LMFDB - Embedded newform 112.4.a.h.1.2

Show commands: Magma / Pari/GP / SageMath

Newspace parameters

comment:Compute space of new eigenforms

gp:[N,k,chi] = [112,4,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, 4); 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, 4, names="a")

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

Newform invariants

comment:select newform

sage:traces = [2,0,2,0,22] 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:	$6.60821392064$
Analytic rank:	$0$
Dimension:	$2$
Coefficient field:	$\Q(\sqrt{57}) $
comment:defining polynomial gp:f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{2} - x - 14 $ x^2 - x - 14
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			$4.27492$ 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)

$f(q)$	$=$	$q+8.54983 q^{3} +18.5498 q^{5} -7.00000 q^{7} +46.0997 q^{9} -63.2990 q^{11} -16.7492 q^{13} +158.598 q^{15} -41.2990 q^{17} -39.1478 q^{19} -59.8488 q^{21} +21.8007 q^{23} +219.096 q^{25} +163.299 q^{27} +138.096 q^{29} +95.6977 q^{31} -541.196 q^{33} -129.849 q^{35} +176.502 q^{37} -143.203 q^{39} -407.897 q^{41} -100.302 q^{43} +855.141 q^{45} +144.302 q^{47} +49.0000 q^{49} -353.100 q^{51} -409.588 q^{53} -1174.19 q^{55} -334.708 q^{57} +0.852152 q^{59} -407.045 q^{61} -322.698 q^{63} -310.694 q^{65} +9.38874 q^{67} +186.392 q^{69} +944.193 q^{71} +86.2060 q^{73} +1873.24 q^{75} +443.093 q^{77} -563.794 q^{79} +151.488 q^{81} +969.842 q^{83} -766.090 q^{85} +1180.70 q^{87} +1505.97 q^{89} +117.244 q^{91} +818.199 q^{93} -726.186 q^{95} -956.488 q^{97} -2918.06 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 2 q + 2 q^{3} + 22 q^{5} - 14 q^{7} + 62 q^{9} - 36 q^{11} + 42 q^{13} + 136 q^{15} + 8 q^{17} + 118 q^{19} - 14 q^{21} + 104 q^{23} + 106 q^{25} + 236 q^{27} - 56 q^{29} - 20 q^{31} - 720 q^{33} - 154 q^{35}+ \cdots - 2484 q^{99}+O(q^{100}) $ 2 * q + 2 * q^3 + 22 * q^5 - 14 * q^7 + 62 * q^9 - 36 * q^11 + 42 * q^13 + 136 * q^15 + 8 * q^17 + 118 * q^19 - 14 * q^21 + 104 * q^23 + 106 * q^25 + 236 * q^27 - 56 * q^29 - 20 * q^31 - 720 * q^33 - 154 * q^35 + 504 * q^37 - 528 * q^39 - 544 * q^41 - 412 * q^43 + 910 * q^45 + 500 * q^47 + 98 * q^49 - 676 * q^51 + 268 * q^53 - 1080 * q^55 - 1364 * q^57 + 198 * q^59 - 346 * q^61 - 434 * q^63 - 108 * q^65 - 1008 * q^67 - 352 * q^69 + 1224 * q^71 + 716 * q^73 + 2614 * q^75 + 252 * q^77 - 584 * q^79 - 754 * q^81 + 1230 * q^83 - 596 * q^85 + 2452 * q^87 + 596 * q^89 - 294 * q^91 + 1576 * q^93 - 184 * q^95 - 856 * q^97 - 2484 * 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
$3$	8.54983	1.64542	0.822708	−	0.568464i	$-0.192462\pi$
$3$	8.54983	1.64542	0.822708	+	0.568464i	$0.192462\pi$
$4$	0	0
$5$	18.5498	1.65915	0.829574	−	0.558397i	$-0.188583\pi$
$5$	18.5498	1.65915	0.829574	+	0.558397i	$0.188583\pi$
$6$	0	0
$7$	−7.00000	−0.377964
$7$	−7.00000	−0.377964
$8$	0	0
$9$	46.0997	1.70740
$10$	0	0
$11$	−63.2990	−1.73503	−0.867517	−	0.497408i	$-0.834285\pi$
$11$	−63.2990	−1.73503	−0.867517	+	0.497408i	$0.834285\pi$
$12$	0	0
$13$	−16.7492	−0.357337	−0.178669	−	0.983909i	$-0.557179\pi$
$13$	−16.7492	−0.357337	−0.178669	+	0.983909i	$0.557179\pi$
$14$	0	0
$15$	158.598	2.72999
$16$	0	0
$17$	−41.2990	−0.589205	−0.294602	−	0.955620i	$-0.595187\pi$
$17$	−41.2990	−0.589205	−0.294602	+	0.955620i	$0.595187\pi$
$18$	0	0
$19$	−39.1478	−0.472691	−0.236346	−	0.971669i	$-0.575950\pi$
$19$	−39.1478	−0.472691	−0.236346	+	0.971669i	$0.575950\pi$
$20$	0	0
$21$	−59.8488	−0.621909
$22$	0	0
$23$	21.8007	0.197641	0.0988207	−	0.995105i	$-0.468493\pi$
$23$	21.8007	0.197641	0.0988207	+	0.995105i	$0.468493\pi$
$24$	0	0
$25$	219.096	1.75277
$26$	0	0
$27$	163.299	1.16396
$28$	0	0
$29$	138.096	0.884271	0.442135	−	0.896948i	$-0.354221\pi$
$29$	138.096	0.884271	0.442135	+	0.896948i	$0.354221\pi$
$30$	0	0
$31$	95.6977	0.554446	0.277223	−	0.960806i	$-0.410586\pi$
$31$	95.6977	0.554446	0.277223	+	0.960806i	$0.410586\pi$
$32$	0	0
$33$	−541.196	−2.85485
$34$	0	0
$35$	−129.849	−0.627099
$36$	0	0
$37$	176.502	0.784235	0.392117	−	0.919915i	$-0.371743\pi$
$37$	176.502	0.784235	0.392117	+	0.919915i	$0.371743\pi$
$38$	0	0
$39$	−143.203	−0.587969
$40$	0	0
$41$	−407.897	−1.55373	−0.776864	−	0.629669i	$-0.783191\pi$
$41$	−407.897	−1.55373	−0.776864	+	0.629669i	$0.783191\pi$
$42$	0	0
$43$	−100.302	−0.355720	−0.177860	−	0.984056i	$-0.556917\pi$
$43$	−100.302	−0.355720	−0.177860	+	0.984056i	$0.556917\pi$
$44$	0	0
$45$	855.141	2.83282
$46$	0	0
$47$	144.302	0.447844	0.223922	−	0.974607i	$-0.428114\pi$
$47$	144.302	0.447844	0.223922	+	0.974607i	$0.428114\pi$
$48$	0	0
$49$	49.0000	0.142857
$50$	0	0
$51$	−353.100	−0.969487
$52$	0	0
$53$	−409.588	−1.06153	−0.530767	−	0.847518i	$-0.678096\pi$
$53$	−409.588	−1.06153	−0.530767	+	0.847518i	$0.678096\pi$
$54$	0	0
$55$	−1174.19	−2.87868
$56$	0	0
$57$	−334.708	−0.777774
$58$	0	0
$59$	0.852152	0.00188035	0.000940176	−	1.00000i	$-0.499701\pi$
$59$	0.852152	0.00188035	0.000940176		1.00000i	$0.499701\pi$
$60$	0	0
$61$	−407.045	−0.854373	−0.427187	−	0.904164i	$-0.640495\pi$
$61$	−407.045	−0.854373	−0.427187	+	0.904164i	$0.640495\pi$
$62$	0	0
$63$	−322.698	−0.645335
$64$	0	0
$65$	−310.694	−0.592875
$66$	0	0
$67$	9.38874	0.0171197	0.00855983	−	0.999963i	$-0.497275\pi$
$67$	9.38874	0.0171197	0.00855983	+	0.999963i	$0.497275\pi$
$68$	0	0
$69$	186.392	0.325202
$70$	0	0
$71$	944.193	1.57824	0.789120	−	0.614239i	$-0.210537\pi$
$71$	944.193	1.57824	0.789120	+	0.614239i	$0.210537\pi$
$72$	0	0
$73$	86.2060	0.138214	0.0691072	−	0.997609i	$-0.477985\pi$
$73$	86.2060	0.138214	0.0691072	+	0.997609i	$0.477985\pi$
$74$	0	0
$75$	1873.24	2.88404
$76$	0	0
$77$	443.093	0.655781
$78$	0	0
$79$	−563.794	−0.802934	−0.401467	−	0.915873i	$-0.631500\pi$
$79$	−563.794	−0.802934	−0.401467	+	0.915873i	$0.631500\pi$
$80$	0	0
$81$	151.488	0.207803
$82$	0	0
$83$	969.842	1.28258	0.641289	−	0.767299i	$-0.278400\pi$
$83$	969.842	1.28258	0.641289	+	0.767299i	$0.278400\pi$
$84$	0	0
$85$	−766.090	−0.977578
$86$	0	0
$87$	1180.70	1.45499
$88$	0	0
$89$	1505.97	1.79363	0.896814	−	0.442408i	$-0.145876\pi$
$89$	1505.97	1.79363	0.896814	+	0.442408i	$0.145876\pi$
$90$	0	0
$91$	117.244	0.135061
$92$	0	0
$93$	818.199	0.912294
$94$	0	0
$95$	−726.186	−0.784264
$96$	0	0
$97$	−956.488	−1.00120	−0.500601	−	0.865678i	$-0.666888\pi$
$97$	−956.488	−1.00120	−0.500601	+	0.865678i	$0.666888\pi$
$98$	0	0
$99$	−2918.06	−2.96239
$100$	0	0
$101$	1641.75	1.61743	0.808715	−	0.588200i	$-0.200163\pi$
$101$	1641.75	1.61743	0.808715	+	0.588200i	$0.200163\pi$
$102$	0	0
$103$	−580.894	−0.555701	−0.277850	−	0.960624i	$-0.589622\pi$
$103$	−580.894	−0.555701	−0.277850	+	0.960624i	$0.589622\pi$
$104$	0	0
$105$	−1110.19	−1.03184
$106$	0	0
$107$	1681.99	1.51967	0.759834	−	0.650117i	$-0.225280\pi$
$107$	1681.99	1.51967	0.759834	+	0.650117i	$0.225280\pi$
$108$	0	0
$109$	246.302	0.216436	0.108218	−	0.994127i	$-0.465486\pi$
$109$	246.302	0.216436	0.108218	+	0.994127i	$0.465486\pi$
$110$	0	0
$111$	1509.06	1.29039
$112$	0	0
$113$	−2387.25	−1.98738	−0.993690	−	0.112165i	$-0.964221\pi$
$113$	−2387.25	−1.98738	−0.993690	+	0.112165i	$0.964221\pi$
$114$	0	0
$115$	404.399	0.327916
$116$	0	0
$117$	−772.131	−0.610116
$118$	0	0
$119$	289.093	0.222698
$120$	0	0
$121$	2675.76	2.01034
$122$	0	0
$123$	−3487.45	−2.55653
$124$	0	0
$125$	1745.47	1.24896
$126$	0	0
$127$	−1616.00	−1.12911	−0.564554	−	0.825396i	$-0.690952\pi$
$127$	−1616.00	−1.12911	−0.564554	+	0.825396i	$0.690952\pi$
$128$	0	0
$129$	−857.568	−0.585308
$130$	0	0
$131$	−462.350	−0.308365	−0.154182	−	0.988042i	$-0.549274\pi$
$131$	−462.350	−0.308365	−0.154182	+	0.988042i	$0.549274\pi$
$132$	0	0
$133$	274.035	0.178660
$134$	0	0
$135$	3029.17	1.93118
$136$	0	0
$137$	679.362	0.423663	0.211832	−	0.977306i	$-0.432057\pi$
$137$	679.362	0.423663	0.211832	+	0.977306i	$0.432057\pi$
$138$	0	0
$139$	−1514.45	−0.924127	−0.462064	−	0.886847i	$-0.652891\pi$
$139$	−1514.45	−0.924127	−0.462064	+	0.886847i	$0.652891\pi$
$140$	0	0
$141$	1233.76	0.736889
$142$	0	0
$143$	1060.21	0.619992
$144$	0	0
$145$	2561.66	1.46714
$146$	0	0
$147$	418.942	0.235059
$148$	0	0
$149$	−1031.39	−0.567078	−0.283539	−	0.958961i	$-0.591509\pi$
$149$	−1031.39	−0.567078	−0.283539	+	0.958961i	$0.591509\pi$
$150$	0	0
$151$	3066.37	1.65256	0.826282	−	0.563256i	$-0.190452\pi$
$151$	3066.37	1.65256	0.826282	+	0.563256i	$0.190452\pi$
$152$	0	0
$153$	−1903.87	−1.00601
$154$	0	0
$155$	1775.18	0.919907
$156$	0	0
$157$	−3196.35	−1.62482	−0.812409	−	0.583087i	$-0.801844\pi$
$157$	−3196.35	−1.62482	−0.812409	+	0.583087i	$0.801844\pi$
$158$	0	0
$159$	−3501.91	−1.74666
$160$	0	0
$161$	−152.605	−0.0747014
$162$	0	0
$163$	308.907	0.148438	0.0742192	−	0.997242i	$-0.476354\pi$
$163$	308.907	0.148438	0.0742192	+	0.997242i	$0.476354\pi$
$164$	0	0
$165$	−10039.1	−4.73662
$166$	0	0
$167$	−1449.46	−0.671632	−0.335816	−	0.941928i	$-0.609012\pi$
$167$	−1449.46	−0.671632	−0.335816	+	0.941928i	$0.609012\pi$
$168$	0	0
$169$	−1916.47	−0.872310
$170$	0	0
$171$	−1804.70	−0.807071
$172$	0	0
$173$	1817.86	0.798899	0.399449	−	0.916755i	$-0.369201\pi$
$173$	1817.86	0.798899	0.399449	+	0.916755i	$0.369201\pi$
$174$	0	0
$175$	−1533.67	−0.662485
$176$	0	0
$177$	7.28576	0.00309396
$178$	0	0
$179$	2145.14	0.895729	0.447865	−	0.894101i	$-0.352185\pi$
$179$	2145.14	0.895729	0.447865	+	0.894101i	$0.352185\pi$
$180$	0	0
$181$	1952.00	0.801606	0.400803	−	0.916164i	$-0.368731\pi$
$181$	1952.00	0.801606	0.400803	+	0.916164i	$0.368731\pi$
$182$	0	0
$183$	−3480.17	−1.40580
$184$	0	0
$185$	3274.08	1.30116
$186$	0	0
$187$	2614.19	1.02229
$188$	0	0
$189$	−1143.09	−0.439935
$190$	0	0
$191$	−2352.55	−0.891229	−0.445614	−	0.895225i	$-0.647015\pi$
$191$	−2352.55	−0.891229	−0.445614	+	0.895225i	$0.647015\pi$
$192$	0	0
$193$	3358.46	1.25257	0.626287	−	0.779592i	$-0.284574\pi$
$193$	3358.46	1.25257	0.626287	+	0.779592i	$0.284574\pi$
$194$	0	0
$195$	−2656.39	−0.975527
$196$	0	0
$197$	2484.16	0.898422	0.449211	−	0.893426i	$-0.351705\pi$
$197$	2484.16	0.898422	0.449211	+	0.893426i	$0.351705\pi$
$198$	0	0
$199$	4181.66	1.48960	0.744800	−	0.667288i	$-0.232545\pi$
$199$	4181.66	1.48960	0.744800	+	0.667288i	$0.232545\pi$
$200$	0	0
$201$	80.2722	0.0281690
$202$	0	0
$203$	−966.675	−0.334223
$204$	0	0
$205$	−7566.42	−2.57786
$206$	0	0
$207$	1005.00	0.337452
$208$	0	0
$209$	2478.02	0.820135
$210$	0	0
$211$	1246.56	0.406714	0.203357	−	0.979105i	$-0.434815\pi$
$211$	1246.56	0.406714	0.203357	+	0.979105i	$0.434815\pi$
$212$	0	0
$213$	8072.69	2.59686
$214$	0	0
$215$	−1860.59	−0.590192
$216$	0	0
$217$	−669.884	−0.209561
$218$	0	0
$219$	737.047	0.227420
$220$	0	0
$221$	691.724	0.210545
$222$	0	0
$223$	−2292.18	−0.688321	−0.344161	−	0.938911i	$-0.611837\pi$
$223$	−2292.18	−0.688321	−0.344161	+	0.938911i	$0.611837\pi$
$224$	0	0
$225$	10100.3	2.99267
$226$	0	0
$227$	4799.95	1.40345	0.701726	−	0.712447i	$-0.252413\pi$
$227$	4799.95	1.40345	0.701726	+	0.712447i	$0.252413\pi$
$228$	0	0
$229$	−1059.75	−0.305808	−0.152904	−	0.988241i	$-0.548862\pi$
$229$	−1059.75	−0.305808	−0.152904	+	0.988241i	$0.548862\pi$
$230$	0	0
$231$	3788.37	1.07903
$232$	0	0
$233$	−1396.74	−0.392718	−0.196359	−	0.980532i	$-0.562912\pi$
$233$	−1396.74	−0.392718	−0.196359	+	0.980532i	$0.562912\pi$
$234$	0	0
$235$	2676.78	0.743039
$236$	0	0
$237$	−4820.35	−1.32116
$238$	0	0
$239$	−299.050	−0.0809369	−0.0404684	−	0.999181i	$-0.512885\pi$
$239$	−299.050	−0.0809369	−0.0404684	+	0.999181i	$0.512885\pi$
$240$	0	0
$241$	−4552.86	−1.21691	−0.608455	−	0.793588i	$-0.708211\pi$
$241$	−4552.86	−1.21691	−0.608455	+	0.793588i	$0.708211\pi$
$242$	0	0
$243$	−3113.87	−0.822037
$244$	0	0
$245$	908.942	0.237021
$246$	0	0
$247$	655.694	0.168910
$248$	0	0
$249$	8291.99	2.11038
$250$	0	0
$251$	4335.56	1.09027	0.545136	−	0.838348i	$-0.316478\pi$
$251$	4335.56	1.09027	0.545136	+	0.838348i	$0.316478\pi$
$252$	0	0
$253$	−1379.96	−0.342914
$254$	0	0
$255$	−6549.94	−1.60852
$256$	0	0
$257$	−1235.22	−0.299810	−0.149905	−	0.988700i	$-0.547897\pi$
$257$	−1235.22	−0.299810	−0.149905	+	0.988700i	$0.547897\pi$
$258$	0	0
$259$	−1235.51	−0.296413
$260$	0	0
$261$	6366.20	1.50980
$262$	0	0
$263$	2030.65	0.476104	0.238052	−	0.971252i	$-0.423491\pi$
$263$	2030.65	0.476104	0.238052	+	0.971252i	$0.423491\pi$
$264$	0	0
$265$	−7597.79	−1.76124
$266$	0	0
$267$	12875.8	2.95126
$268$	0	0
$269$	−2610.12	−0.591606	−0.295803	−	0.955249i	$-0.595587\pi$
$269$	−2610.12	−0.591606	−0.295803	+	0.955249i	$0.595587\pi$
$270$	0	0
$271$	−609.595	−0.136643	−0.0683215	−	0.997663i	$-0.521764\pi$
$271$	−609.595	−0.136643	−0.0683215	+	0.997663i	$0.521764\pi$
$272$	0	0
$273$	1002.42	0.222231
$274$	0	0
$275$	−13868.6	−3.04112
$276$	0	0
$277$	5083.53	1.10267	0.551335	−	0.834284i	$-0.314119\pi$
$277$	5083.53	1.10267	0.551335	+	0.834284i	$0.314119\pi$
$278$	0	0
$279$	4411.63	0.946658
$280$	0	0
$281$	−5131.55	−1.08941	−0.544703	−	0.838629i	$-0.683357\pi$
$281$	−5131.55	−1.08941	−0.544703	+	0.838629i	$0.683357\pi$
$282$	0	0
$283$	1762.83	0.370281	0.185140	−	0.982712i	$-0.440726\pi$
$283$	1762.83	0.370281	0.185140	+	0.982712i	$0.440726\pi$
$284$	0	0
$285$	−6208.77	−1.29044
$286$	0	0
$287$	2855.28	0.587254
$288$	0	0
$289$	−3207.39	−0.652838
$290$	0	0
$291$	−8177.82	−1.64740
$292$	0	0
$293$	458.796	0.0914782	0.0457391	−	0.998953i	$-0.485436\pi$
$293$	458.796	0.0914782	0.0457391	+	0.998953i	$0.485436\pi$
$294$	0	0
$295$	15.8073	0.00311978
$296$	0	0
$297$	−10336.7	−2.01951
$298$	0	0
$299$	−365.143	−0.0706246
$300$	0	0
$301$	702.116	0.134450
$302$	0	0
$303$	14036.7	2.66135
$304$	0	0
$305$	−7550.61	−1.41753
$306$	0	0
$307$	−4858.28	−0.903182	−0.451591	−	0.892225i	$-0.649143\pi$
$307$	−4858.28	−0.903182	−0.451591	+	0.892225i	$0.649143\pi$
$308$	0	0
$309$	−4966.55	−0.914359
$310$	0	0
$311$	−705.289	−0.128596	−0.0642978	−	0.997931i	$-0.520481\pi$
$311$	−705.289	−0.128596	−0.0642978	+	0.997931i	$0.520481\pi$
$312$	0	0
$313$	6210.34	1.12150	0.560749	−	0.827986i	$-0.310513\pi$
$313$	6210.34	1.12150	0.560749	+	0.827986i	$0.310513\pi$
$314$	0	0
$315$	−5985.99	−1.07071
$316$	0	0
$317$	7438.53	1.31795	0.658973	−	0.752166i	$-0.270991\pi$
$317$	7438.53	1.31795	0.658973	+	0.752166i	$0.270991\pi$
$318$	0	0
$319$	−8741.36	−1.53424
$320$	0	0
$321$	14380.8	2.50049
$322$	0	0
$323$	1616.77	0.278512
$324$	0	0
$325$	−3669.68	−0.626330
$326$	0	0
$327$	2105.84	0.356127
$328$	0	0
$329$	−1010.12	−0.169269
$330$	0	0
$331$	−4883.48	−0.810937	−0.405469	−	0.914109i	$-0.632892\pi$
$331$	−4883.48	−0.810937	−0.405469	+	0.914109i	$0.632892\pi$
$332$	0	0
$333$	8136.67	1.33900
$334$	0	0
$335$	174.160	0.0284040
$336$	0	0
$337$	1890.14	0.305526	0.152763	−	0.988263i	$-0.451183\pi$
$337$	1890.14	0.305526	0.152763	+	0.988263i	$0.451183\pi$
$338$	0	0
$339$	−20410.6	−3.27007
$340$	0	0
$341$	−6057.57	−0.961982
$342$	0	0
$343$	−343.000	−0.0539949
$344$	0	0
$345$	3457.54	0.539559
$346$	0	0
$347$	1938.41	0.299883	0.149941	−	0.988695i	$-0.452092\pi$
$347$	1938.41	0.299883	0.149941	+	0.988695i	$0.452092\pi$
$348$	0	0
$349$	5119.42	0.785204	0.392602	−	0.919709i	$-0.371575\pi$
$349$	5119.42	0.785204	0.392602	+	0.919709i	$0.371575\pi$
$350$	0	0
$351$	−2735.12	−0.415926
$352$	0	0
$353$	8124.62	1.22501	0.612507	−	0.790465i	$-0.290161\pi$
$353$	8124.62	1.22501	0.612507	+	0.790465i	$0.290161\pi$
$354$	0	0
$355$	17514.6	2.61853
$356$	0	0
$357$	2471.70	0.366432
$358$	0	0
$359$	−9429.09	−1.38621	−0.693104	−	0.720838i	$-0.743757\pi$
$359$	−9429.09	−1.38621	−0.693104	+	0.720838i	$0.743757\pi$
$360$	0	0
$361$	−5326.45	−0.776563
$362$	0	0
$363$	22877.3	3.30785
$364$	0	0
$365$	1599.11	0.229318
$366$	0	0
$367$	−12307.6	−1.75055	−0.875273	−	0.483629i	$-0.839318\pi$
$367$	−12307.6	−1.75055	−0.875273	+	0.483629i	$0.839318\pi$
$368$	0	0
$369$	−18803.9	−2.65283
$370$	0	0
$371$	2867.12	0.401222
$372$	0	0
$373$	−7212.01	−1.00114	−0.500568	−	0.865697i	$-0.666876\pi$
$373$	−7212.01	−1.00114	−0.500568	+	0.865697i	$0.666876\pi$
$374$	0	0
$375$	14923.5	2.05506
$376$	0	0
$377$	−2313.00	−0.315983
$378$	0	0
$379$	−7967.91	−1.07991	−0.539953	−	0.841695i	$-0.681558\pi$
$379$	−7967.91	−1.07991	−0.539953	+	0.841695i	$0.681558\pi$
$380$	0	0
$381$	−13816.5	−1.85785
$382$	0	0
$383$	−2179.76	−0.290811	−0.145406	−	0.989372i	$-0.546449\pi$
$383$	−2179.76	−0.290811	−0.145406	+	0.989372i	$0.546449\pi$
$384$	0	0
$385$	8219.30	1.08804
$386$	0	0
$387$	−4623.90	−0.607355
$388$	0	0
$389$	3384.86	0.441181	0.220590	−	0.975367i	$-0.429202\pi$
$389$	3384.86	0.441181	0.220590	+	0.975367i	$0.429202\pi$
$390$	0	0
$391$	−900.346	−0.116451
$392$	0	0
$393$	−3953.02	−0.507388
$394$	0	0
$395$	−10458.3	−1.33219
$396$	0	0
$397$	8422.51	1.06477	0.532385	−	0.846502i	$-0.321296\pi$
$397$	8422.51	1.06477	0.532385	+	0.846502i	$0.321296\pi$
$398$	0	0
$399$	2342.95	0.293971
$400$	0	0
$401$	−4994.70	−0.622004	−0.311002	−	0.950409i	$-0.600665\pi$
$401$	−4994.70	−0.622004	−0.311002	+	0.950409i	$0.600665\pi$
$402$	0	0
$403$	−1602.86	−0.198124
$404$	0	0
$405$	2810.08	0.344776
$406$	0	0
$407$	−11172.4	−1.36067
$408$	0	0
$409$	1919.31	0.232039	0.116019	−	0.993247i	$-0.462987\pi$
$409$	1919.31	0.232039	0.116019	+	0.993247i	$0.462987\pi$
$410$	0	0
$411$	5808.43	0.697102
$412$	0	0
$413$	−5.96507	−0.000710706 0
$414$	0	0
$415$	17990.4	2.12799
$416$	0	0
$417$	−12948.3	−1.52057
$418$	0	0
$419$	−8429.07	−0.982785	−0.491392	−	0.870938i	$-0.663512\pi$
$419$	−8429.07	−0.982785	−0.491392	+	0.870938i	$0.663512\pi$
$420$	0	0
$421$	16423.1	1.90122	0.950611	−	0.310385i	$-0.100458\pi$
$421$	16423.1	1.90122	0.950611	+	0.310385i	$0.100458\pi$
$422$	0	0
$423$	6652.29	0.764646
$424$	0	0
$425$	−9048.46	−1.03274
$426$	0	0
$427$	2849.31	0.322923
$428$	0	0
$429$	9064.59	1.02015
$430$	0	0
$431$	8599.47	0.961072	0.480536	−	0.876975i	$-0.340442\pi$
$431$	8599.47	0.961072	0.480536	+	0.876975i	$0.340442\pi$
$432$	0	0
$433$	−1537.77	−0.170671	−0.0853355	−	0.996352i	$-0.527196\pi$
$433$	−1537.77	−0.170671	−0.0853355	+	0.996352i	$0.527196\pi$
$434$	0	0
$435$	21901.8	2.41405
$436$	0	0
$437$	−853.449	−0.0934233
$438$	0	0
$439$	−1026.94	−0.111647	−0.0558235	−	0.998441i	$-0.517778\pi$
$439$	−1026.94	−0.111647	−0.0558235	+	0.998441i	$0.517778\pi$
$440$	0	0
$441$	2258.88	0.243914
$442$	0	0
$443$	10339.5	1.10890	0.554451	−	0.832216i	$-0.312928\pi$
$443$	10339.5	1.10890	0.554451	+	0.832216i	$0.312928\pi$
$444$	0	0
$445$	27935.6	2.97589
$446$	0	0
$447$	−8818.20	−0.933080
$448$	0	0
$449$	−17742.4	−1.86484	−0.932422	−	0.361371i	$-0.882309\pi$
$449$	−17742.4	−1.86484	−0.932422	+	0.361371i	$0.882309\pi$
$450$	0	0
$451$	25819.5	2.69577
$452$	0	0
$453$	26216.9	2.71916
$454$	0	0
$455$	2174.86	0.224086
$456$	0	0
$457$	3876.86	0.396831	0.198416	−	0.980118i	$-0.436420\pi$
$457$	3876.86	0.396831	0.198416	+	0.980118i	$0.436420\pi$
$458$	0	0
$459$	−6744.09	−0.685810
$460$	0	0
$461$	−6479.33	−0.654604	−0.327302	−	0.944920i	$-0.606139\pi$
$461$	−6479.33	−0.654604	−0.327302	+	0.944920i	$0.606139\pi$
$462$	0	0
$463$	1659.44	0.166568	0.0832838	−	0.996526i	$-0.473459\pi$
$463$	1659.44	0.166568	0.0832838	+	0.996526i	$0.473459\pi$
$464$	0	0
$465$	15177.5	1.51363
$466$	0	0
$467$	17058.4	1.69029	0.845147	−	0.534534i	$-0.179513\pi$
$467$	17058.4	1.69029	0.845147	+	0.534534i	$0.179513\pi$
$468$	0	0
$469$	−65.7212	−0.00647062
$470$	0	0
$471$	−27328.3	−2.67350
$472$	0	0
$473$	6349.04	0.617186
$474$	0	0
$475$	−8577.15	−0.828519
$476$	0	0
$477$	−18881.9	−1.81246
$478$	0	0
$479$	5675.58	0.541386	0.270693	−	0.962666i	$-0.412747\pi$
$479$	5675.58	0.541386	0.270693	+	0.962666i	$0.412747\pi$
$480$	0	0
$481$	−2956.26	−0.280236
$482$	0	0
$483$	−1304.74	−0.122915
$484$	0	0
$485$	−17742.7	−1.66114
$486$	0	0
$487$	408.166	0.0379790	0.0189895	−	0.999820i	$-0.493955\pi$
$487$	408.166	0.0379790	0.0189895	+	0.999820i	$0.493955\pi$
$488$	0	0
$489$	2641.10	0.244243
$490$	0	0
$491$	−1766.25	−0.162342	−0.0811709	−	0.996700i	$-0.525866\pi$
$491$	−1766.25	−0.162342	−0.0811709	+	0.996700i	$0.525866\pi$
$492$	0	0
$493$	−5703.24	−0.521016
$494$	0	0
$495$	−54129.6	−4.91504
$496$	0	0
$497$	−6609.35	−0.596519
$498$	0	0
$499$	−19261.8	−1.72801	−0.864006	−	0.503482i	$-0.832052\pi$
$499$	−19261.8	−1.72801	−0.864006	+	0.503482i	$0.832052\pi$
$500$	0	0
$501$	−12392.6	−1.10511
$502$	0	0
$503$	14343.7	1.27148	0.635738	−	0.771905i	$-0.280696\pi$
$503$	14343.7	1.27148	0.635738	+	0.771905i	$0.280696\pi$
$504$	0	0
$505$	30454.2	2.68356
$506$	0	0
$507$	−16385.5	−1.43531
$508$	0	0
$509$	15480.1	1.34802	0.674009	−	0.738723i	$-0.264571\pi$
$509$	15480.1	1.34802	0.674009	+	0.738723i	$0.264571\pi$
$510$	0	0
$511$	−603.442	−0.0522401
$512$	0	0
$513$	−6392.80	−0.550193
$514$	0	0
$515$	−10775.5	−0.921989
$516$	0	0
$517$	−9134.19	−0.777024
$518$	0	0
$519$	15542.4	1.31452
$520$	0	0
$521$	−9939.06	−0.835774	−0.417887	−	0.908499i	$-0.637229\pi$
$521$	−9939.06	−0.835774	−0.417887	+	0.908499i	$0.637229\pi$
$522$	0	0
$523$	−9517.79	−0.795762	−0.397881	−	0.917437i	$-0.630254\pi$
$523$	−9517.79	−0.795762	−0.397881	+	0.917437i	$0.630254\pi$
$524$	0	0
$525$	−13112.7	−1.09006
$526$	0	0
$527$	−3952.22	−0.326682
$528$	0	0
$529$	−11691.7	−0.960938
$530$	0	0
$531$	39.2839	0.00321050
$532$	0	0
$533$	6831.94	0.555205
$534$	0	0
$535$	31200.7	2.52135
$536$	0	0
$537$	18340.6	1.47385
$538$	0	0
$539$	−3101.65	−0.247862
$540$	0	0
$541$	−3607.08	−0.286655	−0.143328	−	0.989675i	$-0.545780\pi$
$541$	−3607.08	−0.286655	−0.143328	+	0.989675i	$0.545780\pi$
$542$	0	0
$543$	16689.2	1.31898
$544$	0	0
$545$	4568.87	0.359099
$546$	0	0
$547$	−21217.9	−1.65852	−0.829262	−	0.558860i	$-0.811239\pi$
$547$	−21217.9	−1.65852	−0.829262	+	0.558860i	$0.811239\pi$
$548$	0	0
$549$	−18764.6	−1.45875
$550$	0	0
$551$	−5406.18	−0.417987
$552$	0	0
$553$	3946.56	0.303481
$554$	0	0
$555$	27992.8	2.14095
$556$	0	0
$557$	10639.5	0.809352	0.404676	−	0.914460i	$-0.367384\pi$
$557$	10639.5	0.809352	0.404676	+	0.914460i	$0.367384\pi$
$558$	0	0
$559$	1679.98	0.127112
$560$	0	0
$561$	22350.9	1.68209
$562$	0	0
$563$	−24299.5	−1.81901	−0.909505	−	0.415692i	$-0.863539\pi$
$563$	−24299.5	−1.81901	−0.909505	+	0.415692i	$0.863539\pi$
$564$	0	0
$565$	−44283.1	−3.29736
$566$	0	0
$567$	−1060.42	−0.0785422
$568$	0	0
$569$	−25194.9	−1.85629	−0.928143	−	0.372225i	$-0.878595\pi$
$569$	−25194.9	−1.85629	−0.928143	+	0.372225i	$0.878595\pi$
$570$	0	0
$571$	−24545.1	−1.79892	−0.899458	−	0.437007i	$-0.856039\pi$
$571$	−24545.1	−1.79892	−0.899458	+	0.437007i	$0.856039\pi$
$572$	0	0
$573$	−20113.9	−1.46644
$574$	0	0
$575$	4776.45	0.346420
$576$	0	0
$577$	956.392	0.0690037	0.0345018	−	0.999405i	$-0.489016\pi$
$577$	956.392	0.0690037	0.0345018	+	0.999405i	$0.489016\pi$
$578$	0	0
$579$	28714.2	2.06101
$580$	0	0
$581$	−6788.90	−0.484769
$582$	0	0
$583$	25926.5	1.84180
$584$	0	0
$585$	−14322.9	−1.01227
$586$	0	0
$587$	7279.01	0.511817	0.255909	−	0.966701i	$-0.417625\pi$
$587$	7279.01	0.511817	0.255909	+	0.966701i	$0.417625\pi$
$588$	0	0
$589$	−3746.36	−0.262082
$590$	0	0
$591$	21239.2	1.47828
$592$	0	0
$593$	4203.35	0.291081	0.145540	−	0.989352i	$-0.453508\pi$
$593$	4203.35	0.291081	0.145540	+	0.989352i	$0.453508\pi$
$594$	0	0
$595$	5362.63	0.369490
$596$	0	0
$597$	35752.5	2.45101
$598$	0	0
$599$	19920.9	1.35884	0.679419	−	0.733750i	$-0.262232\pi$
$599$	19920.9	1.35884	0.679419	+	0.733750i	$0.262232\pi$
$600$	0	0
$601$	19087.3	1.29549	0.647743	−	0.761859i	$-0.275713\pi$
$601$	19087.3	1.29549	0.647743	+	0.761859i	$0.275713\pi$
$602$	0	0
$603$	432.818	0.0292300
$604$	0	0
$605$	49635.0	3.33545
$606$	0	0
$607$	−16035.1	−1.07223	−0.536115	−	0.844145i	$-0.680109\pi$
$607$	−16035.1	−1.07223	−0.536115	+	0.844145i	$0.680109\pi$
$608$	0	0
$609$	−8264.91	−0.549936
$610$	0	0
$611$	−2416.94	−0.160031
$612$	0	0
$613$	−22477.0	−1.48097	−0.740486	−	0.672071i	$-0.765405\pi$
$613$	−22477.0	−1.48097	−0.740486	+	0.672071i	$0.765405\pi$
$614$	0	0
$615$	−64691.7	−4.24166
$616$	0	0
$617$	8128.36	0.530365	0.265183	−	0.964198i	$-0.414568\pi$
$617$	8128.36	0.530365	0.265183	+	0.964198i	$0.414568\pi$
$618$	0	0
$619$	2911.91	0.189078	0.0945391	−	0.995521i	$-0.469862\pi$
$619$	2911.91	0.189078	0.0945391	+	0.995521i	$0.469862\pi$
$620$	0	0
$621$	3560.03	0.230047
$622$	0	0
$623$	−10541.8	−0.677928
$624$	0	0
$625$	4991.17	0.319435
$626$	0	0
$627$	21186.7	1.34946
$628$	0	0
$629$	−7289.34	−0.462075
$630$	0	0
$631$	−11776.5	−0.742970	−0.371485	−	0.928439i	$-0.621151\pi$
$631$	−11776.5	−0.742970	−0.371485	+	0.928439i	$0.621151\pi$
$632$	0	0
$633$	10657.9	0.669214
$634$	0	0
$635$	−29976.5	−1.87336
$636$	0	0
$637$	−820.709	−0.0510482
$638$	0	0
$639$	43527.0	2.69468
$640$	0	0
$641$	−20978.6	−1.29268	−0.646338	−	0.763051i	$-0.723700\pi$
$641$	−20978.6	−1.29268	−0.646338	+	0.763051i	$0.723700\pi$
$642$	0	0
$643$	4503.50	0.276206	0.138103	−	0.990418i	$-0.455899\pi$
$643$	4503.50	0.276206	0.138103	+	0.990418i	$0.455899\pi$
$644$	0	0
$645$	−15907.7	−0.971112
$646$	0	0
$647$	−8644.52	−0.525272	−0.262636	−	0.964895i	$-0.584592\pi$
$647$	−8644.52	−0.525272	−0.262636	+	0.964895i	$0.584592\pi$
$648$	0	0
$649$	−53.9404	−0.00326247
$650$	0	0
$651$	−5727.40	−0.344815
$652$	0	0
$653$	2964.53	0.177659	0.0888294	−	0.996047i	$-0.471687\pi$
$653$	2964.53	0.177659	0.0888294	+	0.996047i	$0.471687\pi$
$654$	0	0
$655$	−8576.53	−0.511622
$656$	0	0
$657$	3974.07	0.235986
$658$	0	0
$659$	−18295.8	−1.08149	−0.540745	−	0.841187i	$-0.681857\pi$
$659$	−18295.8	−1.08149	−0.540745	+	0.841187i	$0.681857\pi$
$660$	0	0
$661$	585.321	0.0344423	0.0172212	−	0.999852i	$-0.494518\pi$
$661$	585.321	0.0344423	0.0172212	+	0.999852i	$0.494518\pi$
$662$	0	0
$663$	5914.13	0.346434
$664$	0	0
$665$	5083.30	0.296424
$666$	0	0
$667$	3010.59	0.174769
$668$	0	0
$669$	−19597.8	−1.13258
$670$	0	0
$671$	25765.5	1.48237
$672$	0	0
$673$	−7131.10	−0.408445	−0.204223	−	0.978924i	$-0.565467\pi$
$673$	−7131.10	−0.408445	−0.204223	+	0.978924i	$0.565467\pi$
$674$	0	0
$675$	35778.2	2.04015
$676$	0	0
$677$	15564.5	0.883591	0.441796	−	0.897116i	$-0.354342\pi$
$677$	15564.5	0.883591	0.441796	+	0.897116i	$0.354342\pi$
$678$	0	0
$679$	6695.42	0.378419
$680$	0	0
$681$	41038.8	2.30926
$682$	0	0
$683$	9546.68	0.534837	0.267418	−	0.963581i	$-0.413829\pi$
$683$	9546.68	0.534837	0.267418	+	0.963581i	$0.413829\pi$
$684$	0	0
$685$	12602.1	0.702920
$686$	0	0
$687$	−9060.65	−0.503181
$688$	0	0
$689$	6860.26	0.379325
$690$	0	0
$691$	25526.7	1.40533	0.702663	−	0.711523i	$-0.251994\pi$
$691$	25526.7	1.40533	0.702663	+	0.711523i	$0.251994\pi$
$692$	0	0
$693$	20426.4	1.11968
$694$	0	0
$695$	−28092.7	−1.53326
$696$	0	0
$697$	16845.7	0.915463
$698$	0	0
$699$	−11941.9	−0.646185
$700$	0	0
$701$	−18823.3	−1.01419	−0.507095	−	0.861890i	$-0.669281\pi$
$701$	−18823.3	−1.01419	−0.507095	+	0.861890i	$0.669281\pi$
$702$	0	0
$703$	−6909.66	−0.370701
$704$	0	0
$705$	22886.1	1.22261
$706$	0	0
$707$	−11492.3	−0.611331
$708$	0	0
$709$	−17323.9	−0.917647	−0.458823	−	0.888527i	$-0.651729\pi$
$709$	−17323.9	−0.917647	−0.458823	+	0.888527i	$0.651729\pi$
$710$	0	0
$711$	−25990.7	−1.37093
$712$	0	0
$713$	2086.27	0.109581
$714$	0	0
$715$	19666.6	1.02866
$716$	0	0
$717$	−2556.83	−0.133175
$718$	0	0
$719$	11462.3	0.594539	0.297269	−	0.954794i	$-0.403924\pi$
$719$	11462.3	0.594539	0.297269	+	0.954794i	$0.403924\pi$
$720$	0	0
$721$	4066.26	0.210035
$722$	0	0
$723$	−38926.2	−2.00233
$724$	0	0
$725$	30256.4	1.54992
$726$	0	0
$727$	−7842.63	−0.400092	−0.200046	−	0.979786i	$-0.564109\pi$
$727$	−7842.63	−0.400092	−0.200046	+	0.979786i	$0.564109\pi$
$728$	0	0
$729$	−30713.3	−1.56040
$730$	0	0
$731$	4142.39	0.209592
$732$	0	0
$733$	−1243.66	−0.0626678	−0.0313339	−	0.999509i	$-0.509976\pi$
$733$	−1243.66	−0.0626678	−0.0313339	+	0.999509i	$0.509976\pi$
$734$	0	0
$735$	7771.30	0.389998
$736$	0	0
$737$	−594.298	−0.0297032
$738$	0	0
$739$	−4611.17	−0.229533	−0.114766	−	0.993393i	$-0.536612\pi$
$739$	−4611.17	−0.229533	−0.114766	+	0.993393i	$0.536612\pi$
$740$	0	0
$741$	5606.08	0.277928
$742$	0	0
$743$	−18041.5	−0.890818	−0.445409	−	0.895327i	$-0.646942\pi$
$743$	−18041.5	−0.890818	−0.445409	+	0.895327i	$0.646942\pi$
$744$	0	0
$745$	−19132.1	−0.940867
$746$	0	0
$747$	44709.4	2.18987
$748$	0	0
$749$	−11774.0	−0.574380
$750$	0	0
$751$	13666.5	0.664044	0.332022	−	0.943272i	$-0.392269\pi$
$751$	13666.5	0.664044	0.332022	+	0.943272i	$0.392269\pi$
$752$	0	0
$753$	37068.3	1.79395
$754$	0	0
$755$	56880.6	2.74185
$756$	0	0
$757$	912.222	0.0437983	0.0218991	−	0.999760i	$-0.493029\pi$
$757$	912.222	0.0437983	0.0218991	+	0.999760i	$0.493029\pi$
$758$	0	0
$759$	−11798.4	−0.564237
$760$	0	0
$761$	−9524.38	−0.453691	−0.226845	−	0.973931i	$-0.572841\pi$
$761$	−9524.38	−0.453691	−0.226845	+	0.973931i	$0.572841\pi$
$762$	0	0
$763$	−1724.12	−0.0818050
$764$	0	0
$765$	−35316.5	−1.66911
$766$	0	0
$767$	−14.2728	−0.000671920 0
$768$	0	0
$769$	−6048.18	−0.283619	−0.141809	−	0.989894i	$-0.545292\pi$
$769$	−6048.18	−0.283619	−0.141809	+	0.989894i	$0.545292\pi$
$770$	0	0
$771$	−10560.9	−0.493312
$772$	0	0
$773$	−4649.01	−0.216317	−0.108159	−	0.994134i	$-0.534495\pi$
$773$	−4649.01	−0.216317	−0.108159	+	0.994134i	$0.534495\pi$
$774$	0	0
$775$	20967.0	0.971816
$776$	0	0
$777$	−10563.4	−0.487723
$778$	0	0
$779$	15968.3	0.734433
$780$	0	0
$781$	−59766.5	−2.73830
$782$	0	0
$783$	22551.0	1.02926
$784$	0	0
$785$	−59291.8	−2.69581
$786$	0	0
$787$	31218.8	1.41401	0.707007	−	0.707207i	$-0.250045\pi$
$787$	31218.8	1.41401	0.707007	+	0.707207i	$0.250045\pi$
$788$	0	0
$789$	17361.7	0.783389
$790$	0	0
$791$	16710.8	0.751159
$792$	0	0
$793$	6817.66	0.305299
$794$	0	0
$795$	−64959.9	−2.89797
$796$	0	0
$797$	26050.4	1.15778	0.578891	−	0.815405i	$-0.303486\pi$
$797$	26050.4	1.15778	0.578891	+	0.815405i	$0.303486\pi$
$798$	0	0
$799$	−5959.54	−0.263872
$800$	0	0
$801$	69424.9	3.06243
$802$	0	0
$803$	−5456.75	−0.239806
$804$	0	0
$805$	−2830.79	−0.123941
$806$	0	0
$807$	−22316.1	−0.973439
$808$	0	0
$809$	24696.0	1.07326	0.536629	−	0.843818i	$-0.319698\pi$
$809$	24696.0	1.07326	0.536629	+	0.843818i	$0.319698\pi$
$810$	0	0
$811$	−9287.73	−0.402141	−0.201070	−	0.979577i	$-0.564442\pi$
$811$	−9287.73	−0.402141	−0.201070	+	0.979577i	$0.564442\pi$
$812$	0	0
$813$	−5211.93	−0.224835
$814$	0	0
$815$	5730.17	0.246281
$816$	0	0
$817$	3926.62	0.168146
$818$	0	0
$819$	5404.92	0.230602
$820$	0	0
$821$	25629.5	1.08950	0.544748	−	0.838600i	$-0.316625\pi$
$821$	25629.5	1.08950	0.544748	+	0.838600i	$0.316625\pi$
$822$	0	0
$823$	−45028.5	−1.90716	−0.953581	−	0.301136i	$-0.902634\pi$
$823$	−45028.5	−1.90716	−0.953581	+	0.301136i	$0.902634\pi$
$824$	0	0
$825$	−118574.	−5.00390
$826$	0	0
$827$	−13210.9	−0.555488	−0.277744	−	0.960655i	$-0.589587\pi$
$827$	−13210.9	−0.555488	−0.277744	+	0.960655i	$0.589587\pi$
$828$	0	0
$829$	−8486.34	−0.355540	−0.177770	−	0.984072i	$-0.556888\pi$
$829$	−8486.34	−0.355540	−0.177770	+	0.984072i	$0.556888\pi$
$830$	0	0
$831$	43463.3	1.81435
$832$	0	0
$833$	−2023.65	−0.0841721
$834$	0	0
$835$	−26887.2	−1.11434
$836$	0	0
$837$	15627.3	0.645352
$838$	0	0
$839$	−40376.6	−1.66145	−0.830724	−	0.556684i	$-0.812073\pi$
$839$	−40376.6	−1.66145	−0.830724	+	0.556684i	$0.812073\pi$
$840$	0	0
$841$	−5318.40	−0.218065
$842$	0	0
$843$	−43873.9	−1.79253
$844$	0	0
$845$	−35550.1	−1.44729
$846$	0	0
$847$	−18730.3	−0.759838
$848$	0	0
$849$	15071.9	0.609266
$850$	0	0
$851$	3847.85	0.154997
$852$	0	0
$853$	−41790.2	−1.67745	−0.838727	−	0.544552i	$-0.816700\pi$
$853$	−41790.2	−1.67745	−0.838727	+	0.544552i	$0.816700\pi$
$854$	0	0
$855$	−33476.9	−1.33905
$856$	0	0
$857$	−20551.3	−0.819159	−0.409580	−	0.912274i	$-0.634325\pi$
$857$	−20551.3	−0.819159	−0.409580	+	0.912274i	$0.634325\pi$
$858$	0	0
$859$	−4486.87	−0.178219	−0.0891095	−	0.996022i	$-0.528402\pi$
$859$	−4486.87	−0.178219	−0.0891095	+	0.996022i	$0.528402\pi$
$860$	0	0
$861$	24412.2	0.966277
$862$	0	0
$863$	14618.5	0.576615	0.288307	−	0.957538i	$-0.406908\pi$
$863$	14618.5	0.576615	0.288307	+	0.957538i	$0.406908\pi$
$864$	0	0
$865$	33721.0	1.32549
$866$	0	0
$867$	−27422.7	−1.07419
$868$	0	0
$869$	35687.6	1.39312
$870$	0	0
$871$	−157.254	−0.00611749
$872$	0	0
$873$	−44093.8	−1.70945
$874$	0	0
$875$	−12218.3	−0.472062
$876$	0	0
$877$	16408.3	0.631779	0.315890	−	0.948796i	$-0.397697\pi$
$877$	16408.3	0.631779	0.315890	+	0.948796i	$0.397697\pi$
$878$	0	0
$879$	3922.63	0.150520
$880$	0	0
$881$	36429.8	1.39313	0.696567	−	0.717492i	$-0.254710\pi$
$881$	36429.8	1.39313	0.696567	+	0.717492i	$0.254710\pi$
$882$	0	0
$883$	27327.2	1.04149	0.520743	−	0.853713i	$-0.325655\pi$
$883$	27327.2	1.04149	0.520743	+	0.853713i	$0.325655\pi$
$884$	0	0
$885$	135.150	0.00513334
$886$	0	0
$887$	−8004.51	−0.303005	−0.151502	−	0.988457i	$-0.548411\pi$
$887$	−8004.51	−0.303005	−0.151502	+	0.988457i	$0.548411\pi$
$888$	0	0
$889$	11312.0	0.426763
$890$	0	0
$891$	−9589.07	−0.360545
$892$	0	0
$893$	−5649.13	−0.211692
$894$	0	0
$895$	39792.0	1.48615
$896$	0	0
$897$	−3121.91	−0.116207
$898$	0	0
$899$	13215.5	0.490280
$900$	0	0
$901$	16915.6	0.625460
$902$	0	0
$903$	6002.98	0.221225
$904$	0	0
$905$	36209.2	1.32998
$906$	0	0
$907$	31869.3	1.16671	0.583354	−	0.812218i	$-0.301740\pi$
$907$	31869.3	1.16671	0.583354	+	0.812218i	$0.301740\pi$
$908$	0	0
$909$	75684.2	2.76159
$910$	0	0
$911$	37037.3	1.34698	0.673492	−	0.739195i	$-0.264794\pi$
$911$	37037.3	1.34698	0.673492	+	0.739195i	$0.264794\pi$
$912$	0	0
$913$	−61390.0	−2.22532
$914$	0	0
$915$	−64556.5	−2.33243
$916$	0	0
$917$	3236.45	0.116551
$918$	0	0
$919$	38519.8	1.38264	0.691322	−	0.722546i	$-0.257028\pi$
$919$	38519.8	1.38264	0.691322	+	0.722546i	$0.257028\pi$
$920$	0	0
$921$	−41537.5	−1.48611
$922$	0	0
$923$	−15814.4	−0.563964
$924$	0	0
$925$	38670.9	1.37458
$926$	0	0
$927$	−26779.0	−0.948800
$928$	0	0
$929$	−14815.5	−0.523228	−0.261614	−	0.965173i	$-0.584255\pi$
$929$	−14815.5	−0.523228	−0.261614	+	0.965173i	$0.584255\pi$
$930$	0	0
$931$	−1918.24	−0.0675273
$932$	0	0
$933$	−6030.10	−0.211593
$934$	0	0
$935$	48492.7	1.69613
$936$	0	0
$937$	32260.7	1.12477	0.562387	−	0.826874i	$-0.309883\pi$
$937$	32260.7	1.12477	0.562387	+	0.826874i	$0.309883\pi$
$938$	0	0
$939$	53097.3	1.84533
$940$	0	0
$941$	49956.4	1.73064	0.865320	−	0.501219i	$-0.167115\pi$
$941$	49956.4	1.73064	0.865320	+	0.501219i	$0.167115\pi$
$942$	0	0
$943$	−8892.43	−0.307081
$944$	0	0
$945$	−21204.2	−0.729918
$946$	0	0
$947$	24080.8	0.826317	0.413159	−	0.910659i	$-0.364426\pi$
$947$	24080.8	0.826317	0.413159	+	0.910659i	$0.364426\pi$
$948$	0	0
$949$	−1443.88	−0.0493891
$950$	0	0
$951$	63598.2	2.16857
$952$	0	0
$953$	3910.48	0.132920	0.0664600	−	0.997789i	$-0.478830\pi$
$953$	3910.48	0.132920	0.0664600	+	0.997789i	$0.478830\pi$
$954$	0	0
$955$	−43639.4	−1.47868
$956$	0	0
$957$	−74737.2	−2.52446
$958$	0	0
$959$	−4755.54	−0.160130
$960$	0	0
$961$	−20633.0	−0.692590
$962$	0	0
$963$	77539.3	2.59467
$964$	0	0
$965$	62298.8	2.07821
$966$	0	0
$967$	14021.9	0.466303	0.233151	−	0.972440i	$-0.425096\pi$
$967$	14021.9	0.466303	0.233151	+	0.972440i	$0.425096\pi$
$968$	0	0
$969$	13823.1	0.458268
$970$	0	0
$971$	−22029.4	−0.728071	−0.364036	−	0.931385i	$-0.618601\pi$
$971$	−22029.4	−0.728071	−0.364036	+	0.931385i	$0.618601\pi$
$972$	0	0
$973$	10601.1	0.349287
$974$	0	0
$975$	−31375.2	−1.03057
$976$	0	0
$977$	26081.5	0.854063	0.427032	−	0.904237i	$-0.359559\pi$
$977$	26081.5	0.854063	0.427032	+	0.904237i	$0.359559\pi$
$978$	0	0
$979$	−95326.6	−3.11200
$980$	0	0
$981$	11354.5	0.369541
$982$	0	0
$983$	26665.7	0.865212	0.432606	−	0.901583i	$-0.357594\pi$
$983$	26665.7	0.865212	0.432606	+	0.901583i	$0.357594\pi$
$984$	0	0
$985$	46080.7	1.49061
$986$	0	0
$987$	−8636.33	−0.278518
$988$	0	0
$989$	−2186.66	−0.0703050
$990$	0	0
$991$	−32351.8	−1.03702	−0.518511	−	0.855071i	$-0.673513\pi$
$991$	−32351.8	−1.03702	−0.518511	+	0.855071i	$0.673513\pi$
$992$	0	0
$993$	−41752.9	−1.33433
$994$	0	0
$995$	77569.2	2.47147
$996$	0	0
$997$	39663.0	1.25992	0.629960	−	0.776628i	$-0.283071\pi$
$997$	39663.0	1.25992	0.629960	+	0.776628i	$0.283071\pi$
$998$	0	0
$999$	28822.5	0.912818

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	112.4.a.h.1.2		2
3.2	odd	2		1008.4.a.x.1.1		2
4.3	odd	2		56.4.a.c.1.1	✓	2
7.6	odd	2		784.4.a.t.1.1		2
8.3	odd	2		448.4.a.s.1.2		2
8.5	even	2		448.4.a.r.1.1		2
12.11	even	2		504.4.a.i.1.1		2
20.3	even	4		1400.4.g.h.449.1		4
20.7	even	4		1400.4.g.h.449.4		4
20.19	odd	2		1400.4.a.i.1.2		2
28.3	even	6		392.4.i.i.177.1		4
28.11	odd	6		392.4.i.l.177.2		4
28.19	even	6		392.4.i.i.361.1		4
28.23	odd	6		392.4.i.l.361.2		4
28.27	even	2		392.4.a.h.1.2		2

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
56.4.a.c.1.1	✓	2	4.3	odd	2
112.4.a.h.1.2		2	1.1	even	1	trivial
392.4.a.h.1.2		2	28.27	even	2
392.4.i.i.177.1		4	28.3	even	6
392.4.i.i.361.1		4	28.19	even	6
392.4.i.l.177.2		4	28.11	odd	6
392.4.i.l.361.2		4	28.23	odd	6
448.4.a.r.1.1		2	8.5	even	2
448.4.a.s.1.2		2	8.3	odd	2
504.4.a.i.1.1		2	12.11	even	2
784.4.a.t.1.1		2	7.6	odd	2
1008.4.a.x.1.1		2	3.2	odd	2
1400.4.a.i.1.2		2	20.19	odd	2
1400.4.g.h.449.1		4	20.3	even	4
1400.4.g.h.449.4		4	20.7	even	4

Overview	Random
Universe	Knowledge

Classical	Maass
Hilbert	Bianchi

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

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

\(f(q)\)	\(=\)	\(q+8.54983 q^{3} +18.5498 q^{5} -7.00000 q^{7} +46.0997 q^{9} -63.2990 q^{11} -16.7492 q^{13} +158.598 q^{15} -41.2990 q^{17} -39.1478 q^{19} -59.8488 q^{21} +21.8007 q^{23} +219.096 q^{25} +163.299 q^{27} +138.096 q^{29} +95.6977 q^{31} -541.196 q^{33} -129.849 q^{35} +176.502 q^{37} -143.203 q^{39} -407.897 q^{41} -100.302 q^{43} +855.141 q^{45} +144.302 q^{47} +49.0000 q^{49} -353.100 q^{51} -409.588 q^{53} -1174.19 q^{55} -334.708 q^{57} +0.852152 q^{59} -407.045 q^{61} -322.698 q^{63} -310.694 q^{65} +9.38874 q^{67} +186.392 q^{69} +944.193 q^{71} +86.2060 q^{73} +1873.24 q^{75} +443.093 q^{77} -563.794 q^{79} +151.488 q^{81} +969.842 q^{83} -766.090 q^{85} +1180.70 q^{87} +1505.97 q^{89} +117.244 q^{91} +818.199 q^{93} -726.186 q^{95} -956.488 q^{97} -2918.06 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 2 q + 2 q^{3} + 22 q^{5} - 14 q^{7} + 62 q^{9} - 36 q^{11} + 42 q^{13} + 136 q^{15} + 8 q^{17} + 118 q^{19} - 14 q^{21} + 104 q^{23} + 106 q^{25} + 236 q^{27} - 56 q^{29} - 20 q^{31} - 720 q^{33} - 154 q^{35}+ \cdots - 2484 q^{99}+O(q^{100}) \) 2 * q + 2 * q^3 + 22 * q^5 - 14 * q^7 + 62 * q^9 - 36 * q^11 + 42 * q^13 + 136 * q^15 + 8 * q^17 + 118 * q^19 - 14 * q^21 + 104 * q^23 + 106 * q^25 + 236 * q^27 - 56 * q^29 - 20 * q^31 - 720 * q^33 - 154 * q^35 + 504 * q^37 - 528 * q^39 - 544 * q^41 - 412 * q^43 + 910 * q^45 + 500 * q^47 + 98 * q^49 - 676 * q^51 + 268 * q^53 - 1080 * q^55 - 1364 * q^57 + 198 * q^59 - 346 * q^61 - 434 * q^63 - 108 * q^65 - 1008 * q^67 - 352 * q^69 + 1224 * q^71 + 716 * q^73 + 2614 * q^75 + 252 * q^77 - 584 * q^79 - 754 * q^81 + 1230 * q^83 - 596 * q^85 + 2452 * q^87 + 596 * q^89 - 294 * q^91 + 1576 * q^93 - 184 * q^95 - 856 * q^97 - 2484 * q^99

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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