LMFDB - Embedded newform 608.3.e.b.417.13

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			417.13
Root			$-4.46201 - 0.741526i$ of defining polynomial
Character	$\chi$	$=$	608.417
Dual form			608.3.e.b.417.7

$q$-expansion

comment:q-expansion

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

gp:mfcoefs(f, 20)

$f(q)$	$=$	$q+1.48305i q^{3} -4.43504 q^{5} -2.39163 q^{7} +6.80056 q^{9} +12.1358 q^{11} +3.28286i q^{13} -6.57739i q^{15} -7.93631 q^{17} +(18.4057 - 4.71509i) q^{19} -3.54691i q^{21} -22.6573 q^{23} -5.33045 q^{25} +23.4330i q^{27} +42.2757i q^{29} +6.86296i q^{31} +17.9980i q^{33} +10.6070 q^{35} +40.8256i q^{37} -4.86865 q^{39} +59.0876i q^{41} +1.93281 q^{43} -30.1607 q^{45} +12.5092 q^{47} -43.2801 q^{49} -11.7700i q^{51} +89.9311i q^{53} -53.8226 q^{55} +(6.99272 + 27.2965i) q^{57} -51.9117i q^{59} +14.1831 q^{61} -16.2644 q^{63} -14.5596i q^{65} +74.7176i q^{67} -33.6020i q^{69} -88.6362i q^{71} +8.33362 q^{73} -7.90534i q^{75} -29.0242 q^{77} -7.24915i q^{79} +26.4526 q^{81} +55.6631 q^{83} +35.1978 q^{85} -62.6970 q^{87} +11.8042i q^{89} -7.85137i q^{91} -10.1781 q^{93} +(-81.6297 + 20.9116i) q^{95} -20.8644i q^{97} +82.5300 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 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
$3$			1.48305i			0.494351i	0.968971	+	0.247175i	$0.0795023\pi$
$3$			1.48305i			0.494351i	−0.968971	+	0.247175i	$0.920498\pi$
$4$		0			0
$5$	−4.43504			−0.887007			−0.443504	−	0.896273i	$-0.646265\pi$
$5$	−4.43504			−0.887007			−0.443504	+	0.896273i	$0.646265\pi$
$6$		0			0
$7$	−2.39163			−0.341661			−0.170831	−	0.985300i	$-0.554645\pi$
$7$	−2.39163			−0.341661			−0.170831	+	0.985300i	$0.554645\pi$
$8$		0			0
$9$	6.80056			0.755618
$10$		0			0
$11$	12.1358			1.10325			0.551626	−	0.834092i	$-0.314008\pi$
$11$	12.1358			1.10325			0.551626	+	0.834092i	$0.314008\pi$
$12$		0			0
$13$			3.28286i			0.252527i	0.991997	+	0.126264i	$0.0402986\pi$
$13$			3.28286i			0.252527i	−0.991997	+	0.126264i	$0.959701\pi$
$14$		0			0
$15$		−	6.57739i		−	0.438492i
$16$		0			0
$17$	−7.93631			−0.466842			−0.233421	−	0.972376i	$-0.574992\pi$
$17$	−7.93631			−0.466842			−0.233421	+	0.972376i	$0.574992\pi$
$18$		0			0
$19$	18.4057	−	4.71509i	0.968718	−	0.248162i
$19$	18.4057	−	4.71509i	0.968718	−	0.248162i
$20$		0			0
$21$		−	3.54691i		−	0.168900i
$22$		0			0
$23$	−22.6573			−0.985101			−0.492551	−	0.870284i	$-0.663935\pi$
$23$	−22.6573			−0.985101			−0.492551	+	0.870284i	$0.663935\pi$
$24$		0			0
$25$	−5.33045			−0.213218
$26$		0			0
$27$			23.4330i			0.867890i
$28$		0			0
$29$			42.2757i			1.45778i	0.684630	+	0.728891i	$0.259964\pi$
$29$			42.2757i			1.45778i	−0.684630	+	0.728891i	$0.740036\pi$
$30$		0			0
$31$			6.86296i			0.221386i	0.993855	+	0.110693i	$0.0353070\pi$
$31$			6.86296i			0.221386i	−0.993855	+	0.110693i	$0.964693\pi$
$32$		0			0
$33$			17.9980i			0.545393i
$34$		0			0
$35$	10.6070			0.303056
$36$		0			0
$37$			40.8256i			1.10340i	0.834044	+	0.551698i	$0.186020\pi$
$37$			40.8256i			1.10340i	−0.834044	+	0.551698i	$0.813980\pi$
$38$		0			0
$39$	−4.86865			−0.124837
$40$		0			0
$41$			59.0876i			1.44116i	0.693371	+	0.720581i	$0.256125\pi$
$41$			59.0876i			1.44116i	−0.693371	+	0.720581i	$0.743875\pi$
$42$		0			0
$43$	1.93281			0.0449492			0.0224746	−	0.999747i	$-0.492846\pi$
$43$	1.93281			0.0449492			0.0224746	+	0.999747i	$0.492846\pi$
$44$		0			0
$45$	−30.1607			−0.670238
$46$		0			0
$47$	12.5092			0.266153			0.133076	−	0.991106i	$-0.457514\pi$
$47$	12.5092			0.266153			0.133076	+	0.991106i	$0.457514\pi$
$48$		0			0
$49$	−43.2801			−0.883268
$50$		0			0
$51$		−	11.7700i		−	0.230783i
$52$		0			0
$53$			89.9311i			1.69681i	0.529345	+	0.848406i	$0.322438\pi$
$53$			89.9311i			1.69681i	−0.529345	+	0.848406i	$0.677562\pi$
$54$		0			0
$55$	−53.8226			−0.978592
$56$		0			0
$57$	6.99272	+	27.2965i	0.122679	+	0.478886i
$58$		0			0
$59$		−	51.9117i		−	0.879859i	−0.898032	−	0.439930i	$-0.855003\pi$
$59$		−	51.9117i		−	0.879859i	0.898032	−	0.439930i	$-0.144997\pi$
$60$		0			0
$61$	14.1831			0.232510			0.116255	−	0.993219i	$-0.462911\pi$
$61$	14.1831			0.232510			0.116255	+	0.993219i	$0.462911\pi$
$62$		0			0
$63$	−16.2644			−0.258165
$64$		0			0
$65$		−	14.5596i		−	0.223994i
$66$		0			0
$67$			74.7176i			1.11519i	0.830114	+	0.557594i	$0.188275\pi$
$67$			74.7176i			1.11519i	−0.830114	+	0.557594i	$0.811725\pi$
$68$		0			0
$69$		−	33.6020i		−	0.486985i
$70$		0			0
$71$		−	88.6362i		−	1.24840i	−0.781265	−	0.624199i	$-0.785425\pi$
$71$		−	88.6362i		−	1.24840i	0.781265	−	0.624199i	$-0.214575\pi$
$72$		0			0
$73$	8.33362			0.114159			0.0570796	−	0.998370i	$-0.481821\pi$
$73$	8.33362			0.114159			0.0570796	+	0.998370i	$0.481821\pi$
$74$		0			0
$75$		−	7.90534i		−	0.105405i
$76$		0			0
$77$	−29.0242			−0.376938
$78$		0			0
$79$		−	7.24915i		−	0.0917614i	−0.998947	−	0.0458807i	$-0.985391\pi$
$79$		−	7.24915i		−	0.0917614i	0.998947	−	0.0458807i	$-0.0146094\pi$
$80$		0			0
$81$	26.4526			0.326575
$82$		0			0
$83$	55.6631			0.670640			0.335320	−	0.942104i	$-0.391156\pi$
$83$	55.6631			0.670640			0.335320	+	0.942104i	$0.391156\pi$
$84$		0			0
$85$	35.1978			0.414092
$86$		0			0
$87$	−62.6970			−0.720655
$88$		0			0
$89$			11.8042i			0.132632i	0.997799	+	0.0663158i	$0.0211245\pi$
$89$			11.8042i			0.132632i	−0.997799	+	0.0663158i	$0.978876\pi$
$90$		0			0
$91$		−	7.85137i		−	0.0862788i
$92$		0			0
$93$	−10.1781			−0.109442
$94$		0			0
$95$	−81.6297	+	20.9116i	−0.859260	+	0.220122i
$96$		0			0
$97$		−	20.8644i		−	0.215097i	−0.994200	−	0.107549i	$-0.965700\pi$
$97$		−	20.8644i		−	0.215097i	0.994200	−	0.107549i	$-0.0343001\pi$
$98$		0			0
$99$	82.5300			0.833636
$100$		0			0
$101$	−72.5000			−0.717822			−0.358911	−	0.933372i	$-0.616852\pi$
$101$	−72.5000			−0.717822			−0.358911	+	0.933372i	$0.616852\pi$
$102$		0			0
$103$			139.063i			1.35013i	0.737759	+	0.675064i	$0.235884\pi$
$103$			139.063i			1.35013i	−0.737759	+	0.675064i	$0.764116\pi$
$104$		0			0
$105$			15.7307i			0.149816i
$106$		0			0
$107$			88.9100i			0.830935i	0.909608	+	0.415467i	$0.136382\pi$
$107$			88.9100i			0.830935i	−0.909608	+	0.415467i	$0.863618\pi$
$108$		0			0
$109$		−	103.963i		−	0.953785i	−0.878962	−	0.476892i	$-0.841763\pi$
$109$		−	103.963i		−	0.953785i	0.878962	−	0.476892i	$-0.158237\pi$
$110$		0			0
$111$	−60.5465			−0.545464
$112$		0			0
$113$			141.795i			1.25482i	0.778688	+	0.627412i	$0.215886\pi$
$113$			141.795i			1.25482i	−0.778688	+	0.627412i	$0.784114\pi$
$114$		0			0
$115$	100.486			0.873792
$116$		0			0
$117$			22.3253i			0.190814i
$118$		0			0
$119$	18.9807			0.159502
$120$		0			0
$121$	26.2768			0.217164
$122$		0			0
$123$	−87.6300			−0.712439
$124$		0			0
$125$	134.517			1.07613
$126$		0			0
$127$		−	127.343i		−	1.00270i	−0.865243	−	0.501352i	$-0.832836\pi$
$127$		−	127.343i		−	1.00270i	0.865243	−	0.501352i	$-0.167164\pi$
$128$		0			0
$129$			2.86646i			0.0222206i
$130$		0			0
$131$	199.741			1.52474			0.762369	−	0.647142i	$-0.224036\pi$
$131$	199.741			1.52474			0.762369	+	0.647142i	$0.224036\pi$
$132$		0			0
$133$	−44.0195	+	11.2767i	−0.330973	+	0.0847875i
$134$		0			0
$135$		−	103.926i		−	0.769825i
$136$		0			0
$137$	−153.434			−1.11996			−0.559979	−	0.828507i	$-0.689191\pi$
$137$	−153.434			−1.11996			−0.559979	+	0.828507i	$0.689191\pi$
$138$		0			0
$139$	12.6618			0.0910920			0.0455460	−	0.998962i	$-0.485497\pi$
$139$	12.6618			0.0910920			0.0455460	+	0.998962i	$0.485497\pi$
$140$		0			0
$141$			18.5518i			0.131573i
$142$		0			0
$143$			39.8400i			0.278601i
$144$		0			0
$145$		−	187.494i		−	1.29306i
$146$		0			0
$147$		−	64.1866i		−	0.436644i
$148$		0			0
$149$	−4.06864			−0.0273063			−0.0136532	−	0.999907i	$-0.504346\pi$
$149$	−4.06864			−0.0273063			−0.0136532	+	0.999907i	$0.504346\pi$
$150$		0			0
$151$		−	126.737i		−	0.839319i	−0.907682	−	0.419660i	$-0.862149\pi$
$151$		−	126.737i		−	0.839319i	0.907682	−	0.419660i	$-0.137851\pi$
$152$		0			0
$153$	−53.9713			−0.352754
$154$		0			0
$155$		−	30.4375i		−	0.196371i
$156$		0			0
$157$	116.234			0.740343			0.370172	−	0.928963i	$-0.379299\pi$
$157$	116.234			0.740343			0.370172	+	0.928963i	$0.379299\pi$
$158$		0			0
$159$	−133.372			−0.838820
$160$		0			0
$161$	54.1879			0.336571
$162$		0			0
$163$	113.662			0.697315			0.348657	−	0.937250i	$-0.386638\pi$
$163$	113.662			0.697315			0.348657	+	0.937250i	$0.386638\pi$
$164$		0			0
$165$		−	79.8216i		−	0.483767i
$166$		0			0
$167$			172.151i			1.03084i	0.856937	+	0.515421i	$0.172364\pi$
$167$			172.151i			1.03084i	−0.856937	+	0.515421i	$0.827636\pi$
$168$		0			0
$169$	158.223			0.936230
$170$		0			0
$171$	125.169	−	32.0652i	0.731981	−	0.187516i
$172$		0			0
$173$		−	69.1853i		−	0.399915i	−0.979805	−	0.199958i	$-0.935920\pi$
$173$		−	69.1853i		−	0.399915i	0.979805	−	0.199958i	$-0.0640804\pi$
$174$		0			0
$175$	12.7485			0.0728484
$176$		0			0
$177$	76.9877			0.434959
$178$		0			0
$179$			347.290i			1.94017i	0.242768	+	0.970084i	$0.421944\pi$
$179$			347.290i			1.94017i	−0.242768	+	0.970084i	$0.578056\pi$
$180$		0			0
$181$		−	259.881i		−	1.43580i	−0.696144	−	0.717902i	$-0.745103\pi$
$181$		−	259.881i		−	1.43580i	0.696144	−	0.717902i	$-0.254897\pi$
$182$		0			0
$183$			21.0342i			0.114941i
$184$		0			0
$185$		−	181.063i		−	0.978719i
$186$		0			0
$187$	−96.3132			−0.515044
$188$		0			0
$189$		−	56.0431i		−	0.296524i
$190$		0			0
$191$	−214.253			−1.12174			−0.560872	−	0.827902i	$-0.689534\pi$
$191$	−214.253			−1.12174			−0.560872	+	0.827902i	$0.689534\pi$
$192$		0			0
$193$		−	172.646i		−	0.894538i	−0.894399	−	0.447269i	$-0.852397\pi$
$193$		−	172.646i		−	0.894538i	0.894399	−	0.447269i	$-0.147603\pi$
$194$		0			0
$195$	21.5926			0.110731
$196$		0			0
$197$	−170.446			−0.865207			−0.432603	−	0.901584i	$-0.642405\pi$
$197$	−170.446			−0.865207			−0.432603	+	0.901584i	$0.642405\pi$
$198$		0			0
$199$	161.905			0.813593			0.406796	−	0.913519i	$-0.366646\pi$
$199$	161.905			0.813593			0.406796	+	0.913519i	$0.366646\pi$
$200$		0			0
$201$	−110.810			−0.551294
$202$		0			0
$203$		−	101.108i		−	0.498068i
$204$		0			0
$205$		−	262.056i		−	1.27832i
$206$		0			0
$207$	−154.082			−0.744360
$208$		0			0
$209$	223.367	−	57.2212i	1.06874	−	0.273786i
$210$		0			0
$211$		−	76.9837i		−	0.364852i	−0.983220	−	0.182426i	$-0.941605\pi$
$211$		−	76.9837i		−	0.364852i	0.983220	−	0.182426i	$-0.0583949\pi$
$212$		0			0
$213$	131.452			0.617146
$214$		0			0
$215$	−8.57210			−0.0398702
$216$		0			0
$217$		−	16.4137i		−	0.0756390i
$218$		0			0
$219$			12.3592i			0.0564347i
$220$		0			0
$221$		−	26.0538i		−	0.117890i
$222$		0			0
$223$		−	124.863i		−	0.559922i	−0.960011	−	0.279961i	$-0.909679\pi$
$223$		−	124.863i		−	0.559922i	0.960011	−	0.279961i	$-0.0903215\pi$
$224$		0			0
$225$	−36.2501			−0.161111
$226$		0			0
$227$		−	232.836i		−	1.02571i	−0.858475	−	0.512855i	$-0.828588\pi$
$227$		−	232.836i		−	1.02571i	0.858475	−	0.512855i	$-0.171412\pi$
$228$		0			0
$229$	76.6562			0.334743			0.167372	−	0.985894i	$-0.446472\pi$
$229$	76.6562			0.334743			0.167372	+	0.985894i	$0.446472\pi$
$230$		0			0
$231$		−	43.0444i		−	0.186340i
$232$		0			0
$233$	−20.3112			−0.0871724			−0.0435862	−	0.999050i	$-0.513878\pi$
$233$	−20.3112			−0.0871724			−0.0435862	+	0.999050i	$0.513878\pi$
$234$		0			0
$235$	−55.4787			−0.236080
$236$		0			0
$237$	10.7509			0.0453623
$238$		0			0
$239$	−112.228			−0.469573			−0.234787	−	0.972047i	$-0.575439\pi$
$239$	−112.228			−0.469573			−0.234787	+	0.972047i	$0.575439\pi$
$240$		0			0
$241$		−	226.606i		−	0.940274i	−0.882593	−	0.470137i	$-0.844204\pi$
$241$		−	226.606i		−	0.940274i	0.882593	−	0.470137i	$-0.155796\pi$
$242$		0			0
$243$			250.128i			1.02933i
$244$		0			0
$245$	191.949			0.783465
$246$		0			0
$247$	15.4790	+	60.4231i	0.0626678	+	0.244628i
$248$		0			0
$249$			82.5513i			0.331531i
$250$		0			0
$251$	−268.070			−1.06801			−0.534004	−	0.845482i	$-0.679313\pi$
$251$	−268.070			−1.06801			−0.534004	+	0.845482i	$0.679313\pi$
$252$		0			0
$253$	−274.964			−1.08681
$254$		0			0
$255$			52.2002i			0.204707i
$256$		0			0
$257$		−	64.4086i		−	0.250617i	−0.992118	−	0.125309i	$-0.960008\pi$
$257$		−	64.4086i		−	0.250617i	0.992118	−	0.125309i	$-0.0399921\pi$
$258$		0			0
$259$		−	97.6397i		−	0.376987i
$260$		0			0
$261$			287.498i			1.10153i
$262$		0			0
$263$	503.756			1.91542			0.957710	−	0.287735i	$-0.0929022\pi$
$263$	503.756			1.91542			0.957710	+	0.287735i	$0.0929022\pi$
$264$		0			0
$265$		−	398.848i		−	1.50509i
$266$		0			0
$267$	−17.5063			−0.0655665
$268$		0			0
$269$			86.2874i			0.320771i	0.987054	+	0.160386i	$0.0512737\pi$
$269$			86.2874i			0.320771i	−0.987054	+	0.160386i	$0.948726\pi$
$270$		0			0
$271$	−1.14933			−0.00424108			−0.00212054	−	0.999998i	$-0.500675\pi$
$271$	−1.14933			−0.00424108			−0.00212054	+	0.999998i	$0.500675\pi$
$272$		0			0
$273$	11.6440			0.0426520
$274$		0			0
$275$	−64.6891			−0.235233
$276$		0			0
$277$	272.800			0.984838			0.492419	−	0.870358i	$-0.336113\pi$
$277$	272.800			0.984838			0.492419	+	0.870358i	$0.336113\pi$
$278$		0			0
$279$			46.6720i			0.167283i
$280$		0			0
$281$			86.7076i			0.308568i	0.988027	+	0.154284i	$0.0493071\pi$
$281$			86.7076i			0.308568i	−0.988027	+	0.154284i	$0.950693\pi$
$282$		0			0
$283$	−355.833			−1.25736			−0.628680	−	0.777664i	$-0.716405\pi$
$283$	−355.833			−1.25736			−0.628680	+	0.777664i	$0.716405\pi$
$284$		0			0
$285$	−31.0129	−	121.061i	−0.108817	−	0.424776i
$286$		0			0
$287$		−	141.316i		−	0.492389i
$288$		0			0
$289$	−226.015			−0.782059
$290$		0			0
$291$	30.9430			0.106333
$292$		0			0
$293$		−	10.2955i		−	0.0351381i	−0.999846	−	0.0175691i	$-0.994407\pi$
$293$		−	10.2955i		−	0.0351381i	0.999846	−	0.0175691i	$-0.00559270\pi$
$294$		0			0
$295$			230.230i			0.780442i
$296$		0			0
$297$			284.378i			0.957501i
$298$		0			0
$299$		−	74.3808i		−	0.248765i
$300$		0			0
$301$	−4.62257			−0.0153574
$302$		0			0
$303$		−	107.521i		−	0.354856i
$304$		0			0
$305$	−62.9025			−0.206238
$306$		0			0
$307$		−	402.839i		−	1.31218i	−0.754683	−	0.656090i	$-0.772209\pi$
$307$		−	402.839i		−	1.31218i	0.754683	−	0.656090i	$-0.227791\pi$
$308$		0			0
$309$	−206.238			−0.667437
$310$		0			0
$311$	394.559			1.26868			0.634340	−	0.773055i	$-0.281272\pi$
$311$	394.559			1.26868			0.634340	+	0.773055i	$0.281272\pi$
$312$		0			0
$313$	375.989			1.20124			0.600621	−	0.799534i	$-0.294920\pi$
$313$	375.989			1.20124			0.600621	+	0.799534i	$0.294920\pi$
$314$		0			0
$315$	72.1332			0.228994
$316$		0			0
$317$			295.081i			0.930856i	0.885086	+	0.465428i	$0.154100\pi$
$317$			295.081i			0.930856i	−0.885086	+	0.465428i	$0.845900\pi$
$318$		0			0
$319$			513.048i			1.60830i
$320$		0			0
$321$	−131.858			−0.410773
$322$		0			0
$323$	−146.073	+	37.4204i	−0.452238	+	0.115853i
$324$		0			0
$325$		−	17.4991i		−	0.0538434i
$326$		0			0
$327$	154.182			0.471504
$328$		0			0
$329$	−29.9173			−0.0909341
$330$		0			0
$331$			290.207i			0.876757i	0.898790	+	0.438378i	$0.144447\pi$
$331$			290.207i			0.876757i	−0.898790	+	0.438378i	$0.855553\pi$
$332$		0			0
$333$			277.637i			0.833745i
$334$		0			0
$335$		−	331.375i		−	0.989180i
$336$		0			0
$337$			467.781i			1.38807i	0.719940	+	0.694037i	$0.244169\pi$
$337$			467.781i			1.38807i	−0.719940	+	0.694037i	$0.755831\pi$
$338$		0			0
$339$	−210.289			−0.620323
$340$		0			0
$341$			83.2873i			0.244244i
$342$		0			0
$343$	220.700			0.643439
$344$		0			0
$345$			149.026i			0.431960i
$346$		0			0
$347$	−391.163			−1.12727			−0.563635	−	0.826024i	$-0.690598\pi$
$347$	−391.163			−1.12727			−0.563635	+	0.826024i	$0.690598\pi$
$348$		0			0
$349$	−485.699			−1.39169			−0.695844	−	0.718193i	$-0.744970\pi$
$349$	−485.699			−1.39169			−0.695844	+	0.718193i	$0.744970\pi$
$350$		0			0
$351$	−76.9273			−0.219166
$352$		0			0
$353$	−227.668			−0.644950			−0.322475	−	0.946578i	$-0.604515\pi$
$353$	−227.668			−0.644950			−0.322475	+	0.946578i	$0.604515\pi$
$354$		0			0
$355$			393.105i			1.10734i
$356$		0			0
$357$			28.1494i			0.0788497i
$358$		0			0
$359$	494.159			1.37649			0.688244	−	0.725479i	$-0.258382\pi$
$359$	494.159			1.37649			0.688244	+	0.725479i	$0.258382\pi$
$360$		0			0
$361$	316.536	−	173.568i	0.876831	−	0.480799i
$362$		0			0
$363$			38.9698i			0.107355i
$364$		0			0
$365$	−36.9599			−0.101260
$366$		0			0
$367$	195.737			0.533344			0.266672	−	0.963787i	$-0.414076\pi$
$367$	195.737			0.533344			0.266672	+	0.963787i	$0.414076\pi$
$368$		0			0
$369$			401.829i			1.08897i
$370$		0			0
$371$		−	215.082i		−	0.579735i
$372$		0			0
$373$			257.888i			0.691390i	0.938347	+	0.345695i	$0.112357\pi$
$373$			257.888i			0.691390i	−0.938347	+	0.345695i	$0.887643\pi$
$374$		0			0
$375$			199.495i			0.531987i
$376$		0			0
$377$	−138.785			−0.368130
$378$		0			0
$379$		−	485.408i		−	1.28076i	−0.768058	−	0.640380i	$-0.778777\pi$
$379$		−	485.408i		−	1.28076i	0.768058	−	0.640380i	$-0.221223\pi$
$380$		0			0
$381$	188.857			0.495687
$382$		0			0
$383$		−	289.386i		−	0.755578i	−0.925892	−	0.377789i	$-0.876685\pi$
$383$		−	289.386i		−	0.755578i	0.925892	−	0.377789i	$-0.123315\pi$
$384$		0			0
$385$	128.724			0.334347
$386$		0			0
$387$	13.1442			0.0339644
$388$		0			0
$389$	−90.1464			−0.231739			−0.115869	−	0.993264i	$-0.536965\pi$
$389$	−90.1464			−0.231739			−0.115869	+	0.993264i	$0.536965\pi$
$390$		0			0
$391$	179.816			0.459886
$392$		0			0
$393$			296.226i			0.753756i
$394$		0			0
$395$			32.1502i			0.0813930i
$396$		0			0
$397$	243.533			0.613433			0.306717	−	0.951801i	$-0.400770\pi$
$397$	243.533			0.613433			0.306717	+	0.951801i	$0.400770\pi$
$398$		0			0
$399$	−16.7240	−	65.2831i	−0.0419147	−	0.163617i
$400$		0			0
$401$		−	42.6638i		−	0.106394i	−0.998584	−	0.0531968i	$-0.983059\pi$
$401$		−	42.6638i		−	0.106394i	0.998584	−	0.0531968i	$-0.0169411\pi$
$402$		0			0
$403$	−22.5301			−0.0559060
$404$		0			0
$405$	−117.318			−0.289675
$406$		0			0
$407$			495.450i			1.21732i
$408$		0			0
$409$			118.559i			0.289874i	0.989441	+	0.144937i	$0.0462980\pi$
$409$			118.559i			0.289874i	−0.989441	+	0.144937i	$0.953702\pi$
$410$		0			0
$411$		−	227.551i		−	0.553652i
$412$		0			0
$413$			124.153i			0.300614i
$414$		0			0
$415$	−246.868			−0.594863
$416$		0			0
$417$			18.7781i			0.0450314i
$418$		0			0
$419$	246.145			0.587459			0.293729	−	0.955889i	$-0.405104\pi$
$419$	246.145			0.587459			0.293729	+	0.955889i	$0.405104\pi$
$420$		0			0
$421$		−	135.369i		−	0.321542i	−0.986992	−	0.160771i	$-0.948602\pi$
$421$		−	135.369i		−	0.321542i	0.986992	−	0.160771i	$-0.0513981\pi$
$422$		0			0
$423$	85.0695			0.201110
$424$		0			0
$425$	42.3041			0.0995391
$426$		0			0
$427$	−33.9207			−0.0794395
$428$		0			0
$429$	−59.0847			−0.137727
$430$		0			0
$431$			526.311i			1.22114i	0.791962	+	0.610570i	$0.209060\pi$
$431$			526.311i			1.22114i	−0.791962	+	0.610570i	$0.790940\pi$
$432$		0			0
$433$		−	340.306i		−	0.785926i	−0.919554	−	0.392963i	$-0.871450\pi$
$433$		−	340.306i		−	0.785926i	0.919554	−	0.392963i	$-0.128550\pi$
$434$		0			0
$435$	278.064			0.639227
$436$		0			0
$437$	−417.023	+	106.831i	−0.954286	+	0.244465i
$438$		0			0
$439$		−	482.555i		−	1.09921i	−0.835423	−	0.549607i	$-0.814777\pi$
$439$		−	482.555i		−	1.09921i	0.835423	−	0.549607i	$-0.185223\pi$
$440$		0			0
$441$	−294.329			−0.667413
$442$		0			0
$443$	−341.370			−0.770586			−0.385293	−	0.922794i	$-0.625900\pi$
$443$	−341.370			−0.770586			−0.385293	+	0.922794i	$0.625900\pi$
$444$		0			0
$445$		−	52.3521i		−	0.117645i
$446$		0			0
$447$		−	6.03401i		−	0.0134989i
$448$		0			0
$449$			605.183i			1.34785i	0.738802	+	0.673923i	$0.235392\pi$
$449$			605.183i			1.34785i	−0.738802	+	0.673923i	$0.764608\pi$
$450$		0			0
$451$			717.074i			1.58996i
$452$		0			0
$453$	187.958			0.414918
$454$		0			0
$455$			34.8211i			0.0765299i
$456$		0			0
$457$	−671.856			−1.47015			−0.735073	−	0.677988i	$-0.762852\pi$
$457$	−671.856			−1.47015			−0.735073	+	0.677988i	$0.762852\pi$
$458$		0			0
$459$		−	185.972i		−	0.405167i
$460$		0			0
$461$	768.509			1.66705			0.833524	−	0.552483i	$-0.186320\pi$
$461$	768.509			1.66705			0.833524	+	0.552483i	$0.186320\pi$
$462$		0			0
$463$	−507.242			−1.09555			−0.547777	−	0.836624i	$-0.684526\pi$
$463$	−507.242			−1.09555			−0.547777	+	0.836624i	$0.684526\pi$
$464$		0			0
$465$	45.1404			0.0970761
$466$		0			0
$467$	−538.017			−1.15207			−0.576035	−	0.817425i	$-0.695401\pi$
$467$	−538.017			−1.15207			−0.576035	+	0.817425i	$0.695401\pi$
$468$		0			0
$469$		−	178.697i		−	0.381017i
$470$		0			0
$471$			172.381i			0.365989i
$472$		0			0
$473$	23.4562			0.0495902
$474$		0			0
$475$	−98.1105	+	25.1336i	−0.206548	+	0.0529127i
$476$		0			0
$477$			611.581i			1.28214i
$478$		0			0
$479$	221.728			0.462897			0.231448	−	0.972847i	$-0.425654\pi$
$479$	221.728			0.462897			0.231448	+	0.972847i	$0.425654\pi$
$480$		0			0
$481$	−134.025			−0.278638
$482$		0			0
$483$			80.3635i			0.166384i
$484$		0			0
$485$			92.5345i			0.190793i
$486$		0			0
$487$		−	771.461i		−	1.58411i	−0.610450	−	0.792055i	$-0.709012\pi$
$487$		−	771.461i		−	1.58411i	0.610450	−	0.792055i	$-0.290988\pi$
$488$		0			0
$489$			168.567i			0.344718i
$490$		0			0
$491$	−302.331			−0.615745			−0.307873	−	0.951428i	$-0.599617\pi$
$491$	−302.331			−0.615745			−0.307873	+	0.951428i	$0.599617\pi$
$492$		0			0
$493$		−	335.513i		−	0.680554i
$494$		0			0
$495$	−366.023			−0.739441
$496$		0			0
$497$			211.985i			0.426529i
$498$		0			0
$499$	663.715			1.33009			0.665045	−	0.746803i	$-0.268412\pi$
$499$	663.715			1.33009			0.665045	+	0.746803i	$0.268412\pi$
$500$		0			0
$501$	−255.308			−0.509597
$502$		0			0
$503$	274.172			0.545073			0.272537	−	0.962145i	$-0.412137\pi$
$503$	274.172			0.545073			0.272537	+	0.962145i	$0.412137\pi$
$504$		0			0
$505$	321.540			0.636713
$506$		0			0
$507$			234.653i			0.462826i
$508$		0			0
$509$			710.634i			1.39614i	0.716030	+	0.698069i	$0.245957\pi$
$509$			710.634i			1.39614i	−0.716030	+	0.698069i	$0.754043\pi$
$510$		0			0
$511$	−19.9309			−0.0390038
$512$		0			0
$513$	110.489	+	431.300i	0.215378	+	0.840742i
$514$		0			0
$515$		−	616.750i		−	1.19757i
$516$		0			0
$517$	151.809			0.293634
$518$		0			0
$519$	102.605			0.197698
$520$		0			0
$521$		−	189.532i		−	0.363784i	−0.983318	−	0.181892i	$-0.941778\pi$
$521$		−	189.532i		−	0.363784i	0.983318	−	0.181892i	$-0.0582222\pi$
$522$		0			0
$523$		−	267.270i		−	0.511032i	−0.966805	−	0.255516i	$-0.917755\pi$
$523$		−	267.270i		−	0.511032i	0.966805	−	0.255516i	$-0.0822454\pi$
$524$		0			0
$525$			18.9066i			0.0360126i
$526$		0			0
$527$		−	54.4666i		−	0.103352i
$528$		0			0
$529$	−15.6453			−0.0295752
$530$		0			0
$531$		−	353.029i		−	0.664837i
$532$		0			0
$533$	−193.976			−0.363933
$534$		0			0
$535$		−	394.319i		−	0.737045i
$536$		0			0
$537$	−515.049			−0.959123
$538$		0			0
$539$	−525.237			−0.974466
$540$		0			0
$541$	−53.1260			−0.0981996			−0.0490998	−	0.998794i	$-0.515635\pi$
$541$	−53.1260			−0.0981996			−0.0490998	+	0.998794i	$0.515635\pi$
$542$		0			0
$543$	385.416			0.709791
$544$		0			0
$545$			461.078i			0.846014i
$546$		0			0
$547$			277.407i			0.507143i	0.967317	+	0.253572i	$0.0816054\pi$
$547$			277.407i			0.507143i	−0.967317	+	0.253572i	$0.918395\pi$
$548$		0			0
$549$	96.4529			0.175688
$550$		0			0
$551$	199.334	+	778.112i	0.361767	+	1.41218i
$552$		0			0
$553$			17.3373i			0.0313513i
$554$		0			0
$555$	268.526			0.483831
$556$		0			0
$557$	−469.822			−0.843487			−0.421744	−	0.906715i	$-0.638582\pi$
$557$	−469.822			−0.843487			−0.421744	+	0.906715i	$0.638582\pi$
$558$		0			0
$559$			6.34515i			0.0113509i
$560$		0			0
$561$		−	142.837i		−	0.254612i
$562$		0			0
$563$		−	351.584i		−	0.624483i	−0.950003	−	0.312242i	$-0.898920\pi$
$563$		−	351.584i		−	0.624483i	0.950003	−	0.312242i	$-0.101080\pi$
$564$		0			0
$565$		−	628.866i		−	1.11304i
$566$		0			0
$567$	−63.2648			−0.111578
$568$		0			0
$569$			684.643i			1.20324i	0.798782	+	0.601620i	$0.205478\pi$
$569$			684.643i			1.20324i	−0.798782	+	0.601620i	$0.794522\pi$
$570$		0			0
$571$	−773.542			−1.35471			−0.677357	−	0.735654i	$-0.736875\pi$
$571$	−773.542			−1.35471			−0.677357	+	0.735654i	$0.736875\pi$
$572$		0			0
$573$		−	317.749i		−	0.554535i
$574$		0			0
$575$	120.774			0.210042
$576$		0			0
$577$	846.880			1.46773			0.733865	−	0.679296i	$-0.237715\pi$
$577$	846.880			1.46773			0.733865	+	0.679296i	$0.237715\pi$
$578$		0			0
$579$	256.043			0.442215
$580$		0			0
$581$	−133.126			−0.229132
$582$		0			0
$583$			1091.38i			1.87201i
$584$		0			0
$585$		−	99.0133i		−	0.169254i
$586$		0			0
$587$	−906.841			−1.54487			−0.772437	−	0.635091i	$-0.780963\pi$
$587$	−906.841			−1.54487			−0.772437	+	0.635091i	$0.780963\pi$
$588$		0			0
$589$	32.3595	+	126.317i	0.0549397	+	0.214461i
$590$		0			0
$591$		−	252.780i		−	0.427715i
$592$		0			0
$593$	−234.274			−0.395066			−0.197533	−	0.980296i	$-0.563293\pi$
$593$	−234.274			−0.395066			−0.197533	+	0.980296i	$0.563293\pi$
$594$		0			0
$595$	−84.1801			−0.141479
$596$		0			0
$597$			240.113i			0.402200i
$598$		0			0
$599$		−	113.042i		−	0.188718i	−0.995538	−	0.0943589i	$-0.969920\pi$
$599$		−	113.042i		−	0.188718i	0.995538	−	0.0943589i	$-0.0300801\pi$
$600$		0			0
$601$		−	1107.97i		−	1.84355i	−0.387730	−	0.921773i	$-0.626741\pi$
$601$		−	1107.97i		−	1.84355i	0.387730	−	0.921773i	$-0.373259\pi$
$602$		0			0
$603$			508.122i			0.842656i
$604$		0			0
$605$	−116.538			−0.192626
$606$		0			0
$607$		−	185.612i		−	0.305786i	−0.988243	−	0.152893i	$-0.951141\pi$
$607$		−	185.612i		−	0.305786i	0.988243	−	0.152893i	$-0.0488590\pi$
$608$		0			0
$609$	149.948			0.246220
$610$		0			0
$611$			41.0659i			0.0672109i
$612$		0			0
$613$	472.424			0.770675			0.385337	−	0.922776i	$-0.374085\pi$
$613$	472.424			0.770675			0.385337	+	0.922776i	$0.374085\pi$
$614$		0			0
$615$	388.642			0.631939
$616$		0			0
$617$	826.340			1.33929			0.669643	−	0.742683i	$-0.266447\pi$
$617$	826.340			1.33929			0.669643	+	0.742683i	$0.266447\pi$
$618$		0			0
$619$	−0.726144			−0.00117309			−0.000586546	−	1.00000i	$-0.500187\pi$
$619$	−0.726144			−0.00117309			−0.000586546		1.00000i	$0.500187\pi$
$620$		0			0
$621$		−	530.930i		−	0.854960i
$622$		0			0
$623$		−	28.2313i		−	0.0453151i
$624$		0			0
$625$	−463.325			−0.741320
$626$		0			0
$627$	84.8620	+	331.264i	0.135346	+	0.528332i
$628$		0			0
$629$		−	324.005i		−	0.515111i
$630$		0			0
$631$	−521.942			−0.827166			−0.413583	−	0.910466i	$-0.635723\pi$
$631$	−521.942			−0.827166			−0.413583	+	0.910466i	$0.635723\pi$
$632$		0			0
$633$	114.171			0.180365
$634$		0			0
$635$			564.773i			0.889406i
$636$		0			0
$637$		−	142.082i		−	0.223049i
$638$		0			0
$639$		−	602.776i		−	0.943311i
$640$		0			0
$641$		−	1225.37i		−	1.91165i	−0.293942	−	0.955823i	$-0.594967\pi$
$641$		−	1225.37i		−	1.91165i	0.293942	−	0.955823i	$-0.405033\pi$
$642$		0			0
$643$	1075.09			1.67200			0.835999	−	0.548731i	$-0.184889\pi$
$643$	1075.09			1.67200			0.835999	+	0.548731i	$0.184889\pi$
$644$		0			0
$645$		−	12.7129i		−	0.0197099i
$646$		0			0
$647$	1026.31			1.58627			0.793133	−	0.609048i	$-0.208448\pi$
$647$	1026.31			1.58627			0.793133	+	0.609048i	$0.208448\pi$
$648$		0			0
$649$		−	629.988i		−	0.970706i
$650$		0			0
$651$	24.3423			0.0373922
$652$		0			0
$653$	−668.344			−1.02350			−0.511749	−	0.859135i	$-0.671002\pi$
$653$	−668.344			−1.02350			−0.511749	+	0.859135i	$0.671002\pi$
$654$		0			0
$655$	−885.858			−1.35245
$656$		0			0
$657$	56.6733			0.0862607
$658$		0			0
$659$			928.692i			1.40924i	0.709582	+	0.704622i	$0.248884\pi$
$659$			928.692i			1.40924i	−0.709582	+	0.704622i	$0.751116\pi$
$660$		0			0
$661$		−	937.827i		−	1.41880i	−0.704806	−	0.709400i	$-0.748966\pi$
$661$		−	937.827i		−	1.41880i	0.704806	−	0.709400i	$-0.251034\pi$
$662$		0			0
$663$	38.6391			0.0582792
$664$		0			0
$665$	195.228	−	50.0127i	0.293576	−	0.0752071i
$666$		0			0
$667$		−	957.854i		−	1.43606i
$668$		0			0
$669$	185.178			0.276798
$670$		0			0
$671$	172.123			0.256517
$672$		0			0
$673$		−	1155.00i		−	1.71620i	−0.513485	−	0.858098i	$-0.671646\pi$
$673$		−	1155.00i		−	1.71620i	0.513485	−	0.858098i	$-0.328354\pi$
$674$		0			0
$675$		−	124.909i		−	0.185050i
$676$		0			0
$677$		−	133.721i		−	0.197520i	−0.995111	−	0.0987600i	$-0.968512\pi$
$677$		−	133.721i		−	0.197520i	0.995111	−	0.0987600i	$-0.0314876\pi$
$678$		0			0
$679$			49.9000i			0.0734904i
$680$		0			0
$681$	345.308			0.507061
$682$		0			0
$683$		−	1232.57i		−	1.80464i	−0.431069	−	0.902319i	$-0.641864\pi$
$683$		−	1232.57i		−	1.80464i	0.431069	−	0.902319i	$-0.358136\pi$
$684$		0			0
$685$	680.487			0.993412
$686$		0			0
$687$			113.685i			0.165481i
$688$		0			0
$689$	−295.231			−0.428492
$690$		0			0
$691$	1205.13			1.74404			0.872020	−	0.489470i	$-0.162810\pi$
$691$	1205.13			1.74404			0.872020	+	0.489470i	$0.162810\pi$
$692$		0			0
$693$	−197.381			−0.284821
$694$		0			0
$695$	−56.1555			−0.0807993
$696$		0			0
$697$		−	468.938i		−	0.672794i
$698$		0			0
$699$		−	30.1225i		−	0.0430937i
$700$		0			0
$701$	329.414			0.469920			0.234960	−	0.972005i	$-0.424504\pi$
$701$	329.414			0.469920			0.234960	+	0.972005i	$0.424504\pi$
$702$		0			0
$703$	192.496	+	751.422i	0.273821	+	1.06888i
$704$		0			0
$705$		−	82.2778i		−	0.116706i
$706$		0			0
$707$	173.393			0.245252
$708$		0			0
$709$	1113.70			1.57080			0.785401	−	0.618988i	$-0.212457\pi$
$709$	1113.70			1.57080			0.785401	+	0.618988i	$0.212457\pi$
$710$		0			0
$711$		−	49.2983i		−	0.0693365i
$712$		0			0
$713$		−	155.496i		−	0.218088i
$714$		0			0
$715$		−	176.692i		−	0.247121i
$716$		0			0
$717$		−	166.440i		−	0.232134i
$718$		0			0
$719$	−715.341			−0.994911			−0.497456	−	0.867489i	$-0.665732\pi$
$719$	−715.341			−0.994911			−0.497456	+	0.867489i	$0.665732\pi$
$720$		0			0
$721$		−	332.588i		−	0.461286i
$722$		0			0
$723$	336.069			0.464825
$724$		0			0
$725$		−	225.349i		−	0.310826i
$726$		0			0
$727$	−1282.98			−1.76476			−0.882381	−	0.470536i	$-0.844061\pi$
$727$	−1282.98			−1.76476			−0.882381	+	0.470536i	$0.844061\pi$
$728$		0			0
$729$	−132.879			−0.182276
$730$		0			0
$731$	−15.3394			−0.0209841
$732$		0			0
$733$	−481.707			−0.657171			−0.328586	−	0.944474i	$-0.606572\pi$
$733$	−481.707			−0.657171			−0.328586	+	0.944474i	$0.606572\pi$
$734$		0			0
$735$			284.670i			0.387306i
$736$		0			0
$737$			906.756i			1.23033i
$738$		0			0
$739$	580.859			0.786007			0.393003	−	0.919537i	$-0.371436\pi$
$739$	580.859			0.786007			0.393003	+	0.919537i	$0.371436\pi$
$740$		0			0
$741$	−89.6106	+	22.9561i	−0.120932	+	0.0309799i
$742$		0			0
$743$		−	1280.64i		−	1.72361i	−0.507242	−	0.861804i	$-0.669335\pi$
$743$		−	1280.64i		−	1.72361i	0.507242	−	0.861804i	$-0.330665\pi$
$744$		0			0
$745$	18.0446			0.0242209
$746$		0			0
$747$	378.540			0.506747
$748$		0			0
$749$		−	212.640i		−	0.283898i
$750$		0			0
$751$		−	611.873i		−	0.814745i	−0.913262	−	0.407372i	$-0.866445\pi$
$751$		−	611.873i		−	0.814745i	0.913262	−	0.407372i	$-0.133555\pi$
$752$		0			0
$753$		−	397.561i		−	0.527970i
$754$		0			0
$755$			562.084i			0.744482i
$756$		0			0
$757$	−322.424			−0.425923			−0.212962	−	0.977061i	$-0.568311\pi$
$757$	−322.424			−0.425923			−0.212962	+	0.977061i	$0.568311\pi$
$758$		0			0
$759$		−	407.786i		−	0.537267i
$760$		0			0
$761$	−53.2368			−0.0699563			−0.0349782	−	0.999388i	$-0.511136\pi$
$761$	−53.2368			−0.0699563			−0.0349782	+	0.999388i	$0.511136\pi$
$762$		0			0
$763$			248.640i			0.325871i
$764$		0			0
$765$	239.365			0.312895
$766$		0			0
$767$	170.419			0.222189
$768$		0			0
$769$	−272.155			−0.353908			−0.176954	−	0.984219i	$-0.556624\pi$
$769$	−272.155			−0.353908			−0.176954	+	0.984219i	$0.556624\pi$
$770$		0			0
$771$	95.5213			0.123893
$772$		0			0
$773$			601.659i			0.778343i	0.921165	+	0.389172i	$0.127239\pi$
$773$			601.659i			0.778343i	−0.921165	+	0.389172i	$0.872761\pi$
$774$		0			0
$775$		−	36.5827i		−	0.0472035i
$776$		0			0
$777$	144.805			0.186364
$778$		0			0
$779$	278.603	+	1087.55i	0.357642	+	1.39608i
$780$		0			0
$781$		−	1075.67i		−	1.37730i
$782$		0			0
$783$	−990.648			−1.26520
$784$		0			0
$785$	−515.502			−0.656690
$786$		0			0
$787$			793.870i			1.00873i	0.863491	+	0.504364i	$0.168273\pi$
$787$			793.870i			1.00873i	−0.863491	+	0.504364i	$0.831727\pi$
$788$		0			0
$789$			747.095i			0.946889i
$790$		0			0
$791$		−	339.121i		−	0.428725i
$792$		0			0
$793$			46.5610i			0.0587151i
$794$		0			0
$795$	591.512			0.744040
$796$		0			0
$797$			597.536i			0.749732i	0.927079	+	0.374866i	$0.122311\pi$
$797$			597.536i			0.749732i	−0.927079	+	0.374866i	$0.877689\pi$
$798$		0			0
$799$	−99.2768			−0.124251
$800$		0			0
$801$			80.2752i			0.100219i
$802$		0			0
$803$	101.135			0.125946
$804$		0			0
$805$	−240.325			−0.298541
$806$		0			0
$807$	−127.969			−0.158573
$808$		0			0
$809$	385.508			0.476524			0.238262	−	0.971201i	$-0.423422\pi$
$809$	385.508			0.476524			0.238262	+	0.971201i	$0.423422\pi$
$810$		0			0
$811$			1258.22i			1.55144i	0.631076	+	0.775721i	$0.282614\pi$
$811$			1258.22i			1.55144i	−0.631076	+	0.775721i	$0.717386\pi$
$812$		0			0
$813$		−	1.70452i		−	0.00209658i
$814$		0			0
$815$	−504.096			−0.618523
$816$		0			0
$817$	35.5747	−	9.11338i	0.0435431	−	0.0111547i
$818$		0			0
$819$		−	53.3937i		−	0.0651938i
$820$		0			0
$821$	804.198			0.979535			0.489768	−	0.871853i	$-0.337082\pi$
$821$	804.198			0.979535			0.489768	+	0.871853i	$0.337082\pi$
$822$		0			0
$823$	−979.035			−1.18959			−0.594797	−	0.803876i	$-0.702767\pi$
$823$	−979.035			−1.18959			−0.594797	+	0.803876i	$0.702767\pi$
$824$		0			0
$825$		−	95.9373i		−	0.116288i
$826$		0			0
$827$		−	726.111i		−	0.878006i	−0.898485	−	0.439003i	$-0.855332\pi$
$827$		−	726.111i		−	0.878006i	0.898485	−	0.439003i	$-0.144668\pi$
$828$		0			0
$829$		−	609.462i		−	0.735177i	−0.929988	−	0.367589i	$-0.880183\pi$
$829$		−	609.462i		−	0.735177i	0.929988	−	0.367589i	$-0.119817\pi$
$830$		0			0
$831$			404.577i			0.486855i
$832$		0			0
$833$	343.484			0.412346
$834$		0			0
$835$		−	763.494i		−	0.914364i
$836$		0			0
$837$	−160.820			−0.192139
$838$		0			0
$839$			898.822i			1.07130i	0.844439	+	0.535651i	$0.179934\pi$
$839$			898.822i			1.07130i	−0.844439	+	0.535651i	$0.820066\pi$
$840$		0			0
$841$	−946.234			−1.12513
$842$		0			0
$843$	−128.592			−0.152541
$844$		0			0
$845$	−701.724			−0.830443
$846$		0			0
$847$	−62.8443			−0.0741963
$848$		0			0
$849$		−	527.719i		−	0.621577i
$850$		0			0
$851$		−	925.000i		−	1.08696i
$852$		0			0
$853$	437.364			0.512737			0.256368	−	0.966579i	$-0.417474\pi$
$853$	437.364			0.512737			0.256368	+	0.966579i	$0.417474\pi$
$854$		0			0
$855$	−555.128	+	142.210i	−0.649272	+	0.166328i
$856$		0			0
$857$		−	668.970i		−	0.780595i	−0.920689	−	0.390298i	$-0.872372\pi$
$857$		−	668.970i		−	0.780595i	0.920689	−	0.390298i	$-0.127628\pi$
$858$		0			0
$859$	833.585			0.970413			0.485207	−	0.874400i	$-0.338744\pi$
$859$	833.585			0.970413			0.485207	+	0.874400i	$0.338744\pi$
$860$		0			0
$861$	209.578			0.243413
$862$		0			0
$863$			948.742i			1.09935i	0.835378	+	0.549676i	$0.185249\pi$
$863$			948.742i			1.09935i	−0.835378	+	0.549676i	$0.814751\pi$
$864$		0			0
$865$			306.839i			0.354728i
$866$		0			0
$867$		−	335.192i		−	0.386611i
$868$		0			0
$869$		−	87.9740i		−	0.101236i
$870$		0			0
$871$	−245.287			−0.281616
$872$		0			0
$873$		−	141.890i		−	0.162531i
$874$		0			0
$875$	−321.714			−0.367673
$876$		0			0
$877$			1167.56i			1.33131i	0.746259	+	0.665656i	$0.231848\pi$
$877$			1167.56i			1.33131i	−0.746259	+	0.665656i	$0.768152\pi$
$878$		0			0
$879$	15.2687			0.0173706
$880$		0			0
$881$	1395.03			1.58346			0.791729	−	0.610873i	$-0.209181\pi$
$881$	1395.03			1.58346			0.791729	+	0.610873i	$0.209181\pi$
$882$		0			0
$883$	839.991			0.951292			0.475646	−	0.879637i	$-0.342214\pi$
$883$	839.991			0.951292			0.475646	+	0.879637i	$0.342214\pi$
$884$		0			0
$885$	−341.443			−0.385812
$886$		0			0
$887$			353.635i			0.398687i	0.979930	+	0.199344i	$0.0638810\pi$
$887$			353.635i			0.398687i	−0.979930	+	0.199344i	$0.936119\pi$
$888$		0			0
$889$			304.558i			0.342585i
$890$		0			0
$891$	321.023			0.360295
$892$		0			0
$893$	230.240	−	58.9819i	0.257827	−	0.0660492i
$894$		0			0
$895$		−	1540.24i		−	1.72094i
$896$		0			0
$897$	110.311			0.122977
$898$		0			0
$899$	−290.137			−0.322733
$900$		0			0
$901$		−	713.721i		−	0.792143i
$902$		0			0
$903$		−	6.85551i		−	0.00759193i
$904$		0			0
$905$			1152.58i			1.27357i
$906$		0			0
$907$			451.706i			0.498022i	0.968501	+	0.249011i	$0.0801055\pi$
$907$			451.706i			0.498022i	−0.968501	+	0.249011i	$0.919894\pi$
$908$		0			0
$909$	−493.040			−0.542399
$910$		0			0
$911$		−	1396.09i		−	1.53248i	−0.642554	−	0.766241i	$-0.722125\pi$
$911$		−	1396.09i		−	1.53248i	0.642554	−	0.766241i	$-0.277875\pi$
$912$		0			0
$913$	675.515			0.739885
$914$		0			0
$915$		−	93.2876i		−	0.101954i
$916$		0			0
$917$	−477.706			−0.520944
$918$		0			0
$919$	−1290.37			−1.40411			−0.702054	−	0.712124i	$-0.747733\pi$
$919$	−1290.37			−1.40411			−0.702054	+	0.712124i	$0.747733\pi$
$920$		0			0
$921$	597.431			0.648677
$922$		0			0
$923$	290.980			0.315255
$924$		0			0
$925$		−	217.619i		−	0.235264i
$926$		0			0
$927$			945.707i			1.02018i
$928$		0			0
$929$	−1383.61			−1.48935			−0.744677	−	0.667425i	$-0.767396\pi$
$929$	−1383.61			−1.48935			−0.744677	+	0.667425i	$0.767396\pi$
$930$		0			0
$931$	−796.599	+	204.069i	−0.855638	+	0.219194i
$932$		0			0
$933$			585.152i			0.627172i
$934$		0			0
$935$	427.152			0.456848
$936$		0			0
$937$	−1047.14			−1.11755			−0.558774	−	0.829320i	$-0.688728\pi$
$937$	−1047.14			−1.11755			−0.558774	+	0.829320i	$0.688728\pi$
$938$		0			0
$939$			557.611i			0.593834i
$940$		0			0
$941$		−	1072.95i		−	1.14022i	−0.821569	−	0.570110i	$-0.806901\pi$
$941$		−	1072.95i		−	1.14022i	0.821569	−	0.570110i	$-0.193099\pi$
$942$		0			0
$943$		−	1338.77i		−	1.41969i
$944$		0			0
$945$			248.553i			0.263019i
$946$		0			0
$947$	−484.472			−0.511586			−0.255793	−	0.966732i	$-0.582337\pi$
$947$	−484.472			−0.511586			−0.255793	+	0.966732i	$0.582337\pi$
$948$		0			0
$949$			27.3581i			0.0288283i
$950$		0			0
$951$	−437.621			−0.460169
$952$		0			0
$953$			416.133i			0.436656i	0.975875	+	0.218328i	$0.0700603\pi$
$953$			416.133i			0.436656i	−0.975875	+	0.218328i	$0.929940\pi$
$954$		0			0
$955$	950.221			0.994996
$956$		0			0
$957$	−760.876			−0.795064
$958$		0			0
$959$	366.958			0.382646
$960$		0			0
$961$	913.900			0.950988
$962$		0			0
$963$			604.638i			0.627869i
$964$		0			0
$965$			765.691i			0.793462i
$966$		0			0
$967$	601.352			0.621874			0.310937	−	0.950431i	$-0.399357\pi$
$967$	601.352			0.621874			0.310937	+	0.950431i	$0.399357\pi$
$968$		0			0
$969$	−55.4964	−	216.634i	−0.0572718	−	0.223564i
$970$		0			0
$971$		−	1501.64i		−	1.54648i	−0.634111	−	0.773242i	$-0.718634\pi$
$971$		−	1501.64i		−	1.54648i	0.634111	−	0.773242i	$-0.281366\pi$
$972$		0			0
$973$	−30.2823			−0.0311226
$974$		0			0
$975$	25.9521			0.0266175
$976$		0			0
$977$			1123.10i			1.14954i	0.818317	+	0.574768i	$0.194908\pi$
$977$			1123.10i			1.14954i	−0.818317	+	0.574768i	$0.805092\pi$
$978$		0			0
$979$			143.253i			0.146326i
$980$		0			0
$981$		−	707.003i		−	0.720697i
$982$		0			0
$983$		−	630.261i		−	0.641161i	−0.947221	−	0.320580i	$-0.896122\pi$
$983$		−	630.261i		−	0.641161i	0.947221	−	0.320580i	$-0.103878\pi$
$984$		0			0
$985$	755.933			0.767444
$986$		0			0
$987$		−	44.3689i		−	0.0449533i
$988$		0			0
$989$	−43.7924			−0.0442795
$990$		0			0
$991$		−	486.447i		−	0.490864i	−0.969414	−	0.245432i	$-0.921070\pi$
$991$		−	486.447i		−	0.490864i	0.969414	−	0.245432i	$-0.0789299\pi$
$992$		0			0
$993$	−430.391			−0.433425
$994$		0			0
$995$	−718.054			−0.721662
$996$		0			0
$997$	−932.988			−0.935795			−0.467898	−	0.883783i	$-0.654988\pi$
$997$	−932.988			−0.935795			−0.467898	+	0.883783i	$0.654988\pi$
$998$		0			0
$999$	−956.669			−0.957626

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	608.3.e.b.417.13	yes	20
4.3	odd	2	inner	608.3.e.b.417.8	yes	20
8.3	odd	2		1216.3.e.p.1025.14		20
8.5	even	2		1216.3.e.p.1025.7		20
19.18	odd	2	inner	608.3.e.b.417.7	✓	20
76.75	even	2	inner	608.3.e.b.417.14	yes	20
152.37	odd	2		1216.3.e.p.1025.13		20
152.75	even	2		1216.3.e.p.1025.8		20

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
608.3.e.b.417.7	✓	20	19.18	odd	2	inner
608.3.e.b.417.8	yes	20	4.3	odd	2	inner
608.3.e.b.417.13	yes	20	1.1	even	1	trivial
608.3.e.b.417.14	yes	20	76.75	even	2	inner
1216.3.e.p.1025.7		20	8.5	even	2
1216.3.e.p.1025.8		20	152.75	even	2
1216.3.e.p.1025.13		20	152.37	odd	2
1216.3.e.p.1025.14		20	8.3	odd	2

Overview	Random
Universe	Knowledge

Classical	Maass
Hilbert	Bianchi

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

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

\(f(q)\)	\(=\)	\(q+1.48305i q^{3} -4.43504 q^{5} -2.39163 q^{7} +6.80056 q^{9} +12.1358 q^{11} +3.28286i q^{13} -6.57739i q^{15} -7.93631 q^{17} +(18.4057 - 4.71509i) q^{19} -3.54691i q^{21} -22.6573 q^{23} -5.33045 q^{25} +23.4330i q^{27} +42.2757i q^{29} +6.86296i q^{31} +17.9980i q^{33} +10.6070 q^{35} +40.8256i q^{37} -4.86865 q^{39} +59.0876i q^{41} +1.93281 q^{43} -30.1607 q^{45} +12.5092 q^{47} -43.2801 q^{49} -11.7700i q^{51} +89.9311i q^{53} -53.8226 q^{55} +(6.99272 + 27.2965i) q^{57} -51.9117i q^{59} +14.1831 q^{61} -16.2644 q^{63} -14.5596i q^{65} +74.7176i q^{67} -33.6020i q^{69} -88.6362i q^{71} +8.33362 q^{73} -7.90534i q^{75} -29.0242 q^{77} -7.24915i q^{79} +26.4526 q^{81} +55.6631 q^{83} +35.1978 q^{85} -62.6970 q^{87} +11.8042i q^{89} -7.85137i q^{91} -10.1781 q^{93} +(-81.6297 + 20.9116i) q^{95} -20.8644i q^{97} +82.5300 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 20 q - 52 q^{9} + 56 q^{17} + 36 q^{25} + 64 q^{45} + 332 q^{49} + 88 q^{57} - 32 q^{61} - 152 q^{73} + 360 q^{77} - 476 q^{81} - 552 q^{85} + 336 q^{93}+O(q^{100}) \) 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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