LMFDB - Embedded newform 112.6.a.i.1.1

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [112,6,Mod(1,112)]

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([0, 0, 0]))

N = Newforms(chi, 6, names="a")

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

chi := DirichletCharacter("112.1");

S:= CuspForms(chi, 6);

N := Newforms(S);

Level:	$ N $	$=$	$ 112 = 2^{4} \cdot 7 $
Weight:	$ k $	$=$	$ 6 $
Character orbit:	$[\chi]$	$=$	112.a (trivial)

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:	yes
Analytic conductor:	$17.9629878191$
Analytic rank:	$0$
Dimension:	$2$
Coefficient field:	$\Q(\sqrt{345}) $
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{2} - x - 86 $ x^2 - x - 86
Coefficient ring:	$\Z[a_1, a_2, a_3]$
Coefficient ring index:	$ 2 $
Twist minimal:	no (minimal twist has level 56)
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.1
Root			$-8.78709$ of defining polynomial
Character	$\chi$	$=$	112.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-15.5742 q^{3} +96.7225 q^{5} +49.0000 q^{7} -0.445054 q^{9} +O(q^{10})$	$q-15.5742 q^{3} +96.7225 q^{5} +49.0000 q^{7} -0.445054 q^{9} -281.445 q^{11} -269.393 q^{13} -1506.37 q^{15} +1719.45 q^{17} +1172.84 q^{19} -763.135 q^{21} -785.341 q^{23} +6230.25 q^{25} +3791.46 q^{27} +6149.46 q^{29} +7006.58 q^{31} +4383.27 q^{33} +4739.40 q^{35} -11499.9 q^{37} +4195.57 q^{39} -13245.3 q^{41} +19824.2 q^{43} -43.0467 q^{45} -6887.96 q^{47} +2401.00 q^{49} -26778.9 q^{51} +8654.97 q^{53} -27222.1 q^{55} -18266.0 q^{57} +47856.6 q^{59} +52762.1 q^{61} -21.8076 q^{63} -26056.4 q^{65} +24040.2 q^{67} +12231.0 q^{69} -12540.9 q^{71} +4079.29 q^{73} -97031.0 q^{75} -13790.8 q^{77} +11919.1 q^{79} -58940.7 q^{81} -81916.2 q^{83} +166309. q^{85} -95772.7 q^{87} -96968.7 q^{89} -13200.2 q^{91} -109122. q^{93} +113440. q^{95} -26410.7 q^{97} +125.258 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 2 q + 6 q^{3} + 82 q^{5} + 98 q^{7} + 222 q^{9}+O(q^{10}) $ 2 * q + 6 * q^3 + 82 * q^5 + 98 * q^7 + 222 * q^9	$ 2 q + 6 q^{3} + 82 q^{5} + 98 q^{7} + 222 q^{9} - 340 q^{11} + 910 q^{13} - 1824 q^{15} + 3216 q^{17} + 674 q^{19} + 294 q^{21} + 1104 q^{23} + 3322 q^{25} + 3348 q^{27} + 8064 q^{29} + 6212 q^{31} + 3120 q^{33} + 4018 q^{35} - 8512 q^{37} + 29640 q^{39} - 1304 q^{41} + 10004 q^{43} - 3318 q^{45} + 12748 q^{47} + 4802 q^{49} + 5508 q^{51} - 11220 q^{53} - 26360 q^{55} - 29028 q^{57} + 12018 q^{59} + 102738 q^{61} + 10878 q^{63} - 43420 q^{65} - 24136 q^{67} + 52992 q^{69} - 89720 q^{71} - 55588 q^{73} - 159774 q^{75} - 16660 q^{77} - 48824 q^{79} - 122562 q^{81} - 35782 q^{83} + 144276 q^{85} - 54468 q^{87} - 18300 q^{89} + 44590 q^{91} - 126264 q^{93} + 120784 q^{95} - 69984 q^{97} - 12900 q^{99}+O(q^{100}) $ 2 * q + 6 * q^3 + 82 * q^5 + 98 * q^7 + 222 * q^9 - 340 * q^11 + 910 * q^13 - 1824 * q^15 + 3216 * q^17 + 674 * q^19 + 294 * q^21 + 1104 * q^23 + 3322 * q^25 + 3348 * q^27 + 8064 * q^29 + 6212 * q^31 + 3120 * q^33 + 4018 * q^35 - 8512 * q^37 + 29640 * q^39 - 1304 * q^41 + 10004 * q^43 - 3318 * q^45 + 12748 * q^47 + 4802 * q^49 + 5508 * q^51 - 11220 * q^53 - 26360 * q^55 - 29028 * q^57 + 12018 * q^59 + 102738 * q^61 + 10878 * q^63 - 43420 * q^65 - 24136 * q^67 + 52992 * q^69 - 89720 * q^71 - 55588 * q^73 - 159774 * q^75 - 16660 * q^77 - 48824 * q^79 - 122562 * q^81 - 35782 * q^83 + 144276 * q^85 - 54468 * q^87 - 18300 * q^89 + 44590 * q^91 - 126264 * q^93 + 120784 * q^95 - 69984 * q^97 - 12900 * q^99

Display 100 coefficients 10 coefficients

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
$3$	−15.5742	−0.999084	−0.499542	−	0.866290i	$-0.666498\pi$
$3$	−15.5742	−0.999084	−0.499542	+	0.866290i	$0.666498\pi$
$4$	0	0
$5$	96.7225	1.73023	0.865113	−	0.501578i	$-0.167247\pi$
$5$	96.7225	1.73023	0.865113	+	0.501578i	$0.167247\pi$
$6$	0	0
$7$	49.0000	0.377964
$7$	49.0000	0.377964
$8$	0	0
$9$	−0.445054	−0.00183150
$10$	0	0
$11$	−281.445	−0.701313	−0.350657	−	0.936504i	$-0.614042\pi$
$11$	−281.445	−0.701313	−0.350657	+	0.936504i	$0.614042\pi$
$12$	0	0
$13$	−269.393	−0.442107	−0.221054	−	0.975262i	$-0.570950\pi$
$13$	−269.393	−0.442107	−0.221054	+	0.975262i	$0.570950\pi$
$14$	0	0
$15$	−1506.37	−1.72864
$16$	0	0
$17$	1719.45	1.44300	0.721499	−	0.692415i	$-0.243453\pi$
$17$	1719.45	1.44300	0.721499	+	0.692415i	$0.243453\pi$
$18$	0	0
$19$	1172.84	0.745339	0.372670	−	0.927964i	$-0.378442\pi$
$19$	1172.84	0.745339	0.372670	+	0.927964i	$0.378442\pi$
$20$	0	0
$21$	−763.135	−0.377618
$22$	0	0
$23$	−785.341	−0.309555	−0.154778	−	0.987949i	$-0.549466\pi$
$23$	−785.341	−0.309555	−0.154778	+	0.987949i	$0.549466\pi$
$24$	0	0
$25$	6230.25	1.99368
$26$	0	0
$27$	3791.46	1.00091
$28$	0	0
$29$	6149.46	1.35782	0.678909	−	0.734222i	$-0.262453\pi$
$29$	6149.46	1.35782	0.678909	+	0.734222i	$0.262453\pi$
$30$	0	0
$31$	7006.58	1.30949	0.654744	−	0.755851i	$-0.272776\pi$
$31$	7006.58	1.30949	0.654744	+	0.755851i	$0.272776\pi$
$32$	0	0
$33$	4383.27	0.700671
$34$	0	0
$35$	4739.40	0.653964
$36$	0	0
$37$	−11499.9	−1.38099	−0.690495	−	0.723337i	$-0.742607\pi$
$37$	−11499.9	−1.38099	−0.690495	+	0.723337i	$0.742607\pi$
$38$	0	0
$39$	4195.57	0.441702
$40$	0	0
$41$	−13245.3	−1.23056	−0.615279	−	0.788310i	$-0.710957\pi$
$41$	−13245.3	−1.23056	−0.615279	+	0.788310i	$0.710957\pi$
$42$	0	0
$43$	19824.2	1.63502	0.817512	−	0.575911i	$-0.195353\pi$
$43$	19824.2	1.63502	0.817512	+	0.575911i	$0.195353\pi$
$44$	0	0
$45$	−43.0467	−0.00316890
$46$	0	0
$47$	−6887.96	−0.454827	−0.227413	−	0.973798i	$-0.573027\pi$
$47$	−6887.96	−0.454827	−0.227413	+	0.973798i	$0.573027\pi$
$48$	0	0
$49$	2401.00	0.142857
$50$	0	0
$51$	−26778.9	−1.44168
$52$	0	0
$53$	8654.97	0.423229	0.211615	−	0.977353i	$-0.432128\pi$
$53$	8654.97	0.423229	0.211615	+	0.977353i	$0.432128\pi$
$54$	0	0
$55$	−27222.1	−1.21343
$56$	0	0
$57$	−18266.0	−0.744656
$58$	0	0
$59$	47856.6	1.78983	0.894915	−	0.446236i	$-0.147236\pi$
$59$	47856.6	1.78983	0.894915	+	0.446236i	$0.147236\pi$
$60$	0	0
$61$	52762.1	1.81550	0.907752	−	0.419507i	$-0.137797\pi$
$61$	52762.1	1.81550	0.907752	+	0.419507i	$0.137797\pi$
$62$	0	0
$63$	−21.8076	−0.000692241 0
$64$	0	0
$65$	−26056.4	−0.764945
$66$	0	0
$67$	24040.2	0.654261	0.327130	−	0.944979i	$-0.393918\pi$
$67$	24040.2	0.654261	0.327130	+	0.944979i	$0.393918\pi$
$68$	0	0
$69$	12231.0	0.309272
$70$	0	0
$71$	−12540.9	−0.295246	−0.147623	−	0.989044i	$-0.547162\pi$
$71$	−12540.9	−0.295246	−0.147623	+	0.989044i	$0.547162\pi$
$72$	0	0
$73$	4079.29	0.0895936	0.0447968	−	0.998996i	$-0.485736\pi$
$73$	4079.29	0.0895936	0.0447968	+	0.998996i	$0.485736\pi$
$74$	0	0
$75$	−97031.0	−1.99185
$76$	0	0
$77$	−13790.8	−0.265071
$78$	0	0
$79$	11919.1	0.214870	0.107435	−	0.994212i	$-0.465736\pi$
$79$	11919.1	0.214870	0.107435	+	0.994212i	$0.465736\pi$
$80$	0	0
$81$	−58940.7	−0.998165
$82$	0	0
$83$	−81916.2	−1.30519	−0.652596	−	0.757706i	$-0.726320\pi$
$83$	−81916.2	−1.30519	−0.652596	+	0.757706i	$0.726320\pi$
$84$	0	0
$85$	166309.	2.49671
$86$	0	0
$87$	−95772.7	−1.35657
$88$	0	0
$89$	−96968.7	−1.29765	−0.648824	−	0.760939i	$-0.724739\pi$
$89$	−96968.7	−1.29765	−0.648824	+	0.760939i	$0.724739\pi$
$90$	0	0
$91$	−13200.2	−0.167101
$92$	0	0
$93$	−109122.	−1.30829
$94$	0	0
$95$	113440.	1.28960
$96$	0	0
$97$	−26410.7	−0.285004	−0.142502	−	0.989795i	$-0.545515\pi$
$97$	−26410.7	−0.285004	−0.142502	+	0.989795i	$0.545515\pi$
$98$	0	0
$99$	125.258	0.00128445
$100$	0	0
$101$	−73137.7	−0.713408	−0.356704	−	0.934217i	$-0.616099\pi$
$101$	−73137.7	−0.713408	−0.356704	+	0.934217i	$0.616099\pi$
$102$	0	0
$103$	−87649.5	−0.814060	−0.407030	−	0.913415i	$-0.633435\pi$
$103$	−87649.5	−0.814060	−0.407030	+	0.913415i	$0.633435\pi$
$104$	0	0
$105$	−73812.3	−0.653365
$106$	0	0
$107$	115438.	0.974743	0.487372	−	0.873195i	$-0.337956\pi$
$107$	115438.	0.974743	0.487372	+	0.873195i	$0.337956\pi$
$108$	0	0
$109$	−117303.	−0.945675	−0.472837	−	0.881150i	$-0.656770\pi$
$109$	−117303.	−0.945675	−0.472837	+	0.881150i	$0.656770\pi$
$110$	0	0
$111$	179102.	1.37973
$112$	0	0
$113$	181535.	1.33741	0.668705	−	0.743528i	$-0.266849\pi$
$113$	181535.	1.33741	0.668705	+	0.743528i	$0.266849\pi$
$114$	0	0
$115$	−75960.1	−0.535601
$116$	0	0
$117$	119.894	0.000809718 0
$118$	0	0
$119$	84252.8	0.545402
$120$	0	0
$121$	−81839.7	−0.508160
$122$	0	0
$123$	206284.	1.22943
$124$	0	0
$125$	300347.	1.71929
$126$	0	0
$127$	−106111.	−0.583780	−0.291890	−	0.956452i	$-0.594284\pi$
$127$	−106111.	−0.583780	−0.291890	+	0.956452i	$0.594284\pi$
$128$	0	0
$129$	−308745.	−1.63353
$130$	0	0
$131$	−300334.	−1.52907	−0.764534	−	0.644584i	$-0.777031\pi$
$131$	−300334.	−1.52907	−0.764534	+	0.644584i	$0.777031\pi$
$132$	0	0
$133$	57469.1	0.281712
$134$	0	0
$135$	366719.	1.73181
$136$	0	0
$137$	−91892.5	−0.418291	−0.209146	−	0.977885i	$-0.567068\pi$
$137$	−91892.5	−0.418291	−0.209146	+	0.977885i	$0.567068\pi$
$138$	0	0
$139$	−153133.	−0.672252	−0.336126	−	0.941817i	$-0.609117\pi$
$139$	−153133.	−0.672252	−0.336126	+	0.941817i	$0.609117\pi$
$140$	0	0
$141$	107274.	0.454410
$142$	0	0
$143$	75819.3	0.310056
$144$	0	0
$145$	594791.	2.34933
$146$	0	0
$147$	−37393.6	−0.142726
$148$	0	0
$149$	498602.	1.83987	0.919937	−	0.392066i	$-0.128240\pi$
$149$	498602.	1.83987	0.919937	+	0.392066i	$0.128240\pi$
$150$	0	0
$151$	−209609.	−0.748114	−0.374057	−	0.927406i	$-0.622034\pi$
$151$	−209609.	−0.748114	−0.374057	+	0.927406i	$0.622034\pi$
$152$	0	0
$153$	−765.245	−0.00264285
$154$	0	0
$155$	677694.	2.26571
$156$	0	0
$157$	−7381.72	−0.0239006	−0.0119503	−	0.999929i	$-0.503804\pi$
$157$	−7381.72	−0.0239006	−0.0119503	+	0.999929i	$0.503804\pi$
$158$	0	0
$159$	−134794.	−0.422842
$160$	0	0
$161$	−38481.7	−0.117001
$162$	0	0
$163$	−388154.	−1.14429	−0.572144	−	0.820153i	$-0.693888\pi$
$163$	−388154.	−1.14429	−0.572144	+	0.820153i	$0.693888\pi$
$164$	0	0
$165$	423961.	1.21232
$166$	0	0
$167$	168240.	0.466808	0.233404	−	0.972380i	$-0.425013\pi$
$167$	168240.	0.466808	0.233404	+	0.972380i	$0.425013\pi$
$168$	0	0
$169$	−298720.	−0.804541
$170$	0	0
$171$	−521.976	−0.00136509
$172$	0	0
$173$	321513.	0.816740	0.408370	−	0.912817i	$-0.366097\pi$
$173$	321513.	0.816740	0.408370	+	0.912817i	$0.366097\pi$
$174$	0	0
$175$	305282.	0.753540
$176$	0	0
$177$	−745327.	−1.78819
$178$	0	0
$179$	611906.	1.42742	0.713710	−	0.700441i	$-0.247013\pi$
$179$	611906.	1.42742	0.713710	+	0.700441i	$0.247013\pi$
$180$	0	0
$181$	261474.	0.593242	0.296621	−	0.954995i	$-0.404140\pi$
$181$	261474.	0.593242	0.296621	+	0.954995i	$0.404140\pi$
$182$	0	0
$183$	−821726.	−1.81384
$184$	0	0
$185$	−1.11230e6	−2.38943
$186$	0	0
$187$	−483929.	−1.01199
$188$	0	0
$189$	185781.	0.378310
$190$	0	0
$191$	−703301.	−1.39495	−0.697474	−	0.716610i	$-0.745693\pi$
$191$	−703301.	−1.39495	−0.697474	+	0.716610i	$0.745693\pi$
$192$	0	0
$193$	516411.	0.997934	0.498967	−	0.866621i	$-0.333713\pi$
$193$	516411.	0.997934	0.498967	+	0.866621i	$0.333713\pi$
$194$	0	0
$195$	405806.	0.764244
$196$	0	0
$197$	142579.	0.261752	0.130876	−	0.991399i	$-0.458221\pi$
$197$	142579.	0.261752	0.130876	+	0.991399i	$0.458221\pi$
$198$	0	0
$199$	−984837.	−1.76292	−0.881458	−	0.472262i	$-0.843438\pi$
$199$	−984837.	−1.76292	−0.881458	+	0.472262i	$0.843438\pi$
$200$	0	0
$201$	−374406.	−0.653662
$202$	0	0
$203$	301323.	0.513207
$204$	0	0
$205$	−1.28112e6	−2.12914
$206$	0	0
$207$	349.519	0.000566950 0
$208$	0	0
$209$	−330089.	−0.522716
$210$	0	0
$211$	242836.	0.375497	0.187748	−	0.982217i	$-0.439881\pi$
$211$	242836.	0.375497	0.187748	+	0.982217i	$0.439881\pi$
$212$	0	0
$213$	195315.	0.294976
$214$	0	0
$215$	1.91745e6	2.82896
$216$	0	0
$217$	343322.	0.494940
$218$	0	0
$219$	−63531.5	−0.0895115
$220$	0	0
$221$	−463206.	−0.637960
$222$	0	0
$223$	797063.	1.07332	0.536661	−	0.843798i	$-0.319685\pi$
$223$	797063.	1.07332	0.536661	+	0.843798i	$0.319685\pi$
$224$	0	0
$225$	−2772.79	−0.00365142
$226$	0	0
$227$	26121.4	0.0336459	0.0168229	−	0.999858i	$-0.494645\pi$
$227$	26121.4	0.0336459	0.0168229	+	0.999858i	$0.494645\pi$
$228$	0	0
$229$	−761592.	−0.959695	−0.479848	−	0.877352i	$-0.659308\pi$
$229$	−761592.	−0.959695	−0.479848	+	0.877352i	$0.659308\pi$
$230$	0	0
$231$	214780.	0.264829
$232$	0	0
$233$	657852.	0.793850	0.396925	−	0.917851i	$-0.370077\pi$
$233$	657852.	0.793850	0.396925	+	0.917851i	$0.370077\pi$
$234$	0	0
$235$	−666221.	−0.786953
$236$	0	0
$237$	−185630.	−0.214673
$238$	0	0
$239$	−965388.	−1.09322	−0.546609	−	0.837388i	$-0.684082\pi$
$239$	−965388.	−1.09322	−0.546609	+	0.837388i	$0.684082\pi$
$240$	0	0
$241$	35064.2	0.0388885	0.0194443	−	0.999811i	$-0.493810\pi$
$241$	35064.2	0.0388885	0.0194443	+	0.999811i	$0.493810\pi$
$242$	0	0
$243$	−3371.72	−0.00366299
$244$	0	0
$245$	232231.	0.247175
$246$	0	0
$247$	−315954.	−0.329520
$248$	0	0
$249$	1.27578e6	1.30400
$250$	0	0
$251$	−1.31610e6	−1.31857	−0.659286	−	0.751892i	$-0.729141\pi$
$251$	−1.31610e6	−1.31857	−0.659286	+	0.751892i	$0.729141\pi$
$252$	0	0
$253$	221030.	0.217095
$254$	0	0
$255$	−2.59013e6	−2.49443
$256$	0	0
$257$	−529802.	−0.500358	−0.250179	−	0.968200i	$-0.580489\pi$
$257$	−529802.	−0.500358	−0.250179	+	0.968200i	$0.580489\pi$
$258$	0	0
$259$	−563496.	−0.521966
$260$	0	0
$261$	−2736.84	−0.00248684
$262$	0	0
$263$	1.29457e6	1.15408	0.577041	−	0.816715i	$-0.304207\pi$
$263$	1.29457e6	1.15408	0.577041	+	0.816715i	$0.304207\pi$
$264$	0	0
$265$	837130.	0.732282
$266$	0	0
$267$	1.51021e6	1.29646
$268$	0	0
$269$	−444336.	−0.374396	−0.187198	−	0.982322i	$-0.559941\pi$
$269$	−444336.	−0.374396	−0.187198	+	0.982322i	$0.559941\pi$
$270$	0	0
$271$	−624757.	−0.516759	−0.258379	−	0.966044i	$-0.583188\pi$
$271$	−624757.	−0.516759	−0.258379	+	0.966044i	$0.583188\pi$
$272$	0	0
$273$	205583.	0.166948
$274$	0	0
$275$	−1.75347e6	−1.39819
$276$	0	0
$277$	−2.00202e6	−1.56772	−0.783860	−	0.620938i	$-0.786752\pi$
$277$	−2.00202e6	−1.56772	−0.783860	+	0.620938i	$0.786752\pi$
$278$	0	0
$279$	−3118.30	−0.00239832
$280$	0	0
$281$	249316.	0.188358	0.0941792	−	0.995555i	$-0.469977\pi$
$281$	249316.	0.188358	0.0941792	+	0.995555i	$0.469977\pi$
$282$	0	0
$283$	−645136.	−0.478835	−0.239417	−	0.970917i	$-0.576956\pi$
$283$	−645136.	−0.478835	−0.239417	+	0.970917i	$0.576956\pi$
$284$	0	0
$285$	−1.76673e6	−1.28842
$286$	0	0
$287$	−649019.	−0.465107
$288$	0	0
$289$	1.53663e6	1.08225
$290$	0	0
$291$	411325.	0.284743
$292$	0	0
$293$	2.00830e6	1.36666	0.683328	−	0.730112i	$-0.260532\pi$
$293$	2.00830e6	1.36666	0.683328	+	0.730112i	$0.260532\pi$
$294$	0	0
$295$	4.62881e6	3.09681
$296$	0	0
$297$	−1.06709e6	−0.701954
$298$	0	0
$299$	211565.	0.136857
$300$	0	0
$301$	971385.	0.617981
$302$	0	0
$303$	1.13906e6	0.712754
$304$	0	0
$305$	5.10328e6	3.14123
$306$	0	0
$307$	1.71113e6	1.03618	0.518092	−	0.855325i	$-0.326642\pi$
$307$	1.71113e6	1.03618	0.518092	+	0.855325i	$0.326642\pi$
$308$	0	0
$309$	1.36507e6	0.813314
$310$	0	0
$311$	602050.	0.352965	0.176483	−	0.984304i	$-0.443528\pi$
$311$	602050.	0.352965	0.176483	+	0.984304i	$0.443528\pi$
$312$	0	0
$313$	−1.34208e6	−0.774318	−0.387159	−	0.922013i	$-0.626543\pi$
$313$	−1.34208e6	−0.774318	−0.387159	+	0.922013i	$0.626543\pi$
$314$	0	0
$315$	−2109.29	−0.00119773
$316$	0	0
$317$	1.01803e6	0.569001	0.284500	−	0.958676i	$-0.408172\pi$
$317$	1.01803e6	0.569001	0.284500	+	0.958676i	$0.408172\pi$
$318$	0	0
$319$	−1.73073e6	−0.952256
$320$	0	0
$321$	−1.79786e6	−0.973850
$322$	0	0
$323$	2.01663e6	1.07552
$324$	0	0
$325$	−1.67838e6	−0.881420
$326$	0	0
$327$	1.82689e6	0.944809
$328$	0	0
$329$	−337510.	−0.171908
$330$	0	0
$331$	532371.	0.267082	0.133541	−	0.991043i	$-0.457365\pi$
$331$	532371.	0.267082	0.133541	+	0.991043i	$0.457365\pi$
$332$	0	0
$333$	5118.09	0.00252928
$334$	0	0
$335$	2.32523e6	1.13202
$336$	0	0
$337$	−234947.	−0.112693	−0.0563463	−	0.998411i	$-0.517945\pi$
$337$	−234947.	−0.112693	−0.0563463	+	0.998411i	$0.517945\pi$
$338$	0	0
$339$	−2.82726e6	−1.33618
$340$	0	0
$341$	−1.97197e6	−0.918361
$342$	0	0
$343$	117649.	0.0539949
$344$	0	0
$345$	1.18302e6	0.535110
$346$	0	0
$347$	1.90054e6	0.847330	0.423665	−	0.905819i	$-0.360743\pi$
$347$	1.90054e6	0.847330	0.423665	+	0.905819i	$0.360743\pi$
$348$	0	0
$349$	341162.	0.149933	0.0749665	−	0.997186i	$-0.476115\pi$
$349$	341162.	0.149933	0.0749665	+	0.997186i	$0.476115\pi$
$350$	0	0
$351$	−1.02139e6	−0.442511
$352$	0	0
$353$	673122.	0.287513	0.143756	−	0.989613i	$-0.454082\pi$
$353$	673122.	0.287513	0.143756	+	0.989613i	$0.454082\pi$
$354$	0	0
$355$	−1.21299e6	−0.510842
$356$	0	0
$357$	−1.31217e6	−0.544903
$358$	0	0
$359$	−1.43301e6	−0.586833	−0.293417	−	0.955985i	$-0.594792\pi$
$359$	−1.43301e6	−0.586833	−0.293417	+	0.955985i	$0.594792\pi$
$360$	0	0
$361$	−1.10055e6	−0.444469
$362$	0	0
$363$	1.27459e6	0.507694
$364$	0	0
$365$	394559.	0.155017
$366$	0	0
$367$	2.04781e6	0.793643	0.396822	−	0.917896i	$-0.370113\pi$
$367$	2.04781e6	0.793643	0.396822	+	0.917896i	$0.370113\pi$
$368$	0	0
$369$	5894.87	0.00225376
$370$	0	0
$371$	424093.	0.159966
$372$	0	0
$373$	−3.33578e6	−1.24144	−0.620719	−	0.784033i	$-0.713159\pi$
$373$	−3.33578e6	−1.24144	−0.620719	+	0.784033i	$0.713159\pi$
$374$	0	0
$375$	−4.67766e6	−1.71771
$376$	0	0
$377$	−1.65662e6	−0.600301
$378$	0	0
$379$	−2.29135e6	−0.819394	−0.409697	−	0.912222i	$-0.634366\pi$
$379$	−2.29135e6	−0.819394	−0.409697	+	0.912222i	$0.634366\pi$
$380$	0	0
$381$	1.65258e6	0.583245
$382$	0	0
$383$	−1.84422e6	−0.642414	−0.321207	−	0.947009i	$-0.604089\pi$
$383$	−1.84422e6	−0.642414	−0.321207	+	0.947009i	$0.604089\pi$
$384$	0	0
$385$	−1.33388e6	−0.458633
$386$	0	0
$387$	−8822.83	−0.00299454
$388$	0	0
$389$	3.66601e6	1.22834	0.614172	−	0.789172i	$-0.289490\pi$
$389$	3.66601e6	1.22834	0.614172	+	0.789172i	$0.289490\pi$
$390$	0	0
$391$	−1.35035e6	−0.446688
$392$	0	0
$393$	4.67746e6	1.52767
$394$	0	0
$395$	1.15284e6	0.371773
$396$	0	0
$397$	−4.59382e6	−1.46284	−0.731421	−	0.681926i	$-0.761142\pi$
$397$	−4.59382e6	−1.46284	−0.731421	+	0.681926i	$0.761142\pi$
$398$	0	0
$399$	−895033.	−0.281454
$400$	0	0
$401$	−1.10984e6	−0.344666	−0.172333	−	0.985039i	$-0.555130\pi$
$401$	−1.10984e6	−0.344666	−0.172333	+	0.985039i	$0.555130\pi$
$402$	0	0
$403$	−1.88752e6	−0.578934
$404$	0	0
$405$	−5.70089e6	−1.72705
$406$	0	0
$407$	3.23660e6	0.968507
$408$	0	0
$409$	3.60689e6	1.06617	0.533083	−	0.846063i	$-0.321033\pi$
$409$	3.60689e6	1.06617	0.533083	+	0.846063i	$0.321033\pi$
$410$	0	0
$411$	1.43115e6	0.417908
$412$	0	0
$413$	2.34497e6	0.676492
$414$	0	0
$415$	−7.92314e6	−2.25828
$416$	0	0
$417$	2.38492e6	0.671636
$418$	0	0
$419$	−5.51790e6	−1.53546	−0.767731	−	0.640773i	$-0.778614\pi$
$419$	−5.51790e6	−1.53546	−0.767731	+	0.640773i	$0.778614\pi$
$420$	0	0
$421$	−2.59092e6	−0.712440	−0.356220	−	0.934402i	$-0.615935\pi$
$421$	−2.59092e6	−0.712440	−0.356220	+	0.934402i	$0.615935\pi$
$422$	0	0
$423$	3065.51	0.000833014 0
$424$	0	0
$425$	1.07126e7	2.87688
$426$	0	0
$427$	2.58534e6	0.686196
$428$	0	0
$429$	−1.18082e6	−0.309772
$430$	0	0
$431$	3.45364e6	0.895537	0.447769	−	0.894150i	$-0.352219\pi$
$431$	3.45364e6	0.895537	0.447769	+	0.894150i	$0.352219\pi$
$432$	0	0
$433$	397522.	0.101892	0.0509461	−	0.998701i	$-0.483776\pi$
$433$	397522.	0.101892	0.0509461	+	0.998701i	$0.483776\pi$
$434$	0	0
$435$	−9.26338e6	−2.34718
$436$	0	0
$437$	−921077.	−0.230724
$438$	0	0
$439$	−1.60903e6	−0.398476	−0.199238	−	0.979951i	$-0.563847\pi$
$439$	−1.60903e6	−0.398476	−0.199238	+	0.979951i	$0.563847\pi$
$440$	0	0
$441$	−1068.57	−0.000261642 0
$442$	0	0
$443$	7.08182e6	1.71449	0.857246	−	0.514906i	$-0.172173\pi$
$443$	7.08182e6	1.71449	0.857246	+	0.514906i	$0.172173\pi$
$444$	0	0
$445$	−9.37906e6	−2.24522
$446$	0	0
$447$	−7.76531e6	−1.83819
$448$	0	0
$449$	−6.46156e6	−1.51259	−0.756295	−	0.654230i	$-0.772993\pi$
$449$	−6.46156e6	−1.51259	−0.756295	+	0.654230i	$0.772993\pi$
$450$	0	0
$451$	3.72782e6	0.863006
$452$	0	0
$453$	3.26449e6	0.747429
$454$	0	0
$455$	−1.27676e6	−0.289122
$456$	0	0
$457$	−3.37073e6	−0.754976	−0.377488	−	0.926015i	$-0.623212\pi$
$457$	−3.37073e6	−0.754976	−0.377488	+	0.926015i	$0.623212\pi$
$458$	0	0
$459$	6.51920e6	1.44432
$460$	0	0
$461$	6.20821e6	1.36055	0.680274	−	0.732958i	$-0.261861\pi$
$461$	6.20821e6	1.36055	0.680274	+	0.732958i	$0.261861\pi$
$462$	0	0
$463$	5.82940e6	1.26378	0.631890	−	0.775058i	$-0.282279\pi$
$463$	5.82940e6	1.26378	0.631890	+	0.775058i	$0.282279\pi$
$464$	0	0
$465$	−1.05545e7	−2.26363
$466$	0	0
$467$	174742.	0.0370770	0.0185385	−	0.999828i	$-0.494099\pi$
$467$	174742.	0.0370770	0.0185385	+	0.999828i	$0.494099\pi$
$468$	0	0
$469$	1.17797e6	0.247287
$470$	0	0
$471$	114964.	0.0238787
$472$	0	0
$473$	−5.57942e6	−1.14666
$474$	0	0
$475$	7.30707e6	1.48597
$476$	0	0
$477$	−3851.93	−0.000775143 0
$478$	0	0
$479$	−371746.	−0.0740299	−0.0370150	−	0.999315i	$-0.511785\pi$
$479$	−371746.	−0.0740299	−0.0370150	+	0.999315i	$0.511785\pi$
$480$	0	0
$481$	3.09800e6	0.610546
$482$	0	0
$483$	599321.	0.116894
$484$	0	0
$485$	−2.55451e6	−0.493121
$486$	0	0
$487$	8.71307e6	1.66475	0.832375	−	0.554213i	$-0.186981\pi$
$487$	8.71307e6	1.66475	0.832375	+	0.554213i	$0.186981\pi$
$488$	0	0
$489$	6.04519e6	1.14324
$490$	0	0
$491$	−6.88530e6	−1.28890	−0.644450	−	0.764646i	$-0.722914\pi$
$491$	−6.88530e6	−1.28890	−0.644450	+	0.764646i	$0.722914\pi$
$492$	0	0
$493$	1.05737e7	1.95933
$494$	0	0
$495$	12115.3	0.00222239
$496$	0	0
$497$	−614506.	−0.111592
$498$	0	0
$499$	45442.7	0.00816982	0.00408491	−	0.999992i	$-0.498700\pi$
$499$	45442.7	0.00816982	0.00408491	+	0.999992i	$0.498700\pi$
$500$	0	0
$501$	−2.62020e6	−0.466381
$502$	0	0
$503$	623417.	0.109865	0.0549324	−	0.998490i	$-0.482506\pi$
$503$	623417.	0.109865	0.0549324	+	0.998490i	$0.482506\pi$
$504$	0	0
$505$	−7.07406e6	−1.23436
$506$	0	0
$507$	4.65233e6	0.803804
$508$	0	0
$509$	7.69092e6	1.31578	0.657891	−	0.753113i	$-0.271449\pi$
$509$	7.69092e6	1.31578	0.657891	+	0.753113i	$0.271449\pi$
$510$	0	0
$511$	199885.	0.0338632
$512$	0	0
$513$	4.44676e6	0.746020
$514$	0	0
$515$	−8.47768e6	−1.40851
$516$	0	0
$517$	1.93858e6	0.318976
$518$	0	0
$519$	−5.00730e6	−0.815991
$520$	0	0
$521$	7.32268e6	1.18189	0.590943	−	0.806713i	$-0.298756\pi$
$521$	7.32268e6	1.18189	0.590943	+	0.806713i	$0.298756\pi$
$522$	0	0
$523$	−5.17976e6	−0.828047	−0.414024	−	0.910266i	$-0.635877\pi$
$523$	−5.17976e6	−0.828047	−0.414024	+	0.910266i	$0.635877\pi$
$524$	0	0
$525$	−4.75452e6	−0.752850
$526$	0	0
$527$	1.20474e7	1.88959
$528$	0	0
$529$	−5.81958e6	−0.904175
$530$	0	0
$531$	−21298.8	−0.00327807
$532$	0	0
$533$	3.56819e6	0.544038
$534$	0	0
$535$	1.11655e7	1.68653
$536$	0	0
$537$	−9.52993e6	−1.42611
$538$	0	0
$539$	−675750.	−0.100188
$540$	0	0
$541$	9.31804e6	1.36877	0.684387	−	0.729119i	$-0.260070\pi$
$541$	9.31804e6	1.36877	0.684387	+	0.729119i	$0.260070\pi$
$542$	0	0
$543$	−4.07224e6	−0.592698
$544$	0	0
$545$	−1.13458e7	−1.63623
$546$	0	0
$547$	−1.26417e7	−1.80650	−0.903250	−	0.429114i	$-0.858826\pi$
$547$	−1.26417e7	−1.80650	−0.903250	+	0.429114i	$0.858826\pi$
$548$	0	0
$549$	−23482.0	−0.00332509
$550$	0	0
$551$	7.21232e6	1.01204
$552$	0	0
$553$	584035.	0.0812131
$554$	0	0
$555$	1.73232e7	2.38724
$556$	0	0
$557$	−3.84693e6	−0.525384	−0.262692	−	0.964880i	$-0.584610\pi$
$557$	−3.84693e6	−0.525384	−0.262692	+	0.964880i	$0.584610\pi$
$558$	0	0
$559$	−5.34050e6	−0.722856
$560$	0	0
$561$	7.53680e6	1.01107
$562$	0	0
$563$	−3.69193e6	−0.490888	−0.245444	−	0.969411i	$-0.578934\pi$
$563$	−3.69193e6	−0.490888	−0.245444	+	0.969411i	$0.578934\pi$
$564$	0	0
$565$	1.75585e7	2.31402
$566$	0	0
$567$	−2.88809e6	−0.377271
$568$	0	0
$569$	−6.64173e6	−0.860005	−0.430002	−	0.902828i	$-0.641487\pi$
$569$	−6.64173e6	−0.860005	−0.430002	+	0.902828i	$0.641487\pi$
$570$	0	0
$571$	−292599.	−0.0375562	−0.0187781	−	0.999824i	$-0.505978\pi$
$571$	−292599.	−0.0375562	−0.0187781	+	0.999824i	$0.505978\pi$
$572$	0	0
$573$	1.09533e7	1.39367
$574$	0	0
$575$	−4.89287e6	−0.617154
$576$	0	0
$577$	3.55569e6	0.444615	0.222307	−	0.974977i	$-0.428641\pi$
$577$	3.55569e6	0.444615	0.222307	+	0.974977i	$0.428641\pi$
$578$	0	0
$579$	−8.04267e6	−0.997020
$580$	0	0
$581$	−4.01389e6	−0.493316
$582$	0	0
$583$	−2.43590e6	−0.296816
$584$	0	0
$585$	11596.5	0.00140099
$586$	0	0
$587$	2.64810e6	0.317204	0.158602	−	0.987343i	$-0.449301\pi$
$587$	2.64810e6	0.317204	0.158602	+	0.987343i	$0.449301\pi$
$588$	0	0
$589$	8.21758e6	0.976013
$590$	0	0
$591$	−2.22055e6	−0.261512
$592$	0	0
$593$	1.05356e6	0.123034	0.0615168	−	0.998106i	$-0.480406\pi$
$593$	1.05356e6	0.123034	0.0615168	+	0.998106i	$0.480406\pi$
$594$	0	0
$595$	8.14914e6	0.943669
$596$	0	0
$597$	1.53380e7	1.76130
$598$	0	0
$599$	−9.19012e6	−1.04654	−0.523268	−	0.852168i	$-0.675287\pi$
$599$	−9.19012e6	−1.04654	−0.523268	+	0.852168i	$0.675287\pi$
$600$	0	0
$601$	3.46924e6	0.391785	0.195893	−	0.980625i	$-0.437240\pi$
$601$	3.46924e6	0.391785	0.195893	+	0.980625i	$0.437240\pi$
$602$	0	0
$603$	−10699.2	−0.00119828
$604$	0	0
$605$	−7.91574e6	−0.879231
$606$	0	0
$607$	744150.	0.0819764	0.0409882	−	0.999160i	$-0.486949\pi$
$607$	744150.	0.0819764	0.0409882	+	0.999160i	$0.486949\pi$
$608$	0	0
$609$	−4.69286e6	−0.512737
$610$	0	0
$611$	1.85557e6	0.201082
$612$	0	0
$613$	1.14578e7	1.23154	0.615772	−	0.787925i	$-0.288844\pi$
$613$	1.14578e7	1.23154	0.615772	+	0.787925i	$0.288844\pi$
$614$	0	0
$615$	1.99524e7	2.12719
$616$	0	0
$617$	−3.90052e6	−0.412486	−0.206243	−	0.978501i	$-0.566124\pi$
$617$	−3.90052e6	−0.412486	−0.206243	+	0.978501i	$0.566124\pi$
$618$	0	0
$619$	−2.65029e6	−0.278014	−0.139007	−	0.990291i	$-0.544391\pi$
$619$	−2.65029e6	−0.278014	−0.139007	+	0.990291i	$0.544391\pi$
$620$	0	0
$621$	−2.97758e6	−0.309838
$622$	0	0
$623$	−4.75147e6	−0.490464
$624$	0	0
$625$	9.58083e6	0.981077
$626$	0	0
$627$	5.14087e6	0.522237
$628$	0	0
$629$	−1.97735e7	−1.99277
$630$	0	0
$631$	8.97599e6	0.897447	0.448723	−	0.893671i	$-0.351879\pi$
$631$	8.97599e6	0.897447	0.448723	+	0.893671i	$0.351879\pi$
$632$	0	0
$633$	−3.78197e6	−0.375153
$634$	0	0
$635$	−1.02633e7	−1.01007
$636$	0	0
$637$	−646812.	−0.0631582
$638$	0	0
$639$	5581.39	0.000540742 0
$640$	0	0
$641$	−2.87152e6	−0.276036	−0.138018	−	0.990430i	$-0.544073\pi$
$641$	−2.87152e6	−0.276036	−0.138018	+	0.990430i	$0.544073\pi$
$642$	0	0
$643$	−1.38761e7	−1.32354	−0.661772	−	0.749705i	$-0.730195\pi$
$643$	−1.38761e7	−1.32354	−0.661772	+	0.749705i	$0.730195\pi$
$644$	0	0
$645$	−2.98626e7	−2.82637
$646$	0	0
$647$	7.24853e6	0.680752	0.340376	−	0.940289i	$-0.389446\pi$
$647$	7.24853e6	0.680752	0.340376	+	0.940289i	$0.389446\pi$
$648$	0	0
$649$	−1.34690e7	−1.25523
$650$	0	0
$651$	−5.34696e6	−0.494487
$652$	0	0
$653$	−1.23719e7	−1.13541	−0.567706	−	0.823231i	$-0.692169\pi$
$653$	−1.23719e7	−1.13541	−0.567706	+	0.823231i	$0.692169\pi$
$654$	0	0
$655$	−2.90491e7	−2.64563
$656$	0	0
$657$	−1815.50	−0.000164090 0
$658$	0	0
$659$	572294.	0.0513341	0.0256670	−	0.999671i	$-0.491829\pi$
$659$	572294.	0.0513341	0.0256670	+	0.999671i	$0.491829\pi$
$660$	0	0
$661$	6.73679e6	0.599721	0.299861	−	0.953983i	$-0.403060\pi$
$661$	6.73679e6	0.599721	0.299861	+	0.953983i	$0.403060\pi$
$662$	0	0
$663$	7.21405e6	0.637376
$664$	0	0
$665$	5.55855e6	0.487425
$666$	0	0
$667$	−4.82942e6	−0.420320
$668$	0	0
$669$	−1.24136e7	−1.07234
$670$	0	0
$671$	−1.48496e7	−1.27324
$672$	0	0
$673$	−1.22608e7	−1.04348	−0.521738	−	0.853106i	$-0.674716\pi$
$673$	−1.22608e7	−1.04348	−0.521738	+	0.853106i	$0.674716\pi$
$674$	0	0
$675$	2.36217e7	1.99550
$676$	0	0
$677$	−462085.	−0.0387481	−0.0193741	−	0.999812i	$-0.506167\pi$
$677$	−462085.	−0.0387481	−0.0193741	+	0.999812i	$0.506167\pi$
$678$	0	0
$679$	−1.29413e6	−0.107721
$680$	0	0
$681$	−406819.	−0.0336151
$682$	0	0
$683$	1.15378e7	0.946392	0.473196	−	0.880957i	$-0.343100\pi$
$683$	1.15378e7	0.946392	0.473196	+	0.880957i	$0.343100\pi$
$684$	0	0
$685$	−8.88807e6	−0.723738
$686$	0	0
$687$	1.18612e7	0.958816
$688$	0	0
$689$	−2.33159e6	−0.187113
$690$	0	0
$691$	−3.09921e6	−0.246920	−0.123460	−	0.992350i	$-0.539399\pi$
$691$	−3.09921e6	−0.246920	−0.123460	+	0.992350i	$0.539399\pi$
$692$	0	0
$693$	6137.65	0.000485477 0
$694$	0	0
$695$	−1.48114e7	−1.16315
$696$	0	0
$697$	−2.27746e7	−1.77569
$698$	0	0
$699$	−1.02455e7	−0.793122
$700$	0	0
$701$	5.83889e6	0.448782	0.224391	−	0.974499i	$-0.427961\pi$
$701$	5.83889e6	0.448782	0.224391	+	0.974499i	$0.427961\pi$
$702$	0	0
$703$	−1.34876e7	−1.02931
$704$	0	0
$705$	1.03758e7	0.786232
$706$	0	0
$707$	−3.58375e6	−0.269643
$708$	0	0
$709$	75435.7	0.00563587	0.00281794	−	0.999996i	$-0.499103\pi$
$709$	75435.7	0.00563587	0.00281794	+	0.999996i	$0.499103\pi$
$710$	0	0
$711$	−5304.63	−0.000393533 0
$712$	0	0
$713$	−5.50255e6	−0.405359
$714$	0	0
$715$	7.33343e6	0.536466
$716$	0	0
$717$	1.50351e7	1.09222
$718$	0	0
$719$	1.30869e7	0.944091	0.472045	−	0.881574i	$-0.343516\pi$
$719$	1.30869e7	0.944091	0.472045	+	0.881574i	$0.343516\pi$
$720$	0	0
$721$	−4.29482e6	−0.307686
$722$	0	0
$723$	−546096.	−0.0388529
$724$	0	0
$725$	3.83126e7	2.70705
$726$	0	0
$727$	3.58160e6	0.251328	0.125664	−	0.992073i	$-0.459894\pi$
$727$	3.58160e6	0.251328	0.125664	+	0.992073i	$0.459894\pi$
$728$	0	0
$729$	1.43751e7	1.00182
$730$	0	0
$731$	3.40866e7	2.35934
$732$	0	0
$733$	−9.23826e6	−0.635083	−0.317541	−	0.948244i	$-0.602857\pi$
$733$	−9.23826e6	−0.635083	−0.317541	+	0.948244i	$0.602857\pi$
$734$	0	0
$735$	−3.61680e6	−0.246949
$736$	0	0
$737$	−6.76599e6	−0.458842
$738$	0	0
$739$	1.90444e7	1.28279	0.641397	−	0.767210i	$-0.278355\pi$
$739$	1.90444e7	1.28279	0.641397	+	0.767210i	$0.278355\pi$
$740$	0	0
$741$	4.92073e6	0.329218
$742$	0	0
$743$	−9.30669e6	−0.618477	−0.309238	−	0.950985i	$-0.600074\pi$
$743$	−9.30669e6	−0.618477	−0.309238	+	0.950985i	$0.600074\pi$
$744$	0	0
$745$	4.82260e7	3.18340
$746$	0	0
$747$	36457.1	0.00239046
$748$	0	0
$749$	5.65647e6	0.368418
$750$	0	0
$751$	−2.56243e7	−1.65788	−0.828939	−	0.559339i	$-0.811055\pi$
$751$	−2.56243e7	−1.65788	−0.828939	+	0.559339i	$0.811055\pi$
$752$	0	0
$753$	2.04971e7	1.31736
$754$	0	0
$755$	−2.02739e7	−1.29441
$756$	0	0
$757$	5.60956e6	0.355786	0.177893	−	0.984050i	$-0.443072\pi$
$757$	5.60956e6	0.355786	0.177893	+	0.984050i	$0.443072\pi$
$758$	0	0
$759$	−3.44236e6	−0.216896
$760$	0	0
$761$	−2.83985e7	−1.77760	−0.888799	−	0.458298i	$-0.848459\pi$
$761$	−2.83985e7	−1.77760	−0.888799	+	0.458298i	$0.848459\pi$
$762$	0	0
$763$	−5.74783e6	−0.357432
$764$	0	0
$765$	−74016.5	−0.00457272
$766$	0	0
$767$	−1.28922e7	−0.791297
$768$	0	0
$769$	−2.41205e6	−0.147086	−0.0735429	−	0.997292i	$-0.523431\pi$
$769$	−2.41205e6	−0.147086	−0.0735429	+	0.997292i	$0.523431\pi$
$770$	0	0
$771$	8.25123e6	0.499899
$772$	0	0
$773$	5.93891e6	0.357485	0.178743	−	0.983896i	$-0.442797\pi$
$773$	5.93891e6	0.357485	0.178743	+	0.983896i	$0.442797\pi$
$774$	0	0
$775$	4.36527e7	2.61070
$776$	0	0
$777$	8.77599e6	0.521487
$778$	0	0
$779$	−1.55346e7	−0.917183
$780$	0	0
$781$	3.52958e6	0.207060
$782$	0	0
$783$	2.33154e7	1.35906
$784$	0	0
$785$	−713979.	−0.0413534
$786$	0	0
$787$	−2.62637e7	−1.51154	−0.755769	−	0.654839i	$-0.772737\pi$
$787$	−2.62637e7	−1.51154	−0.755769	+	0.654839i	$0.772737\pi$
$788$	0	0
$789$	−2.01619e7	−1.15302
$790$	0	0
$791$	8.89522e6	0.505494
$792$	0	0
$793$	−1.42137e7	−0.802648
$794$	0	0
$795$	−1.30376e7	−0.731611
$796$	0	0
$797$	−9.33689e6	−0.520663	−0.260331	−	0.965519i	$-0.583832\pi$
$797$	−9.33689e6	−0.520663	−0.260331	+	0.965519i	$0.583832\pi$
$798$	0	0
$799$	−1.18435e7	−0.656315
$800$	0	0
$801$	43156.3	0.00237664
$802$	0	0
$803$	−1.14809e6	−0.0628332
$804$	0	0
$805$	−3.72205e6	−0.202438
$806$	0	0
$807$	6.92017e6	0.374053
$808$	0	0
$809$	−3.43772e7	−1.84671	−0.923357	−	0.383942i	$-0.874566\pi$
$809$	−3.43772e7	−1.84671	−0.923357	+	0.383942i	$0.874566\pi$
$810$	0	0
$811$	2.02626e7	1.08179	0.540896	−	0.841090i	$-0.318085\pi$
$811$	2.02626e7	1.08179	0.540896	+	0.841090i	$0.318085\pi$
$812$	0	0
$813$	9.73008e6	0.516285
$814$	0	0
$815$	−3.75433e7	−1.97988
$816$	0	0
$817$	2.32506e7	1.21865
$818$	0	0
$819$	5874.82	0.000306045 0
$820$	0	0
$821$	1.73641e7	0.899073	0.449537	−	0.893262i	$-0.351589\pi$
$821$	1.73641e7	0.899073	0.449537	+	0.893262i	$0.351589\pi$
$822$	0	0
$823$	2.65094e7	1.36427	0.682136	−	0.731226i	$-0.261051\pi$
$823$	2.65094e7	1.36427	0.682136	+	0.731226i	$0.261051\pi$
$824$	0	0
$825$	2.73089e7	1.39691
$826$	0	0
$827$	2.30816e6	0.117355	0.0586775	−	0.998277i	$-0.481312\pi$
$827$	2.30816e6	0.117355	0.0586775	+	0.998277i	$0.481312\pi$
$828$	0	0
$829$	−6.14314e6	−0.310459	−0.155230	−	0.987878i	$-0.549612\pi$
$829$	−6.14314e6	−0.310459	−0.155230	+	0.987878i	$0.549612\pi$
$830$	0	0
$831$	3.11798e7	1.56628
$832$	0	0
$833$	4.12839e6	0.206143
$834$	0	0
$835$	1.62726e7	0.807684
$836$	0	0
$837$	2.65651e7	1.31068
$838$	0	0
$839$	−1.04628e7	−0.513150	−0.256575	−	0.966524i	$-0.582594\pi$
$839$	−1.04628e7	−0.513150	−0.256575	+	0.966524i	$0.582594\pi$
$840$	0	0
$841$	1.73047e7	0.843671
$842$	0	0
$843$	−3.88290e6	−0.188186
$844$	0	0
$845$	−2.88930e7	−1.39204
$846$	0	0
$847$	−4.01014e6	−0.192066
$848$	0	0
$849$	1.00475e7	0.478396
$850$	0	0
$851$	9.03136e6	0.427493
$852$	0	0
$853$	−3.45134e7	−1.62411	−0.812054	−	0.583583i	$-0.801650\pi$
$853$	−3.45134e7	−1.62411	−0.812054	+	0.583583i	$0.801650\pi$
$854$	0	0
$855$	−50486.8	−0.00236191
$856$	0	0
$857$	−1.35665e7	−0.630978	−0.315489	−	0.948929i	$-0.602169\pi$
$857$	−1.35665e7	−0.630978	−0.315489	+	0.948929i	$0.602169\pi$
$858$	0	0
$859$	−3.12783e7	−1.44630	−0.723152	−	0.690688i	$-0.757308\pi$
$859$	−3.12783e7	−1.44630	−0.723152	+	0.690688i	$0.757308\pi$
$860$	0	0
$861$	1.01079e7	0.464681
$862$	0	0
$863$	1.98071e7	0.905305	0.452653	−	0.891687i	$-0.350478\pi$
$863$	1.98071e7	0.905305	0.452653	+	0.891687i	$0.350478\pi$
$864$	0	0
$865$	3.10976e7	1.41314
$866$	0	0
$867$	−2.39318e7	−1.08125
$868$	0	0
$869$	−3.35457e6	−0.150691
$870$	0	0
$871$	−6.47626e6	−0.289254
$872$	0	0
$873$	11754.2	0.000521984 0
$874$	0	0
$875$	1.47170e7	0.649830
$876$	0	0
$877$	−1.74647e6	−0.0766765	−0.0383383	−	0.999265i	$-0.512206\pi$
$877$	−1.74647e6	−0.0766765	−0.0383383	+	0.999265i	$0.512206\pi$
$878$	0	0
$879$	−3.12776e7	−1.36540
$880$	0	0
$881$	1.17415e7	0.509663	0.254832	−	0.966985i	$-0.417980\pi$
$881$	1.17415e7	0.509663	0.254832	+	0.966985i	$0.417980\pi$
$882$	0	0
$883$	4.26514e7	1.84091	0.920453	−	0.390854i	$-0.127820\pi$
$883$	4.26514e7	1.84091	0.920453	+	0.390854i	$0.127820\pi$
$884$	0	0
$885$	−7.20899e7	−3.09397
$886$	0	0
$887$	4.28737e7	1.82971	0.914854	−	0.403785i	$-0.132306\pi$
$887$	4.28737e7	1.82971	0.914854	+	0.403785i	$0.132306\pi$
$888$	0	0
$889$	−5.19942e6	−0.220648
$890$	0	0
$891$	1.65886e7	0.700026
$892$	0	0
$893$	−8.07846e6	−0.339000
$894$	0	0
$895$	5.91851e7	2.46976
$896$	0	0
$897$	−3.29495e6	−0.136731
$898$	0	0
$899$	4.30866e7	1.77805
$900$	0	0
$901$	1.48817e7	0.610719
$902$	0	0
$903$	−1.51285e7	−0.617415
$904$	0	0
$905$	2.52904e7	1.02644
$906$	0	0
$907$	1.99065e7	0.803482	0.401741	−	0.915753i	$-0.368405\pi$
$907$	1.99065e7	0.803482	0.401741	+	0.915753i	$0.368405\pi$
$908$	0	0
$909$	32550.2	0.00130660
$910$	0	0
$911$	1.12063e7	0.447371	0.223685	−	0.974661i	$-0.428191\pi$
$911$	1.12063e7	0.447371	0.223685	+	0.974661i	$0.428191\pi$
$912$	0	0
$913$	2.30549e7	0.915348
$914$	0	0
$915$	−7.94794e7	−3.13835
$916$	0	0
$917$	−1.47164e7	−0.577933
$918$	0	0
$919$	−3.38620e7	−1.32259	−0.661293	−	0.750128i	$-0.729992\pi$
$919$	−3.38620e7	−1.32259	−0.661293	+	0.750128i	$0.729992\pi$
$920$	0	0
$921$	−2.66494e7	−1.03523
$922$	0	0
$923$	3.37844e6	0.130530
$924$	0	0
$925$	−7.16474e7	−2.75325
$926$	0	0
$927$	39008.7	0.00149095
$928$	0	0
$929$	−2.29019e7	−0.870627	−0.435313	−	0.900279i	$-0.643362\pi$
$929$	−2.29019e7	−0.870627	−0.435313	+	0.900279i	$0.643362\pi$
$930$	0	0
$931$	2.81598e6	0.106477
$932$	0	0
$933$	−9.37644e6	−0.352642
$934$	0	0
$935$	−4.68069e7	−1.75098
$936$	0	0
$937$	2.97781e7	1.10802	0.554011	−	0.832510i	$-0.313097\pi$
$937$	2.97781e7	1.10802	0.554011	+	0.832510i	$0.313097\pi$
$938$	0	0
$939$	2.09019e7	0.773608
$940$	0	0
$941$	−1.30322e7	−0.479782	−0.239891	−	0.970800i	$-0.577112\pi$
$941$	−1.30322e7	−0.479782	−0.239891	+	0.970800i	$0.577112\pi$
$942$	0	0
$943$	1.04021e7	0.380926
$944$	0	0
$945$	1.79692e7	0.654561
$946$	0	0
$947$	−4.66683e7	−1.69101	−0.845506	−	0.533966i	$-0.820701\pi$
$947$	−4.66683e7	−1.69101	−0.845506	+	0.533966i	$0.820701\pi$
$948$	0	0
$949$	−1.09893e6	−0.0396100
$950$	0	0
$951$	−1.58550e7	−0.568480
$952$	0	0
$953$	2.00948e7	0.716724	0.358362	−	0.933583i	$-0.383335\pi$
$953$	2.00948e7	0.716724	0.358362	+	0.933583i	$0.383335\pi$
$954$	0	0
$955$	−6.80251e7	−2.41357
$956$	0	0
$957$	2.69548e7	0.951383
$958$	0	0
$959$	−4.50273e6	−0.158099
$960$	0	0
$961$	2.04630e7	0.714760
$962$	0	0
$963$	−51376.2	−0.00178524
$964$	0	0
$965$	4.99485e7	1.72665
$966$	0	0
$967$	−7.33633e6	−0.252297	−0.126149	−	0.992011i	$-0.540262\pi$
$967$	−7.33633e6	−0.252297	−0.126149	+	0.992011i	$0.540262\pi$
$968$	0	0
$969$	−3.14074e7	−1.07454
$970$	0	0
$971$	−4.01630e7	−1.36703	−0.683514	−	0.729937i	$-0.739549\pi$
$971$	−4.01630e7	−1.36703	−0.683514	+	0.729937i	$0.739549\pi$
$972$	0	0
$973$	−7.50352e6	−0.254087
$974$	0	0
$975$	2.61394e7	0.880613
$976$	0	0
$977$	2.65602e7	0.890216	0.445108	−	0.895477i	$-0.353165\pi$
$977$	2.65602e7	0.890216	0.445108	+	0.895477i	$0.353165\pi$
$978$	0	0
$979$	2.72914e7	0.910057
$980$	0	0
$981$	52206.0	0.00173200
$982$	0	0
$983$	−2.77362e7	−0.915510	−0.457755	−	0.889078i	$-0.651346\pi$
$983$	−2.77362e7	−0.915510	−0.457755	+	0.889078i	$0.651346\pi$
$984$	0	0
$985$	1.37906e7	0.452890
$986$	0	0
$987$	5.25644e6	0.171751
$988$	0	0
$989$	−1.55687e7	−0.506131
$990$	0	0
$991$	2.35437e7	0.761536	0.380768	−	0.924671i	$-0.375660\pi$
$991$	2.35437e7	0.761536	0.380768	+	0.924671i	$0.375660\pi$
$992$	0	0
$993$	−8.29123e6	−0.266837
$994$	0	0
$995$	−9.52559e7	−3.05024
$996$	0	0
$997$	−4.05582e7	−1.29223	−0.646117	−	0.763239i	$-0.723608\pi$
$997$	−4.05582e7	−1.29223	−0.646117	+	0.763239i	$0.723608\pi$
$998$	0	0
$999$	−4.36015e7	−1.38225

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	112.6.a.i.1.1		2
3.2	odd	2		1008.6.a.bd.1.1		2
4.3	odd	2		56.6.a.e.1.2	✓	2
7.6	odd	2		784.6.a.u.1.2		2
8.3	odd	2		448.6.a.v.1.1		2
8.5	even	2		448.6.a.t.1.2		2
12.11	even	2		504.6.a.i.1.1		2
28.3	even	6		392.6.i.i.177.2		4
28.11	odd	6		392.6.i.j.177.1		4
28.19	even	6		392.6.i.i.361.2		4
28.23	odd	6		392.6.i.j.361.1		4
28.27	even	2		392.6.a.d.1.1		2

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
56.6.a.e.1.2	✓	2	4.3	odd	2
112.6.a.i.1.1		2	1.1	even	1	trivial
392.6.a.d.1.1		2	28.27	even	2
392.6.i.i.177.2		4	28.3	even	6
392.6.i.i.361.2		4	28.19	even	6
392.6.i.j.177.1		4	28.11	odd	6
392.6.i.j.361.1		4	28.23	odd	6
448.6.a.t.1.2		2	8.5	even	2
448.6.a.v.1.1		2	8.3	odd	2
504.6.a.i.1.1		2	12.11	even	2
784.6.a.u.1.2		2	7.6	odd	2
1008.6.a.bd.1.1		2	3.2	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 $\F_{q}$

Level:	\( N \)	\(=\)	\( 112 = 2^{4} \cdot 7 \)
Weight:	\( k \)	\(=\)	\( 6 \)
Character orbit:	\([\chi]\)	\(=\)	112.a (trivial)

\(f(q)\)	\(=\)	\(q-15.5742 q^{3} +96.7225 q^{5} +49.0000 q^{7} -0.445054 q^{9} +O(q^{10})\)	\(q-15.5742 q^{3} +96.7225 q^{5} +49.0000 q^{7} -0.445054 q^{9} -281.445 q^{11} -269.393 q^{13} -1506.37 q^{15} +1719.45 q^{17} +1172.84 q^{19} -763.135 q^{21} -785.341 q^{23} +6230.25 q^{25} +3791.46 q^{27} +6149.46 q^{29} +7006.58 q^{31} +4383.27 q^{33} +4739.40 q^{35} -11499.9 q^{37} +4195.57 q^{39} -13245.3 q^{41} +19824.2 q^{43} -43.0467 q^{45} -6887.96 q^{47} +2401.00 q^{49} -26778.9 q^{51} +8654.97 q^{53} -27222.1 q^{55} -18266.0 q^{57} +47856.6 q^{59} +52762.1 q^{61} -21.8076 q^{63} -26056.4 q^{65} +24040.2 q^{67} +12231.0 q^{69} -12540.9 q^{71} +4079.29 q^{73} -97031.0 q^{75} -13790.8 q^{77} +11919.1 q^{79} -58940.7 q^{81} -81916.2 q^{83} +166309. q^{85} -95772.7 q^{87} -96968.7 q^{89} -13200.2 q^{91} -109122. q^{93} +113440. q^{95} -26410.7 q^{97} +125.258 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 2 q + 6 q^{3} + 82 q^{5} + 98 q^{7} + 222 q^{9}+O(q^{10}) \) 2 * q + 6 * q^3 + 82 * q^5 + 98 * q^7 + 222 * q^9	\( 2 q + 6 q^{3} + 82 q^{5} + 98 q^{7} + 222 q^{9} - 340 q^{11} + 910 q^{13} - 1824 q^{15} + 3216 q^{17} + 674 q^{19} + 294 q^{21} + 1104 q^{23} + 3322 q^{25} + 3348 q^{27} + 8064 q^{29} + 6212 q^{31} + 3120 q^{33} + 4018 q^{35} - 8512 q^{37} + 29640 q^{39} - 1304 q^{41} + 10004 q^{43} - 3318 q^{45} + 12748 q^{47} + 4802 q^{49} + 5508 q^{51} - 11220 q^{53} - 26360 q^{55} - 29028 q^{57} + 12018 q^{59} + 102738 q^{61} + 10878 q^{63} - 43420 q^{65} - 24136 q^{67} + 52992 q^{69} - 89720 q^{71} - 55588 q^{73} - 159774 q^{75} - 16660 q^{77} - 48824 q^{79} - 122562 q^{81} - 35782 q^{83} + 144276 q^{85} - 54468 q^{87} - 18300 q^{89} + 44590 q^{91} - 126264 q^{93} + 120784 q^{95} - 69984 q^{97} - 12900 q^{99}+O(q^{100}) \) 2 * q + 6 * q^3 + 82 * q^5 + 98 * q^7 + 222 * q^9 - 340 * q^11 + 910 * q^13 - 1824 * q^15 + 3216 * q^17 + 674 * q^19 + 294 * q^21 + 1104 * q^23 + 3322 * q^25 + 3348 * q^27 + 8064 * q^29 + 6212 * q^31 + 3120 * q^33 + 4018 * q^35 - 8512 * q^37 + 29640 * q^39 - 1304 * q^41 + 10004 * q^43 - 3318 * q^45 + 12748 * q^47 + 4802 * q^49 + 5508 * q^51 - 11220 * q^53 - 26360 * q^55 - 29028 * q^57 + 12018 * q^59 + 102738 * q^61 + 10878 * q^63 - 43420 * q^65 - 24136 * q^67 + 52992 * q^69 - 89720 * q^71 - 55588 * q^73 - 159774 * q^75 - 16660 * q^77 - 48824 * q^79 - 122562 * q^81 - 35782 * q^83 + 144276 * q^85 - 54468 * q^87 - 18300 * q^89 + 44590 * q^91 - 126264 * q^93 + 120784 * q^95 - 69984 * q^97 - 12900 * q^99

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

Related objects

Downloads

Learn more