LMFDB - Embedded newform 112.10.a.i.1.3

Show commands: Magma / Pari/GP / SageMath

Newspace parameters

comment:Compute space of new eigenforms

gp:[N,k,chi] = [112,10,Mod(1,112)] mf = mfinit([N,k,chi],0) lf = mfeigenbasis(mf)

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, 10, names="a")

magma://Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code chi := DirichletCharacter("112.1"); S:= CuspForms(chi, 10); N := Newforms(S);

Level:	$ N $	$=$	$ 112 = 2^{4} \cdot 7 $
Weight:	$ k $	$=$	$ 10 $
Character orbit:	$[\chi]$	$=$	112.a (trivial)

Newform invariants

comment:select newform

sage:traces = [3,0,92,0,274] 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:	$57.6840136504$
Analytic rank:	$0$
Dimension:	$3$
Coefficient field:	$\mathbb{Q}[x]/(x^{3} - \cdots)$
comment:defining polynomial gp:f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{3} - 823x - 4578 $ x^3 - 823*x - 4578
Coefficient ring:	$\Z[a_1, \ldots, a_{5}]$
Coefficient ring index:	$ 2^{6}\cdot 3 $
Twist minimal:	no (minimal twist has level 56)
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.3
Root			$31.1447$ 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)

$f(q)$	$=$	$q+273.944 q^{3} -187.226 q^{5} +2401.00 q^{7} +55362.2 q^{9} -6322.34 q^{11} +70415.3 q^{13} -51289.4 q^{15} +164999. q^{17} -482060. q^{19} +657739. q^{21} +571132. q^{23} -1.91807e6 q^{25} +9.77410e6 q^{27} +2.49049e6 q^{29} +8.74543e6 q^{31} -1.73196e6 q^{33} -449529. q^{35} +1.74181e7 q^{37} +1.92898e7 q^{39} -3.27790e7 q^{41} -1.01909e7 q^{43} -1.03652e7 q^{45} -2.96277e7 q^{47} +5.76480e6 q^{49} +4.52005e7 q^{51} +5.85308e7 q^{53} +1.18370e6 q^{55} -1.32057e8 q^{57} +1.04666e8 q^{59} -2.06399e8 q^{61} +1.32925e8 q^{63} -1.31836e7 q^{65} -1.79752e8 q^{67} +1.56458e8 q^{69} +1.16985e8 q^{71} +2.81318e8 q^{73} -5.25444e8 q^{75} -1.51799e7 q^{77} +3.44616e8 q^{79} +1.58786e9 q^{81} +2.85676e8 q^{83} -3.08921e7 q^{85} +6.82254e8 q^{87} +9.14231e8 q^{89} +1.69067e8 q^{91} +2.39576e9 q^{93} +9.02541e7 q^{95} -1.47825e9 q^{97} -3.50018e8 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 3 q + 92 q^{3} + 274 q^{5} + 7203 q^{7} + 51871 q^{9} + 45364 q^{11} - 11158 q^{13} + 95584 q^{15} - 55866 q^{17} - 488772 q^{19} + 220892 q^{21} + 253888 q^{23} - 3872619 q^{25} + 10157192 q^{27} - 765318 q^{29}+ \cdots - 168732636 q^{99}+O(q^{100}) $ 3 * q + 92 * q^3 + 274 * q^5 + 7203 * q^7 + 51871 * q^9 + 45364 * q^11 - 11158 * q^13 + 95584 * q^15 - 55866 * q^17 - 488772 * q^19 + 220892 * q^21 + 253888 * q^23 - 3872619 * q^25 + 10157192 * q^27 - 765318 * q^29 + 8189680 * q^31 - 7926240 * q^33 + 657874 * q^35 + 7208194 * q^37 + 44623664 * q^39 - 57891986 * q^41 + 37304708 * q^43 - 45527158 * q^45 + 29210992 * q^47 + 17294403 * q^49 + 142742952 * q^51 + 45140082 * q^53 - 1479800 * q^55 - 77053192 * q^57 + 205334804 * q^59 - 109264870 * q^61 + 124542271 * q^63 + 143056644 * q^65 + 52322348 * q^67 + 425002688 * q^69 + 150048136 * q^71 + 761429998 * q^73 - 260540684 * q^75 + 108918964 * q^77 + 214389152 * q^79 + 1527489907 * q^81 + 36110892 * q^83 + 675044052 * q^85 + 236662680 * q^87 + 1319848398 * q^89 - 26790358 * q^91 + 2846904208 * q^93 + 620260624 * q^95 + 101766310 * q^97 - 168732636 * 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
$3$	273.944	1.95261	0.976306	−	0.216396i	$-0.0694302\pi$
$3$	273.944	1.95261	0.976306	+	0.216396i	$0.0694302\pi$
$4$	0	0
$5$	−187.226	−0.133968	−0.0669840	−	0.997754i	$-0.521338\pi$
$5$	−187.226	−0.133968	−0.0669840	+	0.997754i	$0.521338\pi$
$6$	0	0
$7$	2401.00	0.377964
$7$	2401.00	0.377964
$8$	0	0
$9$	55362.2	2.81269
$10$	0	0
$11$	−6322.34	−0.130200	−0.0650999	−	0.997879i	$-0.520737\pi$
$11$	−6322.34	−0.130200	−0.0650999	+	0.997879i	$0.520737\pi$
$12$	0	0
$13$	70415.3	0.683789	0.341895	−	0.939738i	$-0.388931\pi$
$13$	70415.3	0.683789	0.341895	+	0.939738i	$0.388931\pi$
$14$	0	0
$15$	−51289.4	−0.261587
$16$	0	0
$17$	164999.	0.479139	0.239569	−	0.970879i	$-0.422994\pi$
$17$	164999.	0.479139	0.239569	+	0.970879i	$0.422994\pi$
$18$	0	0
$19$	−482060.	−0.848613	−0.424307	−	0.905519i	$-0.639482\pi$
$19$	−482060.	−0.848613	−0.424307	+	0.905519i	$0.639482\pi$
$20$	0	0
$21$	657739.	0.738018
$22$	0	0
$23$	571132.	0.425560	0.212780	−	0.977100i	$-0.431748\pi$
$23$	571132.	0.425560	0.212780	+	0.977100i	$0.431748\pi$
$24$	0	0
$25$	−1.91807e6	−0.982053
$26$	0	0
$27$	9.77410e6	3.53948
$28$	0	0
$29$	2.49049e6	0.653874	0.326937	−	0.945046i	$-0.393984\pi$
$29$	2.49049e6	0.653874	0.326937	+	0.945046i	$0.393984\pi$
$30$	0	0
$31$	8.74543e6	1.70080	0.850400	−	0.526136i	$-0.176360\pi$
$31$	8.74543e6	1.70080	0.850400	+	0.526136i	$0.176360\pi$
$32$	0	0
$33$	−1.73196e6	−0.254230
$34$	0	0
$35$	−449529.	−0.0506351
$36$	0	0
$37$	1.74181e7	1.52790	0.763948	−	0.645278i	$-0.223258\pi$
$37$	1.74181e7	1.52790	0.763948	+	0.645278i	$0.223258\pi$
$38$	0	0
$39$	1.92898e7	1.33517
$40$	0	0
$41$	−3.27790e7	−1.81163	−0.905813	−	0.423677i	$-0.860739\pi$
$41$	−3.27790e7	−1.81163	−0.905813	+	0.423677i	$0.860739\pi$
$42$	0	0
$43$	−1.01909e7	−0.454576	−0.227288	−	0.973828i	$-0.572986\pi$
$43$	−1.01909e7	−0.454576	−0.227288	+	0.973828i	$0.572986\pi$
$44$	0	0
$45$	−1.03652e7	−0.376810
$46$	0	0
$47$	−2.96277e7	−0.885639	−0.442820	−	0.896611i	$-0.646022\pi$
$47$	−2.96277e7	−0.885639	−0.442820	+	0.896611i	$0.646022\pi$
$48$	0	0
$49$	5.76480e6	0.142857
$50$	0	0
$51$	4.52005e7	0.935572
$52$	0	0
$53$	5.85308e7	1.01893	0.509463	−	0.860492i	$-0.329844\pi$
$53$	5.85308e7	1.01893	0.509463	+	0.860492i	$0.329844\pi$
$54$	0	0
$55$	1.18370e6	0.0174426
$56$	0	0
$57$	−1.32057e8	−1.65701
$58$	0	0
$59$	1.04666e8	1.12453	0.562263	−	0.826958i	$-0.309931\pi$
$59$	1.04666e8	1.12453	0.562263	+	0.826958i	$0.309931\pi$
$60$	0	0
$61$	−2.06399e8	−1.90864	−0.954318	−	0.298792i	$-0.903416\pi$
$61$	−2.06399e8	−1.90864	−0.954318	+	0.298792i	$0.903416\pi$
$62$	0	0
$63$	1.32925e8	1.06310
$64$	0	0
$65$	−1.31836e7	−0.0916058
$66$	0	0
$67$	−1.79752e8	−1.08977	−0.544887	−	0.838510i	$-0.683427\pi$
$67$	−1.79752e8	−1.08977	−0.544887	+	0.838510i	$0.683427\pi$
$68$	0	0
$69$	1.56458e8	0.830954
$70$	0	0
$71$	1.16985e8	0.546345	0.273172	−	0.961965i	$-0.411927\pi$
$71$	1.16985e8	0.546345	0.273172	+	0.961965i	$0.411927\pi$
$72$	0	0
$73$	2.81318e8	1.15943	0.579715	−	0.814819i	$-0.303164\pi$
$73$	2.81318e8	1.15943	0.579715	+	0.814819i	$0.303164\pi$
$74$	0	0
$75$	−5.25444e8	−1.91757
$76$	0	0
$77$	−1.51799e7	−0.0492109
$78$	0	0
$79$	3.44616e8	0.995435	0.497718	−	0.867339i	$-0.334172\pi$
$79$	3.44616e8	0.995435	0.497718	+	0.867339i	$0.334172\pi$
$80$	0	0
$81$	1.58786e9	4.09854
$82$	0	0
$83$	2.85676e8	0.660728	0.330364	−	0.943854i	$-0.392828\pi$
$83$	2.85676e8	0.660728	0.330364	+	0.943854i	$0.392828\pi$
$84$	0	0
$85$	−3.08921e7	−0.0641892
$86$	0	0
$87$	6.82254e8	1.27676
$88$	0	0
$89$	9.14231e8	1.54455	0.772273	−	0.635290i	$-0.219120\pi$
$89$	9.14231e8	1.54455	0.772273	+	0.635290i	$0.219120\pi$
$90$	0	0
$91$	1.69067e8	0.258448
$92$	0	0
$93$	2.39576e9	3.32100
$94$	0	0
$95$	9.02541e7	0.113687
$96$	0	0
$97$	−1.47825e9	−1.69541	−0.847706	−	0.530466i	$-0.822017\pi$
$97$	−1.47825e9	−1.69541	−0.847706	+	0.530466i	$0.822017\pi$
$98$	0	0
$99$	−3.50018e8	−0.366212
$100$	0	0
$101$	−1.89175e9	−1.80891	−0.904455	−	0.426569i	$-0.859722\pi$
$101$	−1.89175e9	−1.80891	−0.904455	+	0.426569i	$0.859722\pi$
$102$	0	0
$103$	3.47776e8	0.304461	0.152231	−	0.988345i	$-0.451354\pi$
$103$	3.47776e8	0.304461	0.152231	+	0.988345i	$0.451354\pi$
$104$	0	0
$105$	−1.23146e8	−0.0988707
$106$	0	0
$107$	−8.95113e8	−0.660162	−0.330081	−	0.943953i	$-0.607076\pi$
$107$	−8.95113e8	−0.660162	−0.330081	+	0.943953i	$0.607076\pi$
$108$	0	0
$109$	4.17983e8	0.283621	0.141811	−	0.989894i	$-0.454708\pi$
$109$	4.17983e8	0.283621	0.141811	+	0.989894i	$0.454708\pi$
$110$	0	0
$111$	4.77159e9	2.98339
$112$	0	0
$113$	−1.23275e9	−0.711249	−0.355625	−	0.934629i	$-0.615732\pi$
$113$	−1.23275e9	−0.711249	−0.355625	+	0.934629i	$0.615732\pi$
$114$	0	0
$115$	−1.06931e8	−0.0570115
$116$	0	0
$117$	3.89835e9	1.92329
$118$	0	0
$119$	3.96163e8	0.181097
$120$	0	0
$121$	−2.31798e9	−0.983048
$122$	0	0
$123$	−8.97961e9	−3.53740
$124$	0	0
$125$	7.24788e8	0.265532
$126$	0	0
$127$	2.59443e9	0.884963	0.442482	−	0.896778i	$-0.354098\pi$
$127$	2.59443e9	0.884963	0.442482	+	0.896778i	$0.354098\pi$
$128$	0	0
$129$	−2.79175e9	−0.887610
$130$	0	0
$131$	4.11259e9	1.22010	0.610048	−	0.792364i	$-0.291150\pi$
$131$	4.11259e9	1.22010	0.610048	+	0.792364i	$0.291150\pi$
$132$	0	0
$133$	−1.15743e9	−0.320746
$134$	0	0
$135$	−1.82996e9	−0.474177
$136$	0	0
$137$	−2.25492e9	−0.546876	−0.273438	−	0.961890i	$-0.588161\pi$
$137$	−2.25492e9	−0.546876	−0.273438	+	0.961890i	$0.588161\pi$
$138$	0	0
$139$	−2.20985e9	−0.502107	−0.251054	−	0.967973i	$-0.580777\pi$
$139$	−2.20985e9	−0.502107	−0.251054	+	0.967973i	$0.580777\pi$
$140$	0	0
$141$	−8.11631e9	−1.72931
$142$	0	0
$143$	−4.45189e8	−0.0890292
$144$	0	0
$145$	−4.66284e8	−0.0875981
$146$	0	0
$147$	1.57923e9	0.278944
$148$	0	0
$149$	−5.08122e9	−0.844559	−0.422280	−	0.906466i	$-0.638770\pi$
$149$	−5.08122e9	−0.844559	−0.422280	+	0.906466i	$0.638770\pi$
$150$	0	0
$151$	4.15987e8	0.0651153	0.0325576	−	0.999470i	$-0.489635\pi$
$151$	4.15987e8	0.0651153	0.0325576	+	0.999470i	$0.489635\pi$
$152$	0	0
$153$	9.13471e9	1.34767
$154$	0	0
$155$	−1.63737e9	−0.227853
$156$	0	0
$157$	−3.01793e9	−0.396425	−0.198212	−	0.980159i	$-0.563514\pi$
$157$	−3.01793e9	−0.396425	−0.198212	+	0.980159i	$0.563514\pi$
$158$	0	0
$159$	1.60341e10	1.98957
$160$	0	0
$161$	1.37129e9	0.160847
$162$	0	0
$163$	9.54190e9	1.05874	0.529371	−	0.848390i	$-0.322428\pi$
$163$	9.54190e9	1.05874	0.529371	+	0.848390i	$0.322428\pi$
$164$	0	0
$165$	3.24269e8	0.0340586
$166$	0	0
$167$	6.92059e9	0.688524	0.344262	−	0.938874i	$-0.388129\pi$
$167$	6.92059e9	0.688524	0.344262	+	0.938874i	$0.388129\pi$
$168$	0	0
$169$	−5.64618e9	−0.532433
$170$	0	0
$171$	−2.66879e10	−2.38689
$172$	0	0
$173$	5.56394e9	0.472254	0.236127	−	0.971722i	$-0.424122\pi$
$173$	5.56394e9	0.472254	0.236127	+	0.971722i	$0.424122\pi$
$174$	0	0
$175$	−4.60529e9	−0.371181
$176$	0	0
$177$	2.86725e10	2.19576
$178$	0	0
$179$	2.41361e9	0.175723	0.0878615	−	0.996133i	$-0.471997\pi$
$179$	2.41361e9	0.175723	0.0878615	+	0.996133i	$0.471997\pi$
$180$	0	0
$181$	2.84953e9	0.197342	0.0986710	−	0.995120i	$-0.468541\pi$
$181$	2.84953e9	0.197342	0.0986710	+	0.995120i	$0.468541\pi$
$182$	0	0
$183$	−5.65417e10	−3.72683
$184$	0	0
$185$	−3.26112e9	−0.204689
$186$	0	0
$187$	−1.04318e9	−0.0623838
$188$	0	0
$189$	2.34676e10	1.33780
$190$	0	0
$191$	−2.57074e10	−1.39768	−0.698839	−	0.715279i	$-0.746300\pi$
$191$	−2.57074e10	−1.39768	−0.698839	+	0.715279i	$0.746300\pi$
$192$	0	0
$193$	−3.61507e10	−1.87547	−0.937733	−	0.347357i	$-0.887079\pi$
$193$	−3.61507e10	−1.87547	−0.937733	+	0.347357i	$0.887079\pi$
$194$	0	0
$195$	−3.61156e9	−0.178871
$196$	0	0
$197$	−2.21880e10	−1.04959	−0.524795	−	0.851229i	$-0.675858\pi$
$197$	−2.21880e10	−1.04959	−0.524795	+	0.851229i	$0.675858\pi$
$198$	0	0
$199$	−1.97412e10	−0.892348	−0.446174	−	0.894946i	$-0.647214\pi$
$199$	−1.97412e10	−0.892348	−0.446174	+	0.894946i	$0.647214\pi$
$200$	0	0
$201$	−4.92418e10	−2.12790
$202$	0	0
$203$	5.97967e9	0.247141
$204$	0	0
$205$	6.13708e9	0.242700
$206$	0	0
$207$	3.16191e10	1.19697
$208$	0	0
$209$	3.04775e9	0.110489
$210$	0	0
$211$	−2.27897e10	−0.791531	−0.395766	−	0.918352i	$-0.629521\pi$
$211$	−2.27897e10	−0.791531	−0.395766	+	0.918352i	$0.629521\pi$
$212$	0	0
$213$	3.20473e10	1.06680
$214$	0	0
$215$	1.90801e9	0.0608986
$216$	0	0
$217$	2.09978e10	0.642842
$218$	0	0
$219$	7.70653e10	2.26392
$220$	0	0
$221$	1.16185e10	0.327630
$222$	0	0
$223$	1.46742e10	0.397357	0.198679	−	0.980065i	$-0.436335\pi$
$223$	1.46742e10	0.397357	0.198679	+	0.980065i	$0.436335\pi$
$224$	0	0
$225$	−1.06189e11	−2.76221
$226$	0	0
$227$	−3.73991e10	−0.934857	−0.467429	−	0.884031i	$-0.654819\pi$
$227$	−3.73991e10	−0.934857	−0.467429	+	0.884031i	$0.654819\pi$
$228$	0	0
$229$	−4.04940e10	−0.973040	−0.486520	−	0.873670i	$-0.661734\pi$
$229$	−4.04940e10	−0.973040	−0.486520	+	0.873670i	$0.661734\pi$
$230$	0	0
$231$	−4.15845e9	−0.0960898
$232$	0	0
$233$	−7.79707e9	−0.173313	−0.0866563	−	0.996238i	$-0.527618\pi$
$233$	−7.79707e9	−0.173313	−0.0866563	+	0.996238i	$0.527618\pi$
$234$	0	0
$235$	5.54706e9	0.118647
$236$	0	0
$237$	9.44053e10	1.94370
$238$	0	0
$239$	−4.99554e10	−0.990357	−0.495178	−	0.868791i	$-0.664897\pi$
$239$	−4.99554e10	−0.990357	−0.495178	+	0.868791i	$0.664897\pi$
$240$	0	0
$241$	−1.44796e10	−0.276490	−0.138245	−	0.990398i	$-0.544146\pi$
$241$	−1.44796e10	−0.276490	−0.138245	+	0.990398i	$0.544146\pi$
$242$	0	0
$243$	2.42600e11	4.46338
$244$	0	0
$245$	−1.07932e9	−0.0191383
$246$	0	0
$247$	−3.39444e10	−0.580273
$248$	0	0
$249$	7.82593e10	1.29015
$250$	0	0
$251$	−7.53524e10	−1.19830	−0.599150	−	0.800637i	$-0.704495\pi$
$251$	−7.53524e10	−1.19830	−0.599150	+	0.800637i	$0.704495\pi$
$252$	0	0
$253$	−3.61089e9	−0.0554079
$254$	0	0
$255$	−8.46270e9	−0.125337
$256$	0	0
$257$	6.97951e10	0.997990	0.498995	−	0.866605i	$-0.333703\pi$
$257$	6.97951e10	0.997990	0.498995	+	0.866605i	$0.333703\pi$
$258$	0	0
$259$	4.18209e10	0.577490
$260$	0	0
$261$	1.37879e11	1.83914
$262$	0	0
$263$	−6.51074e10	−0.839130	−0.419565	−	0.907725i	$-0.637817\pi$
$263$	−6.51074e10	−0.839130	−0.419565	+	0.907725i	$0.637817\pi$
$264$	0	0
$265$	−1.09585e10	−0.136503
$266$	0	0
$267$	2.50448e11	3.01590
$268$	0	0
$269$	−3.07988e10	−0.358632	−0.179316	−	0.983792i	$-0.557388\pi$
$269$	−3.07988e10	−0.358632	−0.179316	+	0.983792i	$0.557388\pi$
$270$	0	0
$271$	2.16598e10	0.243946	0.121973	−	0.992533i	$-0.461078\pi$
$271$	2.16598e10	0.243946	0.121973	+	0.992533i	$0.461078\pi$
$272$	0	0
$273$	4.63149e10	0.504648
$274$	0	0
$275$	1.21267e10	0.127863
$276$	0	0
$277$	7.43648e9	0.0758942	0.0379471	−	0.999280i	$-0.487918\pi$
$277$	7.43648e9	0.0758942	0.0379471	+	0.999280i	$0.487918\pi$
$278$	0	0
$279$	4.84166e11	4.78383
$280$	0	0
$281$	−3.90190e9	−0.0373334	−0.0186667	−	0.999826i	$-0.505942\pi$
$281$	−3.90190e9	−0.0373334	−0.0186667	+	0.999826i	$0.505942\pi$
$282$	0	0
$283$	−1.27084e10	−0.117774	−0.0588872	−	0.998265i	$-0.518755\pi$
$283$	−1.27084e10	−0.117774	−0.0588872	+	0.998265i	$0.518755\pi$
$284$	0	0
$285$	2.47246e10	0.221987
$286$	0	0
$287$	−7.87025e10	−0.684731
$288$	0	0
$289$	−9.13632e10	−0.770426
$290$	0	0
$291$	−4.04958e11	−3.31048
$292$	0	0
$293$	−9.00027e10	−0.713430	−0.356715	−	0.934213i	$-0.616103\pi$
$293$	−9.00027e10	−0.713430	−0.356715	+	0.934213i	$0.616103\pi$
$294$	0	0
$295$	−1.95961e10	−0.150650
$296$	0	0
$297$	−6.17951e10	−0.460840
$298$	0	0
$299$	4.02164e10	0.290994
$300$	0	0
$301$	−2.44685e10	−0.171813
$302$	0	0
$303$	−5.18232e11	−3.53210
$304$	0	0
$305$	3.86432e10	0.255696
$306$	0	0
$307$	−8.38744e10	−0.538898	−0.269449	−	0.963015i	$-0.586842\pi$
$307$	−8.38744e10	−0.538898	−0.269449	+	0.963015i	$0.586842\pi$
$308$	0	0
$309$	9.52711e10	0.594495
$310$	0	0
$311$	−5.69975e10	−0.345489	−0.172744	−	0.984967i	$-0.555263\pi$
$311$	−5.69975e10	−0.345489	−0.172744	+	0.984967i	$0.555263\pi$
$312$	0	0
$313$	−7.69060e9	−0.0452909	−0.0226454	−	0.999744i	$-0.507209\pi$
$313$	−7.69060e9	−0.0452909	−0.0226454	+	0.999744i	$0.507209\pi$
$314$	0	0
$315$	−2.48869e10	−0.142421
$316$	0	0
$317$	2.76038e11	1.53533	0.767665	−	0.640851i	$-0.221418\pi$
$317$	2.76038e11	1.53533	0.767665	+	0.640851i	$0.221418\pi$
$318$	0	0
$319$	−1.57457e10	−0.0851343
$320$	0	0
$321$	−2.45211e11	−1.28904
$322$	0	0
$323$	−7.95395e10	−0.406604
$324$	0	0
$325$	−1.35062e11	−0.671517
$326$	0	0
$327$	1.14504e11	0.553802
$328$	0	0
$329$	−7.11360e10	−0.334740
$330$	0	0
$331$	−5.74005e10	−0.262839	−0.131419	−	0.991327i	$-0.541953\pi$
$331$	−5.74005e10	−0.262839	−0.131419	+	0.991327i	$0.541953\pi$
$332$	0	0
$333$	9.64306e11	4.29750
$334$	0	0
$335$	3.36542e10	0.145995
$336$	0	0
$337$	−3.13050e11	−1.32215	−0.661073	−	0.750321i	$-0.729899\pi$
$337$	−3.13050e11	−1.32215	−0.661073	+	0.750321i	$0.729899\pi$
$338$	0	0
$339$	−3.37704e11	−1.38879
$340$	0	0
$341$	−5.52915e10	−0.221444
$342$	0	0
$343$	1.38413e10	0.0539949
$344$	0	0
$345$	−2.92930e10	−0.111321
$346$	0	0
$347$	−1.24609e11	−0.461389	−0.230695	−	0.973026i	$-0.574100\pi$
$347$	−1.24609e11	−0.461389	−0.230695	+	0.973026i	$0.574100\pi$
$348$	0	0
$349$	−1.79519e11	−0.647734	−0.323867	−	0.946103i	$-0.604983\pi$
$349$	−1.79519e11	−0.647734	−0.323867	+	0.946103i	$0.604983\pi$
$350$	0	0
$351$	6.88246e11	2.42026
$352$	0	0
$353$	−9.79970e8	−0.00335913	−0.00167956	−	0.999999i	$-0.500535\pi$
$353$	−9.79970e8	−0.00335913	−0.00167956	+	0.999999i	$0.500535\pi$
$354$	0	0
$355$	−2.19026e10	−0.0731927
$356$	0	0
$357$	1.08526e11	0.353613
$358$	0	0
$359$	3.73991e11	1.18833	0.594164	−	0.804344i	$-0.297483\pi$
$359$	3.73991e11	1.18833	0.594164	+	0.804344i	$0.297483\pi$
$360$	0	0
$361$	−9.03059e10	−0.279855
$362$	0	0
$363$	−6.34995e11	−1.91951
$364$	0	0
$365$	−5.26700e10	−0.155326
$366$	0	0
$367$	5.35893e11	1.54199	0.770993	−	0.636843i	$-0.219760\pi$
$367$	5.35893e11	1.54199	0.770993	+	0.636843i	$0.219760\pi$
$368$	0	0
$369$	−1.81472e12	−5.09555
$370$	0	0
$371$	1.40532e11	0.385118
$372$	0	0
$373$	−4.04569e11	−1.08219	−0.541094	−	0.840962i	$-0.681990\pi$
$373$	−4.04569e11	−1.08219	−0.541094	+	0.840962i	$0.681990\pi$
$374$	0	0
$375$	1.98551e11	0.518480
$376$	0	0
$377$	1.75369e11	0.447112
$378$	0	0
$379$	7.02688e11	1.74939	0.874693	−	0.484677i	$-0.161063\pi$
$379$	7.02688e11	1.74939	0.874693	+	0.484677i	$0.161063\pi$
$380$	0	0
$381$	7.10728e11	1.72799
$382$	0	0
$383$	−4.69960e11	−1.11601	−0.558003	−	0.829839i	$-0.688432\pi$
$383$	−4.69960e11	−1.11601	−0.558003	+	0.829839i	$0.688432\pi$
$384$	0	0
$385$	2.84208e9	0.00659269
$386$	0	0
$387$	−5.64193e11	−1.27858
$388$	0	0
$389$	−4.73136e10	−0.104764	−0.0523821	−	0.998627i	$-0.516681\pi$
$389$	−4.73136e10	−0.104764	−0.0523821	+	0.998627i	$0.516681\pi$
$390$	0	0
$391$	9.42363e10	0.203903
$392$	0	0
$393$	1.12662e12	2.38237
$394$	0	0
$395$	−6.45210e10	−0.133356
$396$	0	0
$397$	−5.68408e11	−1.14843	−0.574213	−	0.818706i	$-0.694692\pi$
$397$	−5.68408e11	−1.14843	−0.574213	+	0.818706i	$0.694692\pi$
$398$	0	0
$399$	−3.17070e11	−0.626292
$400$	0	0
$401$	7.44652e10	0.143815	0.0719074	−	0.997411i	$-0.477091\pi$
$401$	7.44652e10	0.143815	0.0719074	+	0.997411i	$0.477091\pi$
$402$	0	0
$403$	6.15812e11	1.16299
$404$	0	0
$405$	−2.97288e11	−0.549073
$406$	0	0
$407$	−1.10123e11	−0.198932
$408$	0	0
$409$	−1.18444e11	−0.209295	−0.104648	−	0.994509i	$-0.533372\pi$
$409$	−1.18444e11	−0.209295	−0.104648	+	0.994509i	$0.533372\pi$
$410$	0	0
$411$	−6.17721e11	−1.06784
$412$	0	0
$413$	2.51302e11	0.425031
$414$	0	0
$415$	−5.34860e10	−0.0885164
$416$	0	0
$417$	−6.05375e11	−0.980420
$418$	0	0
$419$	3.16733e11	0.502030	0.251015	−	0.967983i	$-0.419236\pi$
$419$	3.16733e11	0.502030	0.251015	+	0.967983i	$0.419236\pi$
$420$	0	0
$421$	3.05566e11	0.474063	0.237032	−	0.971502i	$-0.423826\pi$
$421$	3.05566e11	0.474063	0.237032	+	0.971502i	$0.423826\pi$
$422$	0	0
$423$	−1.64025e12	−2.49103
$424$	0	0
$425$	−3.16480e11	−0.470540
$426$	0	0
$427$	−4.95564e11	−0.721397
$428$	0	0
$429$	−1.21957e11	−0.173839
$430$	0	0
$431$	1.00084e12	1.39707	0.698535	−	0.715576i	$-0.253836\pi$
$431$	1.00084e12	1.39707	0.698535	+	0.715576i	$0.253836\pi$
$432$	0	0
$433$	−8.50664e11	−1.16295	−0.581477	−	0.813563i	$-0.697525\pi$
$433$	−8.50664e11	−1.16295	−0.581477	+	0.813563i	$0.697525\pi$
$434$	0	0
$435$	−1.27736e11	−0.171045
$436$	0	0
$437$	−2.75320e11	−0.361136
$438$	0	0
$439$	4.13624e11	0.531515	0.265758	−	0.964040i	$-0.414378\pi$
$439$	4.13624e11	0.531515	0.265758	+	0.964040i	$0.414378\pi$
$440$	0	0
$441$	3.19152e11	0.401813
$442$	0	0
$443$	−9.79404e11	−1.20822	−0.604109	−	0.796902i	$-0.706471\pi$
$443$	−9.79404e11	−1.20822	−0.604109	+	0.796902i	$0.706471\pi$
$444$	0	0
$445$	−1.71168e11	−0.206920
$446$	0	0
$447$	−1.39197e12	−1.64910
$448$	0	0
$449$	7.09255e11	0.823558	0.411779	−	0.911284i	$-0.364908\pi$
$449$	7.09255e11	0.823558	0.411779	+	0.911284i	$0.364908\pi$
$450$	0	0
$451$	2.07240e11	0.235874
$452$	0	0
$453$	1.13957e11	0.127145
$454$	0	0
$455$	−3.16538e10	−0.0346237
$456$	0	0
$457$	−8.30945e11	−0.891147	−0.445574	−	0.895245i	$-0.647000\pi$
$457$	−8.30945e11	−0.891147	−0.445574	+	0.895245i	$0.647000\pi$
$458$	0	0
$459$	1.61272e12	1.69590
$460$	0	0
$461$	1.57811e12	1.62736	0.813678	−	0.581315i	$-0.197462\pi$
$461$	1.57811e12	1.62736	0.813678	+	0.581315i	$0.197462\pi$
$462$	0	0
$463$	1.26972e12	1.28409	0.642043	−	0.766669i	$-0.278087\pi$
$463$	1.26972e12	1.28409	0.642043	+	0.766669i	$0.278087\pi$
$464$	0	0
$465$	−4.48547e11	−0.444908
$466$	0	0
$467$	1.19937e12	1.16688	0.583442	−	0.812155i	$-0.301706\pi$
$467$	1.19937e12	1.16688	0.583442	+	0.812155i	$0.301706\pi$
$468$	0	0
$469$	−4.31584e11	−0.411896
$470$	0	0
$471$	−8.26744e11	−0.774064
$472$	0	0
$473$	6.44306e10	0.0591857
$474$	0	0
$475$	9.24625e11	0.833383
$476$	0	0
$477$	3.24039e12	2.86593
$478$	0	0
$479$	−3.35551e11	−0.291239	−0.145619	−	0.989341i	$-0.546517\pi$
$479$	−3.35551e11	−0.291239	−0.145619	+	0.989341i	$0.546517\pi$
$480$	0	0
$481$	1.22650e12	1.04476
$482$	0	0
$483$	3.75656e11	0.314071
$484$	0	0
$485$	2.76767e11	0.227131
$486$	0	0
$487$	−1.44592e12	−1.16483	−0.582417	−	0.812890i	$-0.697893\pi$
$487$	−1.44592e12	−1.16483	−0.582417	+	0.812890i	$0.697893\pi$
$488$	0	0
$489$	2.61394e12	2.06731
$490$	0	0
$491$	1.76921e12	1.37376	0.686882	−	0.726769i	$-0.258979\pi$
$491$	1.76921e12	1.37376	0.686882	+	0.726769i	$0.258979\pi$
$492$	0	0
$493$	4.10929e11	0.313296
$494$	0	0
$495$	6.55325e10	0.0490607
$496$	0	0
$497$	2.80881e11	0.206499
$498$	0	0
$499$	1.01090e12	0.729887	0.364943	−	0.931030i	$-0.381088\pi$
$499$	1.01090e12	0.729887	0.364943	+	0.931030i	$0.381088\pi$
$500$	0	0
$501$	1.89585e12	1.34442
$502$	0	0
$503$	−2.12466e12	−1.47990	−0.739952	−	0.672660i	$-0.765152\pi$
$503$	−2.12466e12	−1.47990	−0.739952	+	0.672660i	$0.765152\pi$
$504$	0	0
$505$	3.54184e11	0.242336
$506$	0	0
$507$	−1.54674e12	−1.03963
$508$	0	0
$509$	2.40756e12	1.58981	0.794907	−	0.606731i	$-0.207519\pi$
$509$	2.40756e12	1.58981	0.794907	+	0.606731i	$0.207519\pi$
$510$	0	0
$511$	6.75444e11	0.438223
$512$	0	0
$513$	−4.71170e12	−3.00365
$514$	0	0
$515$	−6.51127e10	−0.0407881
$516$	0	0
$517$	1.87316e11	0.115310
$518$	0	0
$519$	1.52421e12	0.922128
$520$	0	0
$521$	1.13069e12	0.672317	0.336159	−	0.941805i	$-0.390872\pi$
$521$	1.13069e12	0.672317	0.336159	+	0.941805i	$0.390872\pi$
$522$	0	0
$523$	−1.86264e11	−0.108861	−0.0544303	−	0.998518i	$-0.517334\pi$
$523$	−1.86264e11	−0.108861	−0.0544303	+	0.998518i	$0.517334\pi$
$524$	0	0
$525$	−1.26159e12	−0.724772
$526$	0	0
$527$	1.44299e12	0.814920
$528$	0	0
$529$	−1.47496e12	−0.818898
$530$	0	0
$531$	5.79451e12	3.16295
$532$	0	0
$533$	−2.30815e12	−1.23877
$534$	0	0
$535$	1.67588e11	0.0884406
$536$	0	0
$537$	6.61194e11	0.343119
$538$	0	0
$539$	−3.64470e10	−0.0186000
$540$	0	0
$541$	2.16613e12	1.08717	0.543583	−	0.839355i	$-0.317067\pi$
$541$	2.16613e12	1.08717	0.543583	+	0.839355i	$0.317067\pi$
$542$	0	0
$543$	7.80611e11	0.385332
$544$	0	0
$545$	−7.82572e10	−0.0379962
$546$	0	0
$547$	−2.69320e12	−1.28625	−0.643125	−	0.765762i	$-0.722362\pi$
$547$	−2.69320e12	−1.28625	−0.643125	+	0.765762i	$0.722362\pi$
$548$	0	0
$549$	−1.14267e13	−5.36840
$550$	0	0
$551$	−1.20057e12	−0.554886
$552$	0	0
$553$	8.27422e11	0.376239
$554$	0	0
$555$	−8.93365e11	−0.399678
$556$	0	0
$557$	3.72511e12	1.63980	0.819899	−	0.572508i	$-0.194030\pi$
$557$	3.72511e12	1.63980	0.819899	+	0.572508i	$0.194030\pi$
$558$	0	0
$559$	−7.17599e11	−0.310834
$560$	0	0
$561$	−2.85773e11	−0.121811
$562$	0	0
$563$	2.84268e12	1.19245	0.596226	−	0.802817i	$-0.296666\pi$
$563$	2.84268e12	1.19245	0.596226	+	0.802817i	$0.296666\pi$
$564$	0	0
$565$	2.30803e11	0.0952846
$566$	0	0
$567$	3.81245e12	1.54910
$568$	0	0
$569$	3.65600e12	1.46218	0.731090	−	0.682281i	$-0.239012\pi$
$569$	3.65600e12	1.46218	0.731090	+	0.682281i	$0.239012\pi$
$570$	0	0
$571$	2.69092e12	1.05935	0.529674	−	0.848201i	$-0.322314\pi$
$571$	2.69092e12	1.05935	0.529674	+	0.848201i	$0.322314\pi$
$572$	0	0
$573$	−7.04237e12	−2.72912
$574$	0	0
$575$	−1.09547e12	−0.417923
$576$	0	0
$577$	−2.39332e12	−0.898895	−0.449447	−	0.893307i	$-0.648379\pi$
$577$	−2.39332e12	−0.898895	−0.449447	+	0.893307i	$0.648379\pi$
$578$	0	0
$579$	−9.90327e12	−3.66206
$580$	0	0
$581$	6.85909e11	0.249732
$582$	0	0
$583$	−3.70051e11	−0.132664
$584$	0	0
$585$	−7.29872e11	−0.257659
$586$	0	0
$587$	4.80967e12	1.67203	0.836015	−	0.548707i	$-0.184880\pi$
$587$	4.80967e12	1.67203	0.836015	+	0.548707i	$0.184880\pi$
$588$	0	0
$589$	−4.21582e12	−1.44332
$590$	0	0
$591$	−6.07826e12	−2.04944
$592$	0	0
$593$	4.89458e11	0.162544	0.0812718	−	0.996692i	$-0.474102\pi$
$593$	4.89458e11	0.162544	0.0812718	+	0.996692i	$0.474102\pi$
$594$	0	0
$595$	−7.41719e10	−0.0242613
$596$	0	0
$597$	−5.40797e12	−1.74241
$598$	0	0
$599$	−1.81979e12	−0.577566	−0.288783	−	0.957395i	$-0.593251\pi$
$599$	−1.81979e12	−0.577566	−0.288783	+	0.957395i	$0.593251\pi$
$600$	0	0
$601$	3.86475e12	1.20833	0.604165	−	0.796859i	$-0.293507\pi$
$601$	3.86475e12	1.20833	0.604165	+	0.796859i	$0.293507\pi$
$602$	0	0
$603$	−9.95144e12	−3.06520
$604$	0	0
$605$	4.33985e11	0.131697
$606$	0	0
$607$	−2.84934e12	−0.851914	−0.425957	−	0.904743i	$-0.640063\pi$
$607$	−2.84934e12	−0.851914	−0.425957	+	0.904743i	$0.640063\pi$
$608$	0	0
$609$	1.63809e12	0.482570
$610$	0	0
$611$	−2.08624e12	−0.605590
$612$	0	0
$613$	3.41042e12	0.975519	0.487759	−	0.872978i	$-0.337814\pi$
$613$	3.41042e12	0.975519	0.487759	+	0.872978i	$0.337814\pi$
$614$	0	0
$615$	1.68122e12	0.473899
$616$	0	0
$617$	3.44371e12	0.956630	0.478315	−	0.878188i	$-0.341248\pi$
$617$	3.44371e12	0.956630	0.478315	+	0.878188i	$0.341248\pi$
$618$	0	0
$619$	−1.70276e12	−0.466170	−0.233085	−	0.972456i	$-0.574882\pi$
$619$	−1.70276e12	−0.466170	−0.233085	+	0.972456i	$0.574882\pi$
$620$	0	0
$621$	5.58230e12	1.50626
$622$	0	0
$623$	2.19507e12	0.583784
$624$	0	0
$625$	3.61053e12	0.946480
$626$	0	0
$627$	8.34911e11	0.215743
$628$	0	0
$629$	2.87398e12	0.732074
$630$	0	0
$631$	4.09336e12	1.02789	0.513946	−	0.857823i	$-0.328183\pi$
$631$	4.09336e12	1.02789	0.513946	+	0.857823i	$0.328183\pi$
$632$	0	0
$633$	−6.24310e12	−1.54555
$634$	0	0
$635$	−4.85744e11	−0.118557
$636$	0	0
$637$	4.05930e11	0.0976841
$638$	0	0
$639$	6.47654e12	1.53670
$640$	0	0
$641$	−6.60642e12	−1.54563	−0.772814	−	0.634633i	$-0.781151\pi$
$641$	−6.60642e12	−1.54563	−0.772814	+	0.634633i	$0.781151\pi$
$642$	0	0
$643$	−6.27495e12	−1.44764	−0.723821	−	0.689988i	$-0.757616\pi$
$643$	−6.27495e12	−1.44764	−0.723821	+	0.689988i	$0.757616\pi$
$644$	0	0
$645$	5.22687e11	0.118911
$646$	0	0
$647$	5.06881e12	1.13720	0.568600	−	0.822614i	$-0.307485\pi$
$647$	5.06881e12	1.13720	0.568600	+	0.822614i	$0.307485\pi$
$648$	0	0
$649$	−6.61731e11	−0.146413
$650$	0	0
$651$	5.75221e12	1.25522
$652$	0	0
$653$	−3.49805e12	−0.752865	−0.376432	−	0.926444i	$-0.622849\pi$
$653$	−3.49805e12	−0.752865	−0.376432	+	0.926444i	$0.622849\pi$
$654$	0	0
$655$	−7.69983e11	−0.163454
$656$	0	0
$657$	1.55744e13	3.26112
$658$	0	0
$659$	4.13419e12	0.853897	0.426949	−	0.904276i	$-0.359589\pi$
$659$	4.13419e12	0.853897	0.426949	+	0.904276i	$0.359589\pi$
$660$	0	0
$661$	−4.48641e12	−0.914098	−0.457049	−	0.889442i	$-0.651094\pi$
$661$	−4.48641e12	−0.914098	−0.457049	+	0.889442i	$0.651094\pi$
$662$	0	0
$663$	3.18281e12	0.639734
$664$	0	0
$665$	2.16700e11	0.0429696
$666$	0	0
$667$	1.42240e12	0.278263
$668$	0	0
$669$	4.01989e12	0.775884
$670$	0	0
$671$	1.30492e12	0.248504
$672$	0	0
$673$	−3.49560e12	−0.656832	−0.328416	−	0.944533i	$-0.606515\pi$
$673$	−3.49560e12	−0.656832	−0.328416	+	0.944533i	$0.606515\pi$
$674$	0	0
$675$	−1.87474e13	−3.47596
$676$	0	0
$677$	5.96605e12	1.09153	0.545767	−	0.837937i	$-0.316238\pi$
$677$	5.96605e12	1.09153	0.545767	+	0.837937i	$0.316238\pi$
$678$	0	0
$679$	−3.54928e12	−0.640806
$680$	0	0
$681$	−1.02453e13	−1.82541
$682$	0	0
$683$	3.88138e12	0.682485	0.341243	−	0.939975i	$-0.389152\pi$
$683$	3.88138e12	0.682485	0.341243	+	0.939975i	$0.389152\pi$
$684$	0	0
$685$	4.22179e11	0.0732638
$686$	0	0
$687$	−1.10931e13	−1.89997
$688$	0	0
$689$	4.12146e12	0.696731
$690$	0	0
$691$	1.42915e12	0.238465	0.119233	−	0.992866i	$-0.461957\pi$
$691$	1.42915e12	0.238465	0.119233	+	0.992866i	$0.461957\pi$
$692$	0	0
$693$	−8.40394e11	−0.138415
$694$	0	0
$695$	4.13741e11	0.0672663
$696$	0	0
$697$	−5.40851e12	−0.868021
$698$	0	0
$699$	−2.13596e12	−0.338412
$700$	0	0
$701$	−7.99721e12	−1.25086	−0.625428	−	0.780282i	$-0.715076\pi$
$701$	−7.99721e12	−1.25086	−0.625428	+	0.780282i	$0.715076\pi$
$702$	0	0
$703$	−8.39658e12	−1.29659
$704$	0	0
$705$	1.51958e12	0.231672
$706$	0	0
$707$	−4.54208e12	−0.683704
$708$	0	0
$709$	−9.16623e11	−0.136233	−0.0681166	−	0.997677i	$-0.521699\pi$
$709$	−9.16623e11	−0.136233	−0.0681166	+	0.997677i	$0.521699\pi$
$710$	0	0
$711$	1.90787e13	2.79985
$712$	0	0
$713$	4.99479e12	0.723793
$714$	0	0
$715$	8.33510e10	0.0119271
$716$	0	0
$717$	−1.36850e13	−1.93378
$718$	0	0
$719$	6.02145e12	0.840275	0.420137	−	0.907460i	$-0.361982\pi$
$719$	6.02145e12	0.840275	0.420137	+	0.907460i	$0.361982\pi$
$720$	0	0
$721$	8.35011e11	0.115076
$722$	0	0
$723$	−3.96659e12	−0.539877
$724$	0	0
$725$	−4.77694e12	−0.642138
$726$	0	0
$727$	2.44115e12	0.324108	0.162054	−	0.986782i	$-0.448188\pi$
$727$	2.44115e12	0.324108	0.162054	+	0.986782i	$0.448188\pi$
$728$	0	0
$729$	3.52051e13	4.61670
$730$	0	0
$731$	−1.68150e12	−0.217805
$732$	0	0
$733$	1.88589e12	0.241296	0.120648	−	0.992695i	$-0.461503\pi$
$733$	1.88589e12	0.241296	0.120648	+	0.992695i	$0.461503\pi$
$734$	0	0
$735$	−2.95673e11	−0.0373696
$736$	0	0
$737$	1.13645e12	0.141888
$738$	0	0
$739$	9.41376e11	0.116108	0.0580542	−	0.998313i	$-0.481510\pi$
$739$	9.41376e11	0.116108	0.0580542	+	0.998313i	$0.481510\pi$
$740$	0	0
$741$	−9.29886e12	−1.13305
$742$	0	0
$743$	8.77855e12	1.05675	0.528376	−	0.849011i	$-0.322801\pi$
$743$	8.77855e12	1.05675	0.528376	+	0.849011i	$0.322801\pi$
$744$	0	0
$745$	9.51336e11	0.113144
$746$	0	0
$747$	1.58157e13	1.85842
$748$	0	0
$749$	−2.14917e12	−0.249518
$750$	0	0
$751$	−1.42629e13	−1.63616	−0.818082	−	0.575102i	$-0.804963\pi$
$751$	−1.42629e13	−1.63616	−0.818082	+	0.575102i	$0.804963\pi$
$752$	0	0
$753$	−2.06423e13	−2.33981
$754$	0	0
$755$	−7.78835e10	−0.00872336
$756$	0	0
$757$	4.46366e12	0.494038	0.247019	−	0.969011i	$-0.420549\pi$
$757$	4.46366e12	0.494038	0.247019	+	0.969011i	$0.420549\pi$
$758$	0	0
$759$	−9.89180e11	−0.108190
$760$	0	0
$761$	−5.03864e12	−0.544605	−0.272303	−	0.962212i	$-0.587785\pi$
$761$	−5.03864e12	−0.544605	−0.272303	+	0.962212i	$0.587785\pi$
$762$	0	0
$763$	1.00358e12	0.107199
$764$	0	0
$765$	−1.71025e12	−0.180545
$766$	0	0
$767$	7.37006e12	0.768939
$768$	0	0
$769$	3.65365e12	0.376754	0.188377	−	0.982097i	$-0.439677\pi$
$769$	3.65365e12	0.376754	0.188377	+	0.982097i	$0.439677\pi$
$770$	0	0
$771$	1.91199e13	1.94869
$772$	0	0
$773$	−1.74477e13	−1.75764	−0.878822	−	0.477149i	$-0.841670\pi$
$773$	−1.74477e13	−1.75764	−0.878822	+	0.477149i	$0.841670\pi$
$774$	0	0
$775$	−1.67744e13	−1.67028
$776$	0	0
$777$	1.14566e13	1.12761
$778$	0	0
$779$	1.58015e13	1.53737
$780$	0	0
$781$	−7.39617e11	−0.0711340
$782$	0	0
$783$	2.43423e13	2.31437
$784$	0	0
$785$	5.65035e11	0.0531082
$786$	0	0
$787$	1.38487e13	1.28683	0.643415	−	0.765517i	$-0.277517\pi$
$787$	1.38487e13	1.28683	0.643415	+	0.765517i	$0.277517\pi$
$788$	0	0
$789$	−1.78358e13	−1.63850
$790$	0	0
$791$	−2.95983e12	−0.268827
$792$	0	0
$793$	−1.45336e13	−1.30510
$794$	0	0
$795$	−3.00201e12	−0.266538
$796$	0	0
$797$	9.44767e11	0.0829397	0.0414698	−	0.999140i	$-0.486796\pi$
$797$	9.44767e11	0.0829397	0.0414698	+	0.999140i	$0.486796\pi$
$798$	0	0
$799$	−4.88854e12	−0.424344
$800$	0	0
$801$	5.06139e13	4.34433
$802$	0	0
$803$	−1.77859e12	−0.150958
$804$	0	0
$805$	−2.56741e11	−0.0215483
$806$	0	0
$807$	−8.43714e12	−0.700268
$808$	0	0
$809$	1.11474e13	0.914964	0.457482	−	0.889219i	$-0.348751\pi$
$809$	1.11474e13	0.914964	0.457482	+	0.889219i	$0.348751\pi$
$810$	0	0
$811$	−1.46259e13	−1.18721	−0.593606	−	0.804755i	$-0.702296\pi$
$811$	−1.46259e13	−1.18721	−0.593606	+	0.804755i	$0.702296\pi$
$812$	0	0
$813$	5.93357e12	0.476331
$814$	0	0
$815$	−1.78649e12	−0.141838
$816$	0	0
$817$	4.91265e12	0.385759
$818$	0	0
$819$	9.35993e12	0.726934
$820$	0	0
$821$	2.10452e13	1.61663	0.808313	−	0.588754i	$-0.200381\pi$
$821$	2.10452e13	1.61663	0.808313	+	0.588754i	$0.200381\pi$
$822$	0	0
$823$	−7.02381e12	−0.533671	−0.266835	−	0.963742i	$-0.585978\pi$
$823$	−7.02381e12	−0.533671	−0.266835	+	0.963742i	$0.585978\pi$
$824$	0	0
$825$	3.32203e12	0.249667
$826$	0	0
$827$	2.09001e13	1.55372	0.776862	−	0.629671i	$-0.216810\pi$
$827$	2.09001e13	1.55372	0.776862	+	0.629671i	$0.216810\pi$
$828$	0	0
$829$	−1.83964e12	−0.135281	−0.0676406	−	0.997710i	$-0.521547\pi$
$829$	−1.83964e12	−0.135281	−0.0676406	+	0.997710i	$0.521547\pi$
$830$	0	0
$831$	2.03718e12	0.148192
$832$	0	0
$833$	9.51187e11	0.0684484
$834$	0	0
$835$	−1.29571e12	−0.0922401
$836$	0	0
$837$	8.54786e13	6.01995
$838$	0	0
$839$	−1.33010e13	−0.926735	−0.463367	−	0.886166i	$-0.653359\pi$
$839$	−1.33010e13	−0.926735	−0.463367	+	0.886166i	$0.653359\pi$
$840$	0	0
$841$	−8.30460e12	−0.572449
$842$	0	0
$843$	−1.06890e12	−0.0728976
$844$	0	0
$845$	1.05711e12	0.0713289
$846$	0	0
$847$	−5.56546e12	−0.371557
$848$	0	0
$849$	−3.48138e12	−0.229967
$850$	0	0
$851$	9.94805e12	0.650212
$852$	0	0
$853$	−4.55820e12	−0.294797	−0.147399	−	0.989077i	$-0.547090\pi$
$853$	−4.55820e12	−0.294797	−0.147399	+	0.989077i	$0.547090\pi$
$854$	0	0
$855$	4.99667e12	0.319766
$856$	0	0
$857$	−1.65785e13	−1.04986	−0.524930	−	0.851146i	$-0.675908\pi$
$857$	−1.65785e13	−1.04986	−0.524930	+	0.851146i	$0.675908\pi$
$858$	0	0
$859$	−1.45198e13	−0.909896	−0.454948	−	0.890518i	$-0.650342\pi$
$859$	−1.45198e13	−0.909896	−0.454948	+	0.890518i	$0.650342\pi$
$860$	0	0
$861$	−2.15600e13	−1.33701
$862$	0	0
$863$	−1.56245e13	−0.958862	−0.479431	−	0.877580i	$-0.659157\pi$
$863$	−1.56245e13	−0.958862	−0.479431	+	0.877580i	$0.659157\pi$
$864$	0	0
$865$	−1.04171e12	−0.0632669
$866$	0	0
$867$	−2.50284e13	−1.50434
$868$	0	0
$869$	−2.17878e12	−0.129606
$870$	0	0
$871$	−1.26573e13	−0.745175
$872$	0	0
$873$	−8.18393e13	−4.76867
$874$	0	0
$875$	1.74022e12	0.100361
$876$	0	0
$877$	1.20199e13	0.686122	0.343061	−	0.939313i	$-0.388536\pi$
$877$	1.20199e13	0.686122	0.343061	+	0.939313i	$0.388536\pi$
$878$	0	0
$879$	−2.46557e13	−1.39305
$880$	0	0
$881$	−3.20627e12	−0.179312	−0.0896559	−	0.995973i	$-0.528577\pi$
$881$	−3.20627e12	−0.179312	−0.0896559	+	0.995973i	$0.528577\pi$
$882$	0	0
$883$	1.83352e13	1.01499	0.507497	−	0.861654i	$-0.330571\pi$
$883$	1.83352e13	1.01499	0.507497	+	0.861654i	$0.330571\pi$
$884$	0	0
$885$	−5.36823e12	−0.294162
$886$	0	0
$887$	−3.19067e13	−1.73072	−0.865359	−	0.501153i	$-0.832909\pi$
$887$	−3.19067e13	−1.73072	−0.865359	+	0.501153i	$0.832909\pi$
$888$	0	0
$889$	6.22923e12	0.334485
$890$	0	0
$891$	−1.00390e13	−0.533629
$892$	0	0
$893$	1.42823e13	0.751565
$894$	0	0
$895$	−4.51890e11	−0.0235412
$896$	0	0
$897$	1.10170e13	0.568197
$898$	0	0
$899$	2.17804e13	1.11211
$900$	0	0
$901$	9.65752e12	0.488207
$902$	0	0
$903$	−6.70298e12	−0.335485
$904$	0	0
$905$	−5.33506e11	−0.0264375
$906$	0	0
$907$	−3.39519e13	−1.66583	−0.832917	−	0.553398i	$-0.813331\pi$
$907$	−3.39519e13	−1.66583	−0.832917	+	0.553398i	$0.813331\pi$
$908$	0	0
$909$	−1.04731e14	−5.08790
$910$	0	0
$911$	2.84171e13	1.36693	0.683466	−	0.729982i	$-0.260472\pi$
$911$	2.84171e13	1.36693	0.683466	+	0.729982i	$0.260472\pi$
$912$	0	0
$913$	−1.80614e12	−0.0860267
$914$	0	0
$915$	1.05861e13	0.499275
$916$	0	0
$917$	9.87432e12	0.461153
$918$	0	0
$919$	−1.79635e13	−0.830750	−0.415375	−	0.909650i	$-0.636350\pi$
$919$	−1.79635e13	−0.830750	−0.415375	+	0.909650i	$0.636350\pi$
$920$	0	0
$921$	−2.29769e13	−1.05226
$922$	0	0
$923$	8.23752e12	0.373585
$924$	0	0
$925$	−3.34092e13	−1.50047
$926$	0	0
$927$	1.92537e13	0.856356
$928$	0	0
$929$	−1.69946e10	−0.000748582 0	−0.000374291	−	1.00000i	$-0.500119\pi$
$929$	−1.69946e10	−0.000748582 0	−0.000374291		1.00000i	$0.500119\pi$
$930$	0	0
$931$	−2.77898e12	−0.121230
$932$	0	0
$933$	−1.56141e13	−0.674605
$934$	0	0
$935$	1.95310e11	0.00835743
$936$	0	0
$937$	1.73276e13	0.734362	0.367181	−	0.930149i	$-0.380323\pi$
$937$	1.73276e13	0.734362	0.367181	+	0.930149i	$0.380323\pi$
$938$	0	0
$939$	−2.10679e12	−0.0884355
$940$	0	0
$941$	3.37061e13	1.40138	0.700689	−	0.713467i	$-0.252876\pi$
$941$	3.37061e13	1.40138	0.700689	+	0.713467i	$0.252876\pi$
$942$	0	0
$943$	−1.87212e13	−0.770957
$944$	0	0
$945$	−4.39374e12	−0.179222
$946$	0	0
$947$	−2.02156e13	−0.816793	−0.408396	−	0.912805i	$-0.633912\pi$
$947$	−2.02156e13	−0.816793	−0.408396	+	0.912805i	$0.633912\pi$
$948$	0	0
$949$	1.98091e13	0.792805
$950$	0	0
$951$	7.56188e13	2.99790
$952$	0	0
$953$	1.01564e13	0.398863	0.199431	−	0.979912i	$-0.436091\pi$
$953$	1.01564e13	0.398863	0.199431	+	0.979912i	$0.436091\pi$
$954$	0	0
$955$	4.81308e12	0.187244
$956$	0	0
$957$	−4.31344e12	−0.166234
$958$	0	0
$959$	−5.41406e12	−0.206700
$960$	0	0
$961$	5.00429e13	1.89272
$962$	0	0
$963$	−4.95554e13	−1.85683
$964$	0	0
$965$	6.76835e12	0.251252
$966$	0	0
$967$	−1.01682e13	−0.373960	−0.186980	−	0.982364i	$-0.559870\pi$
$967$	−1.01682e13	−0.373960	−0.186980	+	0.982364i	$0.559870\pi$
$968$	0	0
$969$	−2.17893e13	−0.793939
$970$	0	0
$971$	7.38490e12	0.266599	0.133299	−	0.991076i	$-0.457443\pi$
$971$	7.38490e12	0.266599	0.133299	+	0.991076i	$0.457443\pi$
$972$	0	0
$973$	−5.30585e12	−0.189779
$974$	0	0
$975$	−3.69993e13	−1.31121
$976$	0	0
$977$	−2.59982e13	−0.912889	−0.456444	−	0.889752i	$-0.650877\pi$
$977$	−2.59982e13	−0.912889	−0.456444	+	0.889752i	$0.650877\pi$
$978$	0	0
$979$	−5.78008e12	−0.201100
$980$	0	0
$981$	2.31404e13	0.797739
$982$	0	0
$983$	1.44993e13	0.495287	0.247643	−	0.968851i	$-0.420344\pi$
$983$	1.44993e13	0.495287	0.247643	+	0.968851i	$0.420344\pi$
$984$	0	0
$985$	4.15416e12	0.140611
$986$	0	0
$987$	−1.94873e13	−0.653618
$988$	0	0
$989$	−5.82037e12	−0.193449
$990$	0	0
$991$	2.21059e13	0.728077	0.364039	−	0.931384i	$-0.381398\pi$
$991$	2.21059e13	0.728077	0.364039	+	0.931384i	$0.381398\pi$
$992$	0	0
$993$	−1.57245e13	−0.513222
$994$	0	0
$995$	3.69606e12	0.119546
$996$	0	0
$997$	−4.74598e13	−1.52124	−0.760620	−	0.649198i	$-0.775105\pi$
$997$	−4.74598e13	−1.52124	−0.760620	+	0.649198i	$0.775105\pi$
$998$	0	0
$999$	1.70246e14	5.40796

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	112.10.a.i.1.3		3
4.3	odd	2		56.10.a.a.1.1	✓	3
28.27	even	2		392.10.a.d.1.3		3

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
56.10.a.a.1.1	✓	3	4.3	odd	2
112.10.a.i.1.3		3	1.1	even	1	trivial
392.10.a.d.1.3		3	28.27	even	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:	\( k \)	\(=\)	\( 10 \)
Character orbit:	\([\chi]\)	\(=\)	112.a (trivial)

\(f(q)\)	\(=\)	\(q+273.944 q^{3} -187.226 q^{5} +2401.00 q^{7} +55362.2 q^{9} -6322.34 q^{11} +70415.3 q^{13} -51289.4 q^{15} +164999. q^{17} -482060. q^{19} +657739. q^{21} +571132. q^{23} -1.91807e6 q^{25} +9.77410e6 q^{27} +2.49049e6 q^{29} +8.74543e6 q^{31} -1.73196e6 q^{33} -449529. q^{35} +1.74181e7 q^{37} +1.92898e7 q^{39} -3.27790e7 q^{41} -1.01909e7 q^{43} -1.03652e7 q^{45} -2.96277e7 q^{47} +5.76480e6 q^{49} +4.52005e7 q^{51} +5.85308e7 q^{53} +1.18370e6 q^{55} -1.32057e8 q^{57} +1.04666e8 q^{59} -2.06399e8 q^{61} +1.32925e8 q^{63} -1.31836e7 q^{65} -1.79752e8 q^{67} +1.56458e8 q^{69} +1.16985e8 q^{71} +2.81318e8 q^{73} -5.25444e8 q^{75} -1.51799e7 q^{77} +3.44616e8 q^{79} +1.58786e9 q^{81} +2.85676e8 q^{83} -3.08921e7 q^{85} +6.82254e8 q^{87} +9.14231e8 q^{89} +1.69067e8 q^{91} +2.39576e9 q^{93} +9.02541e7 q^{95} -1.47825e9 q^{97} -3.50018e8 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 3 q + 92 q^{3} + 274 q^{5} + 7203 q^{7} + 51871 q^{9} + 45364 q^{11} - 11158 q^{13} + 95584 q^{15} - 55866 q^{17} - 488772 q^{19} + 220892 q^{21} + 253888 q^{23} - 3872619 q^{25} + 10157192 q^{27} - 765318 q^{29}+ \cdots - 168732636 q^{99}+O(q^{100}) \) 3 * q + 92 * q^3 + 274 * q^5 + 7203 * q^7 + 51871 * q^9 + 45364 * q^11 - 11158 * q^13 + 95584 * q^15 - 55866 * q^17 - 488772 * q^19 + 220892 * q^21 + 253888 * q^23 - 3872619 * q^25 + 10157192 * q^27 - 765318 * q^29 + 8189680 * q^31 - 7926240 * q^33 + 657874 * q^35 + 7208194 * q^37 + 44623664 * q^39 - 57891986 * q^41 + 37304708 * q^43 - 45527158 * q^45 + 29210992 * q^47 + 17294403 * q^49 + 142742952 * q^51 + 45140082 * q^53 - 1479800 * q^55 - 77053192 * q^57 + 205334804 * q^59 - 109264870 * q^61 + 124542271 * q^63 + 143056644 * q^65 + 52322348 * q^67 + 425002688 * q^69 + 150048136 * q^71 + 761429998 * q^73 - 260540684 * q^75 + 108918964 * q^77 + 214389152 * q^79 + 1527489907 * q^81 + 36110892 * q^83 + 675044052 * q^85 + 236662680 * q^87 + 1319848398 * q^89 - 26790358 * q^91 + 2846904208 * q^93 + 620260624 * q^95 + 101766310 * q^97 - 168732636 * q^99

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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