LMFDB - Embedded newform 112.6.a.i.1.2

Show commands: Magma / Pari/GP / SageMath

Newspace parameters

comment:Compute space of new eigenforms

gp:[N,k,chi] = [112,6,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, 6); 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, 6, names="a")

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

Newform invariants

comment:select newform

sage:traces = [2,0,6,0,82] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(5)] == traces)

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			1.2
Root			$9.78709$ 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)

$f(q)$	$=$	$q+21.5742 q^{3} -14.7225 q^{5} +49.0000 q^{7} +222.445 q^{9} -58.5549 q^{11} +1179.39 q^{13} -317.626 q^{15} +1496.55 q^{17} -498.838 q^{19} +1057.13 q^{21} +1889.34 q^{23} -2908.25 q^{25} -443.456 q^{27} +1914.54 q^{29} -794.577 q^{31} -1263.27 q^{33} -721.404 q^{35} +2987.93 q^{37} +25444.4 q^{39} +11941.3 q^{41} -9820.19 q^{43} -3274.95 q^{45} +19636.0 q^{47} +2401.00 q^{49} +32286.9 q^{51} -19875.0 q^{53} +862.077 q^{55} -10762.0 q^{57} -35838.6 q^{59} +49975.9 q^{61} +10899.8 q^{63} -17363.6 q^{65} -48176.2 q^{67} +40761.0 q^{69} -77179.1 q^{71} -59667.3 q^{73} -62743.0 q^{75} -2869.19 q^{77} -60743.1 q^{79} -63621.3 q^{81} +46134.2 q^{83} -22033.1 q^{85} +41304.7 q^{87} +78668.7 q^{89} +57790.2 q^{91} -17142.3 q^{93} +7344.15 q^{95} -43573.3 q^{97} -13025.3 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 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}+ \cdots - 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

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
$3$	21.5742	1.38398	0.691992	−	0.721905i	$-0.256733\pi$
$3$	21.5742	1.38398	0.691992	+	0.721905i	$0.256733\pi$
$4$	0	0
$5$	−14.7225	−0.263365	−0.131682	−	0.991292i	$-0.542038\pi$
$5$	−14.7225	−0.263365	−0.131682	+	0.991292i	$0.542038\pi$
$6$	0	0
$7$	49.0000	0.377964
$7$	49.0000	0.377964
$8$	0	0
$9$	222.445	0.915412
$10$	0	0
$11$	−58.5549	−0.145909	−0.0729545	−	0.997335i	$-0.523243\pi$
$11$	−58.5549	−0.145909	−0.0729545	+	0.997335i	$0.523243\pi$
$12$	0	0
$13$	1179.39	1.93553	0.967765	−	0.251853i	$-0.0810400\pi$
$13$	1179.39	1.93553	0.967765	+	0.251853i	$0.0810400\pi$
$14$	0	0
$15$	−317.626	−0.364492
$16$	0	0
$17$	1496.55	1.25594	0.627972	−	0.778236i	$-0.283885\pi$
$17$	1496.55	1.25594	0.627972	+	0.778236i	$0.283885\pi$
$18$	0	0
$19$	−498.838	−0.317012	−0.158506	−	0.987358i	$-0.550668\pi$
$19$	−498.838	−0.317012	−0.158506	+	0.987358i	$0.550668\pi$
$20$	0	0
$21$	1057.13	0.523097
$22$	0	0
$23$	1889.34	0.744716	0.372358	−	0.928089i	$-0.378549\pi$
$23$	1889.34	0.744716	0.372358	+	0.928089i	$0.378549\pi$
$24$	0	0
$25$	−2908.25	−0.930639
$26$	0	0
$27$	−443.456	−0.117069
$28$	0	0
$29$	1914.54	0.422737	0.211369	−	0.977406i	$-0.432208\pi$
$29$	1914.54	0.422737	0.211369	+	0.977406i	$0.432208\pi$
$30$	0	0
$31$	−794.577	−0.148502	−0.0742509	−	0.997240i	$-0.523657\pi$
$31$	−794.577	−0.148502	−0.0742509	+	0.997240i	$0.523657\pi$
$32$	0	0
$33$	−1263.27	−0.201936
$34$	0	0
$35$	−721.404	−0.0995424
$36$	0	0
$37$	2987.93	0.358811	0.179406	−	0.983775i	$-0.442583\pi$
$37$	2987.93	0.358811	0.179406	+	0.983775i	$0.442583\pi$
$38$	0	0
$39$	25444.4	2.67874
$40$	0	0
$41$	11941.3	1.10941	0.554704	−	0.832047i	$-0.312831\pi$
$41$	11941.3	1.10941	0.554704	+	0.832047i	$0.312831\pi$
$42$	0	0
$43$	−9820.19	−0.809933	−0.404966	−	0.914332i	$-0.632717\pi$
$43$	−9820.19	−0.809933	−0.404966	+	0.914332i	$0.632717\pi$
$44$	0	0
$45$	−3274.95	−0.241087
$46$	0	0
$47$	19636.0	1.29660	0.648302	−	0.761383i	$-0.275479\pi$
$47$	19636.0	1.29660	0.648302	+	0.761383i	$0.275479\pi$
$48$	0	0
$49$	2401.00	0.142857
$50$	0	0
$51$	32286.9	1.73821
$52$	0	0
$53$	−19875.0	−0.971889	−0.485945	−	0.873990i	$-0.661524\pi$
$53$	−19875.0	−0.971889	−0.485945	+	0.873990i	$0.661524\pi$
$54$	0	0
$55$	862.077	0.0384272
$56$	0	0
$57$	−10762.0	−0.438739
$58$	0	0
$59$	−35838.6	−1.34036	−0.670180	−	0.742199i	$-0.733783\pi$
$59$	−35838.6	−1.34036	−0.670180	+	0.742199i	$0.733783\pi$
$60$	0	0
$61$	49975.9	1.71964	0.859818	−	0.510601i	$-0.170577\pi$
$61$	49975.9	1.71964	0.859818	+	0.510601i	$0.170577\pi$
$62$	0	0
$63$	10899.8	0.345993
$64$	0	0
$65$	−17363.6	−0.509750
$66$	0	0
$67$	−48176.2	−1.31113	−0.655565	−	0.755139i	$-0.727569\pi$
$67$	−48176.2	−1.31113	−0.655565	+	0.755139i	$0.727569\pi$
$68$	0	0
$69$	40761.0	1.03068
$70$	0	0
$71$	−77179.1	−1.81699	−0.908497	−	0.417891i	$-0.862770\pi$
$71$	−77179.1	−1.81699	−0.908497	+	0.417891i	$0.862770\pi$
$72$	0	0
$73$	−59667.3	−1.31048	−0.655238	−	0.755422i	$-0.727432\pi$
$73$	−59667.3	−1.31048	−0.655238	+	0.755422i	$0.727432\pi$
$74$	0	0
$75$	−62743.0	−1.28799
$76$	0	0
$77$	−2869.19	−0.0551484
$78$	0	0
$79$	−60743.1	−1.09504	−0.547519	−	0.836793i	$-0.684428\pi$
$79$	−60743.1	−1.09504	−0.547519	+	0.836793i	$0.684428\pi$
$80$	0	0
$81$	−63621.3	−1.07743
$82$	0	0
$83$	46134.2	0.735068	0.367534	−	0.930010i	$-0.380202\pi$
$83$	46134.2	0.735068	0.367534	+	0.930010i	$0.380202\pi$
$84$	0	0
$85$	−22033.1	−0.330771
$86$	0	0
$87$	41304.7	0.585061
$88$	0	0
$89$	78668.7	1.05275	0.526377	−	0.850251i	$-0.323550\pi$
$89$	78668.7	1.05275	0.526377	+	0.850251i	$0.323550\pi$
$90$	0	0
$91$	57790.2	0.731562
$92$	0	0
$93$	−17142.3	−0.205524
$94$	0	0
$95$	7344.15	0.0834897
$96$	0	0
$97$	−43573.3	−0.470209	−0.235104	−	0.971970i	$-0.575543\pi$
$97$	−43573.3	−0.470209	−0.235104	+	0.971970i	$0.575543\pi$
$98$	0	0
$99$	−13025.3	−0.133567
$100$	0	0
$101$	−164188.	−1.60154	−0.800772	−	0.598970i	$-0.795577\pi$
$101$	−164188.	−1.60154	−0.800772	+	0.598970i	$0.795577\pi$
$102$	0	0
$103$	−164547.	−1.52825	−0.764127	−	0.645065i	$-0.776830\pi$
$103$	−164547.	−1.52825	−0.764127	+	0.645065i	$0.776830\pi$
$104$	0	0
$105$	−15563.7	−0.137765
$106$	0	0
$107$	185426.	1.56571	0.782854	−	0.622206i	$-0.213763\pi$
$107$	185426.	1.56571	0.782854	+	0.622206i	$0.213763\pi$
$108$	0	0
$109$	190063.	1.53225	0.766127	−	0.642689i	$-0.222181\pi$
$109$	190063.	1.53225	0.766127	+	0.642689i	$0.222181\pi$
$110$	0	0
$111$	64462.1	0.496589
$112$	0	0
$113$	−116023.	−0.854769	−0.427384	−	0.904070i	$-0.640565\pi$
$113$	−116023.	−0.854769	−0.427384	+	0.904070i	$0.640565\pi$
$114$	0	0
$115$	−27815.9	−0.196132
$116$	0	0
$117$	262350.	1.77181
$118$	0	0
$119$	73331.2	0.474702
$120$	0	0
$121$	−157622.	−0.978711
$122$	0	0
$123$	257624.	1.53540
$124$	0	0
$125$	88824.6	0.508462
$126$	0	0
$127$	−71785.5	−0.394936	−0.197468	−	0.980309i	$-0.563272\pi$
$127$	−71785.5	−0.394936	−0.197468	+	0.980309i	$0.563272\pi$
$128$	0	0
$129$	−211863.	−1.12093
$130$	0	0
$131$	2016.22	0.0102650	0.00513250	−	0.999987i	$-0.498366\pi$
$131$	2016.22	0.0102650	0.00513250	+	0.999987i	$0.498366\pi$
$132$	0	0
$133$	−24443.1	−0.119819
$134$	0	0
$135$	6528.79	0.0308318
$136$	0	0
$137$	−243904.	−1.11024	−0.555120	−	0.831770i	$-0.687328\pi$
$137$	−243904.	−1.11024	−0.555120	+	0.831770i	$0.687328\pi$
$138$	0	0
$139$	−209413.	−0.919319	−0.459660	−	0.888095i	$-0.652029\pi$
$139$	−209413.	−0.919319	−0.459660	+	0.888095i	$0.652029\pi$
$140$	0	0
$141$	423630.	1.79448
$142$	0	0
$143$	−69059.3	−0.282411
$144$	0	0
$145$	−28186.9	−0.111334
$146$	0	0
$147$	51799.6	0.197712
$148$	0	0
$149$	22954.3	0.0847028	0.0423514	−	0.999103i	$-0.486515\pi$
$149$	22954.3	0.0847028	0.0423514	+	0.999103i	$0.486515\pi$
$150$	0	0
$151$	276737.	0.987700	0.493850	−	0.869547i	$-0.335589\pi$
$151$	276737.	0.987700	0.493850	+	0.869547i	$0.335589\pi$
$152$	0	0
$153$	332901.	1.14971
$154$	0	0
$155$	11698.2	0.0391101
$156$	0	0
$157$	276692.	0.895874	0.447937	−	0.894065i	$-0.352159\pi$
$157$	276692.	0.895874	0.447937	+	0.894065i	$0.352159\pi$
$158$	0	0
$159$	−428786.	−1.34508
$160$	0	0
$161$	92577.7	0.281476
$162$	0	0
$163$	−274258.	−0.808518	−0.404259	−	0.914645i	$-0.632471\pi$
$163$	−274258.	−0.808518	−0.404259	+	0.914645i	$0.632471\pi$
$164$	0	0
$165$	18598.6	0.0531827
$166$	0	0
$167$	215716.	0.598537	0.299268	−	0.954169i	$-0.403257\pi$
$167$	215716.	0.598537	0.299268	+	0.954169i	$0.403257\pi$
$168$	0	0
$169$	1.01967e6	2.74628
$170$	0	0
$171$	−110964.	−0.290196
$172$	0	0
$173$	86252.7	0.219108	0.109554	−	0.993981i	$-0.465058\pi$
$173$	86252.7	0.219108	0.109554	+	0.993981i	$0.465058\pi$
$174$	0	0
$175$	−142504.	−0.351749
$176$	0	0
$177$	−773189.	−1.85504
$178$	0	0
$179$	435822.	1.01666	0.508331	−	0.861162i	$-0.330262\pi$
$179$	435822.	1.01666	0.508331	+	0.861162i	$0.330262\pi$
$180$	0	0
$181$	56972.2	0.129261	0.0646303	−	0.997909i	$-0.479413\pi$
$181$	56972.2	0.129261	0.0646303	+	0.997909i	$0.479413\pi$
$182$	0	0
$183$	1.07819e6	2.37995
$184$	0	0
$185$	−43989.9	−0.0944981
$186$	0	0
$187$	−87630.7	−0.183253
$188$	0	0
$189$	−21729.3	−0.0442478
$190$	0	0
$191$	350077.	0.694353	0.347177	−	0.937800i	$-0.387140\pi$
$191$	350077.	0.694353	0.347177	+	0.937800i	$0.387140\pi$
$192$	0	0
$193$	−623227.	−1.20435	−0.602175	−	0.798364i	$-0.705699\pi$
$193$	−623227.	−1.20435	−0.602175	+	0.798364i	$0.705699\pi$
$194$	0	0
$195$	−374606.	−0.705486
$196$	0	0
$197$	223265.	0.409879	0.204939	−	0.978775i	$-0.434300\pi$
$197$	223265.	0.409879	0.204939	+	0.978775i	$0.434300\pi$
$198$	0	0
$199$	−756375.	−1.35396	−0.676978	−	0.736004i	$-0.736711\pi$
$199$	−756375.	−1.35396	−0.676978	+	0.736004i	$0.736711\pi$
$200$	0	0
$201$	−1.03936e6	−1.81458
$202$	0	0
$203$	93812.7	0.159780
$204$	0	0
$205$	−175806.	−0.292179
$206$	0	0
$207$	420274.	0.681722
$208$	0	0
$209$	29209.4	0.0462549
$210$	0	0
$211$	30644.3	0.0473853	0.0236927	−	0.999719i	$-0.492458\pi$
$211$	30644.3	0.0473853	0.0236927	+	0.999719i	$0.492458\pi$
$212$	0	0
$213$	−1.66507e6	−2.51469
$214$	0	0
$215$	144578.	0.213308
$216$	0	0
$217$	−38934.3	−0.0561284
$218$	0	0
$219$	−1.28727e6	−1.81368
$220$	0	0
$221$	1.76503e6	2.43092
$222$	0	0
$223$	−461375.	−0.621286	−0.310643	−	0.950527i	$-0.600544\pi$
$223$	−461375.	−0.621286	−0.310643	+	0.950527i	$0.600544\pi$
$224$	0	0
$225$	−646925.	−0.851918
$226$	0	0
$227$	−346439.	−0.446234	−0.223117	−	0.974792i	$-0.571623\pi$
$227$	−346439.	−0.446234	−0.223117	+	0.974792i	$0.571623\pi$
$228$	0	0
$229$	−521650.	−0.657341	−0.328671	−	0.944445i	$-0.606601\pi$
$229$	−521650.	−0.657341	−0.328671	+	0.944445i	$0.606601\pi$
$230$	0	0
$231$	−61900.5	−0.0763245
$232$	0	0
$233$	154120.	0.185981	0.0929907	−	0.995667i	$-0.470357\pi$
$233$	154120.	0.185981	0.0929907	+	0.995667i	$0.470357\pi$
$234$	0	0
$235$	−289091.	−0.341480
$236$	0	0
$237$	−1.31048e6	−1.51551
$238$	0	0
$239$	478940.	0.542358	0.271179	−	0.962529i	$-0.412586\pi$
$239$	478940.	0.542358	0.271179	+	0.962529i	$0.412586\pi$
$240$	0	0
$241$	86551.8	0.0959917	0.0479958	−	0.998848i	$-0.484717\pi$
$241$	86551.8	0.0959917	0.0479958	+	0.998848i	$0.484717\pi$
$242$	0	0
$243$	−1.26482e6	−1.37408
$244$	0	0
$245$	−35348.8	−0.0376235
$246$	0	0
$247$	−588326.	−0.613586
$248$	0	0
$249$	995307.	1.01732
$250$	0	0
$251$	1.90322e6	1.90679	0.953397	−	0.301719i	$-0.0975605\pi$
$251$	1.90322e6	1.90679	0.953397	+	0.301719i	$0.0975605\pi$
$252$	0	0
$253$	−110630.	−0.108661
$254$	0	0
$255$	−475345.	−0.457782
$256$	0	0
$257$	−569922.	−0.538248	−0.269124	−	0.963105i	$-0.586734\pi$
$257$	−569922.	−0.538248	−0.269124	+	0.963105i	$0.586734\pi$
$258$	0	0
$259$	146408.	0.135618
$260$	0	0
$261$	425881.	0.386978
$262$	0	0
$263$	−1.29853e6	−1.15761	−0.578806	−	0.815465i	$-0.696481\pi$
$263$	−1.29853e6	−1.15761	−0.578806	+	0.815465i	$0.696481\pi$
$264$	0	0
$265$	292610.	0.255961
$266$	0	0
$267$	1.69721e6	1.45699
$268$	0	0
$269$	−53721.5	−0.0452655	−0.0226328	−	0.999744i	$-0.507205\pi$
$269$	−53721.5	−0.0452655	−0.0226328	+	0.999744i	$0.507205\pi$
$270$	0	0
$271$	758053.	0.627013	0.313506	−	0.949586i	$-0.398496\pi$
$271$	758053.	0.627013	0.313506	+	0.949586i	$0.398496\pi$
$272$	0	0
$273$	1.24678e6	1.01247
$274$	0	0
$275$	170292.	0.135789
$276$	0	0
$277$	2.07598e6	1.62564	0.812819	−	0.582516i	$-0.197932\pi$
$277$	2.07598e6	1.62564	0.812819	+	0.582516i	$0.197932\pi$
$278$	0	0
$279$	−176750.	−0.135940
$280$	0	0
$281$	2.61195e6	1.97333	0.986664	−	0.162770i	$-0.0520427\pi$
$281$	2.61195e6	1.97333	0.986664	+	0.162770i	$0.0520427\pi$
$282$	0	0
$283$	−992734.	−0.736829	−0.368414	−	0.929662i	$-0.620099\pi$
$283$	−992734.	−0.736829	−0.368414	+	0.929662i	$0.620099\pi$
$284$	0	0
$285$	158444.	0.115548
$286$	0	0
$287$	585123.	0.419317
$288$	0	0
$289$	819820.	0.577396
$290$	0	0
$291$	−940057.	−0.650762
$292$	0	0
$293$	371728.	0.252962	0.126481	−	0.991969i	$-0.459632\pi$
$293$	371728.	0.252962	0.126481	+	0.991969i	$0.459632\pi$
$294$	0	0
$295$	527635.	0.353003
$296$	0	0
$297$	25966.5	0.0170814
$298$	0	0
$299$	2.22827e6	1.44142
$300$	0	0
$301$	−481189.	−0.306126
$302$	0	0
$303$	−3.54223e6	−2.21651
$304$	0	0
$305$	−735772.	−0.452891
$306$	0	0
$307$	2.83906e6	1.71921	0.859606	−	0.510958i	$-0.170709\pi$
$307$	2.83906e6	1.71921	0.859606	+	0.510958i	$0.170709\pi$
$308$	0	0
$309$	−3.54996e6	−2.11508
$310$	0	0
$311$	−759362.	−0.445193	−0.222596	−	0.974911i	$-0.571453\pi$
$311$	−759362.	−0.445193	−0.222596	+	0.974911i	$0.571453\pi$
$312$	0	0
$313$	−1.80191e6	−1.03961	−0.519806	−	0.854284i	$-0.673996\pi$
$313$	−1.80191e6	−1.03961	−0.519806	+	0.854284i	$0.673996\pi$
$314$	0	0
$315$	−160473.	−0.0911223
$316$	0	0
$317$	−441899.	−0.246987	−0.123494	−	0.992345i	$-0.539410\pi$
$317$	−441899.	−0.246987	−0.123494	+	0.992345i	$0.539410\pi$
$318$	0	0
$319$	−112106.	−0.0616811
$320$	0	0
$321$	4.00041e6	2.16691
$322$	0	0
$323$	−746538.	−0.398149
$324$	0	0
$325$	−3.42997e6	−1.80128
$326$	0	0
$327$	4.10045e6	2.12061
$328$	0	0
$329$	962162.	0.490070
$330$	0	0
$331$	3.15779e6	1.58421	0.792107	−	0.610383i	$-0.208984\pi$
$331$	3.15779e6	1.58421	0.792107	+	0.610383i	$0.208984\pi$
$332$	0	0
$333$	664650.	0.328460
$334$	0	0
$335$	709275.	0.345305
$336$	0	0
$337$	−1.66612e6	−0.799158	−0.399579	−	0.916699i	$-0.630844\pi$
$337$	−1.66612e6	−0.799158	−0.399579	+	0.916699i	$0.630844\pi$
$338$	0	0
$339$	−2.50310e6	−1.18299
$340$	0	0
$341$	46526.4	0.0216677
$342$	0	0
$343$	117649.	0.0539949
$344$	0	0
$345$	−600104.	−0.271443
$346$	0	0
$347$	3.41953e6	1.52455	0.762277	−	0.647251i	$-0.224081\pi$
$347$	3.41953e6	1.52455	0.762277	+	0.647251i	$0.224081\pi$
$348$	0	0
$349$	−3.18396e6	−1.39928	−0.699638	−	0.714497i	$-0.746655\pi$
$349$	−3.18396e6	−1.39928	−0.699638	+	0.714497i	$0.746655\pi$
$350$	0	0
$351$	−523009.	−0.226590
$352$	0	0
$353$	3.02506e6	1.29210	0.646051	−	0.763294i	$-0.276419\pi$
$353$	3.02506e6	1.29210	0.646051	+	0.763294i	$0.276419\pi$
$354$	0	0
$355$	1.13627e6	0.478532
$356$	0	0
$357$	1.58206e6	0.656980
$358$	0	0
$359$	1.12353e6	0.460098	0.230049	−	0.973179i	$-0.426111\pi$
$359$	1.12353e6	0.460098	0.230049	+	0.973179i	$0.426111\pi$
$360$	0	0
$361$	−2.22726e6	−0.899504
$362$	0	0
$363$	−3.40057e6	−1.35452
$364$	0	0
$365$	878453.	0.345133
$366$	0	0
$367$	−3.39115e6	−1.31426	−0.657131	−	0.753777i	$-0.728230\pi$
$367$	−3.39115e6	−1.31426	−0.657131	+	0.753777i	$0.728230\pi$
$368$	0	0
$369$	2.65628e6	1.01557
$370$	0	0
$371$	−973873.	−0.367340
$372$	0	0
$373$	1.33823e6	0.498033	0.249017	−	0.968499i	$-0.419893\pi$
$373$	1.33823e6	0.498033	0.249017	+	0.968499i	$0.419893\pi$
$374$	0	0
$375$	1.91632e6	0.703703
$376$	0	0
$377$	2.25800e6	0.818221
$378$	0	0
$379$	−3.11403e6	−1.11359	−0.556795	−	0.830650i	$-0.687969\pi$
$379$	−3.11403e6	−1.11359	−0.556795	+	0.830650i	$0.687969\pi$
$380$	0	0
$381$	−1.54871e6	−0.546586
$382$	0	0
$383$	334533.	0.116531	0.0582656	−	0.998301i	$-0.481443\pi$
$383$	334533.	0.116531	0.0582656	+	0.998301i	$0.481443\pi$
$384$	0	0
$385$	42241.8	0.0145241
$386$	0	0
$387$	−2.18445e6	−0.741422
$388$	0	0
$389$	1.59871e6	0.535667	0.267834	−	0.963465i	$-0.413692\pi$
$389$	1.59871e6	0.535667	0.267834	+	0.963465i	$0.413692\pi$
$390$	0	0
$391$	2.82750e6	0.935322
$392$	0	0
$393$	43498.2	0.0142066
$394$	0	0
$395$	894292.	0.288394
$396$	0	0
$397$	−2.20789e6	−0.703074	−0.351537	−	0.936174i	$-0.614341\pi$
$397$	−2.20789e6	−0.703074	−0.351537	+	0.936174i	$0.614341\pi$
$398$	0	0
$399$	−527339.	−0.165828
$400$	0	0
$401$	1.89272e6	0.587793	0.293897	−	0.955837i	$-0.405048\pi$
$401$	1.89272e6	0.587793	0.293897	+	0.955837i	$0.405048\pi$
$402$	0	0
$403$	−937118.	−0.287430
$404$	0	0
$405$	936667.	0.283758
$406$	0	0
$407$	−174958.	−0.0523537
$408$	0	0
$409$	1.14418e6	0.338210	0.169105	−	0.985598i	$-0.445912\pi$
$409$	1.14418e6	0.338210	0.169105	+	0.985598i	$0.445912\pi$
$410$	0	0
$411$	−5.26202e6	−1.53655
$412$	0	0
$413$	−1.75609e6	−0.506608
$414$	0	0
$415$	−679212.	−0.193591
$416$	0	0
$417$	−4.51791e6	−1.27232
$418$	0	0
$419$	−5.17298e6	−1.43948	−0.719740	−	0.694244i	$-0.755739\pi$
$419$	−5.17298e6	−1.43948	−0.719740	+	0.694244i	$0.755739\pi$
$420$	0	0
$421$	−1.40960e6	−0.387606	−0.193803	−	0.981040i	$-0.562082\pi$
$421$	−1.40960e6	−0.387606	−0.193803	+	0.981040i	$0.562082\pi$
$422$	0	0
$423$	4.36792e6	1.18693
$424$	0	0
$425$	−4.35235e6	−1.16883
$426$	0	0
$427$	2.44882e6	0.649961
$428$	0	0
$429$	−1.48990e6	−0.390853
$430$	0	0
$431$	−6.17076e6	−1.60009	−0.800047	−	0.599938i	$-0.795192\pi$
$431$	−6.17076e6	−1.60009	−0.800047	+	0.599938i	$0.795192\pi$
$432$	0	0
$433$	387046.	0.0992071	0.0496036	−	0.998769i	$-0.484204\pi$
$433$	387046.	0.0992071	0.0496036	+	0.998769i	$0.484204\pi$
$434$	0	0
$435$	−608110.	−0.154084
$436$	0	0
$437$	−942475.	−0.236084
$438$	0	0
$439$	−3.32974e6	−0.824611	−0.412305	−	0.911046i	$-0.635276\pi$
$439$	−3.32974e6	−0.824611	−0.412305	+	0.911046i	$0.635276\pi$
$440$	0	0
$441$	534091.	0.130773
$442$	0	0
$443$	1.44003e6	0.348627	0.174314	−	0.984690i	$-0.444229\pi$
$443$	1.44003e6	0.348627	0.174314	+	0.984690i	$0.444229\pi$
$444$	0	0
$445$	−1.15820e6	−0.277258
$446$	0	0
$447$	495219.	0.117227
$448$	0	0
$449$	4.28977e6	1.00420	0.502098	−	0.864811i	$-0.332562\pi$
$449$	4.28977e6	1.00420	0.502098	+	0.864811i	$0.332562\pi$
$450$	0	0
$451$	−699222.	−0.161873
$452$	0	0
$453$	5.97037e6	1.36696
$454$	0	0
$455$	−850819.	−0.192667
$456$	0	0
$457$	−5.27889e6	−1.18237	−0.591183	−	0.806537i	$-0.701339\pi$
$457$	−5.27889e6	−1.18237	−0.591183	+	0.806537i	$0.701339\pi$
$458$	0	0
$459$	−663656.	−0.147032
$460$	0	0
$461$	−3.72322e6	−0.815955	−0.407977	−	0.912992i	$-0.633766\pi$
$461$	−3.72322e6	−0.815955	−0.407977	+	0.912992i	$0.633766\pi$
$462$	0	0
$463$	−1.04809e6	−0.227220	−0.113610	−	0.993525i	$-0.536241\pi$
$463$	−1.04809e6	−0.227220	−0.113610	+	0.993525i	$0.536241\pi$
$464$	0	0
$465$	252379.	0.0541278
$466$	0	0
$467$	−6.04244e6	−1.28210	−0.641048	−	0.767501i	$-0.721500\pi$
$467$	−6.04244e6	−1.28210	−0.641048	+	0.767501i	$0.721500\pi$
$468$	0	0
$469$	−2.36063e6	−0.495560
$470$	0	0
$471$	5.96940e6	1.23988
$472$	0	0
$473$	575021.	0.118176
$474$	0	0
$475$	1.45074e6	0.295024
$476$	0	0
$477$	−4.42109e6	−0.889679
$478$	0	0
$479$	2.06868e6	0.411959	0.205980	−	0.978556i	$-0.433962\pi$
$479$	2.06868e6	0.411959	0.205980	+	0.978556i	$0.433962\pi$
$480$	0	0
$481$	3.52394e6	0.694490
$482$	0	0
$483$	1.99729e6	0.389559
$484$	0	0
$485$	641509.	0.123836
$486$	0	0
$487$	2.82966e6	0.540645	0.270323	−	0.962770i	$-0.412870\pi$
$487$	2.82966e6	0.540645	0.270323	+	0.962770i	$0.412870\pi$
$488$	0	0
$489$	−5.91688e6	−1.11898
$490$	0	0
$491$	4.44554e6	0.832186	0.416093	−	0.909322i	$-0.363399\pi$
$491$	4.44554e6	0.832186	0.416093	+	0.909322i	$0.363399\pi$
$492$	0	0
$493$	2.86522e6	0.530934
$494$	0	0
$495$	191765.	0.0351767
$496$	0	0
$497$	−3.78177e6	−0.686759
$498$	0	0
$499$	1.34623e6	0.242029	0.121015	−	0.992651i	$-0.461385\pi$
$499$	1.34623e6	0.242029	0.121015	+	0.992651i	$0.461385\pi$
$500$	0	0
$501$	4.65389e6	0.828365
$502$	0	0
$503$	4.88909e6	0.861604	0.430802	−	0.902446i	$-0.358231\pi$
$503$	4.88909e6	0.861604	0.430802	+	0.902446i	$0.358231\pi$
$504$	0	0
$505$	2.41727e6	0.421790
$506$	0	0
$507$	2.19986e7	3.80081
$508$	0	0
$509$	6.55518e6	1.12148	0.560738	−	0.827993i	$-0.310517\pi$
$509$	6.55518e6	1.12148	0.560738	+	0.827993i	$0.310517\pi$
$510$	0	0
$511$	−2.92370e6	−0.495313
$512$	0	0
$513$	221213.	0.0371122
$514$	0	0
$515$	2.42254e6	0.402488
$516$	0	0
$517$	−1.14978e6	−0.189186
$518$	0	0
$519$	1.86083e6	0.303242
$520$	0	0
$521$	−4.77290e6	−0.770349	−0.385175	−	0.922844i	$-0.625859\pi$
$521$	−4.77290e6	−0.770349	−0.385175	+	0.922844i	$0.625859\pi$
$522$	0	0
$523$	828135.	0.132387	0.0661937	−	0.997807i	$-0.478914\pi$
$523$	828135.	0.132387	0.0661937	+	0.997807i	$0.478914\pi$
$524$	0	0
$525$	−3.07441e6	−0.486814
$526$	0	0
$527$	−1.18913e6	−0.186510
$528$	0	0
$529$	−2.86673e6	−0.445398
$530$	0	0
$531$	−7.97212e6	−1.22698
$532$	0	0
$533$	1.40835e7	2.14730
$534$	0	0
$535$	−2.72994e6	−0.412352
$536$	0	0
$537$	9.40251e6	1.40705
$538$	0	0
$539$	−140590.	−0.0208441
$540$	0	0
$541$	6.93356e6	1.01851	0.509253	−	0.860617i	$-0.329922\pi$
$541$	6.93356e6	1.01851	0.509253	+	0.860617i	$0.329922\pi$
$542$	0	0
$543$	1.22913e6	0.178895
$544$	0	0
$545$	−2.79820e6	−0.403541
$546$	0	0
$547$	−559077.	−0.0798920	−0.0399460	−	0.999202i	$-0.512719\pi$
$547$	−559077.	−0.0798920	−0.0399460	+	0.999202i	$0.512719\pi$
$548$	0	0
$549$	1.11169e7	1.57417
$550$	0	0
$551$	−955047.	−0.134013
$552$	0	0
$553$	−2.97641e6	−0.413885
$554$	0	0
$555$	−949045.	−0.130784
$556$	0	0
$557$	−3.90444e6	−0.533237	−0.266619	−	0.963802i	$-0.585906\pi$
$557$	−3.90444e6	−0.533237	−0.266619	+	0.963802i	$0.585906\pi$
$558$	0	0
$559$	−1.15819e7	−1.56765
$560$	0	0
$561$	−1.89056e6	−0.253620
$562$	0	0
$563$	4.02307e6	0.534918	0.267459	−	0.963569i	$-0.413816\pi$
$563$	4.02307e6	0.534918	0.267459	+	0.963569i	$0.413816\pi$
$564$	0	0
$565$	1.70815e6	0.225116
$566$	0	0
$567$	−3.11745e6	−0.407231
$568$	0	0
$569$	−2.35801e6	−0.305327	−0.152663	−	0.988278i	$-0.548785\pi$
$569$	−2.35801e6	−0.305327	−0.152663	+	0.988278i	$0.548785\pi$
$570$	0	0
$571$	7.76733e6	0.996969	0.498484	−	0.866899i	$-0.333890\pi$
$571$	7.76733e6	0.996969	0.498484	+	0.866899i	$0.333890\pi$
$572$	0	0
$573$	7.55263e6	0.960974
$574$	0	0
$575$	−5.49467e6	−0.693062
$576$	0	0
$577$	1.38363e7	1.73013	0.865066	−	0.501657i	$-0.167276\pi$
$577$	1.38363e7	1.73013	0.865066	+	0.501657i	$0.167276\pi$
$578$	0	0
$579$	−1.34456e7	−1.66680
$580$	0	0
$581$	2.26057e6	0.277830
$582$	0	0
$583$	1.16378e6	0.141807
$584$	0	0
$585$	−3.86246e6	−0.466631
$586$	0	0
$587$	−2.13747e6	−0.256038	−0.128019	−	0.991772i	$-0.540862\pi$
$587$	−2.13747e6	−0.256038	−0.128019	+	0.991772i	$0.540862\pi$
$588$	0	0
$589$	396365.	0.0470768
$590$	0	0
$591$	4.81676e6	0.567265
$592$	0	0
$593$	−1.11381e7	−1.30069	−0.650345	−	0.759639i	$-0.725376\pi$
$593$	−1.11381e7	−1.30069	−0.650345	+	0.759639i	$0.725376\pi$
$594$	0	0
$595$	−1.07962e6	−0.125020
$596$	0	0
$597$	−1.63182e7	−1.87385
$598$	0	0
$599$	−1.10044e7	−1.25314	−0.626572	−	0.779364i	$-0.715543\pi$
$599$	−1.10044e7	−1.25314	−0.626572	+	0.779364i	$0.715543\pi$
$600$	0	0
$601$	1.28046e6	0.144604	0.0723019	−	0.997383i	$-0.476965\pi$
$601$	1.28046e6	0.144604	0.0723019	+	0.997383i	$0.476965\pi$
$602$	0	0
$603$	−1.07166e7	−1.20022
$604$	0	0
$605$	2.32060e6	0.257758
$606$	0	0
$607$	2.08327e6	0.229496	0.114748	−	0.993395i	$-0.463394\pi$
$607$	2.08327e6	0.229496	0.114748	+	0.993395i	$0.463394\pi$
$608$	0	0
$609$	2.02393e6	0.221132
$610$	0	0
$611$	2.31585e7	2.50962
$612$	0	0
$613$	−1.37116e7	−1.47380	−0.736899	−	0.676003i	$-0.763711\pi$
$613$	−1.37116e7	−1.47380	−0.736899	+	0.676003i	$0.763711\pi$
$614$	0	0
$615$	−3.79287e6	−0.404371
$616$	0	0
$617$	−1.65172e7	−1.74672	−0.873361	−	0.487073i	$-0.838064\pi$
$617$	−1.65172e7	−1.74672	−0.873361	+	0.487073i	$0.838064\pi$
$618$	0	0
$619$	−3.51321e6	−0.368534	−0.184267	−	0.982876i	$-0.558991\pi$
$619$	−3.51321e6	−0.368534	−0.184267	+	0.982876i	$0.558991\pi$
$620$	0	0
$621$	−837839.	−0.0871830
$622$	0	0
$623$	3.85477e6	0.397904
$624$	0	0
$625$	7.78055e6	0.796728
$626$	0	0
$627$	630169.	0.0640160
$628$	0	0
$629$	4.47160e6	0.450647
$630$	0	0
$631$	8.54135e6	0.853991	0.426995	−	0.904254i	$-0.359572\pi$
$631$	8.54135e6	0.853991	0.426995	+	0.904254i	$0.359572\pi$
$632$	0	0
$633$	661126.	0.0655805
$634$	0	0
$635$	1.05686e6	0.104012
$636$	0	0
$637$	2.83172e6	0.276504
$638$	0	0
$639$	−1.71681e7	−1.66330
$640$	0	0
$641$	−9.95519e6	−0.956983	−0.478492	−	0.878092i	$-0.658816\pi$
$641$	−9.95519e6	−0.956983	−0.478492	+	0.878092i	$0.658816\pi$
$642$	0	0
$643$	3.98313e6	0.379924	0.189962	−	0.981791i	$-0.439163\pi$
$643$	3.98313e6	0.379924	0.189962	+	0.981791i	$0.439163\pi$
$644$	0	0
$645$	3.11915e6	0.295214
$646$	0	0
$647$	3.84121e6	0.360750	0.180375	−	0.983598i	$-0.442269\pi$
$647$	3.84121e6	0.360750	0.180375	+	0.983598i	$0.442269\pi$
$648$	0	0
$649$	2.09853e6	0.195570
$650$	0	0
$651$	−839975.	−0.0776808
$652$	0	0
$653$	1.84286e7	1.69126	0.845628	−	0.533772i	$-0.179226\pi$
$653$	1.84286e7	1.69126	0.845628	+	0.533772i	$0.179226\pi$
$654$	0	0
$655$	−29683.8	−0.00270344
$656$	0	0
$657$	−1.32727e7	−1.19963
$658$	0	0
$659$	−2.15854e7	−1.93619	−0.968093	−	0.250590i	$-0.919376\pi$
$659$	−2.15854e7	−1.93619	−0.968093	+	0.250590i	$0.919376\pi$
$660$	0	0
$661$	1.44072e7	1.28256	0.641278	−	0.767308i	$-0.278404\pi$
$661$	1.44072e7	1.28256	0.641278	+	0.767308i	$0.278404\pi$
$662$	0	0
$663$	3.80790e7	3.36435
$664$	0	0
$665$	359864.	0.0315561
$666$	0	0
$667$	3.61723e6	0.314819
$668$	0	0
$669$	−9.95378e6	−0.859850
$670$	0	0
$671$	−2.92634e6	−0.250910
$672$	0	0
$673$	−5.55230e6	−0.472536	−0.236268	−	0.971688i	$-0.575924\pi$
$673$	−5.55230e6	−0.472536	−0.236268	+	0.971688i	$0.575924\pi$
$674$	0	0
$675$	1.28968e6	0.108949
$676$	0	0
$677$	−9.92009e6	−0.831848	−0.415924	−	0.909399i	$-0.636542\pi$
$677$	−9.92009e6	−0.831848	−0.415924	+	0.909399i	$0.636542\pi$
$678$	0	0
$679$	−2.13509e6	−0.177722
$680$	0	0
$681$	−7.47414e6	−0.617581
$682$	0	0
$683$	1.01291e7	0.830846	0.415423	−	0.909628i	$-0.363634\pi$
$683$	1.01291e7	0.830846	0.415423	+	0.909628i	$0.363634\pi$
$684$	0	0
$685$	3.59088e6	0.292398
$686$	0	0
$687$	−1.12542e7	−0.909750
$688$	0	0
$689$	−2.34404e7	−1.88112
$690$	0	0
$691$	−1.98747e7	−1.58345	−0.791727	−	0.610875i	$-0.790818\pi$
$691$	−1.98747e7	−1.58345	−0.791727	+	0.610875i	$0.790818\pi$
$692$	0	0
$693$	−638238.	−0.0504835
$694$	0	0
$695$	3.08309e6	0.242116
$696$	0	0
$697$	1.78708e7	1.39336
$698$	0	0
$699$	3.32502e6	0.257395
$700$	0	0
$701$	−2.13147e7	−1.63826	−0.819132	−	0.573605i	$-0.805545\pi$
$701$	−2.13147e7	−1.63826	−0.819132	+	0.573605i	$0.805545\pi$
$702$	0	0
$703$	−1.49049e6	−0.113747
$704$	0	0
$705$	−6.23690e6	−0.472602
$706$	0	0
$707$	−8.04523e6	−0.605327
$708$	0	0
$709$	9.83200e6	0.734559	0.367279	−	0.930111i	$-0.380289\pi$
$709$	9.83200e6	0.734559	0.367279	+	0.930111i	$0.380289\pi$
$710$	0	0
$711$	−1.35120e7	−1.00241
$712$	0	0
$713$	−1.50123e6	−0.110592
$714$	0	0
$715$	1.01673e6	0.0743771
$716$	0	0
$717$	1.03327e7	0.750615
$718$	0	0
$719$	1.80674e7	1.30338	0.651692	−	0.758484i	$-0.274060\pi$
$719$	1.80674e7	1.30338	0.651692	+	0.758484i	$0.274060\pi$
$720$	0	0
$721$	−8.06278e6	−0.577626
$722$	0	0
$723$	1.86728e6	0.132851
$724$	0	0
$725$	−5.56797e6	−0.393416
$726$	0	0
$727$	1.82003e7	1.27715	0.638576	−	0.769559i	$-0.279524\pi$
$727$	1.82003e7	1.27715	0.638576	+	0.769559i	$0.279524\pi$
$728$	0	0
$729$	−1.18274e7	−0.824274
$730$	0	0
$731$	−1.46965e7	−1.01723
$732$	0	0
$733$	6.70629e6	0.461023	0.230512	−	0.973070i	$-0.425960\pi$
$733$	6.70629e6	0.461023	0.230512	+	0.973070i	$0.425960\pi$
$734$	0	0
$735$	−762621.	−0.0520703
$736$	0	0
$737$	2.82095e6	0.191305
$738$	0	0
$739$	−2.16502e7	−1.45831	−0.729156	−	0.684347i	$-0.760087\pi$
$739$	−2.16502e7	−1.45831	−0.729156	+	0.684347i	$0.760087\pi$
$740$	0	0
$741$	−1.26926e7	−0.849193
$742$	0	0
$743$	1.40437e7	0.933276	0.466638	−	0.884448i	$-0.345465\pi$
$743$	1.40437e7	0.933276	0.466638	+	0.884448i	$0.345465\pi$
$744$	0	0
$745$	−337945.	−0.0223077
$746$	0	0
$747$	1.02623e7	0.672890
$748$	0	0
$749$	9.08586e6	0.591782
$750$	0	0
$751$	−8.91381e6	−0.576718	−0.288359	−	0.957522i	$-0.593110\pi$
$751$	−8.91381e6	−0.576718	−0.288359	+	0.957522i	$0.593110\pi$
$752$	0	0
$753$	4.10603e7	2.63897
$754$	0	0
$755$	−4.07427e6	−0.260125
$756$	0	0
$757$	−1.04419e7	−0.662276	−0.331138	−	0.943582i	$-0.607433\pi$
$757$	−1.04419e7	−0.662276	−0.331138	+	0.943582i	$0.607433\pi$
$758$	0	0
$759$	−2.38676e6	−0.150385
$760$	0	0
$761$	2.96214e7	1.85414	0.927072	−	0.374883i	$-0.122317\pi$
$761$	2.96214e7	1.85414	0.927072	+	0.374883i	$0.122317\pi$
$762$	0	0
$763$	9.31307e6	0.579138
$764$	0	0
$765$	−4.90115e6	−0.302792
$766$	0	0
$767$	−4.22678e7	−2.59431
$768$	0	0
$769$	−4.24042e6	−0.258579	−0.129289	−	0.991607i	$-0.541270\pi$
$769$	−4.24042e6	−0.258579	−0.129289	+	0.991607i	$0.541270\pi$
$770$	0	0
$771$	−1.22956e7	−0.744927
$772$	0	0
$773$	−1.37072e7	−0.825086	−0.412543	−	0.910938i	$-0.635359\pi$
$773$	−1.37072e7	−0.825086	−0.412543	+	0.910938i	$0.635359\pi$
$774$	0	0
$775$	2.31083e6	0.138202
$776$	0	0
$777$	3.15864e6	0.187693
$778$	0	0
$779$	−5.95677e6	−0.351696
$780$	0	0
$781$	4.51922e6	0.265116
$782$	0	0
$783$	−849016.	−0.0494893
$784$	0	0
$785$	−4.07360e6	−0.235942
$786$	0	0
$787$	−1.02427e7	−0.589491	−0.294745	−	0.955576i	$-0.595235\pi$
$787$	−1.02427e7	−0.589491	−0.294745	+	0.955576i	$0.595235\pi$
$788$	0	0
$789$	−2.80148e7	−1.60212
$790$	0	0
$791$	−5.68513e6	−0.323072
$792$	0	0
$793$	5.89413e7	3.32841
$794$	0	0
$795$	6.31281e6	0.354246
$796$	0	0
$797$	3.00087e7	1.67340	0.836702	−	0.547658i	$-0.184480\pi$
$797$	3.00087e7	1.67340	0.836702	+	0.547658i	$0.184480\pi$
$798$	0	0
$799$	2.93863e7	1.62846
$800$	0	0
$801$	1.74995e7	0.963704
$802$	0	0
$803$	3.49381e6	0.191210
$804$	0	0
$805$	−1.36298e6	−0.0741309
$806$	0	0
$807$	−1.15900e6	−0.0626468
$808$	0	0
$809$	922081.	0.0495333	0.0247667	−	0.999693i	$-0.492116\pi$
$809$	922081.	0.0495333	0.0247667	+	0.999693i	$0.492116\pi$
$810$	0	0
$811$	2.49323e7	1.33110	0.665549	−	0.746355i	$-0.268198\pi$
$811$	2.49323e7	1.33110	0.665549	+	0.746355i	$0.268198\pi$
$812$	0	0
$813$	1.63544e7	0.867776
$814$	0	0
$815$	4.03776e6	0.212935
$816$	0	0
$817$	4.89868e6	0.256758
$818$	0	0
$819$	1.28552e7	0.669680
$820$	0	0
$821$	−8.06228e6	−0.417446	−0.208723	−	0.977975i	$-0.566931\pi$
$821$	−8.06228e6	−0.417446	−0.208723	+	0.977975i	$0.566931\pi$
$822$	0	0
$823$	5.68168e6	0.292400	0.146200	−	0.989255i	$-0.453296\pi$
$823$	5.68168e6	0.292400	0.146200	+	0.989255i	$0.453296\pi$
$824$	0	0
$825$	3.67392e6	0.187929
$826$	0	0
$827$	976611.	0.0496544	0.0248272	−	0.999692i	$-0.492096\pi$
$827$	976611.	0.0496544	0.0248272	+	0.999692i	$0.492096\pi$
$828$	0	0
$829$	−4.94857e6	−0.250088	−0.125044	−	0.992151i	$-0.539907\pi$
$829$	−4.94857e6	−0.250088	−0.125044	+	0.992151i	$0.539907\pi$
$830$	0	0
$831$	4.47876e7	2.24986
$832$	0	0
$833$	3.59323e6	0.179421
$834$	0	0
$835$	−3.17588e6	−0.157633
$836$	0	0
$837$	352360.	0.0173849
$838$	0	0
$839$	2.03716e7	0.999125	0.499562	−	0.866278i	$-0.333494\pi$
$839$	2.03716e7	0.999125	0.499562	+	0.866278i	$0.333494\pi$
$840$	0	0
$841$	−1.68457e7	−0.821293
$842$	0	0
$843$	5.63507e7	2.73105
$844$	0	0
$845$	−1.50122e7	−0.723273
$846$	0	0
$847$	−7.72349e6	−0.369918
$848$	0	0
$849$	−2.14174e7	−1.01976
$850$	0	0
$851$	5.64521e6	0.267212
$852$	0	0
$853$	1.41579e7	0.666234	0.333117	−	0.942885i	$-0.391900\pi$
$853$	1.41579e7	0.666234	0.333117	+	0.942885i	$0.391900\pi$
$854$	0	0
$855$	1.63367e6	0.0764274
$856$	0	0
$857$	−1.24397e7	−0.578575	−0.289287	−	0.957242i	$-0.593418\pi$
$857$	−1.24397e7	−0.578575	−0.289287	+	0.957242i	$0.593418\pi$
$858$	0	0
$859$	−1.72105e7	−0.795810	−0.397905	−	0.917427i	$-0.630263\pi$
$859$	−1.72105e7	−0.795810	−0.397905	+	0.917427i	$0.630263\pi$
$860$	0	0
$861$	1.26236e7	0.580328
$862$	0	0
$863$	−2.04374e7	−0.934113	−0.467057	−	0.884227i	$-0.654686\pi$
$863$	−2.04374e7	−0.934113	−0.467057	+	0.884227i	$0.654686\pi$
$864$	0	0
$865$	−1.26986e6	−0.0577052
$866$	0	0
$867$	1.76869e7	0.799107
$868$	0	0
$869$	3.55681e6	0.159776
$870$	0	0
$871$	−5.68187e7	−2.53773
$872$	0	0
$873$	−9.69266e6	−0.430435
$874$	0	0
$875$	4.35241e6	0.192181
$876$	0	0
$877$	−1.72398e7	−0.756890	−0.378445	−	0.925624i	$-0.623541\pi$
$877$	−1.72398e7	−0.756890	−0.378445	+	0.925624i	$0.623541\pi$
$878$	0	0
$879$	8.01972e6	0.350096
$880$	0	0
$881$	−3.86589e7	−1.67807	−0.839034	−	0.544080i	$-0.816879\pi$
$881$	−3.86589e7	−1.67807	−0.839034	+	0.544080i	$0.816879\pi$
$882$	0	0
$883$	3.29454e7	1.42198	0.710990	−	0.703203i	$-0.248247\pi$
$883$	3.29454e7	1.42198	0.710990	+	0.703203i	$0.248247\pi$
$884$	0	0
$885$	1.13833e7	0.488551
$886$	0	0
$887$	−3.61545e7	−1.54295	−0.771477	−	0.636258i	$-0.780482\pi$
$887$	−3.61545e7	−1.54295	−0.771477	+	0.636258i	$0.780482\pi$
$888$	0	0
$889$	−3.51749e6	−0.149272
$890$	0	0
$891$	3.72534e6	0.157207
$892$	0	0
$893$	−9.79516e6	−0.411039
$894$	0	0
$895$	−6.41641e6	−0.267753
$896$	0	0
$897$	4.80732e7	1.99490
$898$	0	0
$899$	−1.52125e6	−0.0627772
$900$	0	0
$901$	−2.97440e7	−1.22064
$902$	0	0
$903$	−1.03813e7	−0.423673
$904$	0	0
$905$	−838774.	−0.0340427
$906$	0	0
$907$	4.15620e7	1.67756	0.838781	−	0.544469i	$-0.183269\pi$
$907$	4.15620e7	1.67756	0.838781	+	0.544469i	$0.183269\pi$
$908$	0	0
$909$	−3.65229e7	−1.46607
$910$	0	0
$911$	9.55917e6	0.381614	0.190807	−	0.981628i	$-0.438890\pi$
$911$	9.55917e6	0.381614	0.190807	+	0.981628i	$0.438890\pi$
$912$	0	0
$913$	−2.70138e6	−0.107253
$914$	0	0
$915$	−1.58737e7	−0.626794
$916$	0	0
$917$	98794.6	0.00387980
$918$	0	0
$919$	1.20453e7	0.470469	0.235234	−	0.971939i	$-0.424414\pi$
$919$	1.20453e7	0.470469	0.235234	+	0.971939i	$0.424414\pi$
$920$	0	0
$921$	6.12505e7	2.37936
$922$	0	0
$923$	−9.10244e7	−3.51685
$924$	0	0
$925$	−8.68963e6	−0.333924
$926$	0	0
$927$	−3.66026e7	−1.39898
$928$	0	0
$929$	−2.42707e7	−0.922661	−0.461331	−	0.887228i	$-0.652628\pi$
$929$	−2.42707e7	−0.922661	−0.461331	+	0.887228i	$0.652628\pi$
$930$	0	0
$931$	−1.19771e6	−0.0452874
$932$	0	0
$933$	−1.63826e7	−0.616140
$934$	0	0
$935$	1.29015e6	0.0482625
$936$	0	0
$937$	2.02014e7	0.751679	0.375840	−	0.926685i	$-0.377354\pi$
$937$	2.02014e7	0.751679	0.375840	+	0.926685i	$0.377354\pi$
$938$	0	0
$939$	−3.88747e7	−1.43881
$940$	0	0
$941$	−6.68084e6	−0.245956	−0.122978	−	0.992409i	$-0.539244\pi$
$941$	−6.68084e6	−0.245956	−0.122978	+	0.992409i	$0.539244\pi$
$942$	0	0
$943$	2.25612e7	0.826195
$944$	0	0
$945$	319911.	0.0116533
$946$	0	0
$947$	3.19373e7	1.15724	0.578619	−	0.815598i	$-0.303592\pi$
$947$	3.19373e7	1.15724	0.578619	+	0.815598i	$0.303592\pi$
$948$	0	0
$949$	−7.03712e7	−2.53647
$950$	0	0
$951$	−9.53361e6	−0.341827
$952$	0	0
$953$	2.26242e7	0.806938	0.403469	−	0.914993i	$-0.367804\pi$
$953$	2.26242e7	0.806938	0.403469	+	0.914993i	$0.367804\pi$
$954$	0	0
$955$	−5.15402e6	−0.182868
$956$	0	0
$957$	−2.41859e6	−0.0853657
$958$	0	0
$959$	−1.19513e7	−0.419631
$960$	0	0
$961$	−2.79978e7	−0.977947
$962$	0	0
$963$	4.12470e7	1.43327
$964$	0	0
$965$	9.17547e6	0.317183
$966$	0	0
$967$	2.38201e7	0.819178	0.409589	−	0.912270i	$-0.365672\pi$
$967$	2.38201e7	0.819178	0.409589	+	0.912270i	$0.365672\pi$
$968$	0	0
$969$	−1.61059e7	−0.551032
$970$	0	0
$971$	−1.87074e7	−0.636746	−0.318373	−	0.947965i	$-0.603136\pi$
$971$	−1.87074e7	−0.636746	−0.318373	+	0.947965i	$0.603136\pi$
$972$	0	0
$973$	−1.02612e7	−0.347470
$974$	0	0
$975$	−7.39987e7	−2.49294
$976$	0	0
$977$	1.50783e7	0.505376	0.252688	−	0.967548i	$-0.418685\pi$
$977$	1.50783e7	0.505376	0.252688	+	0.967548i	$0.418685\pi$
$978$	0	0
$979$	−4.60644e6	−0.153606
$980$	0	0
$981$	4.22785e7	1.40264
$982$	0	0
$983$	−2.38312e6	−0.0786614	−0.0393307	−	0.999226i	$-0.512523\pi$
$983$	−2.38312e6	−0.0786614	−0.0393307	+	0.999226i	$0.512523\pi$
$984$	0	0
$985$	−3.28703e6	−0.107948
$986$	0	0
$987$	2.07579e7	0.678250
$988$	0	0
$989$	−1.85537e7	−0.603170
$990$	0	0
$991$	4.86331e7	1.57307	0.786535	−	0.617546i	$-0.211873\pi$
$991$	4.86331e7	1.57307	0.786535	+	0.617546i	$0.211873\pi$
$992$	0	0
$993$	6.81268e7	2.19253
$994$	0	0
$995$	1.11357e7	0.356584
$996$	0	0
$997$	1.73445e7	0.552617	0.276309	−	0.961069i	$-0.410889\pi$
$997$	1.73445e7	0.552617	0.276309	+	0.961069i	$0.410889\pi$
$998$	0	0
$999$	−1.32501e6	−0.0420056

Twists

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

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

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

\(f(q)\)	\(=\)	\(q+21.5742 q^{3} -14.7225 q^{5} +49.0000 q^{7} +222.445 q^{9} -58.5549 q^{11} +1179.39 q^{13} -317.626 q^{15} +1496.55 q^{17} -498.838 q^{19} +1057.13 q^{21} +1889.34 q^{23} -2908.25 q^{25} -443.456 q^{27} +1914.54 q^{29} -794.577 q^{31} -1263.27 q^{33} -721.404 q^{35} +2987.93 q^{37} +25444.4 q^{39} +11941.3 q^{41} -9820.19 q^{43} -3274.95 q^{45} +19636.0 q^{47} +2401.00 q^{49} +32286.9 q^{51} -19875.0 q^{53} +862.077 q^{55} -10762.0 q^{57} -35838.6 q^{59} +49975.9 q^{61} +10899.8 q^{63} -17363.6 q^{65} -48176.2 q^{67} +40761.0 q^{69} -77179.1 q^{71} -59667.3 q^{73} -62743.0 q^{75} -2869.19 q^{77} -60743.1 q^{79} -63621.3 q^{81} +46134.2 q^{83} -22033.1 q^{85} +41304.7 q^{87} +78668.7 q^{89} +57790.2 q^{91} -17142.3 q^{93} +7344.15 q^{95} -43573.3 q^{97} -13025.3 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 2 q + 6 q^{3} + 82 q^{5} + 98 q^{7} + 222 q^{9} - 340 q^{11} + 910 q^{13} - 1824 q^{15} + 3216 q^{17} + 674 q^{19} + 294 q^{21} + 1104 q^{23} + 3322 q^{25} + 3348 q^{27} + 8064 q^{29} + 6212 q^{31} + 3120 q^{33}+ \cdots - 12900 q^{99}+O(q^{100}) \) 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

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