LMFDB - Embedded newform 1232.4.a.p.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:	$2$
Coefficient field:	$\Q(\sqrt{137}) $
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{2} - x - 34 $ x^2 - x - 34
Coefficient ring:	$\Z[a_1, a_2, a_3]$
Coefficient ring index:	$ 1 $
Twist minimal:	no (minimal twist has level 154)
Fricke sign:	$1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.1
Root			$6.35235$ 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)

$f(q)$	$=$	$q-3.35235 q^{3} -9.35235 q^{5} -7.00000 q^{7} -15.7617 q^{9} +O(q^{10})$	$q-3.35235 q^{3} -9.35235 q^{5} -7.00000 q^{7} -15.7617 q^{9} +11.0000 q^{11} -35.5235 q^{13} +31.3523 q^{15} -67.4094 q^{17} -58.9329 q^{19} +23.4664 q^{21} -30.2852 q^{23} -37.5336 q^{25} +143.352 q^{27} -279.638 q^{29} +86.0570 q^{31} -36.8758 q^{33} +65.4664 q^{35} -38.7617 q^{37} +119.087 q^{39} -337.047 q^{41} -49.1812 q^{43} +147.409 q^{45} -223.544 q^{47} +49.0000 q^{49} +225.980 q^{51} -286.953 q^{53} -102.876 q^{55} +197.564 q^{57} -398.017 q^{59} +741.235 q^{61} +110.332 q^{63} +332.228 q^{65} -744.910 q^{67} +101.527 q^{69} -663.044 q^{71} -74.1880 q^{73} +125.826 q^{75} -77.0000 q^{77} +147.886 q^{79} -55.0000 q^{81} +1279.05 q^{83} +630.436 q^{85} +937.443 q^{87} +740.399 q^{89} +248.664 q^{91} -288.493 q^{93} +551.161 q^{95} -704.775 q^{97} -173.379 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 2 q + 5 q^{3} - 7 q^{5} - 14 q^{7} + 27 q^{9}+O(q^{10}) $ 2 * q + 5 * q^3 - 7 * q^5 - 14 * q^7 + 27 * q^9	$ 2 q + 5 q^{3} - 7 q^{5} - 14 q^{7} + 27 q^{9} + 22 q^{11} + 46 q^{13} + 51 q^{15} - 88 q^{17} + 46 q^{19} - 35 q^{21} + 115 q^{23} - 157 q^{25} + 275 q^{27} - 372 q^{29} + 137 q^{31} + 55 q^{33} + 49 q^{35} - 19 q^{37} + 800 q^{39} - 440 q^{41} - 192 q^{43} + 248 q^{45} - 728 q^{47} + 98 q^{49} + 54 q^{51} - 808 q^{53} - 77 q^{55} + 1074 q^{57} + 35 q^{59} + 312 q^{61} - 189 q^{63} + 524 q^{65} + 301 q^{67} + 1315 q^{69} + 137 q^{71} + 788 q^{73} - 872 q^{75} - 154 q^{77} + 366 q^{79} - 110 q^{81} + 2324 q^{83} + 582 q^{85} + 166 q^{87} + 1235 q^{89} - 322 q^{91} + 137 q^{93} + 798 q^{95} + 709 q^{97} + 297 q^{99}+O(q^{100}) $ 2 * q + 5 * q^3 - 7 * q^5 - 14 * q^7 + 27 * q^9 + 22 * q^11 + 46 * q^13 + 51 * q^15 - 88 * q^17 + 46 * q^19 - 35 * q^21 + 115 * q^23 - 157 * q^25 + 275 * q^27 - 372 * q^29 + 137 * q^31 + 55 * q^33 + 49 * q^35 - 19 * q^37 + 800 * q^39 - 440 * q^41 - 192 * q^43 + 248 * q^45 - 728 * q^47 + 98 * q^49 + 54 * q^51 - 808 * q^53 - 77 * q^55 + 1074 * q^57 + 35 * q^59 + 312 * q^61 - 189 * q^63 + 524 * q^65 + 301 * q^67 + 1315 * q^69 + 137 * q^71 + 788 * q^73 - 872 * q^75 - 154 * q^77 + 366 * q^79 - 110 * q^81 + 2324 * q^83 + 582 * q^85 + 166 * q^87 + 1235 * q^89 - 322 * q^91 + 137 * q^93 + 798 * q^95 + 709 * q^97 + 297 * 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
$3$	−3.35235	−0.645160	−0.322580	−	0.946542i	$-0.604550\pi$
$3$	−3.35235	−0.645160	−0.322580	+	0.946542i	$0.604550\pi$
$4$	0	0
$5$	−9.35235	−0.836500	−0.418250	−	0.908332i	$-0.637356\pi$
$5$	−9.35235	−0.836500	−0.418250	+	0.908332i	$0.637356\pi$
$6$	0	0
$7$	−7.00000	−0.377964
$7$	−7.00000	−0.377964
$8$	0	0
$9$	−15.7617	−0.583769
$10$	0	0
$11$	11.0000	0.301511
$11$	11.0000	0.301511
$12$	0	0
$13$	−35.5235	−0.757880	−0.378940	−	0.925421i	$-0.623711\pi$
$13$	−35.5235	−0.757880	−0.378940	+	0.925421i	$0.623711\pi$
$14$	0	0
$15$	31.3523	0.539676
$16$	0	0
$17$	−67.4094	−0.961717	−0.480858	−	0.876798i	$-0.659675\pi$
$17$	−67.4094	−0.961717	−0.480858	+	0.876798i	$0.659675\pi$
$18$	0	0
$19$	−58.9329	−0.711586	−0.355793	−	0.934565i	$-0.615789\pi$
$19$	−58.9329	−0.711586	−0.355793	+	0.934565i	$0.615789\pi$
$20$	0	0
$21$	23.4664	0.243848
$22$	0	0
$23$	−30.2852	−0.274561	−0.137281	−	0.990532i	$-0.543836\pi$
$23$	−30.2852	−0.274561	−0.137281	+	0.990532i	$0.543836\pi$
$24$	0	0
$25$	−37.5336	−0.300268
$26$	0	0
$27$	143.352	1.02178
$28$	0	0
$29$	−279.638	−1.79060	−0.895300	−	0.445464i	$-0.853039\pi$
$29$	−279.638	−1.79060	−0.895300	+	0.445464i	$0.853039\pi$
$30$	0	0
$31$	86.0570	0.498590	0.249295	−	0.968428i	$-0.419801\pi$
$31$	86.0570	0.498590	0.249295	+	0.968428i	$0.419801\pi$
$32$	0	0
$33$	−36.8758	−0.194523
$34$	0	0
$35$	65.4664	0.316167
$36$	0	0
$37$	−38.7617	−0.172227	−0.0861134	−	0.996285i	$-0.527445\pi$
$37$	−38.7617	−0.172227	−0.0861134	+	0.996285i	$0.527445\pi$
$38$	0	0
$39$	119.087	0.488954
$40$	0	0
$41$	−337.047	−1.28385	−0.641926	−	0.766767i	$-0.721864\pi$
$41$	−337.047	−1.28385	−0.641926	+	0.766767i	$0.721864\pi$
$42$	0	0
$43$	−49.1812	−0.174420	−0.0872100	−	0.996190i	$-0.527795\pi$
$43$	−49.1812	−0.174420	−0.0872100	+	0.996190i	$0.527795\pi$
$44$	0	0
$45$	147.409	0.488322
$46$	0	0
$47$	−223.544	−0.693770	−0.346885	−	0.937908i	$-0.612761\pi$
$47$	−223.544	−0.693770	−0.346885	+	0.937908i	$0.612761\pi$
$48$	0	0
$49$	49.0000	0.142857
$50$	0	0
$51$	225.980	0.620461
$52$	0	0
$53$	−286.953	−0.743699	−0.371849	−	0.928293i	$-0.621276\pi$
$53$	−286.953	−0.743699	−0.371849	+	0.928293i	$0.621276\pi$
$54$	0	0
$55$	−102.876	−0.252214
$56$	0	0
$57$	197.564	0.459087
$58$	0	0
$59$	−398.017	−0.878261	−0.439130	−	0.898423i	$-0.644713\pi$
$59$	−398.017	−0.878261	−0.439130	+	0.898423i	$0.644713\pi$
$60$	0	0
$61$	741.235	1.55583	0.777913	−	0.628372i	$-0.216278\pi$
$61$	741.235	1.55583	0.777913	+	0.628372i	$0.216278\pi$
$62$	0	0
$63$	110.332	0.220644
$64$	0	0
$65$	332.228	0.633967
$66$	0	0
$67$	−744.910	−1.35829	−0.679143	−	0.734006i	$-0.737648\pi$
$67$	−744.910	−1.35829	−0.679143	+	0.734006i	$0.737648\pi$
$68$	0	0
$69$	101.527	0.177136
$70$	0	0
$71$	−663.044	−1.10829	−0.554147	−	0.832419i	$-0.686955\pi$
$71$	−663.044	−1.10829	−0.554147	+	0.832419i	$0.686955\pi$
$72$	0	0
$73$	−74.1880	−0.118946	−0.0594729	−	0.998230i	$-0.518942\pi$
$73$	−74.1880	−0.118946	−0.0594729	+	0.998230i	$0.518942\pi$
$74$	0	0
$75$	125.826	0.193721
$76$	0	0
$77$	−77.0000	−0.113961
$78$	0	0
$79$	147.886	0.210613	0.105307	−	0.994440i	$-0.466418\pi$
$79$	147.886	0.210613	0.105307	+	0.994440i	$0.466418\pi$
$80$	0	0
$81$	−55.0000	−0.0754458
$82$	0	0
$83$	1279.05	1.69149	0.845745	−	0.533588i	$-0.179157\pi$
$83$	1279.05	1.69149	0.845745	+	0.533588i	$0.179157\pi$
$84$	0	0
$85$	630.436	0.804475
$86$	0	0
$87$	937.443	1.15522
$88$	0	0
$89$	740.399	0.881822	0.440911	−	0.897551i	$-0.354655\pi$
$89$	740.399	0.881822	0.440911	+	0.897551i	$0.354655\pi$
$90$	0	0
$91$	248.664	0.286452
$92$	0	0
$93$	−288.493	−0.321671
$94$	0	0
$95$	551.161	0.595241
$96$	0	0
$97$	−704.775	−0.737723	−0.368861	−	0.929484i	$-0.620252\pi$
$97$	−704.775	−0.737723	−0.368861	+	0.929484i	$0.620252\pi$
$98$	0	0
$99$	−173.379	−0.176013
$100$	0	0
$101$	62.9867	0.0620536	0.0310268	−	0.999519i	$-0.490122\pi$
$101$	62.9867	0.0620536	0.0310268	+	0.999519i	$0.490122\pi$
$102$	0	0
$103$	−923.503	−0.883451	−0.441726	−	0.897150i	$-0.645634\pi$
$103$	−923.503	−0.883451	−0.441726	+	0.897150i	$0.645634\pi$
$104$	0	0
$105$	−219.466	−0.203978
$106$	0	0
$107$	604.611	0.546261	0.273130	−	0.961977i	$-0.411941\pi$
$107$	604.611	0.546261	0.273130	+	0.961977i	$0.411941\pi$
$108$	0	0
$109$	−1033.08	−0.907809	−0.453905	−	0.891050i	$-0.649969\pi$
$109$	−1033.08	−0.907809	−0.453905	+	0.891050i	$0.649969\pi$
$110$	0	0
$111$	129.943	0.111114
$112$	0	0
$113$	2012.09	1.67506	0.837529	−	0.546393i	$-0.183999\pi$
$113$	2012.09	1.67506	0.837529	+	0.546393i	$0.183999\pi$
$114$	0	0
$115$	283.238	0.229670
$116$	0	0
$117$	559.913	0.442427
$118$	0	0
$119$	471.866	0.363495
$120$	0	0
$121$	121.000	0.0909091
$122$	0	0
$123$	1129.90	0.828289
$124$	0	0
$125$	1520.07	1.08767
$126$	0	0
$127$	733.027	0.512170	0.256085	−	0.966654i	$-0.417567\pi$
$127$	733.027	0.512170	0.256085	+	0.966654i	$0.417567\pi$
$128$	0	0
$129$	164.873	0.112529
$130$	0	0
$131$	1517.30	1.01196	0.505980	−	0.862545i	$-0.331131\pi$
$131$	1517.30	1.01196	0.505980	+	0.862545i	$0.331131\pi$
$132$	0	0
$133$	412.530	0.268954
$134$	0	0
$135$	−1340.68	−0.854722
$136$	0	0
$137$	−2114.30	−1.31852	−0.659258	−	0.751917i	$-0.729130\pi$
$137$	−2114.30	−1.31852	−0.659258	+	0.751917i	$0.729130\pi$
$138$	0	0
$139$	246.060	0.150148	0.0750740	−	0.997178i	$-0.476081\pi$
$139$	246.060	0.150148	0.0750740	+	0.997178i	$0.476081\pi$
$140$	0	0
$141$	749.396	0.447593
$142$	0	0
$143$	−390.758	−0.228510
$144$	0	0
$145$	2615.27	1.49784
$146$	0	0
$147$	−164.265	−0.0921657
$148$	0	0
$149$	−3395.29	−1.86680	−0.933399	−	0.358840i	$-0.883172\pi$
$149$	−3395.29	−1.86680	−0.933399	+	0.358840i	$0.883172\pi$
$150$	0	0
$151$	−970.047	−0.522790	−0.261395	−	0.965232i	$-0.584183\pi$
$151$	−970.047	−0.522790	−0.261395	+	0.965232i	$0.584183\pi$
$152$	0	0
$153$	1062.49	0.561420
$154$	0	0
$155$	−804.836	−0.417071
$156$	0	0
$157$	2373.04	1.20630	0.603150	−	0.797628i	$-0.293912\pi$
$157$	2373.04	1.20630	0.603150	+	0.797628i	$0.293912\pi$
$158$	0	0
$159$	961.967	0.479805
$160$	0	0
$161$	211.997	0.103774
$162$	0	0
$163$	1412.22	0.678609	0.339304	−	0.940677i	$-0.389808\pi$
$163$	1412.22	0.678609	0.339304	+	0.940677i	$0.389808\pi$
$164$	0	0
$165$	344.876	0.162718
$166$	0	0
$167$	−1426.74	−0.661107	−0.330553	−	0.943787i	$-0.607235\pi$
$167$	−1426.74	−0.661107	−0.330553	+	0.943787i	$0.607235\pi$
$168$	0	0
$169$	−935.081	−0.425617
$170$	0	0
$171$	928.886	0.415402
$172$	0	0
$173$	3495.90	1.53635	0.768174	−	0.640241i	$-0.221165\pi$
$173$	3495.90	1.53635	0.768174	+	0.640241i	$0.221165\pi$
$174$	0	0
$175$	262.735	0.113491
$176$	0	0
$177$	1334.29	0.566619
$178$	0	0
$179$	4131.10	1.72499	0.862495	−	0.506065i	$-0.168901\pi$
$179$	4131.10	1.72499	0.862495	+	0.506065i	$0.168901\pi$
$180$	0	0
$181$	1610.37	0.661315	0.330658	−	0.943751i	$-0.392729\pi$
$181$	1610.37	0.661315	0.330658	+	0.943751i	$0.392729\pi$
$182$	0	0
$183$	−2484.88	−1.00376
$184$	0	0
$185$	362.513	0.144068
$186$	0	0
$187$	−741.503	−0.289968
$188$	0	0
$189$	−1003.47	−0.386198
$190$	0	0
$191$	2086.58	0.790470	0.395235	−	0.918580i	$-0.370663\pi$
$191$	2086.58	0.790470	0.395235	+	0.918580i	$0.370663\pi$
$192$	0	0
$193$	−1830.38	−0.682660	−0.341330	−	0.939944i	$-0.610877\pi$
$193$	−1830.38	−0.682660	−0.341330	+	0.939944i	$0.610877\pi$
$194$	0	0
$195$	−1113.75	−0.409010
$196$	0	0
$197$	−2575.49	−0.931452	−0.465726	−	0.884929i	$-0.654207\pi$
$197$	−2575.49	−0.931452	−0.465726	+	0.884929i	$0.654207\pi$
$198$	0	0
$199$	425.208	0.151468	0.0757342	−	0.997128i	$-0.475870\pi$
$199$	425.208	0.151468	0.0757342	+	0.997128i	$0.475870\pi$
$200$	0	0
$201$	2497.20	0.876312
$202$	0	0
$203$	1957.46	0.676783
$204$	0	0
$205$	3152.18	1.07394
$206$	0	0
$207$	477.349	0.160280
$208$	0	0
$209$	−648.262	−0.214551
$210$	0	0
$211$	1307.91	0.426732	0.213366	−	0.976972i	$-0.431557\pi$
$211$	1307.91	0.426732	0.213366	+	0.976972i	$0.431557\pi$
$212$	0	0
$213$	2222.75	0.715026
$214$	0	0
$215$	459.960	0.145902
$216$	0	0
$217$	−602.399	−0.188449
$218$	0	0
$219$	248.704	0.0767391
$220$	0	0
$221$	2394.62	0.728866
$222$	0	0
$223$	−875.473	−0.262897	−0.131448	−	0.991323i	$-0.541963\pi$
$223$	−875.473	−0.262897	−0.131448	+	0.991323i	$0.541963\pi$
$224$	0	0
$225$	591.594	0.175287
$226$	0	0
$227$	−2589.91	−0.757260	−0.378630	−	0.925548i	$-0.623605\pi$
$227$	−2589.91	−0.757260	−0.378630	+	0.925548i	$0.623605\pi$
$228$	0	0
$229$	−525.399	−0.151613	−0.0758064	−	0.997123i	$-0.524153\pi$
$229$	−525.399	−0.151613	−0.0758064	+	0.997123i	$0.524153\pi$
$230$	0	0
$231$	258.131	0.0735228
$232$	0	0
$233$	3778.05	1.06227	0.531134	−	0.847288i	$-0.321766\pi$
$233$	3778.05	1.06227	0.531134	+	0.847288i	$0.321766\pi$
$234$	0	0
$235$	2090.66	0.580338
$236$	0	0
$237$	−495.765	−0.135879
$238$	0	0
$239$	6615.76	1.79053	0.895267	−	0.445529i	$-0.146984\pi$
$239$	6615.76	1.79053	0.895267	+	0.445529i	$0.146984\pi$
$240$	0	0
$241$	−1997.99	−0.534034	−0.267017	−	0.963692i	$-0.586038\pi$
$241$	−1997.99	−0.534034	−0.267017	+	0.963692i	$0.586038\pi$
$242$	0	0
$243$	−3686.13	−0.973110
$244$	0	0
$245$	−458.265	−0.119500
$246$	0	0
$247$	2093.50	0.539297
$248$	0	0
$249$	−4287.81	−1.09128
$250$	0	0
$251$	2227.49	0.560150	0.280075	−	0.959978i	$-0.409641\pi$
$251$	2227.49	0.560150	0.280075	+	0.959978i	$0.409641\pi$
$252$	0	0
$253$	−333.138	−0.0827834
$254$	0	0
$255$	−2113.44	−0.519015
$256$	0	0
$257$	−4077.84	−0.989761	−0.494881	−	0.868961i	$-0.664788\pi$
$257$	−4077.84	−0.989761	−0.494881	+	0.868961i	$0.664788\pi$
$258$	0	0
$259$	271.332	0.0650956
$260$	0	0
$261$	4407.58	1.04530
$262$	0	0
$263$	−3023.82	−0.708961	−0.354480	−	0.935063i	$-0.615342\pi$
$263$	−3023.82	−0.708961	−0.354480	+	0.935063i	$0.615342\pi$
$264$	0	0
$265$	2683.68	0.622104
$266$	0	0
$267$	−2482.08	−0.568916
$268$	0	0
$269$	−2142.40	−0.485593	−0.242797	−	0.970077i	$-0.578065\pi$
$269$	−2142.40	−0.485593	−0.242797	+	0.970077i	$0.578065\pi$
$270$	0	0
$271$	−3887.67	−0.871435	−0.435717	−	0.900084i	$-0.643505\pi$
$271$	−3887.67	−0.871435	−0.435717	+	0.900084i	$0.643505\pi$
$272$	0	0
$273$	−833.610	−0.184807
$274$	0	0
$275$	−412.869	−0.0905343
$276$	0	0
$277$	5834.48	1.26556	0.632779	−	0.774332i	$-0.281914\pi$
$277$	5834.48	1.26556	0.632779	+	0.774332i	$0.281914\pi$
$278$	0	0
$279$	−1356.41	−0.291061
$280$	0	0
$281$	−5911.56	−1.25500	−0.627499	−	0.778617i	$-0.715921\pi$
$281$	−5911.56	−1.25500	−0.627499	+	0.778617i	$0.715921\pi$
$282$	0	0
$283$	3225.30	0.677471	0.338736	−	0.940882i	$-0.390001\pi$
$283$	3225.30	0.677471	0.338736	+	0.940882i	$0.390001\pi$
$284$	0	0
$285$	−1847.68	−0.384026
$286$	0	0
$287$	2359.33	0.485250
$288$	0	0
$289$	−368.973	−0.0751013
$290$	0	0
$291$	2362.65	0.475949
$292$	0	0
$293$	8132.22	1.62146	0.810732	−	0.585417i	$-0.199069\pi$
$293$	8132.22	1.62146	0.810732	+	0.585417i	$0.199069\pi$
$294$	0	0
$295$	3722.39	0.734665
$296$	0	0
$297$	1576.88	0.308080
$298$	0	0
$299$	1075.84	0.208085
$300$	0	0
$301$	344.268	0.0659246
$302$	0	0
$303$	−211.153	−0.0400345
$304$	0	0
$305$	−6932.29	−1.30145
$306$	0	0
$307$	9655.10	1.79494	0.897469	−	0.441078i	$-0.145404\pi$
$307$	9655.10	1.79494	0.897469	+	0.441078i	$0.145404\pi$
$308$	0	0
$309$	3095.91	0.569968
$310$	0	0
$311$	−4529.61	−0.825886	−0.412943	−	0.910757i	$-0.635499\pi$
$311$	−4529.61	−0.825886	−0.412943	+	0.910757i	$0.635499\pi$
$312$	0	0
$313$	−918.419	−0.165853	−0.0829267	−	0.996556i	$-0.526427\pi$
$313$	−918.419	−0.165853	−0.0829267	+	0.996556i	$0.526427\pi$
$314$	0	0
$315$	−1031.87	−0.184568
$316$	0	0
$317$	−8393.78	−1.48720	−0.743598	−	0.668626i	$-0.766883\pi$
$317$	−8393.78	−1.48720	−0.743598	+	0.668626i	$0.766883\pi$
$318$	0	0
$319$	−3076.01	−0.539886
$320$	0	0
$321$	−2026.87	−0.352426
$322$	0	0
$323$	3972.63	0.684344
$324$	0	0
$325$	1333.32	0.227568
$326$	0	0
$327$	3463.25	0.585682
$328$	0	0
$329$	1564.81	0.262220
$330$	0	0
$331$	9412.92	1.56308	0.781542	−	0.623852i	$-0.214433\pi$
$331$	9412.92	1.56308	0.781542	+	0.623852i	$0.214433\pi$
$332$	0	0
$333$	610.953	0.100541
$334$	0	0
$335$	6966.65	1.13621
$336$	0	0
$337$	−3128.50	−0.505698	−0.252849	−	0.967506i	$-0.581368\pi$
$337$	−3128.50	−0.505698	−0.252849	+	0.967506i	$0.581368\pi$
$338$	0	0
$339$	−6745.23	−1.08068
$340$	0	0
$341$	946.628	0.150331
$342$	0	0
$343$	−343.000	−0.0539949
$344$	0	0
$345$	−949.514	−0.148174
$346$	0	0
$347$	−4484.72	−0.693811	−0.346905	−	0.937900i	$-0.612768\pi$
$347$	−4484.72	−0.693811	−0.346905	+	0.937900i	$0.612768\pi$
$348$	0	0
$349$	−11247.9	−1.72518	−0.862591	−	0.505901i	$-0.831160\pi$
$349$	−11247.9	−1.72518	−0.862591	+	0.505901i	$0.831160\pi$
$350$	0	0
$351$	−5092.38	−0.774390
$352$	0	0
$353$	658.950	0.0993551	0.0496776	−	0.998765i	$-0.484181\pi$
$353$	658.950	0.0993551	0.0496776	+	0.998765i	$0.484181\pi$
$354$	0	0
$355$	6201.02	0.927087
$356$	0	0
$357$	−1581.86	−0.234512
$358$	0	0
$359$	−9212.09	−1.35431	−0.677153	−	0.735842i	$-0.736786\pi$
$359$	−9212.09	−1.35431	−0.677153	+	0.735842i	$0.736786\pi$
$360$	0	0
$361$	−3385.91	−0.493645
$362$	0	0
$363$	−405.634	−0.0586509
$364$	0	0
$365$	693.832	0.0994982
$366$	0	0
$367$	−2664.56	−0.378989	−0.189494	−	0.981882i	$-0.560685\pi$
$367$	−2664.56	−0.378989	−0.189494	+	0.981882i	$0.560685\pi$
$368$	0	0
$369$	5312.45	0.749472
$370$	0	0
$371$	2008.67	0.281092
$372$	0	0
$373$	−727.914	−0.101045	−0.0505227	−	0.998723i	$-0.516089\pi$
$373$	−727.914	−0.101045	−0.0505227	+	0.998723i	$0.516089\pi$
$374$	0	0
$375$	−5095.81	−0.701724
$376$	0	0
$377$	9933.71	1.35706
$378$	0	0
$379$	6080.35	0.824080	0.412040	−	0.911166i	$-0.364816\pi$
$379$	6080.35	0.824080	0.412040	+	0.911166i	$0.364816\pi$
$380$	0	0
$381$	−2457.36	−0.330432
$382$	0	0
$383$	−10211.7	−1.36238	−0.681190	−	0.732107i	$-0.738537\pi$
$383$	−10211.7	−1.36238	−0.681190	+	0.732107i	$0.738537\pi$
$384$	0	0
$385$	720.131	0.0953280
$386$	0	0
$387$	775.182	0.101821
$388$	0	0
$389$	−5140.84	−0.670053	−0.335027	−	0.942209i	$-0.608745\pi$
$389$	−5140.84	−0.670053	−0.335027	+	0.942209i	$0.608745\pi$
$390$	0	0
$391$	2041.51	0.264050
$392$	0	0
$393$	−5086.50	−0.652876
$394$	0	0
$395$	−1383.08	−0.176178
$396$	0	0
$397$	824.269	0.104204	0.0521019	−	0.998642i	$-0.483408\pi$
$397$	824.269	0.104204	0.0521019	+	0.998642i	$0.483408\pi$
$398$	0	0
$399$	−1382.95	−0.173519
$400$	0	0
$401$	6020.85	0.749792	0.374896	−	0.927067i	$-0.377678\pi$
$401$	6020.85	0.749792	0.374896	+	0.927067i	$0.377678\pi$
$402$	0	0
$403$	−3057.05	−0.377872
$404$	0	0
$405$	514.379	0.0631104
$406$	0	0
$407$	−426.379	−0.0519283
$408$	0	0
$409$	1852.05	0.223907	0.111954	−	0.993713i	$-0.464289\pi$
$409$	1852.05	0.223907	0.111954	+	0.993713i	$0.464289\pi$
$410$	0	0
$411$	7087.87	0.850654
$412$	0	0
$413$	2786.12	0.331951
$414$	0	0
$415$	−11962.1	−1.41493
$416$	0	0
$417$	−824.880	−0.0968694
$418$	0	0
$419$	292.739	0.0341318	0.0170659	−	0.999854i	$-0.494567\pi$
$419$	292.739	0.0341318	0.0170659	+	0.999854i	$0.494567\pi$
$420$	0	0
$421$	−1268.47	−0.146844	−0.0734221	−	0.997301i	$-0.523392\pi$
$421$	−1268.47	−0.146844	−0.0734221	+	0.997301i	$0.523392\pi$
$422$	0	0
$423$	3523.44	0.405001
$424$	0	0
$425$	2530.11	0.288773
$426$	0	0
$427$	−5188.64	−0.588047
$428$	0	0
$429$	1309.96	0.147425
$430$	0	0
$431$	−11370.4	−1.27075	−0.635373	−	0.772205i	$-0.719154\pi$
$431$	−11370.4	−1.27075	−0.635373	+	0.772205i	$0.719154\pi$
$432$	0	0
$433$	10934.9	1.21362	0.606811	−	0.794846i	$-0.292448\pi$
$433$	10934.9	1.21362	0.606811	+	0.794846i	$0.292448\pi$
$434$	0	0
$435$	−8767.30	−0.966344
$436$	0	0
$437$	1784.80	0.195374
$438$	0	0
$439$	13306.4	1.44665	0.723323	−	0.690510i	$-0.242614\pi$
$439$	13306.4	1.44665	0.723323	+	0.690510i	$0.242614\pi$
$440$	0	0
$441$	−772.326	−0.0833955
$442$	0	0
$443$	−16693.8	−1.79039	−0.895197	−	0.445671i	$-0.852965\pi$
$443$	−16693.8	−1.79039	−0.895197	+	0.445671i	$0.852965\pi$
$444$	0	0
$445$	−6924.47	−0.737644
$446$	0	0
$447$	11382.2	1.20438
$448$	0	0
$449$	−10253.2	−1.07768	−0.538841	−	0.842407i	$-0.681138\pi$
$449$	−10253.2	−1.07768	−0.538841	+	0.842407i	$0.681138\pi$
$450$	0	0
$451$	−3707.52	−0.387096
$452$	0	0
$453$	3251.94	0.337283
$454$	0	0
$455$	−2325.60	−0.239617
$456$	0	0
$457$	−2851.30	−0.291856	−0.145928	−	0.989295i	$-0.546617\pi$
$457$	−2851.30	−0.291856	−0.145928	+	0.989295i	$0.546617\pi$
$458$	0	0
$459$	−9663.30	−0.982667
$460$	0	0
$461$	−11578.6	−1.16978	−0.584889	−	0.811113i	$-0.698862\pi$
$461$	−11578.6	−1.16978	−0.584889	+	0.811113i	$0.698862\pi$
$462$	0	0
$463$	12740.7	1.27886	0.639428	−	0.768851i	$-0.279171\pi$
$463$	12740.7	1.27886	0.639428	+	0.768851i	$0.279171\pi$
$464$	0	0
$465$	2698.09	0.269077
$466$	0	0
$467$	−6002.96	−0.594826	−0.297413	−	0.954749i	$-0.596124\pi$
$467$	−6002.96	−0.594826	−0.297413	+	0.954749i	$0.596124\pi$
$468$	0	0
$469$	5214.37	0.513384
$470$	0	0
$471$	−7955.25	−0.778256
$472$	0	0
$473$	−540.993	−0.0525896
$474$	0	0
$475$	2211.96	0.213667
$476$	0	0
$477$	4522.88	0.434148
$478$	0	0
$479$	−7416.40	−0.707441	−0.353720	−	0.935351i	$-0.615084\pi$
$479$	−7416.40	−0.707441	−0.353720	+	0.935351i	$0.615084\pi$
$480$	0	0
$481$	1376.95	0.130527
$482$	0	0
$483$	−710.687	−0.0669511
$484$	0	0
$485$	6591.31	0.617105
$486$	0	0
$487$	−81.2105	−0.00755647	−0.00377823	−	0.999993i	$-0.501203\pi$
$487$	−81.2105	−0.00755647	−0.00377823	+	0.999993i	$0.501203\pi$
$488$	0	0
$489$	−4734.24	−0.437811
$490$	0	0
$491$	−3763.83	−0.345946	−0.172973	−	0.984927i	$-0.555337\pi$
$491$	−3763.83	−0.345946	−0.172973	+	0.984927i	$0.555337\pi$
$492$	0	0
$493$	18850.2	1.72205
$494$	0	0
$495$	1621.50	0.147235
$496$	0	0
$497$	4641.31	0.418895
$498$	0	0
$499$	−20835.3	−1.86917	−0.934584	−	0.355742i	$-0.884228\pi$
$499$	−20835.3	−1.86917	−0.934584	+	0.355742i	$0.884228\pi$
$500$	0	0
$501$	4782.95	0.426520
$502$	0	0
$503$	−9230.83	−0.818255	−0.409128	−	0.912477i	$-0.634167\pi$
$503$	−9230.83	−0.818255	−0.409128	+	0.912477i	$0.634167\pi$
$504$	0	0
$505$	−589.074	−0.0519078
$506$	0	0
$507$	3134.72	0.274591
$508$	0	0
$509$	−3453.11	−0.300700	−0.150350	−	0.988633i	$-0.548040\pi$
$509$	−3453.11	−0.300700	−0.150350	+	0.988633i	$0.548040\pi$
$510$	0	0
$511$	519.316	0.0449573
$512$	0	0
$513$	−8448.17	−0.727087
$514$	0	0
$515$	8636.93	0.739007
$516$	0	0
$517$	−2458.98	−0.209179
$518$	0	0
$519$	−11719.5	−0.991191
$520$	0	0
$521$	4734.94	0.398161	0.199080	−	0.979983i	$-0.436205\pi$
$521$	4734.94	0.398161	0.199080	+	0.979983i	$0.436205\pi$
$522$	0	0
$523$	1095.22	0.0915691	0.0457846	−	0.998951i	$-0.485421\pi$
$523$	1095.22	0.0915691	0.0457846	+	0.998951i	$0.485421\pi$
$524$	0	0
$525$	−880.779	−0.0732197
$526$	0	0
$527$	−5801.05	−0.479503
$528$	0	0
$529$	−11249.8	−0.924616
$530$	0	0
$531$	6273.44	0.512701
$532$	0	0
$533$	11973.1	0.973006
$534$	0	0
$535$	−5654.53	−0.456947
$536$	0	0
$537$	−13848.9	−1.11289
$538$	0	0
$539$	539.000	0.0430730
$540$	0	0
$541$	−17274.5	−1.37281	−0.686404	−	0.727220i	$-0.740812\pi$
$541$	−17274.5	−1.37281	−0.686404	+	0.727220i	$0.740812\pi$
$542$	0	0
$543$	−5398.53	−0.426654
$544$	0	0
$545$	9661.73	0.759382
$546$	0	0
$547$	24143.3	1.88719	0.943597	−	0.331097i	$-0.107419\pi$
$547$	24143.3	1.88719	0.943597	+	0.331097i	$0.107419\pi$
$548$	0	0
$549$	−11683.2	−0.908243
$550$	0	0
$551$	16479.9	1.27417
$552$	0	0
$553$	−1035.20	−0.0796044
$554$	0	0
$555$	−1215.27	−0.0929467
$556$	0	0
$557$	−20061.9	−1.52612	−0.763059	−	0.646329i	$-0.776304\pi$
$557$	−20061.9	−1.52612	−0.763059	+	0.646329i	$0.776304\pi$
$558$	0	0
$559$	1747.09	0.132190
$560$	0	0
$561$	2485.78	0.187076
$562$	0	0
$563$	−8682.12	−0.649925	−0.324962	−	0.945727i	$-0.605352\pi$
$563$	−8682.12	−0.649925	−0.324962	+	0.945727i	$0.605352\pi$
$564$	0	0
$565$	−18817.8	−1.40119
$566$	0	0
$567$	385.000	0.0285158
$568$	0	0
$569$	11156.9	0.822008	0.411004	−	0.911633i	$-0.365178\pi$
$569$	11156.9	0.822008	0.411004	+	0.911633i	$0.365178\pi$
$570$	0	0
$571$	−9893.85	−0.725122	−0.362561	−	0.931960i	$-0.618098\pi$
$571$	−9893.85	−0.725122	−0.362561	+	0.931960i	$0.618098\pi$
$572$	0	0
$573$	−6994.95	−0.509979
$574$	0	0
$575$	1136.71	0.0824421
$576$	0	0
$577$	11288.6	0.814471	0.407236	−	0.913323i	$-0.366493\pi$
$577$	11288.6	0.814471	0.407236	+	0.913323i	$0.366493\pi$
$578$	0	0
$579$	6136.06	0.440425
$580$	0	0
$581$	−8953.33	−0.639323
$582$	0	0
$583$	−3156.48	−0.224234
$584$	0	0
$585$	−5236.50	−0.370090
$586$	0	0
$587$	−3964.92	−0.278790	−0.139395	−	0.990237i	$-0.544516\pi$
$587$	−3964.92	−0.278790	−0.139395	+	0.990237i	$0.544516\pi$
$588$	0	0
$589$	−5071.59	−0.354790
$590$	0	0
$591$	8633.94	0.600936
$592$	0	0
$593$	−11758.2	−0.814251	−0.407126	−	0.913372i	$-0.633469\pi$
$593$	−11758.2	−0.814251	−0.407126	+	0.913372i	$0.633469\pi$
$594$	0	0
$595$	−4413.05	−0.304063
$596$	0	0
$597$	−1425.45	−0.0977214
$598$	0	0
$599$	2415.65	0.164776	0.0823880	−	0.996600i	$-0.473745\pi$
$599$	2415.65	0.164776	0.0823880	+	0.996600i	$0.473745\pi$
$600$	0	0
$601$	−24568.7	−1.66752	−0.833760	−	0.552128i	$-0.813816\pi$
$601$	−24568.7	−1.66752	−0.833760	+	0.552128i	$0.813816\pi$
$602$	0	0
$603$	11741.1	0.792925
$604$	0	0
$605$	−1131.63	−0.0760454
$606$	0	0
$607$	14506.3	0.970003	0.485001	−	0.874513i	$-0.338819\pi$
$607$	14506.3	0.970003	0.485001	+	0.874513i	$0.338819\pi$
$608$	0	0
$609$	−6562.10	−0.436633
$610$	0	0
$611$	7941.05	0.525795
$612$	0	0
$613$	16722.4	1.10181	0.550907	−	0.834567i	$-0.314282\pi$
$613$	16722.4	1.10181	0.550907	+	0.834567i	$0.314282\pi$
$614$	0	0
$615$	−10567.2	−0.692864
$616$	0	0
$617$	−673.462	−0.0439426	−0.0219713	−	0.999759i	$-0.506994\pi$
$617$	−673.462	−0.0439426	−0.0219713	+	0.999759i	$0.506994\pi$
$618$	0	0
$619$	26427.1	1.71598	0.857991	−	0.513665i	$-0.171712\pi$
$619$	26427.1	1.71598	0.857991	+	0.513665i	$0.171712\pi$
$620$	0	0
$621$	−4341.46	−0.280542
$622$	0	0
$623$	−5182.80	−0.333297
$624$	0	0
$625$	−9524.54	−0.609570
$626$	0	0
$627$	2173.20	0.138420
$628$	0	0
$629$	2612.91	0.165633
$630$	0	0
$631$	−18147.3	−1.14490	−0.572449	−	0.819940i	$-0.694007\pi$
$631$	−18147.3	−1.14490	−0.572449	+	0.819940i	$0.694007\pi$
$632$	0	0
$633$	−4384.58	−0.275310
$634$	0	0
$635$	−6855.52	−0.428430
$636$	0	0
$637$	−1740.65	−0.108269
$638$	0	0
$639$	10450.7	0.646987
$640$	0	0
$641$	−6866.75	−0.423121	−0.211560	−	0.977365i	$-0.567854\pi$
$641$	−6866.75	−0.423121	−0.211560	+	0.977365i	$0.567854\pi$
$642$	0	0
$643$	20569.1	1.26154	0.630768	−	0.775972i	$-0.282740\pi$
$643$	20569.1	1.26154	0.630768	+	0.775972i	$0.282740\pi$
$644$	0	0
$645$	−1541.95	−0.0941303
$646$	0	0
$647$	−29088.2	−1.76751	−0.883753	−	0.467953i	$-0.844992\pi$
$647$	−29088.2	−1.76751	−0.883753	+	0.467953i	$0.844992\pi$
$648$	0	0
$649$	−4378.19	−0.264806
$650$	0	0
$651$	2019.45	0.121580
$652$	0	0
$653$	−17467.0	−1.04677	−0.523383	−	0.852097i	$-0.675330\pi$
$653$	−17467.0	−1.04677	−0.523383	+	0.852097i	$0.675330\pi$
$654$	0	0
$655$	−14190.3	−0.846504
$656$	0	0
$657$	1169.33	0.0694369
$658$	0	0
$659$	2716.17	0.160557	0.0802784	−	0.996772i	$-0.474419\pi$
$659$	2716.17	0.160557	0.0802784	+	0.996772i	$0.474419\pi$
$660$	0	0
$661$	−30321.9	−1.78424	−0.892120	−	0.451798i	$-0.850783\pi$
$661$	−30321.9	−1.78424	−0.892120	+	0.451798i	$0.850783\pi$
$662$	0	0
$663$	−8027.60	−0.470235
$664$	0	0
$665$	−3858.13	−0.224980
$666$	0	0
$667$	8468.89	0.491630
$668$	0	0
$669$	2934.89	0.169611
$670$	0	0
$671$	8153.58	0.469099
$672$	0	0
$673$	25341.7	1.45149	0.725744	−	0.687965i	$-0.241496\pi$
$673$	25341.7	1.45149	0.725744	+	0.687965i	$0.241496\pi$
$674$	0	0
$675$	−5380.52	−0.306810
$676$	0	0
$677$	23282.3	1.32173	0.660865	−	0.750505i	$-0.270190\pi$
$677$	23282.3	1.32173	0.660865	+	0.750505i	$0.270190\pi$
$678$	0	0
$679$	4933.43	0.278833
$680$	0	0
$681$	8682.27	0.488554
$682$	0	0
$683$	13271.9	0.743534	0.371767	−	0.928326i	$-0.378752\pi$
$683$	13271.9	0.743534	0.371767	+	0.928326i	$0.378752\pi$
$684$	0	0
$685$	19773.7	1.10294
$686$	0	0
$687$	1761.32	0.0978146
$688$	0	0
$689$	10193.6	0.563635
$690$	0	0
$691$	16332.0	0.899127	0.449564	−	0.893248i	$-0.351580\pi$
$691$	16332.0	0.899127	0.449564	+	0.893248i	$0.351580\pi$
$692$	0	0
$693$	1213.65	0.0665266
$694$	0	0
$695$	−2301.24	−0.125599
$696$	0	0
$697$	22720.1	1.23470
$698$	0	0
$699$	−12665.4	−0.685333
$700$	0	0
$701$	12513.0	0.674196	0.337098	−	0.941470i	$-0.390555\pi$
$701$	12513.0	0.674196	0.337098	+	0.941470i	$0.390555\pi$
$702$	0	0
$703$	2284.34	0.122554
$704$	0	0
$705$	−7008.62	−0.374411
$706$	0	0
$707$	−440.907	−0.0234540
$708$	0	0
$709$	23162.3	1.22691	0.613454	−	0.789731i	$-0.289780\pi$
$709$	23162.3	1.22691	0.613454	+	0.789731i	$0.289780\pi$
$710$	0	0
$711$	−2330.94	−0.122950
$712$	0	0
$713$	−2606.26	−0.136894
$714$	0	0
$715$	3654.51	0.191148
$716$	0	0
$717$	−22178.3	−1.15518
$718$	0	0
$719$	−425.956	−0.0220938	−0.0110469	−	0.999939i	$-0.503516\pi$
$719$	−425.956	−0.0220938	−0.0110469	+	0.999939i	$0.503516\pi$
$720$	0	0
$721$	6464.52	0.333913
$722$	0	0
$723$	6697.97	0.344537
$724$	0	0
$725$	10495.8	0.537661
$726$	0	0
$727$	−16697.6	−0.851830	−0.425915	−	0.904763i	$-0.640048\pi$
$727$	−16697.6	−0.851830	−0.425915	+	0.904763i	$0.640048\pi$
$728$	0	0
$729$	13842.2	0.703257
$730$	0	0
$731$	3315.28	0.167743
$732$	0	0
$733$	9609.35	0.484215	0.242107	−	0.970249i	$-0.422161\pi$
$733$	9609.35	0.484215	0.242107	+	0.970249i	$0.422161\pi$
$734$	0	0
$735$	1536.27	0.0770966
$736$	0	0
$737$	−8194.00	−0.409539
$738$	0	0
$739$	−16110.7	−0.801949	−0.400974	−	0.916089i	$-0.631328\pi$
$739$	−16110.7	−0.801949	−0.400974	+	0.916089i	$0.631328\pi$
$740$	0	0
$741$	−7018.15	−0.347933
$742$	0	0
$743$	−9112.35	−0.449932	−0.224966	−	0.974367i	$-0.572227\pi$
$743$	−9112.35	−0.449932	−0.224966	+	0.974367i	$0.572227\pi$
$744$	0	0
$745$	31753.9	1.56158
$746$	0	0
$747$	−20160.0	−0.987438
$748$	0	0
$749$	−4232.27	−0.206467
$750$	0	0
$751$	−25753.5	−1.25134	−0.625671	−	0.780087i	$-0.715175\pi$
$751$	−25753.5	−1.25134	−0.625671	+	0.780087i	$0.715175\pi$
$752$	0	0
$753$	−7467.32	−0.361387
$754$	0	0
$755$	9072.22	0.437314
$756$	0	0
$757$	−15242.5	−0.731832	−0.365916	−	0.930648i	$-0.619244\pi$
$757$	−15242.5	−0.731832	−0.365916	+	0.930648i	$0.619244\pi$
$758$	0	0
$759$	1116.79	0.0534085
$760$	0	0
$761$	32014.0	1.52498	0.762489	−	0.647001i	$-0.223977\pi$
$761$	32014.0	1.52498	0.762489	+	0.647001i	$0.223977\pi$
$762$	0	0
$763$	7231.56	0.343120
$764$	0	0
$765$	−9936.78	−0.469627
$766$	0	0
$767$	14139.0	0.665617
$768$	0	0
$769$	12648.9	0.593148	0.296574	−	0.955010i	$-0.404156\pi$
$769$	12648.9	0.593148	0.296574	+	0.955010i	$0.404156\pi$
$770$	0	0
$771$	13670.3	0.638554
$772$	0	0
$773$	10622.5	0.494262	0.247131	−	0.968982i	$-0.420512\pi$
$773$	10622.5	0.494262	0.247131	+	0.968982i	$0.420512\pi$
$774$	0	0
$775$	−3230.03	−0.149711
$776$	0	0
$777$	−909.601	−0.0419971
$778$	0	0
$779$	19863.2	0.913571
$780$	0	0
$781$	−7293.48	−0.334163
$782$	0	0
$783$	−40086.7	−1.82961
$784$	0	0
$785$	−22193.5	−1.00907
$786$	0	0
$787$	34348.0	1.55575	0.777874	−	0.628421i	$-0.216298\pi$
$787$	34348.0	1.55575	0.777874	+	0.628421i	$0.216298\pi$
$788$	0	0
$789$	10136.9	0.457393
$790$	0	0
$791$	−14084.6	−0.633112
$792$	0	0
$793$	−26331.3	−1.17913
$794$	0	0
$795$	−8996.65	−0.401356
$796$	0	0
$797$	−13987.7	−0.621669	−0.310835	−	0.950464i	$-0.600609\pi$
$797$	−13987.7	−0.621669	−0.310835	+	0.950464i	$0.600609\pi$
$798$	0	0
$799$	15068.9	0.667210
$800$	0	0
$801$	−11670.0	−0.514780
$802$	0	0
$803$	−816.068	−0.0358635
$804$	0	0
$805$	−1982.67	−0.0868073
$806$	0	0
$807$	7182.09	0.313285
$808$	0	0
$809$	1663.22	0.0722815	0.0361407	−	0.999347i	$-0.488494\pi$
$809$	1663.22	0.0722815	0.0361407	+	0.999347i	$0.488494\pi$
$810$	0	0
$811$	−15310.3	−0.662906	−0.331453	−	0.943472i	$-0.607539\pi$
$811$	−15310.3	−0.662906	−0.331453	+	0.943472i	$0.607539\pi$
$812$	0	0
$813$	13032.8	0.562215
$814$	0	0
$815$	−13207.5	−0.567656
$816$	0	0
$817$	2898.39	0.124115
$818$	0	0
$819$	−3919.39	−0.167222
$820$	0	0
$821$	−26159.0	−1.11200	−0.556001	−	0.831182i	$-0.687665\pi$
$821$	−26159.0	−1.11200	−0.556001	+	0.831182i	$0.687665\pi$
$822$	0	0
$823$	−32581.5	−1.37998	−0.689988	−	0.723821i	$-0.742384\pi$
$823$	−32581.5	−1.37998	−0.689988	+	0.723821i	$0.742384\pi$
$824$	0	0
$825$	1384.08	0.0584091
$826$	0	0
$827$	1423.69	0.0598629	0.0299315	−	0.999552i	$-0.490471\pi$
$827$	1423.69	0.0598629	0.0299315	+	0.999552i	$0.490471\pi$
$828$	0	0
$829$	−41753.7	−1.74929	−0.874647	−	0.484760i	$-0.838907\pi$
$829$	−41753.7	−1.74929	−0.874647	+	0.484760i	$0.838907\pi$
$830$	0	0
$831$	−19559.2	−0.816488
$832$	0	0
$833$	−3303.06	−0.137388
$834$	0	0
$835$	13343.4	0.553016
$836$	0	0
$837$	12336.5	0.509452
$838$	0	0
$839$	−17346.3	−0.713779	−0.356889	−	0.934147i	$-0.616163\pi$
$839$	−17346.3	−0.713779	−0.356889	+	0.934147i	$0.616163\pi$
$840$	0	0
$841$	53808.2	2.20625
$842$	0	0
$843$	19817.6	0.809674
$844$	0	0
$845$	8745.20	0.356029
$846$	0	0
$847$	−847.000	−0.0343604
$848$	0	0
$849$	−10812.3	−0.437077
$850$	0	0
$851$	1173.91	0.0472868
$852$	0	0
$853$	−14073.7	−0.564916	−0.282458	−	0.959280i	$-0.591150\pi$
$853$	−14073.7	−0.564916	−0.282458	+	0.959280i	$0.591150\pi$
$854$	0	0
$855$	−8687.26	−0.347483
$856$	0	0
$857$	−34427.2	−1.37224	−0.686121	−	0.727488i	$-0.740688\pi$
$857$	−34427.2	−1.37224	−0.686121	+	0.727488i	$0.740688\pi$
$858$	0	0
$859$	44054.9	1.74986	0.874931	−	0.484247i	$-0.160906\pi$
$859$	44054.9	1.74986	0.874931	+	0.484247i	$0.160906\pi$
$860$	0	0
$861$	−7909.30	−0.313064
$862$	0	0
$863$	−39955.4	−1.57601	−0.788006	−	0.615668i	$-0.788886\pi$
$863$	−39955.4	−1.57601	−0.788006	+	0.615668i	$0.788886\pi$
$864$	0	0
$865$	−32694.9	−1.28516
$866$	0	0
$867$	1236.93	0.0484524
$868$	0	0
$869$	1626.74	0.0635023
$870$	0	0
$871$	26461.8	1.02942
$872$	0	0
$873$	11108.5	0.430659
$874$	0	0
$875$	−10640.5	−0.411102
$876$	0	0
$877$	1563.57	0.0602030	0.0301015	−	0.999547i	$-0.490417\pi$
$877$	1563.57	0.0602030	0.0301015	+	0.999547i	$0.490417\pi$
$878$	0	0
$879$	−27262.0	−1.04610
$880$	0	0
$881$	30194.6	1.15469	0.577344	−	0.816501i	$-0.304089\pi$
$881$	30194.6	1.15469	0.577344	+	0.816501i	$0.304089\pi$
$882$	0	0
$883$	27472.4	1.04702	0.523511	−	0.852019i	$-0.324622\pi$
$883$	27472.4	1.04702	0.523511	+	0.852019i	$0.324622\pi$
$884$	0	0
$885$	−12478.8	−0.473976
$886$	0	0
$887$	−7974.31	−0.301862	−0.150931	−	0.988544i	$-0.548227\pi$
$887$	−7974.31	−0.301862	−0.150931	+	0.988544i	$0.548227\pi$
$888$	0	0
$889$	−5131.19	−0.193582
$890$	0	0
$891$	−605.000	−0.0227478
$892$	0	0
$893$	13174.1	0.493677
$894$	0	0
$895$	−38635.5	−1.44295
$896$	0	0
$897$	−3606.59	−0.134248
$898$	0	0
$899$	−24064.8	−0.892776
$900$	0	0
$901$	19343.3	0.715227
$902$	0	0
$903$	−1154.11	−0.0425319
$904$	0	0
$905$	−15060.8	−0.553190
$906$	0	0
$907$	−16772.9	−0.614041	−0.307020	−	0.951703i	$-0.599332\pi$
$907$	−16772.9	−0.614041	−0.307020	+	0.951703i	$0.599332\pi$
$908$	0	0
$909$	−992.781	−0.0362249
$910$	0	0
$911$	18924.6	0.688255	0.344128	−	0.938923i	$-0.388175\pi$
$911$	18924.6	0.688255	0.344128	+	0.938923i	$0.388175\pi$
$912$	0	0
$913$	14069.5	0.510003
$914$	0	0
$915$	23239.5	0.839642
$916$	0	0
$917$	−10621.1	−0.382485
$918$	0	0
$919$	7053.25	0.253172	0.126586	−	0.991956i	$-0.459598\pi$
$919$	7053.25	0.253172	0.126586	+	0.991956i	$0.459598\pi$
$920$	0	0
$921$	−32367.3	−1.15802
$922$	0	0
$923$	23553.6	0.839954
$924$	0	0
$925$	1454.87	0.0517143
$926$	0	0
$927$	14556.0	0.515731
$928$	0	0
$929$	30258.0	1.06860	0.534302	−	0.845294i	$-0.320575\pi$
$929$	30258.0	1.06860	0.534302	+	0.845294i	$0.320575\pi$
$930$	0	0
$931$	−2887.71	−0.101655
$932$	0	0
$933$	15184.8	0.532829
$934$	0	0
$935$	6934.80	0.242558
$936$	0	0
$937$	28135.8	0.980958	0.490479	−	0.871453i	$-0.336822\pi$
$937$	28135.8	0.980958	0.490479	+	0.871453i	$0.336822\pi$
$938$	0	0
$939$	3078.86	0.107002
$940$	0	0
$941$	14928.2	0.517159	0.258580	−	0.965990i	$-0.416746\pi$
$941$	14928.2	0.517159	0.258580	+	0.965990i	$0.416746\pi$
$942$	0	0
$943$	10207.6	0.352496
$944$	0	0
$945$	9384.77	0.323055
$946$	0	0
$947$	49874.2	1.71140	0.855699	−	0.517474i	$-0.173127\pi$
$947$	49874.2	1.71140	0.855699	+	0.517474i	$0.173127\pi$
$948$	0	0
$949$	2635.42	0.0901467
$950$	0	0
$951$	28138.9	0.959480
$952$	0	0
$953$	48082.5	1.63436	0.817180	−	0.576383i	$-0.195536\pi$
$953$	48082.5	1.63436	0.817180	+	0.576383i	$0.195536\pi$
$954$	0	0
$955$	−19514.4	−0.661227
$956$	0	0
$957$	10311.9	0.348313
$958$	0	0
$959$	14800.1	0.498352
$960$	0	0
$961$	−22385.2	−0.751408
$962$	0	0
$963$	−9529.72	−0.318890
$964$	0	0
$965$	17118.3	0.571045
$966$	0	0
$967$	−20698.4	−0.688331	−0.344165	−	0.938909i	$-0.611838\pi$
$967$	−20698.4	−0.688331	−0.344165	+	0.938909i	$0.611838\pi$
$968$	0	0
$969$	−13317.7	−0.441511
$970$	0	0
$971$	−10237.4	−0.338347	−0.169174	−	0.985586i	$-0.554110\pi$
$971$	−10237.4	−0.338347	−0.169174	+	0.985586i	$0.554110\pi$
$972$	0	0
$973$	−1722.42	−0.0567506
$974$	0	0
$975$	−4469.77	−0.146817
$976$	0	0
$977$	−23119.9	−0.757086	−0.378543	−	0.925584i	$-0.623575\pi$
$977$	−23119.9	−0.757086	−0.378543	+	0.925584i	$0.623575\pi$
$978$	0	0
$979$	8144.39	0.265879
$980$	0	0
$981$	16283.2	0.529950
$982$	0	0
$983$	1938.69	0.0629039	0.0314519	−	0.999505i	$-0.489987\pi$
$983$	1938.69	0.0629039	0.0314519	+	0.999505i	$0.489987\pi$
$984$	0	0
$985$	24086.9	0.779159
$986$	0	0
$987$	−5245.77	−0.169174
$988$	0	0
$989$	1489.46	0.0478890
$990$	0	0
$991$	43312.6	1.38837	0.694183	−	0.719799i	$-0.255766\pi$
$991$	43312.6	1.38837	0.694183	+	0.719799i	$0.255766\pi$
$992$	0	0
$993$	−31555.4	−1.00844
$994$	0	0
$995$	−3976.70	−0.126703
$996$	0	0
$997$	−29271.9	−0.929839	−0.464920	−	0.885353i	$-0.653917\pi$
$997$	−29271.9	−0.929839	−0.464920	+	0.885353i	$0.653917\pi$
$998$	0	0
$999$	−5556.59	−0.175979

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	1232.4.a.p.1.1		2
4.3	odd	2		154.4.a.f.1.2	✓	2
12.11	even	2		1386.4.a.ba.1.2		2
28.27	even	2		1078.4.a.j.1.1		2
44.43	even	2		1694.4.a.l.1.2		2

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
154.4.a.f.1.2	✓	2	4.3	odd	2
1078.4.a.j.1.1		2	28.27	even	2
1232.4.a.p.1.1		2	1.1	even	1	trivial
1386.4.a.ba.1.2		2	12.11	even	2
1694.4.a.l.1.2		2	44.43	even	2

Overview	Random
Universe	Knowledge

Classical	Maass
Hilbert	Bianchi

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

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

\(f(q)\)	\(=\)	\(q-3.35235 q^{3} -9.35235 q^{5} -7.00000 q^{7} -15.7617 q^{9} +O(q^{10})\)	\(q-3.35235 q^{3} -9.35235 q^{5} -7.00000 q^{7} -15.7617 q^{9} +11.0000 q^{11} -35.5235 q^{13} +31.3523 q^{15} -67.4094 q^{17} -58.9329 q^{19} +23.4664 q^{21} -30.2852 q^{23} -37.5336 q^{25} +143.352 q^{27} -279.638 q^{29} +86.0570 q^{31} -36.8758 q^{33} +65.4664 q^{35} -38.7617 q^{37} +119.087 q^{39} -337.047 q^{41} -49.1812 q^{43} +147.409 q^{45} -223.544 q^{47} +49.0000 q^{49} +225.980 q^{51} -286.953 q^{53} -102.876 q^{55} +197.564 q^{57} -398.017 q^{59} +741.235 q^{61} +110.332 q^{63} +332.228 q^{65} -744.910 q^{67} +101.527 q^{69} -663.044 q^{71} -74.1880 q^{73} +125.826 q^{75} -77.0000 q^{77} +147.886 q^{79} -55.0000 q^{81} +1279.05 q^{83} +630.436 q^{85} +937.443 q^{87} +740.399 q^{89} +248.664 q^{91} -288.493 q^{93} +551.161 q^{95} -704.775 q^{97} -173.379 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 2 q + 5 q^{3} - 7 q^{5} - 14 q^{7} + 27 q^{9}+O(q^{10}) \) 2 * q + 5 * q^3 - 7 * q^5 - 14 * q^7 + 27 * q^9	\( 2 q + 5 q^{3} - 7 q^{5} - 14 q^{7} + 27 q^{9} + 22 q^{11} + 46 q^{13} + 51 q^{15} - 88 q^{17} + 46 q^{19} - 35 q^{21} + 115 q^{23} - 157 q^{25} + 275 q^{27} - 372 q^{29} + 137 q^{31} + 55 q^{33} + 49 q^{35} - 19 q^{37} + 800 q^{39} - 440 q^{41} - 192 q^{43} + 248 q^{45} - 728 q^{47} + 98 q^{49} + 54 q^{51} - 808 q^{53} - 77 q^{55} + 1074 q^{57} + 35 q^{59} + 312 q^{61} - 189 q^{63} + 524 q^{65} + 301 q^{67} + 1315 q^{69} + 137 q^{71} + 788 q^{73} - 872 q^{75} - 154 q^{77} + 366 q^{79} - 110 q^{81} + 2324 q^{83} + 582 q^{85} + 166 q^{87} + 1235 q^{89} - 322 q^{91} + 137 q^{93} + 798 q^{95} + 709 q^{97} + 297 q^{99}+O(q^{100}) \) 2 * q + 5 * q^3 - 7 * q^5 - 14 * q^7 + 27 * q^9 + 22 * q^11 + 46 * q^13 + 51 * q^15 - 88 * q^17 + 46 * q^19 - 35 * q^21 + 115 * q^23 - 157 * q^25 + 275 * q^27 - 372 * q^29 + 137 * q^31 + 55 * q^33 + 49 * q^35 - 19 * q^37 + 800 * q^39 - 440 * q^41 - 192 * q^43 + 248 * q^45 - 728 * q^47 + 98 * q^49 + 54 * q^51 - 808 * q^53 - 77 * q^55 + 1074 * q^57 + 35 * q^59 + 312 * q^61 - 189 * q^63 + 524 * q^65 + 301 * q^67 + 1315 * q^69 + 137 * q^71 + 788 * q^73 - 872 * q^75 - 154 * q^77 + 366 * q^79 - 110 * q^81 + 2324 * q^83 + 582 * q^85 + 166 * q^87 + 1235 * q^89 - 322 * q^91 + 137 * q^93 + 798 * q^95 + 709 * q^97 + 297 * q^99

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