LMFDB - Embedded newform 1216.3.d.e.191.9

Show commands: Magma / Pari/GP / SageMath

Newspace parameters

comment:Compute space of new eigenforms

gp:[N,k,chi] = [1216,3,Mod(191,1216)] mf = mfinit([N,k,chi],0) lf = mfeigenbasis(mf)

sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(1216, base_ring=CyclotomicField(2)) chi = DirichletCharacter(H, H._module([1, 0, 0])) N = Newforms(chi, 3, names="a")

magma://Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code chi := DirichletCharacter("1216.191"); S:= CuspForms(chi, 3); N := Newforms(S);

Level:	$ N $	$=$	$ 1216 = 2^{6} \cdot 19 $
Weight:	$ k $	$=$	$ 3 $
Character orbit:	$[\chi]$	$=$	1216.d (of order $2$, degree $1$, not minimal)

Newform invariants

comment:select newform

sage:traces = [16] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(1)] == traces)

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

Self dual:	no
Analytic conductor:	$33.1336001462$
Analytic rank:	$0$
Dimension:	$16$
Coefficient field:	$\mathbb{Q}[x]/(x^{16} - \cdots)$
comment:defining polynomial gp:f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{16} - 4 x^{15} - 30 x^{14} + 116 x^{13} + 707 x^{12} - 2372 x^{11} - 7342 x^{10} + 12048 x^{9} + \cdots + 19859428 $ x^16 - 4x^15 - 30x^14 + 116x^13 + 707x^12 - 2372x^11 - 7342x^10 + 12048x^9 + 66793x^8 + 65416x^7 - 419708x^6 - 1201304x^5 + 619800x^4 + 4221760x^3 + 6110408x^2 + 10041872*x + 19859428
Coefficient ring:	$\Z[a_1, \ldots, a_{7}]$
Coefficient ring index:	$ 2^{26} $
Twist minimal:	no (minimal twist has level 608)
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label			191.9
Root			$3.55583 - 0.146051i$ of defining polynomial
Character	$\chi$	$=$	1216.191
Dual form			1216.3.d.e.191.8

$q$-expansion

comment:q-expansion

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

gp:mfcoefs(f, 20)

$f(q)$	$=$	$q+0.292102i q^{3} -6.11167 q^{5} -3.90281i q^{7} +8.91468 q^{9} +9.90522i q^{11} -6.62899 q^{13} -1.78523i q^{15} -9.01742 q^{17} -4.35890i q^{19} +1.14002 q^{21} -20.9308i q^{23} +12.3525 q^{25} +5.23292i q^{27} +5.74476 q^{29} +20.2048i q^{31} -2.89334 q^{33} +23.8527i q^{35} -2.91977 q^{37} -1.93634i q^{39} +31.5606 q^{41} +26.6794i q^{43} -54.4836 q^{45} -1.43072i q^{47} +33.7681 q^{49} -2.63401i q^{51} +4.43992 q^{53} -60.5374i q^{55} +1.27325 q^{57} -61.2354i q^{59} +67.7019 q^{61} -34.7923i q^{63} +40.5142 q^{65} +71.5884i q^{67} +6.11393 q^{69} -47.4052i q^{71} +118.522 q^{73} +3.60820i q^{75} +38.6582 q^{77} -37.1573i q^{79} +78.7035 q^{81} -1.37281i q^{83} +55.1115 q^{85} +1.67806i q^{87} +101.156 q^{89} +25.8717i q^{91} -5.90188 q^{93} +26.6402i q^{95} +147.089 q^{97} +88.3018i q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 16 q + 8 q^{5} - 40 q^{9} - 40 q^{13} - 32 q^{17} + 96 q^{21} + 88 q^{25} - 144 q^{29} + 88 q^{33} + 56 q^{37} - 104 q^{41} - 40 q^{45} - 144 q^{49} + 320 q^{53} - 8 q^{61} + 336 q^{65} - 392 q^{69} + 72 q^{77}+ \cdots - 240 q^{97}+O(q^{100}) $ 16 * q + 8 * q^5 - 40 * q^9 - 40 * q^13 - 32 * q^17 + 96 * q^21 + 88 * q^25 - 144 * q^29 + 88 * q^33 + 56 * q^37 - 104 * q^41 - 40 * q^45 - 144 * q^49 + 320 * q^53 - 8 * q^61 + 336 * q^65 - 392 * q^69 + 72 * q^77 - 272 * q^81 - 552 * q^85 - 136 * q^89 + 752 * q^93 - 240 * q^97

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/1216\mathbb{Z}\right)^\times$.

$n$	$191$	$705$	$837$
$\chi(n)$	$-1$	$1$	$1$

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
$3$	0.292102i	0.0973675i	0.998814	+	0.0486837i	$0.0155026\pi$
$3$	0.292102i	0.0973675i	−0.998814	+	0.0486837i	$0.984497\pi$
$4$	0	0
$5$	−6.11167	−1.22233	−0.611167	−	0.791502i	$-0.709300\pi$
$5$	−6.11167	−1.22233	−0.611167	+	0.791502i	$0.709300\pi$
$6$	0	0
$7$	− 3.90281i	− 0.557544i	−0.960357	−	0.278772i	$-0.910073\pi$
$7$	− 3.90281i	− 0.557544i	0.960357	−	0.278772i	$-0.0899274\pi$
$8$	0	0
$9$	8.91468	0.990520
$10$	0	0
$11$	9.90522i	0.900475i	0.892909	+	0.450237i	$0.148661\pi$
$11$	9.90522i	0.900475i	−0.892909	+	0.450237i	$0.851339\pi$
$12$	0	0
$13$	−6.62899	−0.509922	−0.254961	−	0.966951i	$-0.582063\pi$
$13$	−6.62899	−0.509922	−0.254961	+	0.966951i	$0.582063\pi$
$14$	0	0
$15$	− 1.78523i	− 0.119016i
$16$	0	0
$17$	−9.01742	−0.530436	−0.265218	−	0.964188i	$-0.585444\pi$
$17$	−9.01742	−0.530436	−0.265218	+	0.964188i	$0.585444\pi$
$18$	0	0
$19$	− 4.35890i	− 0.229416i
$19$	− 4.35890i	− 0.229416i
$20$	0	0
$21$	1.14002	0.0542867
$22$	0	0
$23$	− 20.9308i	− 0.910033i	−0.890483	−	0.455016i	$-0.849633\pi$
$23$	− 20.9308i	− 0.910033i	0.890483	−	0.455016i	$-0.150367\pi$
$24$	0	0
$25$	12.3525	0.494100
$26$	0	0
$27$	5.23292i	0.193812i
$28$	0	0
$29$	5.74476	0.198095	0.0990476	−	0.995083i	$-0.468420\pi$
$29$	5.74476	0.198095	0.0990476	+	0.995083i	$0.468420\pi$
$30$	0	0
$31$	20.2048i	0.651769i	0.945410	+	0.325884i	$0.105662\pi$
$31$	20.2048i	0.651769i	−0.945410	+	0.325884i	$0.894338\pi$
$32$	0	0
$33$	−2.89334	−0.0876770
$34$	0	0
$35$	23.8527i	0.681505i
$36$	0	0
$37$	−2.91977	−0.0789127	−0.0394564	−	0.999221i	$-0.512563\pi$
$37$	−2.91977	−0.0789127	−0.0394564	+	0.999221i	$0.512563\pi$
$38$	0	0
$39$	− 1.93634i	− 0.0496498i
$40$	0	0
$41$	31.5606	0.769770	0.384885	−	0.922964i	$-0.374241\pi$
$41$	31.5606	0.769770	0.384885	+	0.922964i	$0.374241\pi$
$42$	0	0
$43$	26.6794i	0.620451i	0.950663	+	0.310226i	$0.100405\pi$
$43$	26.6794i	0.620451i	−0.950663	+	0.310226i	$0.899595\pi$
$44$	0	0
$45$	−54.4836	−1.21075
$46$	0	0
$47$	− 1.43072i	− 0.0304408i	−0.999884	−	0.0152204i	$-0.995155\pi$
$47$	− 1.43072i	− 0.0304408i	0.999884	−	0.0152204i	$-0.00484499\pi$
$48$	0	0
$49$	33.7681	0.689144
$50$	0	0
$51$	− 2.63401i	− 0.0516472i
$52$	0	0
$53$	4.43992	0.0837721	0.0418861	−	0.999122i	$-0.486663\pi$
$53$	4.43992	0.0837721	0.0418861	+	0.999122i	$0.486663\pi$
$54$	0	0
$55$	− 60.5374i	− 1.10068i
$56$	0	0
$57$	1.27325	0.0223376
$58$	0	0
$59$	− 61.2354i	− 1.03789i	−0.854808	−	0.518944i	$-0.826325\pi$
$59$	− 61.2354i	− 1.03789i	0.854808	−	0.518944i	$-0.173675\pi$
$60$	0	0
$61$	67.7019	1.10987	0.554933	−	0.831895i	$-0.312744\pi$
$61$	67.7019	1.10987	0.554933	+	0.831895i	$0.312744\pi$
$62$	0	0
$63$	− 34.7923i	− 0.552259i
$64$	0	0
$65$	40.5142	0.623295
$66$	0	0
$67$	71.5884i	1.06848i	0.845332	+	0.534242i	$0.179403\pi$
$67$	71.5884i	1.06848i	−0.845332	+	0.534242i	$0.820597\pi$
$68$	0	0
$69$	6.11393	0.0886076
$70$	0	0
$71$	− 47.4052i	− 0.667679i	−0.942630	−	0.333839i	$-0.891656\pi$
$71$	− 47.4052i	− 0.667679i	0.942630	−	0.333839i	$-0.108344\pi$
$72$	0	0
$73$	118.522	1.62358	0.811792	−	0.583946i	$-0.198492\pi$
$73$	118.522	1.62358	0.811792	+	0.583946i	$0.198492\pi$
$74$	0	0
$75$	3.60820i	0.0481093i
$76$	0	0
$77$	38.6582	0.502055
$78$	0	0
$79$	− 37.1573i	− 0.470346i	−0.971954	−	0.235173i	$-0.924434\pi$
$79$	− 37.1573i	− 0.470346i	0.971954	−	0.235173i	$-0.0755657\pi$
$80$	0	0
$81$	78.7035	0.971649
$82$	0	0
$83$	− 1.37281i	− 0.0165399i	−0.999966	−	0.00826996i	$-0.997368\pi$
$83$	− 1.37281i	− 0.0165399i	0.999966	−	0.00826996i	$-0.00263244\pi$
$84$	0	0
$85$	55.1115	0.648370
$86$	0	0
$87$	1.67806i	0.0192880i
$88$	0	0
$89$	101.156	1.13658	0.568292	−	0.822827i	$-0.307604\pi$
$89$	101.156	1.13658	0.568292	+	0.822827i	$0.307604\pi$
$90$	0	0
$91$	25.8717i	0.284304i
$92$	0	0
$93$	−5.90188	−0.0634611
$94$	0	0
$95$	26.6402i	0.280423i
$96$	0	0
$97$	147.089	1.51638	0.758189	−	0.652034i	$-0.226084\pi$
$97$	147.089	1.51638	0.758189	+	0.652034i	$0.226084\pi$
$98$	0	0
$99$	88.3018i	0.891938i
$100$	0	0
$101$	72.6814	0.719618	0.359809	−	0.933026i	$-0.382842\pi$
$101$	72.6814	0.719618	0.359809	+	0.933026i	$0.382842\pi$
$102$	0	0
$103$	− 6.19421i	− 0.0601379i	−0.999548	−	0.0300690i	$-0.990427\pi$
$103$	− 6.19421i	− 0.0601379i	0.999548	−	0.0300690i	$-0.00957269\pi$
$104$	0	0
$105$	−6.96743	−0.0663565
$106$	0	0
$107$	− 80.8798i	− 0.755886i	−0.925829	−	0.377943i	$-0.876632\pi$
$107$	− 80.8798i	− 0.755886i	0.925829	−	0.377943i	$-0.123368\pi$
$108$	0	0
$109$	157.605	1.44592	0.722959	−	0.690891i	$-0.242782\pi$
$109$	157.605	1.44592	0.722959	+	0.690891i	$0.242782\pi$
$110$	0	0
$111$	− 0.852872i	− 0.00768353i
$112$	0	0
$113$	−55.9659	−0.495274	−0.247637	−	0.968853i	$-0.579654\pi$
$113$	−55.9659	−0.495274	−0.247637	+	0.968853i	$0.579654\pi$
$114$	0	0
$115$	127.922i	1.11236i
$116$	0	0
$117$	−59.0953	−0.505088
$118$	0	0
$119$	35.1933i	0.295742i
$120$	0	0
$121$	22.8866	0.189145
$122$	0	0
$123$	9.21892i	0.0749506i
$124$	0	0
$125$	77.2973	0.618378
$126$	0	0
$127$	204.396i	1.60942i	0.593669	+	0.804709i	$0.297679\pi$
$127$	204.396i	1.60942i	−0.593669	+	0.804709i	$0.702321\pi$
$128$	0	0
$129$	−7.79312	−0.0604118
$130$	0	0
$131$	11.7719i	0.0898622i	0.998990	+	0.0449311i	$0.0143068\pi$
$131$	11.7719i	0.0898622i	−0.998990	+	0.0449311i	$0.985693\pi$
$132$	0	0
$133$	−17.0120	−0.127909
$134$	0	0
$135$	− 31.9819i	− 0.236903i
$136$	0	0
$137$	−119.745	−0.874049	−0.437024	−	0.899450i	$-0.643968\pi$
$137$	−119.745	−0.874049	−0.437024	+	0.899450i	$0.643968\pi$
$138$	0	0
$139$	− 243.947i	− 1.75501i	−0.479565	−	0.877506i	$-0.659205\pi$
$139$	− 243.947i	− 1.75501i	0.479565	−	0.877506i	$-0.340795\pi$
$140$	0	0
$141$	0.417916	0.00296394
$142$	0	0
$143$	− 65.6616i	− 0.459172i
$144$	0	0
$145$	−35.1101	−0.242138
$146$	0	0
$147$	9.86374i	0.0671003i
$148$	0	0
$149$	−50.4912	−0.338867	−0.169434	−	0.985542i	$-0.554194\pi$
$149$	−50.4912	−0.338867	−0.169434	+	0.985542i	$0.554194\pi$
$150$	0	0
$151$	109.222i	0.723325i	0.932309	+	0.361662i	$0.117791\pi$
$151$	109.222i	0.723325i	−0.932309	+	0.361662i	$0.882209\pi$
$152$	0	0
$153$	−80.3873	−0.525407
$154$	0	0
$155$	− 123.485i	− 0.796679i
$156$	0	0
$157$	124.637	0.793869	0.396934	−	0.917847i	$-0.370074\pi$
$157$	124.637	0.793869	0.396934	+	0.917847i	$0.370074\pi$
$158$	0	0
$159$	1.29691i	0.00815668i
$160$	0	0
$161$	−81.6888	−0.507384
$162$	0	0
$163$	128.491i	0.788289i	0.919049	+	0.394144i	$0.128959\pi$
$163$	128.491i	0.788289i	−0.919049	+	0.394144i	$0.871041\pi$
$164$	0	0
$165$	17.6831	0.107171
$166$	0	0
$167$	323.621i	1.93785i	0.247351	+	0.968926i	$0.420440\pi$
$167$	323.621i	1.93785i	−0.247351	+	0.968926i	$0.579560\pi$
$168$	0	0
$169$	−125.057	−0.739980
$170$	0	0
$171$	− 38.8582i	− 0.227241i
$172$	0	0
$173$	−99.7258	−0.576450	−0.288225	−	0.957563i	$-0.593065\pi$
$173$	−99.7258	−0.576450	−0.288225	+	0.957563i	$0.593065\pi$
$174$	0	0
$175$	− 48.2095i	− 0.275483i
$176$	0	0
$177$	17.8870	0.101057
$178$	0	0
$179$	− 197.680i	− 1.10436i	−0.833726	−	0.552178i	$-0.813797\pi$
$179$	− 197.680i	− 1.10436i	0.833726	−	0.552178i	$-0.186203\pi$
$180$	0	0
$181$	214.024	1.18245	0.591227	−	0.806505i	$-0.298644\pi$
$181$	214.024	1.18245	0.591227	+	0.806505i	$0.298644\pi$
$182$	0	0
$183$	19.7759i	0.108065i
$184$	0	0
$185$	17.8447	0.0964577
$186$	0	0
$187$	− 89.3195i	− 0.477644i
$188$	0	0
$189$	20.4231	0.108059
$190$	0	0
$191$	− 180.062i	− 0.942732i	−0.881938	−	0.471366i	$-0.843761\pi$
$191$	− 180.062i	− 0.942732i	0.881938	−	0.471366i	$-0.156239\pi$
$192$	0	0
$193$	−132.733	−0.687737	−0.343869	−	0.939018i	$-0.611738\pi$
$193$	−132.733	−0.687737	−0.343869	+	0.939018i	$0.611738\pi$
$194$	0	0
$195$	11.8343i	0.0606887i
$196$	0	0
$197$	−38.4534	−0.195195	−0.0975975	−	0.995226i	$-0.531116\pi$
$197$	−38.4534	−0.195195	−0.0975975	+	0.995226i	$0.531116\pi$
$198$	0	0
$199$	193.345i	0.971581i	0.874075	+	0.485790i	$0.161468\pi$
$199$	193.345i	0.971581i	−0.874075	+	0.485790i	$0.838532\pi$
$200$	0	0
$201$	−20.9112	−0.104036
$202$	0	0
$203$	− 22.4207i	− 0.110447i
$204$	0	0
$205$	−192.888	−0.940916
$206$	0	0
$207$	− 186.591i	− 0.901405i
$208$	0	0
$209$	43.1759	0.206583
$210$	0	0
$211$	− 298.214i	− 1.41333i	−0.707546	−	0.706667i	$-0.750198\pi$
$211$	− 298.214i	− 1.41333i	0.707546	−	0.706667i	$-0.249802\pi$
$212$	0	0
$213$	13.8472	0.0650102
$214$	0	0
$215$	− 163.056i	− 0.758399i
$216$	0	0
$217$	78.8556	0.363390
$218$	0	0
$219$	34.6205i	0.158084i
$220$	0	0
$221$	59.7763	0.270481
$222$	0	0
$223$	11.4875i	0.0515134i	0.999668	+	0.0257567i	$0.00819952\pi$
$223$	11.4875i	0.0515134i	−0.999668	+	0.0257567i	$0.991800\pi$
$224$	0	0
$225$	110.119	0.489416
$226$	0	0
$227$	288.249i	1.26982i	0.772586	+	0.634910i	$0.218963\pi$
$227$	288.249i	1.26982i	−0.772586	+	0.634910i	$0.781037\pi$
$228$	0	0
$229$	387.140	1.69057	0.845285	−	0.534316i	$-0.179431\pi$
$229$	387.140	1.69057	0.845285	+	0.534316i	$0.179431\pi$
$230$	0	0
$231$	11.2922i	0.0488838i
$232$	0	0
$233$	−225.342	−0.967134	−0.483567	−	0.875307i	$-0.660659\pi$
$233$	−225.342	−0.967134	−0.483567	+	0.875307i	$0.660659\pi$
$234$	0	0
$235$	8.74407i	0.0372088i
$236$	0	0
$237$	10.8538	0.0457964
$238$	0	0
$239$	− 224.022i	− 0.937330i	−0.883376	−	0.468665i	$-0.844735\pi$
$239$	− 224.022i	− 0.937330i	0.883376	−	0.468665i	$-0.155265\pi$
$240$	0	0
$241$	110.078	0.456756	0.228378	−	0.973573i	$-0.426658\pi$
$241$	110.078	0.456756	0.228378	+	0.973573i	$0.426658\pi$
$242$	0	0
$243$	70.0858i	0.288419i
$244$	0	0
$245$	−206.379	−0.842364
$246$	0	0
$247$	28.8951i	0.116984i
$248$	0	0
$249$	0.401002	0.00161045
$250$	0	0
$251$	2.53572i	0.0101025i	0.999987	+	0.00505123i	$0.00160786\pi$
$251$	2.53572i	0.0101025i	−0.999987	+	0.00505123i	$0.998392\pi$
$252$	0	0
$253$	207.324	0.819461
$254$	0	0
$255$	16.0982i	0.0631302i
$256$	0	0
$257$	221.920	0.863503	0.431752	−	0.901993i	$-0.357896\pi$
$257$	221.920	0.863503	0.431752	+	0.901993i	$0.357896\pi$
$258$	0	0
$259$	11.3953i	0.0439973i
$260$	0	0
$261$	51.2127	0.196217
$262$	0	0
$263$	307.882i	1.17065i	0.810797	+	0.585327i	$0.199034\pi$
$263$	307.882i	1.17065i	−0.810797	+	0.585327i	$0.800966\pi$
$264$	0	0
$265$	−27.1353	−0.102397
$266$	0	0
$267$	29.5479i	0.110666i
$268$	0	0
$269$	−3.41169	−0.0126829	−0.00634143	−	0.999980i	$-0.502019\pi$
$269$	−3.41169	−0.0126829	−0.00634143	+	0.999980i	$0.502019\pi$
$270$	0	0
$271$	22.0634i	0.0814149i	0.999171	+	0.0407075i	$0.0129612\pi$
$271$	22.0634i	0.0814149i	−0.999171	+	0.0407075i	$0.987039\pi$
$272$	0	0
$273$	−7.55718	−0.0276820
$274$	0	0
$275$	122.354i	0.444925i
$276$	0	0
$277$	155.181	0.560220	0.280110	−	0.959968i	$-0.409629\pi$
$277$	155.181	0.560220	0.280110	+	0.959968i	$0.409629\pi$
$278$	0	0
$279$	180.120i	0.645590i
$280$	0	0
$281$	−38.6055	−0.137386	−0.0686930	−	0.997638i	$-0.521883\pi$
$281$	−38.6055	−0.137386	−0.0686930	+	0.997638i	$0.521883\pi$
$282$	0	0
$283$	− 428.155i	− 1.51292i	−0.654042	−	0.756458i	$-0.726928\pi$
$283$	− 428.155i	− 1.51292i	0.654042	−	0.756458i	$-0.273072\pi$
$284$	0	0
$285$	−7.78165	−0.0273041
$286$	0	0
$287$	− 123.175i	− 0.429181i
$288$	0	0
$289$	−207.686	−0.718637
$290$	0	0
$291$	42.9650i	0.147646i
$292$	0	0
$293$	−106.217	−0.362517	−0.181258	−	0.983435i	$-0.558017\pi$
$293$	−106.217	−0.362517	−0.181258	+	0.983435i	$0.558017\pi$
$294$	0	0
$295$	374.251i	1.26865i
$296$	0	0
$297$	−51.8332	−0.174523
$298$	0	0
$299$	138.750i	0.464046i
$300$	0	0
$301$	104.125	0.345929
$302$	0	0
$303$	21.2304i	0.0700674i
$304$	0	0
$305$	−413.772	−1.35663
$306$	0	0
$307$	− 335.513i	− 1.09288i	−0.837500	−	0.546438i	$-0.815983\pi$
$307$	− 335.513i	− 1.09288i	0.837500	−	0.546438i	$-0.184017\pi$
$308$	0	0
$309$	1.80934	0.00585548
$310$	0	0
$311$	486.601i	1.56463i	0.622881	+	0.782316i	$0.285962\pi$
$311$	486.601i	1.56463i	−0.622881	+	0.782316i	$0.714038\pi$
$312$	0	0
$313$	259.056	0.827654	0.413827	−	0.910356i	$-0.364192\pi$
$313$	259.056	0.827654	0.413827	+	0.910356i	$0.364192\pi$
$314$	0	0
$315$	212.639i	0.675044i
$316$	0	0
$317$	−508.795	−1.60503	−0.802515	−	0.596631i	$-0.796505\pi$
$317$	−508.795	−1.60503	−0.802515	+	0.596631i	$0.796505\pi$
$318$	0	0
$319$	56.9031i	0.178380i
$320$	0	0
$321$	23.6252	0.0735987
$322$	0	0
$323$	39.3060i	0.121690i
$324$	0	0
$325$	−81.8846	−0.251953
$326$	0	0
$327$	46.0368i	0.140785i
$328$	0	0
$329$	−5.58381	−0.0169721
$330$	0	0
$331$	− 486.139i	− 1.46870i	−0.678773	−	0.734348i	$-0.737488\pi$
$331$	− 486.139i	− 1.46870i	0.678773	−	0.734348i	$-0.262512\pi$
$332$	0	0
$333$	−26.0288	−0.0781646
$334$	0	0
$335$	− 437.525i	− 1.30604i
$336$	0	0
$337$	−253.342	−0.751758	−0.375879	−	0.926669i	$-0.622659\pi$
$337$	−253.342	−0.751758	−0.375879	+	0.926669i	$0.622659\pi$
$338$	0	0
$339$	− 16.3478i	− 0.0482235i
$340$	0	0
$341$	−200.133	−0.586901
$342$	0	0
$343$	− 323.028i	− 0.941773i
$344$	0	0
$345$	−37.3663	−0.108308
$346$	0	0
$347$	652.459i	1.88028i	0.340783	+	0.940142i	$0.389308\pi$
$347$	652.459i	1.88028i	−0.340783	+	0.940142i	$0.610692\pi$
$348$	0	0
$349$	−16.6947	−0.0478359	−0.0239180	−	0.999714i	$-0.507614\pi$
$349$	−16.6947	−0.0478359	−0.0239180	+	0.999714i	$0.507614\pi$
$350$	0	0
$351$	− 34.6890i	− 0.0988290i
$352$	0	0
$353$	−179.315	−0.507974	−0.253987	−	0.967208i	$-0.581742\pi$
$353$	−179.315	−0.507974	−0.253987	+	0.967208i	$0.581742\pi$
$354$	0	0
$355$	289.725i	0.816126i
$356$	0	0
$357$	−10.2800	−0.0287956
$358$	0	0
$359$	− 450.090i	− 1.25373i	−0.779127	−	0.626867i	$-0.784337\pi$
$359$	− 450.090i	− 1.25373i	0.779127	−	0.626867i	$-0.215663\pi$
$360$	0	0
$361$	−19.0000	−0.0526316
$362$	0	0
$363$	6.68523i	0.0184166i
$364$	0	0
$365$	−724.365	−1.98456
$366$	0	0
$367$	− 302.333i	− 0.823796i	−0.911230	−	0.411898i	$-0.864866\pi$
$367$	− 302.333i	− 0.823796i	0.911230	−	0.411898i	$-0.135134\pi$
$368$	0	0
$369$	281.352	0.762473
$370$	0	0
$371$	− 17.3282i	− 0.0467067i
$372$	0	0
$373$	34.8630	0.0934665	0.0467333	−	0.998907i	$-0.485119\pi$
$373$	34.8630	0.0934665	0.0467333	+	0.998907i	$0.485119\pi$
$374$	0	0
$375$	22.5787i	0.0602099i
$376$	0	0
$377$	−38.0819	−0.101013
$378$	0	0
$379$	132.632i	0.349953i	0.984573	+	0.174976i	$0.0559849\pi$
$379$	132.632i	0.349953i	−0.984573	+	0.174976i	$0.944015\pi$
$380$	0	0
$381$	−59.7046	−0.156705
$382$	0	0
$383$	− 672.616i	− 1.75618i	−0.478498	−	0.878089i	$-0.658818\pi$
$383$	− 672.616i	− 1.75618i	0.478498	−	0.878089i	$-0.341182\pi$
$384$	0	0
$385$	−236.266	−0.613678
$386$	0	0
$387$	237.838i	0.614569i
$388$	0	0
$389$	−721.188	−1.85395	−0.926976	−	0.375120i	$-0.877602\pi$
$389$	−721.188	−1.85395	−0.926976	+	0.375120i	$0.877602\pi$
$390$	0	0
$391$	188.741i	0.482714i
$392$	0	0
$393$	−3.43861	−0.00874966
$394$	0	0
$395$	227.093i	0.574920i
$396$	0	0
$397$	14.8340	0.0373654	0.0186827	−	0.999825i	$-0.494053\pi$
$397$	14.8340	0.0373654	0.0186827	+	0.999825i	$0.494053\pi$
$398$	0	0
$399$	− 4.96923i	− 0.0124542i
$400$	0	0
$401$	−51.9950	−0.129663	−0.0648317	−	0.997896i	$-0.520651\pi$
$401$	−51.9950	−0.129663	−0.0648317	+	0.997896i	$0.520651\pi$
$402$	0	0
$403$	− 133.938i	− 0.332351i
$404$	0	0
$405$	−481.010	−1.18768
$406$	0	0
$407$	− 28.9210i	− 0.0710589i
$408$	0	0
$409$	54.7328	0.133821	0.0669105	−	0.997759i	$-0.478686\pi$
$409$	54.7328	0.133821	0.0669105	+	0.997759i	$0.478686\pi$
$410$	0	0
$411$	− 34.9777i	− 0.0851039i
$412$	0	0
$413$	−238.990	−0.578669
$414$	0	0
$415$	8.39018i	0.0202173i
$416$	0	0
$417$	71.2575	0.170881
$418$	0	0
$419$	257.175i	0.613783i	0.951745	+	0.306891i	$0.0992889\pi$
$419$	257.175i	0.613783i	−0.951745	+	0.306891i	$0.900711\pi$
$420$	0	0
$421$	−194.856	−0.462841	−0.231420	−	0.972854i	$-0.574337\pi$
$421$	−194.856	−0.462841	−0.231420	+	0.972854i	$0.574337\pi$
$422$	0	0
$423$	− 12.7544i	− 0.0301522i
$424$	0	0
$425$	−111.388	−0.262089
$426$	0	0
$427$	− 264.228i	− 0.618800i
$428$	0	0
$429$	19.1799	0.0447084
$430$	0	0
$431$	− 360.926i	− 0.837415i	−0.908121	−	0.418707i	$-0.862483\pi$
$431$	− 360.926i	− 0.837415i	0.908121	−	0.418707i	$-0.137517\pi$
$432$	0	0
$433$	−440.213	−1.01666	−0.508329	−	0.861163i	$-0.669737\pi$
$433$	−440.213	−1.01666	−0.508329	+	0.861163i	$0.669737\pi$
$434$	0	0
$435$	− 10.2557i	− 0.0235764i
$436$	0	0
$437$	−91.2350	−0.208776
$438$	0	0
$439$	− 43.6477i	− 0.0994254i	−0.998764	−	0.0497127i	$-0.984169\pi$
$439$	− 43.6477i	− 0.0994254i	0.998764	−	0.0497127i	$-0.0158306\pi$
$440$	0	0
$441$	301.031	0.682611
$442$	0	0
$443$	− 31.5928i	− 0.0713155i	−0.999364	−	0.0356578i	$-0.988647\pi$
$443$	− 31.5928i	− 0.0713155i	0.999364	−	0.0356578i	$-0.0113526\pi$
$444$	0	0
$445$	−618.232	−1.38928
$446$	0	0
$447$	− 14.7486i	− 0.0329946i
$448$	0	0
$449$	622.909	1.38732	0.693662	−	0.720300i	$-0.255996\pi$
$449$	622.909	1.38732	0.693662	+	0.720300i	$0.255996\pi$
$450$	0	0
$451$	312.615i	0.693159i
$452$	0	0
$453$	−31.9040	−0.0704283
$454$	0	0
$455$	− 158.119i	− 0.347515i
$456$	0	0
$457$	−103.548	−0.226582	−0.113291	−	0.993562i	$-0.536139\pi$
$457$	−103.548	−0.226582	−0.113291	+	0.993562i	$0.536139\pi$
$458$	0	0
$459$	− 47.1874i	− 0.102805i
$460$	0	0
$461$	572.581	1.24204	0.621021	−	0.783794i	$-0.286718\pi$
$461$	572.581	1.24204	0.621021	+	0.783794i	$0.286718\pi$
$462$	0	0
$463$	461.054i	0.995797i	0.867235	+	0.497898i	$0.165895\pi$
$463$	461.054i	0.995797i	−0.867235	+	0.497898i	$0.834105\pi$
$464$	0	0
$465$	36.0704	0.0775706
$466$	0	0
$467$	489.721i	1.04865i	0.851517	+	0.524326i	$0.175683\pi$
$467$	489.721i	1.04865i	−0.851517	+	0.524326i	$0.824317\pi$
$468$	0	0
$469$	279.396	0.595727
$470$	0	0
$471$	36.4069i	0.0772970i
$472$	0	0
$473$	−264.265	−0.558701
$474$	0	0
$475$	− 53.8433i	− 0.113354i
$476$	0	0
$477$	39.5805	0.0829779
$478$	0	0
$479$	650.528i	1.35810i	0.734094	+	0.679048i	$0.237607\pi$
$479$	650.528i	1.35810i	−0.734094	+	0.679048i	$0.762393\pi$
$480$	0	0
$481$	19.3551	0.0402393
$482$	0	0
$483$	− 23.8615i	− 0.0494027i
$484$	0	0
$485$	−898.958	−1.85352
$486$	0	0
$487$	− 168.062i	− 0.345097i	−0.985001	−	0.172549i	$-0.944800\pi$
$487$	− 168.062i	− 0.345097i	0.985001	−	0.172549i	$-0.0552002\pi$
$488$	0	0
$489$	−37.5326	−0.0767537
$490$	0	0
$491$	− 620.584i	− 1.26392i	−0.775001	−	0.631960i	$-0.782251\pi$
$491$	− 620.584i	− 1.26392i	0.775001	−	0.631960i	$-0.217749\pi$
$492$	0	0
$493$	−51.8029	−0.105077
$494$	0	0
$495$	− 539.672i	− 1.09025i
$496$	0	0
$497$	−185.013	−0.372261
$498$	0	0
$499$	413.259i	0.828175i	0.910237	+	0.414088i	$0.135899\pi$
$499$	413.259i	0.828175i	−0.910237	+	0.414088i	$0.864101\pi$
$500$	0	0
$501$	−94.5306	−0.188684
$502$	0	0
$503$	603.658i	1.20012i	0.799957	+	0.600058i	$0.204856\pi$
$503$	603.658i	1.20012i	−0.799957	+	0.600058i	$0.795144\pi$
$504$	0	0
$505$	−444.205	−0.879613
$506$	0	0
$507$	− 36.5293i	− 0.0720500i
$508$	0	0
$509$	−97.7927	−0.192127	−0.0960635	−	0.995375i	$-0.530625\pi$
$509$	−97.7927	−0.192127	−0.0960635	+	0.995375i	$0.530625\pi$
$510$	0	0
$511$	− 462.568i	− 0.905221i
$512$	0	0
$513$	22.8098	0.0444635
$514$	0	0
$515$	37.8569i	0.0735086i
$516$	0	0
$517$	14.1716	0.0274111
$518$	0	0
$519$	− 29.1302i	− 0.0561275i
$520$	0	0
$521$	83.1379	0.159574	0.0797868	−	0.996812i	$-0.474576\pi$
$521$	83.1379	0.159574	0.0797868	+	0.996812i	$0.474576\pi$
$522$	0	0
$523$	− 334.055i	− 0.638729i	−0.947632	−	0.319365i	$-0.896531\pi$
$523$	− 334.055i	− 0.638729i	0.947632	−	0.319365i	$-0.103469\pi$
$524$	0	0
$525$	14.0821	0.0268231
$526$	0	0
$527$	− 182.195i	− 0.345722i
$528$	0	0
$529$	90.9035	0.171840
$530$	0	0
$531$	− 545.894i	− 1.02805i
$532$	0	0
$533$	−209.215	−0.392523
$534$	0	0
$535$	494.311i	0.923945i
$536$	0	0
$537$	57.7427	0.107528
$538$	0	0
$539$	334.480i	0.620557i
$540$	0	0
$541$	374.174	0.691634	0.345817	−	0.938302i	$-0.387602\pi$
$541$	374.174	0.691634	0.345817	+	0.938302i	$0.387602\pi$
$542$	0	0
$543$	62.5170i	0.115133i
$544$	0	0
$545$	−963.230	−1.76739
$546$	0	0
$547$	1008.55i	1.84379i	0.387437	+	0.921896i	$0.373361\pi$
$547$	1008.55i	1.84379i	−0.387437	+	0.921896i	$0.626639\pi$
$548$	0	0
$549$	603.540	1.09934
$550$	0	0
$551$	− 25.0408i	− 0.0454462i
$552$	0	0
$553$	−145.018	−0.262239
$554$	0	0
$555$	5.21247i	0.00939184i
$556$	0	0
$557$	192.766	0.346079	0.173039	−	0.984915i	$-0.444641\pi$
$557$	192.766	0.346079	0.173039	+	0.984915i	$0.444641\pi$
$558$	0	0
$559$	− 176.857i	− 0.316382i
$560$	0	0
$561$	26.0904	0.0465070
$562$	0	0
$563$	413.982i	0.735315i	0.929961	+	0.367658i	$0.119840\pi$
$563$	413.982i	0.735315i	−0.929961	+	0.367658i	$0.880160\pi$
$564$	0	0
$565$	342.045	0.605390
$566$	0	0
$567$	− 307.165i	− 0.541737i
$568$	0	0
$569$	387.260	0.680597	0.340298	−	0.940317i	$-0.389472\pi$
$569$	387.260	0.680597	0.340298	+	0.940317i	$0.389472\pi$
$570$	0	0
$571$	− 893.702i	− 1.56515i	−0.622555	−	0.782576i	$-0.713905\pi$
$571$	− 893.702i	− 1.56515i	0.622555	−	0.782576i	$-0.286095\pi$
$572$	0	0
$573$	52.5965	0.0917915
$574$	0	0
$575$	− 258.547i	− 0.449648i
$576$	0	0
$577$	170.975	0.296317	0.148158	−	0.988964i	$-0.452665\pi$
$577$	170.975	0.296317	0.148158	+	0.988964i	$0.452665\pi$
$578$	0	0
$579$	− 38.7717i	− 0.0669633i
$580$	0	0
$581$	−5.35783	−0.00922173
$582$	0	0
$583$	43.9784i	0.0754347i
$584$	0	0
$585$	361.171	0.617386
$586$	0	0
$587$	− 155.528i	− 0.264954i	−0.991186	−	0.132477i	$-0.957707\pi$
$587$	− 155.528i	− 0.264954i	0.991186	−	0.132477i	$-0.0422931\pi$
$588$	0	0
$589$	88.0708	0.149526
$590$	0	0
$591$	− 11.2323i	− 0.0190056i
$592$	0	0
$593$	1120.14	1.88893	0.944465	−	0.328611i	$-0.106581\pi$
$593$	1120.14	1.88893	0.944465	+	0.328611i	$0.106581\pi$
$594$	0	0
$595$	− 215.090i	− 0.361495i
$596$	0	0
$597$	−56.4764	−0.0946004
$598$	0	0
$599$	780.665i	1.30328i	0.758528	+	0.651640i	$0.225919\pi$
$599$	780.665i	1.30328i	−0.758528	+	0.651640i	$0.774081\pi$
$600$	0	0
$601$	−1164.58	−1.93774	−0.968868	−	0.247576i	$-0.920366\pi$
$601$	−1164.58	−1.93774	−0.968868	+	0.247576i	$0.920366\pi$
$602$	0	0
$603$	638.188i	1.05835i
$604$	0	0
$605$	−139.875	−0.231199
$606$	0	0
$607$	1089.36i	1.79467i	0.441355	+	0.897333i	$0.354498\pi$
$607$	1089.36i	1.79467i	−0.441355	+	0.897333i	$0.645502\pi$
$608$	0	0
$609$	6.54915	0.0107539
$610$	0	0
$611$	9.48420i	0.0155224i
$612$	0	0
$613$	437.827	0.714236	0.357118	−	0.934059i	$-0.383759\pi$
$613$	437.827	0.714236	0.357118	+	0.934059i	$0.383759\pi$
$614$	0	0
$615$	− 56.3430i	− 0.0916147i
$616$	0	0
$617$	290.046	0.470090	0.235045	−	0.971984i	$-0.424476\pi$
$617$	290.046	0.470090	0.235045	+	0.971984i	$0.424476\pi$
$618$	0	0
$619$	− 366.321i	− 0.591795i	−0.955220	−	0.295897i	$-0.904381\pi$
$619$	− 366.321i	− 0.591795i	0.955220	−	0.295897i	$-0.0956186\pi$
$620$	0	0
$621$	109.529	0.176375
$622$	0	0
$623$	− 394.793i	− 0.633696i
$624$	0	0
$625$	−781.228	−1.24997
$626$	0	0
$627$	12.6118i	0.0201145i
$628$	0	0
$629$	26.3288	0.0418582
$630$	0	0
$631$	718.973i	1.13942i	0.821846	+	0.569709i	$0.192944\pi$
$631$	718.973i	1.13942i	−0.821846	+	0.569709i	$0.807056\pi$
$632$	0	0
$633$	87.1089	0.137613
$634$	0	0
$635$	− 1249.20i	− 1.96725i
$636$	0	0
$637$	−223.848	−0.351410
$638$	0	0
$639$	− 422.602i	− 0.661349i
$640$	0	0
$641$	−740.418	−1.15510	−0.577549	−	0.816356i	$-0.695991\pi$
$641$	−740.418	−1.15510	−0.577549	+	0.816356i	$0.695991\pi$
$642$	0	0
$643$	− 312.874i	− 0.486585i	−0.969953	−	0.243293i	$-0.921772\pi$
$643$	− 312.874i	− 0.486585i	0.969953	−	0.243293i	$-0.0782275\pi$
$644$	0	0
$645$	47.6290	0.0738434
$646$	0	0
$647$	− 858.982i	− 1.32764i	−0.747893	−	0.663819i	$-0.768934\pi$
$647$	− 858.982i	− 1.32764i	0.747893	−	0.663819i	$-0.231066\pi$
$648$	0	0
$649$	606.551	0.934593
$650$	0	0
$651$	23.0339i	0.0353824i
$652$	0	0
$653$	151.224	0.231584	0.115792	−	0.993274i	$-0.463059\pi$
$653$	151.224	0.231584	0.115792	+	0.993274i	$0.463059\pi$
$654$	0	0
$655$	− 71.9462i	− 0.109842i
$656$	0	0
$657$	1056.58	1.60819
$658$	0	0
$659$	626.555i	0.950767i	0.879779	+	0.475383i	$0.157691\pi$
$659$	626.555i	0.950767i	−0.879779	+	0.475383i	$0.842309\pi$
$660$	0	0
$661$	959.618	1.45177	0.725884	−	0.687817i	$-0.241431\pi$
$661$	959.618	1.45177	0.725884	+	0.687817i	$0.241431\pi$
$662$	0	0
$663$	17.4608i	0.0263361i
$664$	0	0
$665$	103.971	0.156348
$666$	0	0
$667$	− 120.242i	− 0.180273i
$668$	0	0
$669$	−3.35552	−0.00501573
$670$	0	0
$671$	670.602i	0.999407i
$672$	0	0
$673$	965.229	1.43422	0.717109	−	0.696961i	$-0.245465\pi$
$673$	965.229	1.43422	0.717109	+	0.696961i	$0.245465\pi$
$674$	0	0
$675$	64.6397i	0.0957625i
$676$	0	0
$677$	372.816	0.550688	0.275344	−	0.961346i	$-0.411208\pi$
$677$	372.816	0.550688	0.275344	+	0.961346i	$0.411208\pi$
$678$	0	0
$679$	− 574.059i	− 0.845448i
$680$	0	0
$681$	−84.1982	−0.123639
$682$	0	0
$683$	− 964.540i	− 1.41221i	−0.708107	−	0.706105i	$-0.750450\pi$
$683$	− 964.540i	− 1.41221i	0.708107	−	0.706105i	$-0.249550\pi$
$684$	0	0
$685$	731.840	1.06838
$686$	0	0
$687$	113.085i	0.164607i
$688$	0	0
$689$	−29.4322	−0.0427172
$690$	0	0
$691$	620.354i	0.897762i	0.893591	+	0.448881i	$0.148177\pi$
$691$	620.354i	0.897762i	−0.893591	+	0.448881i	$0.851823\pi$
$692$	0	0
$693$	344.625	0.497295
$694$	0	0
$695$	1490.92i	2.14521i
$696$	0	0
$697$	−284.595	−0.408314
$698$	0	0
$699$	− 65.8230i	− 0.0941674i
$700$	0	0
$701$	945.055	1.34815	0.674076	−	0.738662i	$-0.264542\pi$
$701$	945.055	1.34815	0.674076	+	0.738662i	$0.264542\pi$
$702$	0	0
$703$	12.7270i	0.0181038i
$704$	0	0
$705$	−2.55416	−0.00362293
$706$	0	0
$707$	− 283.662i	− 0.401219i
$708$	0	0
$709$	−475.441	−0.670580	−0.335290	−	0.942115i	$-0.608834\pi$
$709$	−475.441	−0.670580	−0.335290	+	0.942115i	$0.608834\pi$
$710$	0	0
$711$	− 331.246i	− 0.465887i
$712$	0	0
$713$	422.902	0.593131
$714$	0	0
$715$	401.302i	0.561261i
$716$	0	0
$717$	65.4374	0.0912655
$718$	0	0
$719$	− 969.029i	− 1.34775i	−0.738847	−	0.673873i	$-0.764629\pi$
$719$	− 969.029i	− 1.34775i	0.738847	−	0.673873i	$-0.235371\pi$
$720$	0	0
$721$	−24.1748	−0.0335296
$722$	0	0
$723$	32.1541i	0.0444732i
$724$	0	0
$725$	70.9622	0.0978789
$726$	0	0
$727$	− 584.964i	− 0.804627i	−0.915502	−	0.402314i	$-0.868206\pi$
$727$	− 584.964i	− 0.804627i	0.915502	−	0.402314i	$-0.131794\pi$
$728$	0	0
$729$	687.860	0.943566
$730$	0	0
$731$	− 240.579i	− 0.329110i
$732$	0	0
$733$	−90.7544	−0.123812	−0.0619061	−	0.998082i	$-0.519718\pi$
$733$	−90.7544	−0.123812	−0.0619061	+	0.998082i	$0.519718\pi$
$734$	0	0
$735$	− 60.2839i	− 0.0820189i
$736$	0	0
$737$	−709.099	−0.962143
$738$	0	0
$739$	689.042i	0.932398i	0.884680	+	0.466199i	$0.154377\pi$
$739$	689.042i	0.932398i	−0.884680	+	0.466199i	$0.845623\pi$
$740$	0	0
$741$	−8.44032	−0.0113905
$742$	0	0
$743$	505.348i	0.680146i	0.940399	+	0.340073i	$0.110452\pi$
$743$	505.348i	0.680146i	−0.940399	+	0.340073i	$0.889548\pi$
$744$	0	0
$745$	308.586	0.414209
$746$	0	0
$747$	− 12.2382i	− 0.0163831i
$748$	0	0
$749$	−315.659	−0.421440
$750$	0	0
$751$	1275.11i	1.69788i	0.528489	+	0.848940i	$0.322759\pi$
$751$	1275.11i	1.69788i	−0.528489	+	0.848940i	$0.677241\pi$
$752$	0	0
$753$	−0.740690	−0.000983651 0
$754$	0	0
$755$	− 667.529i	− 0.884145i
$756$	0	0
$757$	−142.904	−0.188776	−0.0943882	−	0.995535i	$-0.530089\pi$
$757$	−142.904	−0.188776	−0.0943882	+	0.995535i	$0.530089\pi$
$758$	0	0
$759$	60.5598i	0.0797889i
$760$	0	0
$761$	−35.4430	−0.0465742	−0.0232871	−	0.999729i	$-0.507413\pi$
$761$	−35.4430	−0.0465742	−0.0232871	+	0.999729i	$0.507413\pi$
$762$	0	0
$763$	− 615.103i	− 0.806163i
$764$	0	0
$765$	491.301	0.642223
$766$	0	0
$767$	405.929i	0.529242i
$768$	0	0
$769$	−407.706	−0.530177	−0.265089	−	0.964224i	$-0.585401\pi$
$769$	−407.706	−0.530177	−0.265089	+	0.964224i	$0.585401\pi$
$770$	0	0
$771$	64.8235i	0.0840771i
$772$	0	0
$773$	962.961	1.24575	0.622873	−	0.782323i	$-0.285965\pi$
$773$	962.961	1.24575	0.622873	+	0.782323i	$0.285965\pi$
$774$	0	0
$775$	249.580i	0.322039i
$776$	0	0
$777$	−3.32860	−0.00428391
$778$	0	0
$779$	− 137.569i	− 0.176597i
$780$	0	0
$781$	469.559	0.601228
$782$	0	0
$783$	30.0619i	0.0383932i
$784$	0	0
$785$	−761.743	−0.970373
$786$	0	0
$787$	− 890.213i	− 1.13115i	−0.824698	−	0.565574i	$-0.808655\pi$
$787$	− 890.213i	− 1.13115i	0.824698	−	0.565574i	$-0.191345\pi$
$788$	0	0
$789$	−89.9331	−0.113984
$790$	0	0
$791$	218.424i	0.276137i
$792$	0	0
$793$	−448.795	−0.565946
$794$	0	0
$795$	− 7.92630i	− 0.00997019i
$796$	0	0
$797$	−869.224	−1.09062	−0.545310	−	0.838235i	$-0.683588\pi$
$797$	−869.224	−1.09062	−0.545310	+	0.838235i	$0.683588\pi$
$798$	0	0
$799$	12.9014i	0.0161469i
$800$	0	0
$801$	901.773	1.12581
$802$	0	0
$803$	1173.98i	1.46200i
$804$	0	0
$805$	499.255	0.620192
$806$	0	0
$807$	− 0.996563i	− 0.00123490i
$808$	0	0
$809$	−708.028	−0.875189	−0.437595	−	0.899172i	$-0.644170\pi$
$809$	−708.028	−0.875189	−0.437595	+	0.899172i	$0.644170\pi$
$810$	0	0
$811$	− 179.169i	− 0.220923i	−0.993880	−	0.110462i	$-0.964767\pi$
$811$	− 179.169i	− 0.220923i	0.993880	−	0.110462i	$-0.0352329\pi$
$812$	0	0
$813$	−6.44479	−0.00792717
$814$	0	0
$815$	− 785.295i	− 0.963552i
$816$	0	0
$817$	116.293	0.142341
$818$	0	0
$819$	230.638i	0.281609i
$820$	0	0
$821$	−21.2430	−0.0258746	−0.0129373	−	0.999916i	$-0.504118\pi$
$821$	−21.2430	−0.0258746	−0.0129373	+	0.999916i	$0.504118\pi$
$822$	0	0
$823$	257.671i	0.313088i	0.987671	+	0.156544i	$0.0500353\pi$
$823$	257.671i	0.313088i	−0.987671	+	0.156544i	$0.949965\pi$
$824$	0	0
$825$	−35.7400	−0.0433212
$826$	0	0
$827$	81.0005i	0.0979450i	0.998800	+	0.0489725i	$0.0155947\pi$
$827$	81.0005i	0.0979450i	−0.998800	+	0.0489725i	$0.984405\pi$
$828$	0	0
$829$	209.389	0.252580	0.126290	−	0.991993i	$-0.459693\pi$
$829$	209.389	0.252580	0.126290	+	0.991993i	$0.459693\pi$
$830$	0	0
$831$	45.3287i	0.0545472i
$832$	0	0
$833$	−304.501	−0.365547
$834$	0	0
$835$	− 1977.87i	− 2.36870i
$836$	0	0
$837$	−105.730	−0.126321
$838$	0	0
$839$	271.665i	0.323796i	0.986807	+	0.161898i	$0.0517616\pi$
$839$	271.665i	0.323796i	−0.986807	+	0.161898i	$0.948238\pi$
$840$	0	0
$841$	−807.998	−0.960758
$842$	0	0
$843$	− 11.2768i	− 0.0133769i
$844$	0	0
$845$	764.304	0.904502
$846$	0	0
$847$	− 89.3220i	− 0.105457i
$848$	0	0
$849$	125.065	0.147309
$850$	0	0
$851$	61.1130i	0.0718132i
$852$	0	0
$853$	1320.35	1.54789	0.773943	−	0.633255i	$-0.218282\pi$
$853$	1320.35	1.54789	0.773943	+	0.633255i	$0.218282\pi$
$854$	0	0
$855$	237.488i	0.277764i
$856$	0	0
$857$	−569.084	−0.664042	−0.332021	−	0.943272i	$-0.607730\pi$
$857$	−569.084	−0.664042	−0.332021	+	0.943272i	$0.607730\pi$
$858$	0	0
$859$	− 1146.46i	− 1.33465i	−0.744767	−	0.667325i	$-0.767439\pi$
$859$	− 1146.46i	− 1.33465i	0.744767	−	0.667325i	$-0.232561\pi$
$860$	0	0
$861$	35.9797	0.0417883
$862$	0	0
$863$	352.061i	0.407951i	0.978976	+	0.203975i	$0.0653862\pi$
$863$	352.061i	0.407951i	−0.978976	+	0.203975i	$0.934614\pi$
$864$	0	0
$865$	609.491	0.704614
$866$	0	0
$867$	− 60.6657i	− 0.0699719i
$868$	0	0
$869$	368.052	0.423535
$870$	0	0
$871$	− 474.559i	− 0.544843i
$872$	0	0
$873$	1311.25	1.50200
$874$	0	0
$875$	− 301.677i	− 0.344773i
$876$	0	0
$877$	−1084.47	−1.23657	−0.618285	−	0.785954i	$-0.712172\pi$
$877$	−1084.47	−1.23657	−0.618285	+	0.785954i	$0.712172\pi$
$878$	0	0
$879$	− 31.0264i	− 0.0352974i
$880$	0	0
$881$	−487.061	−0.552851	−0.276425	−	0.961035i	$-0.589150\pi$
$881$	−487.061	−0.552851	−0.276425	+	0.961035i	$0.589150\pi$
$882$	0	0
$883$	1006.35i	1.13969i	0.821753	+	0.569844i	$0.192997\pi$
$883$	1006.35i	1.13969i	−0.821753	+	0.569844i	$0.807003\pi$
$884$	0	0
$885$	−109.320	−0.123525
$886$	0	0
$887$	− 891.277i	− 1.00482i	−0.864629	−	0.502411i	$-0.832447\pi$
$887$	− 891.277i	− 1.00482i	0.864629	−	0.502411i	$-0.167553\pi$
$888$	0	0
$889$	797.720	0.897322
$890$	0	0
$891$	779.576i	0.874945i
$892$	0	0
$893$	−6.23635	−0.00698359
$894$	0	0
$895$	1208.15i	1.34989i
$896$	0	0
$897$	−40.5291	−0.0451830
$898$	0	0
$899$	116.072i	0.129112i
$900$	0	0
$901$	−40.0366	−0.0444358
$902$	0	0
$903$	30.4151i	0.0336823i
$904$	0	0
$905$	−1308.05	−1.44535
$906$	0	0
$907$	747.223i	0.823841i	0.911220	+	0.411920i	$0.135142\pi$
$907$	747.223i	0.823841i	−0.911220	+	0.411920i	$0.864858\pi$
$908$	0	0
$909$	647.931	0.712795
$910$	0	0
$911$	− 1015.04i	− 1.11420i	−0.830444	−	0.557102i	$-0.811913\pi$
$911$	− 1015.04i	− 1.11420i	0.830444	−	0.557102i	$-0.188087\pi$
$912$	0	0
$913$	13.5980	0.0148938
$914$	0	0
$915$	− 120.864i	− 0.132091i
$916$	0	0
$917$	45.9437	0.0501022
$918$	0	0
$919$	− 482.292i	− 0.524801i	−0.964959	−	0.262401i	$-0.915486\pi$
$919$	− 482.292i	− 0.524801i	0.964959	−	0.262401i	$-0.0845142\pi$
$920$	0	0
$921$	98.0041	0.106411
$922$	0	0
$923$	314.248i	0.340464i
$924$	0	0
$925$	−36.0665	−0.0389908
$926$	0	0
$927$	− 55.2193i	− 0.0595678i
$928$	0	0
$929$	196.255	0.211254	0.105627	−	0.994406i	$-0.466315\pi$
$929$	196.255	0.211254	0.105627	+	0.994406i	$0.466315\pi$
$930$	0	0
$931$	− 147.192i	− 0.158101i
$932$	0	0
$933$	−142.137	−0.152344
$934$	0	0
$935$	545.891i	0.583841i
$936$	0	0
$937$	1142.75	1.21958	0.609789	−	0.792564i	$-0.291254\pi$
$937$	1142.75	1.21958	0.609789	+	0.792564i	$0.291254\pi$
$938$	0	0
$939$	75.6708i	0.0805866i
$940$	0	0
$941$	−1256.87	−1.33568	−0.667840	−	0.744305i	$-0.732781\pi$
$941$	−1256.87	−1.33568	−0.667840	+	0.744305i	$0.732781\pi$
$942$	0	0
$943$	− 660.587i	− 0.700516i
$944$	0	0
$945$	−124.819	−0.132084
$946$	0	0
$947$	− 231.526i	− 0.244484i	−0.992500	−	0.122242i	$-0.960992\pi$
$947$	− 231.526i	− 0.244484i	0.992500	−	0.122242i	$-0.0390084\pi$
$948$	0	0
$949$	−785.679	−0.827902
$950$	0	0
$951$	− 148.620i	− 0.156278i
$952$	0	0
$953$	1322.16	1.38736	0.693680	−	0.720283i	$-0.255988\pi$
$953$	1322.16	1.38736	0.693680	+	0.720283i	$0.255988\pi$
$954$	0	0
$955$	1100.48i	1.15233i
$956$	0	0
$957$	−16.6215	−0.0173684
$958$	0	0
$959$	467.341i	0.487321i
$960$	0	0
$961$	552.765	0.575197
$962$	0	0
$963$	− 721.017i	− 0.748720i
$964$	0	0
$965$	811.222	0.840645
$966$	0	0
$967$	− 1726.27i	− 1.78518i	−0.450866	−	0.892592i	$-0.648885\pi$
$967$	− 1726.27i	− 1.78518i	0.450866	−	0.892592i	$-0.351115\pi$
$968$	0	0
$969$	−11.4814	−0.0118487
$970$	0	0
$971$	1585.66i	1.63302i	0.577333	+	0.816509i	$0.304093\pi$
$971$	1585.66i	1.63302i	−0.577333	+	0.816509i	$0.695907\pi$
$972$	0	0
$973$	−952.078	−0.978498
$974$	0	0
$975$	− 23.9187i	− 0.0245320i
$976$	0	0
$977$	1329.32	1.36062	0.680308	−	0.732926i	$-0.261846\pi$
$977$	1329.32	1.36062	0.680308	+	0.732926i	$0.261846\pi$
$978$	0	0
$979$	1001.97i	1.02346i
$980$	0	0
$981$	1405.00	1.43221
$982$	0	0
$983$	− 756.159i	− 0.769236i	−0.923076	−	0.384618i	$-0.874333\pi$
$983$	− 756.159i	− 0.769236i	0.923076	−	0.384618i	$-0.125667\pi$
$984$	0	0
$985$	235.014	0.238593
$986$	0	0
$987$	− 1.63105i	− 0.00165253i
$988$	0	0
$989$	558.420	0.564631
$990$	0	0
$991$	1732.49i	1.74822i	0.485726	+	0.874111i	$0.338555\pi$
$991$	1732.49i	1.74822i	−0.485726	+	0.874111i	$0.661445\pi$
$992$	0	0
$993$	142.002	0.143003
$994$	0	0
$995$	− 1181.66i	− 1.18760i
$996$	0	0
$997$	−1622.76	−1.62764	−0.813821	−	0.581116i	$-0.802616\pi$
$997$	−1622.76	−1.62764	−0.813821	+	0.581116i	$0.802616\pi$
$998$	0	0
$999$	− 15.2789i	− 0.0152942i

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	1216.3.d.e.191.9		16
4.3	odd	2	inner	1216.3.d.e.191.8		16
8.3	odd	2		608.3.d.a.191.9	yes	16
8.5	even	2		608.3.d.a.191.8	✓	16

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
608.3.d.a.191.8	✓	16	8.5	even	2
608.3.d.a.191.9	yes	16	8.3	odd	2
1216.3.d.e.191.8		16	4.3	odd	2	inner
1216.3.d.e.191.9		16	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 $\F_{q}$
Belyi maps

Level:	\( N \)	\(=\)	\( 1216 = 2^{6} \cdot 19 \)
Weight:	\( k \)	\(=\)	\( 3 \)
Character orbit:	\([\chi]\)	\(=\)	1216.d (of order \(2\), degree \(1\), not minimal)

\(f(q)\)	\(=\)	\(q+0.292102i q^{3} -6.11167 q^{5} -3.90281i q^{7} +8.91468 q^{9} +9.90522i q^{11} -6.62899 q^{13} -1.78523i q^{15} -9.01742 q^{17} -4.35890i q^{19} +1.14002 q^{21} -20.9308i q^{23} +12.3525 q^{25} +5.23292i q^{27} +5.74476 q^{29} +20.2048i q^{31} -2.89334 q^{33} +23.8527i q^{35} -2.91977 q^{37} -1.93634i q^{39} +31.5606 q^{41} +26.6794i q^{43} -54.4836 q^{45} -1.43072i q^{47} +33.7681 q^{49} -2.63401i q^{51} +4.43992 q^{53} -60.5374i q^{55} +1.27325 q^{57} -61.2354i q^{59} +67.7019 q^{61} -34.7923i q^{63} +40.5142 q^{65} +71.5884i q^{67} +6.11393 q^{69} -47.4052i q^{71} +118.522 q^{73} +3.60820i q^{75} +38.6582 q^{77} -37.1573i q^{79} +78.7035 q^{81} -1.37281i q^{83} +55.1115 q^{85} +1.67806i q^{87} +101.156 q^{89} +25.8717i q^{91} -5.90188 q^{93} +26.6402i q^{95} +147.089 q^{97} +88.3018i q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 16 q + 8 q^{5} - 40 q^{9} - 40 q^{13} - 32 q^{17} + 96 q^{21} + 88 q^{25} - 144 q^{29} + 88 q^{33} + 56 q^{37} - 104 q^{41} - 40 q^{45} - 144 q^{49} + 320 q^{53} - 8 q^{61} + 336 q^{65} - 392 q^{69} + 72 q^{77}+ \cdots - 240 q^{97}+O(q^{100}) \) 16 * q + 8 * q^5 - 40 * q^9 - 40 * q^13 - 32 * q^17 + 96 * q^21 + 88 * q^25 - 144 * q^29 + 88 * q^33 + 56 * q^37 - 104 * q^41 - 40 * q^45 - 144 * q^49 + 320 * q^53 - 8 * q^61 + 336 * q^65 - 392 * q^69 + 72 * q^77 - 272 * q^81 - 552 * q^85 - 136 * q^89 + 752 * q^93 - 240 * q^97

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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