LMFDB - Embedded newform 112.8.a.i.1.2

Show commands: Magma / Pari/GP / SageMath

Newspace parameters

comment:Compute space of new eigenforms

gp:[N,k,chi] = [112,8,Mod(1,112)] mf = mfinit([N,k,chi],0) lf = mfeigenbasis(mf)

magma://Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code chi := DirichletCharacter("112.1"); S:= CuspForms(chi, 8); N := Newforms(S);

sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(112, base_ring=CyclotomicField(2)) chi = DirichletCharacter(H, H._module([0, 0, 0])) N = Newforms(chi, 8, names="a")

Level:	$ N $	$=$	$ 112 = 2^{4} \cdot 7 $
Weight:	$ k $	$=$	$ 8 $
Character orbit:	$[\chi]$	$=$	112.a (trivial)

Newform invariants

comment:select newform

sage:traces = [2,0,14] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(3)] == traces)

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

Self dual:	yes
Analytic conductor:	$34.9871228542$
Analytic rank:	$1$
Dimension:	$2$
Coefficient field:	$\Q(\sqrt{3529}) $
comment:defining polynomial gp:f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{2} - x - 882 $ x^2 - x - 882
Coefficient ring:	$\Z[a_1, a_2, a_3]$
Coefficient ring index:	$ 2 $
Twist minimal:	no (minimal twist has level 28)
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.2
Root			$-29.2027$ 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)

$f(q)$	$=$	$q+66.4054 q^{3} -157.216 q^{5} -343.000 q^{7} +2222.68 q^{9} -6209.03 q^{11} +5380.35 q^{13} -10440.0 q^{15} -10994.9 q^{17} -11703.1 q^{19} -22777.0 q^{21} -106141. q^{23} -53408.1 q^{25} +2369.04 q^{27} -51562.7 q^{29} +247557. q^{31} -412313. q^{33} +53925.1 q^{35} +433678. q^{37} +357284. q^{39} +322819. q^{41} -878703. q^{43} -349440. q^{45} -655126. q^{47} +117649. q^{49} -730121. q^{51} -444837. q^{53} +976159. q^{55} -777146. q^{57} -2.14545e6 q^{59} +592902. q^{61} -762378. q^{63} -845878. q^{65} -1.72866e6 q^{67} -7.04833e6 q^{69} -1.58060e6 q^{71} -4.33164e6 q^{73} -3.54658e6 q^{75} +2.12970e6 q^{77} +6.08518e6 q^{79} -4.70367e6 q^{81} +8.10357e6 q^{83} +1.72858e6 q^{85} -3.42404e6 q^{87} +9.86032e6 q^{89} -1.84546e6 q^{91} +1.64391e7 q^{93} +1.83991e6 q^{95} -171786. q^{97} -1.38007e7 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 2 q + 14 q^{3} + 42 q^{5} - 686 q^{7} + 2782 q^{9} - 7428 q^{11} + 11830 q^{13} - 20880 q^{15} + 15792 q^{17} - 26614 q^{19} - 4802 q^{21} - 32640 q^{23} - 91846 q^{25} + 87668 q^{27} - 158016 q^{29} + 180740 q^{31}+ \cdots - 14482452 q^{99}+O(q^{100}) $ 2 * q + 14 * q^3 + 42 * q^5 - 686 * q^7 + 2782 * q^9 - 7428 * q^11 + 11830 * q^13 - 20880 * q^15 + 15792 * q^17 - 26614 * q^19 - 4802 * q^21 - 32640 * q^23 - 91846 * q^25 + 87668 * q^27 - 158016 * q^29 + 180740 * q^31 - 348432 * q^33 - 14406 * q^35 - 45824 * q^37 + 19288 * q^39 - 321720 * q^41 - 1023868 * q^43 - 238014 * q^45 - 1665972 * q^47 + 235298 * q^49 - 2133900 * q^51 - 410628 * q^53 + 733320 * q^55 + 4268 * q^57 - 1702134 * q^59 - 547526 * q^61 - 954226 * q^63 + 438996 * q^65 + 2590616 * q^67 - 10900176 * q^69 - 4129272 * q^71 - 8008868 * q^73 - 1532230 * q^75 + 2547804 * q^77 - 2470456 * q^79 - 10397042 * q^81 + 9900786 * q^83 + 7064964 * q^85 + 2154684 * q^87 + 15423492 * q^89 - 4057690 * q^91 + 19940648 * q^93 - 1130592 * q^95 - 17377472 * q^97 - 14482452 * q^99

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
$3$	66.4054	1.41997	0.709985	−	0.704217i	$-0.248702\pi$
$3$	66.4054	1.41997	0.709985	+	0.704217i	$0.248702\pi$
$4$	0	0
$5$	−157.216	−0.562474	−0.281237	−	0.959638i	$-0.590745\pi$
$5$	−157.216	−0.562474	−0.281237	+	0.959638i	$0.590745\pi$
$6$	0	0
$7$	−343.000	−0.377964
$7$	−343.000	−0.377964
$8$	0	0
$9$	2222.68	1.01631
$10$	0	0
$11$	−6209.03	−1.40653	−0.703265	−	0.710928i	$-0.748275\pi$
$11$	−6209.03	−1.40653	−0.703265	+	0.710928i	$0.748275\pi$
$12$	0	0
$13$	5380.35	0.679218	0.339609	−	0.940567i	$-0.389705\pi$
$13$	5380.35	0.679218	0.339609	+	0.940567i	$0.389705\pi$
$14$	0	0
$15$	−10440.0	−0.798695
$16$	0	0
$17$	−10994.9	−0.542776	−0.271388	−	0.962470i	$-0.587483\pi$
$17$	−10994.9	−0.542776	−0.271388	+	0.962470i	$0.587483\pi$
$18$	0	0
$19$	−11703.1	−0.391437	−0.195718	−	0.980660i	$-0.562704\pi$
$19$	−11703.1	−0.391437	−0.195718	+	0.980660i	$0.562704\pi$
$20$	0	0
$21$	−22777.0	−0.536698
$22$	0	0
$23$	−106141.	−1.81901	−0.909506	−	0.415691i	$-0.863540\pi$
$23$	−106141.	−1.81901	−0.909506	+	0.415691i	$0.863540\pi$
$24$	0	0
$25$	−53408.1	−0.683623
$26$	0	0
$27$	2369.04	0.0231632
$28$	0	0
$29$	−51562.7	−0.392593	−0.196297	−	0.980545i	$-0.562892\pi$
$29$	−51562.7	−0.392593	−0.196297	+	0.980545i	$0.562892\pi$
$30$	0	0
$31$	247557.	1.49248	0.746240	−	0.665677i	$-0.231857\pi$
$31$	247557.	1.49248	0.746240	+	0.665677i	$0.231857\pi$
$32$	0	0
$33$	−412313.	−1.99723
$34$	0	0
$35$	53925.1	0.212595
$36$	0	0
$37$	433678.	1.40754	0.703771	−	0.710427i	$-0.251498\pi$
$37$	433678.	1.40754	0.703771	+	0.710427i	$0.251498\pi$
$38$	0	0
$39$	357284.	0.964468
$40$	0	0
$41$	322819.	0.731501	0.365751	−	0.930713i	$-0.380812\pi$
$41$	322819.	0.731501	0.365751	+	0.930713i	$0.380812\pi$
$42$	0	0
$43$	−878703.	−1.68540	−0.842699	−	0.538385i	$-0.819035\pi$
$43$	−878703.	−1.68540	−0.842699	+	0.538385i	$0.819035\pi$
$44$	0	0
$45$	−349440.	−0.571649
$46$	0	0
$47$	−655126.	−0.920413	−0.460206	−	0.887812i	$-0.652225\pi$
$47$	−655126.	−0.920413	−0.460206	+	0.887812i	$0.652225\pi$
$48$	0	0
$49$	117649.	0.142857
$50$	0	0
$51$	−730121.	−0.770725
$52$	0	0
$53$	−444837.	−0.410426	−0.205213	−	0.978717i	$-0.565789\pi$
$53$	−444837.	−0.410426	−0.205213	+	0.978717i	$0.565789\pi$
$54$	0	0
$55$	976159.	0.791136
$56$	0	0
$57$	−777146.	−0.555828
$58$	0	0
$59$	−2.14545e6	−1.35999	−0.679996	−	0.733216i	$-0.738019\pi$
$59$	−2.14545e6	−1.35999	−0.679996	+	0.733216i	$0.738019\pi$
$60$	0	0
$61$	592902.	0.334448	0.167224	−	0.985919i	$-0.446520\pi$
$61$	592902.	0.334448	0.167224	+	0.985919i	$0.446520\pi$
$62$	0	0
$63$	−762378.	−0.384130
$64$	0	0
$65$	−845878.	−0.382042
$66$	0	0
$67$	−1.72866e6	−0.702180	−0.351090	−	0.936342i	$-0.614189\pi$
$67$	−1.72866e6	−0.702180	−0.351090	+	0.936342i	$0.614189\pi$
$68$	0	0
$69$	−7.04833e6	−2.58294
$70$	0	0
$71$	−1.58060e6	−0.524105	−0.262052	−	0.965054i	$-0.584399\pi$
$71$	−1.58060e6	−0.524105	−0.262052	+	0.965054i	$0.584399\pi$
$72$	0	0
$73$	−4.33164e6	−1.30323	−0.651617	−	0.758548i	$-0.725909\pi$
$73$	−4.33164e6	−1.30323	−0.651617	+	0.758548i	$0.725909\pi$
$74$	0	0
$75$	−3.54658e6	−0.970724
$76$	0	0
$77$	2.12970e6	0.531618
$78$	0	0
$79$	6.08518e6	1.38860	0.694302	−	0.719684i	$-0.255713\pi$
$79$	6.08518e6	1.38860	0.694302	+	0.719684i	$0.255713\pi$
$80$	0	0
$81$	−4.70367e6	−0.983421
$82$	0	0
$83$	8.10357e6	1.55562	0.777809	−	0.628500i	$-0.216331\pi$
$83$	8.10357e6	1.55562	0.777809	+	0.628500i	$0.216331\pi$
$84$	0	0
$85$	1.72858e6	0.305297
$86$	0	0
$87$	−3.42404e6	−0.557470
$88$	0	0
$89$	9.86032e6	1.48261	0.741303	−	0.671170i	$-0.234208\pi$
$89$	9.86032e6	1.48261	0.741303	+	0.671170i	$0.234208\pi$
$90$	0	0
$91$	−1.84546e6	−0.256720
$92$	0	0
$93$	1.64391e7	2.11928
$94$	0	0
$95$	1.83991e6	0.220173
$96$	0	0
$97$	−171786.	−0.0191111	−0.00955555	−	0.999954i	$-0.503042\pi$
$97$	−171786.	−0.0191111	−0.00955555	+	0.999954i	$0.503042\pi$
$98$	0	0
$99$	−1.38007e7	−1.42947
$100$	0	0
$101$	1.50137e6	0.144998	0.0724991	−	0.997368i	$-0.476903\pi$
$101$	1.50137e6	0.144998	0.0724991	+	0.997368i	$0.476903\pi$
$102$	0	0
$103$	1.40099e7	1.26330	0.631648	−	0.775256i	$-0.282379\pi$
$103$	1.40099e7	1.26330	0.631648	+	0.775256i	$0.282379\pi$
$104$	0	0
$105$	3.58092e6	0.301878
$106$	0	0
$107$	−2.36510e7	−1.86641	−0.933203	−	0.359348i	$-0.882999\pi$
$107$	−2.36510e7	−1.86641	−0.933203	+	0.359348i	$0.882999\pi$
$108$	0	0
$109$	−1.82455e7	−1.34947	−0.674736	−	0.738059i	$-0.735743\pi$
$109$	−1.82455e7	−1.34947	−0.674736	+	0.738059i	$0.735743\pi$
$110$	0	0
$111$	2.87985e7	1.99867
$112$	0	0
$113$	6.85019e6	0.446610	0.223305	−	0.974749i	$-0.428315\pi$
$113$	6.85019e6	0.446610	0.223305	+	0.974749i	$0.428315\pi$
$114$	0	0
$115$	1.66871e7	1.02315
$116$	0	0
$117$	1.19588e7	0.690297
$118$	0	0
$119$	3.77126e6	0.205150
$120$	0	0
$121$	1.90648e7	0.978328
$122$	0	0
$123$	2.14369e7	1.03871
$124$	0	0
$125$	2.06791e7	0.946994
$126$	0	0
$127$	−2.93488e7	−1.27139	−0.635693	−	0.771942i	$-0.719286\pi$
$127$	−2.93488e7	−1.27139	−0.635693	+	0.771942i	$0.719286\pi$
$128$	0	0
$129$	−5.83506e7	−2.39321
$130$	0	0
$131$	1.03093e7	0.400663	0.200332	−	0.979728i	$-0.435798\pi$
$131$	1.03093e7	0.400663	0.200332	+	0.979728i	$0.435798\pi$
$132$	0	0
$133$	4.01415e6	0.147949
$134$	0	0
$135$	−372451.	−0.0130287
$136$	0	0
$137$	1.13507e7	0.377138	0.188569	−	0.982060i	$-0.439615\pi$
$137$	1.13507e7	0.377138	0.188569	+	0.982060i	$0.439615\pi$
$138$	0	0
$139$	5.67661e7	1.79282	0.896412	−	0.443222i	$-0.146165\pi$
$139$	5.67661e7	1.79282	0.896412	+	0.443222i	$0.146165\pi$
$140$	0	0
$141$	−4.35039e7	−1.30696
$142$	0	0
$143$	−3.34067e7	−0.955340
$144$	0	0
$145$	8.10649e6	0.220823
$146$	0	0
$147$	7.81253e6	0.202853
$148$	0	0
$149$	2.71126e7	0.671458	0.335729	−	0.941959i	$-0.391017\pi$
$149$	2.71126e7	0.671458	0.335729	+	0.941959i	$0.391017\pi$
$150$	0	0
$151$	−3.23586e6	−0.0764839	−0.0382420	−	0.999269i	$-0.512176\pi$
$151$	−3.23586e6	−0.0764839	−0.0382420	+	0.999269i	$0.512176\pi$
$152$	0	0
$153$	−2.44381e7	−0.551630
$154$	0	0
$155$	−3.89199e7	−0.839481
$156$	0	0
$157$	8.43843e7	1.74025	0.870127	−	0.492827i	$-0.164036\pi$
$157$	8.43843e7	1.74025	0.870127	+	0.492827i	$0.164036\pi$
$158$	0	0
$159$	−2.95395e7	−0.582792
$160$	0	0
$161$	3.64063e7	0.687522
$162$	0	0
$163$	6.28796e7	1.13724	0.568622	−	0.822599i	$-0.307477\pi$
$163$	6.28796e7	1.13724	0.568622	+	0.822599i	$0.307477\pi$
$164$	0	0
$165$	6.48222e7	1.12339
$166$	0	0
$167$	−7.14808e7	−1.18763	−0.593816	−	0.804601i	$-0.702379\pi$
$167$	−7.14808e7	−1.18763	−0.593816	+	0.804601i	$0.702379\pi$
$168$	0	0
$169$	−3.38003e7	−0.538663
$170$	0	0
$171$	−2.60121e7	−0.397822
$172$	0	0
$173$	6.60131e7	0.969323	0.484662	−	0.874702i	$-0.338943\pi$
$173$	6.60131e7	0.969323	0.484662	+	0.874702i	$0.338943\pi$
$174$	0	0
$175$	1.83190e7	0.258385
$176$	0	0
$177$	−1.42470e8	−1.93115
$178$	0	0
$179$	9.42206e6	0.122789	0.0613946	−	0.998114i	$-0.480445\pi$
$179$	9.42206e6	0.122789	0.0613946	+	0.998114i	$0.480445\pi$
$180$	0	0
$181$	3.65024e7	0.457558	0.228779	−	0.973478i	$-0.426527\pi$
$181$	3.65024e7	0.457558	0.228779	+	0.973478i	$0.426527\pi$
$182$	0	0
$183$	3.93719e7	0.474906
$184$	0	0
$185$	−6.81812e7	−0.791705
$186$	0	0
$187$	6.82677e7	0.763431
$188$	0	0
$189$	−812581.	−0.00875488
$190$	0	0
$191$	8.40614e7	0.872930	0.436465	−	0.899721i	$-0.356230\pi$
$191$	8.40614e7	0.872930	0.436465	+	0.899721i	$0.356230\pi$
$192$	0	0
$193$	−6.67421e7	−0.668266	−0.334133	−	0.942526i	$-0.608443\pi$
$193$	−6.67421e7	−0.668266	−0.334133	+	0.942526i	$0.608443\pi$
$194$	0	0
$195$	−5.61709e7	−0.542488
$196$	0	0
$197$	−5.11607e7	−0.476765	−0.238383	−	0.971171i	$-0.576617\pi$
$197$	−5.11607e7	−0.476765	−0.238383	+	0.971171i	$0.576617\pi$
$198$	0	0
$199$	−1.18167e8	−1.06294	−0.531471	−	0.847076i	$-0.678361\pi$
$199$	−1.18167e8	−1.06294	−0.531471	+	0.847076i	$0.678361\pi$
$200$	0	0
$201$	−1.14793e8	−0.997075
$202$	0	0
$203$	1.76860e7	0.148386
$204$	0	0
$205$	−5.07523e7	−0.411450
$206$	0	0
$207$	−2.35917e8	−1.84868
$208$	0	0
$209$	7.26646e7	0.550568
$210$	0	0
$211$	8.10808e7	0.594196	0.297098	−	0.954847i	$-0.403981\pi$
$211$	8.10808e7	0.594196	0.297098	+	0.954847i	$0.403981\pi$
$212$	0	0
$213$	−1.04960e8	−0.744212
$214$	0	0
$215$	1.38146e8	0.947992
$216$	0	0
$217$	−8.49119e7	−0.564105
$218$	0	0
$219$	−2.87644e8	−1.85055
$220$	0	0
$221$	−5.91565e7	−0.368663
$222$	0	0
$223$	−2.40244e8	−1.45072	−0.725362	−	0.688368i	$-0.758328\pi$
$223$	−2.40244e8	−1.45072	−0.725362	+	0.688368i	$0.758328\pi$
$224$	0	0
$225$	−1.18709e8	−0.694775
$226$	0	0
$227$	−7.63393e7	−0.433170	−0.216585	−	0.976264i	$-0.569492\pi$
$227$	−7.63393e7	−0.433170	−0.216585	+	0.976264i	$0.569492\pi$
$228$	0	0
$229$	−6.92500e7	−0.381062	−0.190531	−	0.981681i	$-0.561021\pi$
$229$	−6.92500e7	−0.381062	−0.190531	+	0.981681i	$0.561021\pi$
$230$	0	0
$231$	1.41423e8	0.754882
$232$	0	0
$233$	−6.15262e7	−0.318650	−0.159325	−	0.987226i	$-0.550932\pi$
$233$	−6.15262e7	−0.318650	−0.159325	+	0.987226i	$0.550932\pi$
$234$	0	0
$235$	1.02996e8	0.517708
$236$	0	0
$237$	4.04089e8	1.97178
$238$	0	0
$239$	9.93248e7	0.470614	0.235307	−	0.971921i	$-0.424390\pi$
$239$	9.93248e7	0.470614	0.235307	+	0.971921i	$0.424390\pi$
$240$	0	0
$241$	−1.54605e8	−0.711480	−0.355740	−	0.934585i	$-0.615771\pi$
$241$	−1.54605e8	−0.711480	−0.355740	+	0.934585i	$0.615771\pi$
$242$	0	0
$243$	−3.17530e8	−1.41959
$244$	0	0
$245$	−1.84963e7	−0.0803534
$246$	0	0
$247$	−6.29665e7	−0.265871
$248$	0	0
$249$	5.38121e8	2.20893
$250$	0	0
$251$	−8.53238e7	−0.340575	−0.170287	−	0.985394i	$-0.554470\pi$
$251$	−8.53238e7	−0.340575	−0.170287	+	0.985394i	$0.554470\pi$
$252$	0	0
$253$	6.59032e8	2.55850
$254$	0	0
$255$	1.14787e8	0.433513
$256$	0	0
$257$	−4.27723e8	−1.57180	−0.785900	−	0.618354i	$-0.787800\pi$
$257$	−4.27723e8	−1.57180	−0.785900	+	0.618354i	$0.787800\pi$
$258$	0	0
$259$	−1.48751e8	−0.532001
$260$	0	0
$261$	−1.14607e8	−0.398997
$262$	0	0
$263$	4.67109e8	1.58333	0.791667	−	0.610953i	$-0.209213\pi$
$263$	4.67109e8	1.58333	0.791667	+	0.610953i	$0.209213\pi$
$264$	0	0
$265$	6.99355e7	0.230854
$266$	0	0
$267$	6.54778e8	2.10526
$268$	0	0
$269$	2.00003e8	0.626474	0.313237	−	0.949675i	$-0.398587\pi$
$269$	2.00003e8	0.626474	0.313237	+	0.949675i	$0.398587\pi$
$270$	0	0
$271$	2.32905e8	0.710864	0.355432	−	0.934702i	$-0.384334\pi$
$271$	2.32905e8	0.710864	0.355432	+	0.934702i	$0.384334\pi$
$272$	0	0
$273$	−1.22549e8	−0.364535
$274$	0	0
$275$	3.31612e8	0.961537
$276$	0	0
$277$	4.46817e7	0.126314	0.0631569	−	0.998004i	$-0.479883\pi$
$277$	4.46817e7	0.126314	0.0631569	+	0.998004i	$0.479883\pi$
$278$	0	0
$279$	5.50238e8	1.51683
$280$	0	0
$281$	−1.38876e8	−0.373385	−0.186692	−	0.982418i	$-0.559777\pi$
$281$	−1.38876e8	−0.373385	−0.186692	+	0.982418i	$0.559777\pi$
$282$	0	0
$283$	−3.28528e8	−0.861627	−0.430814	−	0.902441i	$-0.641773\pi$
$283$	−3.28528e8	−0.861627	−0.430814	+	0.902441i	$0.641773\pi$
$284$	0	0
$285$	1.22180e8	0.312639
$286$	0	0
$287$	−1.10727e8	−0.276482
$288$	0	0
$289$	−2.89451e8	−0.705394
$290$	0	0
$291$	−1.14075e7	−0.0271372
$292$	0	0
$293$	−1.91796e8	−0.445455	−0.222727	−	0.974881i	$-0.571496\pi$
$293$	−1.91796e8	−0.445455	−0.222727	+	0.974881i	$0.571496\pi$
$294$	0	0
$295$	3.37300e8	0.764960
$296$	0	0
$297$	−1.47094e7	−0.0325798
$298$	0	0
$299$	−5.71076e8	−1.23550
$300$	0	0
$301$	3.01395e8	0.637021
$302$	0	0
$303$	9.96990e7	0.205893
$304$	0	0
$305$	−9.32138e7	−0.188118
$306$	0	0
$307$	−1.98092e8	−0.390734	−0.195367	−	0.980730i	$-0.562590\pi$
$307$	−1.98092e8	−0.390734	−0.195367	+	0.980730i	$0.562590\pi$
$308$	0	0
$309$	9.30333e8	1.79384
$310$	0	0
$311$	−4.00719e8	−0.755403	−0.377701	−	0.925927i	$-0.623285\pi$
$311$	−4.00719e8	−0.755403	−0.377701	+	0.925927i	$0.623285\pi$
$312$	0	0
$313$	5.31343e8	0.979421	0.489711	−	0.871885i	$-0.337102\pi$
$313$	5.31343e8	0.979421	0.489711	+	0.871885i	$0.337102\pi$
$314$	0	0
$315$	1.19858e8	0.216063
$316$	0	0
$317$	−4.38248e7	−0.0772702	−0.0386351	−	0.999253i	$-0.512301\pi$
$317$	−4.38248e7	−0.0772702	−0.0386351	+	0.999253i	$0.512301\pi$
$318$	0	0
$319$	3.20154e8	0.552194
$320$	0	0
$321$	−1.57055e9	−2.65024
$322$	0	0
$323$	1.28674e8	0.212462
$324$	0	0
$325$	−2.87354e8	−0.464329
$326$	0	0
$327$	−1.21160e9	−1.91621
$328$	0	0
$329$	2.24708e8	0.347883
$330$	0	0
$331$	−3.64466e8	−0.552407	−0.276204	−	0.961099i	$-0.589076\pi$
$331$	−3.64466e8	−0.552407	−0.276204	+	0.961099i	$0.589076\pi$
$332$	0	0
$333$	9.63925e8	1.43050
$334$	0	0
$335$	2.71774e8	0.394958
$336$	0	0
$337$	9.00453e8	1.28161	0.640806	−	0.767703i	$-0.278600\pi$
$337$	9.00453e8	1.28161	0.640806	+	0.767703i	$0.278600\pi$
$338$	0	0
$339$	4.54890e8	0.634172
$340$	0	0
$341$	−1.53709e9	−2.09922
$342$	0	0
$343$	−4.03536e7	−0.0539949
$344$	0	0
$345$	1.10811e9	1.45284
$346$	0	0
$347$	2.69630e8	0.346429	0.173215	−	0.984884i	$-0.444585\pi$
$347$	2.69630e8	0.346429	0.173215	+	0.984884i	$0.444585\pi$
$348$	0	0
$349$	6.42732e8	0.809358	0.404679	−	0.914459i	$-0.367383\pi$
$349$	6.42732e8	0.809358	0.404679	+	0.914459i	$0.367383\pi$
$350$	0	0
$351$	1.27463e7	0.0157329
$352$	0	0
$353$	−9.97045e8	−1.20643	−0.603217	−	0.797577i	$-0.706115\pi$
$353$	−9.97045e8	−1.20643	−0.603217	+	0.797577i	$0.706115\pi$
$354$	0	0
$355$	2.48496e8	0.294795
$356$	0	0
$357$	2.50432e8	0.291307
$358$	0	0
$359$	−1.66135e9	−1.89509	−0.947545	−	0.319622i	$-0.896444\pi$
$359$	−1.66135e9	−1.89509	−0.947545	+	0.319622i	$0.896444\pi$
$360$	0	0
$361$	−7.56910e8	−0.846777
$362$	0	0
$363$	1.26601e9	1.38919
$364$	0	0
$365$	6.81004e8	0.733034
$366$	0	0
$367$	−7.20860e8	−0.761237	−0.380618	−	0.924732i	$-0.624289\pi$
$367$	−7.20860e8	−0.761237	−0.380618	+	0.924732i	$0.624289\pi$
$368$	0	0
$369$	7.17521e8	0.743434
$370$	0	0
$371$	1.52579e8	0.155126
$372$	0	0
$373$	−4.78878e8	−0.477798	−0.238899	−	0.971044i	$-0.576786\pi$
$373$	−4.78878e8	−0.477798	−0.238899	+	0.971044i	$0.576786\pi$
$374$	0	0
$375$	1.37321e9	1.34470
$376$	0	0
$377$	−2.77426e8	−0.266656
$378$	0	0
$379$	1.03267e9	0.974373	0.487186	−	0.873298i	$-0.338023\pi$
$379$	1.03267e9	0.974373	0.487186	+	0.873298i	$0.338023\pi$
$380$	0	0
$381$	−1.94892e9	−1.80533
$382$	0	0
$383$	5.93443e8	0.539738	0.269869	−	0.962897i	$-0.413020\pi$
$383$	5.93443e8	0.539738	0.269869	+	0.962897i	$0.413020\pi$
$384$	0	0
$385$	−3.34823e8	−0.299021
$386$	0	0
$387$	−1.95307e9	−1.71289
$388$	0	0
$389$	−2.15324e9	−1.85468	−0.927340	−	0.374220i	$-0.877911\pi$
$389$	−2.15324e9	−1.85468	−0.927340	+	0.374220i	$0.877911\pi$
$390$	0	0
$391$	1.16701e9	0.987316
$392$	0	0
$393$	6.84593e8	0.568930
$394$	0	0
$395$	−9.56688e8	−0.781053
$396$	0	0
$397$	1.88246e8	0.150994	0.0754969	−	0.997146i	$-0.475946\pi$
$397$	1.88246e8	0.150994	0.0754969	+	0.997146i	$0.475946\pi$
$398$	0	0
$399$	2.66561e8	0.210083
$400$	0	0
$401$	−7.58165e8	−0.587162	−0.293581	−	0.955934i	$-0.594847\pi$
$401$	−7.58165e8	−0.587162	−0.293581	+	0.955934i	$0.594847\pi$
$402$	0	0
$403$	1.33194e9	1.01372
$404$	0	0
$405$	7.39494e8	0.553149
$406$	0	0
$407$	−2.69272e9	−1.97975
$408$	0	0
$409$	−1.50707e9	−1.08919	−0.544593	−	0.838701i	$-0.683316\pi$
$409$	−1.50707e9	−1.08919	−0.544593	+	0.838701i	$0.683316\pi$
$410$	0	0
$411$	7.53747e8	0.535524
$412$	0	0
$413$	7.35890e8	0.514029
$414$	0	0
$415$	−1.27401e9	−0.874994
$416$	0	0
$417$	3.76958e9	2.54575
$418$	0	0
$419$	−1.51449e9	−1.00581	−0.502907	−	0.864341i	$-0.667736\pi$
$419$	−1.51449e9	−1.00581	−0.502907	+	0.864341i	$0.667736\pi$
$420$	0	0
$421$	−1.05648e9	−0.690037	−0.345019	−	0.938596i	$-0.612127\pi$
$421$	−1.05648e9	−0.690037	−0.345019	+	0.938596i	$0.612127\pi$
$422$	0	0
$423$	−1.45613e9	−0.935427
$424$	0	0
$425$	5.87217e8	0.371054
$426$	0	0
$427$	−2.03365e8	−0.126409
$428$	0	0
$429$	−2.21839e9	−1.35655
$430$	0	0
$431$	−2.52464e9	−1.51890	−0.759450	−	0.650565i	$-0.774532\pi$
$431$	−2.52464e9	−1.51890	−0.759450	+	0.650565i	$0.774532\pi$
$432$	0	0
$433$	4.25297e8	0.251759	0.125879	−	0.992046i	$-0.459825\pi$
$433$	4.25297e8	0.251759	0.125879	+	0.992046i	$0.459825\pi$
$434$	0	0
$435$	5.38315e8	0.313562
$436$	0	0
$437$	1.24217e9	0.712028
$438$	0	0
$439$	−1.07323e9	−0.605433	−0.302717	−	0.953081i	$-0.597894\pi$
$439$	−1.07323e9	−0.605433	−0.302717	+	0.953081i	$0.597894\pi$
$440$	0	0
$441$	2.61496e8	0.145187
$442$	0	0
$443$	5.79179e8	0.316519	0.158259	−	0.987398i	$-0.449412\pi$
$443$	5.79179e8	0.316519	0.158259	+	0.987398i	$0.449412\pi$
$444$	0	0
$445$	−1.55020e9	−0.833927
$446$	0	0
$447$	1.80042e9	0.953450
$448$	0	0
$449$	−2.44352e9	−1.27396	−0.636978	−	0.770882i	$-0.719816\pi$
$449$	−2.44352e9	−1.27396	−0.636978	+	0.770882i	$0.719816\pi$
$450$	0	0
$451$	−2.00439e9	−1.02888
$452$	0	0
$453$	−2.14878e8	−0.108605
$454$	0	0
$455$	2.90136e8	0.144398
$456$	0	0
$457$	−5.73498e8	−0.281077	−0.140538	−	0.990075i	$-0.544883\pi$
$457$	−5.73498e8	−0.281077	−0.140538	+	0.990075i	$0.544883\pi$
$458$	0	0
$459$	−2.60474e7	−0.0125724
$460$	0	0
$461$	4.58794e8	0.218105	0.109052	−	0.994036i	$-0.465218\pi$
$461$	4.58794e8	0.218105	0.109052	+	0.994036i	$0.465218\pi$
$462$	0	0
$463$	3.14598e8	0.147307	0.0736533	−	0.997284i	$-0.476534\pi$
$463$	3.14598e8	0.147307	0.0736533	+	0.997284i	$0.476534\pi$
$464$	0	0
$465$	−2.58449e9	−1.19204
$466$	0	0
$467$	−9.42124e8	−0.428054	−0.214027	−	0.976828i	$-0.568658\pi$
$467$	−9.42124e8	−0.428054	−0.214027	+	0.976828i	$0.568658\pi$
$468$	0	0
$469$	5.92932e8	0.265399
$470$	0	0
$471$	5.60357e9	2.47111
$472$	0	0
$473$	5.45589e9	2.37056
$474$	0	0
$475$	6.25038e8	0.267595
$476$	0	0
$477$	−9.88727e8	−0.417121
$478$	0	0
$479$	−3.10364e9	−1.29032	−0.645160	−	0.764048i	$-0.723209\pi$
$479$	−3.10364e9	−1.29032	−0.645160	+	0.764048i	$0.723209\pi$
$480$	0	0
$481$	2.33334e9	0.956027
$482$	0	0
$483$	2.41758e9	0.976260
$484$	0	0
$485$	2.70075e7	0.0107495
$486$	0	0
$487$	1.89934e9	0.745162	0.372581	−	0.928000i	$-0.378473\pi$
$487$	1.89934e9	0.745162	0.372581	+	0.928000i	$0.378473\pi$
$488$	0	0
$489$	4.17555e9	1.61485
$490$	0	0
$491$	1.74671e9	0.665940	0.332970	−	0.942937i	$-0.391949\pi$
$491$	1.74671e9	0.665940	0.332970	+	0.942937i	$0.391949\pi$
$492$	0	0
$493$	5.66928e8	0.213090
$494$	0	0
$495$	2.16969e9	0.804042
$496$	0	0
$497$	5.42146e8	0.198093
$498$	0	0
$499$	3.80758e9	1.37182	0.685910	−	0.727687i	$-0.259405\pi$
$499$	3.80758e9	1.37182	0.685910	+	0.727687i	$0.259405\pi$
$500$	0	0
$501$	−4.74671e9	−1.68640
$502$	0	0
$503$	3.25718e8	0.114118	0.0570589	−	0.998371i	$-0.481828\pi$
$503$	3.25718e8	0.114118	0.0570589	+	0.998371i	$0.481828\pi$
$504$	0	0
$505$	−2.36040e8	−0.0815577
$506$	0	0
$507$	−2.24452e9	−0.764886
$508$	0	0
$509$	−3.90181e9	−1.31146	−0.655728	−	0.754997i	$-0.727638\pi$
$509$	−3.90181e9	−1.31146	−0.655728	+	0.754997i	$0.727638\pi$
$510$	0	0
$511$	1.48575e9	0.492576
$512$	0	0
$513$	−2.77250e7	−0.00906694
$514$	0	0
$515$	−2.20258e9	−0.710570
$516$	0	0
$517$	4.06770e9	1.29459
$518$	0	0
$519$	4.38362e9	1.37641
$520$	0	0
$521$	−2.92867e9	−0.907275	−0.453638	−	0.891186i	$-0.649874\pi$
$521$	−2.92867e9	−0.907275	−0.453638	+	0.891186i	$0.649874\pi$
$522$	0	0
$523$	4.49793e9	1.37486	0.687428	−	0.726253i	$-0.258740\pi$
$523$	4.49793e9	1.37486	0.687428	+	0.726253i	$0.258740\pi$
$524$	0	0
$525$	1.21648e9	0.366899
$526$	0	0
$527$	−2.72186e9	−0.810082
$528$	0	0
$529$	7.86107e9	2.30880
$530$	0	0
$531$	−4.76864e9	−1.38218
$532$	0	0
$533$	1.73688e9	0.496849
$534$	0	0
$535$	3.71832e9	1.04980
$536$	0	0
$537$	6.25676e8	0.174357
$538$	0	0
$539$	−7.30486e8	−0.200933
$540$	0	0
$541$	1.17772e9	0.319779	0.159889	−	0.987135i	$-0.448886\pi$
$541$	1.17772e9	0.319779	0.159889	+	0.987135i	$0.448886\pi$
$542$	0	0
$543$	2.42395e9	0.649718
$544$	0	0
$545$	2.86849e9	0.759042
$546$	0	0
$547$	−3.94940e9	−1.03175	−0.515877	−	0.856663i	$-0.672534\pi$
$547$	−3.94940e9	−1.03175	−0.515877	+	0.856663i	$0.672534\pi$
$548$	0	0
$549$	1.31783e9	0.339904
$550$	0	0
$551$	6.03441e8	0.153675
$552$	0	0
$553$	−2.08722e9	−0.524843
$554$	0	0
$555$	−4.52760e9	−1.12420
$556$	0	0
$557$	−4.20743e9	−1.03163	−0.515815	−	0.856700i	$-0.672511\pi$
$557$	−4.20743e9	−1.03163	−0.515815	+	0.856700i	$0.672511\pi$
$558$	0	0
$559$	−4.72773e9	−1.14475
$560$	0	0
$561$	4.53334e9	1.08405
$562$	0	0
$563$	−3.97322e9	−0.938347	−0.469174	−	0.883106i	$-0.655448\pi$
$563$	−3.97322e9	−0.938347	−0.469174	+	0.883106i	$0.655448\pi$
$564$	0	0
$565$	−1.07696e9	−0.251206
$566$	0	0
$567$	1.61336e9	0.371698
$568$	0	0
$569$	5.06019e9	1.15153	0.575763	−	0.817617i	$-0.304705\pi$
$569$	5.06019e9	1.15153	0.575763	+	0.817617i	$0.304705\pi$
$570$	0	0
$571$	−4.24104e8	−0.0953336	−0.0476668	−	0.998863i	$-0.515179\pi$
$571$	−4.24104e8	−0.0953336	−0.0476668	+	0.998863i	$0.515179\pi$
$572$	0	0
$573$	5.58213e9	1.23953
$574$	0	0
$575$	5.66878e9	1.24352
$576$	0	0
$577$	−5.02389e9	−1.08874	−0.544371	−	0.838845i	$-0.683232\pi$
$577$	−5.02389e9	−1.08874	−0.544371	+	0.838845i	$0.683232\pi$
$578$	0	0
$579$	−4.43204e9	−0.948917
$580$	0	0
$581$	−2.77953e9	−0.587969
$582$	0	0
$583$	2.76200e9	0.577277
$584$	0	0
$585$	−1.88011e9	−0.388274
$586$	0	0
$587$	8.56543e9	1.74790	0.873948	−	0.486019i	$-0.161551\pi$
$587$	8.56543e9	1.74790	0.873948	+	0.486019i	$0.161551\pi$
$588$	0	0
$589$	−2.89717e9	−0.584212
$590$	0	0
$591$	−3.39735e9	−0.676992
$592$	0	0
$593$	−5.79211e9	−1.14063	−0.570315	−	0.821426i	$-0.693179\pi$
$593$	−5.79211e9	−1.14063	−0.570315	+	0.821426i	$0.693179\pi$
$594$	0	0
$595$	−5.92902e8	−0.115391
$596$	0	0
$597$	−7.84691e9	−1.50935
$598$	0	0
$599$	−1.74446e9	−0.331641	−0.165820	−	0.986156i	$-0.553027\pi$
$599$	−1.74446e9	−0.331641	−0.165820	+	0.986156i	$0.553027\pi$
$600$	0	0
$601$	−8.05605e9	−1.51378	−0.756888	−	0.653545i	$-0.773281\pi$
$601$	−8.05605e9	−1.51378	−0.756888	+	0.653545i	$0.773281\pi$
$602$	0	0
$603$	−3.84226e9	−0.713635
$604$	0	0
$605$	−2.99730e9	−0.550283
$606$	0	0
$607$	−4.69926e9	−0.852842	−0.426421	−	0.904525i	$-0.640226\pi$
$607$	−4.69926e9	−0.852842	−0.426421	+	0.904525i	$0.640226\pi$
$608$	0	0
$609$	1.17445e9	0.210704
$610$	0	0
$611$	−3.52481e9	−0.625160
$612$	0	0
$613$	2.30662e9	0.404449	0.202225	−	0.979339i	$-0.435183\pi$
$613$	2.30662e9	0.404449	0.202225	+	0.979339i	$0.435183\pi$
$614$	0	0
$615$	−3.37023e9	−0.584247
$616$	0	0
$617$	−8.81255e9	−1.51044	−0.755220	−	0.655471i	$-0.772470\pi$
$617$	−8.81255e9	−1.51044	−0.755220	+	0.655471i	$0.772470\pi$
$618$	0	0
$619$	5.51648e9	0.934857	0.467428	−	0.884031i	$-0.345181\pi$
$619$	5.51648e9	0.934857	0.467428	+	0.884031i	$0.345181\pi$
$620$	0	0
$621$	−2.51452e8	−0.0421342
$622$	0	0
$623$	−3.38209e9	−0.560373
$624$	0	0
$625$	9.21413e8	0.150964
$626$	0	0
$627$	4.82532e9	0.781789
$628$	0	0
$629$	−4.76825e9	−0.763980
$630$	0	0
$631$	−4.00663e9	−0.634857	−0.317429	−	0.948282i	$-0.602819\pi$
$631$	−4.00663e9	−0.634857	−0.317429	+	0.948282i	$0.602819\pi$
$632$	0	0
$633$	5.38420e9	0.843740
$634$	0	0
$635$	4.61411e9	0.715121
$636$	0	0
$637$	6.32993e8	0.0970311
$638$	0	0
$639$	−3.51316e9	−0.532654
$640$	0	0
$641$	7.52357e9	1.12829	0.564145	−	0.825676i	$-0.309206\pi$
$641$	7.52357e9	1.12829	0.564145	+	0.825676i	$0.309206\pi$
$642$	0	0
$643$	1.08744e10	1.61311	0.806557	−	0.591156i	$-0.201328\pi$
$643$	1.08744e10	1.61311	0.806557	+	0.591156i	$0.201328\pi$
$644$	0	0
$645$	9.17366e9	1.34612
$646$	0	0
$647$	3.17336e9	0.460632	0.230316	−	0.973116i	$-0.426024\pi$
$647$	3.17336e9	0.460632	0.230316	+	0.973116i	$0.426024\pi$
$648$	0	0
$649$	1.33212e10	1.91287
$650$	0	0
$651$	−5.63861e9	−0.801011
$652$	0	0
$653$	−7.94853e9	−1.11710	−0.558549	−	0.829472i	$-0.688642\pi$
$653$	−7.94853e9	−1.11710	−0.558549	+	0.829472i	$0.688642\pi$
$654$	0	0
$655$	−1.62079e9	−0.225363
$656$	0	0
$657$	−9.62783e9	−1.32449
$658$	0	0
$659$	1.08219e10	1.47301	0.736504	−	0.676433i	$-0.236475\pi$
$659$	1.08219e10	1.47301	0.736504	+	0.676433i	$0.236475\pi$
$660$	0	0
$661$	9.86108e9	1.32807	0.664033	−	0.747704i	$-0.268844\pi$
$661$	9.86108e9	1.32807	0.664033	+	0.747704i	$0.268844\pi$
$662$	0	0
$663$	−3.92831e9	−0.523490
$664$	0	0
$665$	−6.31089e8	−0.0832175
$666$	0	0
$667$	5.47291e9	0.714132
$668$	0	0
$669$	−1.59535e10	−2.05998
$670$	0	0
$671$	−3.68135e9	−0.470411
$672$	0	0
$673$	1.48659e10	1.87992	0.939959	−	0.341288i	$-0.110863\pi$
$673$	1.48659e10	1.87992	0.939959	+	0.341288i	$0.110863\pi$
$674$	0	0
$675$	−1.26526e8	−0.0158349
$676$	0	0
$677$	−5.71127e9	−0.707412	−0.353706	−	0.935357i	$-0.615079\pi$
$677$	−5.71127e9	−0.707412	−0.353706	+	0.935357i	$0.615079\pi$
$678$	0	0
$679$	5.89225e7	0.00722332
$680$	0	0
$681$	−5.06934e9	−0.615087
$682$	0	0
$683$	−9.40673e9	−1.12971	−0.564854	−	0.825191i	$-0.691068\pi$
$683$	−9.40673e9	−1.12971	−0.564854	+	0.825191i	$0.691068\pi$
$684$	0	0
$685$	−1.78451e9	−0.212130
$686$	0	0
$687$	−4.59857e9	−0.541096
$688$	0	0
$689$	−2.39338e9	−0.278769
$690$	0	0
$691$	−1.63111e10	−1.88066	−0.940331	−	0.340262i	$-0.889484\pi$
$691$	−1.63111e10	−1.88066	−0.940331	+	0.340262i	$0.889484\pi$
$692$	0	0
$693$	4.73362e9	0.540291
$694$	0	0
$695$	−8.92455e9	−1.00842
$696$	0	0
$697$	−3.54936e9	−0.397041
$698$	0	0
$699$	−4.08567e9	−0.452473
$700$	0	0
$701$	1.18926e10	1.30396	0.651979	−	0.758237i	$-0.273939\pi$
$701$	1.18926e10	1.30396	0.651979	+	0.758237i	$0.273939\pi$
$702$	0	0
$703$	−5.07536e9	−0.550963
$704$	0	0
$705$	6.83952e9	0.735129
$706$	0	0
$707$	−5.14970e8	−0.0548042
$708$	0	0
$709$	1.49941e10	1.58000	0.790002	−	0.613104i	$-0.210080\pi$
$709$	1.49941e10	1.58000	0.790002	+	0.613104i	$0.210080\pi$
$710$	0	0
$711$	1.35254e10	1.41126
$712$	0	0
$713$	−2.62759e10	−2.71484
$714$	0	0
$715$	5.25208e9	0.537354
$716$	0	0
$717$	6.59570e9	0.668257
$718$	0	0
$719$	−4.35637e9	−0.437093	−0.218546	−	0.975827i	$-0.570131\pi$
$719$	−4.35637e9	−0.437093	−0.218546	+	0.975827i	$0.570131\pi$
$720$	0	0
$721$	−4.80540e9	−0.477481
$722$	0	0
$723$	−1.02666e10	−1.01028
$724$	0	0
$725$	2.75387e9	0.268386
$726$	0	0
$727$	−4.64655e9	−0.448498	−0.224249	−	0.974532i	$-0.571993\pi$
$727$	−4.64655e9	−0.448498	−0.224249	+	0.974532i	$0.571993\pi$
$728$	0	0
$729$	−1.07988e10	−1.03235
$730$	0	0
$731$	9.66126e9	0.914793
$732$	0	0
$733$	−1.67128e10	−1.56742	−0.783710	−	0.621126i	$-0.786675\pi$
$733$	−1.67128e10	−1.56742	−0.783710	+	0.621126i	$0.786675\pi$
$734$	0	0
$735$	−1.22826e9	−0.114099
$736$	0	0
$737$	1.07333e10	0.987638
$738$	0	0
$739$	1.04179e10	0.949561	0.474781	−	0.880104i	$-0.342527\pi$
$739$	1.04179e10	0.949561	0.474781	+	0.880104i	$0.342527\pi$
$740$	0	0
$741$	−4.18132e9	−0.377528
$742$	0	0
$743$	−1.42061e8	−0.0127061	−0.00635307	−	0.999980i	$-0.502022\pi$
$743$	−1.42061e8	−0.0127061	−0.00635307	+	0.999980i	$0.502022\pi$
$744$	0	0
$745$	−4.26254e9	−0.377678
$746$	0	0
$747$	1.80116e10	1.58099
$748$	0	0
$749$	8.11229e9	0.705436
$750$	0	0
$751$	2.69386e9	0.232078	0.116039	−	0.993245i	$-0.462980\pi$
$751$	2.69386e9	0.232078	0.116039	+	0.993245i	$0.462980\pi$
$752$	0	0
$753$	−5.66596e9	−0.483605
$754$	0	0
$755$	5.08729e8	0.0430202
$756$	0	0
$757$	−4.63791e8	−0.0388585	−0.0194293	−	0.999811i	$-0.506185\pi$
$757$	−4.63791e8	−0.0388585	−0.0194293	+	0.999811i	$0.506185\pi$
$758$	0	0
$759$	4.37633e10	3.63298
$760$	0	0
$761$	−1.71497e10	−1.41062	−0.705311	−	0.708898i	$-0.749193\pi$
$761$	−1.71497e10	−1.41062	−0.705311	+	0.708898i	$0.749193\pi$
$762$	0	0
$763$	6.25822e9	0.510052
$764$	0	0
$765$	3.84207e9	0.310277
$766$	0	0
$767$	−1.15433e10	−0.923731
$768$	0	0
$769$	2.83370e9	0.224705	0.112352	−	0.993668i	$-0.464161\pi$
$769$	2.83370e9	0.224705	0.112352	+	0.993668i	$0.464161\pi$
$770$	0	0
$771$	−2.84031e10	−2.23191
$772$	0	0
$773$	7.20359e9	0.560946	0.280473	−	0.959862i	$-0.409509\pi$
$773$	7.20359e9	0.560946	0.280473	+	0.959862i	$0.409509\pi$
$774$	0	0
$775$	−1.32215e10	−1.02029
$776$	0	0
$777$	−9.87790e9	−0.755424
$778$	0	0
$779$	−3.77796e9	−0.286337
$780$	0	0
$781$	9.81399e9	0.737169
$782$	0	0
$783$	−1.22154e8	−0.00909373
$784$	0	0
$785$	−1.32666e10	−0.978848
$786$	0	0
$787$	6.82062e9	0.498784	0.249392	−	0.968403i	$-0.419769\pi$
$787$	6.82062e9	0.498784	0.249392	+	0.968403i	$0.419769\pi$
$788$	0	0
$789$	3.10185e10	2.24829
$790$	0	0
$791$	−2.34962e9	−0.168803
$792$	0	0
$793$	3.19002e9	0.227163
$794$	0	0
$795$	4.64409e9	0.327805
$796$	0	0
$797$	1.38571e10	0.969544	0.484772	−	0.874641i	$-0.338903\pi$
$797$	1.38571e10	0.969544	0.484772	+	0.874641i	$0.338903\pi$
$798$	0	0
$799$	7.20306e9	0.499578
$800$	0	0
$801$	2.19163e10	1.50679
$802$	0	0
$803$	2.68953e10	1.83304
$804$	0	0
$805$	−5.72367e9	−0.386713
$806$	0	0
$807$	1.32813e10	0.889573
$808$	0	0
$809$	−5.20582e9	−0.345676	−0.172838	−	0.984950i	$-0.555294\pi$
$809$	−5.20582e9	−0.345676	−0.172838	+	0.984950i	$0.555294\pi$
$810$	0	0
$811$	−1.44908e9	−0.0953938	−0.0476969	−	0.998862i	$-0.515188\pi$
$811$	−1.44908e9	−0.0953938	−0.0476969	+	0.998862i	$0.515188\pi$
$812$	0	0
$813$	1.54662e10	1.00941
$814$	0	0
$815$	−9.88570e9	−0.639669
$816$	0	0
$817$	1.02835e10	0.659727
$818$	0	0
$819$	−4.10186e9	−0.260908
$820$	0	0
$821$	2.69442e10	1.69928	0.849638	−	0.527366i	$-0.176820\pi$
$821$	2.69442e10	1.69928	0.849638	+	0.527366i	$0.176820\pi$
$822$	0	0
$823$	1.12034e10	0.700568	0.350284	−	0.936643i	$-0.386085\pi$
$823$	1.12034e10	0.700568	0.350284	+	0.936643i	$0.386085\pi$
$824$	0	0
$825$	2.20208e10	1.36535
$826$	0	0
$827$	7.78707e9	0.478746	0.239373	−	0.970928i	$-0.423058\pi$
$827$	7.78707e9	0.478746	0.239373	+	0.970928i	$0.423058\pi$
$828$	0	0
$829$	−1.08875e10	−0.663722	−0.331861	−	0.943328i	$-0.607677\pi$
$829$	−1.08875e10	−0.663722	−0.331861	+	0.943328i	$0.607677\pi$
$830$	0	0
$831$	2.96711e9	0.179362
$832$	0	0
$833$	−1.29354e9	−0.0775394
$834$	0	0
$835$	1.12379e10	0.668012
$836$	0	0
$837$	5.86472e8	0.0345707
$838$	0	0
$839$	8.20806e9	0.479815	0.239907	−	0.970796i	$-0.422883\pi$
$839$	8.20806e9	0.479815	0.239907	+	0.970796i	$0.422883\pi$
$840$	0	0
$841$	−1.45912e10	−0.845871
$842$	0	0
$843$	−9.22215e9	−0.530195
$844$	0	0
$845$	5.31396e9	0.302984
$846$	0	0
$847$	−6.53924e9	−0.369773
$848$	0	0
$849$	−2.18160e10	−1.22348
$850$	0	0
$851$	−4.60310e10	−2.56033
$852$	0	0
$853$	4.95254e9	0.273216	0.136608	−	0.990625i	$-0.456380\pi$
$853$	4.95254e9	0.273216	0.136608	+	0.990625i	$0.456380\pi$
$854$	0	0
$855$	4.08952e9	0.223764
$856$	0	0
$857$	3.38165e10	1.83525	0.917625	−	0.397446i	$-0.130104\pi$
$857$	3.38165e10	1.83525	0.917625	+	0.397446i	$0.130104\pi$
$858$	0	0
$859$	6.49803e9	0.349788	0.174894	−	0.984587i	$-0.444042\pi$
$859$	6.49803e9	0.349788	0.174894	+	0.984587i	$0.444042\pi$
$860$	0	0
$861$	−7.35286e9	−0.392595
$862$	0	0
$863$	2.14007e10	1.13342	0.566709	−	0.823918i	$-0.308216\pi$
$863$	2.14007e10	1.13342	0.566709	+	0.823918i	$0.308216\pi$
$864$	0	0
$865$	−1.03783e10	−0.545219
$866$	0	0
$867$	−1.92211e10	−1.00164
$868$	0	0
$869$	−3.77830e10	−1.95311
$870$	0	0
$871$	−9.30082e9	−0.476933
$872$	0	0
$873$	−3.81824e8	−0.0194229
$874$	0	0
$875$	−7.09294e9	−0.357930
$876$	0	0
$877$	−2.81094e10	−1.40719	−0.703595	−	0.710601i	$-0.748423\pi$
$877$	−2.81094e10	−1.40719	−0.703595	+	0.710601i	$0.748423\pi$
$878$	0	0
$879$	−1.27363e10	−0.632532
$880$	0	0
$881$	8.82922e9	0.435017	0.217509	−	0.976058i	$-0.430207\pi$
$881$	8.82922e9	0.435017	0.217509	+	0.976058i	$0.430207\pi$
$882$	0	0
$883$	−2.59216e10	−1.26707	−0.633534	−	0.773715i	$-0.718396\pi$
$883$	−2.59216e10	−1.26707	−0.633534	+	0.773715i	$0.718396\pi$
$884$	0	0
$885$	2.23985e10	1.08622
$886$	0	0
$887$	1.89397e10	0.911258	0.455629	−	0.890170i	$-0.349414\pi$
$887$	1.89397e10	0.911258	0.455629	+	0.890170i	$0.349414\pi$
$888$	0	0
$889$	1.00666e10	0.480539
$890$	0	0
$891$	2.92052e10	1.38321
$892$	0	0
$893$	7.66698e9	0.360283
$894$	0	0
$895$	−1.48130e9	−0.0690657
$896$	0	0
$897$	−3.79225e10	−1.75438
$898$	0	0
$899$	−1.27647e10	−0.585938
$900$	0	0
$901$	4.89094e9	0.222769
$902$	0	0
$903$	2.00143e10	0.904550
$904$	0	0
$905$	−5.73876e9	−0.257364
$906$	0	0
$907$	−9.02019e9	−0.401412	−0.200706	−	0.979652i	$-0.564324\pi$
$907$	−9.02019e9	−0.401412	−0.200706	+	0.979652i	$0.564324\pi$
$908$	0	0
$909$	3.33706e9	0.147364
$910$	0	0
$911$	−3.38470e10	−1.48322	−0.741610	−	0.670831i	$-0.765938\pi$
$911$	−3.38470e10	−1.48322	−0.741610	+	0.670831i	$0.765938\pi$
$912$	0	0
$913$	−5.03153e10	−2.18802
$914$	0	0
$915$	−6.18990e9	−0.267122
$916$	0	0
$917$	−3.53609e9	−0.151437
$918$	0	0
$919$	3.50778e10	1.49083	0.745415	−	0.666601i	$-0.232251\pi$
$919$	3.50778e10	1.49083	0.745415	+	0.666601i	$0.232251\pi$
$920$	0	0
$921$	−1.31543e10	−0.554831
$922$	0	0
$923$	−8.50419e9	−0.355981
$924$	0	0
$925$	−2.31619e10	−0.962228
$926$	0	0
$927$	3.11395e10	1.28390
$928$	0	0
$929$	−2.37377e10	−0.971369	−0.485685	−	0.874134i	$-0.661430\pi$
$929$	−2.37377e10	−0.971369	−0.485685	+	0.874134i	$0.661430\pi$
$930$	0	0
$931$	−1.37685e9	−0.0559195
$932$	0	0
$933$	−2.66099e10	−1.07265
$934$	0	0
$935$	−1.07328e10	−0.429410
$936$	0	0
$937$	2.48298e9	0.0986017	0.0493009	−	0.998784i	$-0.484301\pi$
$937$	2.48298e9	0.0986017	0.0493009	+	0.998784i	$0.484301\pi$
$938$	0	0
$939$	3.52840e10	1.39075
$940$	0	0
$941$	1.70549e10	0.667245	0.333623	−	0.942707i	$-0.391729\pi$
$941$	1.70549e10	0.667245	0.333623	+	0.942707i	$0.391729\pi$
$942$	0	0
$943$	−3.42643e10	−1.33061
$944$	0	0
$945$	1.27751e8	0.00492439
$946$	0	0
$947$	−1.95521e10	−0.748114	−0.374057	−	0.927406i	$-0.622034\pi$
$947$	−1.95521e10	−0.748114	−0.374057	+	0.927406i	$0.622034\pi$
$948$	0	0
$949$	−2.33057e10	−0.885179
$950$	0	0
$951$	−2.91020e9	−0.109721
$952$	0	0
$953$	1.99979e10	0.748444	0.374222	−	0.927339i	$-0.377910\pi$
$953$	1.99979e10	0.748444	0.374222	+	0.927339i	$0.377910\pi$
$954$	0	0
$955$	−1.32158e10	−0.491000
$956$	0	0
$957$	2.12600e10	0.784099
$958$	0	0
$959$	−3.89329e9	−0.142545
$960$	0	0
$961$	3.37717e10	1.22750
$962$	0	0
$963$	−5.25685e10	−1.89685
$964$	0	0
$965$	1.04929e10	0.375882
$966$	0	0
$967$	−2.36717e10	−0.841853	−0.420927	−	0.907095i	$-0.638295\pi$
$967$	−2.36717e10	−0.841853	−0.420927	+	0.907095i	$0.638295\pi$
$968$	0	0
$969$	8.54465e9	0.301690
$970$	0	0
$971$	5.49876e10	1.92751	0.963757	−	0.266781i	$-0.0859601\pi$
$971$	5.49876e10	1.92751	0.963757	+	0.266781i	$0.0859601\pi$
$972$	0	0
$973$	−1.94708e10	−0.677624
$974$	0	0
$975$	−1.90819e10	−0.659333
$976$	0	0
$977$	−5.44353e9	−0.186745	−0.0933727	−	0.995631i	$-0.529765\pi$
$977$	−5.44353e9	−0.186745	−0.0933727	+	0.995631i	$0.529765\pi$
$978$	0	0
$979$	−6.12230e10	−2.08533
$980$	0	0
$981$	−4.05539e10	−1.37149
$982$	0	0
$983$	−3.66356e10	−1.23017	−0.615086	−	0.788460i	$-0.710879\pi$
$983$	−3.66356e10	−1.23017	−0.615086	+	0.788460i	$0.710879\pi$
$984$	0	0
$985$	8.04329e9	0.268168
$986$	0	0
$987$	1.49218e10	0.493983
$988$	0	0
$989$	9.32664e10	3.06576
$990$	0	0
$991$	−1.36900e9	−0.0446832	−0.0223416	−	0.999750i	$-0.507112\pi$
$991$	−1.36900e9	−0.0446832	−0.0223416	+	0.999750i	$0.507112\pi$
$992$	0	0
$993$	−2.42025e10	−0.784401
$994$	0	0
$995$	1.85777e10	0.597877
$996$	0	0
$997$	1.36331e10	0.435673	0.217837	−	0.975985i	$-0.430100\pi$
$997$	1.36331e10	0.435673	0.217837	+	0.975985i	$0.430100\pi$
$998$	0	0
$999$	1.02740e9	0.0326032

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	112.8.a.i.1.2		2
4.3	odd	2		28.8.a.a.1.1	✓	2
8.3	odd	2		448.8.a.p.1.2		2
8.5	even	2		448.8.a.n.1.1		2
12.11	even	2		252.8.a.e.1.2		2
28.3	even	6		196.8.e.a.177.1		4
28.11	odd	6		196.8.e.d.177.2		4
28.19	even	6		196.8.e.a.165.1		4
28.23	odd	6		196.8.e.d.165.2		4
28.27	even	2		196.8.a.b.1.2		2

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
28.8.a.a.1.1	✓	2	4.3	odd	2
112.8.a.i.1.2		2	1.1	even	1	trivial
196.8.a.b.1.2		2	28.27	even	2
196.8.e.a.165.1		4	28.19	even	6
196.8.e.a.177.1		4	28.3	even	6
196.8.e.d.165.2		4	28.23	odd	6
196.8.e.d.177.2		4	28.11	odd	6
252.8.a.e.1.2		2	12.11	even	2
448.8.a.n.1.1		2	8.5	even	2
448.8.a.p.1.2		2	8.3	odd	2

Overview	Random
Universe	Knowledge

Classical	Maass
Hilbert	Bianchi

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

Level:	\( N \)	\(=\)	\( 112 = 2^{4} \cdot 7 \)
Weight:	\( k \)	\(=\)	\( 8 \)
Character orbit:	\([\chi]\)	\(=\)	112.a (trivial)

\(f(q)\)	\(=\)	\(q+66.4054 q^{3} -157.216 q^{5} -343.000 q^{7} +2222.68 q^{9} -6209.03 q^{11} +5380.35 q^{13} -10440.0 q^{15} -10994.9 q^{17} -11703.1 q^{19} -22777.0 q^{21} -106141. q^{23} -53408.1 q^{25} +2369.04 q^{27} -51562.7 q^{29} +247557. q^{31} -412313. q^{33} +53925.1 q^{35} +433678. q^{37} +357284. q^{39} +322819. q^{41} -878703. q^{43} -349440. q^{45} -655126. q^{47} +117649. q^{49} -730121. q^{51} -444837. q^{53} +976159. q^{55} -777146. q^{57} -2.14545e6 q^{59} +592902. q^{61} -762378. q^{63} -845878. q^{65} -1.72866e6 q^{67} -7.04833e6 q^{69} -1.58060e6 q^{71} -4.33164e6 q^{73} -3.54658e6 q^{75} +2.12970e6 q^{77} +6.08518e6 q^{79} -4.70367e6 q^{81} +8.10357e6 q^{83} +1.72858e6 q^{85} -3.42404e6 q^{87} +9.86032e6 q^{89} -1.84546e6 q^{91} +1.64391e7 q^{93} +1.83991e6 q^{95} -171786. q^{97} -1.38007e7 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 2 q + 14 q^{3} + 42 q^{5} - 686 q^{7} + 2782 q^{9} - 7428 q^{11} + 11830 q^{13} - 20880 q^{15} + 15792 q^{17} - 26614 q^{19} - 4802 q^{21} - 32640 q^{23} - 91846 q^{25} + 87668 q^{27} - 158016 q^{29} + 180740 q^{31}+ \cdots - 14482452 q^{99}+O(q^{100}) \) 2 * q + 14 * q^3 + 42 * q^5 - 686 * q^7 + 2782 * q^9 - 7428 * q^11 + 11830 * q^13 - 20880 * q^15 + 15792 * q^17 - 26614 * q^19 - 4802 * q^21 - 32640 * q^23 - 91846 * q^25 + 87668 * q^27 - 158016 * q^29 + 180740 * q^31 - 348432 * q^33 - 14406 * q^35 - 45824 * q^37 + 19288 * q^39 - 321720 * q^41 - 1023868 * q^43 - 238014 * q^45 - 1665972 * q^47 + 235298 * q^49 - 2133900 * q^51 - 410628 * q^53 + 733320 * q^55 + 4268 * q^57 - 1702134 * q^59 - 547526 * q^61 - 954226 * q^63 + 438996 * q^65 + 2590616 * q^67 - 10900176 * q^69 - 4129272 * q^71 - 8008868 * q^73 - 1532230 * q^75 + 2547804 * q^77 - 2470456 * q^79 - 10397042 * q^81 + 9900786 * q^83 + 7064964 * q^85 + 2154684 * q^87 + 15423492 * q^89 - 4057690 * q^91 + 19940648 * q^93 - 1130592 * q^95 - 17377472 * q^97 - 14482452 * q^99

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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