LMFDB - Embedded newform 232.4.a.d.1.5

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [232,4,Mod(1,232)]

mf = mfinit([N,k,chi],0)

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(232, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([0, 0, 0]))

N = Newforms(chi, 4, names="a")

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

chi := DirichletCharacter("232.1");

S:= CuspForms(chi, 4);

N := Newforms(S);

Level:	$ N $	$=$	$ 232 = 2^{3} \cdot 29 $
Weight:	$ k $	$=$	$ 4 $
Character orbit:	$[\chi]$	$=$	232.a (trivial)

Newform invariants

comment: select newform

sage: f = N[0] # Warning: the index may be different

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

Self dual:	yes
Analytic conductor:	$13.6884431213$
Analytic rank:	$0$
Dimension:	$5$
Coefficient field:	$\mathbb{Q}[x]/(x^{5} - \cdots)$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{5} - x^{4} - 34x^{3} + 74x^{2} + 94x - 198 $ x^5 - x^4 - 34x^3 + 74x^2 + 94*x - 198
Coefficient ring:	$\Z[a_1, \ldots, a_{7}]$
Coefficient ring index:	$ 2^{3} $
Twist minimal:	yes
Fricke sign:	$1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.5
Root			$4.30242$ of defining polynomial
Character	$\chi$	$=$	232.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+7.82921 q^{3} -7.04208 q^{5} +14.1473 q^{7} +34.2965 q^{9} +O(q^{10})$	$q+7.82921 q^{3} -7.04208 q^{5} +14.1473 q^{7} +34.2965 q^{9} -16.9982 q^{11} +62.0351 q^{13} -55.1339 q^{15} +104.842 q^{17} -9.38323 q^{19} +110.762 q^{21} +173.620 q^{23} -75.4091 q^{25} +57.1255 q^{27} +29.0000 q^{29} -28.2215 q^{31} -133.082 q^{33} -99.6261 q^{35} -171.527 q^{37} +485.685 q^{39} +92.0027 q^{41} -519.014 q^{43} -241.518 q^{45} +109.612 q^{47} -142.855 q^{49} +820.830 q^{51} +140.553 q^{53} +119.703 q^{55} -73.4632 q^{57} +353.254 q^{59} -185.355 q^{61} +485.201 q^{63} -436.856 q^{65} +574.840 q^{67} +1359.30 q^{69} +535.054 q^{71} -508.723 q^{73} -590.393 q^{75} -240.478 q^{77} -61.0156 q^{79} -478.757 q^{81} -826.753 q^{83} -738.306 q^{85} +227.047 q^{87} -938.445 q^{89} +877.626 q^{91} -220.952 q^{93} +66.0775 q^{95} -888.766 q^{97} -582.978 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 5 q + 4 q^{3} + 10 q^{5} + 32 q^{7} + 29 q^{9}+O(q^{10}) $ 5 * q + 4 * q^3 + 10 * q^5 + 32 * q^7 + 29 * q^9	$ 5 q + 4 q^{3} + 10 q^{5} + 32 q^{7} + 29 q^{9} + 36 q^{11} + 26 q^{13} + 88 q^{15} + 82 q^{17} + 156 q^{19} + 72 q^{21} + 336 q^{23} + 151 q^{25} + 352 q^{27} + 145 q^{29} + 432 q^{31} + 108 q^{33} + 600 q^{35} - 18 q^{37} + 688 q^{39} + 82 q^{41} + 340 q^{43} - 146 q^{45} + 680 q^{47} - 115 q^{49} + 608 q^{51} - 102 q^{53} + 736 q^{55} - 576 q^{57} + 924 q^{59} - 618 q^{61} + 584 q^{63} - 704 q^{65} + 44 q^{67} - 1056 q^{69} + 1032 q^{71} - 1078 q^{73} - 468 q^{75} - 888 q^{77} + 200 q^{79} - 1843 q^{81} + 452 q^{83} - 1700 q^{85} + 116 q^{87} - 1790 q^{89} - 1128 q^{91} - 1884 q^{93} + 1024 q^{95} - 2518 q^{97} - 1500 q^{99}+O(q^{100}) $ 5 * q + 4 * q^3 + 10 * q^5 + 32 * q^7 + 29 * q^9 + 36 * q^11 + 26 * q^13 + 88 * q^15 + 82 * q^17 + 156 * q^19 + 72 * q^21 + 336 * q^23 + 151 * q^25 + 352 * q^27 + 145 * q^29 + 432 * q^31 + 108 * q^33 + 600 * q^35 - 18 * q^37 + 688 * q^39 + 82 * q^41 + 340 * q^43 - 146 * q^45 + 680 * q^47 - 115 * q^49 + 608 * q^51 - 102 * q^53 + 736 * q^55 - 576 * q^57 + 924 * q^59 - 618 * q^61 + 584 * q^63 - 704 * q^65 + 44 * q^67 - 1056 * q^69 + 1032 * q^71 - 1078 * q^73 - 468 * q^75 - 888 * q^77 + 200 * q^79 - 1843 * q^81 + 452 * q^83 - 1700 * q^85 + 116 * q^87 - 1790 * q^89 - 1128 * q^91 - 1884 * q^93 + 1024 * q^95 - 2518 * q^97 - 1500 * q^99

Display 100 coefficients 10 coefficients

Coefficient data

For each $n$ we display the coefficients of the $q$-expansion $a_n$, the Satake parameters $\alpha_p$, and the Satake angles $\theta_p = \textrm{Arg}(\alpha_p)$.

Display $a_p$ with $p$ up to: 50 250 1000 (See $a_n$ instead) (See $a_n$ instead) (See $a_n$ instead) Display $a_n$ with $n$ up to: 50 250 1000 (See only $a_p$) (See only $a_p$) (See only $a_p$)

$n$	$a_n$	$a_n / n^{(k-1)/2}$	$ \alpha_n $			$ \theta_n $
$p$	$a_p$	$a_p / p^{(k-1)/2}$	$ \alpha_p$			$ \theta_p $

$2$	0	0
$2$	0	0
$3$	7.82921	1.50673	0.753366	−	0.657602i	$-0.228429\pi$
$3$	7.82921	1.50673	0.753366	+	0.657602i	$0.228429\pi$
$4$	0	0
$5$	−7.04208	−0.629863	−0.314931	−	0.949114i	$-0.601982\pi$
$5$	−7.04208	−0.629863	−0.314931	+	0.949114i	$0.601982\pi$
$6$	0	0
$7$	14.1473	0.763880	0.381940	−	0.924187i	$-0.375256\pi$
$7$	14.1473	0.763880	0.381940	+	0.924187i	$0.375256\pi$
$8$	0	0
$9$	34.2965	1.27024
$10$	0	0
$11$	−16.9982	−0.465922	−0.232961	−	0.972486i	$-0.574842\pi$
$11$	−16.9982	−0.465922	−0.232961	+	0.972486i	$0.574842\pi$
$12$	0	0
$13$	62.0351	1.32349	0.661747	−	0.749727i	$-0.269815\pi$
$13$	62.0351	1.32349	0.661747	+	0.749727i	$0.269815\pi$
$14$	0	0
$15$	−55.1339	−0.949034
$16$	0	0
$17$	104.842	1.49576	0.747881	−	0.663833i	$-0.231071\pi$
$17$	104.842	1.49576	0.747881	+	0.663833i	$0.231071\pi$
$18$	0	0
$19$	−9.38323	−0.113298	−0.0566490	−	0.998394i	$-0.518042\pi$
$19$	−9.38323	−0.113298	−0.0566490	+	0.998394i	$0.518042\pi$
$20$	0	0
$21$	110.762	1.15096
$22$	0	0
$23$	173.620	1.57401	0.787005	−	0.616947i	$-0.211631\pi$
$23$	173.620	1.57401	0.787005	+	0.616947i	$0.211631\pi$
$24$	0	0
$25$	−75.4091	−0.603273
$26$	0	0
$27$	57.1255	0.407178
$28$	0	0
$29$	29.0000	0.185695
$29$	29.0000	0.185695
$30$	0	0
$31$	−28.2215	−0.163507	−0.0817536	−	0.996653i	$-0.526052\pi$
$31$	−28.2215	−0.163507	−0.0817536	+	0.996653i	$0.526052\pi$
$32$	0	0
$33$	−133.082	−0.702020
$34$	0	0
$35$	−99.6261	−0.481140
$36$	0	0
$37$	−171.527	−0.762130	−0.381065	−	0.924548i	$-0.624443\pi$
$37$	−171.527	−0.762130	−0.381065	+	0.924548i	$0.624443\pi$
$38$	0	0
$39$	485.685	1.99415
$40$	0	0
$41$	92.0027	0.350449	0.175225	−	0.984529i	$-0.443935\pi$
$41$	92.0027	0.350449	0.175225	+	0.984529i	$0.443935\pi$
$42$	0	0
$43$	−519.014	−1.84067	−0.920335	−	0.391131i	$-0.872084\pi$
$43$	−519.014	−1.84067	−0.920335	+	0.391131i	$0.872084\pi$
$44$	0	0
$45$	−241.518	−0.800077
$46$	0	0
$47$	109.612	0.340182	0.170091	−	0.985428i	$-0.445594\pi$
$47$	109.612	0.340182	0.170091	+	0.985428i	$0.445594\pi$
$48$	0	0
$49$	−142.855	−0.416487
$50$	0	0
$51$	820.830	2.25371
$52$	0	0
$53$	140.553	0.364273	0.182136	−	0.983273i	$-0.441699\pi$
$53$	140.553	0.364273	0.182136	+	0.983273i	$0.441699\pi$
$54$	0	0
$55$	119.703	0.293467
$56$	0	0
$57$	−73.4632	−0.170710
$58$	0	0
$59$	353.254	0.779487	0.389744	−	0.920923i	$-0.372564\pi$
$59$	353.254	0.779487	0.389744	+	0.920923i	$0.372564\pi$
$60$	0	0
$61$	−185.355	−0.389053	−0.194526	−	0.980897i	$-0.562317\pi$
$61$	−185.355	−0.389053	−0.194526	+	0.980897i	$0.562317\pi$
$62$	0	0
$63$	485.201	0.970311
$64$	0	0
$65$	−436.856	−0.833620
$66$	0	0
$67$	574.840	1.04818	0.524088	−	0.851664i	$-0.324406\pi$
$67$	574.840	1.04818	0.524088	+	0.851664i	$0.324406\pi$
$68$	0	0
$69$	1359.30	2.37161
$70$	0	0
$71$	535.054	0.894355	0.447177	−	0.894445i	$-0.352429\pi$
$71$	535.054	0.894355	0.447177	+	0.894445i	$0.352429\pi$
$72$	0	0
$73$	−508.723	−0.815637	−0.407818	−	0.913063i	$-0.633710\pi$
$73$	−508.723	−0.815637	−0.407818	+	0.913063i	$0.633710\pi$
$74$	0	0
$75$	−590.393	−0.908970
$76$	0	0
$77$	−240.478	−0.355909
$78$	0	0
$79$	−61.0156	−0.0868961	−0.0434481	−	0.999056i	$-0.513834\pi$
$79$	−61.0156	−0.0868961	−0.0434481	+	0.999056i	$0.513834\pi$
$80$	0	0
$81$	−478.757	−0.656731
$82$	0	0
$83$	−826.753	−1.09335	−0.546674	−	0.837345i	$-0.684106\pi$
$83$	−826.753	−1.09335	−0.546674	+	0.837345i	$0.684106\pi$
$84$	0	0
$85$	−738.306	−0.942124
$86$	0	0
$87$	227.047	0.279793
$88$	0	0
$89$	−938.445	−1.11770	−0.558848	−	0.829270i	$-0.688757\pi$
$89$	−938.445	−1.11770	−0.558848	+	0.829270i	$0.688757\pi$
$90$	0	0
$91$	877.626	1.01099
$92$	0	0
$93$	−220.952	−0.246361
$94$	0	0
$95$	66.0775	0.0713621
$96$	0	0
$97$	−888.766	−0.930315	−0.465158	−	0.885228i	$-0.654002\pi$
$97$	−888.766	−0.930315	−0.465158	+	0.885228i	$0.654002\pi$
$98$	0	0
$99$	−582.978	−0.591833
$100$	0	0
$101$	−1551.83	−1.52884	−0.764420	−	0.644719i	$-0.776974\pi$
$101$	−1551.83	−1.52884	−0.764420	+	0.644719i	$0.776974\pi$
$102$	0	0
$103$	−402.438	−0.384984	−0.192492	−	0.981299i	$-0.561657\pi$
$103$	−402.438	−0.384984	−0.192492	+	0.981299i	$0.561657\pi$
$104$	0	0
$105$	−779.994	−0.724948
$106$	0	0
$107$	−821.553	−0.742267	−0.371133	−	0.928580i	$-0.621031\pi$
$107$	−821.553	−0.742267	−0.371133	+	0.928580i	$0.621031\pi$
$108$	0	0
$109$	803.052	0.705674	0.352837	−	0.935685i	$-0.385217\pi$
$109$	803.052	0.705674	0.352837	+	0.935685i	$0.385217\pi$
$110$	0	0
$111$	−1342.92	−1.14832
$112$	0	0
$113$	−735.506	−0.612306	−0.306153	−	0.951982i	$-0.599042\pi$
$113$	−735.506	−0.612306	−0.306153	+	0.951982i	$0.599042\pi$
$114$	0	0
$115$	−1222.64	−0.991410
$116$	0	0
$117$	2127.58	1.68116
$118$	0	0
$119$	1483.23	1.14258
$120$	0	0
$121$	−1042.06	−0.782916
$122$	0	0
$123$	720.308	0.528033
$124$	0	0
$125$	1411.30	1.00984
$126$	0	0
$127$	−1895.16	−1.32416	−0.662078	−	0.749435i	$-0.730325\pi$
$127$	−1895.16	−1.32416	−0.662078	+	0.749435i	$0.730325\pi$
$128$	0	0
$129$	−4063.46	−2.77340
$130$	0	0
$131$	−1145.54	−0.764020	−0.382010	−	0.924158i	$-0.624768\pi$
$131$	−1145.54	−0.764020	−0.382010	+	0.924158i	$0.624768\pi$
$132$	0	0
$133$	−132.747	−0.0865461
$134$	0	0
$135$	−402.283	−0.256466
$136$	0	0
$137$	2871.41	1.79067	0.895334	−	0.445396i	$-0.146937\pi$
$137$	2871.41	1.79067	0.895334	+	0.445396i	$0.146937\pi$
$138$	0	0
$139$	777.734	0.474579	0.237290	−	0.971439i	$-0.423741\pi$
$139$	777.734	0.474579	0.237290	+	0.971439i	$0.423741\pi$
$140$	0	0
$141$	858.174	0.512562
$142$	0	0
$143$	−1054.48	−0.616646
$144$	0	0
$145$	−204.220	−0.116963
$146$	0	0
$147$	−1118.44	−0.627534
$148$	0	0
$149$	2373.29	1.30488	0.652440	−	0.757840i	$-0.273745\pi$
$149$	2373.29	1.30488	0.652440	+	0.757840i	$0.273745\pi$
$150$	0	0
$151$	102.190	0.0550733	0.0275366	−	0.999621i	$-0.491234\pi$
$151$	102.190	0.0550733	0.0275366	+	0.999621i	$0.491234\pi$
$152$	0	0
$153$	3595.71	1.89998
$154$	0	0
$155$	198.738	0.102987
$156$	0	0
$157$	585.462	0.297611	0.148805	−	0.988866i	$-0.452457\pi$
$157$	585.462	0.297611	0.148805	+	0.988866i	$0.452457\pi$
$158$	0	0
$159$	1100.42	0.548861
$160$	0	0
$161$	2456.24	1.20235
$162$	0	0
$163$	1440.36	0.692131	0.346066	−	0.938210i	$-0.387517\pi$
$163$	1440.36	0.692131	0.346066	+	0.938210i	$0.387517\pi$
$164$	0	0
$165$	937.176	0.442176
$166$	0	0
$167$	3338.00	1.54672	0.773359	−	0.633968i	$-0.218575\pi$
$167$	3338.00	1.54672	0.773359	+	0.633968i	$0.218575\pi$
$168$	0	0
$169$	1651.35	0.751638
$170$	0	0
$171$	−321.812	−0.143916
$172$	0	0
$173$	−3612.07	−1.58740	−0.793701	−	0.608308i	$-0.791849\pi$
$173$	−3612.07	−1.58740	−0.793701	+	0.608308i	$0.791849\pi$
$174$	0	0
$175$	−1066.83	−0.460828
$176$	0	0
$177$	2765.70	1.17448
$178$	0	0
$179$	−2415.54	−1.00864	−0.504319	−	0.863517i	$-0.668256\pi$
$179$	−2415.54	−1.00864	−0.504319	+	0.863517i	$0.668256\pi$
$180$	0	0
$181$	−1755.89	−0.721073	−0.360537	−	0.932745i	$-0.617406\pi$
$181$	−1755.89	−0.721073	−0.360537	+	0.932745i	$0.617406\pi$
$182$	0	0
$183$	−1451.18	−0.586198
$184$	0	0
$185$	1207.90	0.480037
$186$	0	0
$187$	−1782.13	−0.696909
$188$	0	0
$189$	808.170	0.311035
$190$	0	0
$191$	1156.10	0.437972	0.218986	−	0.975728i	$-0.429725\pi$
$191$	1156.10	0.437972	0.218986	+	0.975728i	$0.429725\pi$
$192$	0	0
$193$	−3214.73	−1.19897	−0.599485	−	0.800386i	$-0.704628\pi$
$193$	−3214.73	−1.19897	−0.599485	+	0.800386i	$0.704628\pi$
$194$	0	0
$195$	−3420.23	−1.25604
$196$	0	0
$197$	−3924.03	−1.41916	−0.709582	−	0.704623i	$-0.751116\pi$
$197$	−3924.03	−1.41916	−0.709582	+	0.704623i	$0.751116\pi$
$198$	0	0
$199$	4463.80	1.59010	0.795051	−	0.606543i	$-0.207444\pi$
$199$	4463.80	1.59010	0.795051	+	0.606543i	$0.207444\pi$
$200$	0	0
$201$	4500.54	1.57932
$202$	0	0
$203$	410.271	0.141849
$204$	0	0
$205$	−647.890	−0.220735
$206$	0	0
$207$	5954.54	1.99937
$208$	0	0
$209$	159.498	0.0527881
$210$	0	0
$211$	4881.35	1.59263	0.796317	−	0.604880i	$-0.206779\pi$
$211$	4881.35	1.59263	0.796317	+	0.604880i	$0.206779\pi$
$212$	0	0
$213$	4189.05	1.34755
$214$	0	0
$215$	3654.93	1.15937
$216$	0	0
$217$	−399.256	−0.124900
$218$	0	0
$219$	−3982.90	−1.22895
$220$	0	0
$221$	6503.89	1.97963
$222$	0	0
$223$	−1533.22	−0.460414	−0.230207	−	0.973142i	$-0.573940\pi$
$223$	−1533.22	−0.460414	−0.230207	+	0.973142i	$0.573940\pi$
$224$	0	0
$225$	−2586.27	−0.766301
$226$	0	0
$227$	−3110.06	−0.909348	−0.454674	−	0.890658i	$-0.650244\pi$
$227$	−3110.06	−0.909348	−0.454674	+	0.890658i	$0.650244\pi$
$228$	0	0
$229$	−4686.03	−1.35223	−0.676117	−	0.736794i	$-0.736339\pi$
$229$	−4686.03	−1.35223	−0.676117	+	0.736794i	$0.736339\pi$
$230$	0	0
$231$	−1882.75	−0.536259
$232$	0	0
$233$	−908.748	−0.255511	−0.127756	−	0.991806i	$-0.540777\pi$
$233$	−908.748	−0.255511	−0.127756	+	0.991806i	$0.540777\pi$
$234$	0	0
$235$	−771.896	−0.214268
$236$	0	0
$237$	−477.704	−0.130929
$238$	0	0
$239$	−18.3950	−0.00497854	−0.00248927	−	0.999997i	$-0.500792\pi$
$239$	−18.3950	−0.00497854	−0.00248927	+	0.999997i	$0.500792\pi$
$240$	0	0
$241$	6199.96	1.65716	0.828578	−	0.559874i	$-0.189151\pi$
$241$	6199.96	1.65716	0.828578	+	0.559874i	$0.189151\pi$
$242$	0	0
$243$	−5290.68	−1.39670
$244$	0	0
$245$	1006.00	0.262330
$246$	0	0
$247$	−582.089	−0.149949
$248$	0	0
$249$	−6472.82	−1.64738
$250$	0	0
$251$	3325.41	0.836248	0.418124	−	0.908390i	$-0.362688\pi$
$251$	3325.41	0.836248	0.418124	+	0.908390i	$0.362688\pi$
$252$	0	0
$253$	−2951.22	−0.733366
$254$	0	0
$255$	−5780.35	−1.41953
$256$	0	0
$257$	2713.20	0.658541	0.329271	−	0.944236i	$-0.393197\pi$
$257$	2713.20	0.658541	0.329271	+	0.944236i	$0.393197\pi$
$258$	0	0
$259$	−2426.63	−0.582176
$260$	0	0
$261$	994.597	0.235878
$262$	0	0
$263$	6234.94	1.46184	0.730918	−	0.682465i	$-0.239092\pi$
$263$	6234.94	1.46184	0.730918	+	0.682465i	$0.239092\pi$
$264$	0	0
$265$	−989.787	−0.229442
$266$	0	0
$267$	−7347.28	−1.68407
$268$	0	0
$269$	−5745.23	−1.30220	−0.651102	−	0.758990i	$-0.725693\pi$
$269$	−5745.23	−1.30220	−0.651102	+	0.758990i	$0.725693\pi$
$270$	0	0
$271$	8855.53	1.98500	0.992501	−	0.122238i	$-0.0390072\pi$
$271$	8855.53	1.98500	0.992501	+	0.122238i	$0.0390072\pi$
$272$	0	0
$273$	6871.12	1.52329
$274$	0	0
$275$	1281.82	0.281078
$276$	0	0
$277$	−2805.71	−0.608588	−0.304294	−	0.952578i	$-0.598420\pi$
$277$	−2805.71	−0.608588	−0.304294	+	0.952578i	$0.598420\pi$
$278$	0	0
$279$	−967.896	−0.207693
$280$	0	0
$281$	5133.23	1.08976	0.544881	−	0.838514i	$-0.316575\pi$
$281$	5133.23	1.08976	0.544881	+	0.838514i	$0.316575\pi$
$282$	0	0
$283$	2928.86	0.615204	0.307602	−	0.951515i	$-0.400473\pi$
$283$	2928.86	0.615204	0.307602	+	0.951515i	$0.400473\pi$
$284$	0	0
$285$	517.334	0.107524
$286$	0	0
$287$	1301.59	0.267701
$288$	0	0
$289$	6078.87	1.23730
$290$	0	0
$291$	−6958.34	−1.40173
$292$	0	0
$293$	5685.37	1.13359	0.566796	−	0.823858i	$-0.308183\pi$
$293$	5685.37	1.13359	0.566796	+	0.823858i	$0.308183\pi$
$294$	0	0
$295$	−2487.64	−0.490970
$296$	0	0
$297$	−971.031	−0.189713
$298$	0	0
$299$	10770.5	2.08319
$300$	0	0
$301$	−7342.62	−1.40605
$302$	0	0
$303$	−12149.6	−2.30355
$304$	0	0
$305$	1305.28	0.245050
$306$	0	0
$307$	2879.21	0.535262	0.267631	−	0.963522i	$-0.413759\pi$
$307$	2879.21	0.535262	0.267631	+	0.963522i	$0.413759\pi$
$308$	0	0
$309$	−3150.77	−0.580068
$310$	0	0
$311$	513.825	0.0936860	0.0468430	−	0.998902i	$-0.485084\pi$
$311$	513.825	0.0936860	0.0468430	+	0.998902i	$0.485084\pi$
$312$	0	0
$313$	−1508.76	−0.272460	−0.136230	−	0.990677i	$-0.543499\pi$
$313$	−1508.76	−0.272460	−0.136230	+	0.990677i	$0.543499\pi$
$314$	0	0
$315$	−3416.82	−0.611163
$316$	0	0
$317$	5411.89	0.958872	0.479436	−	0.877577i	$-0.340841\pi$
$317$	5411.89	0.958872	0.479436	+	0.877577i	$0.340841\pi$
$318$	0	0
$319$	−492.947	−0.0865196
$320$	0	0
$321$	−6432.11	−1.11840
$322$	0	0
$323$	−983.758	−0.169467
$324$	0	0
$325$	−4678.01	−0.798429
$326$	0	0
$327$	6287.26	1.06326
$328$	0	0
$329$	1550.71	0.259858
$330$	0	0
$331$	10354.1	1.71938	0.859688	−	0.510820i	$-0.170658\pi$
$331$	10354.1	1.71938	0.859688	+	0.510820i	$0.170658\pi$
$332$	0	0
$333$	−5882.75	−0.968087
$334$	0	0
$335$	−4048.07	−0.660208
$336$	0	0
$337$	4707.20	0.760882	0.380441	−	0.924805i	$-0.375772\pi$
$337$	4707.20	0.760882	0.380441	+	0.924805i	$0.375772\pi$
$338$	0	0
$339$	−5758.43	−0.922581
$340$	0	0
$341$	479.714	0.0761817
$342$	0	0
$343$	−6873.52	−1.08203
$344$	0	0
$345$	−9572.33	−1.49379
$346$	0	0
$347$	−6733.27	−1.04167	−0.520837	−	0.853656i	$-0.674380\pi$
$347$	−6733.27	−1.04167	−0.520837	+	0.853656i	$0.674380\pi$
$348$	0	0
$349$	−9272.50	−1.42219	−0.711097	−	0.703094i	$-0.751801\pi$
$349$	−9272.50	−1.42219	−0.711097	+	0.703094i	$0.751801\pi$
$350$	0	0
$351$	3543.79	0.538898
$352$	0	0
$353$	5776.97	0.871041	0.435520	−	0.900179i	$-0.356564\pi$
$353$	5776.97	0.871041	0.435520	+	0.900179i	$0.356564\pi$
$354$	0	0
$355$	−3767.89	−0.563321
$356$	0	0
$357$	11612.5	1.72157
$358$	0	0
$359$	3821.93	0.561877	0.280938	−	0.959726i	$-0.409354\pi$
$359$	3821.93	0.561877	0.280938	+	0.959726i	$0.409354\pi$
$360$	0	0
$361$	−6770.95	−0.987164
$362$	0	0
$363$	−8158.51	−1.17964
$364$	0	0
$365$	3582.47	0.513739
$366$	0	0
$367$	−9481.16	−1.34854	−0.674268	−	0.738487i	$-0.735541\pi$
$367$	−9481.16	−1.34854	−0.674268	+	0.738487i	$0.735541\pi$
$368$	0	0
$369$	3155.37	0.445154
$370$	0	0
$371$	1988.44	0.278261
$372$	0	0
$373$	−11849.6	−1.64490	−0.822450	−	0.568837i	$-0.807394\pi$
$373$	−11849.6	−1.64490	−0.822450	+	0.568837i	$0.807394\pi$
$374$	0	0
$375$	11049.3	1.52156
$376$	0	0
$377$	1799.02	0.245767
$378$	0	0
$379$	−14650.4	−1.98560	−0.992799	−	0.119794i	$-0.961776\pi$
$379$	−14650.4	−1.98560	−0.992799	+	0.119794i	$0.961776\pi$
$380$	0	0
$381$	−14837.6	−1.99515
$382$	0	0
$383$	−2202.45	−0.293837	−0.146919	−	0.989149i	$-0.546936\pi$
$383$	−2202.45	−0.293837	−0.146919	+	0.989149i	$0.546936\pi$
$384$	0	0
$385$	1693.46	0.224174
$386$	0	0
$387$	−17800.3	−2.33809
$388$	0	0
$389$	5614.61	0.731804	0.365902	−	0.930653i	$-0.380760\pi$
$389$	5614.61	0.731804	0.365902	+	0.930653i	$0.380760\pi$
$390$	0	0
$391$	18202.7	2.35434
$392$	0	0
$393$	−8968.70	−1.15117
$394$	0	0
$395$	429.677	0.0547326
$396$	0	0
$397$	1672.32	0.211414	0.105707	−	0.994397i	$-0.466289\pi$
$397$	1672.32	0.211414	0.105707	+	0.994397i	$0.466289\pi$
$398$	0	0
$399$	−1039.30	−0.130402
$400$	0	0
$401$	−1183.01	−0.147324	−0.0736619	−	0.997283i	$-0.523469\pi$
$401$	−1183.01	−0.147324	−0.0736619	+	0.997283i	$0.523469\pi$
$402$	0	0
$403$	−1750.72	−0.216401
$404$	0	0
$405$	3371.45	0.413651
$406$	0	0
$407$	2915.64	0.355093
$408$	0	0
$409$	−6264.74	−0.757388	−0.378694	−	0.925522i	$-0.623627\pi$
$409$	−6264.74	−0.757388	−0.378694	+	0.925522i	$0.623627\pi$
$410$	0	0
$411$	22480.9	2.69805
$412$	0	0
$413$	4997.58	0.595435
$414$	0	0
$415$	5822.06	0.688659
$416$	0	0
$417$	6089.04	0.715063
$418$	0	0
$419$	−4224.75	−0.492584	−0.246292	−	0.969196i	$-0.579212\pi$
$419$	−4224.75	−0.492584	−0.246292	+	0.969196i	$0.579212\pi$
$420$	0	0
$421$	10801.7	1.25046	0.625231	−	0.780440i	$-0.285005\pi$
$421$	10801.7	1.25046	0.625231	+	0.780440i	$0.285005\pi$
$422$	0	0
$423$	3759.30	0.432112
$424$	0	0
$425$	−7906.05	−0.902352
$426$	0	0
$427$	−2622.26	−0.297190
$428$	0	0
$429$	−8255.77	−0.929120
$430$	0	0
$431$	13127.7	1.46715	0.733574	−	0.679610i	$-0.237851\pi$
$431$	13127.7	1.46715	0.733574	+	0.679610i	$0.237851\pi$
$432$	0	0
$433$	−11403.2	−1.26559	−0.632797	−	0.774318i	$-0.718093\pi$
$433$	−11403.2	−1.26559	−0.632797	+	0.774318i	$0.718093\pi$
$434$	0	0
$435$	−1598.88	−0.176231
$436$	0	0
$437$	−1629.11	−0.178332
$438$	0	0
$439$	14656.2	1.59339	0.796697	−	0.604378i	$-0.206578\pi$
$439$	14656.2	1.59339	0.796697	+	0.604378i	$0.206578\pi$
$440$	0	0
$441$	−4899.42	−0.529038
$442$	0	0
$443$	6136.95	0.658184	0.329092	−	0.944298i	$-0.393257\pi$
$443$	6136.95	0.658184	0.329092	+	0.944298i	$0.393257\pi$
$444$	0	0
$445$	6608.61	0.703995
$446$	0	0
$447$	18580.9	1.96610
$448$	0	0
$449$	−18012.1	−1.89319	−0.946594	−	0.322428i	$-0.895501\pi$
$449$	−18012.1	−1.89319	−0.946594	+	0.322428i	$0.895501\pi$
$450$	0	0
$451$	−1563.88	−0.163282
$452$	0	0
$453$	800.063	0.0829807
$454$	0	0
$455$	−6180.31	−0.636786
$456$	0	0
$457$	715.440	0.0732316	0.0366158	−	0.999329i	$-0.488342\pi$
$457$	715.440	0.0732316	0.0366158	+	0.999329i	$0.488342\pi$
$458$	0	0
$459$	5989.16	0.609042
$460$	0	0
$461$	−16471.2	−1.66408	−0.832038	−	0.554719i	$-0.812826\pi$
$461$	−16471.2	−1.66408	−0.832038	+	0.554719i	$0.812826\pi$
$462$	0	0
$463$	3844.07	0.385851	0.192925	−	0.981213i	$-0.438202\pi$
$463$	3844.07	0.385851	0.192925	+	0.981213i	$0.438202\pi$
$464$	0	0
$465$	1555.96	0.155174
$466$	0	0
$467$	15072.5	1.49352	0.746759	−	0.665094i	$-0.231609\pi$
$467$	15072.5	1.49352	0.746759	+	0.665094i	$0.231609\pi$
$468$	0	0
$469$	8132.41	0.800682
$470$	0	0
$471$	4583.70	0.448420
$472$	0	0
$473$	8822.29	0.857610
$474$	0	0
$475$	707.581	0.0683496
$476$	0	0
$477$	4820.48	0.462714
$478$	0	0
$479$	8856.14	0.844776	0.422388	−	0.906415i	$-0.361192\pi$
$479$	8856.14	0.844776	0.422388	+	0.906415i	$0.361192\pi$
$480$	0	0
$481$	−10640.7	−1.00867
$482$	0	0
$483$	19230.4	1.81163
$484$	0	0
$485$	6258.76	0.585971
$486$	0	0
$487$	6998.94	0.651237	0.325618	−	0.945501i	$-0.394428\pi$
$487$	6998.94	0.651237	0.325618	+	0.945501i	$0.394428\pi$
$488$	0	0
$489$	11276.8	1.04286
$490$	0	0
$491$	−2169.60	−0.199415	−0.0997076	−	0.995017i	$-0.531791\pi$
$491$	−2169.60	−0.199415	−0.0997076	+	0.995017i	$0.531791\pi$
$492$	0	0
$493$	3040.42	0.277756
$494$	0	0
$495$	4105.38	0.372774
$496$	0	0
$497$	7569.55	0.683180
$498$	0	0
$499$	−17541.6	−1.57368	−0.786842	−	0.617155i	$-0.788285\pi$
$499$	−17541.6	−1.57368	−0.786842	+	0.617155i	$0.788285\pi$
$500$	0	0
$501$	26133.9	2.33049
$502$	0	0
$503$	−16190.4	−1.43517	−0.717587	−	0.696469i	$-0.754753\pi$
$503$	−16190.4	−1.43517	−0.717587	+	0.696469i	$0.754753\pi$
$504$	0	0
$505$	10928.1	0.962959
$506$	0	0
$507$	12928.8	1.13252
$508$	0	0
$509$	1493.11	0.130021	0.0650106	−	0.997885i	$-0.479292\pi$
$509$	1493.11	0.130021	0.0650106	+	0.997885i	$0.479292\pi$
$510$	0	0
$511$	−7197.03	−0.623049
$512$	0	0
$513$	−536.022	−0.0461325
$514$	0	0
$515$	2834.00	0.242487
$516$	0	0
$517$	−1863.20	−0.158498
$518$	0	0
$519$	−28279.6	−2.39179
$520$	0	0
$521$	−7026.84	−0.590886	−0.295443	−	0.955360i	$-0.595467\pi$
$521$	−7026.84	−0.590886	−0.295443	+	0.955360i	$0.595467\pi$
$522$	0	0
$523$	21924.1	1.83303	0.916514	−	0.400003i	$-0.130991\pi$
$523$	21924.1	1.83303	0.916514	+	0.400003i	$0.130991\pi$
$524$	0	0
$525$	−8352.45	−0.694345
$526$	0	0
$527$	−2958.80	−0.244568
$528$	0	0
$529$	17976.8	1.47750
$530$	0	0
$531$	12115.4	0.990135
$532$	0	0
$533$	5707.39	0.463817
$534$	0	0
$535$	5785.44	0.467526
$536$	0	0
$537$	−18911.8	−1.51975
$538$	0	0
$539$	2428.28	0.194051
$540$	0	0
$541$	−983.325	−0.0781450	−0.0390725	−	0.999236i	$-0.512440\pi$
$541$	−983.325	−0.0781450	−0.0390725	+	0.999236i	$0.512440\pi$
$542$	0	0
$543$	−13747.2	−1.08646
$544$	0	0
$545$	−5655.16	−0.444478
$546$	0	0
$547$	6100.54	0.476856	0.238428	−	0.971160i	$-0.423368\pi$
$547$	6100.54	0.476856	0.238428	+	0.971160i	$0.423368\pi$
$548$	0	0
$549$	−6357.01	−0.494190
$550$	0	0
$551$	−272.114	−0.0210389
$552$	0	0
$553$	−863.204	−0.0663782
$554$	0	0
$555$	9456.93	0.723287
$556$	0	0
$557$	7531.50	0.572926	0.286463	−	0.958091i	$-0.407520\pi$
$557$	7531.50	0.572926	0.286463	+	0.958091i	$0.407520\pi$
$558$	0	0
$559$	−32197.0	−2.43612
$560$	0	0
$561$	−13952.6	−1.05005
$562$	0	0
$563$	−19553.5	−1.46373	−0.731866	−	0.681449i	$-0.761350\pi$
$563$	−19553.5	−1.46373	−0.731866	+	0.681449i	$0.761350\pi$
$564$	0	0
$565$	5179.49	0.385669
$566$	0	0
$567$	−6773.10	−0.501664
$568$	0	0
$569$	23656.9	1.74297	0.871483	−	0.490426i	$-0.163159\pi$
$569$	23656.9	1.74297	0.871483	+	0.490426i	$0.163159\pi$
$570$	0	0
$571$	−5466.08	−0.400610	−0.200305	−	0.979734i	$-0.564193\pi$
$571$	−5466.08	−0.400610	−0.200305	+	0.979734i	$0.564193\pi$
$572$	0	0
$573$	9051.37	0.659907
$574$	0	0
$575$	−13092.5	−0.949557
$576$	0	0
$577$	−653.065	−0.0471187	−0.0235593	−	0.999722i	$-0.507500\pi$
$577$	−653.065	−0.0471187	−0.0235593	+	0.999722i	$0.507500\pi$
$578$	0	0
$579$	−25168.8	−1.80653
$580$	0	0
$581$	−11696.3	−0.835187
$582$	0	0
$583$	−2389.15	−0.169723
$584$	0	0
$585$	−14982.6	−1.05890
$586$	0	0
$587$	−19414.4	−1.36511	−0.682554	−	0.730835i	$-0.739131\pi$
$587$	−19414.4	−1.36511	−0.682554	+	0.730835i	$0.739131\pi$
$588$	0	0
$589$	264.809	0.0185250
$590$	0	0
$591$	−30722.0	−2.13830
$592$	0	0
$593$	18910.6	1.30955	0.654777	−	0.755822i	$-0.272763\pi$
$593$	18910.6	1.30955	0.654777	+	0.755822i	$0.272763\pi$
$594$	0	0
$595$	−10445.0	−0.719670
$596$	0	0
$597$	34948.0	2.39586
$598$	0	0
$599$	−12128.7	−0.827322	−0.413661	−	0.910431i	$-0.635750\pi$
$599$	−12128.7	−0.827322	−0.413661	+	0.910431i	$0.635750\pi$
$600$	0	0
$601$	−25803.3	−1.75131	−0.875656	−	0.482936i	$-0.839570\pi$
$601$	−25803.3	−1.75131	−0.875656	+	0.482936i	$0.839570\pi$
$602$	0	0
$603$	19715.0	1.33144
$604$	0	0
$605$	7338.28	0.493130
$606$	0	0
$607$	12183.6	0.814690	0.407345	−	0.913274i	$-0.366455\pi$
$607$	12183.6	0.814690	0.407345	+	0.913274i	$0.366455\pi$
$608$	0	0
$609$	3212.09	0.213728
$610$	0	0
$611$	6799.78	0.450229
$612$	0	0
$613$	−16519.2	−1.08843	−0.544213	−	0.838947i	$-0.683171\pi$
$613$	−16519.2	−1.08843	−0.544213	+	0.838947i	$0.683171\pi$
$614$	0	0
$615$	−5072.47	−0.332588
$616$	0	0
$617$	−6518.09	−0.425297	−0.212649	−	0.977129i	$-0.568209\pi$
$617$	−6518.09	−0.425297	−0.212649	+	0.977129i	$0.568209\pi$
$618$	0	0
$619$	−4344.67	−0.282111	−0.141056	−	0.990002i	$-0.545050\pi$
$619$	−4344.67	−0.282111	−0.141056	+	0.990002i	$0.545050\pi$
$620$	0	0
$621$	9918.12	0.640902
$622$	0	0
$623$	−13276.4	−0.853787
$624$	0	0
$625$	−512.326	−0.0327889
$626$	0	0
$627$	1248.74	0.0795374
$628$	0	0
$629$	−17983.2	−1.13996
$630$	0	0
$631$	2989.38	0.188598	0.0942989	−	0.995544i	$-0.469939\pi$
$631$	2989.38	0.188598	0.0942989	+	0.995544i	$0.469939\pi$
$632$	0	0
$633$	38217.1	2.39967
$634$	0	0
$635$	13345.8	0.834037
$636$	0	0
$637$	−8862.02	−0.551218
$638$	0	0
$639$	18350.5	1.13604
$640$	0	0
$641$	15375.4	0.947411	0.473706	−	0.880683i	$-0.342916\pi$
$641$	15375.4	0.947411	0.473706	+	0.880683i	$0.342916\pi$
$642$	0	0
$643$	−16966.5	−1.04058	−0.520291	−	0.853989i	$-0.674177\pi$
$643$	−16966.5	−1.04058	−0.520291	+	0.853989i	$0.674177\pi$
$644$	0	0
$645$	28615.2	1.74686
$646$	0	0
$647$	−12379.7	−0.752234	−0.376117	−	0.926572i	$-0.622741\pi$
$647$	−12379.7	−0.752234	−0.376117	+	0.926572i	$0.622741\pi$
$648$	0	0
$649$	−6004.68	−0.363181
$650$	0	0
$651$	−3125.86	−0.188191
$652$	0	0
$653$	30418.3	1.82291	0.911453	−	0.411403i	$-0.134961\pi$
$653$	30418.3	1.82291	0.911453	+	0.411403i	$0.134961\pi$
$654$	0	0
$655$	8067.01	0.481228
$656$	0	0
$657$	−17447.4	−1.03605
$658$	0	0
$659$	−6069.11	−0.358754	−0.179377	−	0.983780i	$-0.557408\pi$
$659$	−6069.11	−0.358754	−0.179377	+	0.983780i	$0.557408\pi$
$660$	0	0
$661$	−24894.6	−1.46489	−0.732443	−	0.680829i	$-0.761620\pi$
$661$	−24894.6	−1.46489	−0.732443	+	0.680829i	$0.761620\pi$
$662$	0	0
$663$	50920.3	2.98277
$664$	0	0
$665$	934.815	0.0545121
$666$	0	0
$667$	5034.97	0.292286
$668$	0	0
$669$	−12003.9	−0.693720
$670$	0	0
$671$	3150.69	0.181268
$672$	0	0
$673$	21677.5	1.24162	0.620808	−	0.783963i	$-0.286805\pi$
$673$	21677.5	1.24162	0.620808	+	0.783963i	$0.286805\pi$
$674$	0	0
$675$	−4307.79	−0.245640
$676$	0	0
$677$	−21322.2	−1.21046	−0.605228	−	0.796052i	$-0.706918\pi$
$677$	−21322.2	−1.21046	−0.605228	+	0.796052i	$0.706918\pi$
$678$	0	0
$679$	−12573.6	−0.710649
$680$	0	0
$681$	−24349.3	−1.37014
$682$	0	0
$683$	38.7459	0.00217068	0.00108534	−	0.999999i	$-0.499655\pi$
$683$	38.7459	0.00217068	0.00108534	+	0.999999i	$0.499655\pi$
$684$	0	0
$685$	−20220.7	−1.12787
$686$	0	0
$687$	−36687.9	−2.03745
$688$	0	0
$689$	8719.23	0.482113
$690$	0	0
$691$	27812.0	1.53114	0.765571	−	0.643351i	$-0.222456\pi$
$691$	27812.0	1.53114	0.765571	+	0.643351i	$0.222456\pi$
$692$	0	0
$693$	−8247.54	−0.452090
$694$	0	0
$695$	−5476.86	−0.298920
$696$	0	0
$697$	9645.76	0.524188
$698$	0	0
$699$	−7114.78	−0.384987
$700$	0	0
$701$	23404.8	1.26104	0.630519	−	0.776174i	$-0.282842\pi$
$701$	23404.8	1.26104	0.630519	+	0.776174i	$0.282842\pi$
$702$	0	0
$703$	1609.47	0.0863477
$704$	0	0
$705$	−6043.33	−0.322844
$706$	0	0
$707$	−21954.1	−1.16785
$708$	0	0
$709$	8525.03	0.451572	0.225786	−	0.974177i	$-0.427505\pi$
$709$	8525.03	0.451572	0.225786	+	0.974177i	$0.427505\pi$
$710$	0	0
$711$	−2092.62	−0.110379
$712$	0	0
$713$	−4899.80	−0.257362
$714$	0	0
$715$	7425.76	0.388402
$716$	0	0
$717$	−144.018	−0.00750132
$718$	0	0
$719$	−13242.3	−0.686862	−0.343431	−	0.939178i	$-0.611589\pi$
$719$	−13242.3	−0.686862	−0.343431	+	0.939178i	$0.611589\pi$
$720$	0	0
$721$	−5693.39	−0.294082
$722$	0	0
$723$	48540.8	2.49689
$724$	0	0
$725$	−2186.86	−0.112025
$726$	0	0
$727$	5299.22	0.270340	0.135170	−	0.990822i	$-0.456842\pi$
$727$	5299.22	0.270340	0.135170	+	0.990822i	$0.456842\pi$
$728$	0	0
$729$	−28495.4	−1.44771
$730$	0	0
$731$	−54414.5	−2.75320
$732$	0	0
$733$	28433.7	1.43278	0.716388	−	0.697702i	$-0.245794\pi$
$733$	28433.7	1.43278	0.716388	+	0.697702i	$0.245794\pi$
$734$	0	0
$735$	7876.15	0.395260
$736$	0	0
$737$	−9771.23	−0.488369
$738$	0	0
$739$	−19450.7	−0.968209	−0.484105	−	0.875010i	$-0.660855\pi$
$739$	−19450.7	−0.968209	−0.484105	+	0.875010i	$0.660855\pi$
$740$	0	0
$741$	−4557.30	−0.225933
$742$	0	0
$743$	−15260.1	−0.753484	−0.376742	−	0.926318i	$-0.622956\pi$
$743$	−15260.1	−0.753484	−0.376742	+	0.926318i	$0.622956\pi$
$744$	0	0
$745$	−16712.9	−0.821896
$746$	0	0
$747$	−28354.7	−1.38881
$748$	0	0
$749$	−11622.7	−0.567003
$750$	0	0
$751$	23290.8	1.13168	0.565841	−	0.824514i	$-0.308551\pi$
$751$	23290.8	1.13168	0.565841	+	0.824514i	$0.308551\pi$
$752$	0	0
$753$	26035.3	1.26000
$754$	0	0
$755$	−719.627	−0.0346886
$756$	0	0
$757$	−22951.5	−1.10196	−0.550981	−	0.834517i	$-0.685747\pi$
$757$	−22951.5	−1.10196	−0.550981	+	0.834517i	$0.685747\pi$
$758$	0	0
$759$	−23105.7	−1.10499
$760$	0	0
$761$	14975.4	0.713349	0.356675	−	0.934229i	$-0.383910\pi$
$761$	14975.4	0.713349	0.356675	+	0.934229i	$0.383910\pi$
$762$	0	0
$763$	11361.0	0.539050
$764$	0	0
$765$	−25321.3	−1.19672
$766$	0	0
$767$	21914.1	1.03165
$768$	0	0
$769$	30907.7	1.44936	0.724682	−	0.689084i	$-0.241987\pi$
$769$	30907.7	1.44936	0.724682	+	0.689084i	$0.241987\pi$
$770$	0	0
$771$	21242.2	0.992244
$772$	0	0
$773$	−3728.71	−0.173496	−0.0867479	−	0.996230i	$-0.527647\pi$
$773$	−3728.71	−0.173496	−0.0867479	+	0.996230i	$0.527647\pi$
$774$	0	0
$775$	2128.16	0.0986395
$776$	0	0
$777$	−18998.6	−0.877182
$778$	0	0
$779$	−863.283	−0.0397051
$780$	0	0
$781$	−9094.94	−0.416700
$782$	0	0
$783$	1656.64	0.0756111
$784$	0	0
$785$	−4122.87	−0.187454
$786$	0	0
$787$	33291.9	1.50791	0.753957	−	0.656923i	$-0.228142\pi$
$787$	33291.9	1.50791	0.753957	+	0.656923i	$0.228142\pi$
$788$	0	0
$789$	48814.7	2.20260
$790$	0	0
$791$	−10405.4	−0.467729
$792$	0	0
$793$	−11498.5	−0.514909
$794$	0	0
$795$	−7749.24	−0.345707
$796$	0	0
$797$	−36436.3	−1.61937	−0.809687	−	0.586862i	$-0.800363\pi$
$797$	−36436.3	−1.61937	−0.809687	+	0.586862i	$0.800363\pi$
$798$	0	0
$799$	11491.9	0.508831
$800$	0	0
$801$	−32185.4	−1.41974
$802$	0	0
$803$	8647.37	0.380024
$804$	0	0
$805$	−17297.1	−0.757318
$806$	0	0
$807$	−44980.6	−1.96207
$808$	0	0
$809$	9776.41	0.424871	0.212435	−	0.977175i	$-0.431861\pi$
$809$	9776.41	0.424871	0.212435	+	0.977175i	$0.431861\pi$
$810$	0	0
$811$	30799.9	1.33358	0.666788	−	0.745248i	$-0.267669\pi$
$811$	30799.9	1.33358	0.666788	+	0.745248i	$0.267669\pi$
$812$	0	0
$813$	69331.8	2.99086
$814$	0	0
$815$	−10143.1	−0.435948
$816$	0	0
$817$	4870.02	0.208544
$818$	0	0
$819$	30099.5	1.28420
$820$	0	0
$821$	−3080.42	−0.130947	−0.0654735	−	0.997854i	$-0.520856\pi$
$821$	−3080.42	−0.130947	−0.0654735	+	0.997854i	$0.520856\pi$
$822$	0	0
$823$	23048.4	0.976204	0.488102	−	0.872787i	$-0.337689\pi$
$823$	23048.4	0.976204	0.488102	+	0.872787i	$0.337689\pi$
$824$	0	0
$825$	10035.6	0.423510
$826$	0	0
$827$	−13260.3	−0.557565	−0.278783	−	0.960354i	$-0.589931\pi$
$827$	−13260.3	−0.557565	−0.278783	+	0.960354i	$0.589931\pi$
$828$	0	0
$829$	−34413.4	−1.44177	−0.720885	−	0.693055i	$-0.756264\pi$
$829$	−34413.4	−1.44177	−0.720885	+	0.693055i	$0.756264\pi$
$830$	0	0
$831$	−21966.5	−0.916978
$832$	0	0
$833$	−14977.2	−0.622965
$834$	0	0
$835$	−23506.5	−0.974221
$836$	0	0
$837$	−1612.17	−0.0665766
$838$	0	0
$839$	5440.08	0.223853	0.111926	−	0.993717i	$-0.464298\pi$
$839$	5440.08	0.223853	0.111926	+	0.993717i	$0.464298\pi$
$840$	0	0
$841$	841.000	0.0344828
$842$	0	0
$843$	40189.1	1.64198
$844$	0	0
$845$	−11628.9	−0.473429
$846$	0	0
$847$	−14742.3	−0.598054
$848$	0	0
$849$	22930.7	0.926947
$850$	0	0
$851$	−29780.4	−1.19960
$852$	0	0
$853$	22303.5	0.895261	0.447631	−	0.894219i	$-0.352268\pi$
$853$	22303.5	0.895261	0.447631	+	0.894219i	$0.352268\pi$
$854$	0	0
$855$	2266.22	0.0906470
$856$	0	0
$857$	13859.0	0.552411	0.276205	−	0.961099i	$-0.410923\pi$
$857$	13859.0	0.552411	0.276205	+	0.961099i	$0.410923\pi$
$858$	0	0
$859$	−24629.1	−0.978270	−0.489135	−	0.872208i	$-0.662687\pi$
$859$	−24629.1	−0.978270	−0.489135	+	0.872208i	$0.662687\pi$
$860$	0	0
$861$	10190.4	0.403354
$862$	0	0
$863$	−33140.6	−1.30721	−0.653604	−	0.756837i	$-0.726744\pi$
$863$	−33140.6	−1.30721	−0.653604	+	0.756837i	$0.726744\pi$
$864$	0	0
$865$	25436.5	0.999846
$866$	0	0
$867$	47592.7	1.86428
$868$	0	0
$869$	1037.16	0.0404869
$870$	0	0
$871$	35660.2	1.38726
$872$	0	0
$873$	−30481.5	−1.18172
$874$	0	0
$875$	19966.0	0.771398
$876$	0	0
$877$	2224.74	0.0856604	0.0428302	−	0.999082i	$-0.486363\pi$
$877$	2224.74	0.0856604	0.0428302	+	0.999082i	$0.486363\pi$
$878$	0	0
$879$	44511.9	1.70802
$880$	0	0
$881$	24242.9	0.927087	0.463544	−	0.886074i	$-0.346578\pi$
$881$	24242.9	0.927087	0.463544	+	0.886074i	$0.346578\pi$
$882$	0	0
$883$	24060.4	0.916984	0.458492	−	0.888698i	$-0.348390\pi$
$883$	24060.4	0.916984	0.458492	+	0.888698i	$0.348390\pi$
$884$	0	0
$885$	−19476.3	−0.739760
$886$	0	0
$887$	9727.38	0.368223	0.184111	−	0.982905i	$-0.441059\pi$
$887$	9727.38	0.368223	0.184111	+	0.982905i	$0.441059\pi$
$888$	0	0
$889$	−26811.3	−1.01150
$890$	0	0
$891$	8138.00	0.305986
$892$	0	0
$893$	−1028.51	−0.0385419
$894$	0	0
$895$	17010.5	0.635304
$896$	0	0
$897$	84324.5	3.13881
$898$	0	0
$899$	−818.422	−0.0303625
$900$	0	0
$901$	14735.9	0.544865
$902$	0	0
$903$	−57486.9	−2.11854
$904$	0	0
$905$	12365.1	0.454177
$906$	0	0
$907$	4523.40	0.165598	0.0827988	−	0.996566i	$-0.473614\pi$
$907$	4523.40	0.165598	0.0827988	+	0.996566i	$0.473614\pi$
$908$	0	0
$909$	−53222.2	−1.94199
$910$	0	0
$911$	24396.1	0.887245	0.443623	−	0.896214i	$-0.353693\pi$
$911$	24396.1	0.887245	0.443623	+	0.896214i	$0.353693\pi$
$912$	0	0
$913$	14053.3	0.509415
$914$	0	0
$915$	10219.3	0.369224
$916$	0	0
$917$	−16206.3	−0.583620
$918$	0	0
$919$	−51701.0	−1.85578	−0.927888	−	0.372858i	$-0.878378\pi$
$919$	−51701.0	−1.85578	−0.927888	+	0.372858i	$0.878378\pi$
$920$	0	0
$921$	22541.9	0.806496
$922$	0	0
$923$	33192.1	1.18367
$924$	0	0
$925$	12934.7	0.459772
$926$	0	0
$927$	−13802.2	−0.489022
$928$	0	0
$929$	−8172.60	−0.288627	−0.144313	−	0.989532i	$-0.546097\pi$
$929$	−8172.60	−0.288627	−0.144313	+	0.989532i	$0.546097\pi$
$930$	0	0
$931$	1340.44	0.0471871
$932$	0	0
$933$	4022.84	0.141160
$934$	0	0
$935$	12549.9	0.438957
$936$	0	0
$937$	40385.8	1.40806	0.704028	−	0.710173i	$-0.251383\pi$
$937$	40385.8	1.40806	0.704028	+	0.710173i	$0.251383\pi$
$938$	0	0
$939$	−11812.4	−0.410525
$940$	0	0
$941$	3540.29	0.122646	0.0613232	−	0.998118i	$-0.480468\pi$
$941$	3540.29	0.122646	0.0613232	+	0.998118i	$0.480468\pi$
$942$	0	0
$943$	15973.5	0.551610
$944$	0	0
$945$	−5691.20	−0.195910
$946$	0	0
$947$	53975.8	1.85214	0.926071	−	0.377349i	$-0.123164\pi$
$947$	53975.8	1.85214	0.926071	+	0.377349i	$0.123164\pi$
$948$	0	0
$949$	−31558.7	−1.07949
$950$	0	0
$951$	42370.8	1.44476
$952$	0	0
$953$	−19500.7	−0.662843	−0.331421	−	0.943483i	$-0.607528\pi$
$953$	−19500.7	−0.662843	−0.331421	+	0.943483i	$0.607528\pi$
$954$	0	0
$955$	−8141.37	−0.275862
$956$	0	0
$957$	−3859.39	−0.130362
$958$	0	0
$959$	40622.6	1.36786
$960$	0	0
$961$	−28994.5	−0.973265
$962$	0	0
$963$	−28176.4	−0.942857
$964$	0	0
$965$	22638.4	0.755186
$966$	0	0
$967$	−18203.2	−0.605354	−0.302677	−	0.953093i	$-0.597880\pi$
$967$	−18203.2	−0.605354	−0.302677	+	0.953093i	$0.597880\pi$
$968$	0	0
$969$	−7702.04	−0.255341
$970$	0	0
$971$	10588.8	0.349961	0.174981	−	0.984572i	$-0.444014\pi$
$971$	10588.8	0.349961	0.174981	+	0.984572i	$0.444014\pi$
$972$	0	0
$973$	11002.8	0.362522
$974$	0	0
$975$	−36625.1	−1.20302
$976$	0	0
$977$	24805.8	0.812290	0.406145	−	0.913809i	$-0.366873\pi$
$977$	24805.8	0.812290	0.406145	+	0.913809i	$0.366873\pi$
$978$	0	0
$979$	15951.9	0.520760
$980$	0	0
$981$	27541.8	0.896375
$982$	0	0
$983$	−2760.27	−0.0895613	−0.0447807	−	0.998997i	$-0.514259\pi$
$983$	−2760.27	−0.0895613	−0.0447807	+	0.998997i	$0.514259\pi$
$984$	0	0
$985$	27633.3	0.893878
$986$	0	0
$987$	12140.8	0.391536
$988$	0	0
$989$	−90111.0	−2.89723
$990$	0	0
$991$	−5609.17	−0.179799	−0.0898997	−	0.995951i	$-0.528655\pi$
$991$	−5609.17	−0.179799	−0.0898997	+	0.995951i	$0.528655\pi$
$992$	0	0
$993$	81064.5	2.59064
$994$	0	0
$995$	−31434.4	−1.00155
$996$	0	0
$997$	−3629.31	−0.115287	−0.0576436	−	0.998337i	$-0.518359\pi$
$997$	−3629.31	−0.115287	−0.0576436	+	0.998337i	$0.518359\pi$
$998$	0	0
$999$	−9798.55	−0.310323

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	232.4.a.d.1.5	✓	5
3.2	odd	2		2088.4.a.f.1.4		5
4.3	odd	2		464.4.a.m.1.1		5
8.3	odd	2		1856.4.a.ba.1.5		5
8.5	even	2		1856.4.a.z.1.1		5

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
232.4.a.d.1.5	✓	5	1.1	even	1	trivial
464.4.a.m.1.1		5	4.3	odd	2
1856.4.a.z.1.1		5	8.5	even	2
1856.4.a.ba.1.5		5	8.3	odd	2
2088.4.a.f.1.4		5	3.2	odd	2

Overview	Random
Universe	Knowledge

Classical	Maass
Hilbert	Bianchi

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

Level:	\( N \)	\(=\)	\( 232 = 2^{3} \cdot 29 \)
Weight:	\( k \)	\(=\)	\( 4 \)
Character orbit:	\([\chi]\)	\(=\)	232.a (trivial)

\(f(q)\)	\(=\)	\(q+7.82921 q^{3} -7.04208 q^{5} +14.1473 q^{7} +34.2965 q^{9} +O(q^{10})\)	\(q+7.82921 q^{3} -7.04208 q^{5} +14.1473 q^{7} +34.2965 q^{9} -16.9982 q^{11} +62.0351 q^{13} -55.1339 q^{15} +104.842 q^{17} -9.38323 q^{19} +110.762 q^{21} +173.620 q^{23} -75.4091 q^{25} +57.1255 q^{27} +29.0000 q^{29} -28.2215 q^{31} -133.082 q^{33} -99.6261 q^{35} -171.527 q^{37} +485.685 q^{39} +92.0027 q^{41} -519.014 q^{43} -241.518 q^{45} +109.612 q^{47} -142.855 q^{49} +820.830 q^{51} +140.553 q^{53} +119.703 q^{55} -73.4632 q^{57} +353.254 q^{59} -185.355 q^{61} +485.201 q^{63} -436.856 q^{65} +574.840 q^{67} +1359.30 q^{69} +535.054 q^{71} -508.723 q^{73} -590.393 q^{75} -240.478 q^{77} -61.0156 q^{79} -478.757 q^{81} -826.753 q^{83} -738.306 q^{85} +227.047 q^{87} -938.445 q^{89} +877.626 q^{91} -220.952 q^{93} +66.0775 q^{95} -888.766 q^{97} -582.978 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 5 q + 4 q^{3} + 10 q^{5} + 32 q^{7} + 29 q^{9}+O(q^{10}) \) 5 * q + 4 * q^3 + 10 * q^5 + 32 * q^7 + 29 * q^9	\( 5 q + 4 q^{3} + 10 q^{5} + 32 q^{7} + 29 q^{9} + 36 q^{11} + 26 q^{13} + 88 q^{15} + 82 q^{17} + 156 q^{19} + 72 q^{21} + 336 q^{23} + 151 q^{25} + 352 q^{27} + 145 q^{29} + 432 q^{31} + 108 q^{33} + 600 q^{35} - 18 q^{37} + 688 q^{39} + 82 q^{41} + 340 q^{43} - 146 q^{45} + 680 q^{47} - 115 q^{49} + 608 q^{51} - 102 q^{53} + 736 q^{55} - 576 q^{57} + 924 q^{59} - 618 q^{61} + 584 q^{63} - 704 q^{65} + 44 q^{67} - 1056 q^{69} + 1032 q^{71} - 1078 q^{73} - 468 q^{75} - 888 q^{77} + 200 q^{79} - 1843 q^{81} + 452 q^{83} - 1700 q^{85} + 116 q^{87} - 1790 q^{89} - 1128 q^{91} - 1884 q^{93} + 1024 q^{95} - 2518 q^{97} - 1500 q^{99}+O(q^{100}) \) 5 * q + 4 * q^3 + 10 * q^5 + 32 * q^7 + 29 * q^9 + 36 * q^11 + 26 * q^13 + 88 * q^15 + 82 * q^17 + 156 * q^19 + 72 * q^21 + 336 * q^23 + 151 * q^25 + 352 * q^27 + 145 * q^29 + 432 * q^31 + 108 * q^33 + 600 * q^35 - 18 * q^37 + 688 * q^39 + 82 * q^41 + 340 * q^43 - 146 * q^45 + 680 * q^47 - 115 * q^49 + 608 * q^51 - 102 * q^53 + 736 * q^55 - 576 * q^57 + 924 * q^59 - 618 * q^61 + 584 * q^63 - 704 * q^65 + 44 * q^67 - 1056 * q^69 + 1032 * q^71 - 1078 * q^73 - 468 * q^75 - 888 * q^77 + 200 * q^79 - 1843 * q^81 + 452 * q^83 - 1700 * q^85 + 116 * q^87 - 1790 * q^89 - 1128 * q^91 - 1884 * q^93 + 1024 * q^95 - 2518 * q^97 - 1500 * q^99

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