LMFDB - Embedded newform 1232.4.a.z.1.1

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [1232,4,Mod(1,1232)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(1232, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([0, 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("1232.1");

S:= CuspForms(chi, 4);

N := Newforms(S);

Level:	$ N $	$=$	$ 1232 = 2^{4} \cdot 7 \cdot 11 $
Weight:	$ k $	$=$	$ 4 $
Character orbit:	$[\chi]$	$=$	1232.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:	$72.6903531271$
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} - 2x^{4} - 105x^{3} + 92x^{2} + 1858x - 2576 $ x^5 - 2x^4 - 105x^3 + 92x^2 + 1858x - 2576
Coefficient ring:	$\Z[a_1, \ldots, a_{5}]$
Coefficient ring index:	$ 2^{3} $
Twist minimal:	no (minimal twist has level 616)
Fricke sign:	$1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.1
Root			$9.98555$ of defining polynomial
Character	$\chi$	$=$	1232.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-9.98555 q^{3} -20.3895 q^{5} +7.00000 q^{7} +72.7112 q^{9} +O(q^{10})$	$q-9.98555 q^{3} -20.3895 q^{5} +7.00000 q^{7} +72.7112 q^{9} +11.0000 q^{11} +32.5684 q^{13} +203.601 q^{15} -125.397 q^{17} +59.5412 q^{19} -69.8988 q^{21} +71.7742 q^{23} +290.733 q^{25} -456.451 q^{27} -299.537 q^{29} -59.6725 q^{31} -109.841 q^{33} -142.727 q^{35} -130.755 q^{37} -325.213 q^{39} -190.116 q^{41} -164.400 q^{43} -1482.55 q^{45} +470.267 q^{47} +49.0000 q^{49} +1252.16 q^{51} -290.346 q^{53} -224.285 q^{55} -594.552 q^{57} -373.889 q^{59} -396.665 q^{61} +508.978 q^{63} -664.055 q^{65} +443.179 q^{67} -716.704 q^{69} -563.889 q^{71} +164.184 q^{73} -2903.13 q^{75} +77.0000 q^{77} +71.1265 q^{79} +2594.71 q^{81} -168.464 q^{83} +2556.78 q^{85} +2991.04 q^{87} -1284.92 q^{89} +227.979 q^{91} +595.863 q^{93} -1214.02 q^{95} -1493.33 q^{97} +799.823 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 5 q - 2 q^{3} - 14 q^{5} + 35 q^{7} + 79 q^{9}+O(q^{10}) $ 5 * q - 2 * q^3 - 14 * q^5 + 35 * q^7 + 79 * q^9	$ 5 q - 2 q^{3} - 14 q^{5} + 35 q^{7} + 79 q^{9} + 55 q^{11} + 194 q^{13} + 118 q^{15} - 248 q^{17} + 190 q^{19} - 14 q^{21} + 34 q^{23} + 529 q^{25} - 254 q^{27} - 334 q^{29} + 172 q^{31} - 22 q^{33} - 98 q^{35} + 360 q^{37} + 360 q^{39} - 852 q^{41} + 1016 q^{43} - 2164 q^{45} + 194 q^{47} + 245 q^{49} + 836 q^{51} - 934 q^{53} - 154 q^{55} - 980 q^{57} + 542 q^{59} + 838 q^{61} + 553 q^{63} - 1780 q^{65} + 1830 q^{67} + 50 q^{69} - 422 q^{71} - 68 q^{73} - 320 q^{75} + 385 q^{77} + 8 q^{79} + 1889 q^{81} + 366 q^{83} + 2056 q^{85} + 4384 q^{87} - 524 q^{89} + 1358 q^{91} - 2242 q^{93} + 2884 q^{95} - 1336 q^{97} + 869 q^{99}+O(q^{100}) $ 5 * q - 2 * q^3 - 14 * q^5 + 35 * q^7 + 79 * q^9 + 55 * q^11 + 194 * q^13 + 118 * q^15 - 248 * q^17 + 190 * q^19 - 14 * q^21 + 34 * q^23 + 529 * q^25 - 254 * q^27 - 334 * q^29 + 172 * q^31 - 22 * q^33 - 98 * q^35 + 360 * q^37 + 360 * q^39 - 852 * q^41 + 1016 * q^43 - 2164 * q^45 + 194 * q^47 + 245 * q^49 + 836 * q^51 - 934 * q^53 - 154 * q^55 - 980 * q^57 + 542 * q^59 + 838 * q^61 + 553 * q^63 - 1780 * q^65 + 1830 * q^67 + 50 * q^69 - 422 * q^71 - 68 * q^73 - 320 * q^75 + 385 * q^77 + 8 * q^79 + 1889 * q^81 + 366 * q^83 + 2056 * q^85 + 4384 * q^87 - 524 * q^89 + 1358 * q^91 - 2242 * q^93 + 2884 * q^95 - 1336 * q^97 + 869 * 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
$3$	−9.98555	−1.92172	−0.960860	−	0.277035i	$-0.910648\pi$
$3$	−9.98555	−1.92172	−0.960860	+	0.277035i	$0.910648\pi$
$4$	0	0
$5$	−20.3895	−1.82370	−0.911848	−	0.410528i	$-0.865344\pi$
$5$	−20.3895	−1.82370	−0.911848	+	0.410528i	$0.865344\pi$
$6$	0	0
$7$	7.00000	0.377964
$7$	7.00000	0.377964
$8$	0	0
$9$	72.7112	2.69301
$10$	0	0
$11$	11.0000	0.301511
$11$	11.0000	0.301511
$12$	0	0
$13$	32.5684	0.694835	0.347417	−	0.937711i	$-0.387059\pi$
$13$	32.5684	0.694835	0.347417	+	0.937711i	$0.387059\pi$
$14$	0	0
$15$	203.601	3.50463
$16$	0	0
$17$	−125.397	−1.78901	−0.894505	−	0.447057i	$-0.852472\pi$
$17$	−125.397	−1.78901	−0.894505	+	0.447057i	$0.852472\pi$
$18$	0	0
$19$	59.5412	0.718931	0.359466	−	0.933158i	$-0.382959\pi$
$19$	59.5412	0.718931	0.359466	+	0.933158i	$0.382959\pi$
$20$	0	0
$21$	−69.8988	−0.726342
$22$	0	0
$23$	71.7742	0.650694	0.325347	−	0.945595i	$-0.394519\pi$
$23$	71.7742	0.650694	0.325347	+	0.945595i	$0.394519\pi$
$24$	0	0
$25$	290.733	2.32587
$26$	0	0
$27$	−456.451	−3.25348
$28$	0	0
$29$	−299.537	−1.91802	−0.959009	−	0.283374i	$-0.908546\pi$
$29$	−299.537	−1.91802	−0.959009	+	0.283374i	$0.908546\pi$
$30$	0	0
$31$	−59.6725	−0.345726	−0.172863	−	0.984946i	$-0.555302\pi$
$31$	−59.6725	−0.345726	−0.172863	+	0.984946i	$0.555302\pi$
$32$	0	0
$33$	−109.841	−0.579420
$34$	0	0
$35$	−142.727	−0.689292
$36$	0	0
$37$	−130.755	−0.580973	−0.290486	−	0.956879i	$-0.593817\pi$
$37$	−130.755	−0.580973	−0.290486	+	0.956879i	$0.593817\pi$
$38$	0	0
$39$	−325.213	−1.33528
$40$	0	0
$41$	−190.116	−0.724173	−0.362087	−	0.932144i	$-0.617936\pi$
$41$	−190.116	−0.724173	−0.362087	+	0.932144i	$0.617936\pi$
$42$	0	0
$43$	−164.400	−0.583042	−0.291521	−	0.956564i	$-0.594161\pi$
$43$	−164.400	−0.583042	−0.291521	+	0.956564i	$0.594161\pi$
$44$	0	0
$45$	−1482.55	−4.91122
$46$	0	0
$47$	470.267	1.45948	0.729739	−	0.683726i	$-0.239642\pi$
$47$	470.267	1.45948	0.729739	+	0.683726i	$0.239642\pi$
$48$	0	0
$49$	49.0000	0.142857
$50$	0	0
$51$	1252.16	3.43798
$52$	0	0
$53$	−290.346	−0.752491	−0.376246	−	0.926520i	$-0.622785\pi$
$53$	−290.346	−0.752491	−0.376246	+	0.926520i	$0.622785\pi$
$54$	0	0
$55$	−224.285	−0.549865
$56$	0	0
$57$	−594.552	−1.38158
$58$	0	0
$59$	−373.889	−0.825019	−0.412510	−	0.910953i	$-0.635348\pi$
$59$	−373.889	−0.825019	−0.412510	+	0.910953i	$0.635348\pi$
$60$	0	0
$61$	−396.665	−0.832585	−0.416293	−	0.909231i	$-0.636671\pi$
$61$	−396.665	−0.832585	−0.416293	+	0.909231i	$0.636671\pi$
$62$	0	0
$63$	508.978	1.01786
$64$	0	0
$65$	−664.055	−1.26717
$66$	0	0
$67$	443.179	0.808104	0.404052	−	0.914736i	$-0.367601\pi$
$67$	443.179	0.808104	0.404052	+	0.914736i	$0.367601\pi$
$68$	0	0
$69$	−716.704	−1.25045
$70$	0	0
$71$	−563.889	−0.942553	−0.471277	−	0.881985i	$-0.656207\pi$
$71$	−563.889	−0.942553	−0.471277	+	0.881985i	$0.656207\pi$
$72$	0	0
$73$	164.184	0.263236	0.131618	−	0.991301i	$-0.457983\pi$
$73$	164.184	0.263236	0.131618	+	0.991301i	$0.457983\pi$
$74$	0	0
$75$	−2903.13	−4.46967
$76$	0	0
$77$	77.0000	0.113961
$78$	0	0
$79$	71.1265	0.101296	0.0506478	−	0.998717i	$-0.483871\pi$
$79$	71.1265	0.101296	0.0506478	+	0.998717i	$0.483871\pi$
$80$	0	0
$81$	2594.71	3.55927
$82$	0	0
$83$	−168.464	−0.222787	−0.111394	−	0.993776i	$-0.535531\pi$
$83$	−168.464	−0.222787	−0.111394	+	0.993776i	$0.535531\pi$
$84$	0	0
$85$	2556.78	3.26261
$86$	0	0
$87$	2991.04	3.68589
$88$	0	0
$89$	−1284.92	−1.53035	−0.765177	−	0.643820i	$-0.777348\pi$
$89$	−1284.92	−1.53035	−0.765177	+	0.643820i	$0.777348\pi$
$90$	0	0
$91$	227.979	0.262623
$92$	0	0
$93$	595.863	0.664388
$94$	0	0
$95$	−1214.02	−1.31111
$96$	0	0
$97$	−1493.33	−1.56314	−0.781571	−	0.623816i	$-0.785581\pi$
$97$	−1493.33	−1.56314	−0.781571	+	0.623816i	$0.785581\pi$
$98$	0	0
$99$	799.823	0.811972
$100$	0	0
$101$	221.294	0.218016	0.109008	−	0.994041i	$-0.465233\pi$
$101$	221.294	0.218016	0.109008	+	0.994041i	$0.465233\pi$
$102$	0	0
$103$	152.370	0.145761	0.0728807	−	0.997341i	$-0.476781\pi$
$103$	152.370	0.145761	0.0728807	+	0.997341i	$0.476781\pi$
$104$	0	0
$105$	1425.21	1.32463
$106$	0	0
$107$	−1136.91	−1.02719	−0.513594	−	0.858034i	$-0.671686\pi$
$107$	−1136.91	−1.02719	−0.513594	+	0.858034i	$0.671686\pi$
$108$	0	0
$109$	1137.42	0.999498	0.499749	−	0.866170i	$-0.333426\pi$
$109$	1137.42	0.999498	0.499749	+	0.866170i	$0.333426\pi$
$110$	0	0
$111$	1305.66	1.11647
$112$	0	0
$113$	1613.61	1.34332	0.671661	−	0.740859i	$-0.265581\pi$
$113$	1613.61	1.34332	0.671661	+	0.740859i	$0.265581\pi$
$114$	0	0
$115$	−1463.44	−1.18667
$116$	0	0
$117$	2368.09	1.87119
$118$	0	0
$119$	−877.777	−0.676183
$120$	0	0
$121$	121.000	0.0909091
$122$	0	0
$123$	1898.41	1.39166
$124$	0	0
$125$	−3379.23	−2.41798
$126$	0	0
$127$	2292.48	1.60177	0.800886	−	0.598817i	$-0.204362\pi$
$127$	2292.48	1.60177	0.800886	+	0.598817i	$0.204362\pi$
$128$	0	0
$129$	1641.63	1.12044
$130$	0	0
$131$	−106.342	−0.0709248	−0.0354624	−	0.999371i	$-0.511290\pi$
$131$	−106.342	−0.0709248	−0.0354624	+	0.999371i	$0.511290\pi$
$132$	0	0
$133$	416.789	0.271730
$134$	0	0
$135$	9306.83	5.93336
$136$	0	0
$137$	2760.97	1.72179	0.860897	−	0.508780i	$-0.169903\pi$
$137$	2760.97	1.72179	0.860897	+	0.508780i	$0.169903\pi$
$138$	0	0
$139$	−1183.63	−0.722263	−0.361131	−	0.932515i	$-0.617609\pi$
$139$	−1183.63	−0.722263	−0.361131	+	0.932515i	$0.617609\pi$
$140$	0	0
$141$	−4695.87	−2.80471
$142$	0	0
$143$	358.253	0.209501
$144$	0	0
$145$	6107.41	3.49788
$146$	0	0
$147$	−489.292	−0.274531
$148$	0	0
$149$	−1343.94	−0.738924	−0.369462	−	0.929246i	$-0.620458\pi$
$149$	−1343.94	−0.738924	−0.369462	+	0.929246i	$0.620458\pi$
$150$	0	0
$151$	314.793	0.169652	0.0848262	−	0.996396i	$-0.472966\pi$
$151$	314.793	0.169652	0.0848262	+	0.996396i	$0.472966\pi$
$152$	0	0
$153$	−9117.75	−4.81782
$154$	0	0
$155$	1216.70	0.630499
$156$	0	0
$157$	1590.35	0.808433	0.404217	−	0.914663i	$-0.367544\pi$
$157$	1590.35	0.808433	0.404217	+	0.914663i	$0.367544\pi$
$158$	0	0
$159$	2899.26	1.44608
$160$	0	0
$161$	502.419	0.245939
$162$	0	0
$163$	−3432.14	−1.64924	−0.824620	−	0.565686i	$-0.808611\pi$
$163$	−3432.14	−1.64924	−0.824620	+	0.565686i	$0.808611\pi$
$164$	0	0
$165$	2239.61	1.05669
$166$	0	0
$167$	−1010.67	−0.468310	−0.234155	−	0.972199i	$-0.575232\pi$
$167$	−1010.67	−0.468310	−0.234155	+	0.972199i	$0.575232\pi$
$168$	0	0
$169$	−1136.30	−0.517205
$170$	0	0
$171$	4329.31	1.93609
$172$	0	0
$173$	2150.96	0.945286	0.472643	−	0.881254i	$-0.343300\pi$
$173$	2150.96	0.945286	0.472643	+	0.881254i	$0.343300\pi$
$174$	0	0
$175$	2035.13	0.879095
$176$	0	0
$177$	3733.48	1.58546
$178$	0	0
$179$	223.531	0.0933379	0.0466689	−	0.998910i	$-0.485139\pi$
$179$	223.531	0.0933379	0.0466689	+	0.998910i	$0.485139\pi$
$180$	0	0
$181$	−2916.05	−1.19750	−0.598751	−	0.800935i	$-0.704336\pi$
$181$	−2916.05	−1.19750	−0.598751	+	0.800935i	$0.704336\pi$
$182$	0	0
$183$	3960.91	1.60000
$184$	0	0
$185$	2666.04	1.05952
$186$	0	0
$187$	−1379.36	−0.539407
$188$	0	0
$189$	−3195.16	−1.22970
$190$	0	0
$191$	−1310.89	−0.496610	−0.248305	−	0.968682i	$-0.579874\pi$
$191$	−1310.89	−0.496610	−0.248305	+	0.968682i	$0.579874\pi$
$192$	0	0
$193$	−835.616	−0.311653	−0.155826	−	0.987784i	$-0.549804\pi$
$193$	−835.616	−0.311653	−0.155826	+	0.987784i	$0.549804\pi$
$194$	0	0
$195$	6630.95	2.43514
$196$	0	0
$197$	−2861.79	−1.03500	−0.517498	−	0.855684i	$-0.673137\pi$
$197$	−2861.79	−1.03500	−0.517498	+	0.855684i	$0.673137\pi$
$198$	0	0
$199$	−5263.73	−1.87505	−0.937527	−	0.347911i	$-0.886891\pi$
$199$	−5263.73	−1.87505	−0.937527	+	0.347911i	$0.886891\pi$
$200$	0	0
$201$	−4425.39	−1.55295
$202$	0	0
$203$	−2096.76	−0.724943
$204$	0	0
$205$	3876.37	1.32067
$206$	0	0
$207$	5218.78	1.75232
$208$	0	0
$209$	654.953	0.216766
$210$	0	0
$211$	1518.34	0.495389	0.247695	−	0.968838i	$-0.420327\pi$
$211$	1518.34	0.495389	0.247695	+	0.968838i	$0.420327\pi$
$212$	0	0
$213$	5630.74	1.81132
$214$	0	0
$215$	3352.05	1.06329
$216$	0	0
$217$	−417.708	−0.130672
$218$	0	0
$219$	−1639.46	−0.505866
$220$	0	0
$221$	−4083.97	−1.24307
$222$	0	0
$223$	−2246.53	−0.674613	−0.337307	−	0.941395i	$-0.609516\pi$
$223$	−2246.53	−0.674613	−0.337307	+	0.941395i	$0.609516\pi$
$224$	0	0
$225$	21139.6	6.26358
$226$	0	0
$227$	4659.35	1.36234	0.681171	−	0.732124i	$-0.261471\pi$
$227$	4659.35	1.36234	0.681171	+	0.732124i	$0.261471\pi$
$228$	0	0
$229$	840.868	0.242647	0.121323	−	0.992613i	$-0.461286\pi$
$229$	840.868	0.242647	0.121323	+	0.992613i	$0.461286\pi$
$230$	0	0
$231$	−768.887	−0.219000
$232$	0	0
$233$	−3429.69	−0.964320	−0.482160	−	0.876083i	$-0.660148\pi$
$233$	−3429.69	−0.964320	−0.482160	+	0.876083i	$0.660148\pi$
$234$	0	0
$235$	−9588.52	−2.66164
$236$	0	0
$237$	−710.237	−0.194662
$238$	0	0
$239$	1594.44	0.431530	0.215765	−	0.976445i	$-0.430776\pi$
$239$	1594.44	0.431530	0.215765	+	0.976445i	$0.430776\pi$
$240$	0	0
$241$	−2949.40	−0.788330	−0.394165	−	0.919040i	$-0.628966\pi$
$241$	−2949.40	−0.788330	−0.394165	+	0.919040i	$0.628966\pi$
$242$	0	0
$243$	−13585.4	−3.58645
$244$	0	0
$245$	−999.088	−0.260528
$246$	0	0
$247$	1939.16	0.499538
$248$	0	0
$249$	1682.21	0.428135
$250$	0	0
$251$	−1128.56	−0.283802	−0.141901	−	0.989881i	$-0.545321\pi$
$251$	−1128.56	−0.283802	−0.141901	+	0.989881i	$0.545321\pi$
$252$	0	0
$253$	789.516	0.196191
$254$	0	0
$255$	−25530.9	−6.26983
$256$	0	0
$257$	701.329	0.170225	0.0851123	−	0.996371i	$-0.472875\pi$
$257$	701.329	0.170225	0.0851123	+	0.996371i	$0.472875\pi$
$258$	0	0
$259$	−915.285	−0.219587
$260$	0	0
$261$	−21779.7	−5.16524
$262$	0	0
$263$	−5744.52	−1.34685	−0.673427	−	0.739254i	$-0.735178\pi$
$263$	−5744.52	−1.34685	−0.673427	+	0.739254i	$0.735178\pi$
$264$	0	0
$265$	5920.01	1.37232
$266$	0	0
$267$	12830.7	2.94091
$268$	0	0
$269$	5830.16	1.32145	0.660727	−	0.750626i	$-0.270248\pi$
$269$	5830.16	1.32145	0.660727	+	0.750626i	$0.270248\pi$
$270$	0	0
$271$	1685.38	0.377785	0.188892	−	0.981998i	$-0.439510\pi$
$271$	1685.38	0.377785	0.188892	+	0.981998i	$0.439510\pi$
$272$	0	0
$273$	−2276.49	−0.504687
$274$	0	0
$275$	3198.07	0.701276
$276$	0	0
$277$	5492.19	1.19131	0.595657	−	0.803239i	$-0.296892\pi$
$277$	5492.19	1.19131	0.595657	+	0.803239i	$0.296892\pi$
$278$	0	0
$279$	−4338.86	−0.931042
$280$	0	0
$281$	−1179.73	−0.250452	−0.125226	−	0.992128i	$-0.539966\pi$
$281$	−1179.73	−0.250452	−0.125226	+	0.992128i	$0.539966\pi$
$282$	0	0
$283$	4991.51	1.04846	0.524231	−	0.851576i	$-0.324353\pi$
$283$	4991.51	1.04846	0.524231	+	0.851576i	$0.324353\pi$
$284$	0	0
$285$	12122.6	2.51959
$286$	0	0
$287$	−1330.81	−0.273712
$288$	0	0
$289$	10811.4	2.20056
$290$	0	0
$291$	14911.7	3.00392
$292$	0	0
$293$	4351.03	0.867542	0.433771	−	0.901023i	$-0.357183\pi$
$293$	4351.03	0.867542	0.433771	+	0.901023i	$0.357183\pi$
$294$	0	0
$295$	7623.42	1.50458
$296$	0	0
$297$	−5020.96	−0.980962
$298$	0	0
$299$	2337.57	0.452124
$300$	0	0
$301$	−1150.80	−0.220369
$302$	0	0
$303$	−2209.74	−0.418965
$304$	0	0
$305$	8087.81	1.51838
$306$	0	0
$307$	8121.78	1.50988	0.754942	−	0.655792i	$-0.227665\pi$
$307$	8121.78	1.50988	0.754942	+	0.655792i	$0.227665\pi$
$308$	0	0
$309$	−1521.49	−0.280113
$310$	0	0
$311$	3320.90	0.605502	0.302751	−	0.953070i	$-0.402095\pi$
$311$	3320.90	0.605502	0.302751	+	0.953070i	$0.402095\pi$
$312$	0	0
$313$	7924.23	1.43100	0.715501	−	0.698611i	$-0.246198\pi$
$313$	7924.23	1.43100	0.715501	+	0.698611i	$0.246198\pi$
$314$	0	0
$315$	−10377.8	−1.85627
$316$	0	0
$317$	−637.824	−0.113009	−0.0565044	−	0.998402i	$-0.517995\pi$
$317$	−637.824	−0.113009	−0.0565044	+	0.998402i	$0.517995\pi$
$318$	0	0
$319$	−3294.90	−0.578304
$320$	0	0
$321$	11352.6	1.97397
$322$	0	0
$323$	−7466.28	−1.28618
$324$	0	0
$325$	9468.73	1.61609
$326$	0	0
$327$	−11357.8	−1.92075
$328$	0	0
$329$	3291.87	0.551630
$330$	0	0
$331$	7363.52	1.22277	0.611383	−	0.791335i	$-0.290614\pi$
$331$	7363.52	1.22277	0.611383	+	0.791335i	$0.290614\pi$
$332$	0	0
$333$	−9507.35	−1.56456
$334$	0	0
$335$	−9036.22	−1.47374
$336$	0	0
$337$	2541.10	0.410750	0.205375	−	0.978683i	$-0.434159\pi$
$337$	2541.10	0.410750	0.205375	+	0.978683i	$0.434159\pi$
$338$	0	0
$339$	−16112.7	−2.58149
$340$	0	0
$341$	−656.398	−0.104240
$342$	0	0
$343$	343.000	0.0539949
$344$	0	0
$345$	14613.3	2.28044
$346$	0	0
$347$	1833.76	0.283694	0.141847	−	0.989889i	$-0.454696\pi$
$347$	1833.76	0.283694	0.141847	+	0.989889i	$0.454696\pi$
$348$	0	0
$349$	−7781.85	−1.19356	−0.596780	−	0.802405i	$-0.703554\pi$
$349$	−7781.85	−1.19356	−0.596780	+	0.802405i	$0.703554\pi$
$350$	0	0
$351$	−14865.9	−2.26063
$352$	0	0
$353$	−5924.03	−0.893213	−0.446607	−	0.894730i	$-0.647368\pi$
$353$	−5924.03	−0.893213	−0.446607	+	0.894730i	$0.647368\pi$
$354$	0	0
$355$	11497.4	1.71893
$356$	0	0
$357$	8765.09	1.29943
$358$	0	0
$359$	2873.17	0.422395	0.211198	−	0.977443i	$-0.432264\pi$
$359$	2873.17	0.422395	0.211198	+	0.977443i	$0.432264\pi$
$360$	0	0
$361$	−3313.84	−0.483138
$362$	0	0
$363$	−1208.25	−0.174702
$364$	0	0
$365$	−3347.63	−0.480063
$366$	0	0
$367$	2006.13	0.285338	0.142669	−	0.989770i	$-0.454432\pi$
$367$	2006.13	0.285338	0.142669	+	0.989770i	$0.454432\pi$
$368$	0	0
$369$	−13823.5	−1.95020
$370$	0	0
$371$	−2032.42	−0.284415
$372$	0	0
$373$	−1098.54	−0.152494	−0.0762470	−	0.997089i	$-0.524294\pi$
$373$	−1098.54	−0.152494	−0.0762470	+	0.997089i	$0.524294\pi$
$374$	0	0
$375$	33743.5	4.64668
$376$	0	0
$377$	−9755.43	−1.33271
$378$	0	0
$379$	−6090.73	−0.825487	−0.412744	−	0.910847i	$-0.635429\pi$
$379$	−6090.73	−0.825487	−0.412744	+	0.910847i	$0.635429\pi$
$380$	0	0
$381$	−22891.7	−3.07816
$382$	0	0
$383$	12555.8	1.67511	0.837557	−	0.546349i	$-0.183983\pi$
$383$	12555.8	1.67511	0.837557	+	0.546349i	$0.183983\pi$
$384$	0	0
$385$	−1569.99	−0.207829
$386$	0	0
$387$	−11953.7	−1.57014
$388$	0	0
$389$	2899.56	0.377926	0.188963	−	0.981984i	$-0.439487\pi$
$389$	2899.56	0.377926	0.188963	+	0.981984i	$0.439487\pi$
$390$	0	0
$391$	−9000.25	−1.16410
$392$	0	0
$393$	1061.88	0.136298
$394$	0	0
$395$	−1450.24	−0.184732
$396$	0	0
$397$	7865.25	0.994322	0.497161	−	0.867658i	$-0.334376\pi$
$397$	7865.25	0.994322	0.497161	+	0.867658i	$0.334376\pi$
$398$	0	0
$399$	−4161.86	−0.522190
$400$	0	0
$401$	−6041.77	−0.752398	−0.376199	−	0.926539i	$-0.622769\pi$
$401$	−6041.77	−0.752398	−0.376199	+	0.926539i	$0.622769\pi$
$402$	0	0
$403$	−1943.44	−0.240222
$404$	0	0
$405$	−52905.0	−6.49104
$406$	0	0
$407$	−1438.31	−0.175170
$408$	0	0
$409$	−8303.92	−1.00392	−0.501959	−	0.864891i	$-0.667387\pi$
$409$	−8303.92	−1.00392	−0.501959	+	0.864891i	$0.667387\pi$
$410$	0	0
$411$	−27569.8	−3.30880
$412$	0	0
$413$	−2617.22	−0.311828
$414$	0	0
$415$	3434.91	0.406297
$416$	0	0
$417$	11819.2	1.38799
$418$	0	0
$419$	8254.19	0.962395	0.481198	−	0.876612i	$-0.340202\pi$
$419$	8254.19	0.962395	0.481198	+	0.876612i	$0.340202\pi$
$420$	0	0
$421$	−2900.94	−0.335827	−0.167914	−	0.985802i	$-0.553703\pi$
$421$	−2900.94	−0.335827	−0.167914	+	0.985802i	$0.553703\pi$
$422$	0	0
$423$	34193.6	3.93038
$424$	0	0
$425$	−36457.0	−4.16100
$426$	0	0
$427$	−2776.65	−0.314688
$428$	0	0
$429$	−3577.35	−0.402601
$430$	0	0
$431$	12148.2	1.35767	0.678836	−	0.734290i	$-0.262485\pi$
$431$	12148.2	1.35767	0.678836	+	0.734290i	$0.262485\pi$
$432$	0	0
$433$	−2142.00	−0.237732	−0.118866	−	0.992910i	$-0.537926\pi$
$433$	−2142.00	−0.237732	−0.118866	+	0.992910i	$0.537926\pi$
$434$	0	0
$435$	−60985.9	−6.72195
$436$	0	0
$437$	4273.52	0.467804
$438$	0	0
$439$	9221.09	1.00250	0.501251	−	0.865302i	$-0.332873\pi$
$439$	9221.09	1.00250	0.501251	+	0.865302i	$0.332873\pi$
$440$	0	0
$441$	3562.85	0.384715
$442$	0	0
$443$	1746.92	0.187356	0.0936781	−	0.995603i	$-0.470138\pi$
$443$	1746.92	0.187356	0.0936781	+	0.995603i	$0.470138\pi$
$444$	0	0
$445$	26199.0	2.79090
$446$	0	0
$447$	13420.0	1.42000
$448$	0	0
$449$	2087.64	0.219425	0.109713	−	0.993963i	$-0.465007\pi$
$449$	2087.64	0.219425	0.109713	+	0.993963i	$0.465007\pi$
$450$	0	0
$451$	−2091.27	−0.218346
$452$	0	0
$453$	−3143.38	−0.326024
$454$	0	0
$455$	−4648.39	−0.478944
$456$	0	0
$457$	−14565.2	−1.49087	−0.745437	−	0.666576i	$-0.767759\pi$
$457$	−14565.2	−1.49087	−0.745437	+	0.666576i	$0.767759\pi$
$458$	0	0
$459$	57237.5	5.82052
$460$	0	0
$461$	−4740.09	−0.478890	−0.239445	−	0.970910i	$-0.576965\pi$
$461$	−4740.09	−0.478890	−0.239445	+	0.970910i	$0.576965\pi$
$462$	0	0
$463$	−6237.18	−0.626061	−0.313031	−	0.949743i	$-0.601344\pi$
$463$	−6237.18	−0.626061	−0.313031	+	0.949743i	$0.601344\pi$
$464$	0	0
$465$	−12149.4	−1.21164
$466$	0	0
$467$	−4852.06	−0.480785	−0.240393	−	0.970676i	$-0.577276\pi$
$467$	−4852.06	−0.480785	−0.240393	+	0.970676i	$0.577276\pi$
$468$	0	0
$469$	3102.26	0.305435
$470$	0	0
$471$	−15880.5	−1.55358
$472$	0	0
$473$	−1808.40	−0.175794
$474$	0	0
$475$	17310.6	1.67214
$476$	0	0
$477$	−21111.4	−2.02646
$478$	0	0
$479$	−6168.48	−0.588403	−0.294201	−	0.955743i	$-0.595054\pi$
$479$	−6168.48	−0.588403	−0.294201	+	0.955743i	$0.595054\pi$
$480$	0	0
$481$	−4258.48	−0.403680
$482$	0	0
$483$	−5016.93	−0.472626
$484$	0	0
$485$	30448.3	2.85070
$486$	0	0
$487$	−5304.08	−0.493533	−0.246767	−	0.969075i	$-0.579368\pi$
$487$	−5304.08	−0.493533	−0.246767	+	0.969075i	$0.579368\pi$
$488$	0	0
$489$	34271.8	3.16938
$490$	0	0
$491$	−13174.3	−1.21089	−0.605447	−	0.795885i	$-0.707006\pi$
$491$	−13174.3	−1.21089	−0.605447	+	0.795885i	$0.707006\pi$
$492$	0	0
$493$	37560.9	3.43136
$494$	0	0
$495$	−16308.0	−1.48079
$496$	0	0
$497$	−3947.22	−0.356252
$498$	0	0
$499$	−1379.38	−0.123747	−0.0618734	−	0.998084i	$-0.519708\pi$
$499$	−1379.38	−0.123747	−0.0618734	+	0.998084i	$0.519708\pi$
$500$	0	0
$501$	10092.1	0.899960
$502$	0	0
$503$	−10154.6	−0.900143	−0.450072	−	0.892992i	$-0.648602\pi$
$503$	−10154.6	−0.900143	−0.450072	+	0.892992i	$0.648602\pi$
$504$	0	0
$505$	−4512.09	−0.397595
$506$	0	0
$507$	11346.6	0.993922
$508$	0	0
$509$	19735.3	1.71857	0.859284	−	0.511498i	$-0.170909\pi$
$509$	19735.3	1.71857	0.859284	+	0.511498i	$0.170909\pi$
$510$	0	0
$511$	1149.28	0.0994939
$512$	0	0
$513$	−27177.6	−2.33903
$514$	0	0
$515$	−3106.75	−0.265825
$516$	0	0
$517$	5172.93	0.440049
$518$	0	0
$519$	−21478.5	−1.81657
$520$	0	0
$521$	8725.80	0.733751	0.366875	−	0.930270i	$-0.380428\pi$
$521$	8725.80	0.733751	0.366875	+	0.930270i	$0.380428\pi$
$522$	0	0
$523$	23069.5	1.92879	0.964395	−	0.264468i	$-0.0851963\pi$
$523$	23069.5	1.92879	0.964395	+	0.264468i	$0.0851963\pi$
$524$	0	0
$525$	−20321.9	−1.68937
$526$	0	0
$527$	7482.74	0.618507
$528$	0	0
$529$	−7015.47	−0.576598
$530$	0	0
$531$	−27185.9	−2.22178
$532$	0	0
$533$	−6191.77	−0.503181
$534$	0	0
$535$	23181.0	1.87328
$536$	0	0
$537$	−2232.08	−0.179369
$538$	0	0
$539$	539.000	0.0430730
$540$	0	0
$541$	18780.3	1.49247	0.746236	−	0.665682i	$-0.231859\pi$
$541$	18780.3	1.49247	0.746236	+	0.665682i	$0.231859\pi$
$542$	0	0
$543$	29118.3	2.30126
$544$	0	0
$545$	−23191.5	−1.82278
$546$	0	0
$547$	−14851.7	−1.16090	−0.580449	−	0.814296i	$-0.697123\pi$
$547$	−14851.7	−1.16090	−0.580449	+	0.814296i	$0.697123\pi$
$548$	0	0
$549$	−28841.9	−2.24216
$550$	0	0
$551$	−17834.8	−1.37892
$552$	0	0
$553$	497.885	0.0382861
$554$	0	0
$555$	−26621.8	−2.03610
$556$	0	0
$557$	12984.5	0.987738	0.493869	−	0.869536i	$-0.335582\pi$
$557$	12984.5	0.987738	0.493869	+	0.869536i	$0.335582\pi$
$558$	0	0
$559$	−5354.26	−0.405118
$560$	0	0
$561$	13773.7	1.03659
$562$	0	0
$563$	13651.7	1.02193	0.510967	−	0.859600i	$-0.329287\pi$
$563$	13651.7	1.02193	0.510967	+	0.859600i	$0.329287\pi$
$564$	0	0
$565$	−32900.7	−2.44981
$566$	0	0
$567$	18163.0	1.34528
$568$	0	0
$569$	11765.7	0.866857	0.433429	−	0.901188i	$-0.357304\pi$
$569$	11765.7	0.866857	0.433429	+	0.901188i	$0.357304\pi$
$570$	0	0
$571$	−15721.9	−1.15226	−0.576129	−	0.817358i	$-0.695438\pi$
$571$	−15721.9	−1.15226	−0.576129	+	0.817358i	$0.695438\pi$
$572$	0	0
$573$	13089.9	0.954345
$574$	0	0
$575$	20867.2	1.51343
$576$	0	0
$577$	−21237.3	−1.53227	−0.766137	−	0.642677i	$-0.777824\pi$
$577$	−21237.3	−1.53227	−0.766137	+	0.642677i	$0.777824\pi$
$578$	0	0
$579$	8344.08	0.598909
$580$	0	0
$581$	−1179.25	−0.0842058
$582$	0	0
$583$	−3193.80	−0.226885
$584$	0	0
$585$	−48284.2	−3.41249
$586$	0	0
$587$	25632.2	1.80230	0.901152	−	0.433502i	$-0.142722\pi$
$587$	25632.2	1.80230	0.901152	+	0.433502i	$0.142722\pi$
$588$	0	0
$589$	−3552.98	−0.248553
$590$	0	0
$591$	28576.6	1.98897
$592$	0	0
$593$	−1421.23	−0.0984199	−0.0492099	−	0.998788i	$-0.515670\pi$
$593$	−1421.23	−0.0984199	−0.0492099	+	0.998788i	$0.515670\pi$
$594$	0	0
$595$	17897.5	1.23315
$596$	0	0
$597$	52561.2	3.60333
$598$	0	0
$599$	−3818.20	−0.260447	−0.130223	−	0.991485i	$-0.541569\pi$
$599$	−3818.20	−0.260447	−0.130223	+	0.991485i	$0.541569\pi$
$600$	0	0
$601$	−2788.48	−0.189259	−0.0946295	−	0.995513i	$-0.530167\pi$
$601$	−2788.48	−0.189259	−0.0946295	+	0.995513i	$0.530167\pi$
$602$	0	0
$603$	32224.1	2.17623
$604$	0	0
$605$	−2467.13	−0.165791
$606$	0	0
$607$	17455.6	1.16722	0.583610	−	0.812034i	$-0.301640\pi$
$607$	17455.6	1.16722	0.583610	+	0.812034i	$0.301640\pi$
$608$	0	0
$609$	20937.3	1.39314
$610$	0	0
$611$	15315.8	1.01410
$612$	0	0
$613$	82.3095	0.00542325	0.00271162	−	0.999996i	$-0.499137\pi$
$613$	82.3095	0.00542325	0.00271162	+	0.999996i	$0.499137\pi$
$614$	0	0
$615$	−38707.7	−2.53796
$616$	0	0
$617$	−29014.4	−1.89315	−0.946576	−	0.322481i	$-0.895483\pi$
$617$	−29014.4	−1.89315	−0.946576	+	0.322481i	$0.895483\pi$
$618$	0	0
$619$	1915.31	0.124366	0.0621832	−	0.998065i	$-0.480194\pi$
$619$	1915.31	0.124366	0.0621832	+	0.998065i	$0.480194\pi$
$620$	0	0
$621$	−32761.4	−2.11702
$622$	0	0
$623$	−8994.46	−0.578419
$624$	0	0
$625$	32559.3	2.08379
$626$	0	0
$627$	−6540.07	−0.416563
$628$	0	0
$629$	16396.3	1.03937
$630$	0	0
$631$	24429.2	1.54123	0.770613	−	0.637304i	$-0.219950\pi$
$631$	24429.2	1.54123	0.770613	+	0.637304i	$0.219950\pi$
$632$	0	0
$633$	−15161.5	−0.952000
$634$	0	0
$635$	−46742.7	−2.92114
$636$	0	0
$637$	1595.85	0.0992621
$638$	0	0
$639$	−41001.0	−2.53830
$640$	0	0
$641$	26372.6	1.62505	0.812524	−	0.582928i	$-0.198093\pi$
$641$	26372.6	1.62505	0.812524	+	0.582928i	$0.198093\pi$
$642$	0	0
$643$	12988.2	0.796584	0.398292	−	0.917259i	$-0.369603\pi$
$643$	12988.2	0.796584	0.398292	+	0.917259i	$0.369603\pi$
$644$	0	0
$645$	−33472.0	−2.04335
$646$	0	0
$647$	−14810.7	−0.899949	−0.449975	−	0.893041i	$-0.648567\pi$
$647$	−14810.7	−0.899949	−0.449975	+	0.893041i	$0.648567\pi$
$648$	0	0
$649$	−4112.77	−0.248753
$650$	0	0
$651$	4171.04	0.251115
$652$	0	0
$653$	16160.0	0.968436	0.484218	−	0.874947i	$-0.339104\pi$
$653$	16160.0	0.968436	0.484218	+	0.874947i	$0.339104\pi$
$654$	0	0
$655$	2168.27	0.129345
$656$	0	0
$657$	11938.0	0.708896
$658$	0	0
$659$	−7338.75	−0.433805	−0.216902	−	0.976193i	$-0.569595\pi$
$659$	−7338.75	−0.433805	−0.216902	+	0.976193i	$0.569595\pi$
$660$	0	0
$661$	272.612	0.0160414	0.00802070	−	0.999968i	$-0.497447\pi$
$661$	272.612	0.0160414	0.00802070	+	0.999968i	$0.497447\pi$
$662$	0	0
$663$	40780.7	2.38883
$664$	0	0
$665$	−8498.13	−0.495554
$666$	0	0
$667$	−21499.0	−1.24804
$668$	0	0
$669$	22432.8	1.29642
$670$	0	0
$671$	−4363.31	−0.251034
$672$	0	0
$673$	10325.2	0.591394	0.295697	−	0.955282i	$-0.404448\pi$
$673$	10325.2	0.591394	0.295697	+	0.955282i	$0.404448\pi$
$674$	0	0
$675$	−132706.	−7.56717
$676$	0	0
$677$	−3379.67	−0.191863	−0.0959316	−	0.995388i	$-0.530583\pi$
$677$	−3379.67	−0.191863	−0.0959316	+	0.995388i	$0.530583\pi$
$678$	0	0
$679$	−10453.3	−0.590812
$680$	0	0
$681$	−46526.1	−2.61804
$682$	0	0
$683$	17592.9	0.985612	0.492806	−	0.870139i	$-0.335971\pi$
$683$	17592.9	0.985612	0.492806	+	0.870139i	$0.335971\pi$
$684$	0	0
$685$	−56294.9	−3.14003
$686$	0	0
$687$	−8396.53	−0.466299
$688$	0	0
$689$	−9456.10	−0.522857
$690$	0	0
$691$	2462.91	0.135591	0.0677955	−	0.997699i	$-0.478403\pi$
$691$	2462.91	0.135591	0.0677955	+	0.997699i	$0.478403\pi$
$692$	0	0
$693$	5598.76	0.306897
$694$	0	0
$695$	24133.8	1.31719
$696$	0	0
$697$	23839.9	1.29555
$698$	0	0
$699$	34247.3	1.85315
$700$	0	0
$701$	−8955.68	−0.482527	−0.241263	−	0.970460i	$-0.577562\pi$
$701$	−8955.68	−0.482527	−0.241263	+	0.970460i	$0.577562\pi$
$702$	0	0
$703$	−7785.31	−0.417680
$704$	0	0
$705$	95746.6	5.11493
$706$	0	0
$707$	1549.06	0.0824023
$708$	0	0
$709$	5644.88	0.299010	0.149505	−	0.988761i	$-0.452232\pi$
$709$	5644.88	0.299010	0.149505	+	0.988761i	$0.452232\pi$
$710$	0	0
$711$	5171.69	0.272790
$712$	0	0
$713$	−4282.95	−0.224962
$714$	0	0
$715$	−7304.61	−0.382065
$716$	0	0
$717$	−15921.3	−0.829279
$718$	0	0
$719$	16683.2	0.865339	0.432669	−	0.901553i	$-0.357572\pi$
$719$	16683.2	0.865339	0.432669	+	0.901553i	$0.357572\pi$
$720$	0	0
$721$	1066.59	0.0550927
$722$	0	0
$723$	29451.4	1.51495
$724$	0	0
$725$	−87085.3	−4.46106
$726$	0	0
$727$	−13713.5	−0.699593	−0.349797	−	0.936826i	$-0.613749\pi$
$727$	−13713.5	−0.699593	−0.349797	+	0.936826i	$0.613749\pi$
$728$	0	0
$729$	65600.8	3.33287
$730$	0	0
$731$	20615.3	1.04307
$732$	0	0
$733$	14301.9	0.720670	0.360335	−	0.932823i	$-0.382662\pi$
$733$	14301.9	0.720670	0.360335	+	0.932823i	$0.382662\pi$
$734$	0	0
$735$	9976.44	0.500662
$736$	0	0
$737$	4874.97	0.243653
$738$	0	0
$739$	21407.1	1.06559	0.532795	−	0.846244i	$-0.321142\pi$
$739$	21407.1	1.06559	0.532795	+	0.846244i	$0.321142\pi$
$740$	0	0
$741$	−19363.6	−0.959973
$742$	0	0
$743$	35504.9	1.75309	0.876546	−	0.481318i	$-0.159842\pi$
$743$	35504.9	1.75309	0.876546	+	0.481318i	$0.159842\pi$
$744$	0	0
$745$	27402.3	1.34757
$746$	0	0
$747$	−12249.2	−0.599968
$748$	0	0
$749$	−7958.35	−0.388240
$750$	0	0
$751$	−13055.2	−0.634343	−0.317172	−	0.948368i	$-0.602733\pi$
$751$	−13055.2	−0.634343	−0.317172	+	0.948368i	$0.602733\pi$
$752$	0	0
$753$	11269.3	0.545388
$754$	0	0
$755$	−6418.49	−0.309394
$756$	0	0
$757$	7612.29	0.365487	0.182743	−	0.983161i	$-0.441502\pi$
$757$	7612.29	0.365487	0.182743	+	0.983161i	$0.441502\pi$
$758$	0	0
$759$	−7883.75	−0.377025
$760$	0	0
$761$	−4254.54	−0.202664	−0.101332	−	0.994853i	$-0.532310\pi$
$761$	−4254.54	−0.202664	−0.101332	+	0.994853i	$0.532310\pi$
$762$	0	0
$763$	7961.95	0.377775
$764$	0	0
$765$	185907.	8.78623
$766$	0	0
$767$	−12177.0	−0.573252
$768$	0	0
$769$	−35767.1	−1.67724	−0.838619	−	0.544718i	$-0.816637\pi$
$769$	−35767.1	−1.67724	−0.838619	+	0.544718i	$0.816637\pi$
$770$	0	0
$771$	−7003.15	−0.327124
$772$	0	0
$773$	−1684.71	−0.0783893	−0.0391947	−	0.999232i	$-0.512479\pi$
$773$	−1684.71	−0.0783893	−0.0391947	+	0.999232i	$0.512479\pi$
$774$	0	0
$775$	−17348.8	−0.804113
$776$	0	0
$777$	9139.62	0.421985
$778$	0	0
$779$	−11319.7	−0.520631
$780$	0	0
$781$	−6202.78	−0.284190
$782$	0	0
$783$	136724.	6.24024
$784$	0	0
$785$	−32426.6	−1.47434
$786$	0	0
$787$	4348.55	0.196962	0.0984811	−	0.995139i	$-0.468602\pi$
$787$	4348.55	0.196962	0.0984811	+	0.995139i	$0.468602\pi$
$788$	0	0
$789$	57362.2	2.58827
$790$	0	0
$791$	11295.2	0.507728
$792$	0	0
$793$	−12918.7	−0.578509
$794$	0	0
$795$	−59114.6	−2.63721
$796$	0	0
$797$	−41401.2	−1.84003	−0.920017	−	0.391879i	$-0.871825\pi$
$797$	−41401.2	−1.84003	−0.920017	+	0.391879i	$0.871825\pi$
$798$	0	0
$799$	−58969.9	−2.61102
$800$	0	0
$801$	−93428.2	−4.12125
$802$	0	0
$803$	1806.02	0.0793686
$804$	0	0
$805$	−10244.1	−0.448518
$806$	0	0
$807$	−58217.4	−2.53947
$808$	0	0
$809$	−16560.3	−0.719689	−0.359844	−	0.933012i	$-0.617170\pi$
$809$	−16560.3	−0.719689	−0.359844	+	0.933012i	$0.617170\pi$
$810$	0	0
$811$	22522.6	0.975185	0.487593	−	0.873071i	$-0.337875\pi$
$811$	22522.6	0.975185	0.487593	+	0.873071i	$0.337875\pi$
$812$	0	0
$813$	−16829.5	−0.725996
$814$	0	0
$815$	69979.9	3.00771
$816$	0	0
$817$	−9788.60	−0.419167
$818$	0	0
$819$	16576.6	0.707245
$820$	0	0
$821$	−19868.8	−0.844612	−0.422306	−	0.906453i	$-0.638779\pi$
$821$	−19868.8	−0.844612	−0.422306	+	0.906453i	$0.638779\pi$
$822$	0	0
$823$	19053.3	0.806994	0.403497	−	0.914981i	$-0.367795\pi$
$823$	19053.3	0.806994	0.403497	+	0.914981i	$0.367795\pi$
$824$	0	0
$825$	−31934.5	−1.34765
$826$	0	0
$827$	21015.2	0.883639	0.441819	−	0.897104i	$-0.354333\pi$
$827$	21015.2	0.883639	0.441819	+	0.897104i	$0.354333\pi$
$828$	0	0
$829$	13862.3	0.580769	0.290384	−	0.956910i	$-0.406217\pi$
$829$	13862.3	0.580769	0.290384	+	0.956910i	$0.406217\pi$
$830$	0	0
$831$	−54842.5	−2.28937
$832$	0	0
$833$	−6144.44	−0.255573
$834$	0	0
$835$	20607.0	0.854055
$836$	0	0
$837$	27237.6	1.12481
$838$	0	0
$839$	25407.6	1.04549	0.522745	−	0.852489i	$-0.324908\pi$
$839$	25407.6	1.04549	0.522745	+	0.852489i	$0.324908\pi$
$840$	0	0
$841$	65333.2	2.67880
$842$	0	0
$843$	11780.3	0.481299
$844$	0	0
$845$	23168.6	0.943224
$846$	0	0
$847$	847.000	0.0343604
$848$	0	0
$849$	−49843.0	−2.01485
$850$	0	0
$851$	−9384.84	−0.378035
$852$	0	0
$853$	44743.3	1.79599	0.897997	−	0.440002i	$-0.145022\pi$
$853$	44743.3	1.79599	0.897997	+	0.440002i	$0.145022\pi$
$854$	0	0
$855$	−88272.7	−3.53083
$856$	0	0
$857$	8596.07	0.342633	0.171316	−	0.985216i	$-0.445198\pi$
$857$	8596.07	0.342633	0.171316	+	0.985216i	$0.445198\pi$
$858$	0	0
$859$	26967.7	1.07116	0.535579	−	0.844485i	$-0.320093\pi$
$859$	26967.7	1.07116	0.535579	+	0.844485i	$0.320093\pi$
$860$	0	0
$861$	13288.9	0.525997
$862$	0	0
$863$	−24400.9	−0.962477	−0.481239	−	0.876590i	$-0.659813\pi$
$863$	−24400.9	−0.962477	−0.481239	+	0.876590i	$0.659813\pi$
$864$	0	0
$865$	−43857.1	−1.72391
$866$	0	0
$867$	−107957.	−4.22886
$868$	0	0
$869$	782.391	0.0305418
$870$	0	0
$871$	14433.6	0.561499
$872$	0	0
$873$	−108582.	−4.20955
$874$	0	0
$875$	−23654.6	−0.913911
$876$	0	0
$877$	−25012.4	−0.963067	−0.481533	−	0.876428i	$-0.659920\pi$
$877$	−25012.4	−0.963067	−0.481533	+	0.876428i	$0.659920\pi$
$878$	0	0
$879$	−43447.4	−1.66717
$880$	0	0
$881$	4631.24	0.177106	0.0885530	−	0.996071i	$-0.471776\pi$
$881$	4631.24	0.177106	0.0885530	+	0.996071i	$0.471776\pi$
$882$	0	0
$883$	−41219.5	−1.57095	−0.785474	−	0.618895i	$-0.787581\pi$
$883$	−41219.5	−1.57095	−0.785474	+	0.618895i	$0.787581\pi$
$884$	0	0
$885$	−76124.0	−2.89139
$886$	0	0
$887$	42252.8	1.59945	0.799724	−	0.600367i	$-0.204979\pi$
$887$	42252.8	1.59945	0.799724	+	0.600367i	$0.204979\pi$
$888$	0	0
$889$	16047.4	0.605413
$890$	0	0
$891$	28541.8	1.07316
$892$	0	0
$893$	28000.2	1.04926
$894$	0	0
$895$	−4557.69	−0.170220
$896$	0	0
$897$	−23341.9	−0.868856
$898$	0	0
$899$	17874.1	0.663109
$900$	0	0
$901$	36408.4	1.34622
$902$	0	0
$903$	11491.4	0.423488
$904$	0	0
$905$	59456.9	2.18388
$906$	0	0
$907$	−12925.6	−0.473195	−0.236598	−	0.971608i	$-0.576032\pi$
$907$	−12925.6	−0.473195	−0.236598	+	0.971608i	$0.576032\pi$
$908$	0	0
$909$	16090.6	0.587118
$910$	0	0
$911$	45887.0	1.66883	0.834415	−	0.551137i	$-0.185806\pi$
$911$	45887.0	1.66883	0.834415	+	0.551137i	$0.185806\pi$
$912$	0	0
$913$	−1853.11	−0.0671730
$914$	0	0
$915$	−80761.2	−2.91791
$916$	0	0
$917$	−744.395	−0.0268071
$918$	0	0
$919$	8834.65	0.317114	0.158557	−	0.987350i	$-0.449316\pi$
$919$	8834.65	0.317114	0.158557	+	0.987350i	$0.449316\pi$
$920$	0	0
$921$	−81100.4	−2.90157
$922$	0	0
$923$	−18365.0	−0.654919
$924$	0	0
$925$	−38014.9	−1.35127
$926$	0	0
$927$	11079.0	0.392536
$928$	0	0
$929$	−30812.7	−1.08819	−0.544097	−	0.839023i	$-0.683128\pi$
$929$	−30812.7	−1.08819	−0.544097	+	0.839023i	$0.683128\pi$
$930$	0	0
$931$	2917.52	0.102704
$932$	0	0
$933$	−33161.0	−1.16360
$934$	0	0
$935$	28124.6	0.983715
$936$	0	0
$937$	−8602.19	−0.299916	−0.149958	−	0.988692i	$-0.547914\pi$
$937$	−8602.19	−0.299916	−0.149958	+	0.988692i	$0.547914\pi$
$938$	0	0
$939$	−79127.8	−2.74999
$940$	0	0
$941$	46833.7	1.62246	0.811230	−	0.584727i	$-0.198798\pi$
$941$	46833.7	1.62246	0.811230	+	0.584727i	$0.198798\pi$
$942$	0	0
$943$	−13645.4	−0.471215
$944$	0	0
$945$	65147.8	2.24260
$946$	0	0
$947$	8080.17	0.277265	0.138633	−	0.990344i	$-0.455729\pi$
$947$	8080.17	0.277265	0.138633	+	0.990344i	$0.455729\pi$
$948$	0	0
$949$	5347.20	0.182906
$950$	0	0
$951$	6369.02	0.217171
$952$	0	0
$953$	−35073.6	−1.19218	−0.596088	−	0.802919i	$-0.703279\pi$
$953$	−35073.6	−1.19218	−0.596088	+	0.802919i	$0.703279\pi$
$954$	0	0
$955$	26728.4	0.905666
$956$	0	0
$957$	32901.4	1.11134
$958$	0	0
$959$	19326.8	0.650777
$960$	0	0
$961$	−26230.2	−0.880474
$962$	0	0
$963$	−82665.9	−2.76622
$964$	0	0
$965$	17037.8	0.568360
$966$	0	0
$967$	−12912.0	−0.429391	−0.214696	−	0.976681i	$-0.568876\pi$
$967$	−12912.0	−0.429391	−0.214696	+	0.976681i	$0.568876\pi$
$968$	0	0
$969$	74554.9	2.47167
$970$	0	0
$971$	−4264.05	−0.140927	−0.0704634	−	0.997514i	$-0.522448\pi$
$971$	−4264.05	−0.140927	−0.0704634	+	0.997514i	$0.522448\pi$
$972$	0	0
$973$	−8285.44	−0.272990
$974$	0	0
$975$	−94550.4	−3.10568
$976$	0	0
$977$	−2924.54	−0.0957671	−0.0478835	−	0.998853i	$-0.515248\pi$
$977$	−2924.54	−0.0957671	−0.0478835	+	0.998853i	$0.515248\pi$
$978$	0	0
$979$	−14134.1	−0.461419
$980$	0	0
$981$	82703.3	2.69165
$982$	0	0
$983$	48646.1	1.57840	0.789201	−	0.614135i	$-0.210495\pi$
$983$	48646.1	1.57840	0.789201	+	0.614135i	$0.210495\pi$
$984$	0	0
$985$	58350.7	1.88752
$986$	0	0
$987$	−32871.1	−1.06008
$988$	0	0
$989$	−11799.7	−0.379382
$990$	0	0
$991$	−7872.50	−0.252349	−0.126175	−	0.992008i	$-0.540270\pi$
$991$	−7872.50	−0.252349	−0.126175	+	0.992008i	$0.540270\pi$
$992$	0	0
$993$	−73528.7	−2.34981
$994$	0	0
$995$	107325.	3.41953
$996$	0	0
$997$	6658.54	0.211513	0.105756	−	0.994392i	$-0.466274\pi$
$997$	6658.54	0.211513	0.105756	+	0.994392i	$0.466274\pi$
$998$	0	0
$999$	59683.3	1.89019

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	1232.4.a.z.1.1		5
4.3	odd	2		616.4.a.g.1.5	✓	5

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
616.4.a.g.1.5	✓	5	4.3	odd	2
1232.4.a.z.1.1		5	1.1	even	1	trivial

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:	\( N \)	\(=\)	\( 1232 = 2^{4} \cdot 7 \cdot 11 \)
Weight:	\( k \)	\(=\)	\( 4 \)
Character orbit:	\([\chi]\)	\(=\)	1232.a (trivial)

\(f(q)\)	\(=\)	\(q-9.98555 q^{3} -20.3895 q^{5} +7.00000 q^{7} +72.7112 q^{9} +O(q^{10})\)	\(q-9.98555 q^{3} -20.3895 q^{5} +7.00000 q^{7} +72.7112 q^{9} +11.0000 q^{11} +32.5684 q^{13} +203.601 q^{15} -125.397 q^{17} +59.5412 q^{19} -69.8988 q^{21} +71.7742 q^{23} +290.733 q^{25} -456.451 q^{27} -299.537 q^{29} -59.6725 q^{31} -109.841 q^{33} -142.727 q^{35} -130.755 q^{37} -325.213 q^{39} -190.116 q^{41} -164.400 q^{43} -1482.55 q^{45} +470.267 q^{47} +49.0000 q^{49} +1252.16 q^{51} -290.346 q^{53} -224.285 q^{55} -594.552 q^{57} -373.889 q^{59} -396.665 q^{61} +508.978 q^{63} -664.055 q^{65} +443.179 q^{67} -716.704 q^{69} -563.889 q^{71} +164.184 q^{73} -2903.13 q^{75} +77.0000 q^{77} +71.1265 q^{79} +2594.71 q^{81} -168.464 q^{83} +2556.78 q^{85} +2991.04 q^{87} -1284.92 q^{89} +227.979 q^{91} +595.863 q^{93} -1214.02 q^{95} -1493.33 q^{97} +799.823 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 5 q - 2 q^{3} - 14 q^{5} + 35 q^{7} + 79 q^{9}+O(q^{10}) \) 5 * q - 2 * q^3 - 14 * q^5 + 35 * q^7 + 79 * q^9	\( 5 q - 2 q^{3} - 14 q^{5} + 35 q^{7} + 79 q^{9} + 55 q^{11} + 194 q^{13} + 118 q^{15} - 248 q^{17} + 190 q^{19} - 14 q^{21} + 34 q^{23} + 529 q^{25} - 254 q^{27} - 334 q^{29} + 172 q^{31} - 22 q^{33} - 98 q^{35} + 360 q^{37} + 360 q^{39} - 852 q^{41} + 1016 q^{43} - 2164 q^{45} + 194 q^{47} + 245 q^{49} + 836 q^{51} - 934 q^{53} - 154 q^{55} - 980 q^{57} + 542 q^{59} + 838 q^{61} + 553 q^{63} - 1780 q^{65} + 1830 q^{67} + 50 q^{69} - 422 q^{71} - 68 q^{73} - 320 q^{75} + 385 q^{77} + 8 q^{79} + 1889 q^{81} + 366 q^{83} + 2056 q^{85} + 4384 q^{87} - 524 q^{89} + 1358 q^{91} - 2242 q^{93} + 2884 q^{95} - 1336 q^{97} + 869 q^{99}+O(q^{100}) \) 5 * q - 2 * q^3 - 14 * q^5 + 35 * q^7 + 79 * q^9 + 55 * q^11 + 194 * q^13 + 118 * q^15 - 248 * q^17 + 190 * q^19 - 14 * q^21 + 34 * q^23 + 529 * q^25 - 254 * q^27 - 334 * q^29 + 172 * q^31 - 22 * q^33 - 98 * q^35 + 360 * q^37 + 360 * q^39 - 852 * q^41 + 1016 * q^43 - 2164 * q^45 + 194 * q^47 + 245 * q^49 + 836 * q^51 - 934 * q^53 - 154 * q^55 - 980 * q^57 + 542 * q^59 + 838 * q^61 + 553 * q^63 - 1780 * q^65 + 1830 * q^67 + 50 * q^69 - 422 * q^71 - 68 * q^73 - 320 * q^75 + 385 * q^77 + 8 * q^79 + 1889 * q^81 + 366 * q^83 + 2056 * q^85 + 4384 * q^87 - 524 * q^89 + 1358 * q^91 - 2242 * q^93 + 2884 * q^95 - 1336 * q^97 + 869 * q^99

Introduction

L-functions

Modular forms

Varieties

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Database

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