LMFDB - Embedded newform 1232.4.a.s.1.3

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:	$1$
Dimension:	$4$
Coefficient field:	4.4.522072.1
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{4} - x^{3} - 12x^{2} + 5x + 1 $ x^4 - x^3 - 12x^2 + 5x + 1
Coefficient ring:	$\Z[a_1, \ldots, a_{5}]$
Coefficient ring index:	$ 2^{2} $
Twist minimal:	no (minimal twist has level 77)
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.3
Root			$-0.148103$ 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)

$f(q)$	$=$	$q-2.77399 q^{3} +1.84418 q^{5} +7.00000 q^{7} -19.3050 q^{9} +11.0000 q^{11} +24.6401 q^{13} -5.11574 q^{15} +17.8800 q^{17} -32.1459 q^{19} -19.4179 q^{21} -14.1248 q^{23} -121.599 q^{25} +128.450 q^{27} -41.5471 q^{29} -175.766 q^{31} -30.5139 q^{33} +12.9093 q^{35} +292.877 q^{37} -68.3513 q^{39} +154.296 q^{41} +277.144 q^{43} -35.6019 q^{45} +52.1450 q^{47} +49.0000 q^{49} -49.5989 q^{51} +82.3907 q^{53} +20.2860 q^{55} +89.1723 q^{57} -712.816 q^{59} -647.078 q^{61} -135.135 q^{63} +45.4408 q^{65} -260.867 q^{67} +39.1821 q^{69} -369.025 q^{71} +1145.77 q^{73} +337.314 q^{75} +77.0000 q^{77} -488.885 q^{79} +164.917 q^{81} -548.982 q^{83} +32.9740 q^{85} +115.251 q^{87} +105.039 q^{89} +172.481 q^{91} +487.572 q^{93} -59.2829 q^{95} -1361.91 q^{97} -212.355 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 4 q - 14 q^{3} + 10 q^{5} + 28 q^{7} + 76 q^{9} + 44 q^{11} + 58 q^{13} - 284 q^{15} + 4 q^{17} - 258 q^{19} - 98 q^{21} - 8 q^{23} + 80 q^{25} - 428 q^{27} - 396 q^{29} + 56 q^{31} - 154 q^{33} + 70 q^{35}+ \cdots + 836 q^{99}+O(q^{100}) $ 4 * q - 14 * q^3 + 10 * q^5 + 28 * q^7 + 76 * q^9 + 44 * q^11 + 58 * q^13 - 284 * q^15 + 4 * q^17 - 258 * q^19 - 98 * q^21 - 8 * q^23 + 80 * q^25 - 428 * q^27 - 396 * q^29 + 56 * q^31 - 154 * q^33 + 70 * q^35 + 84 * q^37 + 412 * q^39 + 52 * q^41 - 408 * q^43 + 826 * q^45 - 8 * q^47 + 196 * q^49 + 388 * q^51 + 624 * q^53 + 110 * q^55 + 48 * q^57 + 238 * q^59 - 162 * q^61 + 532 * q^63 - 32 * q^65 - 1340 * q^67 + 2416 * q^69 - 1788 * q^71 + 1456 * q^73 + 806 * q^75 + 308 * q^77 + 1324 * q^79 + 1444 * q^81 - 450 * q^83 - 1736 * q^85 - 588 * q^87 - 3072 * q^89 + 406 * q^91 - 1264 * q^93 - 24 * q^95 - 652 * q^97 + 836 * q^99

Coefficient data

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

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

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

$2$	0	0
$2$	0	0
$3$	−2.77399	−0.533854	−0.266927	−	0.963717i	$-0.586008\pi$
$3$	−2.77399	−0.533854	−0.266927	+	0.963717i	$0.586008\pi$
$4$	0	0
$5$	1.84418	0.164949	0.0824744	−	0.996593i	$-0.473718\pi$
$5$	1.84418	0.164949	0.0824744	+	0.996593i	$0.473718\pi$
$6$	0	0
$7$	7.00000	0.377964
$7$	7.00000	0.377964
$8$	0	0
$9$	−19.3050	−0.715000
$10$	0	0
$11$	11.0000	0.301511
$11$	11.0000	0.301511
$12$	0	0
$13$	24.6401	0.525687	0.262844	−	0.964838i	$-0.415340\pi$
$13$	24.6401	0.525687	0.262844	+	0.964838i	$0.415340\pi$
$14$	0	0
$15$	−5.11574	−0.0880586
$16$	0	0
$17$	17.8800	0.255090	0.127545	−	0.991833i	$-0.459290\pi$
$17$	17.8800	0.255090	0.127545	+	0.991833i	$0.459290\pi$
$18$	0	0
$19$	−32.1459	−0.388146	−0.194073	−	0.980987i	$-0.562170\pi$
$19$	−32.1459	−0.388146	−0.194073	+	0.980987i	$0.562170\pi$
$20$	0	0
$21$	−19.4179	−0.201778
$22$	0	0
$23$	−14.1248	−0.128054	−0.0640268	−	0.997948i	$-0.520394\pi$
$23$	−14.1248	−0.128054	−0.0640268	+	0.997948i	$0.520394\pi$
$24$	0	0
$25$	−121.599	−0.972792
$26$	0	0
$27$	128.450	0.915560
$28$	0	0
$29$	−41.5471	−0.266038	−0.133019	−	0.991113i	$-0.542467\pi$
$29$	−41.5471	−0.266038	−0.133019	+	0.991113i	$0.542467\pi$
$30$	0	0
$31$	−175.766	−1.01834	−0.509169	−	0.860667i	$-0.670047\pi$
$31$	−175.766	−1.01834	−0.509169	+	0.860667i	$0.670047\pi$
$32$	0	0
$33$	−30.5139	−0.160963
$34$	0	0
$35$	12.9093	0.0623448
$36$	0	0
$37$	292.877	1.30131	0.650657	−	0.759372i	$-0.274494\pi$
$37$	292.877	1.30131	0.650657	+	0.759372i	$0.274494\pi$
$38$	0	0
$39$	−68.3513	−0.280640
$40$	0	0
$41$	154.296	0.587732	0.293866	−	0.955847i	$-0.405058\pi$
$41$	154.296	0.587732	0.293866	+	0.955847i	$0.405058\pi$
$42$	0	0
$43$	277.144	0.982887	0.491443	−	0.870910i	$-0.336470\pi$
$43$	277.144	0.982887	0.491443	+	0.870910i	$0.336470\pi$
$44$	0	0
$45$	−35.6019	−0.117938
$46$	0	0
$47$	52.1450	0.161832	0.0809162	−	0.996721i	$-0.474215\pi$
$47$	52.1450	0.161832	0.0809162	+	0.996721i	$0.474215\pi$
$48$	0	0
$49$	49.0000	0.142857
$50$	0	0
$51$	−49.5989	−0.136181
$52$	0	0
$53$	82.3907	0.213533	0.106766	−	0.994284i	$-0.465950\pi$
$53$	82.3907	0.213533	0.106766	+	0.994284i	$0.465950\pi$
$54$	0	0
$55$	20.2860	0.0497339
$56$	0	0
$57$	89.1723	0.207213
$58$	0	0
$59$	−712.816	−1.57289	−0.786447	−	0.617658i	$-0.788081\pi$
$59$	−712.816	−1.57289	−0.786447	+	0.617658i	$0.788081\pi$
$60$	0	0
$61$	−647.078	−1.35819	−0.679097	−	0.734048i	$-0.737629\pi$
$61$	−647.078	−1.35819	−0.679097	+	0.734048i	$0.737629\pi$
$62$	0	0
$63$	−135.135	−0.270244
$64$	0	0
$65$	45.4408	0.0867114
$66$	0	0
$67$	−260.867	−0.475672	−0.237836	−	0.971305i	$-0.576438\pi$
$67$	−260.867	−0.475672	−0.237836	+	0.971305i	$0.576438\pi$
$68$	0	0
$69$	39.1821	0.0683619
$70$	0	0
$71$	−369.025	−0.616833	−0.308417	−	0.951251i	$-0.599799\pi$
$71$	−369.025	−0.616833	−0.308417	+	0.951251i	$0.599799\pi$
$72$	0	0
$73$	1145.77	1.83702	0.918509	−	0.395401i	$-0.129394\pi$
$73$	1145.77	1.83702	0.918509	+	0.395401i	$0.129394\pi$
$74$	0	0
$75$	337.314	0.519329
$76$	0	0
$77$	77.0000	0.113961
$78$	0	0
$79$	−488.885	−0.696251	−0.348125	−	0.937448i	$-0.613182\pi$
$79$	−488.885	−0.696251	−0.348125	+	0.937448i	$0.613182\pi$
$80$	0	0
$81$	164.917	0.226224
$82$	0	0
$83$	−548.982	−0.726008	−0.363004	−	0.931788i	$-0.618249\pi$
$83$	−548.982	−0.726008	−0.363004	+	0.931788i	$0.618249\pi$
$84$	0	0
$85$	32.9740	0.0420769
$86$	0	0
$87$	115.251	0.142026
$88$	0	0
$89$	105.039	0.125102	0.0625510	−	0.998042i	$-0.480076\pi$
$89$	105.039	0.125102	0.0625510	+	0.998042i	$0.480076\pi$
$90$	0	0
$91$	172.481	0.198691
$92$	0	0
$93$	487.572	0.543644
$94$	0	0
$95$	−59.2829	−0.0640242
$96$	0	0
$97$	−1361.91	−1.42557	−0.712787	−	0.701381i	$-0.752567\pi$
$97$	−1361.91	−1.42557	−0.712787	+	0.701381i	$0.752567\pi$
$98$	0	0
$99$	−212.355	−0.215580
$100$	0	0
$101$	−1610.32	−1.58647	−0.793234	−	0.608917i	$-0.791604\pi$
$101$	−1610.32	−1.58647	−0.793234	+	0.608917i	$0.791604\pi$
$102$	0	0
$103$	123.044	0.117708	0.0588540	−	0.998267i	$-0.481255\pi$
$103$	123.044	0.117708	0.0588540	+	0.998267i	$0.481255\pi$
$104$	0	0
$105$	−35.8102	−0.0332830
$106$	0	0
$107$	1740.90	1.57289	0.786446	−	0.617660i	$-0.211919\pi$
$107$	1740.90	1.57289	0.786446	+	0.617660i	$0.211919\pi$
$108$	0	0
$109$	248.938	0.218752	0.109376	−	0.994000i	$-0.465115\pi$
$109$	248.938	0.218752	0.109376	+	0.994000i	$0.465115\pi$
$110$	0	0
$111$	−812.436	−0.694712
$112$	0	0
$113$	−494.465	−0.411641	−0.205820	−	0.978590i	$-0.565986\pi$
$113$	−494.465	−0.411641	−0.205820	+	0.978590i	$0.565986\pi$
$114$	0	0
$115$	−26.0488	−0.0211223
$116$	0	0
$117$	−475.677	−0.375866
$118$	0	0
$119$	125.160	0.0964151
$120$	0	0
$121$	121.000	0.0909091
$122$	0	0
$123$	−428.016	−0.313763
$124$	0	0
$125$	−454.774	−0.325410
$126$	0	0
$127$	979.104	0.684106	0.342053	−	0.939681i	$-0.388878\pi$
$127$	979.104	0.684106	0.342053	+	0.939681i	$0.388878\pi$
$128$	0	0
$129$	−768.795	−0.524718
$130$	0	0
$131$	1660.85	1.10770	0.553851	−	0.832616i	$-0.313158\pi$
$131$	1660.85	1.10770	0.553851	+	0.832616i	$0.313158\pi$
$132$	0	0
$133$	−225.021	−0.146705
$134$	0	0
$135$	236.884	0.151020
$136$	0	0
$137$	−1618.17	−1.00912	−0.504559	−	0.863377i	$-0.668345\pi$
$137$	−1618.17	−1.00912	−0.504559	+	0.863377i	$0.668345\pi$
$138$	0	0
$139$	−695.736	−0.424544	−0.212272	−	0.977211i	$-0.568086\pi$
$139$	−695.736	−0.424544	−0.212272	+	0.977211i	$0.568086\pi$
$140$	0	0
$141$	−144.650	−0.0863950
$142$	0	0
$143$	271.041	0.158501
$144$	0	0
$145$	−76.6205	−0.0438826
$146$	0	0
$147$	−135.925	−0.0762649
$148$	0	0
$149$	−2081.84	−1.14464	−0.572318	−	0.820032i	$-0.693956\pi$
$149$	−2081.84	−1.14464	−0.572318	+	0.820032i	$0.693956\pi$
$150$	0	0
$151$	−2679.28	−1.44395	−0.721975	−	0.691919i	$-0.756766\pi$
$151$	−2679.28	−1.44395	−0.721975	+	0.691919i	$0.756766\pi$
$152$	0	0
$153$	−345.173	−0.182390
$154$	0	0
$155$	−324.144	−0.167973
$156$	0	0
$157$	2410.32	1.22525	0.612627	−	0.790372i	$-0.290113\pi$
$157$	2410.32	1.22525	0.612627	+	0.790372i	$0.290113\pi$
$158$	0	0
$159$	−228.551	−0.113995
$160$	0	0
$161$	−98.8738	−0.0483997
$162$	0	0
$163$	−3629.08	−1.74388	−0.871938	−	0.489616i	$-0.837137\pi$
$163$	−3629.08	−1.74388	−0.871938	+	0.489616i	$0.837137\pi$
$164$	0	0
$165$	−56.2732	−0.0265507
$166$	0	0
$167$	3826.04	1.77286	0.886432	−	0.462859i	$-0.153176\pi$
$167$	3826.04	1.77286	0.886432	+	0.462859i	$0.153176\pi$
$168$	0	0
$169$	−1589.87	−0.723653
$170$	0	0
$171$	620.576	0.277524
$172$	0	0
$173$	−4310.67	−1.89442	−0.947209	−	0.320617i	$-0.896110\pi$
$173$	−4310.67	−1.89442	−0.947209	+	0.320617i	$0.896110\pi$
$174$	0	0
$175$	−851.193	−0.367681
$176$	0	0
$177$	1977.34	0.839696
$178$	0	0
$179$	−2491.12	−1.04019	−0.520097	−	0.854107i	$-0.674104\pi$
$179$	−2491.12	−1.04019	−0.520097	+	0.854107i	$0.674104\pi$
$180$	0	0
$181$	−4315.49	−1.77220	−0.886098	−	0.463498i	$-0.846594\pi$
$181$	−4315.49	−1.77220	−0.886098	+	0.463498i	$0.846594\pi$
$182$	0	0
$183$	1794.99	0.725078
$184$	0	0
$185$	540.118	0.214650
$186$	0	0
$187$	196.680	0.0769127
$188$	0	0
$189$	899.147	0.346049
$190$	0	0
$191$	−2840.41	−1.07605	−0.538024	−	0.842930i	$-0.680829\pi$
$191$	−2840.41	−1.07605	−0.538024	+	0.842930i	$0.680829\pi$
$192$	0	0
$193$	−1734.68	−0.646969	−0.323485	−	0.946233i	$-0.604854\pi$
$193$	−1734.68	−0.646969	−0.323485	+	0.946233i	$0.604854\pi$
$194$	0	0
$195$	−126.052	−0.0462913
$196$	0	0
$197$	−3098.42	−1.12057	−0.560287	−	0.828298i	$-0.689309\pi$
$197$	−3098.42	−1.12057	−0.560287	+	0.828298i	$0.689309\pi$
$198$	0	0
$199$	−4497.38	−1.60206	−0.801032	−	0.598622i	$-0.795715\pi$
$199$	−4497.38	−1.60206	−0.801032	+	0.598622i	$0.795715\pi$
$200$	0	0
$201$	723.643	0.253940
$202$	0	0
$203$	−290.830	−0.100553
$204$	0	0
$205$	284.550	0.0969456
$206$	0	0
$207$	272.680	0.0915582
$208$	0	0
$209$	−353.605	−0.117030
$210$	0	0
$211$	−1262.32	−0.411857	−0.205929	−	0.978567i	$-0.566021\pi$
$211$	−1262.32	−0.411857	−0.205929	+	0.978567i	$0.566021\pi$
$212$	0	0
$213$	1023.67	0.329299
$214$	0	0
$215$	511.105	0.162126
$216$	0	0
$217$	−1230.36	−0.384895
$218$	0	0
$219$	−3178.35	−0.980700
$220$	0	0
$221$	440.565	0.134098
$222$	0	0
$223$	−2931.38	−0.880268	−0.440134	−	0.897932i	$-0.645069\pi$
$223$	−2931.38	−0.880268	−0.440134	+	0.897932i	$0.645069\pi$
$224$	0	0
$225$	2347.47	0.695546
$226$	0	0
$227$	−4298.37	−1.25680	−0.628399	−	0.777891i	$-0.716289\pi$
$227$	−4298.37	−1.25680	−0.628399	+	0.777891i	$0.716289\pi$
$228$	0	0
$229$	698.500	0.201564	0.100782	−	0.994909i	$-0.467866\pi$
$229$	698.500	0.201564	0.100782	+	0.994909i	$0.467866\pi$
$230$	0	0
$231$	−213.597	−0.0608383
$232$	0	0
$233$	1887.78	0.530783	0.265391	−	0.964141i	$-0.414499\pi$
$233$	1887.78	0.530783	0.265391	+	0.964141i	$0.414499\pi$
$234$	0	0
$235$	96.1649	0.0266941
$236$	0	0
$237$	1356.16	0.371697
$238$	0	0
$239$	6449.93	1.74565	0.872827	−	0.488030i	$-0.162284\pi$
$239$	6449.93	1.74565	0.872827	+	0.488030i	$0.162284\pi$
$240$	0	0
$241$	4636.98	1.23939	0.619697	−	0.784841i	$-0.287256\pi$
$241$	4636.98	1.23939	0.619697	+	0.784841i	$0.287256\pi$
$242$	0	0
$243$	−3925.62	−1.03633
$244$	0	0
$245$	90.3650	0.0235641
$246$	0	0
$247$	−792.078	−0.204043
$248$	0	0
$249$	1522.87	0.387582
$250$	0	0
$251$	−2194.11	−0.551758	−0.275879	−	0.961192i	$-0.588969\pi$
$251$	−2194.11	−0.551758	−0.275879	+	0.961192i	$0.588969\pi$
$252$	0	0
$253$	−155.373	−0.0386096
$254$	0	0
$255$	−91.4695	−0.0224629
$256$	0	0
$257$	−2966.97	−0.720133	−0.360067	−	0.932927i	$-0.617246\pi$
$257$	−2966.97	−0.720133	−0.360067	+	0.932927i	$0.617246\pi$
$258$	0	0
$259$	2050.14	0.491850
$260$	0	0
$261$	802.066	0.190217
$262$	0	0
$263$	−915.810	−0.214720	−0.107360	−	0.994220i	$-0.534240\pi$
$263$	−915.810	−0.214720	−0.107360	+	0.994220i	$0.534240\pi$
$264$	0	0
$265$	151.944	0.0352220
$266$	0	0
$267$	−291.376	−0.0667863
$268$	0	0
$269$	−164.462	−0.0372768	−0.0186384	−	0.999826i	$-0.505933\pi$
$269$	−164.462	−0.0372768	−0.0186384	+	0.999826i	$0.505933\pi$
$270$	0	0
$271$	−1502.60	−0.336815	−0.168407	−	0.985718i	$-0.553862\pi$
$271$	−1502.60	−0.336815	−0.168407	+	0.985718i	$0.553862\pi$
$272$	0	0
$273$	−478.459	−0.106072
$274$	0	0
$275$	−1337.59	−0.293308
$276$	0	0
$277$	−7500.11	−1.62685	−0.813426	−	0.581669i	$-0.802400\pi$
$277$	−7500.11	−1.62685	−0.813426	+	0.581669i	$0.802400\pi$
$278$	0	0
$279$	3393.15	0.728111
$280$	0	0
$281$	−2945.12	−0.625235	−0.312618	−	0.949879i	$-0.601206\pi$
$281$	−2945.12	−0.625235	−0.312618	+	0.949879i	$0.601206\pi$
$282$	0	0
$283$	−5215.29	−1.09547	−0.547733	−	0.836653i	$-0.684509\pi$
$283$	−5215.29	−1.09547	−0.547733	+	0.836653i	$0.684509\pi$
$284$	0	0
$285$	164.450	0.0341796
$286$	0	0
$287$	1080.07	0.222142
$288$	0	0
$289$	−4593.31	−0.934929
$290$	0	0
$291$	3777.91	0.761049
$292$	0	0
$293$	7407.99	1.47706	0.738531	−	0.674219i	$-0.235520\pi$
$293$	7407.99	1.47706	0.738531	+	0.674219i	$0.235520\pi$
$294$	0	0
$295$	−1314.56	−0.259447
$296$	0	0
$297$	1412.94	0.276052
$298$	0	0
$299$	−348.037	−0.0673161
$300$	0	0
$301$	1940.01	0.371496
$302$	0	0
$303$	4467.02	0.846942
$304$	0	0
$305$	−1193.33	−0.224033
$306$	0	0
$307$	6850.01	1.27345	0.636727	−	0.771089i	$-0.280288\pi$
$307$	6850.01	1.27345	0.636727	+	0.771089i	$0.280288\pi$
$308$	0	0
$309$	−341.324	−0.0628390
$310$	0	0
$311$	5538.62	1.00986	0.504929	−	0.863161i	$-0.331519\pi$
$311$	5538.62	1.00986	0.504929	+	0.863161i	$0.331519\pi$
$312$	0	0
$313$	9361.13	1.69049	0.845243	−	0.534382i	$-0.179456\pi$
$313$	9361.13	1.69049	0.845243	+	0.534382i	$0.179456\pi$
$314$	0	0
$315$	−249.214	−0.0445765
$316$	0	0
$317$	219.221	0.0388413	0.0194206	−	0.999811i	$-0.493818\pi$
$317$	219.221	0.0388413	0.0194206	+	0.999811i	$0.493818\pi$
$318$	0	0
$319$	−457.018	−0.0802135
$320$	0	0
$321$	−4829.24	−0.839695
$322$	0	0
$323$	−574.769	−0.0990123
$324$	0	0
$325$	−2996.21	−0.511384
$326$	0	0
$327$	−690.551	−0.116781
$328$	0	0
$329$	365.015	0.0611669
$330$	0	0
$331$	−3377.63	−0.560880	−0.280440	−	0.959872i	$-0.590480\pi$
$331$	−3377.63	−0.560880	−0.280440	+	0.959872i	$0.590480\pi$
$332$	0	0
$333$	−5653.98	−0.930439
$334$	0	0
$335$	−481.087	−0.0784615
$336$	0	0
$337$	7755.93	1.25369	0.626843	−	0.779145i	$-0.284347\pi$
$337$	7755.93	1.25369	0.626843	+	0.779145i	$0.284347\pi$
$338$	0	0
$339$	1371.64	0.219756
$340$	0	0
$341$	−1933.42	−0.307040
$342$	0	0
$343$	343.000	0.0539949
$344$	0	0
$345$	72.2590	0.0112762
$346$	0	0
$347$	831.044	0.128567	0.0642835	−	0.997932i	$-0.479524\pi$
$347$	831.044	0.128567	0.0642835	+	0.997932i	$0.479524\pi$
$348$	0	0
$349$	1840.94	0.282359	0.141180	−	0.989984i	$-0.454911\pi$
$349$	1840.94	0.282359	0.141180	+	0.989984i	$0.454911\pi$
$350$	0	0
$351$	3165.01	0.481298
$352$	0	0
$353$	3409.11	0.514019	0.257009	−	0.966409i	$-0.417263\pi$
$353$	3409.11	0.514019	0.257009	+	0.966409i	$0.417263\pi$
$354$	0	0
$355$	−680.549	−0.101746
$356$	0	0
$357$	−347.193	−0.0514716
$358$	0	0
$359$	−2199.75	−0.323394	−0.161697	−	0.986840i	$-0.551697\pi$
$359$	−2199.75	−0.323394	−0.161697	+	0.986840i	$0.551697\pi$
$360$	0	0
$361$	−5825.64	−0.849343
$362$	0	0
$363$	−335.653	−0.0485322
$364$	0	0
$365$	2113.01	0.303014
$366$	0	0
$367$	855.008	0.121611	0.0608053	−	0.998150i	$-0.480633\pi$
$367$	855.008	0.121611	0.0608053	+	0.998150i	$0.480633\pi$
$368$	0	0
$369$	−2978.69	−0.420228
$370$	0	0
$371$	576.735	0.0807078
$372$	0	0
$373$	3193.40	0.443293	0.221646	−	0.975127i	$-0.428857\pi$
$373$	3193.40	0.443293	0.221646	+	0.975127i	$0.428857\pi$
$374$	0	0
$375$	1261.54	0.173721
$376$	0	0
$377$	−1023.72	−0.139853
$378$	0	0
$379$	5614.48	0.760940	0.380470	−	0.924793i	$-0.375762\pi$
$379$	5614.48	0.760940	0.380470	+	0.924793i	$0.375762\pi$
$380$	0	0
$381$	−2716.02	−0.365213
$382$	0	0
$383$	1736.86	0.231721	0.115861	−	0.993265i	$-0.463037\pi$
$383$	1736.86	0.231721	0.115861	+	0.993265i	$0.463037\pi$
$384$	0	0
$385$	142.002	0.0187977
$386$	0	0
$387$	−5350.27	−0.702763
$388$	0	0
$389$	8710.78	1.13536	0.567679	−	0.823250i	$-0.307841\pi$
$389$	8710.78	1.13536	0.567679	+	0.823250i	$0.307841\pi$
$390$	0	0
$391$	−252.552	−0.0326652
$392$	0	0
$393$	−4607.17	−0.591352
$394$	0	0
$395$	−901.593	−0.114846
$396$	0	0
$397$	11731.6	1.48311	0.741553	−	0.670894i	$-0.234089\pi$
$397$	11731.6	1.48311	0.741553	+	0.670894i	$0.234089\pi$
$398$	0	0
$399$	624.206	0.0783193
$400$	0	0
$401$	−14408.8	−1.79437	−0.897183	−	0.441659i	$-0.854390\pi$
$401$	−14408.8	−1.79437	−0.897183	+	0.441659i	$0.854390\pi$
$402$	0	0
$403$	−4330.88	−0.535327
$404$	0	0
$405$	304.138	0.0373154
$406$	0	0
$407$	3221.64	0.392361
$408$	0	0
$409$	−2155.54	−0.260598	−0.130299	−	0.991475i	$-0.541594\pi$
$409$	−2155.54	−0.260598	−0.130299	+	0.991475i	$0.541594\pi$
$410$	0	0
$411$	4488.77	0.538722
$412$	0	0
$413$	−4989.71	−0.594498
$414$	0	0
$415$	−1012.42	−0.119754
$416$	0	0
$417$	1929.96	0.226644
$418$	0	0
$419$	−8443.41	−0.984458	−0.492229	−	0.870466i	$-0.663818\pi$
$419$	−8443.41	−0.984458	−0.492229	+	0.870466i	$0.663818\pi$
$420$	0	0
$421$	−3070.28	−0.355430	−0.177715	−	0.984082i	$-0.556871\pi$
$421$	−3070.28	−0.355430	−0.177715	+	0.984082i	$0.556871\pi$
$422$	0	0
$423$	−1006.66	−0.115710
$424$	0	0
$425$	−2174.19	−0.248150
$426$	0	0
$427$	−4529.55	−0.513349
$428$	0	0
$429$	−751.865	−0.0846163
$430$	0	0
$431$	7004.40	0.782808	0.391404	−	0.920219i	$-0.371990\pi$
$431$	7004.40	0.782808	0.391404	+	0.920219i	$0.371990\pi$
$432$	0	0
$433$	7486.00	0.830841	0.415421	−	0.909629i	$-0.363634\pi$
$433$	7486.00	0.830841	0.415421	+	0.909629i	$0.363634\pi$
$434$	0	0
$435$	212.544	0.0234269
$436$	0	0
$437$	454.055	0.0497034
$438$	0	0
$439$	13046.2	1.41836	0.709179	−	0.705029i	$-0.249066\pi$
$439$	13046.2	1.41836	0.709179	+	0.705029i	$0.249066\pi$
$440$	0	0
$441$	−945.944	−0.102143
$442$	0	0
$443$	11781.3	1.26354	0.631768	−	0.775157i	$-0.282329\pi$
$443$	11781.3	1.26354	0.631768	+	0.775157i	$0.282329\pi$
$444$	0	0
$445$	193.711	0.0206354
$446$	0	0
$447$	5774.99	0.611069
$448$	0	0
$449$	7576.58	0.796349	0.398175	−	0.917310i	$-0.369644\pi$
$449$	7576.58	0.796349	0.398175	+	0.917310i	$0.369644\pi$
$450$	0	0
$451$	1697.26	0.177208
$452$	0	0
$453$	7432.28	0.770859
$454$	0	0
$455$	318.086	0.0327738
$456$	0	0
$457$	11793.0	1.20712	0.603560	−	0.797318i	$-0.293748\pi$
$457$	11793.0	1.20712	0.603560	+	0.797318i	$0.293748\pi$
$458$	0	0
$459$	2296.68	0.233551
$460$	0	0
$461$	4228.32	0.427185	0.213592	−	0.976923i	$-0.431484\pi$
$461$	4228.32	0.427185	0.213592	+	0.976923i	$0.431484\pi$
$462$	0	0
$463$	−14448.8	−1.45031	−0.725154	−	0.688586i	$-0.758232\pi$
$463$	−14448.8	−1.45031	−0.725154	+	0.688586i	$0.758232\pi$
$464$	0	0
$465$	899.172	0.0896733
$466$	0	0
$467$	−16547.5	−1.63967	−0.819836	−	0.572599i	$-0.805935\pi$
$467$	−16547.5	−1.63967	−0.819836	+	0.572599i	$0.805935\pi$
$468$	0	0
$469$	−1826.07	−0.179787
$470$	0	0
$471$	−6686.21	−0.654107
$472$	0	0
$473$	3048.59	0.296351
$474$	0	0
$475$	3908.91	0.377585
$476$	0	0
$477$	−1590.55	−0.152676
$478$	0	0
$479$	2989.34	0.285149	0.142575	−	0.989784i	$-0.454462\pi$
$479$	2989.34	0.285149	0.142575	+	0.989784i	$0.454462\pi$
$480$	0	0
$481$	7216.51	0.684084
$482$	0	0
$483$	274.275	0.0258384
$484$	0	0
$485$	−2511.60	−0.235147
$486$	0	0
$487$	5549.61	0.516379	0.258190	−	0.966094i	$-0.416874\pi$
$487$	5549.61	0.516379	0.258190	+	0.966094i	$0.416874\pi$
$488$	0	0
$489$	10067.0	0.930976
$490$	0	0
$491$	7751.20	0.712438	0.356219	−	0.934403i	$-0.384066\pi$
$491$	7751.20	0.712438	0.356219	+	0.934403i	$0.384066\pi$
$492$	0	0
$493$	−742.862	−0.0678638
$494$	0	0
$495$	−391.621	−0.0355597
$496$	0	0
$497$	−2583.17	−0.233141
$498$	0	0
$499$	6841.83	0.613792	0.306896	−	0.951743i	$-0.400710\pi$
$499$	6841.83	0.613792	0.306896	+	0.951743i	$0.400710\pi$
$500$	0	0
$501$	−10613.4	−0.946451
$502$	0	0
$503$	11518.5	1.02105	0.510523	−	0.859864i	$-0.329452\pi$
$503$	11518.5	1.02105	0.510523	+	0.859864i	$0.329452\pi$
$504$	0	0
$505$	−2969.73	−0.261686
$506$	0	0
$507$	4410.27	0.386325
$508$	0	0
$509$	8292.40	0.722110	0.361055	−	0.932545i	$-0.382417\pi$
$509$	8292.40	0.722110	0.361055	+	0.932545i	$0.382417\pi$
$510$	0	0
$511$	8020.39	0.694327
$512$	0	0
$513$	−4129.12	−0.355371
$514$	0	0
$515$	226.917	0.0194158
$516$	0	0
$517$	573.595	0.0487943
$518$	0	0
$519$	11957.8	1.01134
$520$	0	0
$521$	10891.6	0.915868	0.457934	−	0.888986i	$-0.348590\pi$
$521$	10891.6	0.915868	0.457934	+	0.888986i	$0.348590\pi$
$522$	0	0
$523$	−14662.0	−1.22586	−0.612928	−	0.790139i	$-0.710008\pi$
$523$	−14662.0	−1.22586	−0.612928	+	0.790139i	$0.710008\pi$
$524$	0	0
$525$	2361.20	0.196288
$526$	0	0
$527$	−3142.69	−0.259768
$528$	0	0
$529$	−11967.5	−0.983602
$530$	0	0
$531$	13760.9	1.12462
$532$	0	0
$533$	3801.87	0.308963
$534$	0	0
$535$	3210.54	0.259446
$536$	0	0
$537$	6910.33	0.555313
$538$	0	0
$539$	539.000	0.0430730
$540$	0	0
$541$	−19825.8	−1.57556	−0.787778	−	0.615960i	$-0.788768\pi$
$541$	−19825.8	−1.57556	−0.787778	+	0.615960i	$0.788768\pi$
$542$	0	0
$543$	11971.1	0.946094
$544$	0	0
$545$	459.087	0.0360828
$546$	0	0
$547$	−12706.1	−0.993187	−0.496593	−	0.867983i	$-0.665416\pi$
$547$	−12706.1	−0.993187	−0.496593	+	0.867983i	$0.665416\pi$
$548$	0	0
$549$	12491.8	0.971109
$550$	0	0
$551$	1335.57	0.103262
$552$	0	0
$553$	−3422.19	−0.263158
$554$	0	0
$555$	−1498.28	−0.114592
$556$	0	0
$557$	−12599.1	−0.958421	−0.479211	−	0.877700i	$-0.659077\pi$
$557$	−12599.1	−0.958421	−0.479211	+	0.877700i	$0.659077\pi$
$558$	0	0
$559$	6828.86	0.516691
$560$	0	0
$561$	−545.588	−0.0410602
$562$	0	0
$563$	−6004.47	−0.449482	−0.224741	−	0.974419i	$-0.572154\pi$
$563$	−6004.47	−0.449482	−0.224741	+	0.974419i	$0.572154\pi$
$564$	0	0
$565$	−911.884	−0.0678996
$566$	0	0
$567$	1154.42	0.0855046
$568$	0	0
$569$	−3145.89	−0.231779	−0.115890	−	0.993262i	$-0.536972\pi$
$569$	−3145.89	−0.231779	−0.115890	+	0.993262i	$0.536972\pi$
$570$	0	0
$571$	−23549.1	−1.72592	−0.862960	−	0.505273i	$-0.831392\pi$
$571$	−23549.1	−1.72592	−0.862960	+	0.505273i	$0.831392\pi$
$572$	0	0
$573$	7879.27	0.574453
$574$	0	0
$575$	1717.57	0.124569
$576$	0	0
$577$	327.335	0.0236172	0.0118086	−	0.999930i	$-0.496241\pi$
$577$	327.335	0.0236172	0.0118086	+	0.999930i	$0.496241\pi$
$578$	0	0
$579$	4811.98	0.345387
$580$	0	0
$581$	−3842.88	−0.274405
$582$	0	0
$583$	906.298	0.0643825
$584$	0	0
$585$	−877.235	−0.0619986
$586$	0	0
$587$	13270.8	0.933123	0.466562	−	0.884489i	$-0.345493\pi$
$587$	13270.8	0.933123	0.466562	+	0.884489i	$0.345493\pi$
$588$	0	0
$589$	5650.14	0.395263
$590$	0	0
$591$	8594.98	0.598224
$592$	0	0
$593$	5098.92	0.353099	0.176549	−	0.984292i	$-0.443506\pi$
$593$	5098.92	0.353099	0.176549	+	0.984292i	$0.443506\pi$
$594$	0	0
$595$	230.818	0.0159036
$596$	0	0
$597$	12475.7	0.855268
$598$	0	0
$599$	−19358.2	−1.32046	−0.660230	−	0.751064i	$-0.729541\pi$
$599$	−19358.2	−1.32046	−0.660230	+	0.751064i	$0.729541\pi$
$600$	0	0
$601$	1238.87	0.0840841	0.0420420	−	0.999116i	$-0.486614\pi$
$601$	1238.87	0.0840841	0.0420420	+	0.999116i	$0.486614\pi$
$602$	0	0
$603$	5036.04	0.340105
$604$	0	0
$605$	223.146	0.0149953
$606$	0	0
$607$	14175.6	0.947888	0.473944	−	0.880555i	$-0.342830\pi$
$607$	14175.6	0.947888	0.473944	+	0.880555i	$0.342830\pi$
$608$	0	0
$609$	806.758	0.0536806
$610$	0	0
$611$	1284.86	0.0850732
$612$	0	0
$613$	6906.67	0.455070	0.227535	−	0.973770i	$-0.426933\pi$
$613$	6906.67	0.455070	0.227535	+	0.973770i	$0.426933\pi$
$614$	0	0
$615$	−789.339	−0.0517549
$616$	0	0
$617$	12104.3	0.789789	0.394894	−	0.918727i	$-0.370781\pi$
$617$	12104.3	0.789789	0.394894	+	0.918727i	$0.370781\pi$
$618$	0	0
$619$	−14945.2	−0.970437	−0.485218	−	0.874393i	$-0.661260\pi$
$619$	−14945.2	−0.970437	−0.485218	+	0.874393i	$0.661260\pi$
$620$	0	0
$621$	−1814.33	−0.117241
$622$	0	0
$623$	735.271	0.0472841
$624$	0	0
$625$	14361.2	0.919116
$626$	0	0
$627$	980.895	0.0624772
$628$	0	0
$629$	5236.63	0.331953
$630$	0	0
$631$	17711.9	1.11743	0.558717	−	0.829358i	$-0.311294\pi$
$631$	17711.9	1.11743	0.558717	+	0.829358i	$0.311294\pi$
$632$	0	0
$633$	3501.67	0.219872
$634$	0	0
$635$	1805.65	0.112842
$636$	0	0
$637$	1207.36	0.0750982
$638$	0	0
$639$	7124.02	0.441036
$640$	0	0
$641$	−17994.4	−1.10879	−0.554396	−	0.832253i	$-0.687051\pi$
$641$	−17994.4	−1.10879	−0.554396	+	0.832253i	$0.687051\pi$
$642$	0	0
$643$	−27947.8	−1.71408	−0.857039	−	0.515251i	$-0.827699\pi$
$643$	−27947.8	−1.71408	−0.857039	+	0.515251i	$0.827699\pi$
$644$	0	0
$645$	−1417.80	−0.0865516
$646$	0	0
$647$	14336.2	0.871122	0.435561	−	0.900159i	$-0.356550\pi$
$647$	14336.2	0.871122	0.435561	+	0.900159i	$0.356550\pi$
$648$	0	0
$649$	−7840.97	−0.474245
$650$	0	0
$651$	3413.00	0.205478
$652$	0	0
$653$	4315.79	0.258637	0.129318	−	0.991603i	$-0.458721\pi$
$653$	4315.79	0.258637	0.129318	+	0.991603i	$0.458721\pi$
$654$	0	0
$655$	3062.91	0.182714
$656$	0	0
$657$	−22119.1	−1.31347
$658$	0	0
$659$	−4002.23	−0.236578	−0.118289	−	0.992979i	$-0.537741\pi$
$659$	−4002.23	−0.236578	−0.118289	+	0.992979i	$0.537741\pi$
$660$	0	0
$661$	−12223.4	−0.719268	−0.359634	−	0.933094i	$-0.617098\pi$
$661$	−12223.4	−0.719268	−0.359634	+	0.933094i	$0.617098\pi$
$662$	0	0
$663$	−1222.12	−0.0715887
$664$	0	0
$665$	−414.980	−0.0241989
$666$	0	0
$667$	586.846	0.0340671
$668$	0	0
$669$	8131.61	0.469935
$670$	0	0
$671$	−7117.86	−0.409511
$672$	0	0
$673$	−6121.53	−0.350621	−0.175310	−	0.984513i	$-0.556093\pi$
$673$	−6121.53	−0.350621	−0.175310	+	0.984513i	$0.556093\pi$
$674$	0	0
$675$	−15619.3	−0.890649
$676$	0	0
$677$	9626.46	0.546492	0.273246	−	0.961944i	$-0.411903\pi$
$677$	9626.46	0.546492	0.273246	+	0.961944i	$0.411903\pi$
$678$	0	0
$679$	−9533.34	−0.538816
$680$	0	0
$681$	11923.6	0.670947
$682$	0	0
$683$	−11554.4	−0.647315	−0.323658	−	0.946174i	$-0.604913\pi$
$683$	−11554.4	−0.647315	−0.323658	+	0.946174i	$0.604913\pi$
$684$	0	0
$685$	−2984.19	−0.166453
$686$	0	0
$687$	−1937.63	−0.107606
$688$	0	0
$689$	2030.12	0.112251
$690$	0	0
$691$	−2991.45	−0.164689	−0.0823444	−	0.996604i	$-0.526241\pi$
$691$	−2991.45	−0.164689	−0.0823444	+	0.996604i	$0.526241\pi$
$692$	0	0
$693$	−1486.48	−0.0814818
$694$	0	0
$695$	−1283.06	−0.0700279
$696$	0	0
$697$	2758.82	0.149925
$698$	0	0
$699$	−5236.67	−0.283361
$700$	0	0
$701$	30772.8	1.65802	0.829010	−	0.559234i	$-0.188905\pi$
$701$	30772.8	1.65802	0.829010	+	0.559234i	$0.188905\pi$
$702$	0	0
$703$	−9414.77	−0.505099
$704$	0	0
$705$	−266.760	−0.0142507
$706$	0	0
$707$	−11272.3	−0.599628
$708$	0	0
$709$	−21698.3	−1.14936	−0.574681	−	0.818377i	$-0.694874\pi$
$709$	−21698.3	−1.14936	−0.574681	+	0.818377i	$0.694874\pi$
$710$	0	0
$711$	9437.91	0.497819
$712$	0	0
$713$	2482.66	0.130402
$714$	0	0
$715$	499.849	0.0261445
$716$	0	0
$717$	−17892.0	−0.931925
$718$	0	0
$719$	12364.2	0.641318	0.320659	−	0.947195i	$-0.396096\pi$
$719$	12364.2	0.641318	0.320659	+	0.947195i	$0.396096\pi$
$720$	0	0
$721$	861.311	0.0444895
$722$	0	0
$723$	−12862.9	−0.661656
$724$	0	0
$725$	5052.09	0.258800
$726$	0	0
$727$	16017.2	0.817119	0.408559	−	0.912732i	$-0.366031\pi$
$727$	16017.2	0.817119	0.408559	+	0.912732i	$0.366031\pi$
$728$	0	0
$729$	6436.85	0.327026
$730$	0	0
$731$	4955.34	0.250725
$732$	0	0
$733$	10564.4	0.532340	0.266170	−	0.963926i	$-0.414242\pi$
$733$	10564.4	0.532340	0.266170	+	0.963926i	$0.414242\pi$
$734$	0	0
$735$	−250.671	−0.0125798
$736$	0	0
$737$	−2869.54	−0.143420
$738$	0	0
$739$	−1760.32	−0.0876242	−0.0438121	−	0.999040i	$-0.513950\pi$
$739$	−1760.32	−0.0876242	−0.0438121	+	0.999040i	$0.513950\pi$
$740$	0	0
$741$	2197.21	0.108929
$742$	0	0
$743$	−11383.4	−0.562067	−0.281034	−	0.959698i	$-0.590677\pi$
$743$	−11383.4	−0.562067	−0.281034	+	0.959698i	$0.590677\pi$
$744$	0	0
$745$	−3839.29	−0.188806
$746$	0	0
$747$	10598.1	0.519095
$748$	0	0
$749$	12186.3	0.594497
$750$	0	0
$751$	15058.3	0.731670	0.365835	−	0.930680i	$-0.380783\pi$
$751$	15058.3	0.731670	0.365835	+	0.930680i	$0.380783\pi$
$752$	0	0
$753$	6086.45	0.294558
$754$	0	0
$755$	−4941.08	−0.238178
$756$	0	0
$757$	38073.4	1.82801	0.914004	−	0.405706i	$-0.132974\pi$
$757$	38073.4	1.82801	0.914004	+	0.405706i	$0.132974\pi$
$758$	0	0
$759$	431.003	0.0206119
$760$	0	0
$761$	15232.1	0.725577	0.362788	−	0.931872i	$-0.381825\pi$
$761$	15232.1	0.725577	0.362788	+	0.931872i	$0.381825\pi$
$762$	0	0
$763$	1742.57	0.0826803
$764$	0	0
$765$	−636.563	−0.0300849
$766$	0	0
$767$	−17563.8	−0.826850
$768$	0	0
$769$	12013.1	0.563332	0.281666	−	0.959513i	$-0.409113\pi$
$769$	12013.1	0.563332	0.281666	+	0.959513i	$0.409113\pi$
$770$	0	0
$771$	8230.33	0.384446
$772$	0	0
$773$	−14258.5	−0.663443	−0.331721	−	0.943377i	$-0.607629\pi$
$773$	−14258.5	−0.663443	−0.331721	+	0.943377i	$0.607629\pi$
$774$	0	0
$775$	21372.9	0.990630
$776$	0	0
$777$	−5687.05	−0.262576
$778$	0	0
$779$	−4959.99	−0.228126
$780$	0	0
$781$	−4059.27	−0.185982
$782$	0	0
$783$	−5336.70	−0.243574
$784$	0	0
$785$	4445.08	0.202104
$786$	0	0
$787$	14377.9	0.651227	0.325613	−	0.945503i	$-0.394429\pi$
$787$	14377.9	0.651227	0.325613	+	0.945503i	$0.394429\pi$
$788$	0	0
$789$	2540.45	0.114629
$790$	0	0
$791$	−3461.26	−0.155585
$792$	0	0
$793$	−15944.1	−0.713986
$794$	0	0
$795$	−421.490	−0.0188034
$796$	0	0
$797$	38515.9	1.71180	0.855899	−	0.517144i	$-0.173005\pi$
$797$	38515.9	1.71180	0.855899	+	0.517144i	$0.173005\pi$
$798$	0	0
$799$	932.352	0.0412819
$800$	0	0
$801$	−2027.77	−0.0894479
$802$	0	0
$803$	12603.5	0.553882
$804$	0	0
$805$	−182.341	−0.00798347
$806$	0	0
$807$	456.217	0.0199004
$808$	0	0
$809$	11438.6	0.497106	0.248553	−	0.968618i	$-0.420045\pi$
$809$	11438.6	0.497106	0.248553	+	0.968618i	$0.420045\pi$
$810$	0	0
$811$	−15872.9	−0.687265	−0.343632	−	0.939104i	$-0.611657\pi$
$811$	−15872.9	−0.687265	−0.343632	+	0.939104i	$0.611657\pi$
$812$	0	0
$813$	4168.21	0.179810
$814$	0	0
$815$	−6692.70	−0.287650
$816$	0	0
$817$	−8909.05	−0.381503
$818$	0	0
$819$	−3329.74	−0.142064
$820$	0	0
$821$	−11988.8	−0.509638	−0.254819	−	0.966989i	$-0.582016\pi$
$821$	−11988.8	−0.509638	−0.254819	+	0.966989i	$0.582016\pi$
$822$	0	0
$823$	222.224	0.00941219	0.00470610	−	0.999989i	$-0.498502\pi$
$823$	222.224	0.00941219	0.00470610	+	0.999989i	$0.498502\pi$
$824$	0	0
$825$	3710.46	0.156584
$826$	0	0
$827$	−7524.36	−0.316382	−0.158191	−	0.987409i	$-0.550566\pi$
$827$	−7524.36	−0.316382	−0.158191	+	0.987409i	$0.550566\pi$
$828$	0	0
$829$	−11807.1	−0.494663	−0.247332	−	0.968931i	$-0.579554\pi$
$829$	−11807.1	−0.494663	−0.247332	+	0.968931i	$0.579554\pi$
$830$	0	0
$831$	20805.2	0.868502
$832$	0	0
$833$	876.120	0.0364415
$834$	0	0
$835$	7055.93	0.292432
$836$	0	0
$837$	−22577.0	−0.932349
$838$	0	0
$839$	29112.9	1.19796	0.598980	−	0.800764i	$-0.295573\pi$
$839$	29112.9	1.19796	0.598980	+	0.800764i	$0.295573\pi$
$840$	0	0
$841$	−22662.8	−0.929224
$842$	0	0
$843$	8169.73	0.333785
$844$	0	0
$845$	−2932.00	−0.119366
$846$	0	0
$847$	847.000	0.0343604
$848$	0	0
$849$	14467.2	0.584819
$850$	0	0
$851$	−4136.83	−0.166638
$852$	0	0
$853$	−24892.1	−0.999169	−0.499584	−	0.866265i	$-0.666514\pi$
$853$	−24892.1	−0.999169	−0.499584	+	0.866265i	$0.666514\pi$
$854$	0	0
$855$	1144.46	0.0457773
$856$	0	0
$857$	−27299.2	−1.08812	−0.544062	−	0.839045i	$-0.683114\pi$
$857$	−27299.2	−1.08812	−0.544062	+	0.839045i	$0.683114\pi$
$858$	0	0
$859$	6975.13	0.277053	0.138526	−	0.990359i	$-0.455763\pi$
$859$	6975.13	0.277053	0.138526	+	0.990359i	$0.455763\pi$
$860$	0	0
$861$	−2996.11	−0.118591
$862$	0	0
$863$	4481.56	0.176772	0.0883858	−	0.996086i	$-0.471829\pi$
$863$	4481.56	0.176772	0.0883858	+	0.996086i	$0.471829\pi$
$864$	0	0
$865$	−7949.67	−0.312482
$866$	0	0
$867$	12741.8	0.499116
$868$	0	0
$869$	−5377.73	−0.209928
$870$	0	0
$871$	−6427.80	−0.250055
$872$	0	0
$873$	26291.6	1.01928
$874$	0	0
$875$	−3183.42	−0.122993
$876$	0	0
$877$	−7207.96	−0.277532	−0.138766	−	0.990325i	$-0.544314\pi$
$877$	−7207.96	−0.277532	−0.138766	+	0.990325i	$0.544314\pi$
$878$	0	0
$879$	−20549.7	−0.788536
$880$	0	0
$881$	−38413.7	−1.46900	−0.734502	−	0.678607i	$-0.762584\pi$
$881$	−38413.7	−1.46900	−0.734502	+	0.678607i	$0.762584\pi$
$882$	0	0
$883$	8705.04	0.331764	0.165882	−	0.986146i	$-0.446953\pi$
$883$	8705.04	0.331764	0.165882	+	0.986146i	$0.446953\pi$
$884$	0	0
$885$	3646.58	0.138507
$886$	0	0
$887$	44100.0	1.66937	0.834687	−	0.550725i	$-0.185649\pi$
$887$	44100.0	1.66937	0.834687	+	0.550725i	$0.185649\pi$
$888$	0	0
$889$	6853.73	0.258568
$890$	0	0
$891$	1814.09	0.0682091
$892$	0	0
$893$	−1676.25	−0.0628146
$894$	0	0
$895$	−4594.08	−0.171579
$896$	0	0
$897$	965.451	0.0359370
$898$	0	0
$899$	7302.55	0.270916
$900$	0	0
$901$	1473.15	0.0544702
$902$	0	0
$903$	−5381.57	−0.198325
$904$	0	0
$905$	−7958.54	−0.292322
$906$	0	0
$907$	−19921.2	−0.729296	−0.364648	−	0.931145i	$-0.618811\pi$
$907$	−19921.2	−0.729296	−0.364648	+	0.931145i	$0.618811\pi$
$908$	0	0
$909$	31087.3	1.13432
$910$	0	0
$911$	14446.7	0.525402	0.262701	−	0.964877i	$-0.415387\pi$
$911$	14446.7	0.525402	0.262701	+	0.964877i	$0.415387\pi$
$912$	0	0
$913$	−6038.81	−0.218900
$914$	0	0
$915$	3310.29	0.119601
$916$	0	0
$917$	11625.9	0.418672
$918$	0	0
$919$	−44381.1	−1.59303	−0.796516	−	0.604617i	$-0.793326\pi$
$919$	−44381.1	−1.59303	−0.796516	+	0.604617i	$0.793326\pi$
$920$	0	0
$921$	−19001.8	−0.679839
$922$	0	0
$923$	−9092.80	−0.324261
$924$	0	0
$925$	−35613.5	−1.26591
$926$	0	0
$927$	−2375.37	−0.0841612
$928$	0	0
$929$	−3455.30	−0.122029	−0.0610145	−	0.998137i	$-0.519434\pi$
$929$	−3455.30	−0.122029	−0.0610145	+	0.998137i	$0.519434\pi$
$930$	0	0
$931$	−1575.15	−0.0554494
$932$	0	0
$933$	−15364.1	−0.539117
$934$	0	0
$935$	362.714	0.0126866
$936$	0	0
$937$	−14962.8	−0.521678	−0.260839	−	0.965382i	$-0.583999\pi$
$937$	−14962.8	−0.521678	−0.260839	+	0.965382i	$0.583999\pi$
$938$	0	0
$939$	−25967.7	−0.902474
$940$	0	0
$941$	−6116.95	−0.211909	−0.105955	−	0.994371i	$-0.533790\pi$
$941$	−6116.95	−0.211909	−0.105955	+	0.994371i	$0.533790\pi$
$942$	0	0
$943$	−2179.41	−0.0752611
$944$	0	0
$945$	1658.19	0.0570804
$946$	0	0
$947$	−53343.7	−1.83045	−0.915225	−	0.402943i	$-0.867987\pi$
$947$	−53343.7	−1.83045	−0.915225	+	0.402943i	$0.867987\pi$
$948$	0	0
$949$	28231.9	0.965696
$950$	0	0
$951$	−608.117	−0.0207356
$952$	0	0
$953$	1979.62	0.0672887	0.0336443	−	0.999434i	$-0.489289\pi$
$953$	1979.62	0.0672887	0.0336443	+	0.999434i	$0.489289\pi$
$954$	0	0
$955$	−5238.24	−0.177493
$956$	0	0
$957$	1267.76	0.0428223
$958$	0	0
$959$	−11327.2	−0.381411
$960$	0	0
$961$	1102.58	0.0370104
$962$	0	0
$963$	−33608.1	−1.12462
$964$	0	0
$965$	−3199.07	−0.106717
$966$	0	0
$967$	38892.0	1.29336	0.646681	−	0.762761i	$-0.276157\pi$
$967$	38892.0	1.29336	0.646681	+	0.762761i	$0.276157\pi$
$968$	0	0
$969$	1594.40	0.0528582
$970$	0	0
$971$	47826.1	1.58065	0.790325	−	0.612688i	$-0.209912\pi$
$971$	47826.1	1.58065	0.790325	+	0.612688i	$0.209912\pi$
$972$	0	0
$973$	−4870.15	−0.160462
$974$	0	0
$975$	8311.45	0.273005
$976$	0	0
$977$	20840.5	0.682442	0.341221	−	0.939983i	$-0.389160\pi$
$977$	20840.5	0.682442	0.341221	+	0.939983i	$0.389160\pi$
$978$	0	0
$979$	1155.43	0.0377197
$980$	0	0
$981$	−4805.74	−0.156407
$982$	0	0
$983$	54430.5	1.76609	0.883044	−	0.469290i	$-0.155490\pi$
$983$	54430.5	1.76609	0.883044	+	0.469290i	$0.155490\pi$
$984$	0	0
$985$	−5714.05	−0.184837
$986$	0	0
$987$	−1012.55	−0.0326542
$988$	0	0
$989$	−3914.62	−0.125862
$990$	0	0
$991$	35161.9	1.12710	0.563550	−	0.826082i	$-0.309435\pi$
$991$	35161.9	1.12710	0.563550	+	0.826082i	$0.309435\pi$
$992$	0	0
$993$	9369.50	0.299428
$994$	0	0
$995$	−8293.99	−0.264258
$996$	0	0
$997$	26961.6	0.856451	0.428225	−	0.903672i	$-0.359139\pi$
$997$	26961.6	0.856451	0.428225	+	0.903672i	$0.359139\pi$
$998$	0	0
$999$	37619.8	1.19143

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	1232.4.a.s.1.3		4
4.3	odd	2		77.4.a.d.1.4	✓	4
12.11	even	2		693.4.a.l.1.1		4
20.19	odd	2		1925.4.a.p.1.1		4
28.27	even	2		539.4.a.g.1.4		4
44.43	even	2		847.4.a.d.1.1		4

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
77.4.a.d.1.4	✓	4	4.3	odd	2
539.4.a.g.1.4		4	28.27	even	2
693.4.a.l.1.1		4	12.11	even	2
847.4.a.d.1.1		4	44.43	even	2
1232.4.a.s.1.3		4	1.1	even	1	trivial
1925.4.a.p.1.1		4	20.19	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:	\( N \)	\(=\)	\( 1232 = 2^{4} \cdot 7 \cdot 11 \)
Weight:	\( k \)	\(=\)	\( 4 \)
Character orbit:	\([\chi]\)	\(=\)	1232.a (trivial)

\(f(q)\)	\(=\)	\(q-2.77399 q^{3} +1.84418 q^{5} +7.00000 q^{7} -19.3050 q^{9} +11.0000 q^{11} +24.6401 q^{13} -5.11574 q^{15} +17.8800 q^{17} -32.1459 q^{19} -19.4179 q^{21} -14.1248 q^{23} -121.599 q^{25} +128.450 q^{27} -41.5471 q^{29} -175.766 q^{31} -30.5139 q^{33} +12.9093 q^{35} +292.877 q^{37} -68.3513 q^{39} +154.296 q^{41} +277.144 q^{43} -35.6019 q^{45} +52.1450 q^{47} +49.0000 q^{49} -49.5989 q^{51} +82.3907 q^{53} +20.2860 q^{55} +89.1723 q^{57} -712.816 q^{59} -647.078 q^{61} -135.135 q^{63} +45.4408 q^{65} -260.867 q^{67} +39.1821 q^{69} -369.025 q^{71} +1145.77 q^{73} +337.314 q^{75} +77.0000 q^{77} -488.885 q^{79} +164.917 q^{81} -548.982 q^{83} +32.9740 q^{85} +115.251 q^{87} +105.039 q^{89} +172.481 q^{91} +487.572 q^{93} -59.2829 q^{95} -1361.91 q^{97} -212.355 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 4 q - 14 q^{3} + 10 q^{5} + 28 q^{7} + 76 q^{9} + 44 q^{11} + 58 q^{13} - 284 q^{15} + 4 q^{17} - 258 q^{19} - 98 q^{21} - 8 q^{23} + 80 q^{25} - 428 q^{27} - 396 q^{29} + 56 q^{31} - 154 q^{33} + 70 q^{35}+ \cdots + 836 q^{99}+O(q^{100}) \) 4 * q - 14 * q^3 + 10 * q^5 + 28 * q^7 + 76 * q^9 + 44 * q^11 + 58 * q^13 - 284 * q^15 + 4 * q^17 - 258 * q^19 - 98 * q^21 - 8 * q^23 + 80 * q^25 - 428 * q^27 - 396 * q^29 + 56 * q^31 - 154 * q^33 + 70 * q^35 + 84 * q^37 + 412 * q^39 + 52 * q^41 - 408 * q^43 + 826 * q^45 - 8 * q^47 + 196 * q^49 + 388 * q^51 + 624 * q^53 + 110 * q^55 + 48 * q^57 + 238 * q^59 - 162 * q^61 + 532 * q^63 - 32 * q^65 - 1340 * q^67 + 2416 * q^69 - 1788 * q^71 + 1456 * q^73 + 806 * q^75 + 308 * q^77 + 1324 * q^79 + 1444 * q^81 - 450 * q^83 - 1736 * q^85 - 588 * q^87 - 3072 * q^89 + 406 * q^91 - 1264 * q^93 - 24 * q^95 - 652 * q^97 + 836 * q^99

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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