LMFDB - Embedded newform 608.3.e.b.417.9

Show commands: Magma / Pari/GP / SageMath

Newspace parameters

comment:Compute space of new eigenforms

gp:[N,k,chi] = [608,3,Mod(417,608)] mf = mfinit([N,k,chi],0) lf = mfeigenbasis(mf)

sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(608, base_ring=CyclotomicField(2)) chi = DirichletCharacter(H, H._module([0, 0, 1])) N = Newforms(chi, 3, names="a")

magma://Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code chi := DirichletCharacter("608.417"); S:= CuspForms(chi, 3); N := Newforms(S);

Level:	$ N $	$=$	$ 608 = 2^{5} \cdot 19 $
Weight:	$ k $	$=$	$ 3 $
Character orbit:	$[\chi]$	$=$	608.e (of order $2$, degree $1$, minimal)

Newform invariants

comment:select newform

sage:traces = [20,0,0,0,0,0,0,0,-52,0,0,0,0,0,0,0,56] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(17)] == traces)

gp:f = lf[1] \\ Warning: the index may be different

Self dual:	no
Analytic conductor:	$16.5668000731$
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} - 8 x^{19} - 196 x^{18} + 1676 x^{17} + 16346 x^{16} - 161824 x^{15} - 667200 x^{14} + \cdots + 1135285065792 $ x^20 - 8x^19 - 196x^18 + 1676x^17 + 16346x^16 - 161824x^15 - 667200x^14 + 8850712x^13 + 9918758x^12 - 288078272x^11 + 224256138x^10 + 5383493580x^9 - 11116604703x^8 - 53755955160x^7 + 166723956672x^6 + 262105513072x^5 - 1053812084224x^4 - 855143026944x^3 + 3318150008320x^2 + 3262428344448*x + 1135285065792
Coefficient ring:	$\Z[a_1, \ldots, a_{19}]$
Coefficient ring index:	$ 2^{27} $
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label			417.9
Root			$-0.454273 + 0.341165i$ of defining polynomial
Character	$\chi$	$=$	608.417
Dual form			608.3.e.b.417.11

$q$-expansion

comment:q-expansion

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

gp:mfcoefs(f, 20)

$f(q)$	$=$	$q-0.682329i q^{3} +6.57489 q^{5} -8.44839 q^{7} +8.53443 q^{9} -2.26367 q^{11} +19.5893i q^{13} -4.48624i q^{15} +7.76481 q^{17} +(7.24637 + 17.5639i) q^{19} +5.76459i q^{21} +27.5478 q^{23} +18.2292 q^{25} -11.9643i q^{27} -48.3430i q^{29} +18.1890i q^{31} +1.54457i q^{33} -55.5473 q^{35} +42.8469i q^{37} +13.3663 q^{39} +19.0376i q^{41} +74.5673 q^{43} +56.1129 q^{45} -38.1195 q^{47} +22.3754 q^{49} -5.29816i q^{51} +8.32862i q^{53} -14.8834 q^{55} +(11.9844 - 4.94441i) q^{57} -66.1249i q^{59} +50.5024 q^{61} -72.1022 q^{63} +128.797i q^{65} -69.4618i q^{67} -18.7967i q^{69} +48.1973i q^{71} +71.7521 q^{73} -12.4383i q^{75} +19.1243 q^{77} +147.535i q^{79} +68.6463 q^{81} +103.693 q^{83} +51.0528 q^{85} -32.9859 q^{87} -143.373i q^{89} -165.498i q^{91} +12.4109 q^{93} +(47.6441 + 115.481i) q^{95} +49.3349i q^{97} -19.3191 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 20 q - 52 q^{9} + 56 q^{17} + 36 q^{25} + 64 q^{45} + 332 q^{49} + 88 q^{57} - 32 q^{61} - 152 q^{73} + 360 q^{77} - 476 q^{81} - 552 q^{85} + 336 q^{93}+O(q^{100}) $ 20 * q - 52 * q^9 + 56 * q^17 + 36 * q^25 + 64 * q^45 + 332 * q^49 + 88 * q^57 - 32 * q^61 - 152 * q^73 + 360 * q^77 - 476 * q^81 - 552 * q^85 + 336 * q^93

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/608\mathbb{Z}\right)^\times$.

$n$	$97$	$191$	$229$
$\chi(n)$	$-1$	$1$	$1$

Coefficient data

For each $n$ we display the coefficients of the $q$-expansion $a_n$, the Satake parameters $\alpha_p$, and the Satake angles $\theta_p = \textrm{Arg}(\alpha_p)$.

Display $a_p$ with $p$ up to: 50 250 1000 (See $a_n$ instead) (See $a_n$ instead) (See $a_n$ instead) Display $a_n$ with $n$ up to: 50 250 1000 (See only $a_p$) (See only $a_p$) (See only $a_p$)

$n$	$a_n$			$a_n / n^{(k-1)/2}$			$ \alpha_n $			$ \theta_n $
$p$	$a_p$			$a_p / p^{(k-1)/2}$			$ \alpha_p$			$ \theta_p $

$2$		0			0
$2$		0			0
$3$		−	0.682329i		−	0.227443i	−0.993513	−	0.113722i	$-0.963723\pi$
$3$		−	0.682329i		−	0.227443i	0.993513	−	0.113722i	$-0.0362772\pi$
$4$		0			0
$5$	6.57489			1.31498			0.657489	−	0.753464i	$-0.271619\pi$
$5$	6.57489			1.31498			0.657489	+	0.753464i	$0.271619\pi$
$6$		0			0
$7$	−8.44839			−1.20691			−0.603457	−	0.797396i	$-0.706210\pi$
$7$	−8.44839			−1.20691			−0.603457	+	0.797396i	$0.706210\pi$
$8$		0			0
$9$	8.53443			0.948270
$10$		0			0
$11$	−2.26367			−0.205788			−0.102894	−	0.994692i	$-0.532810\pi$
$11$	−2.26367			−0.205788			−0.102894	+	0.994692i	$0.532810\pi$
$12$		0			0
$13$			19.5893i			1.50687i	0.657524	+	0.753434i	$0.271604\pi$
$13$			19.5893i			1.50687i	−0.657524	+	0.753434i	$0.728396\pi$
$14$		0			0
$15$		−	4.48624i		−	0.299083i
$16$		0			0
$17$	7.76481			0.456754			0.228377	−	0.973573i	$-0.426658\pi$
$17$	7.76481			0.456754			0.228377	+	0.973573i	$0.426658\pi$
$18$		0			0
$19$	7.24637	+	17.5639i	0.381388	+	0.924415i
$19$	7.24637	+	17.5639i	0.381388	+	0.924415i
$20$		0			0
$21$			5.76459i			0.274504i
$22$		0			0
$23$	27.5478			1.19773			0.598865	−	0.800850i	$-0.295619\pi$
$23$	27.5478			1.19773			0.598865	+	0.800850i	$0.295619\pi$
$24$		0			0
$25$	18.2292			0.729166
$26$		0			0
$27$		−	11.9643i		−	0.443121i
$28$		0			0
$29$		−	48.3430i		−	1.66700i	−0.552519	−	0.833500i	$-0.686333\pi$
$29$		−	48.3430i		−	1.66700i	0.552519	−	0.833500i	$-0.313667\pi$
$30$		0			0
$31$			18.1890i			0.586742i	0.955999	+	0.293371i	$0.0947771\pi$
$31$			18.1890i			0.586742i	−0.955999	+	0.293371i	$0.905223\pi$
$32$		0			0
$33$			1.54457i			0.0468050i
$34$		0			0
$35$	−55.5473			−1.58706
$36$		0			0
$37$			42.8469i			1.15802i	0.815319	+	0.579012i	$0.196562\pi$
$37$			42.8469i			1.15802i	−0.815319	+	0.579012i	$0.803438\pi$
$38$		0			0
$39$	13.3663			0.342727
$40$		0			0
$41$			19.0376i			0.464332i	0.972676	+	0.232166i	$0.0745813\pi$
$41$			19.0376i			0.464332i	−0.972676	+	0.232166i	$0.925419\pi$
$42$		0			0
$43$	74.5673			1.73412			0.867062	−	0.498201i	$-0.166006\pi$
$43$	74.5673			1.73412			0.867062	+	0.498201i	$0.166006\pi$
$44$		0			0
$45$	56.1129			1.24695
$46$		0			0
$47$	−38.1195			−0.811052			−0.405526	−	0.914083i	$-0.632912\pi$
$47$	−38.1195			−0.811052			−0.405526	+	0.914083i	$0.632912\pi$
$48$		0			0
$49$	22.3754			0.456640
$50$		0			0
$51$		−	5.29816i		−	0.103886i
$52$		0			0
$53$			8.32862i			0.157144i	0.996908	+	0.0785719i	$0.0250360\pi$
$53$			8.32862i			0.157144i	−0.996908	+	0.0785719i	$0.974964\pi$
$54$		0			0
$55$	−14.8834			−0.270606
$56$		0			0
$57$	11.9844	−	4.94441i	0.210252	−	0.0867440i
$58$		0			0
$59$		−	66.1249i		−	1.12076i	−0.828235	−	0.560381i	$-0.810655\pi$
$59$		−	66.1249i		−	1.12076i	0.828235	−	0.560381i	$-0.189345\pi$
$60$		0			0
$61$	50.5024			0.827907			0.413954	−	0.910298i	$-0.364148\pi$
$61$	50.5024			0.827907			0.413954	+	0.910298i	$0.364148\pi$
$62$		0			0
$63$	−72.1022			−1.14448
$64$		0			0
$65$			128.797i			1.98150i
$66$		0			0
$67$		−	69.4618i		−	1.03674i	−0.855155	−	0.518372i	$-0.826538\pi$
$67$		−	69.4618i		−	1.03674i	0.855155	−	0.518372i	$-0.173462\pi$
$68$		0			0
$69$		−	18.7967i		−	0.272415i
$70$		0			0
$71$			48.1973i			0.678835i	0.940636	+	0.339418i	$0.110230\pi$
$71$			48.1973i			0.678835i	−0.940636	+	0.339418i	$0.889770\pi$
$72$		0			0
$73$	71.7521			0.982906			0.491453	−	0.870904i	$-0.336466\pi$
$73$	71.7521			0.982906			0.491453	+	0.870904i	$0.336466\pi$
$74$		0			0
$75$		−	12.4383i		−	0.165844i
$76$		0			0
$77$	19.1243			0.248368
$78$		0			0
$79$			147.535i			1.86753i	0.357891	+	0.933763i	$0.383496\pi$
$79$			147.535i			1.86753i	−0.357891	+	0.933763i	$0.616504\pi$
$80$		0			0
$81$	68.6463			0.847485
$82$		0			0
$83$	103.693			1.24932			0.624658	−	0.780899i	$-0.285239\pi$
$83$	103.693			1.24932			0.624658	+	0.780899i	$0.285239\pi$
$84$		0			0
$85$	51.0528			0.600621
$86$		0			0
$87$	−32.9859			−0.379148
$88$		0			0
$89$		−	143.373i		−	1.61093i	−0.592641	−	0.805466i	$-0.701915\pi$
$89$		−	143.373i		−	1.61093i	0.592641	−	0.805466i	$-0.298085\pi$
$90$		0			0
$91$		−	165.498i		−	1.81866i
$92$		0			0
$93$	12.4109			0.133450
$94$		0			0
$95$	47.6441	+	115.481i	0.501516	+	1.21559i
$96$		0			0
$97$			49.3349i			0.508607i	0.967124	+	0.254304i	$0.0818463\pi$
$97$			49.3349i			0.508607i	−0.967124	+	0.254304i	$0.918154\pi$
$98$		0			0
$99$	−19.3191			−0.195142
$100$		0			0
$101$	13.6012			0.134665			0.0673327	−	0.997731i	$-0.478551\pi$
$101$	13.6012			0.134665			0.0673327	+	0.997731i	$0.478551\pi$
$102$		0			0
$103$		−	11.5637i		−	0.112269i	−0.998423	−	0.0561344i	$-0.982122\pi$
$103$		−	11.5637i		−	0.112269i	0.998423	−	0.0561344i	$-0.0178775\pi$
$104$		0			0
$105$			37.9015i			0.360967i
$106$		0			0
$107$		−	51.7552i		−	0.483693i	−0.970315	−	0.241847i	$-0.922247\pi$
$107$		−	51.7552i		−	0.483693i	0.970315	−	0.241847i	$-0.0777531\pi$
$108$		0			0
$109$			69.7610i			0.640009i	0.947416	+	0.320005i	$0.103684\pi$
$109$			69.7610i			0.640009i	−0.947416	+	0.320005i	$0.896316\pi$
$110$		0			0
$111$	29.2357			0.263385
$112$		0			0
$113$			3.31598i			0.0293449i	0.999892	+	0.0146725i	$0.00467056\pi$
$113$			3.31598i			0.0293449i	−0.999892	+	0.0146725i	$0.995329\pi$
$114$		0			0
$115$	181.124			1.57499
$116$		0			0
$117$			167.183i			1.42892i
$118$		0			0
$119$	−65.6002			−0.551262
$120$		0			0
$121$	−115.876			−0.957651
$122$		0			0
$123$	12.9899			0.105609
$124$		0			0
$125$	−44.5175			−0.356140
$126$		0			0
$127$			220.929i			1.73960i	0.493403	+	0.869801i	$0.335753\pi$
$127$			220.929i			1.73960i	−0.493403	+	0.869801i	$0.664247\pi$
$128$		0			0
$129$		−	50.8795i		−	0.394415i
$130$		0			0
$131$	−169.477			−1.29372			−0.646859	−	0.762609i	$-0.723918\pi$
$131$	−169.477			−1.29372			−0.646859	+	0.762609i	$0.723918\pi$
$132$		0			0
$133$	−61.2202	−	148.387i	−0.460302	−	1.11569i
$134$		0			0
$135$		−	78.6636i		−	0.582694i
$136$		0			0
$137$	−55.9106			−0.408107			−0.204053	−	0.978960i	$-0.565412\pi$
$137$	−55.9106			−0.408107			−0.204053	+	0.978960i	$0.565412\pi$
$138$		0			0
$139$	−151.801			−1.09210			−0.546048	−	0.837754i	$-0.683868\pi$
$139$	−151.801			−1.09210			−0.546048	+	0.837754i	$0.683868\pi$
$140$		0			0
$141$			26.0100i			0.184468i
$142$		0			0
$143$		−	44.3436i		−	0.310095i
$144$		0			0
$145$		−	317.850i		−	2.19207i
$146$		0			0
$147$		−	15.2674i		−	0.103860i
$148$		0			0
$149$	−154.521			−1.03706			−0.518528	−	0.855061i	$-0.673520\pi$
$149$	−154.521			−1.03706			−0.518528	+	0.855061i	$0.673520\pi$
$150$		0			0
$151$		−	33.0833i		−	0.219095i	−0.993982	−	0.109547i	$-0.965060\pi$
$151$		−	33.0833i		−	0.219095i	0.993982	−	0.109547i	$-0.0349402\pi$
$152$		0			0
$153$	66.2682			0.433126
$154$		0			0
$155$			119.591i			0.771552i
$156$		0			0
$157$	−229.071			−1.45905			−0.729524	−	0.683955i	$-0.760258\pi$
$157$	−229.071			−1.45905			−0.729524	+	0.683955i	$0.760258\pi$
$158$		0			0
$159$	5.68286			0.0357413
$160$		0			0
$161$	−232.735			−1.44556
$162$		0			0
$163$	89.5191			0.549197			0.274598	−	0.961559i	$-0.411455\pi$
$163$	89.5191			0.549197			0.274598	+	0.961559i	$0.411455\pi$
$164$		0			0
$165$			10.1554i			0.0615476i
$166$		0			0
$167$		−	212.489i		−	1.27239i	−0.771530	−	0.636193i	$-0.780508\pi$
$167$		−	212.489i		−	1.27239i	0.771530	−	0.636193i	$-0.219492\pi$
$168$		0			0
$169$	−214.740			−1.27065
$170$		0			0
$171$	61.8436	+	149.898i	0.361658	+	0.876595i
$172$		0			0
$173$		−	284.987i		−	1.64732i	−0.567083	−	0.823661i	$-0.691928\pi$
$173$		−	284.987i		−	1.64732i	0.567083	−	0.823661i	$-0.308072\pi$
$174$		0			0
$175$	−154.007			−0.880041
$176$		0			0
$177$	−45.1190			−0.254910
$178$		0			0
$179$		−	186.577i		−	1.04233i	−0.853456	−	0.521165i	$-0.825498\pi$
$179$		−	186.577i		−	1.04233i	0.853456	−	0.521165i	$-0.174502\pi$
$180$		0			0
$181$		−	278.532i		−	1.53885i	−0.638736	−	0.769426i	$-0.720542\pi$
$181$		−	278.532i		−	1.53885i	0.638736	−	0.769426i	$-0.279458\pi$
$182$		0			0
$183$		−	34.4592i		−	0.188302i
$184$		0			0
$185$			281.714i			1.52278i
$186$		0			0
$187$	−17.5769			−0.0939944
$188$		0			0
$189$			101.079i			0.534808i
$190$		0			0
$191$	−321.592			−1.68373			−0.841865	−	0.539689i	$-0.818542\pi$
$191$	−321.592			−1.68373			−0.841865	+	0.539689i	$0.818542\pi$
$192$		0			0
$193$			158.345i			0.820442i	0.911986	+	0.410221i	$0.134548\pi$
$193$			158.345i			0.820442i	−0.911986	+	0.410221i	$0.865452\pi$
$194$		0			0
$195$	87.8822			0.450678
$196$		0			0
$197$	128.416			0.651855			0.325928	−	0.945395i	$-0.394323\pi$
$197$	128.416			0.651855			0.325928	+	0.945395i	$0.394323\pi$
$198$		0			0
$199$	82.0731			0.412427			0.206214	−	0.978507i	$-0.433886\pi$
$199$	82.0731			0.412427			0.206214	+	0.978507i	$0.433886\pi$
$200$		0			0
$201$	−47.3959			−0.235800
$202$		0			0
$203$			408.421i			2.01192i
$204$		0			0
$205$			125.170i			0.610586i
$206$		0			0
$207$	235.105			1.13577
$208$		0			0
$209$	−16.4034	−	39.7588i	−0.0784850	−	0.190233i
$210$		0			0
$211$		−	254.865i		−	1.20789i	−0.797026	−	0.603945i	$-0.793595\pi$
$211$		−	254.865i		−	1.20789i	0.797026	−	0.603945i	$-0.206405\pi$
$212$		0			0
$213$	32.8864			0.154396
$214$		0			0
$215$	490.272			2.28033
$216$		0			0
$217$		−	153.668i		−	0.708147i
$218$		0			0
$219$		−	48.9586i		−	0.223555i
$220$		0			0
$221$			152.107i			0.688268i
$222$		0			0
$223$			293.721i			1.31714i	0.752521	+	0.658568i	$0.228837\pi$
$223$			293.721i			1.31714i	−0.752521	+	0.658568i	$0.771163\pi$
$224$		0			0
$225$	155.575			0.691446
$226$		0			0
$227$		−	39.0553i		−	0.172050i	−0.996293	−	0.0860249i	$-0.972584\pi$
$227$		−	39.0553i		−	0.172050i	0.996293	−	0.0860249i	$-0.0274165\pi$
$228$		0			0
$229$	−336.404			−1.46901			−0.734507	−	0.678602i	$-0.762586\pi$
$229$	−336.404			−1.46901			−0.734507	+	0.678602i	$0.762586\pi$
$230$		0			0
$231$		−	13.0491i		−	0.0564896i
$232$		0			0
$233$	−0.150576			−0.000646248			−0.000323124	−	1.00000i	$-0.500103\pi$
$233$	−0.150576			−0.000646248			−0.000323124		1.00000i	$0.500103\pi$
$234$		0			0
$235$	−250.631			−1.06652
$236$		0			0
$237$	100.667			0.424756
$238$		0			0
$239$	101.554			0.424910			0.212455	−	0.977171i	$-0.431854\pi$
$239$	101.554			0.424910			0.212455	+	0.977171i	$0.431854\pi$
$240$		0			0
$241$			22.2112i			0.0921625i	0.998938	+	0.0460812i	$0.0146733\pi$
$241$			22.2112i			0.0921625i	−0.998938	+	0.0460812i	$0.985327\pi$
$242$		0			0
$243$		−	154.518i		−	0.635875i
$244$		0			0
$245$	147.116			0.600472
$246$		0			0
$247$	−344.064	+	141.951i	−1.39297	+	0.574701i
$248$		0			0
$249$		−	70.7529i		−	0.284148i
$250$		0			0
$251$	279.017			1.11162			0.555811	−	0.831309i	$-0.312408\pi$
$251$	279.017			1.11162			0.555811	+	0.831309i	$0.312408\pi$
$252$		0			0
$253$	−62.3590			−0.246478
$254$		0			0
$255$		−	34.8348i		−	0.136607i
$256$		0			0
$257$		−	124.658i		−	0.485051i	−0.970145	−	0.242525i	$-0.922024\pi$
$257$		−	124.658i		−	0.485051i	0.970145	−	0.242525i	$-0.0779758\pi$
$258$		0			0
$259$		−	361.988i		−	1.39764i
$260$		0			0
$261$		−	412.580i		−	1.58077i
$262$		0			0
$263$	132.853			0.505145			0.252572	−	0.967578i	$-0.418723\pi$
$263$	132.853			0.505145			0.252572	+	0.967578i	$0.418723\pi$
$264$		0			0
$265$			54.7597i			0.206641i
$266$		0			0
$267$	−97.8276			−0.366396
$268$		0			0
$269$		−	97.2652i		−	0.361581i	−0.983522	−	0.180790i	$-0.942134\pi$
$269$		−	97.2652i		−	0.361581i	0.983522	−	0.180790i	$-0.0578656\pi$
$270$		0			0
$271$	−286.837			−1.05844			−0.529220	−	0.848485i	$-0.677515\pi$
$271$	−286.837			−1.05844			−0.529220	+	0.848485i	$0.677515\pi$
$272$		0			0
$273$	−112.924			−0.413642
$274$		0			0
$275$	−41.2647			−0.150054
$276$		0			0
$277$	−186.839			−0.674508			−0.337254	−	0.941414i	$-0.609498\pi$
$277$	−186.839			−0.674508			−0.337254	+	0.941414i	$0.609498\pi$
$278$		0			0
$279$			155.233i			0.556389i
$280$		0			0
$281$		−	67.5878i		−	0.240526i	−0.992742	−	0.120263i	$-0.961626\pi$
$281$		−	67.5878i		−	0.240526i	0.992742	−	0.120263i	$-0.0383738\pi$
$282$		0			0
$283$	178.824			0.631889			0.315944	−	0.948778i	$-0.397679\pi$
$283$	178.824			0.631889			0.315944	+	0.948778i	$0.397679\pi$
$284$		0			0
$285$	78.7958	−	32.5089i	0.276477	−	0.114066i
$286$		0			0
$287$		−	160.837i		−	0.560409i
$288$		0			0
$289$	−228.708			−0.791376
$290$		0			0
$291$	33.6627			0.115679
$292$		0			0
$293$			391.555i			1.33636i	0.743998	+	0.668182i	$0.232927\pi$
$293$			391.555i			1.33636i	−0.743998	+	0.668182i	$0.767073\pi$
$294$		0			0
$295$		−	434.764i		−	1.47378i
$296$		0			0
$297$			27.0831i			0.0911888i
$298$		0			0
$299$			539.641i			1.80482i
$300$		0			0
$301$	−629.974			−2.09294
$302$		0			0
$303$		−	9.28050i		−	0.0306287i
$304$		0			0
$305$	332.047			1.08868
$306$		0			0
$307$		−	80.7622i		−	0.263069i	−0.991312	−	0.131534i	$-0.958010\pi$
$307$		−	80.7622i		−	0.263069i	0.991312	−	0.131534i	$-0.0419904\pi$
$308$		0			0
$309$	−7.89025			−0.0255348
$310$		0			0
$311$	−389.789			−1.25334			−0.626670	−	0.779284i	$-0.715583\pi$
$311$	−389.789			−1.25334			−0.626670	+	0.779284i	$0.715583\pi$
$312$		0			0
$313$	−448.692			−1.43352			−0.716761	−	0.697319i	$-0.754376\pi$
$313$	−448.692			−1.43352			−0.716761	+	0.697319i	$0.754376\pi$
$314$		0			0
$315$	−474.064			−1.50496
$316$		0			0
$317$		−	191.082i		−	0.602783i	−0.953500	−	0.301392i	$-0.902549\pi$
$317$		−	191.082i		−	0.602783i	0.953500	−	0.301392i	$-0.0974511\pi$
$318$		0			0
$319$			109.432i			0.343048i
$320$		0			0
$321$	−35.3141			−0.110013
$322$		0			0
$323$	56.2667	+	136.380i	0.174200	+	0.422230i
$324$		0			0
$325$			357.096i			1.09876i
$326$		0			0
$327$	47.6000			0.145566
$328$		0			0
$329$	322.048			0.978870
$330$		0			0
$331$			285.538i			0.862652i	0.902196	+	0.431326i	$0.141954\pi$
$331$			285.538i			0.862652i	−0.902196	+	0.431326i	$0.858046\pi$
$332$		0			0
$333$			365.674i			1.09812i
$334$		0			0
$335$		−	456.704i		−	1.36330i
$336$		0			0
$337$		−	366.245i		−	1.08678i	−0.839480	−	0.543390i	$-0.817140\pi$
$337$		−	366.245i		−	1.08678i	0.839480	−	0.543390i	$-0.182860\pi$
$338$		0			0
$339$	2.26259			0.00667430
$340$		0			0
$341$		−	41.1738i		−	0.120744i
$342$		0			0
$343$	224.935			0.655788
$344$		0			0
$345$		−	123.586i		−	0.358220i
$346$		0			0
$347$	−101.528			−0.292589			−0.146295	−	0.989241i	$-0.546735\pi$
$347$	−101.528			−0.292589			−0.146295	+	0.989241i	$0.546735\pi$
$348$		0			0
$349$	417.572			1.19648			0.598241	−	0.801316i	$-0.295867\pi$
$349$	417.572			1.19648			0.598241	+	0.801316i	$0.295867\pi$
$350$		0			0
$351$	234.371			0.667724
$352$		0			0
$353$	481.539			1.36413			0.682067	−	0.731290i	$-0.261081\pi$
$353$	481.539			1.36413			0.682067	+	0.731290i	$0.261081\pi$
$354$		0			0
$355$			316.892i			0.892653i
$356$		0			0
$357$			44.7610i			0.125381i
$358$		0			0
$359$	276.977			0.771522			0.385761	−	0.922599i	$-0.373939\pi$
$359$	276.977			0.771522			0.385761	+	0.922599i	$0.373939\pi$
$360$		0			0
$361$	−255.980	+	254.549i	−0.709087	+	0.705121i
$362$		0			0
$363$			79.0655i			0.217811i
$364$		0			0
$365$	471.762			1.29250
$366$		0			0
$367$	123.376			0.336174			0.168087	−	0.985772i	$-0.446241\pi$
$367$	123.376			0.336174			0.168087	+	0.985772i	$0.446241\pi$
$368$		0			0
$369$			162.475i			0.440312i
$370$		0			0
$371$		−	70.3635i		−	0.189659i
$372$		0			0
$373$		−	276.569i		−	0.741472i	−0.928738	−	0.370736i	$-0.879105\pi$
$373$		−	276.569i		−	0.741472i	0.928738	−	0.370736i	$-0.120895\pi$
$374$		0			0
$375$			30.3756i			0.0810016i
$376$		0			0
$377$	947.005			2.51195
$378$		0			0
$379$		−	47.4253i		−	0.125133i	−0.998041	−	0.0625664i	$-0.980071\pi$
$379$		−	47.4253i		−	0.125133i	0.998041	−	0.0625664i	$-0.0199285\pi$
$380$		0			0
$381$	150.747			0.395661
$382$		0			0
$383$		−	598.140i		−	1.56172i	−0.624704	−	0.780862i	$-0.714780\pi$
$383$		−	598.140i		−	1.56172i	0.624704	−	0.780862i	$-0.285220\pi$
$384$		0			0
$385$	125.740			0.326599
$386$		0			0
$387$	636.389			1.64442
$388$		0			0
$389$	579.247			1.48907			0.744534	−	0.667585i	$-0.232672\pi$
$389$	579.247			1.48907			0.744534	+	0.667585i	$0.232672\pi$
$390$		0			0
$391$	213.903			0.547068
$392$		0			0
$393$			115.639i			0.294247i
$394$		0			0
$395$			970.024i			2.45576i
$396$		0			0
$397$	34.3786			0.0865960			0.0432980	−	0.999062i	$-0.486214\pi$
$397$	34.3786			0.0865960			0.0432980	+	0.999062i	$0.486214\pi$
$398$		0			0
$399$	−101.249	+	41.7723i	−0.253756	+	0.104693i
$400$		0			0
$401$			498.060i			1.24204i	0.783793	+	0.621022i	$0.213282\pi$
$401$			498.060i			1.24204i	−0.783793	+	0.621022i	$0.786718\pi$
$402$		0			0
$403$	−356.309			−0.884142
$404$		0			0
$405$	451.342			1.11442
$406$		0			0
$407$		−	96.9911i		−	0.238307i
$408$		0			0
$409$			184.323i			0.450667i	0.974282	+	0.225334i	$0.0723472\pi$
$409$			184.323i			0.450667i	−0.974282	+	0.225334i	$0.927653\pi$
$410$		0			0
$411$			38.1495i			0.0928211i
$412$		0			0
$413$			558.650i			1.35266i
$414$		0			0
$415$	681.771			1.64282
$416$		0			0
$417$			103.578i			0.248390i
$418$		0			0
$419$	−389.280			−0.929068			−0.464534	−	0.885555i	$-0.653778\pi$
$419$	−389.280			−0.929068			−0.464534	+	0.885555i	$0.653778\pi$
$420$		0			0
$421$		−	363.146i		−	0.862580i	−0.902213	−	0.431290i	$-0.858059\pi$
$421$		−	363.146i		−	0.862580i	0.902213	−	0.431290i	$-0.141941\pi$
$422$		0			0
$423$	−325.328			−0.769096
$424$		0			0
$425$	141.546			0.333049
$426$		0			0
$427$	−426.664			−0.999213
$428$		0			0
$429$	−30.2569			−0.0705290
$430$		0			0
$431$			475.965i			1.10433i	0.833736	+	0.552164i	$0.186198\pi$
$431$			475.965i			1.10433i	−0.833736	+	0.552164i	$0.813802\pi$
$432$		0			0
$433$		−	157.342i		−	0.363376i	−0.983356	−	0.181688i	$-0.941844\pi$
$433$		−	157.342i		−	0.363376i	0.983356	−	0.181688i	$-0.0581560\pi$
$434$		0			0
$435$	−216.878			−0.498571
$436$		0			0
$437$	199.621	+	483.846i	0.456799	+	1.10720i
$438$		0			0
$439$		−	438.927i		−	0.999835i	−0.866073	−	0.499917i	$-0.833364\pi$
$439$		−	438.927i		−	0.999835i	0.866073	−	0.499917i	$-0.166636\pi$
$440$		0			0
$441$	190.961			0.433018
$442$		0			0
$443$	−406.102			−0.916709			−0.458354	−	0.888770i	$-0.651561\pi$
$443$	−406.102			−0.916709			−0.458354	+	0.888770i	$0.651561\pi$
$444$		0			0
$445$		−	942.662i		−	2.11834i
$446$		0			0
$447$			105.434i			0.235871i
$448$		0			0
$449$			734.618i			1.63612i	0.575132	+	0.818060i	$0.304951\pi$
$449$			734.618i			1.63612i	−0.575132	+	0.818060i	$0.695049\pi$
$450$		0			0
$451$		−	43.0948i		−	0.0955539i
$452$		0			0
$453$	−22.5737			−0.0498316
$454$		0			0
$455$		−	1088.13i		−	2.39150i
$456$		0			0
$457$	259.189			0.567154			0.283577	−	0.958950i	$-0.408479\pi$
$457$	259.189			0.567154			0.283577	+	0.958950i	$0.408479\pi$
$458$		0			0
$459$		−	92.9002i		−	0.202397i
$460$		0			0
$461$	236.836			0.513743			0.256872	−	0.966446i	$-0.417308\pi$
$461$	236.836			0.513743			0.256872	+	0.966446i	$0.417308\pi$
$462$		0			0
$463$	130.707			0.282305			0.141153	−	0.989988i	$-0.454919\pi$
$463$	130.707			0.282305			0.141153	+	0.989988i	$0.454919\pi$
$464$		0			0
$465$	81.6002			0.175484
$466$		0			0
$467$	442.841			0.948267			0.474134	−	0.880453i	$-0.342761\pi$
$467$	442.841			0.948267			0.474134	+	0.880453i	$0.342761\pi$
$468$		0			0
$469$			586.841i			1.25126i
$470$		0			0
$471$			156.302i			0.331850i
$472$		0			0
$473$	−168.796			−0.356862
$474$		0			0
$475$	132.095	+	320.175i	0.278095	+	0.674052i
$476$		0			0
$477$			71.0800i			0.149015i
$478$		0			0
$479$	−316.258			−0.660246			−0.330123	−	0.943938i	$-0.607090\pi$
$479$	−316.258			−0.660246			−0.330123	+	0.943938i	$0.607090\pi$
$480$		0			0
$481$	−839.341			−1.74499
$482$		0			0
$483$			158.802i			0.328782i
$484$		0			0
$485$			324.372i			0.668807i
$486$		0			0
$487$		−	547.237i		−	1.12369i	−0.827243	−	0.561845i	$-0.810092\pi$
$487$		−	547.237i		−	1.12369i	0.827243	−	0.561845i	$-0.189908\pi$
$488$		0			0
$489$		−	61.0815i		−	0.124911i
$490$		0			0
$491$	−543.520			−1.10697			−0.553483	−	0.832861i	$-0.686701\pi$
$491$	−543.520			−1.10697			−0.553483	+	0.832861i	$0.686701\pi$
$492$		0			0
$493$		−	375.374i		−	0.761409i
$494$		0			0
$495$	−127.021			−0.256608
$496$		0			0
$497$		−	407.190i		−	0.819295i
$498$		0			0
$499$	620.150			1.24279			0.621393	−	0.783499i	$-0.286567\pi$
$499$	620.150			1.24279			0.621393	+	0.783499i	$0.286567\pi$
$500$		0			0
$501$	−144.987			−0.289396
$502$		0			0
$503$	362.663			0.721000			0.360500	−	0.932759i	$-0.382606\pi$
$503$	362.663			0.721000			0.360500	+	0.932759i	$0.382606\pi$
$504$		0			0
$505$	89.4264			0.177082
$506$		0			0
$507$			146.523i			0.289001i
$508$		0			0
$509$		−	209.519i		−	0.411629i	−0.978591	−	0.205814i	$-0.934016\pi$
$509$		−	209.519i		−	0.411629i	0.978591	−	0.205814i	$-0.0659843\pi$
$510$		0			0
$511$	−606.190			−1.18628
$512$		0			0
$513$	210.139	−	86.6974i	0.409627	−	0.169001i
$514$		0			0
$515$		−	76.0300i		−	0.147631i
$516$		0			0
$517$	86.2897			0.166905
$518$		0			0
$519$	−194.455			−0.374672
$520$		0			0
$521$		−	459.602i		−	0.882154i	−0.897469	−	0.441077i	$-0.854597\pi$
$521$		−	459.602i		−	0.882154i	0.897469	−	0.441077i	$-0.145403\pi$
$522$		0			0
$523$			286.558i			0.547912i	0.961742	+	0.273956i	$0.0883323\pi$
$523$			286.558i			0.547912i	−0.961742	+	0.273956i	$0.911668\pi$
$524$		0			0
$525$			105.084i			0.200159i
$526$		0			0
$527$			141.234i			0.267997i
$528$		0			0
$529$	229.880			0.434556
$530$		0			0
$531$		−	564.338i		−	1.06278i
$532$		0			0
$533$	−372.933			−0.699687
$534$		0			0
$535$		−	340.284i		−	0.636046i
$536$		0			0
$537$	−127.307			−0.237071
$538$		0			0
$539$	−50.6504			−0.0939710
$540$		0			0
$541$	−462.578			−0.855042			−0.427521	−	0.904005i	$-0.640613\pi$
$541$	−462.578			−0.855042			−0.427521	+	0.904005i	$0.640613\pi$
$542$		0			0
$543$	−190.051			−0.350001
$544$		0			0
$545$			458.671i			0.841598i
$546$		0			0
$547$			672.215i			1.22891i	0.788951	+	0.614456i	$0.210624\pi$
$547$			672.215i			1.22891i	−0.788951	+	0.614456i	$0.789376\pi$
$548$		0			0
$549$	431.009			0.785079
$550$		0			0
$551$	849.091	−	350.311i	1.54100	−	0.635773i
$552$		0			0
$553$		−	1246.43i		−	2.25394i
$554$		0			0
$555$	192.222			0.346345
$556$		0			0
$557$	705.453			1.26652			0.633262	−	0.773938i	$-0.281716\pi$
$557$	705.453			1.26652			0.633262	+	0.773938i	$0.281716\pi$
$558$		0			0
$559$			1460.72i			2.61310i
$560$		0			0
$561$			11.9933i			0.0213784i
$562$		0			0
$563$		−	698.282i		−	1.24029i	−0.784488	−	0.620144i	$-0.787074\pi$
$563$		−	698.282i		−	1.24029i	0.784488	−	0.620144i	$-0.212926\pi$
$564$		0			0
$565$			21.8022i			0.0385879i
$566$		0			0
$567$	−579.951			−1.02284
$568$		0			0
$569$		−	598.261i		−	1.05142i	−0.850662	−	0.525712i	$-0.823799\pi$
$569$		−	598.261i		−	1.05142i	0.850662	−	0.525712i	$-0.176201\pi$
$570$		0			0
$571$	1112.73			1.94874			0.974368	−	0.224959i	$-0.0722249\pi$
$571$	1112.73			1.94874			0.974368	+	0.224959i	$0.0722249\pi$
$572$		0			0
$573$			219.432i			0.382953i
$574$		0			0
$575$	502.173			0.873344
$576$		0			0
$577$	−347.087			−0.601537			−0.300769	−	0.953697i	$-0.597243\pi$
$577$	−347.087			−0.601537			−0.300769	+	0.953697i	$0.597243\pi$
$578$		0			0
$579$	108.044			0.186604
$580$		0			0
$581$	−876.041			−1.50782
$582$		0			0
$583$		−	18.8532i		−	0.0323383i
$584$		0			0
$585$			1099.21i			1.87899i
$586$		0			0
$587$	−633.519			−1.07925			−0.539625	−	0.841906i	$-0.681434\pi$
$587$	−633.519			−1.07925			−0.539625	+	0.841906i	$0.681434\pi$
$588$		0			0
$589$	−319.469	+	131.804i	−0.542393	+	0.223776i
$590$		0			0
$591$		−	87.6217i		−	0.148260i
$592$		0			0
$593$	−232.227			−0.391614			−0.195807	−	0.980642i	$-0.562733\pi$
$593$	−232.227			−0.391614			−0.195807	+	0.980642i	$0.562733\pi$
$594$		0			0
$595$	−431.314			−0.724898
$596$		0			0
$597$		−	56.0009i		−	0.0938038i
$598$		0			0
$599$			168.706i			0.281645i	0.990035	+	0.140823i	$0.0449748\pi$
$599$			168.706i			0.281645i	−0.990035	+	0.140823i	$0.955025\pi$
$600$		0			0
$601$			823.410i			1.37007i	0.728512	+	0.685033i	$0.240212\pi$
$601$			823.410i			1.37007i	−0.728512	+	0.685033i	$0.759788\pi$
$602$		0			0
$603$		−	592.817i		−	0.983113i
$604$		0			0
$605$	−761.871			−1.25929
$606$		0			0
$607$		−	495.883i		−	0.816941i	−0.912771	−	0.408471i	$-0.866062\pi$
$607$		−	495.883i		−	0.816941i	0.912771	−	0.408471i	$-0.133938\pi$
$608$		0			0
$609$	278.677			0.457599
$610$		0			0
$611$		−	746.733i		−	1.22215i
$612$		0			0
$613$	−312.047			−0.509049			−0.254524	−	0.967066i	$-0.581919\pi$
$613$	−312.047			−0.509049			−0.254524	+	0.967066i	$0.581919\pi$
$614$		0			0
$615$	85.4073			0.138874
$616$		0			0
$617$	64.3994			0.104375			0.0521875	−	0.998637i	$-0.483381\pi$
$617$	64.3994			0.104375			0.0521875	+	0.998637i	$0.483381\pi$
$618$		0			0
$619$	259.565			0.419329			0.209665	−	0.977773i	$-0.432763\pi$
$619$	259.565			0.419329			0.209665	+	0.977773i	$0.432763\pi$
$620$		0			0
$621$		−	329.589i		−	0.530739i
$622$		0			0
$623$			1211.27i			1.94426i
$624$		0			0
$625$	−748.427			−1.19748
$626$		0			0
$627$	−27.1286	+	11.1925i	−0.0432673	+	0.0178509i
$628$		0			0
$629$			332.698i			0.528932i
$630$		0			0
$631$	1054.58			1.67128			0.835640	−	0.549277i	$-0.185097\pi$
$631$	1054.58			1.67128			0.835640	+	0.549277i	$0.185097\pi$
$632$		0			0
$633$	−173.902			−0.274726
$634$		0			0
$635$			1452.59i			2.28754i
$636$		0			0
$637$			438.317i			0.688096i
$638$		0			0
$639$			411.336i			0.643719i
$640$		0			0
$641$		−	894.640i		−	1.39569i	−0.716247	−	0.697847i	$-0.754141\pi$
$641$		−	894.640i		−	1.39569i	0.716247	−	0.697847i	$-0.245859\pi$
$642$		0			0
$643$	−24.9915			−0.0388671			−0.0194335	−	0.999811i	$-0.506186\pi$
$643$	−24.9915			−0.0388671			−0.0194335	+	0.999811i	$0.506186\pi$
$644$		0			0
$645$		−	334.527i		−	0.518646i
$646$		0			0
$647$	147.497			0.227970			0.113985	−	0.993482i	$-0.463638\pi$
$647$	147.497			0.227970			0.113985	+	0.993482i	$0.463638\pi$
$648$		0			0
$649$			149.685i			0.230639i
$650$		0			0
$651$	−104.852			−0.161063
$652$		0			0
$653$	355.032			0.543694			0.271847	−	0.962340i	$-0.412366\pi$
$653$	355.032			0.543694			0.271847	+	0.962340i	$0.412366\pi$
$654$		0			0
$655$	−1114.29			−1.70121
$656$		0			0
$657$	612.363			0.932060
$658$		0			0
$659$		−	18.9531i		−	0.0287605i	−0.999897	−	0.0143802i	$-0.995422\pi$
$659$		−	18.9531i		−	0.0287605i	0.999897	−	0.0143802i	$-0.00457753\pi$
$660$		0			0
$661$		−	988.059i		−	1.49479i	−0.664378	−	0.747397i	$-0.731303\pi$
$661$		−	988.059i		−	1.49479i	0.664378	−	0.747397i	$-0.268697\pi$
$662$		0			0
$663$	103.787			0.156542
$664$		0			0
$665$	−402.516	−	975.626i	−0.605287	−	1.46711i
$666$		0			0
$667$		−	1331.74i		−	1.99662i
$668$		0			0
$669$	200.415			0.299574
$670$		0			0
$671$	−114.320			−0.170373
$672$		0			0
$673$		−	330.989i		−	0.491811i	−0.969294	−	0.245905i	$-0.920915\pi$
$673$		−	330.989i		−	0.491811i	0.969294	−	0.245905i	$-0.0790853\pi$
$674$		0			0
$675$		−	218.098i		−	0.323109i
$676$		0			0
$677$			409.135i			0.604335i	0.953255	+	0.302168i	$0.0977103\pi$
$677$			409.135i			0.604335i	−0.953255	+	0.302168i	$0.902290\pi$
$678$		0			0
$679$		−	416.801i		−	0.613845i
$680$		0			0
$681$	−26.6486			−0.0391315
$682$		0			0
$683$		−	599.751i		−	0.878112i	−0.898460	−	0.439056i	$-0.855313\pi$
$683$		−	599.751i		−	0.878112i	0.898460	−	0.439056i	$-0.144687\pi$
$684$		0			0
$685$	−367.606			−0.536651
$686$		0			0
$687$			229.538i			0.334117i
$688$		0			0
$689$	−163.152			−0.236795
$690$		0			0
$691$	−709.461			−1.02672			−0.513358	−	0.858174i	$-0.671599\pi$
$691$	−709.461			−1.02672			−0.513358	+	0.858174i	$0.671599\pi$
$692$		0			0
$693$	163.215			0.235520
$694$		0			0
$695$	−998.076			−1.43608
$696$		0			0
$697$			147.824i			0.212085i
$698$		0			0
$699$			0.102742i			0.000146985i
$700$		0			0
$701$	−328.829			−0.469085			−0.234543	−	0.972106i	$-0.575359\pi$
$701$	−328.829			−0.469085			−0.234543	+	0.972106i	$0.575359\pi$
$702$		0			0
$703$	−752.559	+	310.485i	−1.07050	+	0.441657i
$704$		0			0
$705$			171.013i			0.242572i
$706$		0			0
$707$	−114.908			−0.162529
$708$		0			0
$709$	222.996			0.314522			0.157261	−	0.987557i	$-0.449734\pi$
$709$	222.996			0.314522			0.157261	+	0.987557i	$0.449734\pi$
$710$		0			0
$711$			1259.12i			1.77092i
$712$		0			0
$713$			501.066i			0.702758i
$714$		0			0
$715$		−	291.554i		−	0.407768i
$716$		0			0
$717$		−	69.2930i		−	0.0966429i
$718$		0			0
$719$	−1268.76			−1.76462			−0.882312	−	0.470666i	$-0.844014\pi$
$719$	−1268.76			−1.76462			−0.882312	+	0.470666i	$0.844014\pi$
$720$		0			0
$721$			97.6946i			0.135499i
$722$		0			0
$723$	15.1553			0.0209617
$724$		0			0
$725$		−	881.252i		−	1.21552i
$726$		0			0
$727$	818.050			1.12524			0.562620	−	0.826715i	$-0.309793\pi$
$727$	818.050			1.12524			0.562620	+	0.826715i	$0.309793\pi$
$728$		0			0
$729$	512.385			0.702859
$730$		0			0
$731$	579.001			0.792067
$732$		0			0
$733$	−973.604			−1.32825			−0.664123	−	0.747623i	$-0.731195\pi$
$733$	−973.604			−1.32825			−0.664123	+	0.747623i	$0.731195\pi$
$734$		0			0
$735$		−	100.381i		−	0.136573i
$736$		0			0
$737$			157.238i			0.213349i
$738$		0			0
$739$	907.317			1.22776			0.613882	−	0.789398i	$-0.289607\pi$
$739$	907.317			1.22776			0.613882	+	0.789398i	$0.289607\pi$
$740$		0			0
$741$	96.8575	+	234.765i	0.130712	+	0.316822i
$742$		0			0
$743$		−	1106.90i		−	1.48978i	−0.667189	−	0.744888i	$-0.732503\pi$
$743$		−	1106.90i		−	1.48978i	0.667189	−	0.744888i	$-0.267497\pi$
$744$		0			0
$745$	−1015.96			−1.36371
$746$		0			0
$747$	884.962			1.18469
$748$		0			0
$749$			437.248i			0.583776i
$750$		0			0
$751$			939.575i			1.25110i	0.780185	+	0.625549i	$0.215125\pi$
$751$			939.575i			1.25110i	−0.780185	+	0.625549i	$0.784875\pi$
$752$		0			0
$753$		−	190.382i		−	0.252831i
$754$		0			0
$755$		−	217.519i		−	0.288105i
$756$		0			0
$757$	−460.148			−0.607857			−0.303928	−	0.952695i	$-0.598298\pi$
$757$	−460.148			−0.607857			−0.303928	+	0.952695i	$0.598298\pi$
$758$		0			0
$759$			42.5494i			0.0560598i
$760$		0			0
$761$	−910.830			−1.19689			−0.598443	−	0.801165i	$-0.704214\pi$
$761$	−910.830			−1.19689			−0.598443	+	0.801165i	$0.704214\pi$
$762$		0			0
$763$		−	589.368i		−	0.772436i
$764$		0			0
$765$	435.706			0.569551
$766$		0			0
$767$	1295.34			1.68884
$768$		0			0
$769$	177.692			0.231068			0.115534	−	0.993304i	$-0.463142\pi$
$769$	177.692			0.231068			0.115534	+	0.993304i	$0.463142\pi$
$770$		0			0
$771$	−85.0578			−0.110321
$772$		0			0
$773$		−	1025.65i		−	1.32684i	−0.748246	−	0.663421i	$-0.769104\pi$
$773$		−	1025.65i		−	1.32684i	0.748246	−	0.663421i	$-0.230896\pi$
$774$		0			0
$775$			331.570i			0.427832i
$776$		0			0
$777$	−246.995			−0.317883
$778$		0			0
$779$	−334.375	+	137.954i	−0.429236	+	0.177091i
$780$		0			0
$781$		−	109.103i		−	0.139696i
$782$		0			0
$783$	−578.388			−0.738682
$784$		0			0
$785$	−1506.11			−1.91862
$786$		0			0
$787$			1380.31i			1.75389i	0.480591	+	0.876945i	$0.340422\pi$
$787$			1380.31i			1.75389i	−0.480591	+	0.876945i	$0.659578\pi$
$788$		0			0
$789$		−	90.6496i		−	0.114892i
$790$		0			0
$791$		−	28.0147i		−	0.0354168i
$792$		0			0
$793$			989.305i			1.24755i
$794$		0			0
$795$	37.3642			0.0469990
$796$		0			0
$797$		−	332.943i		−	0.417745i	−0.977943	−	0.208873i	$-0.933021\pi$
$797$		−	332.943i		−	0.417745i	0.977943	−	0.208873i	$-0.0669794\pi$
$798$		0			0
$799$	−295.990			−0.370451
$800$		0			0
$801$		−	1223.61i		−	1.52760i
$802$		0			0
$803$	−162.423			−0.202270
$804$		0			0
$805$	−1530.20			−1.90087
$806$		0			0
$807$	−66.3669			−0.0822391
$808$		0			0
$809$	865.271			1.06956			0.534778	−	0.844993i	$-0.320395\pi$
$809$	865.271			1.06956			0.534778	+	0.844993i	$0.320395\pi$
$810$		0			0
$811$		−	188.051i		−	0.231876i	−0.993256	−	0.115938i	$-0.963013\pi$
$811$		−	188.051i		−	0.231876i	0.993256	−	0.115938i	$-0.0369874\pi$
$812$		0			0
$813$			195.717i			0.240735i
$814$		0			0
$815$	588.578			0.722182
$816$		0			0
$817$	540.342	+	1309.69i	0.661374	+	1.60305i
$818$		0			0
$819$		−	1412.43i		−	1.72458i
$820$		0			0
$821$	−511.728			−0.623298			−0.311649	−	0.950197i	$-0.600881\pi$
$821$	−511.728			−0.623298			−0.311649	+	0.950197i	$0.600881\pi$
$822$		0			0
$823$	−1373.85			−1.66932			−0.834661	−	0.550765i	$-0.814336\pi$
$823$	−1373.85			−1.66932			−0.834661	+	0.550765i	$0.814336\pi$
$824$		0			0
$825$			28.1561i			0.0341287i
$826$		0			0
$827$			897.171i			1.08485i	0.840104	+	0.542425i	$0.182494\pi$
$827$			897.171i			1.08485i	−0.840104	+	0.542425i	$0.817506\pi$
$828$		0			0
$829$		−	868.594i		−	1.04776i	−0.851792	−	0.523880i	$-0.824484\pi$
$829$		−	868.594i		−	1.04776i	0.851792	−	0.523880i	$-0.175516\pi$
$830$		0			0
$831$			127.486i			0.153412i
$832$		0			0
$833$	173.741			0.208572
$834$		0			0
$835$		−	1397.09i		−	1.67316i
$836$		0			0
$837$	217.618			0.259997
$838$		0			0
$839$		−	1252.04i		−	1.49230i	−0.665780	−	0.746148i	$-0.731901\pi$
$839$		−	1252.04i		−	1.49230i	0.665780	−	0.746148i	$-0.268099\pi$
$840$		0			0
$841$	−1496.05			−1.77889
$842$		0			0
$843$	−46.1172			−0.0547060
$844$		0			0
$845$	−1411.89			−1.67088
$846$		0			0
$847$	978.965			1.15580
$848$		0			0
$849$		−	122.017i		−	0.143719i
$850$		0			0
$851$			1180.34i			1.38700i
$852$		0			0
$853$	1430.27			1.67675			0.838375	−	0.545094i	$-0.183506\pi$
$853$	1430.27			1.67675			0.838375	+	0.545094i	$0.183506\pi$
$854$		0			0
$855$	406.615	+	985.561i	0.475573	+	1.15270i
$856$		0			0
$857$			1296.98i			1.51339i	0.653768	+	0.756695i	$0.273187\pi$
$857$			1296.98i			1.51339i	−0.653768	+	0.756695i	$0.726813\pi$
$858$		0			0
$859$	66.2630			0.0771396			0.0385698	−	0.999256i	$-0.487720\pi$
$859$	66.2630			0.0771396			0.0385698	+	0.999256i	$0.487720\pi$
$860$		0			0
$861$	−109.744			−0.127461
$862$		0			0
$863$			1187.91i			1.37648i	0.725481	+	0.688242i	$0.241617\pi$
$863$			1187.91i			1.37648i	−0.725481	+	0.688242i	$0.758383\pi$
$864$		0			0
$865$		−	1873.75i		−	2.16619i
$866$		0			0
$867$			156.054i			0.179993i
$868$		0			0
$869$		−	333.969i		−	0.384314i
$870$		0			0
$871$	1360.71			1.56224
$872$		0			0
$873$			421.045i			0.482297i
$874$		0			0
$875$	376.102			0.429830
$876$		0			0
$877$			1159.25i			1.32184i	0.750458	+	0.660918i	$0.229833\pi$
$877$			1159.25i			1.32184i	−0.750458	+	0.660918i	$0.770167\pi$
$878$		0			0
$879$	267.169			0.303947
$880$		0			0
$881$	106.538			0.120928			0.0604642	−	0.998170i	$-0.480742\pi$
$881$	106.538			0.120928			0.0604642	+	0.998170i	$0.480742\pi$
$882$		0			0
$883$	−236.402			−0.267726			−0.133863	−	0.991000i	$-0.542738\pi$
$883$	−236.402			−0.267726			−0.133863	+	0.991000i	$0.542738\pi$
$884$		0			0
$885$	−296.652			−0.335200
$886$		0			0
$887$			450.343i			0.507714i	0.967242	+	0.253857i	$0.0816993\pi$
$887$			450.343i			0.507714i	−0.967242	+	0.253857i	$0.918301\pi$
$888$		0			0
$889$		−	1866.50i		−	2.09955i
$890$		0			0
$891$	−155.392			−0.174402
$892$		0			0
$893$	−276.228	−	669.526i	−0.309325	−	0.749749i
$894$		0			0
$895$		−	1226.72i		−	1.37064i
$896$		0			0
$897$	368.213			0.410494
$898$		0			0
$899$	879.311			0.978099
$900$		0			0
$901$			64.6702i			0.0717760i
$902$		0			0
$903$			429.850i			0.476024i
$904$		0			0
$905$		−	1831.32i		−	2.02356i
$906$		0			0
$907$		−	1191.72i		−	1.31391i	−0.753930	−	0.656955i	$-0.771844\pi$
$907$		−	1191.72i		−	1.31391i	0.753930	−	0.656955i	$-0.228156\pi$
$908$		0			0
$909$	116.078			0.127699
$910$		0			0
$911$		−	1211.92i		−	1.33031i	−0.746704	−	0.665157i	$-0.768365\pi$
$911$		−	1211.92i		−	1.33031i	0.746704	−	0.665157i	$-0.231635\pi$
$912$		0			0
$913$	−234.727			−0.257094
$914$		0			0
$915$		−	226.566i		−	0.247613i
$916$		0			0
$917$	1431.81			1.56141
$918$		0			0
$919$	−1360.64			−1.48056			−0.740282	−	0.672297i	$-0.765308\pi$
$919$	−1360.64			−1.48056			−0.740282	+	0.672297i	$0.765308\pi$
$920$		0			0
$921$	−55.1064			−0.0598332
$922$		0			0
$923$	−944.150			−1.02291
$924$		0			0
$925$			781.063i			0.844393i
$926$		0			0
$927$		−	98.6895i		−	0.106461i
$928$		0			0
$929$	198.540			0.213714			0.106857	−	0.994274i	$-0.465921\pi$
$929$	198.540			0.213714			0.106857	+	0.994274i	$0.465921\pi$
$930$		0			0
$931$	162.140	+	392.998i	0.174157	+	0.422125i
$932$		0			0
$933$			265.965i			0.285064i
$934$		0			0
$935$	−115.566			−0.123601
$936$		0			0
$937$	1537.12			1.64047			0.820237	−	0.572023i	$-0.193841\pi$
$937$	1537.12			1.64047			0.820237	+	0.572023i	$0.193841\pi$
$938$		0			0
$939$			306.156i			0.326045i
$940$		0			0
$941$		−	30.8049i		−	0.0327363i	−0.999866	−	0.0163681i	$-0.994790\pi$
$941$		−	30.8049i		−	0.0327363i	0.999866	−	0.0163681i	$-0.00521038\pi$
$942$		0			0
$943$			524.444i			0.556144i
$944$		0			0
$945$			664.581i			0.703261i
$946$		0			0
$947$	1205.66			1.27313			0.636567	−	0.771221i	$-0.280354\pi$
$947$	1205.66			1.27313			0.636567	+	0.771221i	$0.280354\pi$
$948$		0			0
$949$			1405.57i			1.48111i
$950$		0			0
$951$	−130.381			−0.137099
$952$		0			0
$953$			1851.45i			1.94276i	0.237538	+	0.971378i	$0.423660\pi$
$953$			1851.45i			1.94276i	−0.237538	+	0.971378i	$0.576340\pi$
$954$		0			0
$955$	−2114.43			−2.21407
$956$		0			0
$957$	74.6690			0.0780240
$958$		0			0
$959$	472.355			0.492550
$960$		0			0
$961$	630.160			0.655734
$962$		0			0
$963$		−	441.701i		−	0.458672i
$964$		0			0
$965$			1041.10i			1.07886i
$966$		0			0
$967$	−1472.55			−1.52281			−0.761404	−	0.648278i	$-0.775489\pi$
$967$	−1472.55			−1.52281			−0.761404	+	0.648278i	$0.775489\pi$
$968$		0			0
$969$	93.0563	−	38.3924i	0.0960333	−	0.0396207i
$970$		0			0
$971$		−	773.553i		−	0.796656i	−0.917243	−	0.398328i	$-0.869591\pi$
$971$		−	773.553i		−	0.796656i	0.917243	−	0.398328i	$-0.130409\pi$
$972$		0			0
$973$	1282.48			1.31806
$974$		0			0
$975$	243.657			0.249905
$976$		0			0
$977$			1073.46i			1.09873i	0.835582	+	0.549366i	$0.185131\pi$
$977$			1073.46i			1.09873i	−0.835582	+	0.549366i	$0.814869\pi$
$978$		0			0
$979$			324.549i			0.331510i
$980$		0			0
$981$			595.370i			0.606901i
$982$		0			0
$983$			600.675i			0.611063i	0.952182	+	0.305532i	$0.0988342\pi$
$983$			600.675i			0.611063i	−0.952182	+	0.305532i	$0.901166\pi$
$984$		0			0
$985$	844.318			0.857175
$986$		0			0
$987$		−	219.743i		−	0.222637i
$988$		0			0
$989$	2054.16			2.07701
$990$		0			0
$991$		−	91.1601i		−	0.0919880i	−0.998942	−	0.0459940i	$-0.985354\pi$
$991$		−	91.1601i		−	0.0919880i	0.998942	−	0.0459940i	$-0.0146455\pi$
$992$		0			0
$993$	194.831			0.196204
$994$		0			0
$995$	539.621			0.542333
$996$		0			0
$997$	863.571			0.866170			0.433085	−	0.901353i	$-0.357425\pi$
$997$	863.571			0.866170			0.433085	+	0.901353i	$0.357425\pi$
$998$		0			0
$999$	512.632			0.513145

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	608.3.e.b.417.9	✓	20
4.3	odd	2	inner	608.3.e.b.417.12	yes	20
8.3	odd	2		1216.3.e.p.1025.10		20
8.5	even	2		1216.3.e.p.1025.11		20
19.18	odd	2	inner	608.3.e.b.417.11	yes	20
76.75	even	2	inner	608.3.e.b.417.10	yes	20
152.37	odd	2		1216.3.e.p.1025.9		20
152.75	even	2		1216.3.e.p.1025.12		20

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
608.3.e.b.417.9	✓	20	1.1	even	1	trivial
608.3.e.b.417.10	yes	20	76.75	even	2	inner
608.3.e.b.417.11	yes	20	19.18	odd	2	inner
608.3.e.b.417.12	yes	20	4.3	odd	2	inner
1216.3.e.p.1025.9		20	152.37	odd	2
1216.3.e.p.1025.10		20	8.3	odd	2
1216.3.e.p.1025.11		20	8.5	even	2
1216.3.e.p.1025.12		20	152.75	even	2

Overview	Random
Universe	Knowledge

Classical	Maass
Hilbert	Bianchi

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

Level:	\( N \)	\(=\)	\( 608 = 2^{5} \cdot 19 \)
Weight:	\( k \)	\(=\)	\( 3 \)
Character orbit:	\([\chi]\)	\(=\)	608.e (of order \(2\), degree \(1\), minimal)

\(f(q)\)	\(=\)	\(q-0.682329i q^{3} +6.57489 q^{5} -8.44839 q^{7} +8.53443 q^{9} -2.26367 q^{11} +19.5893i q^{13} -4.48624i q^{15} +7.76481 q^{17} +(7.24637 + 17.5639i) q^{19} +5.76459i q^{21} +27.5478 q^{23} +18.2292 q^{25} -11.9643i q^{27} -48.3430i q^{29} +18.1890i q^{31} +1.54457i q^{33} -55.5473 q^{35} +42.8469i q^{37} +13.3663 q^{39} +19.0376i q^{41} +74.5673 q^{43} +56.1129 q^{45} -38.1195 q^{47} +22.3754 q^{49} -5.29816i q^{51} +8.32862i q^{53} -14.8834 q^{55} +(11.9844 - 4.94441i) q^{57} -66.1249i q^{59} +50.5024 q^{61} -72.1022 q^{63} +128.797i q^{65} -69.4618i q^{67} -18.7967i q^{69} +48.1973i q^{71} +71.7521 q^{73} -12.4383i q^{75} +19.1243 q^{77} +147.535i q^{79} +68.6463 q^{81} +103.693 q^{83} +51.0528 q^{85} -32.9859 q^{87} -143.373i q^{89} -165.498i q^{91} +12.4109 q^{93} +(47.6441 + 115.481i) q^{95} +49.3349i q^{97} -19.3191 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 20 q - 52 q^{9} + 56 q^{17} + 36 q^{25} + 64 q^{45} + 332 q^{49} + 88 q^{57} - 32 q^{61} - 152 q^{73} + 360 q^{77} - 476 q^{81} - 552 q^{85} + 336 q^{93}+O(q^{100}) \) 20 * q - 52 * q^9 + 56 * q^17 + 36 * q^25 + 64 * q^45 + 332 * q^49 + 88 * q^57 - 32 * q^61 - 152 * q^73 + 360 * q^77 - 476 * q^81 - 552 * q^85 + 336 * q^93

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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