LMFDB - Embedded newform 112.3.d.a.15.1

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [112,3,Mod(15,112)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(112, base_ring=CyclotomicField(2))

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

N = Newforms(chi, 3, names="a")

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

chi := DirichletCharacter("112.15");

S:= CuspForms(chi, 3);

N := Newforms(S);

Level:	$ N $	$=$	$ 112 = 2^{4} \cdot 7 $
Weight:	$ k $	$=$	$ 3 $
Character orbit:	$[\chi]$	$=$	112.d (of order $2$, degree $1$, minimal)

Newform invariants

comment: select newform

sage: f = N[0] # Warning: the index may be different

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

Self dual:	no
Analytic conductor:	$3.05177896084$
Analytic rank:	$0$
Dimension:	$2$
Coefficient field:	$\Q(\sqrt{-7}) $
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{2} - x + 2 $ x^2 - x + 2
Coefficient ring:	$\Z[a_1, \ldots, a_{7}]$
Coefficient ring index:	$ 2 $
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label			15.1
Root			$0.500000 + 1.32288i$ of defining polynomial
Character	$\chi$	$=$	112.15
Dual form			112.3.d.a.15.2

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-5.29150i q^{3} +8.00000 q^{5} +2.64575i q^{7} -19.0000 q^{9} +O(q^{10})$	$q-5.29150i q^{3} +8.00000 q^{5} +2.64575i q^{7} -19.0000 q^{9} -10.5830i q^{11} -4.00000 q^{13} -42.3320i q^{15} -2.00000 q^{17} +26.4575i q^{19} +14.0000 q^{21} +21.1660i q^{23} +39.0000 q^{25} +52.9150i q^{27} +14.0000 q^{29} -31.7490i q^{31} -56.0000 q^{33} +21.1660i q^{35} +14.0000 q^{37} +21.1660i q^{39} +46.0000 q^{41} +10.5830i q^{43} -152.000 q^{45} +31.7490i q^{47} -7.00000 q^{49} +10.5830i q^{51} -22.0000 q^{53} -84.6640i q^{55} +140.000 q^{57} +89.9555i q^{59} +48.0000 q^{61} -50.2693i q^{63} -32.0000 q^{65} -63.4980i q^{67} +112.000 q^{69} +84.6640i q^{71} -110.000 q^{73} -206.369i q^{75} +28.0000 q^{77} -126.996i q^{79} +109.000 q^{81} -37.0405i q^{83} -16.0000 q^{85} -74.0810i q^{87} -134.000 q^{89} -10.5830i q^{91} -168.000 q^{93} +211.660i q^{95} -178.000 q^{97} +201.077i q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 2 q + 16 q^{5} - 38 q^{9}+O(q^{10}) $ 2 * q + 16 * q^5 - 38 * q^9	$ 2 q + 16 q^{5} - 38 q^{9} - 8 q^{13} - 4 q^{17} + 28 q^{21} + 78 q^{25} + 28 q^{29} - 112 q^{33} + 28 q^{37} + 92 q^{41} - 304 q^{45} - 14 q^{49} - 44 q^{53} + 280 q^{57} + 96 q^{61} - 64 q^{65} + 224 q^{69} - 220 q^{73} + 56 q^{77} + 218 q^{81} - 32 q^{85} - 268 q^{89} - 336 q^{93} - 356 q^{97}+O(q^{100}) $ 2 * q + 16 * q^5 - 38 * q^9 - 8 * q^13 - 4 * q^17 + 28 * q^21 + 78 * q^25 + 28 * q^29 - 112 * q^33 + 28 * q^37 + 92 * q^41 - 304 * q^45 - 14 * q^49 - 44 * q^53 + 280 * q^57 + 96 * q^61 - 64 * q^65 + 224 * q^69 - 220 * q^73 + 56 * q^77 + 218 * q^81 - 32 * q^85 - 268 * q^89 - 336 * q^93 - 356 * q^97

Display 100 coefficients 10 coefficients

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/112\mathbb{Z}\right)^\times$.

$n$	$15$	$17$	$85$
$\chi(n)$	$-1$	$1$	$1$

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
$3$	− 5.29150i	− 1.76383i	−0.471405	−	0.881917i	$-0.656253\pi$
$3$	− 5.29150i	− 1.76383i	0.471405	−	0.881917i	$-0.343747\pi$
$4$	0	0
$5$	8.00000	1.60000	0.800000	−	0.600000i	$-0.204833\pi$
$5$	8.00000	1.60000	0.800000	+	0.600000i	$0.204833\pi$
$6$	0	0
$7$	2.64575i	0.377964i
$7$	2.64575i	0.377964i
$8$	0	0
$9$	−19.0000	−2.11111
$10$	0	0
$11$	− 10.5830i	− 0.962091i	−0.876696	−	0.481046i	$-0.840257\pi$
$11$	− 10.5830i	− 0.962091i	0.876696	−	0.481046i	$-0.159743\pi$
$12$	0	0
$13$	−4.00000	−0.307692	−0.153846	−	0.988095i	$-0.549166\pi$
$13$	−4.00000	−0.307692	−0.153846	+	0.988095i	$0.549166\pi$
$14$	0	0
$15$	− 42.3320i	− 2.82213i
$16$	0	0
$17$	−2.00000	−0.117647	−0.0588235	−	0.998268i	$-0.518735\pi$
$17$	−2.00000	−0.117647	−0.0588235	+	0.998268i	$0.518735\pi$
$18$	0	0
$19$	26.4575i	1.39250i	0.717799	+	0.696250i	$0.245150\pi$
$19$	26.4575i	1.39250i	−0.717799	+	0.696250i	$0.754850\pi$
$20$	0	0
$21$	14.0000	0.666667
$22$	0	0
$23$	21.1660i	0.920261i	0.887851	+	0.460131i	$0.152197\pi$
$23$	21.1660i	0.920261i	−0.887851	+	0.460131i	$0.847803\pi$
$24$	0	0
$25$	39.0000	1.56000
$26$	0	0
$27$	52.9150i	1.95982i
$28$	0	0
$29$	14.0000	0.482759	0.241379	−	0.970431i	$-0.422400\pi$
$29$	14.0000	0.482759	0.241379	+	0.970431i	$0.422400\pi$
$30$	0	0
$31$	− 31.7490i	− 1.02416i	−0.858937	−	0.512081i	$-0.828875\pi$
$31$	− 31.7490i	− 1.02416i	0.858937	−	0.512081i	$-0.171125\pi$
$32$	0	0
$33$	−56.0000	−1.69697
$34$	0	0
$35$	21.1660i	0.604743i
$36$	0	0
$37$	14.0000	0.378378	0.189189	−	0.981941i	$-0.439414\pi$
$37$	14.0000	0.378378	0.189189	+	0.981941i	$0.439414\pi$
$38$	0	0
$39$	21.1660i	0.542718i
$40$	0	0
$41$	46.0000	1.12195	0.560976	−	0.827832i	$-0.310426\pi$
$41$	46.0000	1.12195	0.560976	+	0.827832i	$0.310426\pi$
$42$	0	0
$43$	10.5830i	0.246116i	0.992399	+	0.123058i	$0.0392702\pi$
$43$	10.5830i	0.246116i	−0.992399	+	0.123058i	$0.960730\pi$
$44$	0	0
$45$	−152.000	−3.37778
$46$	0	0
$47$	31.7490i	0.675511i	0.941234	+	0.337755i	$0.109668\pi$
$47$	31.7490i	0.675511i	−0.941234	+	0.337755i	$0.890332\pi$
$48$	0	0
$49$	−7.00000	−0.142857
$50$	0	0
$51$	10.5830i	0.207510i
$52$	0	0
$53$	−22.0000	−0.415094	−0.207547	−	0.978225i	$-0.566548\pi$
$53$	−22.0000	−0.415094	−0.207547	+	0.978225i	$0.566548\pi$
$54$	0	0
$55$	− 84.6640i	− 1.53935i
$56$	0	0
$57$	140.000	2.45614
$58$	0	0
$59$	89.9555i	1.52467i	0.647182	+	0.762335i	$0.275947\pi$
$59$	89.9555i	1.52467i	−0.647182	+	0.762335i	$0.724053\pi$
$60$	0	0
$61$	48.0000	0.786885	0.393443	−	0.919349i	$-0.371284\pi$
$61$	48.0000	0.786885	0.393443	+	0.919349i	$0.371284\pi$
$62$	0	0
$63$	− 50.2693i	− 0.797925i
$64$	0	0
$65$	−32.0000	−0.492308
$66$	0	0
$67$	− 63.4980i	− 0.947732i	−0.880597	−	0.473866i	$-0.842858\pi$
$67$	− 63.4980i	− 0.947732i	0.880597	−	0.473866i	$-0.157142\pi$
$68$	0	0
$69$	112.000	1.62319
$70$	0	0
$71$	84.6640i	1.19245i	0.802817	+	0.596226i	$0.203334\pi$
$71$	84.6640i	1.19245i	−0.802817	+	0.596226i	$0.796666\pi$
$72$	0	0
$73$	−110.000	−1.50685	−0.753425	−	0.657534i	$-0.771599\pi$
$73$	−110.000	−1.50685	−0.753425	+	0.657534i	$0.771599\pi$
$74$	0	0
$75$	− 206.369i	− 2.75158i
$76$	0	0
$77$	28.0000	0.363636
$78$	0	0
$79$	− 126.996i	− 1.60755i	−0.594937	−	0.803773i	$-0.702823\pi$
$79$	− 126.996i	− 1.60755i	0.594937	−	0.803773i	$-0.297177\pi$
$80$	0	0
$81$	109.000	1.34568
$82$	0	0
$83$	− 37.0405i	− 0.446271i	−0.974787	−	0.223136i	$-0.928371\pi$
$83$	− 37.0405i	− 0.446271i	0.974787	−	0.223136i	$-0.0716293\pi$
$84$	0	0
$85$	−16.0000	−0.188235
$86$	0	0
$87$	− 74.0810i	− 0.851506i
$88$	0	0
$89$	−134.000	−1.50562	−0.752809	−	0.658239i	$-0.771302\pi$
$89$	−134.000	−1.50562	−0.752809	+	0.658239i	$0.771302\pi$
$90$	0	0
$91$	− 10.5830i	− 0.116297i
$92$	0	0
$93$	−168.000	−1.80645
$94$	0	0
$95$	211.660i	2.22800i
$96$	0	0
$97$	−178.000	−1.83505	−0.917526	−	0.397676i	$-0.869817\pi$
$97$	−178.000	−1.83505	−0.917526	+	0.397676i	$0.869817\pi$
$98$	0	0
$99$	201.077i	2.03108i
$100$	0	0
$101$	40.0000	0.396040	0.198020	−	0.980198i	$-0.436549\pi$
$101$	40.0000	0.396040	0.198020	+	0.980198i	$0.436549\pi$
$102$	0	0
$103$	10.5830i	0.102748i	0.998679	+	0.0513738i	$0.0163600\pi$
$103$	10.5830i	0.102748i	−0.998679	+	0.0513738i	$0.983640\pi$
$104$	0	0
$105$	112.000	1.06667
$106$	0	0
$107$	− 84.6640i	− 0.791253i	−0.918412	−	0.395626i	$-0.870528\pi$
$107$	− 84.6640i	− 0.791253i	0.918412	−	0.395626i	$-0.129472\pi$
$108$	0	0
$109$	182.000	1.66972	0.834862	−	0.550459i	$-0.185547\pi$
$109$	182.000	1.66972	0.834862	+	0.550459i	$0.185547\pi$
$110$	0	0
$111$	− 74.0810i	− 0.667397i
$112$	0	0
$113$	−58.0000	−0.513274	−0.256637	−	0.966508i	$-0.582615\pi$
$113$	−58.0000	−0.513274	−0.256637	+	0.966508i	$0.582615\pi$
$114$	0	0
$115$	169.328i	1.47242i
$116$	0	0
$117$	76.0000	0.649573
$118$	0	0
$119$	− 5.29150i	− 0.0444664i
$120$	0	0
$121$	9.00000	0.0743802
$122$	0	0
$123$	− 243.409i	− 1.97894i
$124$	0	0
$125$	112.000	0.896000
$126$	0	0
$127$	− 84.6640i	− 0.666646i	−0.942813	−	0.333323i	$-0.891830\pi$
$127$	− 84.6640i	− 0.666646i	0.942813	−	0.333323i	$-0.108170\pi$
$128$	0	0
$129$	56.0000	0.434109
$130$	0	0
$131$	153.454i	1.17140i	0.810527	+	0.585701i	$0.199181\pi$
$131$	153.454i	1.17140i	−0.810527	+	0.585701i	$0.800819\pi$
$132$	0	0
$133$	−70.0000	−0.526316
$134$	0	0
$135$	423.320i	3.13571i
$136$	0	0
$137$	−70.0000	−0.510949	−0.255474	−	0.966816i	$-0.582232\pi$
$137$	−70.0000	−0.510949	−0.255474	+	0.966816i	$0.582232\pi$
$138$	0	0
$139$	121.705i	0.875572i	0.899079	+	0.437786i	$0.144237\pi$
$139$	121.705i	0.875572i	−0.899079	+	0.437786i	$0.855763\pi$
$140$	0	0
$141$	168.000	1.19149
$142$	0	0
$143$	42.3320i	0.296028i
$144$	0	0
$145$	112.000	0.772414
$146$	0	0
$147$	37.0405i	0.251976i
$148$	0	0
$149$	−158.000	−1.06040	−0.530201	−	0.847872i	$-0.677884\pi$
$149$	−158.000	−1.06040	−0.530201	+	0.847872i	$0.677884\pi$
$150$	0	0
$151$	169.328i	1.12138i	0.828026	+	0.560689i	$0.189464\pi$
$151$	169.328i	1.12138i	−0.828026	+	0.560689i	$0.810536\pi$
$152$	0	0
$153$	38.0000	0.248366
$154$	0	0
$155$	− 253.992i	− 1.63866i
$156$	0	0
$157$	−180.000	−1.14650	−0.573248	−	0.819382i	$-0.694317\pi$
$157$	−180.000	−1.14650	−0.573248	+	0.819382i	$0.694317\pi$
$158$	0	0
$159$	116.413i	0.732158i
$160$	0	0
$161$	−56.0000	−0.347826
$162$	0	0
$163$	− 201.077i	− 1.23360i	−0.787119	−	0.616801i	$-0.788428\pi$
$163$	− 201.077i	− 1.23360i	0.787119	−	0.616801i	$-0.211572\pi$
$164$	0	0
$165$	−448.000	−2.71515
$166$	0	0
$167$	201.077i	1.20405i	0.798476	+	0.602027i	$0.205640\pi$
$167$	201.077i	1.20405i	−0.798476	+	0.602027i	$0.794360\pi$
$168$	0	0
$169$	−153.000	−0.905325
$170$	0	0
$171$	− 502.693i	− 2.93972i
$172$	0	0
$173$	36.0000	0.208092	0.104046	−	0.994572i	$-0.466821\pi$
$173$	36.0000	0.208092	0.104046	+	0.994572i	$0.466821\pi$
$174$	0	0
$175$	103.184i	0.589625i
$176$	0	0
$177$	476.000	2.68927
$178$	0	0
$179$	− 42.3320i	− 0.236492i	−0.992984	−	0.118246i	$-0.962273\pi$
$179$	− 42.3320i	− 0.236492i	0.992984	−	0.118246i	$-0.0377271\pi$
$180$	0	0
$181$	−108.000	−0.596685	−0.298343	−	0.954459i	$-0.596434\pi$
$181$	−108.000	−0.596685	−0.298343	+	0.954459i	$0.596434\pi$
$182$	0	0
$183$	− 253.992i	− 1.38794i
$184$	0	0
$185$	112.000	0.605405
$186$	0	0
$187$	21.1660i	0.113187i
$188$	0	0
$189$	−140.000	−0.740741
$190$	0	0
$191$	− 253.992i	− 1.32980i	−0.746932	−	0.664901i	$-0.768474\pi$
$191$	− 253.992i	− 1.32980i	0.746932	−	0.664901i	$-0.231526\pi$
$192$	0	0
$193$	174.000	0.901554	0.450777	−	0.892636i	$-0.351147\pi$
$193$	174.000	0.901554	0.450777	+	0.892636i	$0.351147\pi$
$194$	0	0
$195$	169.328i	0.868349i
$196$	0	0
$197$	114.000	0.578680	0.289340	−	0.957226i	$-0.406564\pi$
$197$	114.000	0.578680	0.289340	+	0.957226i	$0.406564\pi$
$198$	0	0
$199$	− 10.5830i	− 0.0531809i	−0.999646	−	0.0265905i	$-0.991535\pi$
$199$	− 10.5830i	− 0.0531809i	0.999646	−	0.0265905i	$-0.00846501\pi$
$200$	0	0
$201$	−336.000	−1.67164
$202$	0	0
$203$	37.0405i	0.182466i
$204$	0	0
$205$	368.000	1.79512
$206$	0	0
$207$	− 402.154i	− 1.94277i
$208$	0	0
$209$	280.000	1.33971
$210$	0	0
$211$	− 359.822i	− 1.70532i	−0.522468	−	0.852659i	$-0.674988\pi$
$211$	− 359.822i	− 1.70532i	0.522468	−	0.852659i	$-0.325012\pi$
$212$	0	0
$213$	448.000	2.10329
$214$	0	0
$215$	84.6640i	0.393786i
$216$	0	0
$217$	84.0000	0.387097
$218$	0	0
$219$	582.065i	2.65783i
$220$	0	0
$221$	8.00000	0.0361991
$222$	0	0
$223$	− 232.826i	− 1.04406i	−0.852926	−	0.522032i	$-0.825174\pi$
$223$	− 232.826i	− 1.04406i	0.852926	−	0.522032i	$-0.174826\pi$
$224$	0	0
$225$	−741.000	−3.29333
$226$	0	0
$227$	195.786i	0.862492i	0.902234	+	0.431246i	$0.141926\pi$
$227$	195.786i	0.862492i	−0.902234	+	0.431246i	$0.858074\pi$
$228$	0	0
$229$	92.0000	0.401747	0.200873	−	0.979617i	$-0.435622\pi$
$229$	92.0000	0.401747	0.200873	+	0.979617i	$0.435622\pi$
$230$	0	0
$231$	− 148.162i	− 0.641394i
$232$	0	0
$233$	−14.0000	−0.0600858	−0.0300429	−	0.999549i	$-0.509564\pi$
$233$	−14.0000	−0.0600858	−0.0300429	+	0.999549i	$0.509564\pi$
$234$	0	0
$235$	253.992i	1.08082i
$236$	0	0
$237$	−672.000	−2.83544
$238$	0	0
$239$	148.162i	0.619925i	0.950749	+	0.309962i	$0.100317\pi$
$239$	148.162i	0.619925i	−0.950749	+	0.309962i	$0.899683\pi$
$240$	0	0
$241$	334.000	1.38589	0.692946	−	0.720989i	$-0.256312\pi$
$241$	334.000	1.38589	0.692946	+	0.720989i	$0.256312\pi$
$242$	0	0
$243$	− 100.539i	− 0.413739i
$244$	0	0
$245$	−56.0000	−0.228571
$246$	0	0
$247$	− 105.830i	− 0.428462i
$248$	0	0
$249$	−196.000	−0.787149
$250$	0	0
$251$	− 428.612i	− 1.70762i	−0.520588	−	0.853808i	$-0.674287\pi$
$251$	− 428.612i	− 1.70762i	0.520588	−	0.853808i	$-0.325713\pi$
$252$	0	0
$253$	224.000	0.885375
$254$	0	0
$255$	84.6640i	0.332016i
$256$	0	0
$257$	−62.0000	−0.241245	−0.120623	−	0.992698i	$-0.538489\pi$
$257$	−62.0000	−0.241245	−0.120623	+	0.992698i	$0.538489\pi$
$258$	0	0
$259$	37.0405i	0.143014i
$260$	0	0
$261$	−266.000	−1.01916
$262$	0	0
$263$	− 211.660i	− 0.804791i	−0.915466	−	0.402396i	$-0.868178\pi$
$263$	− 211.660i	− 0.804791i	0.915466	−	0.402396i	$-0.131822\pi$
$264$	0	0
$265$	−176.000	−0.664151
$266$	0	0
$267$	709.061i	2.65566i
$268$	0	0
$269$	292.000	1.08550	0.542751	−	0.839894i	$-0.317383\pi$
$269$	292.000	1.08550	0.542751	+	0.839894i	$0.317383\pi$
$270$	0	0
$271$	− 42.3320i	− 0.156207i	−0.996945	−	0.0781034i	$-0.975114\pi$
$271$	− 42.3320i	− 0.156207i	0.996945	−	0.0781034i	$-0.0248864\pi$
$272$	0	0
$273$	−56.0000	−0.205128
$274$	0	0
$275$	− 412.737i	− 1.50086i
$276$	0	0
$277$	−454.000	−1.63899	−0.819495	−	0.573087i	$-0.805746\pi$
$277$	−454.000	−1.63899	−0.819495	+	0.573087i	$0.805746\pi$
$278$	0	0
$279$	603.231i	2.16212i
$280$	0	0
$281$	−350.000	−1.24555	−0.622776	−	0.782400i	$-0.713995\pi$
$281$	−350.000	−1.24555	−0.622776	+	0.782400i	$0.713995\pi$
$282$	0	0
$283$	174.620i	0.617030i	0.951219	+	0.308515i	$0.0998321\pi$
$283$	174.620i	0.617030i	−0.951219	+	0.308515i	$0.900168\pi$
$284$	0	0
$285$	1120.00	3.92982
$286$	0	0
$287$	121.705i	0.424058i
$288$	0	0
$289$	−285.000	−0.986159
$290$	0	0
$291$	941.887i	3.23673i
$292$	0	0
$293$	416.000	1.41980	0.709898	−	0.704305i	$-0.248741\pi$
$293$	416.000	1.41980	0.709898	+	0.704305i	$0.248741\pi$
$294$	0	0
$295$	719.644i	2.43947i
$296$	0	0
$297$	560.000	1.88552
$298$	0	0
$299$	− 84.6640i	− 0.283157i
$300$	0	0
$301$	−28.0000	−0.0930233
$302$	0	0
$303$	− 211.660i	− 0.698548i
$304$	0	0
$305$	384.000	1.25902
$306$	0	0
$307$	− 269.867i	− 0.879044i	−0.898232	−	0.439522i	$-0.855148\pi$
$307$	− 269.867i	− 0.879044i	0.898232	−	0.439522i	$-0.144852\pi$
$308$	0	0
$309$	56.0000	0.181230
$310$	0	0
$311$	63.4980i	0.204174i	0.994775	+	0.102087i	$0.0325520\pi$
$311$	63.4980i	0.204174i	−0.994775	+	0.102087i	$0.967448\pi$
$312$	0	0
$313$	78.0000	0.249201	0.124601	−	0.992207i	$-0.460235\pi$
$313$	78.0000	0.249201	0.124601	+	0.992207i	$0.460235\pi$
$314$	0	0
$315$	− 402.154i	− 1.27668i
$316$	0	0
$317$	−318.000	−1.00315	−0.501577	−	0.865113i	$-0.667247\pi$
$317$	−318.000	−1.00315	−0.501577	+	0.865113i	$0.667247\pi$
$318$	0	0
$319$	− 148.162i	− 0.464458i
$320$	0	0
$321$	−448.000	−1.39564
$322$	0	0
$323$	− 52.9150i	− 0.163824i
$324$	0	0
$325$	−156.000	−0.480000
$326$	0	0
$327$	− 963.053i	− 2.94512i
$328$	0	0
$329$	−84.0000	−0.255319
$330$	0	0
$331$	306.907i	0.927212i	0.886041	+	0.463606i	$0.153445\pi$
$331$	306.907i	0.927212i	−0.886041	+	0.463606i	$0.846555\pi$
$332$	0	0
$333$	−266.000	−0.798799
$334$	0	0
$335$	− 507.984i	− 1.51637i
$336$	0	0
$337$	238.000	0.706231	0.353116	−	0.935580i	$-0.385122\pi$
$337$	238.000	0.706231	0.353116	+	0.935580i	$0.385122\pi$
$338$	0	0
$339$	306.907i	0.905331i
$340$	0	0
$341$	−336.000	−0.985337
$342$	0	0
$343$	− 18.5203i	− 0.0539949i
$344$	0	0
$345$	896.000	2.59710
$346$	0	0
$347$	− 285.741i	− 0.823462i	−0.911306	−	0.411731i	$-0.864924\pi$
$347$	− 285.741i	− 0.823462i	0.911306	−	0.411731i	$-0.135076\pi$
$348$	0	0
$349$	4.00000	0.0114613	0.00573066	−	0.999984i	$-0.498176\pi$
$349$	4.00000	0.0114613	0.00573066	+	0.999984i	$0.498176\pi$
$350$	0	0
$351$	− 211.660i	− 0.603020i
$352$	0	0
$353$	−110.000	−0.311615	−0.155807	−	0.987787i	$-0.549798\pi$
$353$	−110.000	−0.311615	−0.155807	+	0.987787i	$0.549798\pi$
$354$	0	0
$355$	677.312i	1.90792i
$356$	0	0
$357$	−28.0000	−0.0784314
$358$	0	0
$359$	− 84.6640i	− 0.235833i	−0.993024	−	0.117916i	$-0.962378\pi$
$359$	− 84.6640i	− 0.235833i	0.993024	−	0.117916i	$-0.0376215\pi$
$360$	0	0
$361$	−339.000	−0.939058
$362$	0	0
$363$	− 47.6235i	− 0.131194i
$364$	0	0
$365$	−880.000	−2.41096
$366$	0	0
$367$	486.818i	1.32648i	0.748407	+	0.663240i	$0.230819\pi$
$367$	486.818i	1.32648i	−0.748407	+	0.663240i	$0.769181\pi$
$368$	0	0
$369$	−874.000	−2.36856
$370$	0	0
$371$	− 58.2065i	− 0.156891i
$372$	0	0
$373$	290.000	0.777480	0.388740	−	0.921348i	$-0.372910\pi$
$373$	290.000	0.777480	0.388740	+	0.921348i	$0.372910\pi$
$374$	0	0
$375$	− 592.648i	− 1.58040i
$376$	0	0
$377$	−56.0000	−0.148541
$378$	0	0
$379$	− 95.2470i	− 0.251311i	−0.992074	−	0.125656i	$-0.959897\pi$
$379$	− 95.2470i	− 0.251311i	0.992074	−	0.125656i	$-0.0401035\pi$
$380$	0	0
$381$	−448.000	−1.17585
$382$	0	0
$383$	− 264.575i	− 0.690797i	−0.938456	−	0.345398i	$-0.887744\pi$
$383$	− 264.575i	− 0.690797i	0.938456	−	0.345398i	$-0.112256\pi$
$384$	0	0
$385$	224.000	0.581818
$386$	0	0
$387$	− 201.077i	− 0.519579i
$388$	0	0
$389$	574.000	1.47558	0.737789	−	0.675031i	$-0.235870\pi$
$389$	574.000	1.47558	0.737789	+	0.675031i	$0.235870\pi$
$390$	0	0
$391$	− 42.3320i	− 0.108266i
$392$	0	0
$393$	812.000	2.06616
$394$	0	0
$395$	− 1015.97i	− 2.57207i
$396$	0	0
$397$	−428.000	−1.07809	−0.539043	−	0.842278i	$-0.681214\pi$
$397$	−428.000	−1.07809	−0.539043	+	0.842278i	$0.681214\pi$
$398$	0	0
$399$	370.405i	0.928334i
$400$	0	0
$401$	238.000	0.593516	0.296758	−	0.954953i	$-0.404094\pi$
$401$	238.000	0.593516	0.296758	+	0.954953i	$0.404094\pi$
$402$	0	0
$403$	126.996i	0.315127i
$404$	0	0
$405$	872.000	2.15309
$406$	0	0
$407$	− 148.162i	− 0.364035i
$408$	0	0
$409$	198.000	0.484108	0.242054	−	0.970263i	$-0.422179\pi$
$409$	198.000	0.484108	0.242054	+	0.970263i	$0.422179\pi$
$410$	0	0
$411$	370.405i	0.901229i
$412$	0	0
$413$	−238.000	−0.576271
$414$	0	0
$415$	− 296.324i	− 0.714034i
$416$	0	0
$417$	644.000	1.54436
$418$	0	0
$419$	− 68.7895i	− 0.164175i	−0.996625	−	0.0820877i	$-0.973841\pi$
$419$	− 68.7895i	− 0.164175i	0.996625	−	0.0820877i	$-0.0261588\pi$
$420$	0	0
$421$	170.000	0.403800	0.201900	−	0.979406i	$-0.435288\pi$
$421$	170.000	0.403800	0.201900	+	0.979406i	$0.435288\pi$
$422$	0	0
$423$	− 603.231i	− 1.42608i
$424$	0	0
$425$	−78.0000	−0.183529
$426$	0	0
$427$	126.996i	0.297415i
$428$	0	0
$429$	224.000	0.522145
$430$	0	0
$431$	613.814i	1.42416i	0.702097	+	0.712082i	$0.252247\pi$
$431$	613.814i	1.42416i	−0.702097	+	0.712082i	$0.747753\pi$
$432$	0	0
$433$	−74.0000	−0.170901	−0.0854503	−	0.996342i	$-0.527233\pi$
$433$	−74.0000	−0.170901	−0.0854503	+	0.996342i	$0.527233\pi$
$434$	0	0
$435$	− 592.648i	− 1.36241i
$436$	0	0
$437$	−560.000	−1.28146
$438$	0	0
$439$	190.494i	0.433927i	0.976180	+	0.216964i	$0.0696153\pi$
$439$	190.494i	0.433927i	−0.976180	+	0.216964i	$0.930385\pi$
$440$	0	0
$441$	133.000	0.301587
$442$	0	0
$443$	571.482i	1.29003i	0.764171	+	0.645014i	$0.223148\pi$
$443$	571.482i	1.29003i	−0.764171	+	0.645014i	$0.776852\pi$
$444$	0	0
$445$	−1072.00	−2.40899
$446$	0	0
$447$	836.057i	1.87037i
$448$	0	0
$449$	306.000	0.681514	0.340757	−	0.940151i	$-0.389317\pi$
$449$	306.000	0.681514	0.340757	+	0.940151i	$0.389317\pi$
$450$	0	0
$451$	− 486.818i	− 1.07942i
$452$	0	0
$453$	896.000	1.97792
$454$	0	0
$455$	− 84.6640i	− 0.186075i
$456$	0	0
$457$	−354.000	−0.774617	−0.387309	−	0.921950i	$-0.626595\pi$
$457$	−354.000	−0.774617	−0.387309	+	0.921950i	$0.626595\pi$
$458$	0	0
$459$	− 105.830i	− 0.230567i
$460$	0	0
$461$	508.000	1.10195	0.550976	−	0.834521i	$-0.314256\pi$
$461$	508.000	1.10195	0.550976	+	0.834521i	$0.314256\pi$
$462$	0	0
$463$	− 402.154i	− 0.868584i	−0.900772	−	0.434292i	$-0.856999\pi$
$463$	− 402.154i	− 0.868584i	0.900772	−	0.434292i	$-0.143001\pi$
$464$	0	0
$465$	−1344.00	−2.89032
$466$	0	0
$467$	5.29150i	0.0113308i	0.999984	+	0.00566542i	$0.00180337\pi$
$467$	5.29150i	0.0113308i	−0.999984	+	0.00566542i	$0.998197\pi$
$468$	0	0
$469$	168.000	0.358209
$470$	0	0
$471$	952.470i	2.02223i
$472$	0	0
$473$	112.000	0.236786
$474$	0	0
$475$	1031.84i	2.17230i
$476$	0	0
$477$	418.000	0.876310
$478$	0	0
$479$	941.887i	1.96636i	0.182633	+	0.983181i	$0.441538\pi$
$479$	941.887i	1.96636i	−0.182633	+	0.983181i	$0.558462\pi$
$480$	0	0
$481$	−56.0000	−0.116424
$482$	0	0
$483$	296.324i	0.613508i
$484$	0	0
$485$	−1424.00	−2.93608
$486$	0	0
$487$	− 867.806i	− 1.78194i	−0.454059	−	0.890972i	$-0.650024\pi$
$487$	− 867.806i	− 1.78194i	0.454059	−	0.890972i	$-0.349976\pi$
$488$	0	0
$489$	−1064.00	−2.17587
$490$	0	0
$491$	− 867.806i	− 1.76743i	−0.468029	−	0.883713i	$-0.655036\pi$
$491$	− 867.806i	− 1.76743i	0.468029	−	0.883713i	$-0.344964\pi$
$492$	0	0
$493$	−28.0000	−0.0567951
$494$	0	0
$495$	1608.62i	3.24973i
$496$	0	0
$497$	−224.000	−0.450704
$498$	0	0
$499$	− 148.162i	− 0.296918i	−0.988919	−	0.148459i	$-0.952569\pi$
$499$	− 148.162i	− 0.296918i	0.988919	−	0.148459i	$-0.0474313\pi$
$500$	0	0
$501$	1064.00	2.12375
$502$	0	0
$503$	634.980i	1.26239i	0.775626	+	0.631193i	$0.217434\pi$
$503$	634.980i	1.26239i	−0.775626	+	0.631193i	$0.782566\pi$
$504$	0	0
$505$	320.000	0.633663
$506$	0	0
$507$	809.600i	1.59684i
$508$	0	0
$509$	580.000	1.13949	0.569745	−	0.821822i	$-0.307042\pi$
$509$	580.000	1.13949	0.569745	+	0.821822i	$0.307042\pi$
$510$	0	0
$511$	− 291.033i	− 0.569536i
$512$	0	0
$513$	−1400.00	−2.72904
$514$	0	0
$515$	84.6640i	0.164396i
$516$	0	0
$517$	336.000	0.649903
$518$	0	0
$519$	− 190.494i	− 0.367041i
$520$	0	0
$521$	478.000	0.917466	0.458733	−	0.888574i	$-0.348303\pi$
$521$	478.000	0.917466	0.458733	+	0.888574i	$0.348303\pi$
$522$	0	0
$523$	− 608.523i	− 1.16352i	−0.813359	−	0.581762i	$-0.802364\pi$
$523$	− 608.523i	− 1.16352i	0.813359	−	0.581762i	$-0.197636\pi$
$524$	0	0
$525$	546.000	1.04000
$526$	0	0
$527$	63.4980i	0.120490i
$528$	0	0
$529$	81.0000	0.153119
$530$	0	0
$531$	− 1709.16i	− 3.21875i
$532$	0	0
$533$	−184.000	−0.345216
$534$	0	0
$535$	− 677.312i	− 1.26600i
$536$	0	0
$537$	−224.000	−0.417132
$538$	0	0
$539$	74.0810i	0.137442i
$540$	0	0
$541$	514.000	0.950092	0.475046	−	0.879961i	$-0.342431\pi$
$541$	514.000	0.950092	0.475046	+	0.879961i	$0.342431\pi$
$542$	0	0
$543$	571.482i	1.05245i
$544$	0	0
$545$	1456.00	2.67156
$546$	0	0
$547$	243.409i	0.444989i	0.974934	+	0.222495i	$0.0714200\pi$
$547$	243.409i	0.444989i	−0.974934	+	0.222495i	$0.928580\pi$
$548$	0	0
$549$	−912.000	−1.66120
$550$	0	0
$551$	370.405i	0.672242i
$552$	0	0
$553$	336.000	0.607595
$554$	0	0
$555$	− 592.648i	− 1.06783i
$556$	0	0
$557$	946.000	1.69838	0.849192	−	0.528084i	$-0.177089\pi$
$557$	946.000	1.69838	0.849192	+	0.528084i	$0.177089\pi$
$558$	0	0
$559$	− 42.3320i	− 0.0757281i
$560$	0	0
$561$	112.000	0.199643
$562$	0	0
$563$	− 428.612i	− 0.761300i	−0.924719	−	0.380650i	$-0.875700\pi$
$563$	− 428.612i	− 0.761300i	0.924719	−	0.380650i	$-0.124300\pi$
$564$	0	0
$565$	−464.000	−0.821239
$566$	0	0
$567$	288.387i	0.508619i
$568$	0	0
$569$	126.000	0.221441	0.110721	−	0.993852i	$-0.464684\pi$
$569$	126.000	0.221441	0.110721	+	0.993852i	$0.464684\pi$
$570$	0	0
$571$	328.073i	0.574559i	0.957847	+	0.287279i	$0.0927508\pi$
$571$	328.073i	0.574559i	−0.957847	+	0.287279i	$0.907249\pi$
$572$	0	0
$573$	−1344.00	−2.34555
$574$	0	0
$575$	825.474i	1.43561i
$576$	0	0
$577$	530.000	0.918544	0.459272	−	0.888296i	$-0.348110\pi$
$577$	530.000	0.918544	0.459272	+	0.888296i	$0.348110\pi$
$578$	0	0
$579$	− 920.721i	− 1.59019i
$580$	0	0
$581$	98.0000	0.168675
$582$	0	0
$583$	232.826i	0.399359i
$584$	0	0
$585$	608.000	1.03932
$586$	0	0
$587$	1116.51i	1.90206i	0.309104	+	0.951028i	$0.399971\pi$
$587$	1116.51i	1.90206i	−0.309104	+	0.951028i	$0.600029\pi$
$588$	0	0
$589$	840.000	1.42615
$590$	0	0
$591$	− 603.231i	− 1.02070i
$592$	0	0
$593$	−694.000	−1.17032	−0.585160	−	0.810918i	$-0.698968\pi$
$593$	−694.000	−1.17032	−0.585160	+	0.810918i	$0.698968\pi$
$594$	0	0
$595$	− 42.3320i	− 0.0711463i
$596$	0	0
$597$	−56.0000	−0.0938023
$598$	0	0
$599$	− 63.4980i	− 0.106007i	−0.998594	−	0.0530034i	$-0.983121\pi$
$599$	− 63.4980i	− 0.106007i	0.998594	−	0.0530034i	$-0.0168794\pi$
$600$	0	0
$601$	−262.000	−0.435940	−0.217970	−	0.975955i	$-0.569944\pi$
$601$	−262.000	−0.435940	−0.217970	+	0.975955i	$0.569944\pi$
$602$	0	0
$603$	1206.46i	2.00077i
$604$	0	0
$605$	72.0000	0.119008
$606$	0	0
$607$	− 973.636i	− 1.60401i	−0.597315	−	0.802007i	$-0.703766\pi$
$607$	− 973.636i	− 1.60401i	0.597315	−	0.802007i	$-0.296234\pi$
$608$	0	0
$609$	196.000	0.321839
$610$	0	0
$611$	− 126.996i	− 0.207850i
$612$	0	0
$613$	−434.000	−0.707993	−0.353997	−	0.935247i	$-0.615178\pi$
$613$	−434.000	−0.707993	−0.353997	+	0.935247i	$0.615178\pi$
$614$	0	0
$615$	− 1947.27i	− 3.16630i
$616$	0	0
$617$	−1106.00	−1.79254	−0.896272	−	0.443504i	$-0.853735\pi$
$617$	−1106.00	−1.79254	−0.896272	+	0.443504i	$0.853735\pi$
$618$	0	0
$619$	− 37.0405i	− 0.0598393i	−0.999552	−	0.0299196i	$-0.990475\pi$
$619$	− 37.0405i	− 0.0598393i	0.999552	−	0.0299196i	$-0.00952514\pi$
$620$	0	0
$621$	−1120.00	−1.80354
$622$	0	0
$623$	− 354.531i	− 0.569070i
$624$	0	0
$625$	−79.0000	−0.126400
$626$	0	0
$627$	− 1481.62i	− 2.36303i
$628$	0	0
$629$	−28.0000	−0.0445151
$630$	0	0
$631$	− 126.996i	− 0.201262i	−0.994924	−	0.100631i	$-0.967914\pi$
$631$	− 126.996i	− 0.201262i	0.994924	−	0.100631i	$-0.0320861\pi$
$632$	0	0
$633$	−1904.00	−3.00790
$634$	0	0
$635$	− 677.312i	− 1.06663i
$636$	0	0
$637$	28.0000	0.0439560
$638$	0	0
$639$	− 1608.62i	− 2.51740i
$640$	0	0
$641$	350.000	0.546022	0.273011	−	0.962011i	$-0.411980\pi$
$641$	350.000	0.546022	0.273011	+	0.962011i	$0.411980\pi$
$642$	0	0
$643$	− 89.9555i	− 0.139900i	−0.997551	−	0.0699499i	$-0.977716\pi$
$643$	− 89.9555i	− 0.139900i	0.997551	−	0.0699499i	$-0.0222839\pi$
$644$	0	0
$645$	448.000	0.694574
$646$	0	0
$647$	709.061i	1.09592i	0.836504	+	0.547961i	$0.184596\pi$
$647$	709.061i	1.09592i	−0.836504	+	0.547961i	$0.815404\pi$
$648$	0	0
$649$	952.000	1.46687
$650$	0	0
$651$	− 444.486i	− 0.682775i
$652$	0	0
$653$	70.0000	0.107198	0.0535988	−	0.998563i	$-0.482931\pi$
$653$	70.0000	0.107198	0.0535988	+	0.998563i	$0.482931\pi$
$654$	0	0
$655$	1227.63i	1.87424i
$656$	0	0
$657$	2090.00	3.18113
$658$	0	0
$659$	751.393i	1.14020i	0.821574	+	0.570101i	$0.193096\pi$
$659$	751.393i	1.14020i	−0.821574	+	0.570101i	$0.806904\pi$
$660$	0	0
$661$	−432.000	−0.653555	−0.326778	−	0.945101i	$-0.605963\pi$
$661$	−432.000	−0.653555	−0.326778	+	0.945101i	$0.605963\pi$
$662$	0	0
$663$	− 42.3320i	− 0.0638492i
$664$	0	0
$665$	−560.000	−0.842105
$666$	0	0
$667$	296.324i	0.444264i
$668$	0	0
$669$	−1232.00	−1.84155
$670$	0	0
$671$	− 507.984i	− 0.757056i
$672$	0	0
$673$	490.000	0.728083	0.364042	−	0.931383i	$-0.381397\pi$
$673$	490.000	0.728083	0.364042	+	0.931383i	$0.381397\pi$
$674$	0	0
$675$	2063.69i	3.05731i
$676$	0	0
$677$	−636.000	−0.939439	−0.469719	−	0.882816i	$-0.655645\pi$
$677$	−636.000	−0.939439	−0.469719	+	0.882816i	$0.655645\pi$
$678$	0	0
$679$	− 470.944i	− 0.693584i
$680$	0	0
$681$	1036.00	1.52129
$682$	0	0
$683$	− 529.150i	− 0.774744i	−0.921923	−	0.387372i	$-0.873383\pi$
$683$	− 529.150i	− 0.774744i	0.921923	−	0.387372i	$-0.126617\pi$
$684$	0	0
$685$	−560.000	−0.817518
$686$	0	0
$687$	− 486.818i	− 0.708615i
$688$	0	0
$689$	88.0000	0.127721
$690$	0	0
$691$	492.110i	0.712170i	0.934453	+	0.356085i	$0.115889\pi$
$691$	492.110i	0.712170i	−0.934453	+	0.356085i	$0.884111\pi$
$692$	0	0
$693$	−532.000	−0.767677
$694$	0	0
$695$	973.636i	1.40092i
$696$	0	0
$697$	−92.0000	−0.131994
$698$	0	0
$699$	74.0810i	0.105981i
$700$	0	0
$701$	1078.00	1.53780	0.768902	−	0.639367i	$-0.220804\pi$
$701$	1078.00	1.53780	0.768902	+	0.639367i	$0.220804\pi$
$702$	0	0
$703$	370.405i	0.526892i
$704$	0	0
$705$	1344.00	1.90638
$706$	0	0
$707$	105.830i	0.149689i
$708$	0	0
$709$	798.000	1.12553	0.562764	−	0.826617i	$-0.309738\pi$
$709$	798.000	1.12553	0.562764	+	0.826617i	$0.309738\pi$
$710$	0	0
$711$	2412.93i	3.39371i
$712$	0	0
$713$	672.000	0.942496
$714$	0	0
$715$	338.656i	0.473645i
$716$	0	0
$717$	784.000	1.09344
$718$	0	0
$719$	− 624.397i	− 0.868425i	−0.900811	−	0.434212i	$-0.857027\pi$
$719$	− 624.397i	− 0.868425i	0.900811	−	0.434212i	$-0.142973\pi$
$720$	0	0
$721$	−28.0000	−0.0388350
$722$	0	0
$723$	− 1767.36i	− 2.44448i
$724$	0	0
$725$	546.000	0.753103
$726$	0	0
$727$	− 243.409i	− 0.334813i	−0.985888	−	0.167407i	$-0.946461\pi$
$727$	− 243.409i	− 0.334813i	0.985888	−	0.167407i	$-0.0535392\pi$
$728$	0	0
$729$	449.000	0.615912
$730$	0	0
$731$	− 21.1660i	− 0.0289549i
$732$	0	0
$733$	736.000	1.00409	0.502046	−	0.864841i	$-0.332581\pi$
$733$	736.000	1.00409	0.502046	+	0.864841i	$0.332581\pi$
$734$	0	0
$735$	296.324i	0.403162i
$736$	0	0
$737$	−672.000	−0.911805
$738$	0	0
$739$	− 1386.37i	− 1.87601i	−0.346618	−	0.938007i	$-0.612670\pi$
$739$	− 1386.37i	− 1.87601i	0.346618	−	0.938007i	$-0.387330\pi$
$740$	0	0
$741$	−560.000	−0.755735
$742$	0	0
$743$	1354.62i	1.82318i	0.411098	+	0.911591i	$0.365146\pi$
$743$	1354.62i	1.82318i	−0.411098	+	0.911591i	$0.634854\pi$
$744$	0	0
$745$	−1264.00	−1.69664
$746$	0	0
$747$	703.770i	0.942128i
$748$	0	0
$749$	224.000	0.299065
$750$	0	0
$751$	1248.79i	1.66284i	0.555643	+	0.831421i	$0.312472\pi$
$751$	1248.79i	1.66284i	−0.555643	+	0.831421i	$0.687528\pi$
$752$	0	0
$753$	−2268.00	−3.01195
$754$	0	0
$755$	1354.62i	1.79420i
$756$	0	0
$757$	−1218.00	−1.60898	−0.804491	−	0.593964i	$-0.797562\pi$
$757$	−1218.00	−1.60898	−0.804491	+	0.593964i	$0.797562\pi$
$758$	0	0
$759$	− 1185.30i	− 1.56166i
$760$	0	0
$761$	−218.000	−0.286465	−0.143233	−	0.989689i	$-0.545750\pi$
$761$	−218.000	−0.286465	−0.143233	+	0.989689i	$0.545750\pi$
$762$	0	0
$763$	481.527i	0.631097i
$764$	0	0
$765$	304.000	0.397386
$766$	0	0
$767$	− 359.822i	− 0.469129i
$768$	0	0
$769$	−970.000	−1.26138	−0.630689	−	0.776036i	$-0.717228\pi$
$769$	−970.000	−1.26138	−0.630689	+	0.776036i	$0.717228\pi$
$770$	0	0
$771$	328.073i	0.425516i
$772$	0	0
$773$	800.000	1.03493	0.517464	−	0.855705i	$-0.326876\pi$
$773$	800.000	1.03493	0.517464	+	0.855705i	$0.326876\pi$
$774$	0	0
$775$	− 1238.21i	− 1.59769i
$776$	0	0
$777$	196.000	0.252252
$778$	0	0
$779$	1217.05i	1.56232i
$780$	0	0
$781$	896.000	1.14725
$782$	0	0
$783$	740.810i	0.946118i
$784$	0	0
$785$	−1440.00	−1.83439
$786$	0	0
$787$	− 841.349i	− 1.06906i	−0.845150	−	0.534529i	$-0.820489\pi$
$787$	− 841.349i	− 1.06906i	0.845150	−	0.534529i	$-0.179511\pi$
$788$	0	0
$789$	−1120.00	−1.41952
$790$	0	0
$791$	− 153.454i	− 0.193999i
$792$	0	0
$793$	−192.000	−0.242119
$794$	0	0
$795$	931.304i	1.17145i
$796$	0	0
$797$	1060.00	1.32999	0.664994	−	0.746849i	$-0.268434\pi$
$797$	1060.00	1.32999	0.664994	+	0.746849i	$0.268434\pi$
$798$	0	0
$799$	− 63.4980i	− 0.0794719i
$800$	0	0
$801$	2546.00	3.17853
$802$	0	0
$803$	1164.13i	1.44973i
$804$	0	0
$805$	−448.000	−0.556522
$806$	0	0
$807$	− 1545.12i	− 1.91465i
$808$	0	0
$809$	−370.000	−0.457355	−0.228677	−	0.973502i	$-0.573440\pi$
$809$	−370.000	−0.457355	−0.228677	+	0.973502i	$0.573440\pi$
$810$	0	0
$811$	− 650.855i	− 0.802534i	−0.915961	−	0.401267i	$-0.868570\pi$
$811$	− 650.855i	− 0.802534i	0.915961	−	0.401267i	$-0.131430\pi$
$812$	0	0
$813$	−224.000	−0.275523
$814$	0	0
$815$	− 1608.62i	− 1.97376i
$816$	0	0
$817$	−280.000	−0.342717
$818$	0	0
$819$	201.077i	0.245515i
$820$	0	0
$821$	−598.000	−0.728380	−0.364190	−	0.931325i	$-0.618654\pi$
$821$	−598.000	−0.728380	−0.364190	+	0.931325i	$0.618654\pi$
$822$	0	0
$823$	− 444.486i	− 0.540080i	−0.962849	−	0.270040i	$-0.912963\pi$
$823$	− 444.486i	− 0.540080i	0.962849	−	0.270040i	$-0.0870370\pi$
$824$	0	0
$825$	−2184.00	−2.64727
$826$	0	0
$827$	− 380.988i	− 0.460687i	−0.973109	−	0.230344i	$-0.926015\pi$
$827$	− 380.988i	− 0.460687i	0.973109	−	0.230344i	$-0.0739850\pi$
$828$	0	0
$829$	656.000	0.791315	0.395657	−	0.918398i	$-0.370517\pi$
$829$	656.000	0.791315	0.395657	+	0.918398i	$0.370517\pi$
$830$	0	0
$831$	2402.34i	2.89091i
$832$	0	0
$833$	14.0000	0.0168067
$834$	0	0
$835$	1608.62i	1.92649i
$836$	0	0
$837$	1680.00	2.00717
$838$	0	0
$839$	− 709.061i	− 0.845127i	−0.906333	−	0.422563i	$-0.861130\pi$
$839$	− 709.061i	− 0.845127i	0.906333	−	0.422563i	$-0.138870\pi$
$840$	0	0
$841$	−645.000	−0.766944
$842$	0	0
$843$	1852.03i	2.19695i
$844$	0	0
$845$	−1224.00	−1.44852
$846$	0	0
$847$	23.8118i	0.0281131i
$848$	0	0
$849$	924.000	1.08834
$850$	0	0
$851$	296.324i	0.348207i
$852$	0	0
$853$	−500.000	−0.586166	−0.293083	−	0.956087i	$-0.594681\pi$
$853$	−500.000	−0.586166	−0.293083	+	0.956087i	$0.594681\pi$
$854$	0	0
$855$	− 4021.54i	− 4.70356i
$856$	0	0
$857$	502.000	0.585764	0.292882	−	0.956149i	$-0.405386\pi$
$857$	502.000	0.585764	0.292882	+	0.956149i	$0.405386\pi$
$858$	0	0
$859$	− 640.272i	− 0.745369i	−0.927958	−	0.372684i	$-0.878437\pi$
$859$	− 640.272i	− 0.745369i	0.927958	−	0.372684i	$-0.121563\pi$
$860$	0	0
$861$	644.000	0.747967
$862$	0	0
$863$	973.636i	1.12820i	0.825707	+	0.564100i	$0.190777\pi$
$863$	973.636i	1.12820i	−0.825707	+	0.564100i	$0.809223\pi$
$864$	0	0
$865$	288.000	0.332948
$866$	0	0
$867$	1508.08i	1.73942i
$868$	0	0
$869$	−1344.00	−1.54661
$870$	0	0
$871$	253.992i	0.291610i
$872$	0	0
$873$	3382.00	3.87400
$874$	0	0
$875$	296.324i	0.338656i
$876$	0	0
$877$	238.000	0.271380	0.135690	−	0.990751i	$-0.456675\pi$
$877$	238.000	0.271380	0.135690	+	0.990751i	$0.456675\pi$
$878$	0	0
$879$	− 2201.27i	− 2.50428i
$880$	0	0
$881$	234.000	0.265607	0.132804	−	0.991142i	$-0.457602\pi$
$881$	234.000	0.265607	0.132804	+	0.991142i	$0.457602\pi$
$882$	0	0
$883$	− 1523.95i	− 1.72588i	−0.505306	−	0.862940i	$-0.668620\pi$
$883$	− 1523.95i	− 1.72588i	0.505306	−	0.862940i	$-0.331380\pi$
$884$	0	0
$885$	3808.00	4.30282
$886$	0	0
$887$	941.887i	1.06188i	0.847409	+	0.530940i	$0.178161\pi$
$887$	941.887i	1.06188i	−0.847409	+	0.530940i	$0.821839\pi$
$888$	0	0
$889$	224.000	0.251969
$890$	0	0
$891$	− 1153.55i	− 1.29467i
$892$	0	0
$893$	−840.000	−0.940649
$894$	0	0
$895$	− 338.656i	− 0.378387i
$896$	0	0
$897$	−448.000	−0.499443
$898$	0	0
$899$	− 444.486i	− 0.494423i
$900$	0	0
$901$	44.0000	0.0488346
$902$	0	0
$903$	148.162i	0.164078i
$904$	0	0
$905$	−864.000	−0.954696
$906$	0	0
$907$	− 1058.30i	− 1.16681i	−0.812180	−	0.583407i	$-0.801719\pi$
$907$	− 1058.30i	− 1.16681i	0.812180	−	0.583407i	$-0.198281\pi$
$908$	0	0
$909$	−760.000	−0.836084
$910$	0	0
$911$	− 931.304i	− 1.02229i	−0.859495	−	0.511144i	$-0.829222\pi$
$911$	− 931.304i	− 1.02229i	0.859495	−	0.511144i	$-0.170778\pi$
$912$	0	0
$913$	−392.000	−0.429354
$914$	0	0
$915$	− 2031.94i	− 2.22070i
$916$	0	0
$917$	−406.000	−0.442748
$918$	0	0
$919$	− 994.802i	− 1.08248i	−0.840867	−	0.541242i	$-0.817954\pi$
$919$	− 994.802i	− 1.08248i	0.840867	−	0.541242i	$-0.182046\pi$
$920$	0	0
$921$	−1428.00	−1.55049
$922$	0	0
$923$	− 338.656i	− 0.366908i
$924$	0	0
$925$	546.000	0.590270
$926$	0	0
$927$	− 201.077i	− 0.216912i
$928$	0	0
$929$	734.000	0.790097	0.395048	−	0.918660i	$-0.370728\pi$
$929$	734.000	0.790097	0.395048	+	0.918660i	$0.370728\pi$
$930$	0	0
$931$	− 185.203i	− 0.198929i
$932$	0	0
$933$	336.000	0.360129
$934$	0	0
$935$	169.328i	0.181100i
$936$	0	0
$937$	−1438.00	−1.53469	−0.767343	−	0.641237i	$-0.778421\pi$
$937$	−1438.00	−1.53469	−0.767343	+	0.641237i	$0.778421\pi$
$938$	0	0
$939$	− 412.737i	− 0.439550i
$940$	0	0
$941$	−912.000	−0.969182	−0.484591	−	0.874741i	$-0.661032\pi$
$941$	−912.000	−0.969182	−0.484591	+	0.874741i	$0.661032\pi$
$942$	0	0
$943$	973.636i	1.03249i
$944$	0	0
$945$	−1120.00	−1.18519
$946$	0	0
$947$	539.733i	0.569940i	0.958536	+	0.284970i	$0.0919837\pi$
$947$	539.733i	0.569940i	−0.958536	+	0.284970i	$0.908016\pi$
$948$	0	0
$949$	440.000	0.463646
$950$	0	0
$951$	1682.70i	1.76940i
$952$	0	0
$953$	1794.00	1.88248	0.941238	−	0.337743i	$-0.109664\pi$
$953$	1794.00	1.88248	0.941238	+	0.337743i	$0.109664\pi$
$954$	0	0
$955$	− 2031.94i	− 2.12768i
$956$	0	0
$957$	−784.000	−0.819227
$958$	0	0
$959$	− 185.203i	− 0.193121i
$960$	0	0
$961$	−47.0000	−0.0489074
$962$	0	0
$963$	1608.62i	1.67042i
$964$	0	0
$965$	1392.00	1.44249
$966$	0	0
$967$	1756.78i	1.81673i	0.418177	+	0.908365i	$0.362669\pi$
$967$	1756.78i	1.81673i	−0.418177	+	0.908365i	$0.637331\pi$
$968$	0	0
$969$	−280.000	−0.288958
$970$	0	0
$971$	− 37.0405i	− 0.0381468i	−0.999818	−	0.0190734i	$-0.993928\pi$
$971$	− 37.0405i	− 0.0381468i	0.999818	−	0.0190734i	$-0.00607162\pi$
$972$	0	0
$973$	−322.000	−0.330935
$974$	0	0
$975$	825.474i	0.846640i
$976$	0	0
$977$	−1862.00	−1.90583	−0.952917	−	0.303231i	$-0.901935\pi$
$977$	−1862.00	−1.90583	−0.952917	+	0.303231i	$0.901935\pi$
$978$	0	0
$979$	1418.12i	1.44854i
$980$	0	0
$981$	−3458.00	−3.52497
$982$	0	0
$983$	− 497.401i	− 0.506003i	−0.967466	−	0.253002i	$-0.918582\pi$
$983$	− 497.401i	− 0.506003i	0.967466	−	0.253002i	$-0.0814178\pi$
$984$	0	0
$985$	912.000	0.925888
$986$	0	0
$987$	444.486i	0.450341i
$988$	0	0
$989$	−224.000	−0.226491
$990$	0	0
$991$	529.150i	0.533956i	0.963703	+	0.266978i	$0.0860251\pi$
$991$	529.150i	0.533956i	−0.963703	+	0.266978i	$0.913975\pi$
$992$	0	0
$993$	1624.00	1.63545
$994$	0	0
$995$	− 84.6640i	− 0.0850895i
$996$	0	0
$997$	−320.000	−0.320963	−0.160481	−	0.987039i	$-0.551305\pi$
$997$	−320.000	−0.320963	−0.160481	+	0.987039i	$0.551305\pi$
$998$	0	0
$999$	740.810i	0.741552i

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	112.3.d.a.15.1	✓	2
3.2	odd	2		1008.3.m.a.127.2		2
4.3	odd	2	inner	112.3.d.a.15.2	yes	2
7.2	even	3		784.3.r.k.655.2		4
7.3	odd	6		784.3.r.m.79.2		4
7.4	even	3		784.3.r.k.79.1		4
7.5	odd	6		784.3.r.m.655.1		4
7.6	odd	2		784.3.d.d.687.2		2
8.3	odd	2		448.3.d.a.127.1		2
8.5	even	2		448.3.d.a.127.2		2
12.11	even	2		1008.3.m.a.127.1		2
16.3	odd	4		1792.3.g.b.127.1		4
16.5	even	4		1792.3.g.b.127.2		4
16.11	odd	4		1792.3.g.b.127.4		4
16.13	even	4		1792.3.g.b.127.3		4
28.3	even	6		784.3.r.m.79.1		4
28.11	odd	6		784.3.r.k.79.2		4
28.19	even	6		784.3.r.m.655.2		4
28.23	odd	6		784.3.r.k.655.1		4
28.27	even	2		784.3.d.d.687.1		2

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
112.3.d.a.15.1	✓	2	1.1	even	1	trivial
112.3.d.a.15.2	yes	2	4.3	odd	2	inner
448.3.d.a.127.1		2	8.3	odd	2
448.3.d.a.127.2		2	8.5	even	2
784.3.d.d.687.1		2	28.27	even	2
784.3.d.d.687.2		2	7.6	odd	2
784.3.r.k.79.1		4	7.4	even	3
784.3.r.k.79.2		4	28.11	odd	6
784.3.r.k.655.1		4	28.23	odd	6
784.3.r.k.655.2		4	7.2	even	3
784.3.r.m.79.1		4	28.3	even	6
784.3.r.m.79.2		4	7.3	odd	6
784.3.r.m.655.1		4	7.5	odd	6
784.3.r.m.655.2		4	28.19	even	6
1008.3.m.a.127.1		2	12.11	even	2
1008.3.m.a.127.2		2	3.2	odd	2
1792.3.g.b.127.1		4	16.3	odd	4
1792.3.g.b.127.2		4	16.5	even	4
1792.3.g.b.127.3		4	16.13	even	4
1792.3.g.b.127.4		4	16.11	odd	4

Overview	Random
Universe	Knowledge

Classical	Maass
Hilbert	Bianchi

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

Level:	\( N \)	\(=\)	\( 112 = 2^{4} \cdot 7 \)
Weight:	\( k \)	\(=\)	\( 3 \)
Character orbit:	\([\chi]\)	\(=\)	112.d (of order \(2\), degree \(1\), minimal)

\(f(q)\)	\(=\)	\(q-5.29150i q^{3} +8.00000 q^{5} +2.64575i q^{7} -19.0000 q^{9} +O(q^{10})\)	\(q-5.29150i q^{3} +8.00000 q^{5} +2.64575i q^{7} -19.0000 q^{9} -10.5830i q^{11} -4.00000 q^{13} -42.3320i q^{15} -2.00000 q^{17} +26.4575i q^{19} +14.0000 q^{21} +21.1660i q^{23} +39.0000 q^{25} +52.9150i q^{27} +14.0000 q^{29} -31.7490i q^{31} -56.0000 q^{33} +21.1660i q^{35} +14.0000 q^{37} +21.1660i q^{39} +46.0000 q^{41} +10.5830i q^{43} -152.000 q^{45} +31.7490i q^{47} -7.00000 q^{49} +10.5830i q^{51} -22.0000 q^{53} -84.6640i q^{55} +140.000 q^{57} +89.9555i q^{59} +48.0000 q^{61} -50.2693i q^{63} -32.0000 q^{65} -63.4980i q^{67} +112.000 q^{69} +84.6640i q^{71} -110.000 q^{73} -206.369i q^{75} +28.0000 q^{77} -126.996i q^{79} +109.000 q^{81} -37.0405i q^{83} -16.0000 q^{85} -74.0810i q^{87} -134.000 q^{89} -10.5830i q^{91} -168.000 q^{93} +211.660i q^{95} -178.000 q^{97} +201.077i q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 2 q + 16 q^{5} - 38 q^{9}+O(q^{10}) \) 2 * q + 16 * q^5 - 38 * q^9	\( 2 q + 16 q^{5} - 38 q^{9} - 8 q^{13} - 4 q^{17} + 28 q^{21} + 78 q^{25} + 28 q^{29} - 112 q^{33} + 28 q^{37} + 92 q^{41} - 304 q^{45} - 14 q^{49} - 44 q^{53} + 280 q^{57} + 96 q^{61} - 64 q^{65} + 224 q^{69} - 220 q^{73} + 56 q^{77} + 218 q^{81} - 32 q^{85} - 268 q^{89} - 336 q^{93} - 356 q^{97}+O(q^{100}) \) 2 * q + 16 * q^5 - 38 * q^9 - 8 * q^13 - 4 * q^17 + 28 * q^21 + 78 * q^25 + 28 * q^29 - 112 * q^33 + 28 * q^37 + 92 * q^41 - 304 * q^45 - 14 * q^49 - 44 * q^53 + 280 * q^57 + 96 * q^61 - 64 * q^65 + 224 * q^69 - 220 * q^73 + 56 * q^77 + 218 * q^81 - 32 * q^85 - 268 * q^89 - 336 * q^93 - 356 * q^97

Introduction

L-functions

Modular forms

Varieties

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Representations

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Database

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