LMFDB - Embedded newform 232.4.a.e.1.3

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:	$1$
Dimension:	$6$
Coefficient field:	$\mathbb{Q}[x]/(x^{6} - \cdots)$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{6} - 40x^{4} - 88x^{3} - 8x^{2} + 48x - 9 $ x^6 - 40x^4 - 88x^3 - 8x^2 + 48x - 9
Coefficient ring:	$\Z[a_1, \ldots, a_{5}]$
Coefficient ring index:	$ 2^{7} $
Twist minimal:	yes
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.3
Root			$-1.32108$ 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)

$f(q)$	$=$	$q-0.116789 q^{3} -14.4485 q^{5} +30.0998 q^{7} -26.9864 q^{9} +O(q^{10})$	$q-0.116789 q^{3} -14.4485 q^{5} +30.0998 q^{7} -26.9864 q^{9} +19.6760 q^{11} +28.1437 q^{13} +1.68742 q^{15} -105.778 q^{17} -141.581 q^{19} -3.51532 q^{21} -68.5580 q^{23} +83.7592 q^{25} +6.30499 q^{27} -29.0000 q^{29} -302.329 q^{31} -2.29793 q^{33} -434.897 q^{35} +193.709 q^{37} -3.28687 q^{39} -412.636 q^{41} +347.292 q^{43} +389.912 q^{45} -147.203 q^{47} +563.000 q^{49} +12.3537 q^{51} -125.506 q^{53} -284.289 q^{55} +16.5351 q^{57} +293.208 q^{59} -480.509 q^{61} -812.285 q^{63} -406.635 q^{65} +685.288 q^{67} +8.00679 q^{69} -844.181 q^{71} -509.039 q^{73} -9.78212 q^{75} +592.245 q^{77} -561.456 q^{79} +727.895 q^{81} -274.580 q^{83} +1528.33 q^{85} +3.38687 q^{87} +866.738 q^{89} +847.122 q^{91} +35.3085 q^{93} +2045.64 q^{95} +582.748 q^{97} -530.984 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 6 q - 5 q^{3} - 5 q^{5} - 38 q^{7} + 47 q^{9}+O(q^{10}) $ 6 * q - 5 * q^3 - 5 * q^5 - 38 * q^7 + 47 * q^9	$ 6 q - 5 q^{3} - 5 q^{5} - 38 q^{7} + 47 q^{9} - 19 q^{11} + 13 q^{13} - 191 q^{15} - 218 q^{17} - 290 q^{19} - 266 q^{21} - 196 q^{23} - 13 q^{25} - 437 q^{27} - 174 q^{29} - 675 q^{31} + 291 q^{33} - 466 q^{35} - 238 q^{37} - 1297 q^{39} - 464 q^{41} - 579 q^{43} - 148 q^{45} - 975 q^{47} + 914 q^{49} - 576 q^{51} + 515 q^{53} - 1605 q^{55} - 340 q^{57} - 108 q^{59} + 1158 q^{61} - 1136 q^{63} + 1239 q^{65} - 80 q^{67} + 2568 q^{69} + 438 q^{71} + 262 q^{73} + 1766 q^{75} + 194 q^{77} - 237 q^{79} + 2554 q^{81} - 1288 q^{83} + 3112 q^{85} + 145 q^{87} - 252 q^{89} + 2450 q^{91} + 2131 q^{93} + 180 q^{95} + 380 q^{97} + 2264 q^{99}+O(q^{100}) $ 6 * q - 5 * q^3 - 5 * q^5 - 38 * q^7 + 47 * q^9 - 19 * q^11 + 13 * q^13 - 191 * q^15 - 218 * q^17 - 290 * q^19 - 266 * q^21 - 196 * q^23 - 13 * q^25 - 437 * q^27 - 174 * q^29 - 675 * q^31 + 291 * q^33 - 466 * q^35 - 238 * q^37 - 1297 * q^39 - 464 * q^41 - 579 * q^43 - 148 * q^45 - 975 * q^47 + 914 * q^49 - 576 * q^51 + 515 * q^53 - 1605 * q^55 - 340 * q^57 - 108 * q^59 + 1158 * q^61 - 1136 * q^63 + 1239 * q^65 - 80 * q^67 + 2568 * q^69 + 438 * q^71 + 262 * q^73 + 1766 * q^75 + 194 * q^77 - 237 * q^79 + 2554 * q^81 - 1288 * q^83 + 3112 * q^85 + 145 * q^87 - 252 * q^89 + 2450 * q^91 + 2131 * q^93 + 180 * q^95 + 380 * q^97 + 2264 * 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
$3$	−0.116789	−0.0224760	−0.0112380	−	0.999937i	$-0.503577\pi$
$3$	−0.116789	−0.0224760	−0.0112380	+	0.999937i	$0.503577\pi$
$4$	0	0
$5$	−14.4485	−1.29231	−0.646157	−	0.763205i	$-0.723625\pi$
$5$	−14.4485	−1.29231	−0.646157	+	0.763205i	$0.723625\pi$
$6$	0	0
$7$	30.0998	1.62524	0.812619	−	0.582795i	$-0.198041\pi$
$7$	30.0998	1.62524	0.812619	+	0.582795i	$0.198041\pi$
$8$	0	0
$9$	−26.9864	−0.999495
$10$	0	0
$11$	19.6760	0.539322	0.269661	−	0.962955i	$-0.413088\pi$
$11$	19.6760	0.539322	0.269661	+	0.962955i	$0.413088\pi$
$12$	0	0
$13$	28.1437	0.600436	0.300218	−	0.953871i	$-0.402941\pi$
$13$	28.1437	0.600436	0.300218	+	0.953871i	$0.402941\pi$
$14$	0	0
$15$	1.68742	0.0290460
$16$	0	0
$17$	−105.778	−1.50911	−0.754557	−	0.656234i	$-0.772148\pi$
$17$	−105.778	−1.50911	−0.754557	+	0.656234i	$0.772148\pi$
$18$	0	0
$19$	−141.581	−1.70952	−0.854762	−	0.519020i	$-0.826297\pi$
$19$	−141.581	−1.70952	−0.854762	+	0.519020i	$0.826297\pi$
$20$	0	0
$21$	−3.51532	−0.0365288
$22$	0	0
$23$	−68.5580	−0.621536	−0.310768	−	0.950486i	$-0.600586\pi$
$23$	−68.5580	−0.621536	−0.310768	+	0.950486i	$0.600586\pi$
$24$	0	0
$25$	83.7592	0.670074
$26$	0	0
$27$	6.30499	0.0449406
$28$	0	0
$29$	−29.0000	−0.185695
$29$	−29.0000	−0.185695
$30$	0	0
$31$	−302.329	−1.75161	−0.875803	−	0.482668i	$-0.839668\pi$
$31$	−302.329	−1.75161	−0.875803	+	0.482668i	$0.839668\pi$
$32$	0	0
$33$	−2.29793	−0.0121218
$34$	0	0
$35$	−434.897	−2.10032
$36$	0	0
$37$	193.709	0.860692	0.430346	−	0.902664i	$-0.358391\pi$
$37$	193.709	0.860692	0.430346	+	0.902664i	$0.358391\pi$
$38$	0	0
$39$	−3.28687	−0.0134954
$40$	0	0
$41$	−412.636	−1.57178	−0.785889	−	0.618368i	$-0.787794\pi$
$41$	−412.636	−1.57178	−0.785889	+	0.618368i	$0.787794\pi$
$42$	0	0
$43$	347.292	1.23166	0.615832	−	0.787878i	$-0.288820\pi$
$43$	347.292	1.23166	0.615832	+	0.787878i	$0.288820\pi$
$44$	0	0
$45$	389.912	1.29166
$46$	0	0
$47$	−147.203	−0.456846	−0.228423	−	0.973562i	$-0.573357\pi$
$47$	−147.203	−0.456846	−0.228423	+	0.973562i	$0.573357\pi$
$48$	0	0
$49$	563.000	1.64140
$50$	0	0
$51$	12.3537	0.0339188
$52$	0	0
$53$	−125.506	−0.325274	−0.162637	−	0.986686i	$-0.552000\pi$
$53$	−125.506	−0.325274	−0.162637	+	0.986686i	$0.552000\pi$
$54$	0	0
$55$	−284.289	−0.696973
$56$	0	0
$57$	16.5351	0.0384232
$58$	0	0
$59$	293.208	0.646991	0.323495	−	0.946230i	$-0.395142\pi$
$59$	293.208	0.646991	0.323495	+	0.946230i	$0.395142\pi$
$60$	0	0
$61$	−480.509	−1.00857	−0.504286	−	0.863537i	$-0.668244\pi$
$61$	−480.509	−1.00857	−0.504286	+	0.863537i	$0.668244\pi$
$62$	0	0
$63$	−812.285	−1.62442
$64$	0	0
$65$	−406.635	−0.775952
$66$	0	0
$67$	685.288	1.24957	0.624785	−	0.780797i	$-0.285187\pi$
$67$	685.288	1.24957	0.624785	+	0.780797i	$0.285187\pi$
$68$	0	0
$69$	8.00679	0.0139696
$70$	0	0
$71$	−844.181	−1.41107	−0.705534	−	0.708676i	$-0.749293\pi$
$71$	−844.181	−1.41107	−0.705534	+	0.708676i	$0.749293\pi$
$72$	0	0
$73$	−509.039	−0.816143	−0.408072	−	0.912950i	$-0.633799\pi$
$73$	−509.039	−0.816143	−0.408072	+	0.912950i	$0.633799\pi$
$74$	0	0
$75$	−9.78212	−0.0150606
$76$	0	0
$77$	592.245	0.876527
$78$	0	0
$79$	−561.456	−0.799605	−0.399802	−	0.916601i	$-0.630921\pi$
$79$	−561.456	−0.799605	−0.399802	+	0.916601i	$0.630921\pi$
$80$	0	0
$81$	727.895	0.998485
$82$	0	0
$83$	−274.580	−0.363121	−0.181561	−	0.983380i	$-0.558115\pi$
$83$	−274.580	−0.363121	−0.181561	+	0.983380i	$0.558115\pi$
$84$	0	0
$85$	1528.33	1.95025
$86$	0	0
$87$	3.38687	0.00417368
$88$	0	0
$89$	866.738	1.03229	0.516146	−	0.856501i	$-0.327366\pi$
$89$	866.738	1.03229	0.516146	+	0.856501i	$0.327366\pi$
$90$	0	0
$91$	847.122	0.975852
$92$	0	0
$93$	35.3085	0.0393691
$94$	0	0
$95$	2045.64	2.20924
$96$	0	0
$97$	582.748	0.609991	0.304995	−	0.952354i	$-0.401345\pi$
$97$	582.748	0.609991	0.304995	+	0.952354i	$0.401345\pi$
$98$	0	0
$99$	−530.984	−0.539050
$100$	0	0
$101$	−821.853	−0.809677	−0.404839	−	0.914388i	$-0.632672\pi$
$101$	−821.853	−0.809677	−0.404839	+	0.914388i	$0.632672\pi$
$102$	0	0
$103$	1509.59	1.44412	0.722062	−	0.691829i	$-0.243195\pi$
$103$	1509.59	1.44412	0.722062	+	0.691829i	$0.243195\pi$
$104$	0	0
$105$	50.7910	0.0472066
$106$	0	0
$107$	−4.54904	−0.00411002	−0.00205501	−	0.999998i	$-0.500654\pi$
$107$	−4.54904	−0.00411002	−0.00205501	+	0.999998i	$0.500654\pi$
$108$	0	0
$109$	1691.96	1.48679	0.743396	−	0.668851i	$-0.233214\pi$
$109$	1691.96	1.48679	0.743396	+	0.668851i	$0.233214\pi$
$110$	0	0
$111$	−22.6230	−0.0193449
$112$	0	0
$113$	1319.79	1.09872	0.549362	−	0.835585i	$-0.314871\pi$
$113$	1319.79	1.09872	0.549362	+	0.835585i	$0.314871\pi$
$114$	0	0
$115$	990.560	0.803219
$116$	0	0
$117$	−759.497	−0.600133
$118$	0	0
$119$	−3183.90	−2.45267
$120$	0	0
$121$	−943.854	−0.709131
$122$	0	0
$123$	48.1911	0.0353272
$124$	0	0
$125$	595.868	0.426368
$126$	0	0
$127$	−570.013	−0.398271	−0.199136	−	0.979972i	$-0.563813\pi$
$127$	−570.013	−0.398271	−0.199136	+	0.979972i	$0.563813\pi$
$128$	0	0
$129$	−40.5597	−0.0276828
$130$	0	0
$131$	26.0927	0.0174025	0.00870126	−	0.999962i	$-0.497230\pi$
$131$	26.0927	0.0174025	0.00870126	+	0.999962i	$0.497230\pi$
$132$	0	0
$133$	−4261.57	−2.77838
$134$	0	0
$135$	−91.0976	−0.0580773
$136$	0	0
$137$	2496.40	1.55680	0.778402	−	0.627766i	$-0.216031\pi$
$137$	2496.40	1.55680	0.778402	+	0.627766i	$0.216031\pi$
$138$	0	0
$139$	−1357.19	−0.828166	−0.414083	−	0.910239i	$-0.635898\pi$
$139$	−1357.19	−0.828166	−0.414083	+	0.910239i	$0.635898\pi$
$140$	0	0
$141$	17.1916	0.0102681
$142$	0	0
$143$	553.757	0.323829
$144$	0	0
$145$	419.007	0.239977
$146$	0	0
$147$	−65.7519	−0.0368920
$148$	0	0
$149$	−2560.80	−1.40798	−0.703990	−	0.710210i	$-0.748600\pi$
$149$	−2560.80	−1.40798	−0.703990	+	0.710210i	$0.748600\pi$
$150$	0	0
$151$	−720.358	−0.388225	−0.194112	−	0.980979i	$-0.562183\pi$
$151$	−720.358	−0.388225	−0.194112	+	0.980979i	$0.562183\pi$
$152$	0	0
$153$	2854.56	1.50835
$154$	0	0
$155$	4368.19	2.26362
$156$	0	0
$157$	3741.33	1.90185	0.950925	−	0.309422i	$-0.100135\pi$
$157$	3741.33	1.90185	0.950925	+	0.309422i	$0.100135\pi$
$158$	0	0
$159$	14.6576	0.00731086
$160$	0	0
$161$	−2063.58	−1.01014
$162$	0	0
$163$	−2734.41	−1.31396	−0.656981	−	0.753907i	$-0.728167\pi$
$163$	−2734.41	−1.31396	−0.656981	+	0.753907i	$0.728167\pi$
$164$	0	0
$165$	33.2017	0.0156652
$166$	0	0
$167$	2248.16	1.04173	0.520863	−	0.853640i	$-0.325610\pi$
$167$	2248.16	1.04173	0.520863	+	0.853640i	$0.325610\pi$
$168$	0	0
$169$	−1404.93	−0.639476
$170$	0	0
$171$	3820.76	1.70866
$172$	0	0
$173$	−3884.49	−1.70712	−0.853562	−	0.520992i	$-0.825562\pi$
$173$	−3884.49	−1.70712	−0.853562	+	0.520992i	$0.825562\pi$
$174$	0	0
$175$	2521.14	1.08903
$176$	0	0
$177$	−34.2434	−0.0145417
$178$	0	0
$179$	−1837.35	−0.767207	−0.383604	−	0.923498i	$-0.625317\pi$
$179$	−1837.35	−0.767207	−0.383604	+	0.923498i	$0.625317\pi$
$180$	0	0
$181$	−903.126	−0.370878	−0.185439	−	0.982656i	$-0.559371\pi$
$181$	−903.126	−0.370878	−0.185439	+	0.982656i	$0.559371\pi$
$182$	0	0
$183$	56.1180	0.0226686
$184$	0	0
$185$	−2798.81	−1.11228
$186$	0	0
$187$	−2081.29	−0.813899
$188$	0	0
$189$	189.779	0.0730391
$190$	0	0
$191$	−905.635	−0.343086	−0.171543	−	0.985177i	$-0.554875\pi$
$191$	−905.635	−0.343086	−0.171543	+	0.985177i	$0.554875\pi$
$192$	0	0
$193$	−4398.31	−1.64040	−0.820201	−	0.572076i	$-0.806138\pi$
$193$	−4398.31	−1.64040	−0.820201	+	0.572076i	$0.806138\pi$
$194$	0	0
$195$	47.4903	0.0174403
$196$	0	0
$197$	−745.810	−0.269730	−0.134865	−	0.990864i	$-0.543060\pi$
$197$	−745.810	−0.269730	−0.134865	+	0.990864i	$0.543060\pi$
$198$	0	0
$199$	−2952.19	−1.05163	−0.525817	−	0.850598i	$-0.676240\pi$
$199$	−2952.19	−1.05163	−0.525817	+	0.850598i	$0.676240\pi$
$200$	0	0
$201$	−80.0338	−0.0280853
$202$	0	0
$203$	−872.895	−0.301799
$204$	0	0
$205$	5961.97	2.03123
$206$	0	0
$207$	1850.13	0.621222
$208$	0	0
$209$	−2785.76	−0.921984
$210$	0	0
$211$	949.166	0.309684	0.154842	−	0.987939i	$-0.450513\pi$
$211$	949.166	0.309684	0.154842	+	0.987939i	$0.450513\pi$
$212$	0	0
$213$	98.5907	0.0317151
$214$	0	0
$215$	−5017.85	−1.59169
$216$	0	0
$217$	−9100.04	−2.84678
$218$	0	0
$219$	59.4499	0.0183436
$220$	0	0
$221$	−2976.99	−0.906127
$222$	0	0
$223$	−1247.64	−0.374654	−0.187327	−	0.982298i	$-0.559982\pi$
$223$	−1247.64	−0.374654	−0.187327	+	0.982298i	$0.559982\pi$
$224$	0	0
$225$	−2260.36	−0.669735
$226$	0	0
$227$	2641.84	0.772445	0.386223	−	0.922406i	$-0.373780\pi$
$227$	2641.84	0.772445	0.386223	+	0.922406i	$0.373780\pi$
$228$	0	0
$229$	4130.76	1.19200	0.596001	−	0.802984i	$-0.296755\pi$
$229$	4130.76	1.19200	0.596001	+	0.802984i	$0.296755\pi$
$230$	0	0
$231$	−69.1674	−0.0197008
$232$	0	0
$233$	2772.87	0.779643	0.389822	−	0.920890i	$-0.372537\pi$
$233$	2772.87	0.779643	0.389822	+	0.920890i	$0.372537\pi$
$234$	0	0
$235$	2126.86	0.590389
$236$	0	0
$237$	65.5717	0.0179719
$238$	0	0
$239$	6458.17	1.74788	0.873941	−	0.486031i	$-0.161556\pi$
$239$	6458.17	1.74788	0.873941	+	0.486031i	$0.161556\pi$
$240$	0	0
$241$	3776.61	1.00943	0.504716	−	0.863286i	$-0.331597\pi$
$241$	3776.61	1.00943	0.504716	+	0.863286i	$0.331597\pi$
$242$	0	0
$243$	−255.245	−0.0673825
$244$	0	0
$245$	−8134.50	−2.12120
$246$	0	0
$247$	−3984.63	−1.02646
$248$	0	0
$249$	32.0678	0.00816150
$250$	0	0
$251$	1570.69	0.394985	0.197492	−	0.980304i	$-0.436720\pi$
$251$	1570.69	0.394985	0.197492	+	0.980304i	$0.436720\pi$
$252$	0	0
$253$	−1348.95	−0.335208
$254$	0	0
$255$	−178.492	−0.0438337
$256$	0	0
$257$	−1015.18	−0.246402	−0.123201	−	0.992382i	$-0.539316\pi$
$257$	−1015.18	−0.246402	−0.123201	+	0.992382i	$0.539316\pi$
$258$	0	0
$259$	5830.62	1.39883
$260$	0	0
$261$	782.604	0.185602
$262$	0	0
$263$	−3103.58	−0.727661	−0.363830	−	0.931465i	$-0.618531\pi$
$263$	−3103.58	−0.727661	−0.363830	+	0.931465i	$0.618531\pi$
$264$	0	0
$265$	1813.37	0.420357
$266$	0	0
$267$	−101.225	−0.0232018
$268$	0	0
$269$	−1449.67	−0.328581	−0.164290	−	0.986412i	$-0.552533\pi$
$269$	−1449.67	−0.328581	−0.164290	+	0.986412i	$0.552533\pi$
$270$	0	0
$271$	−2573.95	−0.576960	−0.288480	−	0.957486i	$-0.593150\pi$
$271$	−2573.95	−0.576960	−0.288480	+	0.957486i	$0.593150\pi$
$272$	0	0
$273$	−98.9341	−0.0219332
$274$	0	0
$275$	1648.05	0.361386
$276$	0	0
$277$	6992.04	1.51665	0.758323	−	0.651878i	$-0.226019\pi$
$277$	6992.04	1.51665	0.758323	+	0.651878i	$0.226019\pi$
$278$	0	0
$279$	8158.75	1.75072
$280$	0	0
$281$	−4719.79	−1.00199	−0.500994	−	0.865451i	$-0.667032\pi$
$281$	−4719.79	−1.00199	−0.500994	+	0.865451i	$0.667032\pi$
$282$	0	0
$283$	4706.84	0.988667	0.494334	−	0.869272i	$-0.335412\pi$
$283$	4706.84	0.988667	0.494334	+	0.869272i	$0.335412\pi$
$284$	0	0
$285$	−238.907	−0.0496548
$286$	0	0
$287$	−12420.3	−2.55451
$288$	0	0
$289$	6275.99	1.27743
$290$	0	0
$291$	−68.0583	−0.0137101
$292$	0	0
$293$	−987.186	−0.196833	−0.0984164	−	0.995145i	$-0.531378\pi$
$293$	−987.186	−0.196833	−0.0984164	+	0.995145i	$0.531378\pi$
$294$	0	0
$295$	−4236.42	−0.836115
$296$	0	0
$297$	124.057	0.0242375
$298$	0	0
$299$	−1929.48	−0.373193
$300$	0	0
$301$	10453.4	2.00175
$302$	0	0
$303$	95.9830	0.0181983
$304$	0	0
$305$	6942.64	1.30339
$306$	0	0
$307$	326.127	0.0606289	0.0303144	−	0.999540i	$-0.490349\pi$
$307$	326.127	0.0606289	0.0303144	+	0.999540i	$0.490349\pi$
$308$	0	0
$309$	−176.303	−0.0324581
$310$	0	0
$311$	−7616.01	−1.38863	−0.694316	−	0.719670i	$-0.744293\pi$
$311$	−7616.01	−1.38863	−0.694316	+	0.719670i	$0.744293\pi$
$312$	0	0
$313$	3697.12	0.667647	0.333823	−	0.942636i	$-0.391661\pi$
$313$	3697.12	0.667647	0.333823	+	0.942636i	$0.391661\pi$
$314$	0	0
$315$	11736.3	2.09926
$316$	0	0
$317$	−238.354	−0.0422312	−0.0211156	−	0.999777i	$-0.506722\pi$
$317$	−238.354	−0.0422312	−0.0211156	+	0.999777i	$0.506722\pi$
$318$	0	0
$319$	−570.605	−0.100150
$320$	0	0
$321$	0.531276	9.23767e−5 0
$322$	0	0
$323$	14976.2	2.57987
$324$	0	0
$325$	2357.30	0.402337
$326$	0	0
$327$	−197.602	−0.0334171
$328$	0	0
$329$	−4430.79	−0.742484
$330$	0	0
$331$	−1039.73	−0.172655	−0.0863274	−	0.996267i	$-0.527513\pi$
$331$	−1039.73	−0.172655	−0.0863274	+	0.996267i	$0.527513\pi$
$332$	0	0
$333$	−5227.51	−0.860258
$334$	0	0
$335$	−9901.38	−1.61484
$336$	0	0
$337$	1299.60	0.210071	0.105036	−	0.994468i	$-0.466504\pi$
$337$	1299.60	0.210071	0.105036	+	0.994468i	$0.466504\pi$
$338$	0	0
$339$	−154.137	−0.0246949
$340$	0	0
$341$	−5948.63	−0.944681
$342$	0	0
$343$	6621.95	1.04243
$344$	0	0
$345$	−115.686	−0.0180531
$346$	0	0
$347$	−5381.28	−0.832513	−0.416256	−	0.909247i	$-0.636658\pi$
$347$	−5381.28	−0.832513	−0.416256	+	0.909247i	$0.636658\pi$
$348$	0	0
$349$	−375.125	−0.0575358	−0.0287679	−	0.999586i	$-0.509158\pi$
$349$	−375.125	−0.0575358	−0.0287679	+	0.999586i	$0.509158\pi$
$350$	0	0
$351$	177.446	0.0269840
$352$	0	0
$353$	−661.715	−0.0997720	−0.0498860	−	0.998755i	$-0.515886\pi$
$353$	−661.715	−0.0997720	−0.0498860	+	0.998755i	$0.515886\pi$
$354$	0	0
$355$	12197.2	1.82354
$356$	0	0
$357$	371.843	0.0551261
$358$	0	0
$359$	−911.647	−0.134025	−0.0670124	−	0.997752i	$-0.521347\pi$
$359$	−911.647	−0.134025	−0.0670124	+	0.997752i	$0.521347\pi$
$360$	0	0
$361$	13186.2	1.92247
$362$	0	0
$363$	110.231	0.0159384
$364$	0	0
$365$	7354.85	1.05471
$366$	0	0
$367$	−3611.47	−0.513671	−0.256836	−	0.966455i	$-0.582680\pi$
$367$	−3611.47	−0.513671	−0.256836	+	0.966455i	$0.582680\pi$
$368$	0	0
$369$	11135.5	1.57098
$370$	0	0
$371$	−3777.70	−0.528648
$372$	0	0
$373$	−9866.50	−1.36962	−0.684810	−	0.728722i	$-0.740115\pi$
$373$	−9866.50	−1.36962	−0.684810	+	0.728722i	$0.740115\pi$
$374$	0	0
$375$	−69.5905	−0.00958304
$376$	0	0
$377$	−816.169	−0.111498
$378$	0	0
$379$	−1657.64	−0.224663	−0.112331	−	0.993671i	$-0.535832\pi$
$379$	−1657.64	−0.224663	−0.112331	+	0.993671i	$0.535832\pi$
$380$	0	0
$381$	66.5710	0.00895153
$382$	0	0
$383$	−10547.0	−1.40712	−0.703558	−	0.710638i	$-0.748406\pi$
$383$	−10547.0	−1.40712	−0.703558	+	0.710638i	$0.748406\pi$
$384$	0	0
$385$	−8557.05	−1.13275
$386$	0	0
$387$	−9372.15	−1.23104
$388$	0	0
$389$	8547.97	1.11414	0.557069	−	0.830466i	$-0.311926\pi$
$389$	8547.97	1.11414	0.557069	+	0.830466i	$0.311926\pi$
$390$	0	0
$391$	7251.93	0.937969
$392$	0	0
$393$	−3.04733	−0.000391139 0
$394$	0	0
$395$	8112.20	1.03334
$396$	0	0
$397$	−5667.79	−0.716520	−0.358260	−	0.933622i	$-0.616630\pi$
$397$	−5667.79	−0.716520	−0.358260	+	0.933622i	$0.616630\pi$
$398$	0	0
$399$	497.702	0.0624468
$400$	0	0
$401$	4736.41	0.589838	0.294919	−	0.955522i	$-0.404707\pi$
$401$	4736.41	0.589838	0.294919	+	0.955522i	$0.404707\pi$
$402$	0	0
$403$	−8508.66	−1.05173
$404$	0	0
$405$	−10517.0	−1.29036
$406$	0	0
$407$	3811.43	0.464191
$408$	0	0
$409$	14590.9	1.76399	0.881994	−	0.471260i	$-0.156201\pi$
$409$	14590.9	1.76399	0.881994	+	0.471260i	$0.156201\pi$
$410$	0	0
$411$	−291.551	−0.0349907
$412$	0	0
$413$	8825.52	1.05151
$414$	0	0
$415$	3967.27	0.469266
$416$	0	0
$417$	158.504	0.0186138
$418$	0	0
$419$	9609.04	1.12036	0.560182	−	0.828370i	$-0.310731\pi$
$419$	9609.04	1.12036	0.560182	+	0.828370i	$0.310731\pi$
$420$	0	0
$421$	4289.46	0.496569	0.248284	−	0.968687i	$-0.420133\pi$
$421$	4289.46	0.496569	0.248284	+	0.968687i	$0.420133\pi$
$422$	0	0
$423$	3972.48	0.456615
$424$	0	0
$425$	−8859.88	−1.01122
$426$	0	0
$427$	−14463.2	−1.63917
$428$	0	0
$429$	−64.6725	−0.00727836
$430$	0	0
$431$	2395.95	0.267770	0.133885	−	0.990997i	$-0.457255\pi$
$431$	2395.95	0.267770	0.133885	+	0.990997i	$0.457255\pi$
$432$	0	0
$433$	−2438.13	−0.270599	−0.135299	−	0.990805i	$-0.543200\pi$
$433$	−2438.13	−0.270599	−0.135299	+	0.990805i	$0.543200\pi$
$434$	0	0
$435$	−48.9352	−0.00539371
$436$	0	0
$437$	9706.52	1.06253
$438$	0	0
$439$	−9826.54	−1.06833	−0.534163	−	0.845381i	$-0.679373\pi$
$439$	−9826.54	−1.06833	−0.534163	+	0.845381i	$0.679373\pi$
$440$	0	0
$441$	−15193.3	−1.64057
$442$	0	0
$443$	2522.41	0.270527	0.135263	−	0.990810i	$-0.456812\pi$
$443$	2522.41	0.270527	0.135263	+	0.990810i	$0.456812\pi$
$444$	0	0
$445$	−12523.1	−1.33404
$446$	0	0
$447$	299.072	0.0316457
$448$	0	0
$449$	9247.85	0.972011	0.486005	−	0.873956i	$-0.338454\pi$
$449$	9247.85	0.972011	0.486005	+	0.873956i	$0.338454\pi$
$450$	0	0
$451$	−8119.04	−0.847695
$452$	0	0
$453$	84.1296	0.00872573
$454$	0	0
$455$	−12239.6	−1.26111
$456$	0	0
$457$	−3055.11	−0.312718	−0.156359	−	0.987700i	$-0.549976\pi$
$457$	−3055.11	−0.312718	−0.156359	+	0.987700i	$0.549976\pi$
$458$	0	0
$459$	−666.929	−0.0678205
$460$	0	0
$461$	−570.464	−0.0576338	−0.0288169	−	0.999585i	$-0.509174\pi$
$461$	−570.464	−0.0576338	−0.0288169	+	0.999585i	$0.509174\pi$
$462$	0	0
$463$	−8477.42	−0.850927	−0.425463	−	0.904976i	$-0.639889\pi$
$463$	−8477.42	−0.850927	−0.425463	+	0.904976i	$0.639889\pi$
$464$	0	0
$465$	−510.155	−0.0508772
$466$	0	0
$467$	−12125.2	−1.20147	−0.600737	−	0.799447i	$-0.705126\pi$
$467$	−12125.2	−1.20147	−0.600737	+	0.799447i	$0.705126\pi$
$468$	0	0
$469$	20627.0	2.03085
$470$	0	0
$471$	−436.944	−0.0427459
$472$	0	0
$473$	6833.33	0.664264
$474$	0	0
$475$	−11858.7	−1.14551
$476$	0	0
$477$	3386.94	0.325110
$478$	0	0
$479$	−17826.1	−1.70041	−0.850205	−	0.526452i	$-0.823522\pi$
$479$	−17826.1	−1.70041	−0.850205	+	0.526452i	$0.823522\pi$
$480$	0	0
$481$	5451.71	0.516791
$482$	0	0
$483$	241.003	0.0227040
$484$	0	0
$485$	−8419.84	−0.788299
$486$	0	0
$487$	18459.7	1.71763	0.858816	−	0.512283i	$-0.171200\pi$
$487$	18459.7	1.71763	0.858816	+	0.512283i	$0.171200\pi$
$488$	0	0
$489$	319.348	0.0295326
$490$	0	0
$491$	−5036.59	−0.462929	−0.231465	−	0.972843i	$-0.574352\pi$
$491$	−5036.59	−0.462929	−0.231465	+	0.972843i	$0.574352\pi$
$492$	0	0
$493$	3067.56	0.280235
$494$	0	0
$495$	7671.93	0.696621
$496$	0	0
$497$	−25409.7	−2.29332
$498$	0	0
$499$	13469.7	1.20839	0.604193	−	0.796838i	$-0.293496\pi$
$499$	13469.7	1.20839	0.604193	+	0.796838i	$0.293496\pi$
$500$	0	0
$501$	−262.560	−0.0234138
$502$	0	0
$503$	−11864.4	−1.05170	−0.525851	−	0.850577i	$-0.676253\pi$
$503$	−11864.4	−1.05170	−0.525851	+	0.850577i	$0.676253\pi$
$504$	0	0
$505$	11874.5	1.04636
$506$	0	0
$507$	164.080	0.0143728
$508$	0	0
$509$	−20536.1	−1.78830	−0.894152	−	0.447763i	$-0.852221\pi$
$509$	−20536.1	−1.78830	−0.894152	+	0.447763i	$0.852221\pi$
$510$	0	0
$511$	−15322.0	−1.32643
$512$	0	0
$513$	−892.668	−0.0768270
$514$	0	0
$515$	−21811.4	−1.86626
$516$	0	0
$517$	−2896.37	−0.246387
$518$	0	0
$519$	453.664	0.0383693
$520$	0	0
$521$	16562.1	1.39271	0.696353	−	0.717699i	$-0.254805\pi$
$521$	16562.1	1.39271	0.696353	+	0.717699i	$0.254805\pi$
$522$	0	0
$523$	−18345.2	−1.53380	−0.766902	−	0.641764i	$-0.778203\pi$
$523$	−18345.2	−1.53380	−0.766902	+	0.641764i	$0.778203\pi$
$524$	0	0
$525$	−294.440	−0.0244770
$526$	0	0
$527$	31979.7	2.64337
$528$	0	0
$529$	−7466.80	−0.613693
$530$	0	0
$531$	−7912.62	−0.646664
$532$	0	0
$533$	−11613.1	−0.943752
$534$	0	0
$535$	65.7268	0.00531143
$536$	0	0
$537$	214.582	0.0172437
$538$	0	0
$539$	11077.6	0.885243
$540$	0	0
$541$	−2376.85	−0.188889	−0.0944444	−	0.995530i	$-0.530107\pi$
$541$	−2376.85	−0.188889	−0.0944444	+	0.995530i	$0.530107\pi$
$542$	0	0
$543$	105.475	0.00833583
$544$	0	0
$545$	−24446.3	−1.92140
$546$	0	0
$547$	9907.40	0.774424	0.387212	−	0.921991i	$-0.373438\pi$
$547$	9907.40	0.774424	0.387212	+	0.921991i	$0.373438\pi$
$548$	0	0
$549$	12967.2	1.00806
$550$	0	0
$551$	4105.85	0.317451
$552$	0	0
$553$	−16899.7	−1.29955
$554$	0	0
$555$	326.869	0.0249997
$556$	0	0
$557$	17898.2	1.36153	0.680763	−	0.732504i	$-0.261648\pi$
$557$	17898.2	1.36153	0.680763	+	0.732504i	$0.261648\pi$
$558$	0	0
$559$	9774.10	0.739535
$560$	0	0
$561$	243.071	0.0182932
$562$	0	0
$563$	15841.8	1.18588	0.592940	−	0.805247i	$-0.297967\pi$
$563$	15841.8	1.18588	0.592940	+	0.805247i	$0.297967\pi$
$564$	0	0
$565$	−19069.0	−1.41989
$566$	0	0
$567$	21909.5	1.62278
$568$	0	0
$569$	3522.45	0.259524	0.129762	−	0.991545i	$-0.458579\pi$
$569$	3522.45	0.259524	0.129762	+	0.991545i	$0.458579\pi$
$570$	0	0
$571$	−4721.87	−0.346067	−0.173033	−	0.984916i	$-0.555357\pi$
$571$	−4721.87	−0.346067	−0.173033	+	0.984916i	$0.555357\pi$
$572$	0	0
$573$	105.768	0.00771119
$574$	0	0
$575$	−5742.36	−0.416475
$576$	0	0
$577$	−12130.3	−0.875204	−0.437602	−	0.899169i	$-0.644172\pi$
$577$	−12130.3	−0.875204	−0.437602	+	0.899169i	$0.644172\pi$
$578$	0	0
$579$	513.673	0.0368696
$580$	0	0
$581$	−8264.81	−0.590158
$582$	0	0
$583$	−2469.46	−0.175428
$584$	0	0
$585$	10973.6	0.775560
$586$	0	0
$587$	19992.7	1.40577	0.702886	−	0.711302i	$-0.251894\pi$
$587$	19992.7	1.40577	0.702886	+	0.711302i	$0.251894\pi$
$588$	0	0
$589$	42804.0	2.99441
$590$	0	0
$591$	87.1020	0.00606244
$592$	0	0
$593$	1898.31	0.131457	0.0657286	−	0.997838i	$-0.479063\pi$
$593$	1898.31	0.131457	0.0657286	+	0.997838i	$0.479063\pi$
$594$	0	0
$595$	46002.6	3.16962
$596$	0	0
$597$	344.782	0.0236365
$598$	0	0
$599$	3755.17	0.256147	0.128073	−	0.991765i	$-0.459121\pi$
$599$	3755.17	0.256147	0.128073	+	0.991765i	$0.459121\pi$
$600$	0	0
$601$	−15502.4	−1.05217	−0.526085	−	0.850432i	$-0.676341\pi$
$601$	−15502.4	−1.05217	−0.526085	+	0.850432i	$0.676341\pi$
$602$	0	0
$603$	−18493.4	−1.24894
$604$	0	0
$605$	13637.3	0.916420
$606$	0	0
$607$	−2966.48	−0.198362	−0.0991811	−	0.995069i	$-0.531622\pi$
$607$	−2966.48	−0.198362	−0.0991811	+	0.995069i	$0.531622\pi$
$608$	0	0
$609$	101.944	0.00678323
$610$	0	0
$611$	−4142.85	−0.274307
$612$	0	0
$613$	−3947.81	−0.260115	−0.130058	−	0.991506i	$-0.541516\pi$
$613$	−3947.81	−0.260115	−0.130058	+	0.991506i	$0.541516\pi$
$614$	0	0
$615$	−696.290	−0.0456538
$616$	0	0
$617$	−16131.9	−1.05258	−0.526292	−	0.850304i	$-0.676418\pi$
$617$	−16131.9	−1.05258	−0.526292	+	0.850304i	$0.676418\pi$
$618$	0	0
$619$	−27235.2	−1.76845	−0.884227	−	0.467057i	$-0.845314\pi$
$619$	−27235.2	−1.76845	−0.884227	+	0.467057i	$0.845314\pi$
$620$	0	0
$621$	−432.257	−0.0279322
$622$	0	0
$623$	26088.7	1.67772
$624$	0	0
$625$	−19079.3	−1.22107
$626$	0	0
$627$	325.344	0.0207225
$628$	0	0
$629$	−20490.2	−1.29888
$630$	0	0
$631$	−1451.22	−0.0915567	−0.0457783	−	0.998952i	$-0.514577\pi$
$631$	−1451.22	−0.0915567	−0.0457783	+	0.998952i	$0.514577\pi$
$632$	0	0
$633$	−110.852	−0.00696045
$634$	0	0
$635$	8235.83	0.514691
$636$	0	0
$637$	15844.9	0.985555
$638$	0	0
$639$	22781.4	1.41036
$640$	0	0
$641$	−26135.3	−1.61042	−0.805211	−	0.592988i	$-0.797948\pi$
$641$	−26135.3	−1.61042	−0.805211	+	0.592988i	$0.797948\pi$
$642$	0	0
$643$	−28223.4	−1.73099	−0.865493	−	0.500921i	$-0.832995\pi$
$643$	−28223.4	−1.73099	−0.865493	+	0.500921i	$0.832995\pi$
$644$	0	0
$645$	586.027	0.0357749
$646$	0	0
$647$	−6173.73	−0.375138	−0.187569	−	0.982251i	$-0.560061\pi$
$647$	−6173.73	−0.375138	−0.187569	+	0.982251i	$0.560061\pi$
$648$	0	0
$649$	5769.17	0.348937
$650$	0	0
$651$	1062.78	0.0639841
$652$	0	0
$653$	−8302.01	−0.497523	−0.248762	−	0.968565i	$-0.580024\pi$
$653$	−8302.01	−0.497523	−0.248762	+	0.968565i	$0.580024\pi$
$654$	0	0
$655$	−377.001	−0.0224895
$656$	0	0
$657$	13737.1	0.815731
$658$	0	0
$659$	13416.1	0.793045	0.396522	−	0.918025i	$-0.370217\pi$
$659$	13416.1	0.793045	0.396522	+	0.918025i	$0.370217\pi$
$660$	0	0
$661$	3115.05	0.183300	0.0916502	−	0.995791i	$-0.470786\pi$
$661$	3115.05	0.183300	0.0916502	+	0.995791i	$0.470786\pi$
$662$	0	0
$663$	347.678	0.0203661
$664$	0	0
$665$	61573.3	3.59054
$666$	0	0
$667$	1988.18	0.115416
$668$	0	0
$669$	145.710	0.00842072
$670$	0	0
$671$	−9454.51	−0.543946
$672$	0	0
$673$	−25717.3	−1.47300	−0.736500	−	0.676438i	$-0.763523\pi$
$673$	−25717.3	−1.47300	−0.736500	+	0.676438i	$0.763523\pi$
$674$	0	0
$675$	528.101	0.0301135
$676$	0	0
$677$	1609.63	0.0913783	0.0456892	−	0.998956i	$-0.485452\pi$
$677$	1609.63	0.0913783	0.0456892	+	0.998956i	$0.485452\pi$
$678$	0	0
$679$	17540.6	0.991380
$680$	0	0
$681$	−308.537	−0.0173615
$682$	0	0
$683$	16031.9	0.898159	0.449079	−	0.893492i	$-0.351752\pi$
$683$	16031.9	0.898159	0.449079	+	0.893492i	$0.351752\pi$
$684$	0	0
$685$	−36069.3	−2.01188
$686$	0	0
$687$	−482.426	−0.0267914
$688$	0	0
$689$	−3532.20	−0.195307
$690$	0	0
$691$	14272.1	0.785726	0.392863	−	0.919597i	$-0.371485\pi$
$691$	14272.1	0.785726	0.392863	+	0.919597i	$0.371485\pi$
$692$	0	0
$693$	−15982.5	−0.876084
$694$	0	0
$695$	19609.3	1.07025
$696$	0	0
$697$	43647.8	2.37199
$698$	0	0
$699$	−323.840	−0.0175232
$700$	0	0
$701$	−3420.31	−0.184284	−0.0921421	−	0.995746i	$-0.529371\pi$
$701$	−3420.31	−0.184284	−0.0921421	+	0.995746i	$0.529371\pi$
$702$	0	0
$703$	−27425.6	−1.47137
$704$	0	0
$705$	−248.393	−0.0132696
$706$	0	0
$707$	−24737.6	−1.31592
$708$	0	0
$709$	17171.7	0.909588	0.454794	−	0.890597i	$-0.349713\pi$
$709$	17171.7	0.909588	0.454794	+	0.890597i	$0.349713\pi$
$710$	0	0
$711$	15151.7	0.799201
$712$	0	0
$713$	20727.0	1.08869
$714$	0	0
$715$	−8000.96	−0.418488
$716$	0	0
$717$	−754.240	−0.0392854
$718$	0	0
$719$	2823.74	0.146464	0.0732320	−	0.997315i	$-0.476669\pi$
$719$	2823.74	0.146464	0.0732320	+	0.997315i	$0.476669\pi$
$720$	0	0
$721$	45438.5	2.34704
$722$	0	0
$723$	−441.065	−0.0226879
$724$	0	0
$725$	−2429.02	−0.124430
$726$	0	0
$727$	6563.47	0.334836	0.167418	−	0.985886i	$-0.446457\pi$
$727$	6563.47	0.334836	0.167418	+	0.985886i	$0.446457\pi$
$728$	0	0
$729$	−19623.4	−0.996970
$730$	0	0
$731$	−36735.9	−1.85872
$732$	0	0
$733$	27199.9	1.37060	0.685301	−	0.728260i	$-0.259671\pi$
$733$	27199.9	1.37060	0.685301	+	0.728260i	$0.259671\pi$
$734$	0	0
$735$	950.017	0.0476760
$736$	0	0
$737$	13483.7	0.673921
$738$	0	0
$739$	−1914.96	−0.0953219	−0.0476610	−	0.998864i	$-0.515177\pi$
$739$	−1914.96	−0.0953219	−0.0476610	+	0.998864i	$0.515177\pi$
$740$	0	0
$741$	465.359	0.0230707
$742$	0	0
$743$	16605.3	0.819904	0.409952	−	0.912107i	$-0.365546\pi$
$743$	16605.3	0.819904	0.409952	+	0.912107i	$0.365546\pi$
$744$	0	0
$745$	36999.7	1.81955
$746$	0	0
$747$	7409.91	0.362938
$748$	0	0
$749$	−136.925	−0.00667976
$750$	0	0
$751$	−11668.2	−0.566950	−0.283475	−	0.958980i	$-0.591487\pi$
$751$	−11668.2	−0.566950	−0.283475	+	0.958980i	$0.591487\pi$
$752$	0	0
$753$	−183.439	−0.00887766
$754$	0	0
$755$	10408.1	0.501708
$756$	0	0
$757$	−15707.6	−0.754163	−0.377082	−	0.926180i	$-0.623072\pi$
$757$	−15707.6	−0.754163	−0.377082	+	0.926180i	$0.623072\pi$
$758$	0	0
$759$	157.542	0.00753413
$760$	0	0
$761$	26445.7	1.25973	0.629865	−	0.776704i	$-0.283110\pi$
$761$	26445.7	1.25973	0.629865	+	0.776704i	$0.283110\pi$
$762$	0	0
$763$	50927.7	2.41639
$764$	0	0
$765$	−41244.2	−1.94926
$766$	0	0
$767$	8251.98	0.388477
$768$	0	0
$769$	−22270.9	−1.04436	−0.522178	−	0.852837i	$-0.674880\pi$
$769$	−22270.9	−1.04436	−0.522178	+	0.852837i	$0.674880\pi$
$770$	0	0
$771$	118.561	0.00553811
$772$	0	0
$773$	−6235.87	−0.290154	−0.145077	−	0.989420i	$-0.546343\pi$
$773$	−6235.87	−0.290154	−0.145077	+	0.989420i	$0.546343\pi$
$774$	0	0
$775$	−25322.8	−1.17371
$776$	0	0
$777$	−680.949	−0.0314401
$778$	0	0
$779$	58421.5	2.68699
$780$	0	0
$781$	−16610.1	−0.761021
$782$	0	0
$783$	−182.845	−0.00834526
$784$	0	0
$785$	−54056.6	−2.45779
$786$	0	0
$787$	−17385.7	−0.787463	−0.393732	−	0.919225i	$-0.628816\pi$
$787$	−17385.7	−0.787463	−0.393732	+	0.919225i	$0.628816\pi$
$788$	0	0
$789$	362.462	0.0163549
$790$	0	0
$791$	39725.6	1.78569
$792$	0	0
$793$	−13523.3	−0.605583
$794$	0	0
$795$	−211.781	−0.00944792
$796$	0	0
$797$	−9728.96	−0.432393	−0.216197	−	0.976350i	$-0.569365\pi$
$797$	−9728.96	−0.432393	−0.216197	+	0.976350i	$0.569365\pi$
$798$	0	0
$799$	15570.8	0.689433
$800$	0	0
$801$	−23390.1	−1.03177
$802$	0	0
$803$	−10015.9	−0.440164
$804$	0	0
$805$	29815.7	1.30542
$806$	0	0
$807$	169.305	0.00738517
$808$	0	0
$809$	−16514.4	−0.717693	−0.358847	−	0.933397i	$-0.616830\pi$
$809$	−16514.4	−0.717693	−0.358847	+	0.933397i	$0.616830\pi$
$810$	0	0
$811$	−16295.2	−0.705551	−0.352776	−	0.935708i	$-0.614762\pi$
$811$	−16295.2	−0.705551	−0.352776	+	0.935708i	$0.614762\pi$
$812$	0	0
$813$	300.608	0.0129677
$814$	0	0
$815$	39508.2	1.69805
$816$	0	0
$817$	−49170.0	−2.10556
$818$	0	0
$819$	−22860.7	−0.975359
$820$	0	0
$821$	−34079.3	−1.44869	−0.724345	−	0.689438i	$-0.757858\pi$
$821$	−34079.3	−1.44869	−0.724345	+	0.689438i	$0.757858\pi$
$822$	0	0
$823$	−38508.0	−1.63099	−0.815495	−	0.578764i	$-0.803535\pi$
$823$	−38508.0	−1.63099	−0.815495	+	0.578764i	$0.803535\pi$
$824$	0	0
$825$	−192.473	−0.00812249
$826$	0	0
$827$	−18818.4	−0.791270	−0.395635	−	0.918408i	$-0.629475\pi$
$827$	−18818.4	−0.791270	−0.395635	+	0.918408i	$0.629475\pi$
$828$	0	0
$829$	−4623.29	−0.193696	−0.0968478	−	0.995299i	$-0.530876\pi$
$829$	−4623.29	−0.193696	−0.0968478	+	0.995299i	$0.530876\pi$
$830$	0	0
$831$	−816.591	−0.0340881
$832$	0	0
$833$	−59553.0	−2.47706
$834$	0	0
$835$	−32482.6	−1.34624
$836$	0	0
$837$	−1906.18	−0.0787182
$838$	0	0
$839$	24726.3	1.01746	0.508728	−	0.860927i	$-0.330116\pi$
$839$	24726.3	1.01746	0.508728	+	0.860927i	$0.330116\pi$
$840$	0	0
$841$	841.000	0.0344828
$842$	0	0
$843$	551.217	0.0225207
$844$	0	0
$845$	20299.1	0.826404
$846$	0	0
$847$	−28409.8	−1.15251
$848$	0	0
$849$	−549.705	−0.0222212
$850$	0	0
$851$	−13280.3	−0.534951
$852$	0	0
$853$	−13087.6	−0.525335	−0.262667	−	0.964886i	$-0.584602\pi$
$853$	−13087.6	−0.525335	−0.262667	+	0.964886i	$0.584602\pi$
$854$	0	0
$855$	−55204.3	−2.20812
$856$	0	0
$857$	19184.1	0.764663	0.382332	−	0.924025i	$-0.375121\pi$
$857$	19184.1	0.764663	0.382332	+	0.924025i	$0.375121\pi$
$858$	0	0
$859$	−38866.7	−1.54379	−0.771894	−	0.635751i	$-0.780690\pi$
$859$	−38866.7	−1.54379	−0.771894	+	0.635751i	$0.780690\pi$
$860$	0	0
$861$	1450.55	0.0574152
$862$	0	0
$863$	12078.2	0.476415	0.238207	−	0.971214i	$-0.423440\pi$
$863$	12078.2	0.476415	0.238207	+	0.971214i	$0.423440\pi$
$864$	0	0
$865$	56125.1	2.20614
$866$	0	0
$867$	−732.964	−0.0287114
$868$	0	0
$869$	−11047.2	−0.431245
$870$	0	0
$871$	19286.6	0.750287
$872$	0	0
$873$	−15726.3	−0.609683
$874$	0	0
$875$	17935.5	0.692950
$876$	0	0
$877$	45369.0	1.74687	0.873434	−	0.486943i	$-0.161888\pi$
$877$	45369.0	1.74687	0.873434	+	0.486943i	$0.161888\pi$
$878$	0	0
$879$	115.292	0.00442401
$880$	0	0
$881$	41537.5	1.58846	0.794230	−	0.607617i	$-0.207874\pi$
$881$	41537.5	1.58846	0.794230	+	0.607617i	$0.207874\pi$
$882$	0	0
$883$	−33042.6	−1.25931	−0.629656	−	0.776874i	$-0.716804\pi$
$883$	−33042.6	−1.25931	−0.629656	+	0.776874i	$0.716804\pi$
$884$	0	0
$885$	494.765	0.0187925
$886$	0	0
$887$	−8165.01	−0.309080	−0.154540	−	0.987986i	$-0.549390\pi$
$887$	−8165.01	−0.309080	−0.154540	+	0.987986i	$0.549390\pi$
$888$	0	0
$889$	−17157.3	−0.647286
$890$	0	0
$891$	14322.1	0.538505
$892$	0	0
$893$	20841.2	0.780990
$894$	0	0
$895$	26547.0	0.991472
$896$	0	0
$897$	225.341	0.00838787
$898$	0	0
$899$	8767.53	0.325265
$900$	0	0
$901$	13275.8	0.490876
$902$	0	0
$903$	−1220.84	−0.0449912
$904$	0	0
$905$	13048.8	0.479290
$906$	0	0
$907$	4263.93	0.156099	0.0780493	−	0.996949i	$-0.475131\pi$
$907$	4263.93	0.156099	0.0780493	+	0.996949i	$0.475131\pi$
$908$	0	0
$909$	22178.8	0.809268
$910$	0	0
$911$	−25204.8	−0.916656	−0.458328	−	0.888783i	$-0.651551\pi$
$911$	−25204.8	−0.916656	−0.458328	+	0.888783i	$0.651551\pi$
$912$	0	0
$913$	−5402.64	−0.195839
$914$	0	0
$915$	−810.821	−0.0292950
$916$	0	0
$917$	785.386	0.0282832
$918$	0	0
$919$	−44763.2	−1.60675	−0.803374	−	0.595474i	$-0.796964\pi$
$919$	−44763.2	−1.60675	−0.803374	+	0.595474i	$0.796964\pi$
$920$	0	0
$921$	−38.0879	−0.00136269
$922$	0	0
$923$	−23758.4	−0.847257
$924$	0	0
$925$	16224.9	0.576727
$926$	0	0
$927$	−40738.4	−1.44339
$928$	0	0
$929$	17083.5	0.603326	0.301663	−	0.953415i	$-0.402458\pi$
$929$	17083.5	0.603326	0.301663	+	0.953415i	$0.402458\pi$
$930$	0	0
$931$	−79710.2	−2.80601
$932$	0	0
$933$	889.463	0.0312108
$934$	0	0
$935$	30071.5	1.05181
$936$	0	0
$937$	42461.9	1.48044	0.740219	−	0.672366i	$-0.234722\pi$
$937$	42461.9	1.48044	0.740219	+	0.672366i	$0.234722\pi$
$938$	0	0
$939$	−431.781	−0.0150060
$940$	0	0
$941$	8151.35	0.282387	0.141194	−	0.989982i	$-0.454906\pi$
$941$	8151.35	0.282387	0.141194	+	0.989982i	$0.454906\pi$
$942$	0	0
$943$	28289.5	0.976916
$944$	0	0
$945$	−2742.02	−0.0943894
$946$	0	0
$947$	−17101.8	−0.586836	−0.293418	−	0.955984i	$-0.594793\pi$
$947$	−17101.8	−0.586836	−0.293418	+	0.955984i	$0.594793\pi$
$948$	0	0
$949$	−14326.3	−0.490042
$950$	0	0
$951$	27.8370	0.000949188 0
$952$	0	0
$953$	−7680.08	−0.261052	−0.130526	−	0.991445i	$-0.541667\pi$
$953$	−7680.08	−0.261052	−0.130526	+	0.991445i	$0.541667\pi$
$954$	0	0
$955$	13085.1	0.443375
$956$	0	0
$957$	66.6401	0.00225096
$958$	0	0
$959$	75141.3	2.53018
$960$	0	0
$961$	61611.5	2.06813
$962$	0	0
$963$	122.762	0.00410794
$964$	0	0
$965$	63549.0	2.11991
$966$	0	0
$967$	−24538.3	−0.816026	−0.408013	−	0.912976i	$-0.633778\pi$
$967$	−24538.3	−0.816026	−0.408013	+	0.912976i	$0.633778\pi$
$968$	0	0
$969$	−1749.05	−0.0579850
$970$	0	0
$971$	−56442.0	−1.86541	−0.932703	−	0.360646i	$-0.882556\pi$
$971$	−56442.0	−1.86541	−0.932703	+	0.360646i	$0.882556\pi$
$972$	0	0
$973$	−40851.1	−1.34597
$974$	0	0
$975$	−275.305	−0.00904290
$976$	0	0
$977$	−42436.6	−1.38963	−0.694815	−	0.719188i	$-0.744514\pi$
$977$	−42436.6	−1.38963	−0.694815	+	0.719188i	$0.744514\pi$
$978$	0	0
$979$	17054.0	0.556738
$980$	0	0
$981$	−45659.9	−1.48604
$982$	0	0
$983$	−55695.2	−1.80712	−0.903561	−	0.428460i	$-0.859056\pi$
$983$	−55695.2	−1.80712	−0.903561	+	0.428460i	$0.859056\pi$
$984$	0	0
$985$	10775.8	0.348575
$986$	0	0
$987$	517.465	0.0166880
$988$	0	0
$989$	−23809.6	−0.765523
$990$	0	0
$991$	−16023.3	−0.513621	−0.256810	−	0.966462i	$-0.582672\pi$
$991$	−16023.3	−0.513621	−0.256810	+	0.966462i	$0.582672\pi$
$992$	0	0
$993$	121.429	0.00388058
$994$	0	0
$995$	42654.7	1.35904
$996$	0	0
$997$	−27236.3	−0.865179	−0.432590	−	0.901591i	$-0.642400\pi$
$997$	−27236.3	−0.865179	−0.432590	+	0.901591i	$0.642400\pi$
$998$	0	0
$999$	1221.34	0.0386800

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	232.4.a.e.1.3	✓	6
3.2	odd	2		2088.4.a.l.1.6		6
4.3	odd	2		464.4.a.n.1.4		6
8.3	odd	2		1856.4.a.bc.1.3		6
8.5	even	2		1856.4.a.bd.1.4		6

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
232.4.a.e.1.3	✓	6	1.1	even	1	trivial
464.4.a.n.1.4		6	4.3	odd	2
1856.4.a.bc.1.3		6	8.3	odd	2
1856.4.a.bd.1.4		6	8.5	even	2
2088.4.a.l.1.6		6	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)

\(f(q)\)	\(=\)	\(q-0.116789 q^{3} -14.4485 q^{5} +30.0998 q^{7} -26.9864 q^{9} +O(q^{10})\)	\(q-0.116789 q^{3} -14.4485 q^{5} +30.0998 q^{7} -26.9864 q^{9} +19.6760 q^{11} +28.1437 q^{13} +1.68742 q^{15} -105.778 q^{17} -141.581 q^{19} -3.51532 q^{21} -68.5580 q^{23} +83.7592 q^{25} +6.30499 q^{27} -29.0000 q^{29} -302.329 q^{31} -2.29793 q^{33} -434.897 q^{35} +193.709 q^{37} -3.28687 q^{39} -412.636 q^{41} +347.292 q^{43} +389.912 q^{45} -147.203 q^{47} +563.000 q^{49} +12.3537 q^{51} -125.506 q^{53} -284.289 q^{55} +16.5351 q^{57} +293.208 q^{59} -480.509 q^{61} -812.285 q^{63} -406.635 q^{65} +685.288 q^{67} +8.00679 q^{69} -844.181 q^{71} -509.039 q^{73} -9.78212 q^{75} +592.245 q^{77} -561.456 q^{79} +727.895 q^{81} -274.580 q^{83} +1528.33 q^{85} +3.38687 q^{87} +866.738 q^{89} +847.122 q^{91} +35.3085 q^{93} +2045.64 q^{95} +582.748 q^{97} -530.984 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 6 q - 5 q^{3} - 5 q^{5} - 38 q^{7} + 47 q^{9}+O(q^{10}) \) 6 * q - 5 * q^3 - 5 * q^5 - 38 * q^7 + 47 * q^9	\( 6 q - 5 q^{3} - 5 q^{5} - 38 q^{7} + 47 q^{9} - 19 q^{11} + 13 q^{13} - 191 q^{15} - 218 q^{17} - 290 q^{19} - 266 q^{21} - 196 q^{23} - 13 q^{25} - 437 q^{27} - 174 q^{29} - 675 q^{31} + 291 q^{33} - 466 q^{35} - 238 q^{37} - 1297 q^{39} - 464 q^{41} - 579 q^{43} - 148 q^{45} - 975 q^{47} + 914 q^{49} - 576 q^{51} + 515 q^{53} - 1605 q^{55} - 340 q^{57} - 108 q^{59} + 1158 q^{61} - 1136 q^{63} + 1239 q^{65} - 80 q^{67} + 2568 q^{69} + 438 q^{71} + 262 q^{73} + 1766 q^{75} + 194 q^{77} - 237 q^{79} + 2554 q^{81} - 1288 q^{83} + 3112 q^{85} + 145 q^{87} - 252 q^{89} + 2450 q^{91} + 2131 q^{93} + 180 q^{95} + 380 q^{97} + 2264 q^{99}+O(q^{100}) \) 6 * q - 5 * q^3 - 5 * q^5 - 38 * q^7 + 47 * q^9 - 19 * q^11 + 13 * q^13 - 191 * q^15 - 218 * q^17 - 290 * q^19 - 266 * q^21 - 196 * q^23 - 13 * q^25 - 437 * q^27 - 174 * q^29 - 675 * q^31 + 291 * q^33 - 466 * q^35 - 238 * q^37 - 1297 * q^39 - 464 * q^41 - 579 * q^43 - 148 * q^45 - 975 * q^47 + 914 * q^49 - 576 * q^51 + 515 * q^53 - 1605 * q^55 - 340 * q^57 - 108 * q^59 + 1158 * q^61 - 1136 * q^63 + 1239 * q^65 - 80 * q^67 + 2568 * q^69 + 438 * q^71 + 262 * q^73 + 1766 * q^75 + 194 * q^77 - 237 * q^79 + 2554 * q^81 - 1288 * q^83 + 3112 * q^85 + 145 * q^87 - 252 * q^89 + 2450 * q^91 + 2131 * q^93 + 180 * q^95 + 380 * q^97 + 2264 * q^99

Introduction

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