	$-0.221535\pi$
$840$		0			0
$841$	−837.442			−0.995769
$842$		0			0
$843$		−	1150.63i		−	1.36493i
$844$		0			0
$845$		−	377.787i		−	0.447086i
$846$		0			0
$847$		−	996.126i		−	1.17606i
$848$		0			0
$849$			990.515i			1.16668i
$850$		0			0
$851$	95.2003	+	437.991i	0.111869	+	0.514678i
$852$		0			0
$853$	−127.475			−0.149443			−0.0747214	−	0.997204i	$-0.523807\pi$
$853$	−127.475			−0.149443			−0.0747214	+	0.997204i	$0.523807\pi$
$854$		0			0
$855$	11.7475			0.0137398
$856$		0			0
$857$	−618.354			−0.721533			−0.360767	−	0.932656i	$-0.617485\pi$
$857$	−618.354			−0.721533			−0.360767	+	0.932656i	$0.617485\pi$
$858$		0			0
$859$	745.538			0.867914			0.433957	−	0.900934i	$-0.357117\pi$
$859$	745.538			0.867914			0.433957	+	0.900934i	$0.357117\pi$
$860$		0			0
$861$			686.986i			0.797894i
$862$		0			0
$863$	−64.5791			−0.0748310			−0.0374155	−	0.999300i	$-0.511912\pi$
$863$	−64.5791			−0.0748310			−0.0374155	+	0.999300i	$0.511912\pi$
$864$		0			0
$865$			545.727i			0.630899i
$866$		0			0
$867$	835.814			0.964030
$868$		0			0
$869$	−27.9428			−0.0321552
$870$		0			0
$871$			3.42560i			0.00393295i
$872$		0			0
$873$			294.075i			0.336855i
$874$		0			0
$875$	−92.3132			−0.105501
$876$		0			0
$877$	601.465			0.685821			0.342910	−	0.939368i	$-0.388587\pi$
$877$	601.465			0.685821			0.342910	+	0.939368i	$0.388587\pi$
$878$		0			0
$879$			82.3317i			0.0936652i
$880$		0			0
$881$			863.232i			0.979832i	0.871770	+	0.489916i	$0.162973\pi$
$881$			863.232i			0.979832i	−0.871770	+	0.489916i	$0.837027\pi$
$882$		0			0
$883$	−933.043			−1.05667			−0.528337	−	0.849035i	$-0.677184\pi$
$883$	−933.043			−1.05667			−0.528337	+	0.849035i	$0.677184\pi$
$884$		0			0
$885$		−	152.336i		−	0.172131i
$886$		0			0
$887$	55.1069			0.0621273			0.0310637	−	0.999517i	$-0.490111\pi$
$887$	55.1069			0.0621273			0.0310637	+	0.999517i	$0.490111\pi$
$888$		0			0
$889$			1557.26i			1.75169i
$890$		0			0
$891$			27.7612i			0.0311573i
$892$		0			0
$893$		−	134.195i		−	0.150274i
$894$		0			0
$895$			456.573i			0.510138i
$896$		0			0
$897$	12.2068	−	2.65323i	0.0136085	−	0.00295789i
$898$		0			0
$899$	35.5642			0.0395597
$900$		0			0
$901$	−453.792			−0.503654
$902$		0			0
$903$	395.063			0.437501
$904$		0			0
$905$	501.181			0.553791
$906$		0			0
$907$			1046.63i			1.15395i	0.816763	+	0.576974i	$0.195766\pi$
$907$			1046.63i			1.15395i	−0.816763	+	0.576974i	$0.804234\pi$
$908$		0			0
$909$	−356.240			−0.391903
$910$		0			0
$911$		−	842.705i		−	0.925033i	−0.886611	−	0.462516i	$-0.846947\pi$
$911$		−	842.705i		−	0.925033i	0.886611	−	0.462516i	$-0.153053\pi$
$912$		0			0
$913$	−6.91349			−0.00757228
$914$		0			0
$915$	−207.939			−0.227255
$916$		0			0
$917$		−	1301.17i		−	1.41894i
$918$		0			0
$919$			981.227i			1.06771i	0.845576	+	0.533856i	$0.179257\pi$
$919$			981.227i			1.06771i	−0.845576	+	0.533856i	$0.820743\pi$
$920$		0			0
$921$	1010.11			1.09676
$922$		0			0
$923$	10.9668			0.0118817
$924$		0			0
$925$		−	97.4387i		−	0.105339i
$926$		0			0
$927$		−	77.5925i		−	0.0837028i
$928$		0			0
$929$	745.480			0.802454			0.401227	−	0.915979i	$-0.368584\pi$
$929$	745.480			0.802454			0.401227	+	0.915979i	$0.368584\pi$
$930$		0			0
$931$		−	34.7591i		−	0.0373352i
$932$		0			0
$933$	194.922			0.208920
$934$		0			0
$935$			33.4232i			0.0357467i
$936$		0			0
$937$			847.258i			0.904224i	0.891961	+	0.452112i	$0.149329\pi$
$937$			847.258i			0.904224i	−0.891961	+	0.452112i	$0.850671\pi$
$938$		0			0
$939$		−	1042.31i		−	1.11002i
$940$		0			0
$941$		−	1395.97i		−	1.48349i	−0.670681	−	0.741746i	$-0.733998\pi$
$941$		−	1395.97i		−	1.48349i	0.670681	−	0.741746i	$-0.266002\pi$
$942$		0			0
$943$	757.021	−	164.544i	0.802780	−	0.174490i
$944$		0			0
$945$	542.630			0.574212
$946$		0			0
$947$	54.4962			0.0575461			0.0287730	−	0.999586i	$-0.490840\pi$
$947$	54.4962			0.0575461			0.0287730	+	0.999586i	$0.490840\pi$
$948$		0			0
$949$	3.04600			0.00320970
$950$		0			0
$951$	−5.93409			−0.00623985
$952$		0			0
$953$			126.158i			0.132380i	0.997807	+	0.0661901i	$0.0210844\pi$
$953$			126.158i			0.132380i	−0.997807	+	0.0661901i	$0.978916\pi$
$954$		0			0
$955$	262.872			0.275259
$956$		0			0
$957$		−	2.78077i		−	0.00290572i
$958$		0			0
$959$	−1993.54			−2.07877
$960$		0			0
$961$	−605.553			−0.630128
$962$		0			0
$963$			290.722i			0.301892i
$964$		0			0
$965$			458.151i			0.474767i
$966$		0			0
$967$	−462.481			−0.478263			−0.239132	−	0.970987i	$-0.576863\pi$
$967$	−462.481			−0.478263			−0.239132	+	0.970987i	$0.576863\pi$
$968$		0			0
$969$	112.164			0.115752
$970$		0			0
$971$			737.368i			0.759390i	0.925112	+	0.379695i	$0.123971\pi$
$971$			737.368i			0.759390i	−0.925112	+	0.379695i	$0.876029\pi$
$972$		0			0
$973$		−	641.218i		−	0.659011i
$974$		0			0
$975$	−2.71561			−0.00278524
$976$		0			0
$977$		−	915.226i		−	0.936771i	−0.883524	−	0.468386i	$-0.844836\pi$
$977$		−	915.226i		−	0.936771i	0.883524	−	0.468386i	$-0.155164\pi$
$978$		0			0
$979$	19.3845			0.0198003
$980$		0			0
$981$			429.093i			0.437404i
$982$		0			0
$983$		−	674.233i		−	0.685893i	−0.939355	−	0.342946i	$-0.888575\pi$
$983$		−	674.233i		−	0.685893i	0.939355	−	0.342946i	$-0.111425\pi$
$984$		0			0
$985$		−	280.909i		−	0.285187i
$986$		0			0
$987$		−	1509.80i		−	1.52969i
$988$		0			0
$989$	−94.6236	−	435.338i	−0.0956760	−	0.440180i
$990$		0			0
$991$	970.677			0.979493			0.489746	−	0.871865i	$-0.337089\pi$
$991$	970.677			0.979493			0.489746	+	0.871865i	$0.337089\pi$
$992$		0			0
$993$	−913.697			−0.920138
$994$		0			0
$995$	−268.029			−0.269376
$996$		0			0
$997$	653.389			0.655355			0.327678	−	0.944790i	$-0.393734\pi$
$997$	653.389			0.655355			0.327678	+	0.944790i	$0.393734\pi$
$998$		0			0
$999$			572.758i			0.573332i

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	1840.3.k.c.321.6		16
4.3	odd	2		460.3.f.a.321.12	yes	16
12.11	even	2		4140.3.d.a.2161.8		16
20.3	even	4		2300.3.d.b.1149.28		32
20.7	even	4		2300.3.d.b.1149.5		32
20.19	odd	2		2300.3.f.e.1701.5		16
23.22	odd	2	inner	1840.3.k.c.321.5		16
92.91	even	2		460.3.f.a.321.11	✓	16
276.275	odd	2		4140.3.d.a.2161.9		16
460.183	odd	4		2300.3.d.b.1149.6		32
460.367	odd	4		2300.3.d.b.1149.27		32
460.459	even	2		2300.3.f.e.1701.6		16

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
460.3.f.a.321.11	✓	16	92.91	even	2
460.3.f.a.321.12	yes	16	4.3	odd	2
1840.3.k.c.321.5		16	23.22	odd	2	inner
1840.3.k.c.321.6		16	1.1	even	1	trivial
2300.3.d.b.1149.5		32	20.7	even	4
2300.3.d.b.1149.6		32	460.183	odd	4
2300.3.d.b.1149.27		32	460.367	odd	4
2300.3.d.b.1149.28		32	20.3	even	4
2300.3.f.e.1701.5		16	20.19	odd	2
2300.3.f.e.1701.6		16	460.459	even	2
4140.3.d.a.2161.8		16	12.11	even	2
4140.3.d.a.2161.9		16	276.275	odd	2

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:	\( N \)	\(=\)	\( 1840 = 2^{4} \cdot 5 \cdot 23 \)
Weight:	\( k \)	\(=\)	\( 3 \)
Character orbit:	\([\chi]\)	\(=\)	1840.k (of order \(2\), degree \(1\), not minimal)

\(f(q)\)	\(=\)	\(q-2.47022 q^{3} +2.23607i q^{5} -8.25674i q^{7} -2.89803 q^{9} +O(q^{10})\)	\(q-2.47022 q^{3} +2.23607i q^{5} -8.25674i q^{7} -2.89803 q^{9} -0.596769i q^{11} -0.219868 q^{13} -5.52357i q^{15} +25.0471i q^{17} +1.81284i q^{19} +20.3959i q^{21} +(22.4752 - 4.88514i) q^{23} -5.00000 q^{25} +29.3907 q^{27} -1.88636 q^{29} -18.8533 q^{31} +1.47415i q^{33} +18.4626 q^{35} +19.4877i q^{37} +0.543122 q^{39} +33.6825 q^{41} -19.3697i q^{43} -6.48019i q^{45} -74.0247 q^{47} -19.1738 q^{49} -61.8717i q^{51} +18.1176i q^{53} +1.33441 q^{55} -4.47811i q^{57} +27.5793 q^{59} -37.6457i q^{61} +23.9283i q^{63} -0.491641i q^{65} -15.5802i q^{67} +(-55.5187 + 12.0674i) q^{69} -49.8789 q^{71} -13.8538 q^{73} +12.3511 q^{75} -4.92737 q^{77} -46.8236i q^{79} -46.5192 q^{81} -11.5849i q^{83} -56.0070 q^{85} +4.65972 q^{87} +32.4824i q^{89} +1.81540i q^{91} +46.5718 q^{93} -4.05363 q^{95} -101.474i q^{97} +1.72945i q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 16 q + 64 q^{9}+O(q^{10}) \) 16 * q + 64 * q^9	\( 16 q + 64 q^{9} - 12 q^{13} + 14 q^{23} - 80 q^{25} - 48 q^{27} + 90 q^{29} - 10 q^{31} - 30 q^{35} - 20 q^{39} + 186 q^{41} + 320 q^{47} + 2 q^{49} + 120 q^{55} + 90 q^{59} - 232 q^{69} + 238 q^{71} - 280 q^{73} + 324 q^{77} + 704 q^{81} - 30 q^{85} - 724 q^{87} - 380 q^{93} - 80 q^{95}+O(q^{100}) \) 16 * q + 64 * q^9 - 12 * q^13 + 14 * q^23 - 80 * q^25 - 48 * q^27 + 90 * q^29 - 10 * q^31 - 30 * q^35 - 20 * q^39 + 186 * q^41 + 320 * q^47 + 2 * q^49 + 120 * q^55 + 90 * q^59 - 232 * q^69 + 238 * q^71 - 280 * q^73 + 324 * q^77 + 704 * q^81 - 30 * q^85 - 724 * q^87 - 380 * q^93 - 80 * q^95

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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