LMFDB - Embedded newform 1764.4.t.b.521.8

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [1764,4,Mod(521,1764)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(1764, base_ring=CyclotomicField(6))

chi = DirichletCharacter(H, H._module([0, 3, 1]))

N = Newforms(chi, 4, names="a")

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

chi := DirichletCharacter("1764.521");

S:= CuspForms(chi, 4);

N := Newforms(S);

Level:	$ N $	$=$	$ 1764 = 2^{2} \cdot 3^{2} \cdot 7^{2} $
Weight:	$ k $	$=$	$ 4 $
Character orbit:	$[\chi]$	$=$	1764.t (of order $6$, degree $2$, 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:	$104.079369250$
Analytic rank:	$0$
Dimension:	$16$
Relative dimension:	$8$ over $\Q(\zeta_{6})$
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} - 4 x^{15} - 290 x^{14} + 1728 x^{13} + 29275 x^{12} - 246984 x^{11} - 955194 x^{10} + \cdots + 7375227456 $ x^16 - 4x^15 - 290x^14 + 1728x^13 + 29275x^12 - 246984x^11 - 955194x^10 + 14344616x^9 - 18123280x^8 - 273588032x^7 + 1239640536x^6 - 1407381792x^5 - 1961185792x^4 + 4297169408x^3 + 2991779296x^2 - 11217342336*x + 7375227456
Coefficient ring:	$\Z[a_1, \ldots, a_{19}]$
Coefficient ring index:	$ 2^{6}\cdot 3^{18} $
Twist minimal:	no (minimal twist has level 252)
Sato-Tate group:	$\mathrm{SU}(2)[C_{6}]$

Embedding invariants

Embedding label			521.8
Root			$2.20073 + 0.707107i$ of defining polynomial
Character	$\chi$	$=$	1764.521
Dual form			1764.4.t.b.1097.8

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+(10.1259 + 17.5386i) q^{5} +O(q^{10})$	$q+(10.1259 + 17.5386i) q^{5} +(-15.4532 - 8.92191i) q^{11} -33.1829i q^{13} +(22.9077 - 39.6772i) q^{17} +(35.0091 - 20.2125i) q^{19} +(-69.7804 + 40.2877i) q^{23} +(-142.568 + 246.935i) q^{25} -233.844i q^{29} +(195.398 + 112.813i) q^{31} +(135.456 + 234.617i) q^{37} +154.432 q^{41} +367.102 q^{43} +(263.852 + 457.006i) q^{47} +(78.8892 + 45.5467i) q^{53} -361.370i q^{55} +(312.768 - 541.729i) q^{59} +(78.8343 - 45.5150i) q^{61} +(581.982 - 336.007i) q^{65} +(-431.509 + 747.395i) q^{67} +303.596i q^{71} +(999.127 + 576.846i) q^{73} +(3.48120 + 6.02961i) q^{79} -815.694 q^{83} +927.844 q^{85} +(-155.494 - 269.323i) q^{89} +(708.998 + 409.340i) q^{95} +1832.92i q^{97} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 16 q+O(q^{10}) $ 16 * q	$ 16 q + 72 q^{19} - 212 q^{25} + 708 q^{31} + 76 q^{37} + 1408 q^{43} + 1632 q^{61} - 1528 q^{67} + 2700 q^{73} - 364 q^{79} + 7392 q^{85}+O(q^{100}) $ 16 * q + 72 * q^19 - 212 * q^25 + 708 * q^31 + 76 * q^37 + 1408 * q^43 + 1632 * q^61 - 1528 * q^67 + 2700 * q^73 - 364 * q^79 + 7392 * q^85

Display 100 coefficients 10 coefficients

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/1764\mathbb{Z}\right)^\times$.

$n$	$785$	$883$	$1081$
$\chi(n)$	$-1$	$1$	$e\left(\frac{1}{6}\right)$

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
$3$		0			0
$3$		0			0
$4$		0			0
$5$	10.1259	+	17.5386i	0.905689	+	1.56870i	0.819990	+	0.572378i	$0.193979\pi$
$5$	10.1259	+	17.5386i	0.905689	+	1.56870i	0.0856984	+	0.996321i	$0.472688\pi$
$6$		0			0
$7$		0			0
$7$		0			0
$8$		0			0
$9$		0			0
$10$		0			0
$11$	−15.4532	−	8.92191i	−0.423574	−	0.244551i	0.273031	−	0.962005i	$-0.411974\pi$
$11$	−15.4532	−	8.92191i	−0.423574	−	0.244551i	−0.696605	+	0.717455i	$0.745307\pi$
$12$		0			0
$13$		−	33.1829i		−	0.707945i	−0.935256	−	0.353973i	$-0.884831\pi$
$13$		−	33.1829i		−	0.707945i	0.935256	−	0.353973i	$-0.115169\pi$
$14$		0			0
$15$		0			0
$16$		0			0
$17$	22.9077	−	39.6772i	0.326819	−	0.566067i	−0.655060	−	0.755577i	$-0.727357\pi$
$17$	22.9077	−	39.6772i	0.326819	−	0.566067i	0.981879	+	0.189510i	$0.0606899\pi$
$18$		0			0
$19$	35.0091	−	20.2125i	0.422718	−	0.244056i	−0.273522	−	0.961866i	$-0.588189\pi$
$19$	35.0091	−	20.2125i	0.422718	−	0.244056i	0.696239	+	0.717810i	$0.254855\pi$
$20$		0			0
$21$		0			0
$22$		0			0
$23$	−69.7804	+	40.2877i	−0.632618	+	0.365242i	−0.781765	−	0.623573i	$-0.785680\pi$
$23$	−69.7804	+	40.2877i	−0.632618	+	0.365242i	0.149147	+	0.988815i	$0.452347\pi$
$24$		0			0
$25$	−142.568	+	246.935i	−1.14054	+	1.97548i
$26$		0			0
$27$		0			0
$28$		0			0
$29$		−	233.844i		−	1.49737i	−0.662926	−	0.748685i	$-0.730686\pi$
$29$		−	233.844i		−	1.49737i	0.662926	−	0.748685i	$-0.269314\pi$
$30$		0			0
$31$	195.398	+	112.813i	1.13208	+	0.653607i	0.944457	−	0.328635i	$-0.106588\pi$
$31$	195.398	+	112.813i	1.13208	+	0.653607i	0.187623	+	0.982241i	$0.439922\pi$
$32$		0			0
$33$		0			0
$34$		0			0
$35$		0			0
$36$		0			0
$37$	135.456	+	234.617i	0.601860	+	1.04245i	0.992539	+	0.121925i	$0.0389069\pi$
$37$	135.456	+	234.617i	0.601860	+	1.04245i	−0.390679	+	0.920527i	$0.627760\pi$
$38$		0			0
$39$		0			0
$40$		0			0
$41$	154.432			0.588248			0.294124	−	0.955767i	$-0.404972\pi$
$41$	154.432			0.588248			0.294124	+	0.955767i	$0.404972\pi$
$42$		0			0
$43$	367.102			1.30192			0.650960	−	0.759112i	$-0.274366\pi$
$43$	367.102			1.30192			0.650960	+	0.759112i	$0.274366\pi$
$44$		0			0
$45$		0			0
$46$		0			0
$47$	263.852	+	457.006i	0.818869	+	1.41832i	0.906517	+	0.422170i	$0.138732\pi$
$47$	263.852	+	457.006i	0.818869	+	1.41832i	−0.0876480	+	0.996152i	$0.527935\pi$
$48$		0			0
$49$		0			0
$50$		0			0
$51$		0			0
$52$		0			0
$53$	78.8892	+	45.5467i	0.204458	+	0.118044i	0.598733	−	0.800949i	$-0.295671\pi$
$53$	78.8892	+	45.5467i	0.204458	+	0.118044i	−0.394275	+	0.918992i	$0.629004\pi$
$54$		0			0
$55$		−	361.370i		−	0.885947i
$56$		0			0
$57$		0			0
$58$		0			0
$59$	312.768	−	541.729i	0.690150	−	1.19538i	−0.281638	−	0.959521i	$-0.590878\pi$
$59$	312.768	−	541.729i	0.690150	−	1.19538i	0.971788	−	0.235855i	$-0.0757889\pi$
$60$		0			0
$61$	78.8343	−	45.5150i	0.165470	−	0.0955344i	−0.414978	−	0.909832i	$-0.636211\pi$
$61$	78.8343	−	45.5150i	0.165470	−	0.0955344i	0.580448	+	0.814297i	$0.302877\pi$
$62$		0			0
$63$		0			0
$64$		0			0
$65$	581.982	−	336.007i	1.11055	−	0.641178i
$66$		0			0
$67$	−431.509	+	747.395i	−0.786823	+	1.36282i	0.141080	+	0.989998i	$0.454942\pi$
$67$	−431.509	+	747.395i	−0.786823	+	1.36282i	−0.927904	+	0.372820i	$0.878391\pi$
$68$		0			0
$69$		0			0
$70$		0			0
$71$			303.596i			0.507468i	0.967274	+	0.253734i	$0.0816587\pi$
$71$			303.596i			0.507468i	−0.967274	+	0.253734i	$0.918341\pi$
$72$		0			0
$73$	999.127	+	576.846i	1.60190	+	0.924859i	0.991106	+	0.133072i	$0.0424841\pi$
$73$	999.127	+	576.846i	1.60190	+	0.924859i	0.610797	+	0.791787i	$0.290849\pi$
$74$		0			0
$75$		0			0
$76$		0			0
$77$		0			0
$78$		0			0
$79$	3.48120	+	6.02961i	0.00495779	+	0.00858715i	0.868494	−	0.495700i	$-0.165089\pi$
$79$	3.48120	+	6.02961i	0.00495779	+	0.00858715i	−0.863536	+	0.504287i	$0.831755\pi$
$80$		0			0
$81$		0			0
$82$		0			0
$83$	−815.694			−1.07872			−0.539362	−	0.842074i	$-0.681334\pi$
$83$	−815.694			−1.07872			−0.539362	+	0.842074i	$0.681334\pi$
$84$		0			0
$85$	927.844			1.18399
$86$		0			0
$87$		0			0
$88$		0			0
$89$	−155.494	−	269.323i	−0.185194	−	0.320766i	0.758448	−	0.651734i	$-0.225958\pi$
$89$	−155.494	−	269.323i	−0.185194	−	0.320766i	−0.943642	+	0.330968i	$0.892625\pi$
$90$		0			0
$91$		0			0
$92$		0			0
$93$		0			0
$94$		0			0
$95$	708.998	+	409.340i	0.765701	+	0.442078i
$96$		0			0
$97$			1832.92i			1.91861i	0.282370	+	0.959305i	$0.408879\pi$
$97$			1832.92i			1.91861i	−0.282370	+	0.959305i	$0.591121\pi$
$98$		0			0
$99$		0			0
$100$		0			0
$101$	−721.053	+	1248.90i	−0.710371	+	1.23040i	0.254347	+	0.967113i	$0.418139\pi$
$101$	−721.053	+	1248.90i	−0.710371	+	1.23040i	−0.964718	+	0.263285i	$0.915194\pi$
$102$		0			0
$103$	−928.361	+	535.989i	−0.888098	+	0.512744i	−0.873320	−	0.487147i	$-0.838038\pi$
$103$	−928.361	+	535.989i	−0.888098	+	0.512744i	−0.0147781	+	0.999891i	$0.504704\pi$
$104$		0			0
$105$		0			0
$106$		0			0
$107$	−722.125	+	416.919i	−0.652434	+	0.376683i	−0.789388	−	0.613894i	$-0.789602\pi$
$107$	−722.125	+	416.919i	−0.652434	+	0.376683i	0.136954	+	0.990577i	$0.456269\pi$
$108$		0			0
$109$	922.514	−	1597.84i	0.810650	−	1.40409i	−0.101760	−	0.994809i	$-0.532447\pi$
$109$	922.514	−	1597.84i	0.810650	−	1.40409i	0.912410	−	0.409278i	$-0.134219\pi$
$110$		0			0
$111$		0			0
$112$		0			0
$113$			1388.90i			1.15626i	0.815946	+	0.578128i	$0.196216\pi$
$113$			1388.90i			1.15626i	−0.815946	+	0.578128i	$0.803784\pi$
$114$		0			0
$115$	−1413.18	−	815.899i	−1.14591	−	0.661591i
$116$		0			0
$117$		0			0
$118$		0			0
$119$		0			0
$120$		0			0
$121$	−506.299	−	876.936i	−0.380390	−	0.658855i
$122$		0			0
$123$		0			0
$124$		0			0
$125$	−3243.04			−2.32053
$126$		0			0
$127$	−838.992			−0.586208			−0.293104	−	0.956080i	$-0.594688\pi$
$127$	−838.992			−0.586208			−0.293104	+	0.956080i	$0.594688\pi$
$128$		0			0
$129$		0			0
$130$		0			0
$131$	195.038	+	337.815i	0.130080	+	0.225306i	0.923707	−	0.383099i	$-0.125143\pi$
$131$	195.038	+	337.815i	0.130080	+	0.225306i	−0.793627	+	0.608405i	$0.791810\pi$
$132$		0			0
$133$		0			0
$134$		0			0
$135$		0			0
$136$		0			0
$137$	−1191.72	−	688.039i	−0.743178	−	0.429074i	0.0800456	−	0.996791i	$-0.474493\pi$
$137$	−1191.72	−	688.039i	−0.743178	−	0.429074i	−0.823224	+	0.567717i	$0.807827\pi$
$138$		0			0
$139$			953.705i			0.581958i	0.956729	+	0.290979i	$0.0939810\pi$
$139$			953.705i			0.581958i	−0.956729	+	0.290979i	$0.906019\pi$
$140$		0			0
$141$		0			0
$142$		0			0
$143$	−296.055	+	512.782i	−0.173128	+	0.299867i
$144$		0			0
$145$	4101.29	−	2367.88i	2.34892	−	1.35615i
$146$		0			0
$147$		0			0
$148$		0			0
$149$	2198.94	−	1269.56i	1.20902	−	0.698029i	0.246476	−	0.969149i	$-0.420727\pi$
$149$	2198.94	−	1269.56i	1.20902	−	0.698029i	0.962546	+	0.271120i	$0.0873939\pi$
$150$		0			0
$151$	515.001	−	892.008i	0.277551	−	0.480733i	−0.693225	−	0.720722i	$-0.743811\pi$
$151$	515.001	−	892.008i	0.277551	−	0.480733i	0.970776	+	0.239989i	$0.0771439\pi$
$152$		0			0
$153$		0			0
$154$		0			0
$155$			4569.33i			2.36786i
$156$		0			0
$157$	−1558.78	−	899.959i	−0.792381	−	0.457481i	0.0484190	−	0.998827i	$-0.484582\pi$
$157$	−1558.78	−	899.959i	−0.792381	−	0.457481i	−0.840800	+	0.541346i	$0.817915\pi$
$158$		0			0
$159$		0			0
$160$		0			0
$161$		0			0
$162$		0			0
$163$	−1312.03	−	2272.50i	−0.630466	−	1.09200i	−0.987457	−	0.157891i	$-0.949531\pi$
$163$	−1312.03	−	2272.50i	−0.630466	−	1.09200i	0.356991	−	0.934108i	$-0.383803\pi$
$164$		0			0
$165$		0			0
$166$		0			0
$167$	714.138			0.330908			0.165454	−	0.986217i	$-0.447091\pi$
$167$	714.138			0.330908			0.165454	+	0.986217i	$0.447091\pi$
$168$		0			0
$169$	1095.89			0.498813
$170$		0			0
$171$		0			0
$172$		0			0
$173$	373.785	+	647.415i	0.164268	+	0.284520i	0.936395	−	0.350948i	$-0.114141\pi$
$173$	373.785	+	647.415i	0.164268	+	0.284520i	−0.772127	+	0.635468i	$0.780807\pi$
$174$		0			0
$175$		0			0
$176$		0			0
$177$		0			0
$178$		0			0
$179$	−3797.55	−	2192.52i	−1.58571	−	0.915511i	−0.994002	−	0.109360i	$-0.965120\pi$
$179$	−3797.55	−	2192.52i	−1.58571	−	0.915511i	−0.591710	−	0.806151i	$-0.701547\pi$
$180$		0			0
$181$			2143.42i			0.880215i	0.897945	+	0.440107i	$0.145060\pi$
$181$			2143.42i			0.880215i	−0.897945	+	0.440107i	$0.854940\pi$
$182$		0			0
$183$		0			0
$184$		0			0
$185$	−2743.23	+	4751.41i	−1.09020	+	1.88827i
$186$		0			0
$187$	−707.993	+	408.760i	−0.276864	+	0.159848i
$188$		0			0
$189$		0			0
$190$		0			0
$191$	−4030.20	+	2326.84i	−1.52678	+	0.881486i	−0.527285	+	0.849688i	$0.676790\pi$
$191$	−4030.20	+	2326.84i	−1.52678	+	0.881486i	−0.999494	+	0.0317981i	$0.989877\pi$
$192$		0			0
$193$	1052.69	−	1823.32i	0.392614	−	0.680028i	−0.600179	−	0.799866i	$-0.704904\pi$
$193$	1052.69	−	1823.32i	0.392614	−	0.680028i	0.992793	+	0.119838i	$0.0382374\pi$
$194$		0			0
$195$		0			0
$196$		0			0
$197$			707.475i			0.255866i	0.991783	+	0.127933i	$0.0408342\pi$
$197$			707.475i			0.255866i	−0.991783	+	0.127933i	$0.959166\pi$
$198$		0			0
$199$	1115.28	+	643.907i	0.397287	+	0.229374i	0.685313	−	0.728249i	$-0.259666\pi$
$199$	1115.28	+	643.907i	0.397287	+	0.229374i	−0.288026	+	0.957623i	$0.592999\pi$
$200$		0			0
$201$		0			0
$202$		0			0
$203$		0			0
$204$		0			0
$205$	1563.76	+	2708.51i	0.532770	+	0.922785i
$206$		0			0
$207$		0			0
$208$		0			0
$209$	−721.336			−0.238736
$210$		0			0
$211$	1193.83			0.389509			0.194755	−	0.980852i	$-0.437609\pi$
$211$	1193.83			0.389509			0.194755	+	0.980852i	$0.437609\pi$
$212$		0			0
$213$		0			0
$214$		0			0
$215$	3717.24	+	6438.46i	1.17913	+	2.04232i
$216$		0			0
$217$		0			0
$218$		0			0
$219$		0			0
$220$		0			0
$221$	−1316.61	−	760.144i	−0.400745	−	0.231370i
$222$		0			0
$223$		−	420.089i		−	0.126149i	−0.998009	−	0.0630745i	$-0.979909\pi$
$223$		−	420.089i		−	0.126149i	0.998009	−	0.0630745i	$-0.0200906\pi$
$224$		0			0
$225$		0			0
$226$		0			0
$227$	483.829	−	838.016i	0.141466	−	0.245027i	−0.786583	−	0.617485i	$-0.788152\pi$
$227$	483.829	−	838.016i	0.141466	−	0.245027i	0.928049	+	0.372458i	$0.121485\pi$
$228$		0			0
$229$	3018.69	−	1742.84i	0.871096	−	0.502927i	0.00338359	−	0.999994i	$-0.498923\pi$
$229$	3018.69	−	1742.84i	0.871096	−	0.502927i	0.867712	+	0.497067i	$0.165590\pi$
$230$		0			0
$231$		0			0
$232$		0			0
$233$	1814.94	−	1047.86i	0.510305	−	0.294625i	−0.222654	−	0.974897i	$-0.571472\pi$
$233$	1814.94	−	1047.86i	0.510305	−	0.294625i	0.732959	+	0.680273i	$0.238139\pi$
$234$		0			0
$235$	−5343.49	+	9255.19i	−1.48328	+	2.56912i
$236$		0			0
$237$		0			0
$238$		0			0
$239$			6892.95i			1.86555i	0.360453	+	0.932777i	$0.382622\pi$
$239$			6892.95i			1.86555i	−0.360453	+	0.932777i	$0.617378\pi$
$240$		0			0
$241$	2348.99	+	1356.19i	0.627850	+	0.362489i	0.779919	−	0.625881i	$-0.215260\pi$
$241$	2348.99	+	1356.19i	0.627850	+	0.362489i	−0.152069	+	0.988370i	$0.548594\pi$
$242$		0			0
$243$		0			0
$244$		0			0
$245$		0			0
$246$		0			0
$247$	−670.710	−	1161.70i	−0.172778	−	0.299261i
$248$		0			0
$249$		0			0
$250$		0			0
$251$	−6334.19			−1.59287			−0.796435	−	0.604724i	$-0.793284\pi$
$251$	−6334.19			−1.59287			−0.796435	+	0.604724i	$0.793284\pi$
$252$		0			0
$253$	1437.77			0.357281
$254$		0			0
$255$		0			0
$256$		0			0
$257$	2395.68	+	4149.44i	0.581472	+	1.00714i	0.995305	+	0.0967867i	$0.0308565\pi$
$257$	2395.68	+	4149.44i	0.581472	+	1.00714i	−0.413833	+	0.910353i	$0.635810\pi$
$258$		0			0
$259$		0			0
$260$		0			0
$261$		0			0
$262$		0			0
$263$	−200.111	−	115.534i	−0.0469177	−	0.0270879i	0.476358	−	0.879252i	$-0.341957\pi$
$263$	−200.111	−	115.534i	−0.0469177	−	0.0270879i	−0.523275	+	0.852164i	$0.675290\pi$
$264$		0			0
$265$			1844.81i			0.427644i
$266$		0			0
$267$		0			0
$268$		0			0
$269$	2356.86	−	4082.21i	0.534203	−	0.925266i	−0.464999	−	0.885311i	$-0.653945\pi$
$269$	2356.86	−	4082.21i	0.534203	−	0.925266i	0.999201	−	0.0399551i	$-0.0127215\pi$
$270$		0			0
$271$	3251.50	−	1877.26i	0.728836	−	0.420794i	−0.0891599	−	0.996017i	$-0.528418\pi$
$271$	3251.50	−	1877.26i	0.728836	−	0.420794i	0.817996	+	0.575223i	$0.195085\pi$
$272$		0			0
$273$		0			0
$274$		0			0
$275$	4406.26	−	2543.96i	0.966209	−	0.557841i
$276$		0			0
$277$	757.120	−	1311.37i	0.164227	−	0.284450i	−0.772153	−	0.635436i	$-0.780820\pi$
$277$	757.120	−	1311.37i	0.164227	−	0.284450i	0.936381	+	0.350986i	$0.114154\pi$
$278$		0			0
$279$		0			0
$280$		0			0
$281$		−	4418.40i		−	0.938005i	−0.883197	−	0.469003i	$-0.844613\pi$
$281$		−	4418.40i		−	0.938005i	0.883197	−	0.469003i	$-0.155387\pi$
$282$		0			0
$283$	4219.93	+	2436.38i	0.886391	+	0.511758i	0.872760	−	0.488149i	$-0.162328\pi$
$283$	4219.93	+	2436.38i	0.886391	+	0.511758i	0.0136304	+	0.999907i	$0.495661\pi$
$284$		0			0
$285$		0			0
$286$		0			0
$287$		0			0
$288$		0			0
$289$	1406.98	+	2436.96i	0.286378	+	0.496022i
$290$		0			0
$291$		0			0
$292$		0			0
$293$	−340.271			−0.0678458			−0.0339229	−	0.999424i	$-0.510800\pi$
$293$	−340.271			−0.0678458			−0.0339229	+	0.999424i	$0.510800\pi$
$294$		0			0
$295$	12668.2			2.50025
$296$		0			0
$297$		0			0
$298$		0			0
$299$	1336.86	+	2315.52i	0.258571	+	0.447859i
$300$		0			0
$301$		0			0
$302$		0			0
$303$		0			0
$304$		0			0
$305$	1596.54	+	921.761i	0.299729	+	0.173049i
$306$		0			0
$307$			7302.13i			1.35751i	0.734366	+	0.678754i	$0.237480\pi$
$307$			7302.13i			1.35751i	−0.734366	+	0.678754i	$0.762520\pi$
$308$		0			0
$309$		0			0
$310$		0			0
$311$	−372.538	+	645.255i	−0.0679251	+	0.117650i	−0.897988	−	0.440020i	$-0.854971\pi$
$311$	−372.538	+	645.255i	−0.0679251	+	0.117650i	0.830063	+	0.557670i	$0.188305\pi$
$312$		0			0
$313$	−689.742	+	398.223i	−0.124558	+	0.0719133i	−0.560984	−	0.827827i	$-0.689577\pi$
$313$	−689.742	+	398.223i	−0.124558	+	0.0719133i	0.436427	+	0.899740i	$0.356244\pi$
$314$		0			0
$315$		0			0
$316$		0			0
$317$	974.487	−	562.621i	0.172658	−	0.0996843i	−0.411180	−	0.911554i	$-0.634883\pi$
$317$	974.487	−	562.621i	0.172658	−	0.0996843i	0.583839	+	0.811870i	$0.301550\pi$
$318$		0			0
$319$	−2086.33	+	3613.64i	−0.366183	+	0.634247i
$320$		0			0
$321$		0			0
$322$		0			0
$323$		−	1852.09i		−	0.319049i
$324$		0			0
$325$	8194.03	+	4730.82i	1.39853	+	0.807443i
$326$		0			0
$327$		0			0
$328$		0			0
$329$		0			0
$330$		0			0
$331$	−11.1216	−	19.2632i	−0.00184683	−	0.00319880i	0.865101	−	0.501599i	$-0.167255\pi$
$331$	−11.1216	−	19.2632i	−0.00184683	−	0.00319880i	−0.866947	+	0.498400i	$0.833921\pi$
$332$		0			0
$333$		0			0
$334$		0			0
$335$	−17477.7			−2.85047
$336$		0			0
$337$	9570.35			1.54697			0.773487	−	0.633812i	$-0.218511\pi$
$337$	9570.35			1.54697			0.773487	+	0.633812i	$0.218511\pi$
$338$		0			0
$339$		0			0
$340$		0			0
$341$	−2013.01	−	3486.64i	−0.319680	−	0.553702i
$342$		0			0
$343$		0			0
$344$		0			0
$345$		0			0
$346$		0			0
$347$	9927.17	+	5731.45i	1.53579	+	0.886688i	0.999078	+	0.0429203i	$0.0136662\pi$
$347$	9927.17	+	5731.45i	1.53579	+	0.886688i	0.536709	+	0.843767i	$0.319667\pi$
$348$		0			0
$349$			1575.13i			0.241590i	0.992677	+	0.120795i	$0.0385443\pi$
$349$			1575.13i			0.241590i	−0.992677	+	0.120795i	$0.961456\pi$
$350$		0			0
$351$		0			0
$352$		0			0
$353$	−2316.74	+	4012.71i	−0.349313	+	0.605028i	−0.986128	−	0.165989i	$-0.946918\pi$
$353$	−2316.74	+	4012.71i	−0.349313	+	0.605028i	0.636814	+	0.771017i	$0.280252\pi$
$354$		0			0
$355$	−5324.64	+	3074.18i	−0.796064	+	0.459608i
$356$		0			0
$357$		0			0
$358$		0			0
$359$	−2062.64	+	1190.86i	−0.303236	+	0.175073i	−0.643896	−	0.765113i	$-0.722683\pi$
$359$	−2062.64	+	1190.86i	−0.303236	+	0.175073i	0.340660	+	0.940187i	$0.389350\pi$
$360$		0			0
$361$	−2612.41	+	4524.83i	−0.380873	+	0.659692i
$362$		0			0
$363$		0			0
$364$		0			0
$365$			23364.4i			3.35054i
$366$		0			0
$367$	990.766	+	572.019i	0.140920	+	0.0813601i	0.568802	−	0.822474i	$-0.307407\pi$
$367$	990.766	+	572.019i	0.140920	+	0.0813601i	−0.427882	+	0.903834i	$0.640740\pi$
$368$		0			0
$369$		0			0
$370$		0			0
$371$		0			0
$372$		0			0
$373$	205.226	+	355.461i	0.0284884	+	0.0493434i	0.879918	−	0.475125i	$-0.157597\pi$
$373$	205.226	+	355.461i	0.0284884	+	0.0493434i	−0.851430	+	0.524469i	$0.824264\pi$
$374$		0			0
$375$		0			0
$376$		0			0
$377$	−7759.63			−1.06006
$378$		0			0
$379$	−6400.00			−0.867403			−0.433702	−	0.901057i	$-0.642793\pi$
$379$	−6400.00			−0.867403			−0.433702	+	0.901057i	$0.642793\pi$
$380$		0			0
$381$		0			0
$382$		0			0
$383$	−1742.19	−	3017.57i	−0.232433	−	0.402586i	0.726090	−	0.687599i	$-0.241335\pi$
$383$	−1742.19	−	3017.57i	−0.232433	−	0.402586i	−0.958524	+	0.285013i	$0.908002\pi$
$384$		0			0
$385$		0			0
$386$		0			0
$387$		0			0
$388$		0			0
$389$	3641.68	+	2102.53i	0.474654	+	0.274042i	0.718186	−	0.695851i	$-0.244973\pi$
$389$	3641.68	+	2102.53i	0.474654	+	0.274042i	−0.243532	+	0.969893i	$0.578306\pi$
$390$		0			0
$391$			3691.59i			0.477472i
$392$		0			0
$393$		0			0
$394$		0			0
$395$	−70.5006	+	122.111i	−0.00898043	+	0.0155546i
$396$		0			0
$397$	4699.38	−	2713.19i	0.594093	−	0.343000i	−0.172621	−	0.984988i	$-0.555224\pi$
$397$	4699.38	−	2713.19i	0.594093	−	0.343000i	0.766714	+	0.641989i	$0.221890\pi$
$398$		0			0
$399$		0			0
$400$		0			0
$401$	7357.88	−	4248.07i	0.916297	−	0.529024i	0.0338449	−	0.999427i	$-0.489225\pi$
$401$	7357.88	−	4248.07i	0.916297	−	0.529024i	0.882452	+	0.470403i	$0.155891\pi$
$402$		0			0
$403$	3743.47	−	6483.87i	0.462718	−	0.801451i
$404$		0			0
$405$		0			0
$406$		0			0
$407$		−	4834.10i		−	0.588741i
$408$		0			0
$409$	−5581.42	−	3222.44i	−0.674776	−	0.389582i	0.123108	−	0.992393i	$-0.460714\pi$
$409$	−5581.42	−	3222.44i	−0.674776	−	0.389582i	−0.797884	+	0.602811i	$0.794047\pi$
$410$		0			0
$411$		0			0
$412$		0			0
$413$		0			0
$414$		0			0
$415$	−8259.64	−	14306.1i	−0.976987	−	1.69219i
$416$		0			0
$417$		0			0
$418$		0			0
$419$	14198.4			1.65546			0.827728	−	0.561130i	$-0.189633\pi$
$419$	14198.4			1.65546			0.827728	+	0.561130i	$0.189633\pi$
$420$		0			0
$421$	6986.18			0.808754			0.404377	−	0.914592i	$-0.367488\pi$
$421$	6986.18			0.808754			0.404377	+	0.914592i	$0.367488\pi$
$422$		0			0
$423$		0			0
$424$		0			0
$425$	6531.80	+	11313.4i	0.745503	+	1.29125i
$426$		0			0
$427$		0			0
$428$		0			0
$429$		0			0
$430$		0			0
$431$	−12104.1	−	6988.29i	−1.35274	−	0.781007i	−0.364111	−	0.931356i	$-0.618627\pi$
$431$	−12104.1	−	6988.29i	−1.35274	−	0.781007i	−0.988633	+	0.150349i	$0.951960\pi$
$432$		0			0
$433$		−	4418.70i		−	0.490414i	−0.969471	−	0.245207i	$-0.921144\pi$
$433$		−	4418.70i		−	0.490414i	0.969471	−	0.245207i	$-0.0788559\pi$
$434$		0			0
$435$		0			0
$436$		0			0
$437$	−1628.63	+	2820.87i	−0.178279	+	0.308789i
$438$		0			0
$439$	−2497.13	+	1441.72i	−0.271484	+	0.156742i	−0.629562	−	0.776950i	$-0.716766\pi$
$439$	−2497.13	+	1441.72i	−0.271484	+	0.156742i	0.358078	+	0.933692i	$0.383432\pi$
$440$		0			0
$441$		0			0
$442$		0			0
$443$	11013.8	−	6358.80i	1.18122	−	0.681976i	0.224922	−	0.974377i	$-0.427787\pi$
$443$	11013.8	−	6358.80i	1.18122	−	0.681976i	0.956296	+	0.292400i	$0.0944540\pi$
$444$		0			0
$445$	3149.03	−	5454.28i	0.335457	−	0.581029i
$446$		0			0
$447$		0			0
$448$		0			0
$449$		−	5375.10i		−	0.564959i	−0.959273	−	0.282479i	$-0.908843\pi$
$449$		−	5375.10i		−	0.564959i	0.959273	−	0.282479i	$-0.0911569\pi$
$450$		0			0
$451$	−2386.46	−	1377.83i	−0.249167	−	0.143856i
$452$		0			0
$453$		0			0
$454$		0			0
$455$		0			0
$456$		0			0
$457$	2060.59	+	3569.05i	0.210920	+	0.365324i	0.952003	−	0.306090i	$-0.0990208\pi$
$457$	2060.59	+	3569.05i	0.210920	+	0.365324i	−0.741083	+	0.671414i	$0.765687\pi$
$458$		0			0
$459$		0			0
$460$		0			0
$461$	6019.06			0.608103			0.304052	−	0.952656i	$-0.401660\pi$
$461$	6019.06			0.608103			0.304052	+	0.952656i	$0.401660\pi$
$462$		0			0
$463$	−3263.90			−0.327616			−0.163808	−	0.986492i	$-0.552378\pi$
$463$	−3263.90			−0.327616			−0.163808	+	0.986492i	$0.552378\pi$
$464$		0			0
$465$		0			0
$466$		0			0
$467$	−660.961	−	1144.82i	−0.0654938	−	0.113439i	0.831419	−	0.555646i	$-0.187529\pi$
$467$	−660.961	−	1144.82i	−0.0654938	−	0.113439i	−0.896913	+	0.442207i	$0.854196\pi$
$468$		0			0
$469$		0			0
$470$		0			0
$471$		0			0
$472$		0			0
$473$	−5672.90	−	3275.25i	−0.551460	−	0.318385i
$474$		0			0
$475$			11526.6i			1.11343i
$476$		0			0
$477$		0			0
$478$		0			0
$479$	1887.69	−	3269.58i	0.180064	−	0.311881i	−0.761838	−	0.647768i	$-0.775703\pi$
$479$	1887.69	−	3269.58i	0.180064	−	0.311881i	0.941902	+	0.335887i	$0.109036\pi$
$480$		0			0
$481$	7785.27	−	4494.83i	0.737999	−	0.426084i
$482$		0			0
$483$		0			0
$484$		0			0
$485$	−32146.9	+	18560.0i	−3.00972	+	1.73766i
$486$		0			0
$487$	6119.80	−	10599.8i	0.569434	−	0.986289i	−0.427187	−	0.904163i	$-0.640496\pi$
$487$	6119.80	−	10599.8i	0.569434	−	0.986289i	0.996622	−	0.0821263i	$-0.0261711\pi$
$488$		0			0
$489$		0			0
$490$		0			0
$491$			13018.0i			1.19652i	0.801301	+	0.598261i	$0.204142\pi$
$491$			13018.0i			1.19652i	−0.801301	+	0.598261i	$0.795858\pi$
$492$		0			0
$493$	−9278.28	−	5356.82i	−0.847612	−	0.489369i
$494$		0			0
$495$		0			0
$496$		0			0
$497$		0			0
$498$		0			0
$499$	8603.67	+	14902.0i	0.771850	+	1.33688i	0.936548	+	0.350540i	$0.114002\pi$
$499$	8603.67	+	14902.0i	0.771850	+	1.33688i	−0.164697	+	0.986344i	$0.552665\pi$
$500$		0			0
$501$		0			0
$502$		0			0
$503$	−19760.5			−1.75164			−0.875821	−	0.482635i	$-0.839680\pi$
$503$	−19760.5			−1.75164			−0.875821	+	0.482635i	$0.839680\pi$
$504$		0			0
$505$	−29205.3			−2.57350
$506$		0			0
$507$		0			0
$508$		0			0
$509$	−2351.82	−	4073.48i	−0.204799	−	0.354722i	0.745270	−	0.666763i	$-0.232321\pi$
$509$	−2351.82	−	4073.48i	−0.204799	−	0.354722i	−0.950069	+	0.312041i	$0.898987\pi$
$510$		0			0
$511$		0			0
$512$		0			0
$513$		0			0
$514$		0			0
$515$	−18801.0	−	10854.8i	−1.60868	−	0.928772i
$516$		0			0
$517$		−	9416.26i		−	0.801019i
$518$		0			0
$519$		0			0
$520$		0			0
$521$	−7971.12	+	13806.4i	−0.670290	+	1.16098i	0.307532	+	0.951538i	$0.400497\pi$
$521$	−7971.12	+	13806.4i	−0.670290	+	1.16098i	−0.977822	+	0.209438i	$0.932836\pi$
$522$		0			0
$523$	17362.2	−	10024.0i	1.45161	−	0.838090i	0.453041	−	0.891490i	$-0.350339\pi$
$523$	17362.2	−	10024.0i	1.45161	−	0.838090i	0.998573	+	0.0533995i	$0.0170057\pi$
$524$		0			0
$525$		0			0
$526$		0			0
$527$	8952.22	−	5168.56i	0.739971	−	0.427222i
$528$		0			0
$529$	−2837.30	+	4914.35i	−0.233196	+	0.403908i
$530$		0			0
$531$		0			0
$532$		0			0
$533$		−	5124.50i		−	0.416448i
$534$		0			0
$535$	−14624.3	−	8443.37i	−1.18180	−	0.682315i
$536$		0			0
$537$		0			0
$538$		0			0
$539$		0			0
$540$		0			0
$541$	−1931.85	−	3346.06i	−0.153524	−	0.265912i	0.778997	−	0.627028i	$-0.215729\pi$
$541$	−1931.85	−	3346.06i	−0.153524	−	0.265912i	−0.932521	+	0.361117i	$0.882396\pi$
$542$		0			0
$543$		0			0
$544$		0			0
$545$	37365.2			2.93678
$546$		0			0
$547$	−18674.5			−1.45972			−0.729859	−	0.683598i	$-0.760414\pi$
$547$	−18674.5			−1.45972			−0.729859	+	0.683598i	$0.760414\pi$
$548$		0			0
$549$		0			0
$550$		0			0
$551$	−4726.57	−	8186.66i	−0.365442	−	0.632965i
$552$		0			0
$553$		0			0
$554$		0			0
$555$		0			0
$556$		0			0
$557$	1489.14	+	859.753i	0.113280	+	0.0654020i	0.555569	−	0.831470i	$-0.312500\pi$
$557$	1489.14	+	859.753i	0.113280	+	0.0654020i	−0.442290	+	0.896872i	$0.645834\pi$
$558$		0			0
$559$		−	12181.5i		−	0.921689i
$560$		0			0
$561$		0			0
$562$		0			0
$563$	−6596.37	+	11425.3i	−0.493790	+	0.855270i	−0.999974	−	0.00715535i	$-0.997722\pi$
$563$	−6596.37	+	11425.3i	−0.493790	+	0.855270i	0.506184	+	0.862426i	$0.331056\pi$
$564$		0			0
$565$	−24359.4	+	14063.9i	−1.81382	+	1.04721i
$566$		0			0
$567$		0			0
$568$		0			0
$569$	−21405.5	+	12358.4i	−1.57709	+	0.910533i	−0.581826	+	0.813313i	$0.697661\pi$
$569$	−21405.5	+	12358.4i	−1.57709	+	0.910533i	−0.995263	+	0.0972193i	$0.969005\pi$
$570$		0			0
$571$	−1032.73	+	1788.73i	−0.0756887	+	0.131097i	−0.901386	−	0.433017i	$-0.857449\pi$
$571$	−1032.73	+	1788.73i	−0.0756887	+	0.131097i	0.825697	+	0.564114i	$0.190782\pi$
$572$		0			0
$573$		0			0
$574$		0			0
$575$		−	22974.9i		−	1.66630i
$576$		0			0
$577$	6759.61	+	3902.66i	0.487706	+	0.281577i	0.723622	−	0.690196i	$-0.242476\pi$
$577$	6759.61	+	3902.66i	0.487706	+	0.281577i	−0.235916	+	0.971773i	$0.575809\pi$
$578$		0			0
$579$		0			0
$580$		0			0
$581$		0			0
$582$		0			0
$583$	−812.727	−	1407.68i	−0.0577353	−	0.100001i
$584$		0			0
$585$		0			0
$586$		0			0
$587$	6297.78			0.442823			0.221412	−	0.975180i	$-0.428934\pi$
$587$	6297.78			0.442823			0.221412	+	0.975180i	$0.428934\pi$
$588$		0			0
$589$	9120.93			0.638067
$590$		0			0
$591$		0			0
$592$		0			0
$593$	9115.60	+	15788.7i	0.631253	+	1.09336i	0.987296	+	0.158893i	$0.0507923\pi$
$593$	9115.60	+	15788.7i	0.631253	+	1.09336i	−0.356043	+	0.934470i	$0.615874\pi$
$594$		0			0
$595$		0			0
$596$		0			0
$597$		0			0
$598$		0			0
$599$	−7161.55	−	4134.72i	−0.488503	−	0.282037i	0.235450	−	0.971886i	$-0.424343\pi$
$599$	−7161.55	−	4134.72i	−0.488503	−	0.282037i	−0.723953	+	0.689849i	$0.757677\pi$
$600$		0			0
$601$		−	11106.1i		−	0.753788i	−0.926256	−	0.376894i	$-0.876992\pi$
$601$		−	11106.1i		−	0.753788i	0.926256	−	0.376894i	$-0.123008\pi$
$602$		0			0
$603$		0			0
$604$		0			0
$605$	10253.5	−	17759.5i	0.689030	−	1.19343i
$606$		0			0
$607$	10414.3	−	6012.71i	0.696382	−	0.402056i	−0.109616	−	0.993974i	$-0.534962\pi$
$607$	10414.3	−	6012.71i	0.696382	−	0.402056i	0.805999	+	0.591917i	$0.201629\pi$
$608$		0			0
$609$		0			0
$610$		0			0
$611$	15164.8	−	8755.40i	1.00409	−	0.579714i
$612$		0			0
$613$	2027.75	−	3512.16i	0.133605	−	0.231411i	−0.791459	−	0.611223i	$-0.790678\pi$
$613$	2027.75	−	3512.16i	0.133605	−	0.231411i	0.925064	+	0.379812i	$0.124011\pi$
$614$		0			0
$615$		0			0
$616$		0			0
$617$		−	7236.59i		−	0.472178i	−0.971731	−	0.236089i	$-0.924134\pi$
$617$		−	7236.59i		−	0.472178i	0.971731	−	0.236089i	$-0.0758658\pi$
$618$		0			0
$619$	−13803.6	−	7969.54i	−0.896309	−	0.517484i	−0.0203081	−	0.999794i	$-0.506465\pi$
$619$	−13803.6	−	7969.54i	−0.896309	−	0.517484i	−0.876001	+	0.482310i	$0.839798\pi$
$620$		0			0
$621$		0			0
$622$		0			0
$623$		0			0
$624$		0			0
$625$	−15017.8	−	26011.5i	−0.961136	−	1.66474i
$626$		0			0
$627$		0			0
$628$		0			0
$629$	12411.9			0.786798
$630$		0			0
$631$	25310.1			1.59680			0.798399	−	0.602129i	$-0.205681\pi$
$631$	25310.1			1.59680			0.798399	+	0.602129i	$0.205681\pi$
$632$		0			0
$633$		0			0
$634$		0			0
$635$	−8495.55	−	14714.7i	−0.530922	−	0.919584i
$636$		0			0
$637$		0			0
$638$		0			0
$639$		0			0
$640$		0			0
$641$	21593.9	+	12467.2i	1.33059	+	0.768216i	0.985390	−	0.170314i	$-0.0544782\pi$
$641$	21593.9	+	12467.2i	1.33059	+	0.768216i	0.345199	+	0.938530i	$0.387812\pi$
$642$		0			0
$643$			12466.1i			0.764563i	0.924046	+	0.382282i	$0.124862\pi$
$643$			12466.1i			0.764563i	−0.924046	+	0.382282i	$0.875138\pi$
$644$		0			0
$645$		0			0
$646$		0			0
$647$	12339.6	−	21372.8i	0.749799	−	1.29869i	−0.198119	−	0.980178i	$-0.563483\pi$
$647$	12339.6	−	21372.8i	0.749799	−	1.29869i	0.947919	−	0.318513i	$-0.103183\pi$
$648$		0			0
$649$	−9666.52	+	5580.97i	−0.584659	+	0.337553i
$650$		0			0
$651$		0			0
$652$		0			0
$653$	16498.1	−	9525.16i	0.988697	−	0.570825i	0.0838127	−	0.996482i	$-0.473290\pi$
$653$	16498.1	−	9525.16i	0.988697	−	0.570825i	0.904885	+	0.425657i	$0.139957\pi$
$654$		0			0
$655$	−3949.87	+	6841.37i	−0.235625	+	0.408114i
$656$		0			0
$657$		0			0
$658$		0			0
$659$			5983.49i			0.353693i	0.984238	+	0.176847i	$0.0565897\pi$
$659$			5983.49i			0.353693i	−0.984238	+	0.176847i	$0.943410\pi$
$660$		0			0
$661$	12000.7	+	6928.61i	0.706162	+	0.407703i	0.809638	−	0.586929i	$-0.199663\pi$
$661$	12000.7	+	6928.61i	0.706162	+	0.407703i	−0.103476	+	0.994632i	$0.532997\pi$
$662$		0			0
$663$		0			0
$664$		0			0
$665$		0			0
$666$		0			0
$667$	9421.03	+	16317.7i	0.546902	+	0.947263i
$668$		0			0
$669$		0			0
$670$		0			0
$671$	−1624.32			−0.0934519
$672$		0			0
$673$	−21469.4			−1.22970			−0.614849	−	0.788645i	$-0.710783\pi$
$673$	−21469.4			−1.22970			−0.614849	+	0.788645i	$0.710783\pi$
$674$		0			0
$675$		0			0
$676$		0			0
$677$	−10221.4	−	17703.9i	−0.580265	−	1.00505i	−0.995448	−	0.0953099i	$-0.969616\pi$
$677$	−10221.4	−	17703.9i	−0.580265	−	1.00505i	0.415183	−	0.909738i	$-0.363718\pi$
$678$		0			0
$679$		0			0
$680$		0			0
$681$		0			0
$682$		0			0
$683$	16958.6	+	9791.08i	0.950080	+	0.548529i	0.893106	−	0.449847i	$-0.148521\pi$
$683$	16958.6	+	9791.08i	0.950080	+	0.548529i	0.0569740	+	0.998376i	$0.481855\pi$
$684$		0			0
$685$		−	27868.1i		−	1.55443i
$686$		0			0
$687$		0			0
$688$		0			0
$689$	1511.37	−	2617.77i	0.0835685	−	0.144745i
$690$		0			0
$691$	−10137.3	+	5852.75i	−0.558089	+	0.322213i	−0.752378	−	0.658731i	$-0.771093\pi$
$691$	−10137.3	+	5852.75i	−0.558089	+	0.322213i	0.194289	+	0.980944i	$0.437760\pi$
$692$		0			0
$693$		0			0
$694$		0			0
$695$	−16726.6	+	9657.13i	−0.912917	+	0.527073i
$696$		0			0
$697$	3537.67	−	6127.43i	0.192251	−	0.332988i
$698$		0			0
$699$		0			0
$700$		0			0
$701$		−	17998.6i		−	0.969752i	−0.874583	−	0.484876i	$-0.838865\pi$
$701$		−	17998.6i		−	0.969752i	0.874583	−	0.484876i	$-0.161135\pi$
$702$		0			0
$703$	9484.38	+	5475.81i	0.508834	+	0.293775i
$704$		0			0
$705$		0			0
$706$		0			0
$707$		0			0
$708$		0			0
$709$	11891.5	+	20596.7i	0.629895	+	1.09101i	0.987572	+	0.157166i	$0.0502356\pi$
$709$	11891.5	+	20596.7i	0.629895	+	1.09101i	−0.357677	+	0.933845i	$0.616431\pi$
$710$		0			0
$711$		0			0
$712$		0			0
$713$	−18179.9			−0.954899
$714$		0			0
$715$	−11991.3			−0.627202
$716$		0			0
$717$		0			0
$718$		0			0
$719$	−16513.5	−	28602.2i	−0.856536	−	1.48356i	−0.875213	−	0.483738i	$-0.839279\pi$
$719$	−16513.5	−	28602.2i	−0.856536	−	1.48356i	0.0186774	−	0.999826i	$-0.494054\pi$
$720$		0			0
$721$		0			0
$722$		0			0
$723$		0			0
$724$		0			0
$725$	57744.2	+	33338.6i	2.95802	+	1.70782i
$726$		0			0
$727$		−	6816.11i		−	0.347724i	−0.984770	−	0.173862i	$-0.944375\pi$
$727$		−	6816.11i		−	0.347724i	0.984770	−	0.173862i	$-0.0556247\pi$
$728$		0			0
$729$		0			0
$730$		0			0
$731$	8409.46	−	14565.6i	0.425493	−	0.736975i
$732$		0			0
$733$	−20701.4	+	11951.9i	−1.04314	+	0.602258i	−0.920722	−	0.390220i	$-0.872399\pi$
$733$	−20701.4	+	11951.9i	−1.04314	+	0.602258i	−0.122420	+	0.992478i	$0.539066\pi$
$734$		0			0
$735$		0			0
$736$		0			0
$737$	13336.4	−	7699.76i	0.666556	−	0.384836i
$738$		0			0
$739$	−4558.07	+	7894.81i	−0.226889	+	0.392984i	−0.956885	−	0.290468i	$-0.906189\pi$
$739$	−4558.07	+	7894.81i	−0.226889	+	0.392984i	0.729995	+	0.683452i	$0.239522\pi$
$740$		0			0
$741$		0			0
$742$		0			0
$743$		−	32818.5i		−	1.62045i	−0.586121	−	0.810224i	$-0.699346\pi$
$743$		−	32818.5i		−	1.62045i	0.586121	−	0.810224i	$-0.300654\pi$
$744$		0			0
$745$	44532.5	+	25710.9i	2.18999	+	1.26439i
$746$		0			0
$747$		0			0
$748$		0			0
$749$		0			0
$750$		0			0
$751$	4620.07	+	8002.20i	0.224486	+	0.388821i	0.956165	−	0.292828i	$-0.0945964\pi$
$751$	4620.07	+	8002.20i	0.224486	+	0.388821i	−0.731679	+	0.681649i	$0.761263\pi$
$752$		0			0
$753$		0			0
$754$		0			0
$755$	20859.4			1.00550
$756$		0			0
$757$	16610.7			0.797523			0.398761	−	0.917055i	$-0.369440\pi$
$757$	16610.7			0.797523			0.398761	+	0.917055i	$0.369440\pi$
$758$		0			0
$759$		0			0
$760$		0			0
$761$	−1121.42	−	1942.35i	−0.0534184	−	0.0925234i	0.838080	−	0.545548i	$-0.183678\pi$
$761$	−1121.42	−	1942.35i	−0.0534184	−	0.0925234i	−0.891498	+	0.453024i	$0.850345\pi$
$762$		0			0
$763$		0			0
$764$		0			0
$765$		0			0
$766$		0			0
$767$	−17976.2	−	10378.5i	−0.846261	−	0.488589i
$768$		0			0
$769$		−	9182.73i		−	0.430608i	−0.976547	−	0.215304i	$-0.930926\pi$
$769$		−	9182.73i		−	0.430608i	0.976547	−	0.215304i	$-0.0690743\pi$
$770$		0			0
$771$		0			0
$772$		0			0
$773$	−4272.40	+	7400.02i	−0.198794	+	0.344321i	−0.948138	−	0.317860i	$-0.897036\pi$
$773$	−4272.40	+	7400.02i	−0.198794	+	0.344321i	0.749344	+	0.662181i	$0.230369\pi$
$774$		0			0
$775$	−55714.9	+	32167.0i	−2.58237	+	1.49093i
$776$		0			0
$777$		0			0
$778$		0			0
$779$	5406.51	−	3121.45i	0.248663	−	0.143566i
$780$		0			0
$781$	2708.65	−	4691.53i	0.124101	−	0.214950i
$782$		0			0
$783$		0			0
$784$		0			0
$785$		−	36451.6i		−	1.65734i
$786$		0			0
$787$	−20236.4	−	11683.5i	−0.916583	−	0.529190i	−0.0340400	−	0.999420i	$-0.510837\pi$
$787$	−20236.4	−	11683.5i	−0.916583	−	0.529190i	−0.882544	+	0.470231i	$0.844171\pi$
$788$		0			0
$789$		0			0
$790$		0			0
$791$		0			0
$792$		0			0
$793$	−1510.32	−	2615.95i	−0.0676331	−	0.117144i
$794$		0			0
$795$		0			0
$796$		0			0
$797$	8319.18			0.369737			0.184869	−	0.982763i	$-0.440814\pi$
$797$	8319.18			0.369737			0.184869	+	0.982763i	$0.440814\pi$
$798$		0			0
$799$	24177.0			1.07049
$800$		0			0
$801$		0			0
$802$		0			0
$803$	−10293.1	−	17828.2i	−0.452350	−	0.783493i
$804$		0			0
$805$		0			0
$806$		0			0
$807$		0			0
$808$		0			0
$809$	7769.07	+	4485.47i	0.337634	+	0.194933i	0.659225	−	0.751946i	$-0.270884\pi$
$809$	7769.07	+	4485.47i	0.337634	+	0.194933i	−0.321591	+	0.946879i	$0.604218\pi$
$810$		0			0
$811$			29055.1i			1.25803i	0.777393	+	0.629015i	$0.216542\pi$
$811$			29055.1i			1.25803i	−0.777393	+	0.629015i	$0.783458\pi$
$812$		0			0
$813$		0			0
$814$		0			0
$815$	26570.9	−	46022.2i	1.14201	−	1.97802i
$816$		0			0
$817$	12851.9	−	7420.06i	0.550345	−	0.317742i
$818$		0			0
$819$		0			0
$820$		0			0
$821$	13864.8	−	8004.83i	0.589384	−	0.340281i	−0.175470	−	0.984485i	$-0.556144\pi$
$821$	13864.8	−	8004.83i	0.589384	−	0.340281i	0.764854	+	0.644204i	$0.222811\pi$
$822$		0			0
$823$	−3875.80	+	6713.08i	−0.164158	+	0.284330i	−0.936356	−	0.351052i	$-0.885824\pi$
$823$	−3875.80	+	6713.08i	−0.164158	+	0.284330i	0.772198	+	0.635382i	$0.219157\pi$
$824$		0			0
$825$		0			0
$826$		0			0
$827$		−	4705.84i		−	0.197870i	−0.995094	−	0.0989348i	$-0.968456\pi$
$827$		−	4705.84i		−	0.197870i	0.995094	−	0.0989348i	$-0.0315435\pi$
$828$		0			0
$829$	−33842.3	−	19538.9i	−1.41784	−	0.818593i	−0.421735	−	0.906719i	$-0.638579\pi$
$829$	−33842.3	−	19538.9i	−1.41784	−	0.818593i	−0.996109	+	0.0881261i	$0.971912\pi$
$830$		0			0
$831$		0			0
$832$		0			0
$833$		0			0
$834$		0			0
$835$	7231.29	+	12525.0i	0.299700	+	0.519095i
$836$		0			0
$837$		0			0
$838$		0			0
$839$	24699.3			1.01634			0.508172	−	0.861255i	$-0.330321\pi$
$839$	24699.3			1.01634			0.508172	+	0.861255i	$0.330321\pi$
$840$		0			0
$841$	−30294.0			−1.24212
$842$		0			0
$843$		0			0
$844$		0			0
$845$	11096.9	+	19220.4i	0.451769	+	0.782488i
$846$		0			0
$847$		0			0
$848$		0			0
$849$		0			0
$850$		0			0
$851$	−18904.3	−	10914.4i	−0.761495	−	0.439649i
$852$		0			0
$853$			4313.27i			0.173134i	0.996246	+	0.0865671i	$0.0275897\pi$
$853$			4313.27i			0.173134i	−0.996246	+	0.0865671i	$0.972410\pi$
$854$		0			0
$855$		0			0
$856$		0			0
$857$	−13238.5	+	22929.8i	−0.527678	+	0.913965i	0.471801	+	0.881705i	$0.343604\pi$
$857$	−13238.5	+	22929.8i	−0.527678	+	0.913965i	−0.999479	+	0.0322604i	$0.989729\pi$
$858$		0			0
$859$	33468.0	−	19322.8i	1.32935	−	0.767502i	0.344154	−	0.938913i	$-0.388166\pi$
$859$	33468.0	−	19322.8i	1.32935	−	0.767502i	0.985200	+	0.171411i	$0.0548326\pi$
$860$		0			0
$861$		0			0
$862$		0			0
$863$	−10813.1	+	6242.96i	−0.426516	+	0.246249i	−0.697861	−	0.716233i	$-0.745865\pi$
$863$	−10813.1	+	6242.96i	−0.426516	+	0.246249i	0.271345	+	0.962482i	$0.412531\pi$
$864$		0			0
$865$	−7569.82	+	13111.3i	−0.297551	+	0.515374i
$866$		0			0
$867$		0			0
$868$		0			0
$869$		−	124.236i		−	0.00484972i
$870$		0			0
$871$	24800.8	+	14318.7i	0.964801	+	0.557028i
$872$		0			0
$873$		0			0
$874$		0			0
$875$		0			0
$876$		0			0
$877$	−11426.5	−	19791.2i	−0.439959	−	0.762032i	0.557727	−	0.830025i	$-0.311674\pi$
$877$	−11426.5	−	19791.2i	−0.439959	−	0.762032i	−0.997686	+	0.0679931i	$0.978340\pi$
$878$		0			0
$879$		0			0
$880$		0			0
$881$	20304.1			0.776463			0.388231	−	0.921562i	$-0.373086\pi$
$881$	20304.1			0.776463			0.388231	+	0.921562i	$0.373086\pi$
$882$		0			0
$883$	−12868.9			−0.490456			−0.245228	−	0.969465i	$-0.578863\pi$
$883$	−12868.9			−0.490456			−0.245228	+	0.969465i	$0.578863\pi$
$884$		0			0
$885$		0			0
$886$		0			0
$887$	−5090.81	−	8817.55i	−0.192709	−	0.333782i	0.753438	−	0.657519i	$-0.228394\pi$
$887$	−5090.81	−	8817.55i	−0.192709	−	0.333782i	−0.946147	+	0.323737i	$0.895061\pi$
$888$		0			0
$889$		0			0
$890$		0			0
$891$		0			0
$892$		0			0
$893$	18474.5	+	10666.2i	0.692300	+	0.399700i
$894$		0			0
$895$		−	88805.0i		−	3.31667i
$896$		0			0
$897$		0			0
$898$		0			0
$899$	26380.6	−	45692.6i	0.978691	−	1.69514i
$900$		0			0
$901$	3614.33	−	2086.74i	0.133641	−	0.0771579i
$902$		0			0
$903$		0			0
$904$		0			0
$905$	−37592.5	+	21704.0i	−1.38079	+	0.797200i
$906$		0			0
$907$	−4590.66	+	7951.26i	−0.168060	+	0.291088i	−0.937738	−	0.347344i	$-0.887084\pi$
$907$	−4590.66	+	7951.26i	−0.168060	+	0.291088i	0.769678	+	0.638433i	$0.220417\pi$
$908$		0			0
$909$		0			0
$910$		0			0
$911$		−	18656.7i		−	0.678511i	−0.940694	−	0.339255i	$-0.889825\pi$
$911$		−	18656.7i		−	0.678511i	0.940694	−	0.339255i	$-0.110175\pi$
$912$		0			0
$913$	12605.1	+	7277.54i	0.456919	+	0.263802i
$914$		0			0
$915$		0			0
$916$		0			0
$917$		0			0
$918$		0			0
$919$	2394.27	+	4147.00i	0.0859410	+	0.148854i	0.905792	−	0.423723i	$-0.139277\pi$
$919$	2394.27	+	4147.00i	0.0859410	+	0.148854i	−0.819851	+	0.572577i	$0.805944\pi$
$920$		0			0
$921$		0			0
$922$		0			0
$923$	10074.2			0.359259
$924$		0			0
$925$	−77246.7			−2.74579
$926$		0			0
$927$		0			0
$928$		0			0
$929$	698.963	+	1210.64i	0.0246848	+	0.0427554i	0.878104	−	0.478470i	$-0.158808\pi$
$929$	698.963	+	1210.64i	0.0246848	+	0.0427554i	−0.853419	+	0.521225i	$0.825475\pi$
$930$		0			0
$931$		0			0
$932$		0			0
$933$		0			0
$934$		0			0
$935$	−14338.2	−	8278.13i	−0.501506	−	0.289544i
$936$		0			0
$937$		−	13520.0i		−	0.471376i	−0.971829	−	0.235688i	$-0.924266\pi$
$937$		−	13520.0i		−	0.471376i	0.971829	−	0.235688i	$-0.0757343\pi$
$938$		0			0
$939$		0			0
$940$		0			0
$941$	−20604.4	+	35687.9i	−0.713800	+	1.23634i	0.249621	+	0.968344i	$0.419694\pi$
$941$	−20604.4	+	35687.9i	−0.713800	+	1.23634i	−0.963421	+	0.267994i	$0.913639\pi$
$942$		0			0
$943$	−10776.3	+	6221.70i	−0.372136	+	0.214853i
$944$		0			0
$945$		0			0
$946$		0			0
$947$	−6629.68	+	3827.65i	−0.227493	+	0.131343i	−0.609415	−	0.792852i	$-0.708596\pi$
$947$	−6629.68	+	3827.65i	−0.227493	+	0.131343i	0.381922	+	0.924194i	$0.375262\pi$
$948$		0			0
$949$	19141.4	−	33154.0i	0.654750	−	1.13406i
$950$		0			0
$951$		0			0
$952$		0			0
$953$		−	22854.9i		−	0.776854i	−0.921479	−	0.388427i	$-0.873019\pi$
$953$		−	22854.9i		−	0.776854i	0.921479	−	0.388427i	$-0.126981\pi$
$954$		0			0
$955$	−81618.8	−	47122.6i	−2.76557	−	1.59670i
$956$		0			0
$957$		0			0
$958$		0			0
$959$		0			0
$960$		0			0
$961$	10558.0	+	18287.0i	0.354403	+	0.613845i
$962$		0			0
$963$		0			0
$964$		0			0
$965$	42637.9			1.42234
$966$		0			0
$967$	−11432.2			−0.380180			−0.190090	−	0.981767i	$-0.560878\pi$
$967$	−11432.2			−0.380180			−0.190090	+	0.981767i	$0.560878\pi$
$968$		0			0
$969$		0			0
$970$		0			0
$971$	−9403.48	−	16287.3i	−0.310785	−	0.538295i	0.667748	−	0.744388i	$-0.267259\pi$
$971$	−9403.48	−	16287.3i	−0.310785	−	0.538295i	−0.978533	+	0.206093i	$0.933925\pi$
$972$		0			0
$973$		0			0
$974$		0			0
$975$		0			0
$976$		0			0
$977$	−32098.8	−	18532.3i	−1.05111	−	0.606857i	−0.128148	−	0.991755i	$-0.540903\pi$
$977$	−32098.8	−	18532.3i	−1.05111	−	0.606857i	−0.922959	+	0.384898i	$0.874237\pi$
$978$		0			0
$979$			5549.20i			0.181158i
$980$		0			0
$981$		0			0
$982$		0			0
$983$	−23541.5	+	40775.0i	−0.763841	+	1.32301i	0.177016	+	0.984208i	$0.443356\pi$
$983$	−23541.5	+	40775.0i	−0.763841	+	1.32301i	−0.940857	+	0.338804i	$0.889978\pi$
$984$		0			0
$985$	−12408.1	+	7163.83i	−0.401376	+	0.231735i
$986$		0			0
$987$		0			0
$988$		0			0
$989$	−25616.5	+	14789.7i	−0.823618	+	0.475516i
$990$		0			0
$991$	17117.9	−	29649.0i	0.548705	−	0.950386i	−0.449658	−	0.893201i	$-0.648454\pi$
$991$	17117.9	−	29649.0i	0.548705	−	0.950386i	0.998364	−	0.0571849i	$-0.0182124\pi$
$992$		0			0
$993$		0			0
$994$		0			0
$995$			26080.6i			0.830964i
$996$		0			0
$997$	7484.38	+	4321.11i	0.237746	+	0.137263i	0.614140	−	0.789197i	$-0.289503\pi$
$997$	7484.38	+	4321.11i	0.237746	+	0.137263i	−0.376394	+	0.926460i	$0.622836\pi$
$998$		0			0
$999$		0			0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	1764.4.t.b.521.8		16
3.2	odd	2	inner	1764.4.t.b.521.1		16
7.2	even	3		252.4.t.a.89.8	yes	16
7.3	odd	6		1764.4.f.a.881.16		16
7.4	even	3		1764.4.f.a.881.2		16
7.5	odd	6	inner	1764.4.t.b.1097.1		16
7.6	odd	2		252.4.t.a.17.1	✓	16
21.2	odd	6		252.4.t.a.89.1	yes	16
21.5	even	6	inner	1764.4.t.b.1097.8		16
21.11	odd	6		1764.4.f.a.881.15		16
21.17	even	6		1764.4.f.a.881.1		16
21.20	even	2		252.4.t.a.17.8	yes	16
28.23	odd	6		1008.4.bt.b.593.8		16
28.27	even	2		1008.4.bt.b.17.1		16
84.23	even	6		1008.4.bt.b.593.1		16
84.83	odd	2		1008.4.bt.b.17.8		16

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
252.4.t.a.17.1	✓	16	7.6	odd	2
252.4.t.a.17.8	yes	16	21.20	even	2
252.4.t.a.89.1	yes	16	21.2	odd	6
252.4.t.a.89.8	yes	16	7.2	even	3
1008.4.bt.b.17.1		16	28.27	even	2
1008.4.bt.b.17.8		16	84.83	odd	2
1008.4.bt.b.593.1		16	84.23	even	6
1008.4.bt.b.593.8		16	28.23	odd	6
1764.4.f.a.881.1		16	21.17	even	6
1764.4.f.a.881.2		16	7.4	even	3
1764.4.f.a.881.15		16	21.11	odd	6
1764.4.f.a.881.16		16	7.3	odd	6
1764.4.t.b.521.1		16	3.2	odd	2	inner
1764.4.t.b.521.8		16	1.1	even	1	trivial
1764.4.t.b.1097.1		16	7.5	odd	6	inner
1764.4.t.b.1097.8		16	21.5	even	6	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:	\( N \)	\(=\)	\( 1764 = 2^{2} \cdot 3^{2} \cdot 7^{2} \)
Weight:	\( k \)	\(=\)	\( 4 \)
Character orbit:	\([\chi]\)	\(=\)	1764.t (of order \(6\), degree \(2\), not minimal)

\(f(q)\)	\(=\)	\(q+(10.1259 + 17.5386i) q^{5} +O(q^{10})\)	\(q+(10.1259 + 17.5386i) q^{5} +(-15.4532 - 8.92191i) q^{11} -33.1829i q^{13} +(22.9077 - 39.6772i) q^{17} +(35.0091 - 20.2125i) q^{19} +(-69.7804 + 40.2877i) q^{23} +(-142.568 + 246.935i) q^{25} -233.844i q^{29} +(195.398 + 112.813i) q^{31} +(135.456 + 234.617i) q^{37} +154.432 q^{41} +367.102 q^{43} +(263.852 + 457.006i) q^{47} +(78.8892 + 45.5467i) q^{53} -361.370i q^{55} +(312.768 - 541.729i) q^{59} +(78.8343 - 45.5150i) q^{61} +(581.982 - 336.007i) q^{65} +(-431.509 + 747.395i) q^{67} +303.596i q^{71} +(999.127 + 576.846i) q^{73} +(3.48120 + 6.02961i) q^{79} -815.694 q^{83} +927.844 q^{85} +(-155.494 - 269.323i) q^{89} +(708.998 + 409.340i) q^{95} +1832.92i q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 16 q+O(q^{10}) \) 16 * q	\( 16 q + 72 q^{19} - 212 q^{25} + 708 q^{31} + 76 q^{37} + 1408 q^{43} + 1632 q^{61} - 1528 q^{67} + 2700 q^{73} - 364 q^{79} + 7392 q^{85}+O(q^{100}) \) 16 * q + 72 * q^19 - 212 * q^25 + 708 * q^31 + 76 * q^37 + 1408 * q^43 + 1632 * q^61 - 1528 * q^67 + 2700 * q^73 - 364 * q^79 + 7392 * q^85

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

Related objects

Downloads

Learn more