LMFDB - Embedded newform 531.3.c.c.235.15

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [531,3,Mod(235,531)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(531, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([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("531.235");

S:= CuspForms(chi, 3);

N := Newforms(S);

Level:	$ N $	$=$	$ 531 = 3^{2} \cdot 59 $
Weight:	$ k $	$=$	$ 3 $
Character orbit:	$[\chi]$	$=$	531.c (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:	$14.4687020375$
Analytic rank:	$0$
Dimension:	$20$
Coefficient field:	$\mathbb{Q}[x]/(x^{20} + \cdots)$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{20} + 60 x^{18} + 1522 x^{16} + 21286 x^{14} + 179593 x^{12} + 941588 x^{10} + 3057025 x^{8} + \cdots + 570861 $ x^20 + 60x^18 + 1522x^16 + 21286x^14 + 179593x^12 + 941588x^10 + 3057025x^8 + 5962230x^6 + 6517782x^4 + 3428244*x^2 + 570861
Coefficient ring:	$\Z[a_1, \ldots, a_{7}]$
Coefficient ring index:	$ 2^{11}\cdot 3 $
Twist minimal:	no (minimal twist has level 177)
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label			235.15
Root			$1.86235i$ of defining polynomial
Character	$\chi$	$=$	531.235
Dual form			531.3.c.c.235.6

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+1.86235i q^{2} +0.531642 q^{4} +9.04540 q^{5} +1.29987 q^{7} +8.43952i q^{8} +O(q^{10})$	\(q+1.86235i q^{2} +0.531642 q^{4} +9.04540 q^{5} +1.29987 q^{7} +8.43952i q^{8} +16.8457i q^{10} -12.8790i q^{11} +23.5709i q^{13} +2.42081i q^{14} -13.5908 q^{16} +10.1284 q^{17} -23.3600 q^{19} +4.80892 q^{20} +23.9853 q^{22} -9.25801i q^{23} +56.8193 q^{25} -43.8973 q^{26} +0.691065 q^{28} +25.9681 q^{29} -28.4380i q^{31} +8.44722i q^{32} +18.8627i q^{34} +11.7578 q^{35} +22.3933i q^{37} -43.5046i q^{38} +76.3388i q^{40} +44.4678 q^{41} -42.8166i q^{43} -6.84704i q^{44} +17.2417 q^{46} +30.6792i q^{47} -47.3103 q^{49} +105.818i q^{50} +12.5313i q^{52} -41.9742 q^{53} -116.496i q^{55} +10.9703i q^{56} +48.3618i q^{58} +(2.69759 - 58.9383i) q^{59} -31.9823i q^{61} +52.9616 q^{62} -70.0949 q^{64} +213.208i q^{65} +94.0998i q^{67} +5.38470 q^{68} +21.8972i q^{70} -5.78244 q^{71} +41.5942i q^{73} -41.7043 q^{74} -12.4192 q^{76} -16.7411i q^{77} -105.782 q^{79} -122.934 q^{80} +82.8148i q^{82} +3.14451i q^{83} +91.6157 q^{85} +79.7397 q^{86} +108.693 q^{88} +21.8316i q^{89} +30.6391i q^{91} -4.92194i q^{92} -57.1355 q^{94} -211.301 q^{95} -126.206i q^{97} -88.1085i q^{98} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 20 q - 40 q^{4} - 8 q^{7}+O(q^{10}) $ 20 * q - 40 * q^4 - 8 * q^7	$ 20 q - 40 q^{4} - 8 q^{7} - 8 q^{16} - 16 q^{17} - 60 q^{19} + 164 q^{20} + 40 q^{22} + 100 q^{25} + 156 q^{26} + 200 q^{28} + 60 q^{29} + 32 q^{35} - 28 q^{41} + 180 q^{46} + 284 q^{49} + 8 q^{53} + 152 q^{59} + 8 q^{62} + 204 q^{64} - 384 q^{68} - 92 q^{71} - 104 q^{74} + 120 q^{76} - 420 q^{79} - 376 q^{80} - 348 q^{85} - 232 q^{86} - 212 q^{88} + 152 q^{94} - 788 q^{95}+O(q^{100}) $ 20 * q - 40 * q^4 - 8 * q^7 - 8 * q^16 - 16 * q^17 - 60 * q^19 + 164 * q^20 + 40 * q^22 + 100 * q^25 + 156 * q^26 + 200 * q^28 + 60 * q^29 + 32 * q^35 - 28 * q^41 + 180 * q^46 + 284 * q^49 + 8 * q^53 + 152 * q^59 + 8 * q^62 + 204 * q^64 - 384 * q^68 - 92 * q^71 - 104 * q^74 + 120 * q^76 - 420 * q^79 - 376 * q^80 - 348 * q^85 - 232 * q^86 - 212 * q^88 + 152 * q^94 - 788 * q^95

Display 100 coefficients 10 coefficients

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/531\mathbb{Z}\right)^\times$.

$n$	$119$	$415$
$\chi(n)$	$1$	$-1$

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 $

$2$			1.86235i			0.931176i	0.885001	+	0.465588i	$0.154157\pi$
$2$			1.86235i			0.931176i	−0.885001	+	0.465588i	$0.845843\pi$
$3$		0			0
$3$		0			0
$4$	0.531642			0.132910
$5$	9.04540			1.80908			0.904540	−	0.426388i	$-0.140214\pi$
$5$	9.04540			1.80908			0.904540	+	0.426388i	$0.140214\pi$
$6$		0			0
$7$	1.29987			0.185696			0.0928478	−	0.995680i	$-0.470403\pi$
$7$	1.29987			0.185696			0.0928478	+	0.995680i	$0.470403\pi$
$8$			8.43952i			1.05494i
$9$		0			0
$10$			16.8457i			1.68457i
$11$		−	12.8790i		−	1.17082i	−0.810736	−	0.585411i	$-0.800933\pi$
$11$		−	12.8790i		−	1.17082i	0.810736	−	0.585411i	$-0.199067\pi$
$12$		0			0
$13$			23.5709i			1.81315i	0.422050	+	0.906573i	$0.361311\pi$
$13$			23.5709i			1.81315i	−0.422050	+	0.906573i	$0.638689\pi$
$14$			2.42081i			0.172915i
$15$		0			0
$16$	−13.5908			−0.849424
$17$	10.1284			0.595790			0.297895	−	0.954599i	$-0.403715\pi$
$17$	10.1284			0.595790			0.297895	+	0.954599i	$0.403715\pi$
$18$		0			0
$19$	−23.3600			−1.22948			−0.614738	−	0.788731i	$-0.710738\pi$
$19$	−23.3600			−1.22948			−0.614738	+	0.788731i	$0.710738\pi$
$20$	4.80892			0.240446
$21$		0			0
$22$	23.9853			1.09024
$23$		−	9.25801i		−	0.402522i	−0.979538	−	0.201261i	$-0.935496\pi$
$23$		−	9.25801i		−	0.402522i	0.979538	−	0.201261i	$-0.0645039\pi$
$24$		0			0
$25$	56.8193			2.27277
$26$	−43.8973			−1.68836
$27$		0			0
$28$	0.691065			0.0246809
$29$	25.9681			0.895452			0.447726	−	0.894171i	$-0.352234\pi$
$29$	25.9681			0.895452			0.447726	+	0.894171i	$0.352234\pi$
$30$		0			0
$31$		−	28.4380i		−	0.917355i	−0.888603	−	0.458678i	$-0.848323\pi$
$31$		−	28.4380i		−	0.917355i	0.888603	−	0.458678i	$-0.151677\pi$
$32$			8.44722i			0.263976i
$33$		0			0
$34$			18.8627i			0.554786i
$35$	11.7578			0.335938
$36$		0			0
$37$			22.3933i			0.605225i	0.953114	+	0.302613i	$0.0978588\pi$
$37$			22.3933i			0.605225i	−0.953114	+	0.302613i	$0.902141\pi$
$38$		−	43.5046i		−	1.14486i
$39$		0			0
$40$			76.3388i			1.90847i
$41$	44.4678			1.08458			0.542290	−	0.840191i	$-0.317557\pi$
$41$	44.4678			1.08458			0.542290	+	0.840191i	$0.317557\pi$
$42$		0			0
$43$		−	42.8166i		−	0.995736i	−0.867253	−	0.497868i	$-0.834116\pi$
$43$		−	42.8166i		−	0.995736i	0.867253	−	0.497868i	$-0.165884\pi$
$44$		−	6.84704i		−	0.155615i
$45$		0			0
$46$	17.2417			0.374819
$47$			30.6792i			0.652749i	0.945241	+	0.326374i	$0.105827\pi$
$47$			30.6792i			0.652749i	−0.945241	+	0.326374i	$0.894173\pi$
$48$		0			0
$49$	−47.3103			−0.965517
$50$			105.818i			2.11635i
$51$		0			0
$52$			12.5313i			0.240986i
$53$	−41.9742			−0.791967			−0.395983	−	0.918258i	$-0.629596\pi$
$53$	−41.9742			−0.791967			−0.395983	+	0.918258i	$0.629596\pi$
$54$		0			0
$55$		−	116.496i		−	2.11811i
$56$			10.9703i			0.195898i
$57$		0			0
$58$			48.3618i			0.833824i
$59$	2.69759	−	58.9383i	0.0457219	−	0.998954i
$59$	2.69759	−	58.9383i	0.0457219	−	0.998954i
$60$		0			0
$61$		−	31.9823i		−	0.524300i	−0.965027	−	0.262150i	$-0.915569\pi$
$61$		−	31.9823i		−	0.524300i	0.965027	−	0.262150i	$-0.0844315\pi$
$62$	52.9616			0.854219
$63$		0			0
$64$	−70.0949			−1.09523
$65$			213.208i			3.28013i
$66$		0			0
$67$			94.0998i			1.40447i	0.711943	+	0.702237i	$0.247815\pi$
$67$			94.0998i			1.40447i	−0.711943	+	0.702237i	$0.752185\pi$
$68$	5.38470			0.0791868
$69$		0			0
$70$			21.8972i			0.312818i
$71$	−5.78244			−0.0814428			−0.0407214	−	0.999171i	$-0.512966\pi$
$71$	−5.78244			−0.0814428			−0.0407214	+	0.999171i	$0.512966\pi$
$72$		0			0
$73$			41.5942i			0.569783i	0.958560	+	0.284892i	$0.0919577\pi$
$73$			41.5942i			0.569783i	−0.958560	+	0.284892i	$0.908042\pi$
$74$	−41.7043			−0.563571
$75$		0			0
$76$	−12.4192			−0.163410
$77$		−	16.7411i		−	0.217417i
$78$		0			0
$79$	−105.782			−1.33901			−0.669503	−	0.742809i	$-0.733493\pi$
$79$	−105.782			−1.33901			−0.669503	+	0.742809i	$0.733493\pi$
$80$	−122.934			−1.53668
$81$		0			0
$82$			82.8148i			1.00994i
$83$			3.14451i			0.0378857i	0.999821	+	0.0189428i	$0.00603006\pi$
$83$			3.14451i			0.0378857i	−0.999821	+	0.0189428i	$0.993970\pi$
$84$		0			0
$85$	91.6157			1.07783
$86$	79.7397			0.927206
$87$		0			0
$88$	108.693			1.23515
$89$			21.8316i			0.245299i	0.992450	+	0.122649i	$0.0391391\pi$
$89$			21.8316i			0.245299i	−0.992450	+	0.122649i	$0.960861\pi$
$90$		0			0
$91$			30.6391i			0.336693i
$92$		−	4.92194i		−	0.0534994i
$93$		0			0
$94$	−57.1355			−0.607824
$95$	−211.301			−2.22422
$96$		0			0
$97$		−	126.206i		−	1.30109i	−0.759468	−	0.650544i	$-0.774541\pi$
$97$		−	126.206i		−	1.30109i	0.759468	−	0.650544i	$-0.225459\pi$
$98$		−	88.1085i		−	0.899067i
$99$		0			0
$100$	30.2075			0.302075
$101$		−	71.4450i		−	0.707376i	−0.935363	−	0.353688i	$-0.884927\pi$
$101$		−	71.4450i		−	0.707376i	0.935363	−	0.353688i	$-0.115073\pi$
$102$		0			0
$103$			166.064i			1.61228i	0.591728	+	0.806138i	$0.298446\pi$
$103$			166.064i			1.61228i	−0.591728	+	0.806138i	$0.701554\pi$
$104$	−198.927			−1.91276
$105$		0			0
$106$		−	78.1708i		−	0.737461i
$107$	−27.6484			−0.258396			−0.129198	−	0.991619i	$-0.541240\pi$
$107$	−27.6484			−0.258396			−0.129198	+	0.991619i	$0.541240\pi$
$108$		0			0
$109$		−	110.232i		−	1.01130i	−0.862738	−	0.505651i	$-0.831252\pi$
$109$		−	110.232i		−	1.01130i	0.862738	−	0.505651i	$-0.168748\pi$
$110$	216.957			1.97234
$111$		0			0
$112$	−17.6662			−0.157734
$113$		−	69.5553i		−	0.615534i	−0.951462	−	0.307767i	$-0.900418\pi$
$113$		−	69.5553i		−	0.615534i	0.951462	−	0.307767i	$-0.0995817\pi$
$114$		0			0
$115$		−	83.7424i		−	0.728195i
$116$	13.8057			0.119015
$117$		0			0
$118$	109.764	+	5.02387i	0.930203	+	0.0425752i
$119$	13.1656			0.110636
$120$		0			0
$121$	−44.8699			−0.370826
$122$	59.5623			0.488215
$123$		0			0
$124$		−	15.1188i		−	0.121926i
$125$	287.818			2.30255
$126$		0			0
$127$	218.788			1.72274			0.861370	−	0.507977i	$-0.169607\pi$
$127$	218.788			1.72274			0.861370	+	0.507977i	$0.169607\pi$
$128$		−	96.7525i		−	0.755879i
$129$		0			0
$130$	−397.069			−3.05438
$131$		−	245.998i		−	1.87785i	−0.344127	−	0.938923i	$-0.611825\pi$
$131$		−	245.998i		−	1.87785i	0.344127	−	0.938923i	$-0.388175\pi$
$132$		0			0
$133$	−30.3650			−0.228308
$134$	−175.247			−1.30781
$135$		0			0
$136$			85.4791i			0.628522i
$137$	−250.848			−1.83101			−0.915504	−	0.402308i	$-0.868208\pi$
$137$	−250.848			−1.83101			−0.915504	+	0.402308i	$0.868208\pi$
$138$		0			0
$139$	−14.5923			−0.104981			−0.0524903	−	0.998621i	$-0.516716\pi$
$139$	−14.5923			−0.104981			−0.0524903	+	0.998621i	$0.516716\pi$
$140$	6.25096			0.0446497
$141$		0			0
$142$		−	10.7689i		−	0.0758376i
$143$	303.571			2.12287
$144$		0			0
$145$	234.892			1.61995
$146$	−77.4631			−0.530569
$147$		0			0
$148$			11.9052i			0.0804408i
$149$			182.069i			1.22194i	0.791653	+	0.610971i	$0.209221\pi$
$149$			182.069i			1.22194i	−0.791653	+	0.610971i	$0.790779\pi$
$150$		0			0
$151$		−	31.7495i		−	0.210262i	−0.994458	−	0.105131i	$-0.966474\pi$
$151$		−	31.7495i		−	0.210262i	0.994458	−	0.105131i	$-0.0335262\pi$
$152$		−	197.147i		−	1.29702i
$153$		0			0
$154$	31.1778			0.202453
$155$		−	257.233i		−	1.65957i
$156$		0			0
$157$		−	243.661i		−	1.55198i	−0.630744	−	0.775991i	$-0.717250\pi$
$157$		−	243.661i		−	1.55198i	0.630744	−	0.775991i	$-0.282750\pi$
$158$		−	197.002i		−	1.24685i
$159$		0			0
$160$			76.4085i			0.477553i
$161$		−	12.0342i		−	0.0747465i
$162$		0			0
$163$	−9.01747			−0.0553219			−0.0276610	−	0.999617i	$-0.508806\pi$
$163$	−9.01747			−0.0553219			−0.0276610	+	0.999617i	$0.508806\pi$
$164$	23.6410			0.144152
$165$		0			0
$166$	−5.85619			−0.0352783
$167$	134.849			0.807482			0.403741	−	0.914873i	$-0.367710\pi$
$167$	134.849			0.807482			0.403741	+	0.914873i	$0.367710\pi$
$168$		0			0
$169$	−386.587			−2.28750
$170$			170.621i			1.00365i
$171$		0			0
$172$		−	22.7631i		−	0.132344i
$173$			273.027i			1.57819i	0.614271	+	0.789095i	$0.289450\pi$
$173$			273.027i			1.57819i	−0.614271	+	0.789095i	$0.710550\pi$
$174$		0			0
$175$	73.8576			0.422044
$176$			175.036i			0.994525i
$177$		0			0
$178$	−40.6581			−0.228417
$179$		−	110.268i		−	0.616025i	−0.951382	−	0.308012i	$-0.900336\pi$
$179$		−	110.268i		−	0.616025i	0.951382	−	0.308012i	$-0.0996638\pi$
$180$		0			0
$181$	21.7461			0.120144			0.0600721	−	0.998194i	$-0.480867\pi$
$181$	21.7461			0.120144			0.0600721	+	0.998194i	$0.480867\pi$
$182$	−57.0607			−0.313521
$183$		0			0
$184$	78.1331			0.424636
$185$			202.557i			1.09490i
$186$		0			0
$187$		−	130.445i		−	0.697564i
$188$			16.3104i			0.0867572i
$189$		0			0
$190$		−	393.517i		−	2.07114i
$191$			25.4664i			0.133332i	0.997775	+	0.0666660i	$0.0212362\pi$
$191$			25.4664i			0.133332i	−0.997775	+	0.0666660i	$0.978764\pi$
$192$		0			0
$193$	−140.722			−0.729130			−0.364565	−	0.931178i	$-0.618782\pi$
$193$	−140.722			−0.729130			−0.364565	+	0.931178i	$0.618782\pi$
$194$	235.039			1.21154
$195$		0			0
$196$	−25.1522			−0.128327
$197$	221.365			1.12368			0.561841	−	0.827245i	$-0.310093\pi$
$197$	221.365			1.12368			0.561841	+	0.827245i	$0.310093\pi$
$198$		0			0
$199$	282.321			1.41870			0.709348	−	0.704858i	$-0.248989\pi$
$199$	282.321			1.41870			0.709348	+	0.704858i	$0.248989\pi$
$200$			479.527i			2.39764i
$201$		0			0
$202$	133.056			0.658692
$203$	33.7551			0.166282
$204$		0			0
$205$	402.229			1.96209
$206$	−309.270			−1.50131
$207$		0			0
$208$		−	320.347i		−	1.54013i
$209$			300.855i			1.43950i
$210$		0			0
$211$		−	2.24194i		−	0.0106253i	−0.999986	−	0.00531265i	$-0.998309\pi$
$211$		−	2.24194i		−	0.0106253i	0.999986	−	0.00531265i	$-0.00169108\pi$
$212$	−22.3153			−0.105261
$213$		0			0
$214$		−	51.4910i		−	0.240612i
$215$		−	387.294i		−	1.80137i
$216$		0			0
$217$		−	36.9657i		−	0.170349i
$218$	205.291			0.941701
$219$		0			0
$220$		−	61.9343i		−	0.281519i
$221$			238.736i			1.08025i
$222$		0			0
$223$	−313.196			−1.40447			−0.702234	−	0.711946i	$-0.747814\pi$
$223$	−313.196			−1.40447			−0.702234	+	0.711946i	$0.747814\pi$
$224$			10.9803i			0.0490191i
$225$		0			0
$226$	129.537			0.573170
$227$		−	398.887i		−	1.75721i	−0.477548	−	0.878605i	$-0.658474\pi$
$227$		−	398.887i		−	1.75721i	0.477548	−	0.878605i	$-0.341526\pi$
$228$		0			0
$229$		−	52.2219i		−	0.228043i	−0.993478	−	0.114022i	$-0.963627\pi$
$229$		−	52.2219i		−	0.228043i	0.993478	−	0.114022i	$-0.0363733\pi$
$230$	155.958			0.678078
$231$		0			0
$232$			219.158i			0.944648i
$233$		−	420.216i		−	1.80350i	−0.432255	−	0.901751i	$-0.642282\pi$
$233$		−	420.216i		−	1.80350i	0.432255	−	0.901751i	$-0.357718\pi$
$234$		0			0
$235$			277.506i			1.18088i
$236$	1.43415	−	31.3341i	0.00607692	−	0.132771i
$237$		0			0
$238$			24.5191i			0.103021i
$239$	157.064			0.657171			0.328585	−	0.944474i	$-0.393428\pi$
$239$	157.064			0.657171			0.328585	+	0.944474i	$0.393428\pi$
$240$		0			0
$241$	97.2511			0.403531			0.201766	−	0.979434i	$-0.435332\pi$
$241$	97.2511			0.403531			0.201766	+	0.979434i	$0.435332\pi$
$242$		−	83.5636i		−	0.345304i
$243$		0			0
$244$		−	17.0031i		−	0.0696849i
$245$	−427.941			−1.74670
$246$		0			0
$247$		−	550.617i		−	2.22922i
$248$	240.003			0.967754
$249$		0			0
$250$			536.019i			2.14408i
$251$	−298.241			−1.18821			−0.594105	−	0.804387i	$-0.702494\pi$
$251$	−298.241			−1.18821			−0.594105	+	0.804387i	$0.702494\pi$
$252$		0			0
$253$	−119.234			−0.471282
$254$			407.461i			1.60418i
$255$		0			0
$256$	−100.192			−0.391376
$257$	−398.043			−1.54881			−0.774403	−	0.632692i	$-0.781950\pi$
$257$	−398.043			−1.54881			−0.774403	+	0.632692i	$0.781950\pi$
$258$		0			0
$259$			29.1084i			0.112388i
$260$			113.350i			0.435963i
$261$		0			0
$262$	458.135			1.74861
$263$	−205.829			−0.782620			−0.391310	−	0.920259i	$-0.627978\pi$
$263$	−205.829			−0.782620			−0.391310	+	0.920259i	$0.627978\pi$
$264$		0			0
$265$	−379.674			−1.43273
$266$		−	56.5503i		−	0.212595i
$267$		0			0
$268$			50.0274i			0.186669i
$269$		−	161.097i		−	0.598874i	−0.954116	−	0.299437i	$-0.903201\pi$
$269$		−	161.097i		−	0.598874i	0.954116	−	0.299437i	$-0.0967989\pi$
$270$		0			0
$271$	−222.809			−0.822173			−0.411086	−	0.911596i	$-0.634851\pi$
$271$	−222.809			−0.822173			−0.411086	+	0.911596i	$0.634851\pi$
$272$	−137.653			−0.506079
$273$		0			0
$274$		−	467.168i		−	1.70499i
$275$		−	731.779i		−	2.66101i
$276$		0			0
$277$	−407.546			−1.47128			−0.735642	−	0.677371i	$-0.763119\pi$
$277$	−407.546			−1.47128			−0.735642	+	0.677371i	$0.763119\pi$
$278$		−	27.1760i		−	0.0977555i
$279$		0			0
$280$			99.2304i			0.354394i
$281$	374.820			1.33388			0.666940	−	0.745112i	$-0.267604\pi$
$281$	374.820			1.33388			0.666940	+	0.745112i	$0.267604\pi$
$282$		0			0
$283$			361.855i			1.27864i	0.768941	+	0.639320i	$0.220784\pi$
$283$			361.855i			1.27864i	−0.768941	+	0.639320i	$0.779216\pi$
$284$	−3.07419			−0.0108246
$285$		0			0
$286$			565.356i			1.97677i
$287$	57.8023			0.201402
$288$		0			0
$289$	−186.415			−0.645034
$290$			437.452i			1.50845i
$291$		0			0
$292$			22.1132i			0.0757302i
$293$	−299.087			−1.02078			−0.510388	−	0.859944i	$-0.670498\pi$
$293$	−299.087			−1.02078			−0.510388	+	0.859944i	$0.670498\pi$
$294$		0			0
$295$	24.4008	−	533.121i	0.0827146	−	1.80719i
$296$	−188.989			−0.638476
$297$		0			0
$298$	−339.078			−1.13784
$299$	218.219			0.729831
$300$		0			0
$301$		−	55.6560i		−	0.184904i
$302$	59.1288			0.195791
$303$		0			0
$304$	317.481			1.04435
$305$		−	289.293i		−	0.948500i
$306$		0			0
$307$	183.459			0.597585			0.298793	−	0.954318i	$-0.403416\pi$
$307$	183.459			0.597585			0.298793	+	0.954318i	$0.403416\pi$
$308$		−	8.90026i		−	0.0288969i
$309$		0			0
$310$	479.059			1.54535
$311$	198.361			0.637816			0.318908	−	0.947786i	$-0.396684\pi$
$311$	198.361			0.637816			0.318908	+	0.947786i	$0.396684\pi$
$312$		0			0
$313$		−	81.4573i		−	0.260247i	−0.991498	−	0.130124i	$-0.958463\pi$
$313$		−	81.4573i		−	0.260247i	0.991498	−	0.130124i	$-0.0415374\pi$
$314$	453.783			1.44517
$315$		0			0
$316$	−56.2379			−0.177968
$317$	272.467			0.859516			0.429758	−	0.902944i	$-0.358599\pi$
$317$	272.467			0.859516			0.429758	+	0.902944i	$0.358599\pi$
$318$		0			0
$319$		−	334.445i		−	1.04842i
$320$	−634.036			−1.98136
$321$		0			0
$322$	22.4119			0.0696022
$323$	−236.601			−0.732509
$324$		0			0
$325$			1339.28i			4.12087i
$326$		−	16.7937i		−	0.0515145i
$327$		0			0
$328$			375.287i			1.14417i
$329$			39.8789i			0.121213i
$330$		0			0
$331$	−145.462			−0.439463			−0.219732	−	0.975560i	$-0.570518\pi$
$331$	−145.462			−0.439463			−0.219732	+	0.975560i	$0.570518\pi$
$332$			1.67175i			0.00503541i
$333$		0			0
$334$			251.137i			0.751908i
$335$			851.170i			2.54081i
$336$		0			0
$337$			174.373i			0.517427i	0.965954	+	0.258714i	$0.0832986\pi$
$337$			174.373i			0.517427i	−0.965954	+	0.258714i	$0.916701\pi$
$338$		−	719.961i		−	2.13006i
$339$		0			0
$340$	48.7068			0.143255
$341$	−366.255			−1.07406
$342$		0			0
$343$	−125.191			−0.364988
$344$	361.352			1.05044
$345$		0			0
$346$	−508.472			−1.46957
$347$			366.795i			1.05705i	0.848919	+	0.528524i	$0.177254\pi$
$347$			366.795i			1.05705i	−0.848919	+	0.528524i	$0.822746\pi$
$348$		0			0
$349$			208.270i			0.596763i	0.954447	+	0.298382i	$0.0964468\pi$
$349$			208.270i			0.596763i	−0.954447	+	0.298382i	$0.903553\pi$
$350$			137.549i			0.392997i
$351$		0			0
$352$	108.792			0.309069
$353$		−	174.980i		−	0.495695i	−0.968799	−	0.247847i	$-0.920277\pi$
$353$		−	174.980i		−	0.495695i	0.968799	−	0.247847i	$-0.0797231\pi$
$354$		0			0
$355$	−52.3045			−0.147337
$356$			11.6066i			0.0326028i
$357$		0			0
$358$	205.359			0.573628
$359$	−203.293			−0.566275			−0.283137	−	0.959079i	$-0.591375\pi$
$359$	−203.293			−0.566275			−0.283137	+	0.959079i	$0.591375\pi$
$360$		0			0
$361$	184.691			0.511610
$362$			40.4989i			0.111876i
$363$		0			0
$364$			16.2890i			0.0447500i
$365$			376.236i			1.03078i
$366$		0			0
$367$			514.435i			1.40173i	0.713293	+	0.700866i	$0.247203\pi$
$367$			514.435i			1.40173i	−0.713293	+	0.700866i	$0.752797\pi$
$368$			125.824i			0.341912i
$369$		0			0
$370$	−377.232			−1.01955
$371$	−54.5610			−0.147065
$372$		0			0
$373$	631.405			1.69278			0.846388	−	0.532567i	$-0.178773\pi$
$373$	631.405			1.69278			0.846388	+	0.532567i	$0.178773\pi$
$374$	242.934			0.649556
$375$		0			0
$376$	−258.918			−0.688611
$377$			612.092i			1.62359i
$378$		0			0
$379$	196.674			0.518930			0.259465	−	0.965753i	$-0.416454\pi$
$379$	196.674			0.518930			0.259465	+	0.965753i	$0.416454\pi$
$380$	−112.336			−0.295622
$381$		0			0
$382$	−47.4275			−0.124156
$383$	127.877			0.333883			0.166941	−	0.985967i	$-0.446611\pi$
$383$	127.877			0.333883			0.166941	+	0.985967i	$0.446611\pi$
$384$		0			0
$385$		−	151.430i		−	0.393324i
$386$		−	262.074i		−	0.678949i
$387$		0			0
$388$		−	67.0962i		−	0.172928i
$389$	−188.964			−0.485770			−0.242885	−	0.970055i	$-0.578094\pi$
$389$	−188.964			−0.485770			−0.242885	+	0.970055i	$0.578094\pi$
$390$		0			0
$391$		−	93.7691i		−	0.239819i
$392$		−	399.276i		−	1.01856i
$393$		0			0
$394$			412.261i			1.04635i
$395$	−956.836			−2.42237
$396$		0			0
$397$		−	507.775i		−	1.27903i	−0.768779	−	0.639515i	$-0.779135\pi$
$397$		−	507.775i		−	1.27903i	0.768779	−	0.639515i	$-0.220865\pi$
$398$			525.780i			1.32106i
$399$		0			0
$400$	−772.219			−1.93055
$401$		−	427.106i		−	1.06510i	−0.846398	−	0.532551i	$-0.821233\pi$
$401$		−	427.106i		−	1.06510i	0.846398	−	0.532551i	$-0.178767\pi$
$402$		0			0
$403$	670.309			1.66330
$404$		−	37.9832i		−	0.0940177i
$405$		0			0
$406$			62.8640i			0.154837i
$407$	288.405			0.708611
$408$		0			0
$409$		−	154.410i		−	0.377530i	−0.982022	−	0.188765i	$-0.939551\pi$
$409$		−	154.410i		−	0.377530i	0.982022	−	0.188765i	$-0.0604485\pi$
$410$			749.093i			1.82706i
$411$		0			0
$412$			88.2868i			0.214288i
$413$	3.50652	−	76.6121i	0.00849035	−	0.185501i
$414$		0			0
$415$			28.4434i			0.0685383i
$416$	−199.108			−0.478626
$417$		0			0
$418$	−560.298			−1.34043
$419$			381.674i			0.910917i	0.890257	+	0.455458i	$0.150525\pi$
$419$			381.674i			0.910917i	−0.890257	+	0.455458i	$0.849475\pi$
$420$		0			0
$421$			223.524i			0.530935i	0.964120	+	0.265468i	$0.0855263\pi$
$421$			223.524i			0.530935i	−0.964120	+	0.265468i	$0.914474\pi$
$422$	4.17528			0.00989403
$423$		0			0
$424$		−	354.242i		−	0.835477i
$425$	575.490			1.35410
$426$		0			0
$427$		−	41.5728i		−	0.0973601i
$428$	−14.6990			−0.0343435
$429$		0			0
$430$	721.278			1.67739
$431$			59.8456i			0.138853i	0.997587	+	0.0694264i	$0.0221169\pi$
$431$			59.8456i			0.138853i	−0.997587	+	0.0694264i	$0.977883\pi$
$432$		0			0
$433$	654.088			1.51060			0.755298	−	0.655381i	$-0.227492\pi$
$433$	654.088			1.51060			0.755298	+	0.655381i	$0.227492\pi$
$434$	68.8431			0.158625
$435$		0			0
$436$		−	58.6039i		−	0.134413i
$437$			216.267i			0.494891i
$438$		0			0
$439$	308.578			0.702910			0.351455	−	0.936205i	$-0.385687\pi$
$439$	308.578			0.702910			0.351455	+	0.936205i	$0.385687\pi$
$440$	983.171			2.23448
$441$		0			0
$442$	−444.611			−1.00591
$443$		−	94.0704i		−	0.212348i	−0.994348	−	0.106174i	$-0.966140\pi$
$443$		−	94.0704i		−	0.212348i	0.994348	−	0.106174i	$-0.0338601\pi$
$444$		0			0
$445$			197.476i			0.443765i
$446$		−	583.282i		−	1.30781i
$447$		0			0
$448$	−91.1141			−0.203380
$449$	447.832			0.997399			0.498699	−	0.866775i	$-0.333811\pi$
$449$	447.832			0.997399			0.498699	+	0.866775i	$0.333811\pi$
$450$		0			0
$451$		−	572.703i		−	1.26985i
$452$		−	36.9785i		−	0.0818109i
$453$		0			0
$454$	742.868			1.63627
$455$			277.143i			0.609105i
$456$		0			0
$457$		−	351.865i		−	0.769946i	−0.922928	−	0.384973i	$-0.874211\pi$
$457$		−	351.865i		−	0.769946i	0.922928	−	0.384973i	$-0.125789\pi$
$458$	97.2555			0.212348
$459$		0			0
$460$		−	44.5210i		−	0.0967847i
$461$	350.814			0.760984			0.380492	−	0.924784i	$-0.375755\pi$
$461$	350.814			0.760984			0.380492	+	0.924784i	$0.375755\pi$
$462$		0			0
$463$			321.469i			0.694318i	0.937806	+	0.347159i	$0.112854\pi$
$463$			321.469i			0.694318i	−0.937806	+	0.347159i	$0.887146\pi$
$464$	−352.927			−0.760619
$465$		0			0
$466$	782.591			1.67938
$467$		−	631.306i		−	1.35183i	−0.736978	−	0.675916i	$-0.763748\pi$
$467$		−	631.306i		−	1.35183i	0.736978	−	0.675916i	$-0.236252\pi$
$468$		0			0
$469$			122.317i			0.260805i
$470$	−516.814			−1.09960
$471$		0			0
$472$	497.411	+	22.7664i	1.05384	+	0.0482338i
$473$	−551.438			−1.16583
$474$		0			0
$475$	−1327.30			−2.79432
$476$	6.99940			0.0147046
$477$		0			0
$478$			292.508i			0.611942i
$479$	134.615			0.281032			0.140516	−	0.990078i	$-0.455124\pi$
$479$	134.615			0.281032			0.140516	+	0.990078i	$0.455124\pi$
$480$		0			0
$481$	−527.831			−1.09736
$482$			181.116i			0.375759i
$483$		0			0
$484$	−23.8547			−0.0492866
$485$		−	1141.58i		−	2.35377i
$486$		0			0
$487$	−739.718			−1.51893			−0.759464	−	0.650549i	$-0.774539\pi$
$487$	−739.718			−1.51893			−0.759464	+	0.650549i	$0.774539\pi$
$488$	269.915			0.553104
$489$		0			0
$490$		−	796.977i		−	1.62648i
$491$	424.454			0.864469			0.432234	−	0.901761i	$-0.357725\pi$
$491$	424.454			0.864469			0.432234	+	0.901761i	$0.357725\pi$
$492$		0			0
$493$	263.016			0.533502
$494$	1025.44			2.07580
$495$		0			0
$496$			386.495i			0.779224i
$497$	−7.51641			−0.0151236
$498$		0			0
$499$	10.2031			0.0204471			0.0102236	−	0.999948i	$-0.496746\pi$
$499$	10.2031			0.0204471			0.0102236	+	0.999948i	$0.496746\pi$
$500$	153.016			0.306033
$501$		0			0
$502$		−	555.429i		−	1.10643i
$503$			188.722i			0.375193i	0.982246	+	0.187596i	$0.0600697\pi$
$503$			188.722i			0.375193i	−0.982246	+	0.187596i	$0.939930\pi$
$504$		0			0
$505$		−	646.249i		−	1.27970i
$506$		−	222.056i		−	0.438847i
$507$		0			0
$508$	116.317			0.228970
$509$			939.054i			1.84490i	0.386116	+	0.922450i	$0.373816\pi$
$509$			939.054i			1.84490i	−0.386116	+	0.922450i	$0.626184\pi$
$510$		0			0
$511$			54.0670i			0.105806i
$512$		−	573.603i		−	1.12032i
$513$		0			0
$514$		−	741.297i		−	1.44221i
$515$			1502.12i			2.91674i
$516$		0			0
$517$	395.119			0.764253
$518$	−54.2101			−0.104653
$519$		0			0
$520$	−1799.37			−3.46033
$521$	7.53069			0.0144543			0.00722715	−	0.999974i	$-0.497700\pi$
$521$	7.53069			0.0144543			0.00722715	+	0.999974i	$0.497700\pi$
$522$		0			0
$523$	−843.214			−1.61226			−0.806132	−	0.591736i	$-0.798443\pi$
$523$	−843.214			−1.61226			−0.806132	+	0.591736i	$0.798443\pi$
$524$		−	130.783i		−	0.249585i
$525$		0			0
$526$		−	383.326i		−	0.728757i
$527$		−	288.032i		−	0.546551i
$528$		0			0
$529$	443.289			0.837976
$530$		−	707.087i		−	1.33413i
$531$		0			0
$532$	−16.1433			−0.0303445
$533$			1048.15i			1.96650i
$534$		0			0
$535$	−250.091			−0.467459
$536$	−794.156			−1.48164
$537$		0			0
$538$	300.020			0.557658
$539$			609.312i			1.13045i
$540$		0			0
$541$			895.843i			1.65590i	0.560800	+	0.827951i	$0.310494\pi$
$541$			895.843i			1.65590i	−0.560800	+	0.827951i	$0.689506\pi$
$542$		−	414.949i		−	0.765588i
$543$		0			0
$544$			85.5571i			0.157274i
$545$		−	997.092i		−	1.82953i
$546$		0			0
$547$	217.435			0.397504			0.198752	−	0.980050i	$-0.436311\pi$
$547$	217.435			0.397504			0.198752	+	0.980050i	$0.436311\pi$
$548$	−133.361			−0.243360
$549$		0			0
$550$	1362.83			2.47787
$551$	−606.616			−1.10094
$552$		0			0
$553$	−137.502			−0.248648
$554$		−	758.994i		−	1.37002i
$555$		0			0
$556$	−7.75788			−0.0139530
$557$	946.289			1.69890			0.849451	−	0.527667i	$-0.176933\pi$
$557$	946.289			1.69890			0.849451	+	0.527667i	$0.176933\pi$
$558$		0			0
$559$	1009.23			1.80541
$560$	−159.798			−0.285354
$561$		0			0
$562$			698.047i			1.24208i
$563$		−	273.838i		−	0.486391i	−0.969977	−	0.243195i	$-0.921804\pi$
$563$		−	273.838i		−	0.486391i	0.969977	−	0.243195i	$-0.0781956\pi$
$564$		0			0
$565$		−	629.156i		−	1.11355i
$566$	−673.902			−1.19064
$567$		0			0
$568$		−	48.8010i		−	0.0859173i
$569$			652.571i			1.14687i	0.819250	+	0.573437i	$0.194390\pi$
$569$			652.571i			1.14687i	−0.819250	+	0.573437i	$0.805610\pi$
$570$		0			0
$571$			373.051i			0.653329i	0.945140	+	0.326664i	$0.105925\pi$
$571$			373.051i			0.653329i	−0.945140	+	0.326664i	$0.894075\pi$
$572$	161.391			0.282152
$573$		0			0
$574$			107.648i			0.187541i
$575$		−	526.033i		−	0.914841i
$576$		0			0
$577$	−565.048			−0.979286			−0.489643	−	0.871923i	$-0.662873\pi$
$577$	−565.048			−0.979286			−0.489643	+	0.871923i	$0.662873\pi$
$578$		−	347.170i		−	0.600641i
$579$		0			0
$580$	124.878			0.215308
$581$			4.08745i			0.00703520i
$582$		0			0
$583$			540.588i			0.927253i
$584$	−351.035			−0.601087
$585$		0			0
$586$		−	557.006i		−	0.950523i
$587$		−	34.5560i		−	0.0588688i	−0.999567	−	0.0294344i	$-0.990629\pi$
$587$		−	34.5560i		−	0.0588688i	0.999567	−	0.0294344i	$-0.00937062\pi$
$588$		0			0
$589$			664.313i			1.12787i
$590$	992.859	+	45.4429i	1.68281	+	0.0770219i
$591$		0			0
$592$		−	304.343i		−	0.514093i
$593$	−179.049			−0.301937			−0.150968	−	0.988539i	$-0.548239\pi$
$593$	−179.049			−0.301937			−0.150968	+	0.988539i	$0.548239\pi$
$594$		0			0
$595$	119.088			0.200149
$596$			96.7958i			0.162409i
$597$		0			0
$598$			406.402i			0.679601i
$599$	−41.4253			−0.0691575			−0.0345787	−	0.999402i	$-0.511009\pi$
$599$	−41.4253			−0.0691575			−0.0345787	+	0.999402i	$0.511009\pi$
$600$		0			0
$601$			107.487i			0.178847i	0.995994	+	0.0894236i	$0.0285025\pi$
$601$			107.487i			0.178847i	−0.995994	+	0.0894236i	$0.971498\pi$
$602$	103.651			0.172178
$603$		0			0
$604$		−	16.8794i		−	0.0279460i
$605$	−405.866			−0.670853
$606$		0			0
$607$	594.804			0.979908			0.489954	−	0.871748i	$-0.337014\pi$
$607$	594.804			0.979908			0.489954	+	0.871748i	$0.337014\pi$
$608$		−	197.327i		−	0.324552i
$609$		0			0
$610$	538.765			0.883221
$611$	−723.136			−1.18353
$612$		0			0
$613$			898.671i			1.46602i	0.680217	+	0.733011i	$0.261885\pi$
$613$			898.671i			1.46602i	−0.680217	+	0.733011i	$0.738115\pi$
$614$			341.665i			0.556457i
$615$		0			0
$616$	141.287			0.229361
$617$	−479.889			−0.777778			−0.388889	−	0.921285i	$-0.627141\pi$
$617$	−479.889			−0.777778			−0.388889	+	0.921285i	$0.627141\pi$
$618$		0			0
$619$	−622.793			−1.00613			−0.503064	−	0.864249i	$-0.667794\pi$
$619$	−622.793			−1.00613			−0.503064	+	0.864249i	$0.667794\pi$
$620$		−	136.756i		−	0.220574i
$621$		0			0
$622$			369.418i			0.593920i
$623$			28.3782i			0.0455509i
$624$		0			0
$625$	1182.95			1.89272
$626$	151.702			0.242336
$627$		0			0
$628$		−	129.540i		−	0.206275i
$629$			226.809i			0.360587i
$630$		0			0
$631$	739.694			1.17226			0.586128	−	0.810218i	$-0.300652\pi$
$631$	739.694			1.17226			0.586128	+	0.810218i	$0.300652\pi$
$632$		−	892.745i		−	1.41257i
$633$		0			0
$634$			507.429i			0.800361i
$635$	1979.03			3.11658
$636$		0			0
$637$		−	1115.15i		−	1.75062i
$638$	622.854			0.976260
$639$		0			0
$640$		−	875.165i		−	1.36745i
$641$	−509.503			−0.794857			−0.397428	−	0.917633i	$-0.630097\pi$
$641$	−509.503			−0.794857			−0.397428	+	0.917633i	$0.630097\pi$
$642$		0			0
$643$	−1038.83			−1.61560			−0.807802	−	0.589454i	$-0.799343\pi$
$643$	−1038.83			−1.61560			−0.807802	+	0.589454i	$0.799343\pi$
$644$		−	6.39788i		−	0.00993460i
$645$		0			0
$646$		−	440.634i		−	0.682095i
$647$	−113.373			−0.175228			−0.0876142	−	0.996154i	$-0.527924\pi$
$647$	−113.373			−0.175228			−0.0876142	+	0.996154i	$0.527924\pi$
$648$		0			0
$649$	−759.069	−	34.7424i	−1.16960	−	0.0535322i
$650$	−2494.21			−3.83725
$651$		0			0
$652$	−4.79407			−0.00735286
$653$	196.978			0.301652			0.150826	−	0.988560i	$-0.451807\pi$
$653$	196.978			0.301652			0.150826	+	0.988560i	$0.451807\pi$
$654$		0			0
$655$		−	2225.15i		−	3.39717i
$656$	−604.353			−0.921269
$657$		0			0
$658$	−74.2686			−0.112870
$659$		−	12.7484i		−	0.0193451i	−0.999953	−	0.00967254i	$-0.996921\pi$
$659$		−	12.7484i		−	0.0193451i	0.999953	−	0.00967254i	$-0.00307891\pi$
$660$		0			0
$661$	−617.452			−0.934118			−0.467059	−	0.884226i	$-0.654686\pi$
$661$	−617.452			−0.934118			−0.467059	+	0.884226i	$0.654686\pi$
$662$		−	270.902i		−	0.409218i
$663$		0			0
$664$	−26.5382			−0.0399671
$665$	−274.663			−0.413028
$666$		0			0
$667$		−	240.413i		−	0.360439i
$668$	71.6917			0.107323
$669$		0			0
$670$	−1585.18			−2.36594
$671$	−411.901			−0.613862
$672$		0			0
$673$		−	525.509i		−	0.780845i	−0.920636	−	0.390423i	$-0.872329\pi$
$673$		−	525.509i		−	0.780845i	0.920636	−	0.390423i	$-0.127671\pi$
$674$	−324.744			−0.481816
$675$		0			0
$676$	−205.526			−0.304032
$677$	373.054			0.551040			0.275520	−	0.961295i	$-0.411150\pi$
$677$	373.054			0.551040			0.275520	+	0.961295i	$0.411150\pi$
$678$		0			0
$679$		−	164.051i		−	0.241606i
$680$			773.192i			1.13705i
$681$		0			0
$682$		−	682.095i		−	1.00014i
$683$		−	602.186i		−	0.881679i	−0.897586	−	0.440839i	$-0.854681\pi$
$683$		−	602.186i		−	0.881679i	0.897586	−	0.440839i	$-0.145319\pi$
$684$		0			0
$685$	−2269.02			−3.31244
$686$		−	233.149i		−	0.339868i
$687$		0			0
$688$			581.912i			0.845802i
$689$		−	989.370i		−	1.43595i
$690$		0			0
$691$			1077.20i			1.55889i	0.626469	+	0.779447i	$0.284500\pi$
$691$			1077.20i			1.55889i	−0.626469	+	0.779447i	$0.715500\pi$
$692$			145.153i			0.209758i
$693$		0			0
$694$	−683.103			−0.984298
$695$	−131.993			−0.189918
$696$		0			0
$697$	450.389			0.646182
$698$	−387.873			−0.555692
$699$		0			0
$700$	39.2658			0.0560940
$701$		−	228.502i		−	0.325965i	−0.986629	−	0.162983i	$-0.947889\pi$
$701$		−	228.502i		−	0.325965i	0.986629	−	0.162983i	$-0.0521115\pi$
$702$		0			0
$703$		−	523.109i		−	0.744109i
$704$			902.755i			1.28232i
$705$		0			0
$706$	325.875			0.461579
$707$		−	92.8691i		−	0.131357i
$708$		0			0
$709$	−162.647			−0.229403			−0.114701	−	0.993400i	$-0.536591\pi$
$709$	−162.647			−0.229403			−0.114701	+	0.993400i	$0.536591\pi$
$710$		−	97.4094i		−	0.137196i
$711$		0			0
$712$	−184.248			−0.258775
$713$	−263.279			−0.369256
$714$		0			0
$715$	2745.92			3.84045
$716$		−	58.6233i		−	0.0818762i
$717$		0			0
$718$		−	378.603i		−	0.527302i
$719$		−	719.737i		−	1.00102i	−0.865729	−	0.500512i	$-0.833145\pi$
$719$		−	719.737i		−	1.00102i	0.865729	−	0.500512i	$-0.166855\pi$
$720$		0			0
$721$			215.862i			0.299392i
$722$			343.961i			0.476400i
$723$		0			0
$724$	11.5611			0.0159684
$725$	1475.49			2.03516
$726$		0			0
$727$	−484.589			−0.666559			−0.333280	−	0.942828i	$-0.608155\pi$
$727$	−484.589			−0.666559			−0.333280	+	0.942828i	$0.608155\pi$
$728$	−258.579			−0.355191
$729$		0			0
$730$	−700.685			−0.959842
$731$		−	433.665i		−	0.593250i
$732$		0			0
$733$	1164.71			1.58896			0.794480	−	0.607290i	$-0.207743\pi$
$733$	1164.71			1.58896			0.794480	+	0.607290i	$0.207743\pi$
$734$	−958.060			−1.30526
$735$		0			0
$736$	78.2044			0.106256
$737$	1211.92			1.64439
$738$		0			0
$739$		−	157.927i		−	0.213703i	−0.994275	−	0.106852i	$-0.965923\pi$
$739$		−	157.927i		−	0.213703i	0.994275	−	0.106852i	$-0.0340770\pi$
$740$			107.688i			0.145524i
$741$		0			0
$742$		−	101.612i		−	0.136943i
$743$	−700.568			−0.942891			−0.471445	−	0.881895i	$-0.656268\pi$
$743$	−700.568			−0.942891			−0.471445	+	0.881895i	$0.656268\pi$
$744$		0			0
$745$			1646.89i			2.21059i
$746$			1175.90i			1.57627i
$747$		0			0
$748$		−	69.3498i		−	0.0927136i
$749$	−35.9393			−0.0479830
$750$		0			0
$751$			984.786i			1.31130i	0.755065	+	0.655650i	$0.227605\pi$
$751$			984.786i			1.31130i	−0.755065	+	0.655650i	$0.772395\pi$
$752$		−	416.955i		−	0.554461i
$753$		0			0
$754$	−1139.93			−1.51184
$755$		−	287.187i		−	0.380381i
$756$		0			0
$757$	−288.319			−0.380870			−0.190435	−	0.981700i	$-0.560990\pi$
$757$	−288.319			−0.380870			−0.190435	+	0.981700i	$0.560990\pi$
$758$			366.277i			0.483215i
$759$		0			0
$760$		−	1783.28i		−	2.34642i
$761$	462.140			0.607280			0.303640	−	0.952787i	$-0.401798\pi$
$761$	462.140			0.607280			0.303640	+	0.952787i	$0.401798\pi$
$762$		0			0
$763$		−	143.287i		−	0.187794i
$764$			13.5390i			0.0177212i
$765$		0			0
$766$			238.152i			0.310904i
$767$	1389.23	+	63.5846i	1.81125	+	0.0829005i
$768$		0			0
$769$			569.738i			0.740881i	0.928856	+	0.370441i	$0.120793\pi$
$769$			569.738i			0.740881i	−0.928856	+	0.370441i	$0.879207\pi$
$770$	282.016			0.366254
$771$		0			0
$772$	−74.8138			−0.0969090
$773$		−	401.753i		−	0.519732i	−0.965645	−	0.259866i	$-0.916322\pi$
$773$		−	401.753i		−	0.519732i	0.965645	−	0.259866i	$-0.0836784\pi$
$774$		0			0
$775$		−	1615.83i		−	2.08494i
$776$	1065.11			1.37257
$777$		0			0
$778$		−	351.918i		−	0.452337i
$779$	−1038.77			−1.33347
$780$		0			0
$781$			74.4723i			0.0953551i
$782$	174.631			0.223313
$783$		0			0
$784$	642.985			0.820134
$785$		−	2204.01i		−	2.80766i
$786$		0			0
$787$	454.102			0.577004			0.288502	−	0.957479i	$-0.406843\pi$
$787$	454.102			0.577004			0.288502	+	0.957479i	$0.406843\pi$
$788$	117.687			0.149349
$789$		0			0
$790$		−	1781.97i		−	2.25565i
$791$		−	90.4128i		−	0.114302i
$792$		0			0
$793$	753.851			0.950631
$794$	945.656			1.19100
$795$		0			0
$796$	150.093			0.188560
$797$			1446.19i			1.81454i	0.420545	+	0.907272i	$0.361839\pi$
$797$			1446.19i			1.81454i	−0.420545	+	0.907272i	$0.638161\pi$
$798$		0			0
$799$			310.732i			0.388901i
$800$			479.965i			0.599957i
$801$		0			0
$802$	795.423			0.991799
$803$	535.694			0.667115
$804$		0			0
$805$		−	108.854i		−	0.135222i
$806$			1248.35i			1.54882i
$807$		0			0
$808$	602.961			0.746239
$809$			763.887i			0.944236i	0.881535	+	0.472118i	$0.156510\pi$
$809$			763.887i			0.944236i	−0.881535	+	0.472118i	$0.843490\pi$
$810$		0			0
$811$			109.080i			0.134500i	0.997736	+	0.0672500i	$0.0214225\pi$
$811$			109.080i			0.134500i	−0.997736	+	0.0672500i	$0.978577\pi$
$812$	17.9457			0.0221006
$813$		0			0
$814$			537.111i			0.659842i
$815$	−81.5667			−0.100082
$816$		0			0
$817$			1000.20i			1.22423i
$818$	287.566			0.351547
$819$		0			0
$820$	213.842			0.260783
$821$			561.607i			0.684052i	0.939690	+	0.342026i	$0.111113\pi$
$821$			561.607i			0.684052i	−0.939690	+	0.342026i	$0.888887\pi$
$822$		0			0
$823$			817.432i			0.993235i	0.867970	+	0.496617i	$0.165425\pi$
$823$			817.432i			0.993235i	−0.867970	+	0.496617i	$0.834575\pi$
$824$	−1401.50			−1.70085
$825$		0			0
$826$	142.679	+	6.53037i	0.172734	+	0.00790602i
$827$	430.868			0.521001			0.260501	−	0.965474i	$-0.416112\pi$
$827$	430.868			0.521001			0.260501	+	0.965474i	$0.416112\pi$
$828$		0			0
$829$	−144.179			−0.173919			−0.0869594	−	0.996212i	$-0.527715\pi$
$829$	−144.179			−0.173919			−0.0869594	+	0.996212i	$0.527715\pi$
$830$	−52.9716			−0.0638212
$831$		0			0
$832$		−	1652.20i		−	1.98581i
$833$	−479.180			−0.575246
$834$		0			0
$835$	1219.77			1.46080
$836$			159.947i			0.191324i
$837$		0			0
$838$	−710.812			−0.848224
$839$			1284.54i			1.53104i	0.643415	+	0.765518i	$0.277517\pi$
$839$			1284.54i			1.53104i	−0.643415	+	0.765518i	$0.722483\pi$
$840$		0			0
$841$	−166.657			−0.198165
$842$	−416.280			−0.494394
$843$		0			0
$844$		−	1.19191i		−	0.00141221i
$845$	−3496.83			−4.13826
$846$		0			0
$847$	−58.3250			−0.0688607
$848$	570.463			0.672716
$849$		0			0
$850$			1071.77i			1.26090i
$851$	207.318			0.243616
$852$		0			0
$853$	1449.21			1.69896			0.849480	−	0.527620i	$-0.176916\pi$
$853$	1449.21			1.69896			0.849480	+	0.527620i	$0.176916\pi$
$854$	77.4231			0.0906594
$855$		0			0
$856$		−	233.339i		−	0.272592i
$857$		−	468.180i		−	0.546301i	−0.961971	−	0.273150i	$-0.911934\pi$
$857$		−	468.180i		−	0.546301i	0.961971	−	0.273150i	$-0.0880656\pi$
$858$		0			0
$859$			387.001i			0.450525i	0.974298	+	0.225262i	$0.0723239\pi$
$859$			387.001i			0.450525i	−0.974298	+	0.225262i	$0.927676\pi$
$860$		−	205.902i		−	0.239420i
$861$		0			0
$862$	−111.454			−0.129296
$863$			1197.58i			1.38769i	0.720123	+	0.693846i	$0.244085\pi$
$863$			1197.58i			1.38769i	−0.720123	+	0.693846i	$0.755915\pi$
$864$		0			0
$865$			2469.64i			2.85507i
$866$			1218.14i			1.40663i
$867$		0			0
$868$		−	19.6525i		−	0.0226411i
$869$			1362.37i			1.56774i
$870$		0			0
$871$	−2218.01			−2.54652
$872$	930.304			1.06686
$873$		0			0
$874$	−402.766			−0.460831
$875$	374.126			0.427573
$876$		0			0
$877$	−1017.97			−1.16074			−0.580371	−	0.814352i	$-0.697092\pi$
$877$	−1017.97			−1.16074			−0.580371	+	0.814352i	$0.697092\pi$
$878$			574.680i			0.654533i
$879$		0			0
$880$			1583.27i			1.79918i
$881$			4.59455i			0.00521515i	0.999997	+	0.00260758i	$0.000830018\pi$
$881$			4.59455i			0.00521515i	−0.999997	+	0.00260758i	$0.999170\pi$
$882$		0			0
$883$	−423.083			−0.479143			−0.239571	−	0.970879i	$-0.577007\pi$
$883$	−423.083			−0.479143			−0.239571	+	0.970879i	$0.577007\pi$
$884$			126.922i			0.143577i
$885$		0			0
$886$	175.192			0.197734
$887$			698.508i			0.787495i	0.919219	+	0.393747i	$0.128822\pi$
$887$			698.508i			0.787495i	−0.919219	+	0.393747i	$0.871178\pi$
$888$		0			0
$889$	284.396			0.319905
$890$	−367.769			−0.413224
$891$		0			0
$892$	−166.508			−0.186668
$893$		−	716.667i		−	0.802539i
$894$		0			0
$895$		−	997.422i		−	1.11444i
$896$		−	125.766i		−	0.140363i
$897$		0			0
$898$			834.021i			0.928754i
$899$		−	738.482i		−	0.821448i
$900$		0			0
$901$	−425.133			−0.471846
$902$	1066.58			1.18246
$903$		0			0
$904$	587.013			0.649351
$905$	196.702			0.217351
$906$		0			0
$907$	−564.844			−0.622761			−0.311381	−	0.950285i	$-0.600791\pi$
$907$	−564.844			−0.622761			−0.311381	+	0.950285i	$0.600791\pi$
$908$		−	212.065i		−	0.233552i
$909$		0			0
$910$	−516.137			−0.567184
$911$	70.7219			0.0776311			0.0388156	−	0.999246i	$-0.487642\pi$
$911$	70.7219			0.0776311			0.0388156	+	0.999246i	$0.487642\pi$
$912$		0			0
$913$	40.4983			0.0443574
$914$	655.297			0.716955
$915$		0			0
$916$		−	27.7633i		−	0.0303093i
$917$		−	319.765i		−	0.348708i
$918$		0			0
$919$			1102.02i			1.19915i	0.800318	+	0.599576i	$0.204664\pi$
$919$			1102.02i			1.19915i	−0.800318	+	0.599576i	$0.795336\pi$
$920$	706.745			0.768201
$921$		0			0
$922$			653.339i			0.708610i
$923$		−	136.297i		−	0.147668i
$924$		0			0
$925$			1272.37i			1.37554i
$926$	−598.689			−0.646533
$927$		0			0
$928$			219.358i			0.236378i
$929$		−	1042.64i		−	1.12233i	−0.827705	−	0.561164i	$-0.810354\pi$
$929$		−	1042.64i		−	1.12233i	0.827705	−	0.561164i	$-0.189646\pi$
$930$		0			0
$931$	1105.17			1.18708
$932$		−	223.405i		−	0.239704i
$933$		0			0
$934$	1175.71			1.25879
$935$		−	1179.92i		−	1.26195i
$936$		0			0
$937$		−	567.136i		−	0.605268i	−0.953107	−	0.302634i	$-0.902134\pi$
$937$		−	567.136i		−	0.605268i	0.953107	−	0.302634i	$-0.0978660\pi$
$938$	−227.798			−0.242855
$939$		0			0
$940$			147.534i			0.156951i
$941$		−	105.988i		−	0.112633i	−0.998413	−	0.0563167i	$-0.982064\pi$
$941$		−	105.988i		−	0.112633i	0.998413	−	0.0563167i	$-0.0179356\pi$
$942$		0			0
$943$		−	411.683i		−	0.436568i
$944$	−36.6624	+	801.018i	−0.0388373	+	0.848536i
$945$		0			0
$946$		−	1026.97i		−	1.08559i
$947$	600.889			0.634518			0.317259	−	0.948339i	$-0.397237\pi$
$947$	600.889			0.634518			0.317259	+	0.948339i	$0.397237\pi$
$948$		0			0
$949$	−980.412			−1.03310
$950$		−	2471.90i		−	2.60200i
$951$		0			0
$952$			111.112i			0.116714i
$953$	265.279			0.278362			0.139181	−	0.990267i	$-0.455553\pi$
$953$	265.279			0.278362			0.139181	+	0.990267i	$0.455553\pi$
$954$		0			0
$955$			230.354i			0.241208i
$956$	83.5017			0.0873449
$957$		0			0
$958$			250.700i			0.261691i
$959$	−326.070			−0.340010
$960$		0			0
$961$	152.280			0.158460
$962$		−	983.007i		−	1.02184i
$963$		0			0
$964$	51.7027			0.0536336
$965$	−1272.89			−1.31905
$966$		0			0
$967$		−	268.791i		−	0.277964i	−0.990295	−	0.138982i	$-0.955617\pi$
$967$		−	268.791i		−	0.277964i	0.990295	−	0.138982i	$-0.0443830\pi$
$968$		−	378.680i		−	0.391199i
$969$		0			0
$970$	2126.03			2.19178
$971$	1302.30			1.34120			0.670598	−	0.741821i	$-0.266038\pi$
$971$	1302.30			1.34120			0.670598	+	0.741821i	$0.266038\pi$
$972$		0			0
$973$	−18.9681			−0.0194944
$974$		−	1377.62i		−	1.41439i
$975$		0			0
$976$			434.664i			0.445353i
$977$			657.742i			0.673226i	0.941643	+	0.336613i	$0.109281\pi$
$977$			657.742i			0.673226i	−0.941643	+	0.336613i	$0.890719\pi$
$978$		0			0
$979$	281.170			0.287201
$980$	−227.511			−0.232155
$981$		0			0
$982$			790.483i			0.804973i
$983$			1066.97i			1.08542i	0.839920	+	0.542710i	$0.182602\pi$
$983$			1066.97i			1.08542i	−0.839920	+	0.542710i	$0.817398\pi$
$984$		0			0
$985$	2002.34			2.03283
$986$			489.829i			0.496784i
$987$		0			0
$988$		−	292.731i		−	0.296286i
$989$	−396.397			−0.400806
$990$		0			0
$991$		−	823.329i		−	0.830806i	−0.909637	−	0.415403i	$-0.863640\pi$
$991$		−	823.329i		−	0.830806i	0.909637	−	0.415403i	$-0.136360\pi$
$992$	240.222			0.242159
$993$		0			0
$994$		−	13.9982i		−	0.0140827i
$995$	2553.70			2.56654
$996$		0			0
$997$	−135.805			−0.136214			−0.0681068	−	0.997678i	$-0.521696\pi$
$997$	−135.805			−0.136214			−0.0681068	+	0.997678i	$0.521696\pi$
$998$			19.0018i			0.0190399i
$999$		0			0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	531.3.c.c.235.15		20
3.2	odd	2		177.3.c.a.58.6	✓	20
59.58	odd	2	inner	531.3.c.c.235.6		20
177.176	even	2		177.3.c.a.58.15	yes	20

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
177.3.c.a.58.6	✓	20	3.2	odd	2
177.3.c.a.58.15	yes	20	177.176	even	2
531.3.c.c.235.6		20	59.58	odd	2	inner
531.3.c.c.235.15		20	1.1	even	1	trivial

Overview	Random
Universe	Knowledge

Classical	Maass
Hilbert	Bianchi

Elliptic curves over $\Q$
Elliptic curves over $\Q(\alpha)$
Genus 2 curves over $\Q$
Higher genus families
Abelian varieties over $\F_{q}$

Level:	\( N \)	\(=\)	\( 531 = 3^{2} \cdot 59 \)
Weight:	\( k \)	\(=\)	\( 3 \)
Character orbit:	\([\chi]\)	\(=\)	531.c (of order \(2\), degree \(1\), minimal)

\(f(q)\)	\(=\)	\(q+1.86235i q^{2} +0.531642 q^{4} +9.04540 q^{5} +1.29987 q^{7} +8.43952i q^{8} +O(q^{10})\)	\(q+1.86235i q^{2} +0.531642 q^{4} +9.04540 q^{5} +1.29987 q^{7} +8.43952i q^{8} +16.8457i q^{10} -12.8790i q^{11} +23.5709i q^{13} +2.42081i q^{14} -13.5908 q^{16} +10.1284 q^{17} -23.3600 q^{19} +4.80892 q^{20} +23.9853 q^{22} -9.25801i q^{23} +56.8193 q^{25} -43.8973 q^{26} +0.691065 q^{28} +25.9681 q^{29} -28.4380i q^{31} +8.44722i q^{32} +18.8627i q^{34} +11.7578 q^{35} +22.3933i q^{37} -43.5046i q^{38} +76.3388i q^{40} +44.4678 q^{41} -42.8166i q^{43} -6.84704i q^{44} +17.2417 q^{46} +30.6792i q^{47} -47.3103 q^{49} +105.818i q^{50} +12.5313i q^{52} -41.9742 q^{53} -116.496i q^{55} +10.9703i q^{56} +48.3618i q^{58} +(2.69759 - 58.9383i) q^{59} -31.9823i q^{61} +52.9616 q^{62} -70.0949 q^{64} +213.208i q^{65} +94.0998i q^{67} +5.38470 q^{68} +21.8972i q^{70} -5.78244 q^{71} +41.5942i q^{73} -41.7043 q^{74} -12.4192 q^{76} -16.7411i q^{77} -105.782 q^{79} -122.934 q^{80} +82.8148i q^{82} +3.14451i q^{83} +91.6157 q^{85} +79.7397 q^{86} +108.693 q^{88} +21.8316i q^{89} +30.6391i q^{91} -4.92194i q^{92} -57.1355 q^{94} -211.301 q^{95} -126.206i q^{97} -88.1085i q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 20 q - 40 q^{4} - 8 q^{7}+O(q^{10}) \) 20 * q - 40 * q^4 - 8 * q^7	\( 20 q - 40 q^{4} - 8 q^{7} - 8 q^{16} - 16 q^{17} - 60 q^{19} + 164 q^{20} + 40 q^{22} + 100 q^{25} + 156 q^{26} + 200 q^{28} + 60 q^{29} + 32 q^{35} - 28 q^{41} + 180 q^{46} + 284 q^{49} + 8 q^{53} + 152 q^{59} + 8 q^{62} + 204 q^{64} - 384 q^{68} - 92 q^{71} - 104 q^{74} + 120 q^{76} - 420 q^{79} - 376 q^{80} - 348 q^{85} - 232 q^{86} - 212 q^{88} + 152 q^{94} - 788 q^{95}+O(q^{100}) \) 20 * q - 40 * q^4 - 8 * q^7 - 8 * q^16 - 16 * q^17 - 60 * q^19 + 164 * q^20 + 40 * q^22 + 100 * q^25 + 156 * q^26 + 200 * q^28 + 60 * q^29 + 32 * q^35 - 28 * q^41 + 180 * q^46 + 284 * q^49 + 8 * q^53 + 152 * q^59 + 8 * q^62 + 204 * q^64 - 384 * q^68 - 92 * q^71 - 104 * q^74 + 120 * q^76 - 420 * q^79 - 376 * q^80 - 348 * q^85 - 232 * q^86 - 212 * q^88 + 152 * q^94 - 788 * q^95

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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