LMFDB - Embedded newform 136.4.a.c.1.1

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [136,4,Mod(1,136)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(136, 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("136.1");

S:= CuspForms(chi, 4);

N := Newforms(S);

Level:	$ N $	$=$	$ 136 = 2^{3} \cdot 17 $
Weight:	$ k $	$=$	$ 4 $
Character orbit:	$[\chi]$	$=$	136.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:	$8.02425976078$
Analytic rank:	$0$
Dimension:	$3$
Coefficient field:	3.3.1556.1
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{3} - x^{2} - 9x + 11 $ x^3 - x^2 - 9*x + 11
Coefficient ring:	$\Z[a_1, \ldots, a_{5}]$
Coefficient ring index:	$ 2^{3} $
Twist minimal:	yes
Fricke sign:	$1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.1
Root			$2.80979$ of defining polynomial
Character	$\chi$	$=$	136.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-5.40937 q^{3} -9.23914 q^{5} +2.17022 q^{7} +2.26124 q^{9} +O(q^{10})$	$q-5.40937 q^{3} -9.23914 q^{5} +2.17022 q^{7} +2.26124 q^{9} +51.8877 q^{11} +10.4204 q^{13} +49.9779 q^{15} +17.0000 q^{17} +122.094 q^{19} -11.7395 q^{21} +48.3286 q^{23} -39.6382 q^{25} +133.821 q^{27} +160.746 q^{29} +12.2365 q^{31} -280.679 q^{33} -20.0510 q^{35} -103.125 q^{37} -56.3678 q^{39} +113.350 q^{41} +76.1363 q^{43} -20.8919 q^{45} -289.695 q^{47} -338.290 q^{49} -91.9592 q^{51} +447.549 q^{53} -479.397 q^{55} -660.449 q^{57} +480.106 q^{59} -308.075 q^{61} +4.90739 q^{63} -96.2757 q^{65} -56.4052 q^{67} -261.427 q^{69} +332.832 q^{71} +35.2262 q^{73} +214.418 q^{75} +112.608 q^{77} +1266.17 q^{79} -784.940 q^{81} -471.438 q^{83} -157.065 q^{85} -869.532 q^{87} +559.380 q^{89} +22.6146 q^{91} -66.1917 q^{93} -1128.04 q^{95} +869.391 q^{97} +117.330 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 3 q + 8 q^{3} + 2 q^{5} + 12 q^{7} + 63 q^{9}+O(q^{10}) $ 3 * q + 8 * q^3 + 2 * q^5 + 12 * q^7 + 63 * q^9	$ 3 q + 8 q^{3} + 2 q^{5} + 12 q^{7} + 63 q^{9} + 72 q^{11} + 50 q^{13} + 64 q^{15} + 51 q^{17} + 124 q^{19} - 32 q^{21} + 60 q^{23} - 75 q^{25} + 512 q^{27} - 54 q^{29} + 300 q^{31} - 48 q^{33} + 248 q^{35} - 542 q^{37} + 320 q^{39} + 30 q^{41} + 52 q^{43} - 502 q^{45} + 16 q^{47} - 677 q^{49} + 136 q^{51} - 518 q^{53} - 608 q^{55} - 896 q^{57} + 132 q^{59} - 614 q^{61} - 852 q^{63} - 148 q^{65} - 332 q^{67} - 1280 q^{69} - 268 q^{71} - 578 q^{73} - 712 q^{75} - 128 q^{77} - 668 q^{79} + 2427 q^{81} + 876 q^{83} + 34 q^{85} - 2400 q^{87} + 1790 q^{89} - 168 q^{91} + 2592 q^{93} - 504 q^{95} + 838 q^{97} + 2040 q^{99}+O(q^{100}) $ 3 * q + 8 * q^3 + 2 * q^5 + 12 * q^7 + 63 * q^9 + 72 * q^11 + 50 * q^13 + 64 * q^15 + 51 * q^17 + 124 * q^19 - 32 * q^21 + 60 * q^23 - 75 * q^25 + 512 * q^27 - 54 * q^29 + 300 * q^31 - 48 * q^33 + 248 * q^35 - 542 * q^37 + 320 * q^39 + 30 * q^41 + 52 * q^43 - 502 * q^45 + 16 * q^47 - 677 * q^49 + 136 * q^51 - 518 * q^53 - 608 * q^55 - 896 * q^57 + 132 * q^59 - 614 * q^61 - 852 * q^63 - 148 * q^65 - 332 * q^67 - 1280 * q^69 - 268 * q^71 - 578 * q^73 - 712 * q^75 - 128 * q^77 - 668 * q^79 + 2427 * q^81 + 876 * q^83 + 34 * q^85 - 2400 * q^87 + 1790 * q^89 - 168 * q^91 + 2592 * q^93 - 504 * q^95 + 838 * q^97 + 2040 * 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
$3$	−5.40937	−1.04103	−0.520516	−	0.853852i	$-0.674261\pi$
$3$	−5.40937	−1.04103	−0.520516	+	0.853852i	$0.674261\pi$
$4$	0	0
$5$	−9.23914	−0.826374	−0.413187	−	0.910646i	$-0.635584\pi$
$5$	−9.23914	−0.826374	−0.413187	+	0.910646i	$0.635584\pi$
$6$	0	0
$7$	2.17022	0.117181	0.0585905	−	0.998282i	$-0.481339\pi$
$7$	2.17022	0.117181	0.0585905	+	0.998282i	$0.481339\pi$
$8$	0	0
$9$	2.26124	0.0837495
$10$	0	0
$11$	51.8877	1.42225	0.711123	−	0.703067i	$-0.248187\pi$
$11$	51.8877	1.42225	0.711123	+	0.703067i	$0.248187\pi$
$12$	0	0
$13$	10.4204	0.222316	0.111158	−	0.993803i	$-0.464544\pi$
$13$	10.4204	0.222316	0.111158	+	0.993803i	$0.464544\pi$
$14$	0	0
$15$	49.9779	0.860283
$16$	0	0
$17$	17.0000	0.242536
$17$	17.0000	0.242536
$18$	0	0
$19$	122.094	1.47422	0.737111	−	0.675772i	$-0.236190\pi$
$19$	122.094	1.47422	0.737111	+	0.675772i	$0.236190\pi$
$20$	0	0
$21$	−11.7395	−0.121989
$22$	0	0
$23$	48.3286	0.438140	0.219070	−	0.975709i	$-0.429698\pi$
$23$	48.3286	0.438140	0.219070	+	0.975709i	$0.429698\pi$
$24$	0	0
$25$	−39.6382	−0.317106
$26$	0	0
$27$	133.821	0.953847
$28$	0	0
$29$	160.746	1.02930	0.514650	−	0.857400i	$-0.327922\pi$
$29$	160.746	1.02930	0.514650	+	0.857400i	$0.327922\pi$
$30$	0	0
$31$	12.2365	0.0708949	0.0354474	−	0.999372i	$-0.488714\pi$
$31$	12.2365	0.0708949	0.0354474	+	0.999372i	$0.488714\pi$
$32$	0	0
$33$	−280.679	−1.48061
$34$	0	0
$35$	−20.0510	−0.0968353
$36$	0	0
$37$	−103.125	−0.458206	−0.229103	−	0.973402i	$-0.573579\pi$
$37$	−103.125	−0.458206	−0.229103	+	0.973402i	$0.573579\pi$
$38$	0	0
$39$	−56.3678	−0.231438
$40$	0	0
$41$	113.350	0.431762	0.215881	−	0.976420i	$-0.430738\pi$
$41$	113.350	0.431762	0.215881	+	0.976420i	$0.430738\pi$
$42$	0	0
$43$	76.1363	0.270016	0.135008	−	0.990845i	$-0.456894\pi$
$43$	76.1363	0.270016	0.135008	+	0.990845i	$0.456894\pi$
$44$	0	0
$45$	−20.8919	−0.0692084
$46$	0	0
$47$	−289.695	−0.899072	−0.449536	−	0.893262i	$-0.648411\pi$
$47$	−289.695	−0.899072	−0.449536	+	0.893262i	$0.648411\pi$
$48$	0	0
$49$	−338.290	−0.986269
$50$	0	0
$51$	−91.9592	−0.252488
$52$	0	0
$53$	447.549	1.15992	0.579958	−	0.814646i	$-0.303069\pi$
$53$	447.549	1.15992	0.579958	+	0.814646i	$0.303069\pi$
$54$	0	0
$55$	−479.397	−1.17531
$56$	0	0
$57$	−660.449	−1.53471
$58$	0	0
$59$	480.106	1.05940	0.529699	−	0.848186i	$-0.322305\pi$
$59$	480.106	1.05940	0.529699	+	0.848186i	$0.322305\pi$
$60$	0	0
$61$	−308.075	−0.646640	−0.323320	−	0.946290i	$-0.604799\pi$
$61$	−308.075	−0.646640	−0.323320	+	0.946290i	$0.604799\pi$
$62$	0	0
$63$	4.90739	0.00981385
$64$	0	0
$65$	−96.2757	−0.183716
$66$	0	0
$67$	−56.4052	−0.102851	−0.0514253	−	0.998677i	$-0.516376\pi$
$67$	−56.4052	−0.102851	−0.0514253	+	0.998677i	$0.516376\pi$
$68$	0	0
$69$	−261.427	−0.456118
$70$	0	0
$71$	332.832	0.556336	0.278168	−	0.960532i	$-0.410273\pi$
$71$	332.832	0.556336	0.278168	+	0.960532i	$0.410273\pi$
$72$	0	0
$73$	35.2262	0.0564782	0.0282391	−	0.999601i	$-0.491010\pi$
$73$	35.2262	0.0564782	0.0282391	+	0.999601i	$0.491010\pi$
$74$	0	0
$75$	214.418	0.330118
$76$	0	0
$77$	112.608	0.166660
$78$	0	0
$79$	1266.17	1.80323	0.901613	−	0.432543i	$-0.142384\pi$
$79$	1266.17	1.80323	0.901613	+	0.432543i	$0.142384\pi$
$80$	0	0
$81$	−784.940	−1.07674
$82$	0	0
$83$	−471.438	−0.623459	−0.311729	−	0.950171i	$-0.600908\pi$
$83$	−471.438	−0.623459	−0.311729	+	0.950171i	$0.600908\pi$
$84$	0	0
$85$	−157.065	−0.200425
$86$	0	0
$87$	−869.532	−1.07154
$88$	0	0
$89$	559.380	0.666227	0.333113	−	0.942887i	$-0.391901\pi$
$89$	559.380	0.666227	0.333113	+	0.942887i	$0.391901\pi$
$90$	0	0
$91$	22.6146	0.0260512
$92$	0	0
$93$	−66.1917	−0.0738039
$94$	0	0
$95$	−1128.04	−1.21826
$96$	0	0
$97$	869.391	0.910033	0.455017	−	0.890483i	$-0.349633\pi$
$97$	869.391	0.910033	0.455017	+	0.890483i	$0.349633\pi$
$98$	0	0
$99$	117.330	0.119112
$100$	0	0
$101$	−723.176	−0.712462	−0.356231	−	0.934398i	$-0.615938\pi$
$101$	−723.176	−0.712462	−0.356231	+	0.934398i	$0.615938\pi$
$102$	0	0
$103$	−1535.73	−1.46912	−0.734562	−	0.678541i	$-0.762612\pi$
$103$	−1535.73	−1.46912	−0.734562	+	0.678541i	$0.762612\pi$
$104$	0	0
$105$	108.463	0.100809
$106$	0	0
$107$	1402.53	1.26718	0.633589	−	0.773670i	$-0.281581\pi$
$107$	1402.53	1.26718	0.633589	+	0.773670i	$0.281581\pi$
$108$	0	0
$109$	1666.55	1.46446	0.732231	−	0.681057i	$-0.238479\pi$
$109$	1666.55	1.46446	0.732231	+	0.681057i	$0.238479\pi$
$110$	0	0
$111$	557.840	0.477008
$112$	0	0
$113$	1658.68	1.38084	0.690422	−	0.723407i	$-0.257425\pi$
$113$	1658.68	1.38084	0.690422	+	0.723407i	$0.257425\pi$
$114$	0	0
$115$	−446.515	−0.362067
$116$	0	0
$117$	23.5630	0.0186188
$118$	0	0
$119$	36.8938	0.0284206
$120$	0	0
$121$	1361.33	1.02279
$122$	0	0
$123$	−613.149	−0.449478
$124$	0	0
$125$	1521.12	1.08842
$126$	0	0
$127$	123.318	0.0861632	0.0430816	−	0.999072i	$-0.486282\pi$
$127$	123.318	0.0861632	0.0430816	+	0.999072i	$0.486282\pi$
$128$	0	0
$129$	−411.849	−0.281095
$130$	0	0
$131$	259.534	0.173096	0.0865480	−	0.996248i	$-0.472416\pi$
$131$	259.534	0.173096	0.0865480	+	0.996248i	$0.472416\pi$
$132$	0	0
$133$	264.970	0.172751
$134$	0	0
$135$	−1236.39	−0.788234
$136$	0	0
$137$	3029.98	1.88955	0.944775	−	0.327718i	$-0.106280\pi$
$137$	3029.98	1.88955	0.944775	+	0.327718i	$0.106280\pi$
$138$	0	0
$139$	1431.50	0.873510	0.436755	−	0.899580i	$-0.356128\pi$
$139$	1431.50	0.873510	0.436755	+	0.899580i	$0.356128\pi$
$140$	0	0
$141$	1567.07	0.935964
$142$	0	0
$143$	540.691	0.316188
$144$	0	0
$145$	−1485.15	−0.850587
$146$	0	0
$147$	1829.94	1.02674
$148$	0	0
$149$	883.713	0.485883	0.242941	−	0.970041i	$-0.421888\pi$
$149$	883.713	0.485883	0.242941	+	0.970041i	$0.421888\pi$
$150$	0	0
$151$	−1702.64	−0.917606	−0.458803	−	0.888538i	$-0.651722\pi$
$151$	−1702.64	−0.917606	−0.458803	+	0.888538i	$0.651722\pi$
$152$	0	0
$153$	38.4410	0.0203122
$154$	0	0
$155$	−113.055	−0.0585857
$156$	0	0
$157$	−3690.32	−1.87592	−0.937960	−	0.346743i	$-0.887288\pi$
$157$	−3690.32	−1.87592	−0.937960	+	0.346743i	$0.887288\pi$
$158$	0	0
$159$	−2420.96	−1.20751
$160$	0	0
$161$	104.884	0.0513417
$162$	0	0
$163$	−2079.61	−0.999310	−0.499655	−	0.866224i	$-0.666540\pi$
$163$	−2079.61	−0.999310	−0.499655	+	0.866224i	$0.666540\pi$
$164$	0	0
$165$	2593.24	1.22353
$166$	0	0
$167$	−2701.73	−1.25189	−0.625947	−	0.779866i	$-0.715287\pi$
$167$	−2701.73	−1.25189	−0.625947	+	0.779866i	$0.715287\pi$
$168$	0	0
$169$	−2088.42	−0.950576
$170$	0	0
$171$	276.083	0.123465
$172$	0	0
$173$	−3787.93	−1.66469	−0.832343	−	0.554261i	$-0.813001\pi$
$173$	−3787.93	−1.66469	−0.832343	+	0.554261i	$0.813001\pi$
$174$	0	0
$175$	−86.0237	−0.0371588
$176$	0	0
$177$	−2597.07	−1.10287
$178$	0	0
$179$	4145.32	1.73093	0.865464	−	0.500972i	$-0.167024\pi$
$179$	4145.32	1.73093	0.865464	+	0.500972i	$0.167024\pi$
$180$	0	0
$181$	−3305.46	−1.35742	−0.678710	−	0.734406i	$-0.737461\pi$
$181$	−3305.46	−1.35742	−0.678710	+	0.734406i	$0.737461\pi$
$182$	0	0
$183$	1666.49	0.673173
$184$	0	0
$185$	952.786	0.378650
$186$	0	0
$187$	882.090	0.344946
$188$	0	0
$189$	290.421	0.111773
$190$	0	0
$191$	−2439.99	−0.924352	−0.462176	−	0.886788i	$-0.652931\pi$
$191$	−2439.99	−0.924352	−0.462176	+	0.886788i	$0.652931\pi$
$192$	0	0
$193$	−2133.07	−0.795553	−0.397777	−	0.917482i	$-0.630218\pi$
$193$	−2133.07	−0.795553	−0.397777	+	0.917482i	$0.630218\pi$
$194$	0	0
$195$	520.790	0.191254
$196$	0	0
$197$	−2834.36	−1.02508	−0.512538	−	0.858664i	$-0.671295\pi$
$197$	−2834.36	−1.02508	−0.512538	+	0.858664i	$0.671295\pi$
$198$	0	0
$199$	408.709	0.145591	0.0727954	−	0.997347i	$-0.476808\pi$
$199$	408.709	0.145591	0.0727954	+	0.997347i	$0.476808\pi$
$200$	0	0
$201$	305.116	0.107071
$202$	0	0
$203$	348.854	0.120614
$204$	0	0
$205$	−1047.25	−0.356797
$206$	0	0
$207$	109.282	0.0366940
$208$	0	0
$209$	6335.15	2.09671
$210$	0	0
$211$	1490.16	0.486193	0.243096	−	0.970002i	$-0.421837\pi$
$211$	1490.16	0.486193	0.243096	+	0.970002i	$0.421837\pi$
$212$	0	0
$213$	−1800.41	−0.579164
$214$	0	0
$215$	−703.434	−0.223134
$216$	0	0
$217$	26.5559	0.00830753
$218$	0	0
$219$	−190.551	−0.0587957
$220$	0	0
$221$	177.147	0.0539194
$222$	0	0
$223$	−3373.77	−1.01311	−0.506557	−	0.862207i	$-0.669082\pi$
$223$	−3373.77	−1.01311	−0.506557	+	0.862207i	$0.669082\pi$
$224$	0	0
$225$	−89.6314	−0.0265575
$226$	0	0
$227$	229.065	0.0669761	0.0334880	−	0.999439i	$-0.489338\pi$
$227$	229.065	0.0669761	0.0334880	+	0.999439i	$0.489338\pi$
$228$	0	0
$229$	−87.2044	−0.0251643	−0.0125822	−	0.999921i	$-0.504005\pi$
$229$	−87.2044	−0.0251643	−0.0125822	+	0.999921i	$0.504005\pi$
$230$	0	0
$231$	−609.136	−0.173499
$232$	0	0
$233$	3296.12	0.926764	0.463382	−	0.886159i	$-0.346636\pi$
$233$	3296.12	0.926764	0.463382	+	0.886159i	$0.346636\pi$
$234$	0	0
$235$	2676.54	0.742970
$236$	0	0
$237$	−6849.16	−1.87722
$238$	0	0
$239$	1859.99	0.503402	0.251701	−	0.967805i	$-0.419010\pi$
$239$	1859.99	0.503402	0.251701	+	0.967805i	$0.419010\pi$
$240$	0	0
$241$	−3758.94	−1.00471	−0.502354	−	0.864662i	$-0.667533\pi$
$241$	−3758.94	−1.00471	−0.502354	+	0.864662i	$0.667533\pi$
$242$	0	0
$243$	632.861	0.167070
$244$	0	0
$245$	3125.51	0.815027
$246$	0	0
$247$	1272.27	0.327742
$248$	0	0
$249$	2550.18	0.649041
$250$	0	0
$251$	6478.07	1.62905	0.814526	−	0.580126i	$-0.196997\pi$
$251$	6478.07	1.62905	0.814526	+	0.580126i	$0.196997\pi$
$252$	0	0
$253$	2507.66	0.623143
$254$	0	0
$255$	849.624	0.208649
$256$	0	0
$257$	−6935.59	−1.68339	−0.841693	−	0.539957i	$-0.818441\pi$
$257$	−6935.59	−1.68339	−0.841693	+	0.539957i	$0.818441\pi$
$258$	0	0
$259$	−223.804	−0.0536931
$260$	0	0
$261$	363.484	0.0862034
$262$	0	0
$263$	332.528	0.0779642	0.0389821	−	0.999240i	$-0.487588\pi$
$263$	332.528	0.0779642	0.0389821	+	0.999240i	$0.487588\pi$
$264$	0	0
$265$	−4134.97	−0.958525
$266$	0	0
$267$	−3025.89	−0.693564
$268$	0	0
$269$	4216.46	0.955695	0.477848	−	0.878443i	$-0.341417\pi$
$269$	4216.46	0.955695	0.477848	+	0.878443i	$0.341417\pi$
$270$	0	0
$271$	2919.43	0.654402	0.327201	−	0.944955i	$-0.393895\pi$
$271$	2919.43	0.654402	0.327201	+	0.944955i	$0.393895\pi$
$272$	0	0
$273$	−122.331	−0.0271201
$274$	0	0
$275$	−2056.73	−0.451003
$276$	0	0
$277$	3562.75	0.772798	0.386399	−	0.922332i	$-0.373719\pi$
$277$	3562.75	0.772798	0.386399	+	0.922332i	$0.373719\pi$
$278$	0	0
$279$	27.6696	0.00593741
$280$	0	0
$281$	−8233.37	−1.74791	−0.873953	−	0.486010i	$-0.838452\pi$
$281$	−8233.37	−1.74791	−0.873953	+	0.486010i	$0.838452\pi$
$282$	0	0
$283$	6679.14	1.40295	0.701473	−	0.712696i	$-0.252526\pi$
$283$	6679.14	1.40295	0.701473	+	0.712696i	$0.252526\pi$
$284$	0	0
$285$	6101.99	1.26825
$286$	0	0
$287$	245.994	0.0505942
$288$	0	0
$289$	289.000	0.0588235
$290$	0	0
$291$	−4702.85	−0.947375
$292$	0	0
$293$	5920.46	1.18047	0.590233	−	0.807233i	$-0.299036\pi$
$293$	5920.46	1.18047	0.590233	+	0.807233i	$0.299036\pi$
$294$	0	0
$295$	−4435.77	−0.875459
$296$	0	0
$297$	6943.66	1.35661
$298$	0	0
$299$	503.604	0.0974053
$300$	0	0
$301$	165.233	0.0316407
$302$	0	0
$303$	3911.92	0.741696
$304$	0	0
$305$	2846.35	0.534366
$306$	0	0
$307$	−8353.80	−1.55302	−0.776509	−	0.630107i	$-0.783011\pi$
$307$	−8353.80	−1.55302	−0.776509	+	0.630107i	$0.783011\pi$
$308$	0	0
$309$	8307.32	1.52941
$310$	0	0
$311$	9857.98	1.79741	0.898706	−	0.438553i	$-0.144509\pi$
$311$	9857.98	1.79741	0.898706	+	0.438553i	$0.144509\pi$
$312$	0	0
$313$	−2244.74	−0.405368	−0.202684	−	0.979244i	$-0.564966\pi$
$313$	−2244.74	−0.405368	−0.202684	+	0.979244i	$0.564966\pi$
$314$	0	0
$315$	−45.3400	−0.00810991
$316$	0	0
$317$	−6370.07	−1.12864	−0.564319	−	0.825556i	$-0.690861\pi$
$317$	−6370.07	−1.12864	−0.564319	+	0.825556i	$0.690861\pi$
$318$	0	0
$319$	8340.71	1.46392
$320$	0	0
$321$	−7586.81	−1.31917
$322$	0	0
$323$	2075.59	0.357551
$324$	0	0
$325$	−413.047	−0.0704976
$326$	0	0
$327$	−9014.96	−1.52455
$328$	0	0
$329$	−628.703	−0.105354
$330$	0	0
$331$	11099.0	1.84307	0.921534	−	0.388298i	$-0.126937\pi$
$331$	11099.0	1.84307	0.921534	+	0.388298i	$0.126937\pi$
$332$	0	0
$333$	−233.190	−0.0383746
$334$	0	0
$335$	521.136	0.0849931
$336$	0	0
$337$	1215.94	0.196547	0.0982734	−	0.995159i	$-0.468668\pi$
$337$	1215.94	0.196547	0.0982734	+	0.995159i	$0.468668\pi$
$338$	0	0
$339$	−8972.40	−1.43750
$340$	0	0
$341$	634.923	0.100830
$342$	0	0
$343$	−1478.55	−0.232753
$344$	0	0
$345$	2415.36	0.376924
$346$	0	0
$347$	−5034.11	−0.778805	−0.389403	−	0.921068i	$-0.627319\pi$
$347$	−5034.11	−0.778805	−0.389403	+	0.921068i	$0.627319\pi$
$348$	0	0
$349$	−3706.80	−0.568539	−0.284270	−	0.958744i	$-0.591751\pi$
$349$	−3706.80	−0.568539	−0.284270	+	0.958744i	$0.591751\pi$
$350$	0	0
$351$	1394.47	0.212055
$352$	0	0
$353$	6140.83	0.925901	0.462951	−	0.886384i	$-0.346791\pi$
$353$	6140.83	0.925901	0.462951	+	0.886384i	$0.346791\pi$
$354$	0	0
$355$	−3075.08	−0.459742
$356$	0	0
$357$	−199.572	−0.0295867
$358$	0	0
$359$	−3241.70	−0.476575	−0.238288	−	0.971195i	$-0.576586\pi$
$359$	−3241.70	−0.476575	−0.238288	+	0.971195i	$0.576586\pi$
$360$	0	0
$361$	8047.86	1.17333
$362$	0	0
$363$	−7363.92	−1.06475
$364$	0	0
$365$	−325.460	−0.0466722
$366$	0	0
$367$	10587.7	1.50592	0.752959	−	0.658067i	$-0.228626\pi$
$367$	10587.7	1.50592	0.752959	+	0.658067i	$0.228626\pi$
$368$	0	0
$369$	256.310	0.0361598
$370$	0	0
$371$	971.281	0.135920
$372$	0	0
$373$	−11693.8	−1.62327	−0.811636	−	0.584164i	$-0.801422\pi$
$373$	−11693.8	−1.62327	−0.811636	+	0.584164i	$0.801422\pi$
$374$	0	0
$375$	−8228.27	−1.13308
$376$	0	0
$377$	1675.04	0.228829
$378$	0	0
$379$	3260.51	0.441903	0.220952	−	0.975285i	$-0.429084\pi$
$379$	3260.51	0.441903	0.220952	+	0.975285i	$0.429084\pi$
$380$	0	0
$381$	−667.073	−0.0896987
$382$	0	0
$383$	−5031.37	−0.671256	−0.335628	−	0.941995i	$-0.608948\pi$
$383$	−5031.37	−0.671256	−0.335628	+	0.941995i	$0.608948\pi$
$384$	0	0
$385$	−1040.40	−0.137724
$386$	0	0
$387$	172.162	0.0226137
$388$	0	0
$389$	9833.16	1.28165	0.640824	−	0.767688i	$-0.278593\pi$
$389$	9833.16	1.28165	0.640824	+	0.767688i	$0.278593\pi$
$390$	0	0
$391$	821.587	0.106265
$392$	0	0
$393$	−1403.91	−0.180199
$394$	0	0
$395$	−11698.3	−1.49014
$396$	0	0
$397$	−13696.1	−1.73146	−0.865730	−	0.500512i	$-0.833145\pi$
$397$	−13696.1	−1.73146	−0.865730	+	0.500512i	$0.833145\pi$
$398$	0	0
$399$	−1433.32	−0.179839
$400$	0	0
$401$	−11471.4	−1.42856	−0.714279	−	0.699861i	$-0.753245\pi$
$401$	−11471.4	−1.42856	−0.714279	+	0.699861i	$0.753245\pi$
$402$	0	0
$403$	127.509	0.0157610
$404$	0	0
$405$	7252.18	0.889786
$406$	0	0
$407$	−5350.91	−0.651682
$408$	0	0
$409$	−5132.03	−0.620446	−0.310223	−	0.950664i	$-0.600404\pi$
$409$	−5132.03	−0.620446	−0.310223	+	0.950664i	$0.600404\pi$
$410$	0	0
$411$	−16390.3	−1.96708
$412$	0	0
$413$	1041.94	0.124141
$414$	0	0
$415$	4355.69	0.515210
$416$	0	0
$417$	−7743.49	−0.909353
$418$	0	0
$419$	−10083.4	−1.17567	−0.587835	−	0.808981i	$-0.700019\pi$
$419$	−10083.4	−1.17567	−0.587835	+	0.808981i	$0.700019\pi$
$420$	0	0
$421$	9537.86	1.10415	0.552075	−	0.833795i	$-0.313836\pi$
$421$	9537.86	1.10415	0.552075	+	0.833795i	$0.313836\pi$
$422$	0	0
$423$	−655.070	−0.0752969
$424$	0	0
$425$	−673.850	−0.0769095
$426$	0	0
$427$	−668.592	−0.0757739
$428$	0	0
$429$	−2924.79	−0.329162
$430$	0	0
$431$	−9160.34	−1.02375	−0.511877	−	0.859059i	$-0.671050\pi$
$431$	−9160.34	−1.02375	−0.511877	+	0.859059i	$0.671050\pi$
$432$	0	0
$433$	8890.93	0.986769	0.493384	−	0.869811i	$-0.335760\pi$
$433$	8890.93	0.986769	0.493384	+	0.869811i	$0.335760\pi$
$434$	0	0
$435$	8033.73	0.885489
$436$	0	0
$437$	5900.62	0.645915
$438$	0	0
$439$	2321.21	0.252358	0.126179	−	0.992007i	$-0.459729\pi$
$439$	2321.21	0.252358	0.126179	+	0.992007i	$0.459729\pi$
$440$	0	0
$441$	−764.954	−0.0825995
$442$	0	0
$443$	3588.54	0.384868	0.192434	−	0.981310i	$-0.438362\pi$
$443$	3588.54	0.384868	0.192434	+	0.981310i	$0.438362\pi$
$444$	0	0
$445$	−5168.19	−0.550552
$446$	0	0
$447$	−4780.32	−0.505820
$448$	0	0
$449$	7239.39	0.760908	0.380454	−	0.924800i	$-0.375768\pi$
$449$	7239.39	0.760908	0.380454	+	0.924800i	$0.375768\pi$
$450$	0	0
$451$	5881.44	0.614072
$452$	0	0
$453$	9210.18	0.955258
$454$	0	0
$455$	−208.940	−0.0215280
$456$	0	0
$457$	9086.74	0.930109	0.465055	−	0.885282i	$-0.346035\pi$
$457$	9086.74	0.930109	0.465055	+	0.885282i	$0.346035\pi$
$458$	0	0
$459$	2274.96	0.231342
$460$	0	0
$461$	1356.81	0.137078	0.0685392	−	0.997648i	$-0.478166\pi$
$461$	1356.81	0.137078	0.0685392	+	0.997648i	$0.478166\pi$
$462$	0	0
$463$	2926.20	0.293719	0.146860	−	0.989157i	$-0.453083\pi$
$463$	2926.20	0.293719	0.146860	+	0.989157i	$0.453083\pi$
$464$	0	0
$465$	611.555	0.0609896
$466$	0	0
$467$	−7184.18	−0.711872	−0.355936	−	0.934510i	$-0.615838\pi$
$467$	−7184.18	−0.711872	−0.355936	+	0.934510i	$0.615838\pi$
$468$	0	0
$469$	−122.412	−0.0120521
$470$	0	0
$471$	19962.3	1.95290
$472$	0	0
$473$	3950.53	0.384029
$474$	0	0
$475$	−4839.58	−0.467484
$476$	0	0
$477$	1012.01	0.0971424
$478$	0	0
$479$	−9445.28	−0.900973	−0.450486	−	0.892783i	$-0.648749\pi$
$479$	−9445.28	−0.900973	−0.450486	+	0.892783i	$0.648749\pi$
$480$	0	0
$481$	−1074.60	−0.101866
$482$	0	0
$483$	−567.355	−0.0534484
$484$	0	0
$485$	−8032.42	−0.752028
$486$	0	0
$487$	14596.7	1.35819	0.679095	−	0.734050i	$-0.262372\pi$
$487$	14596.7	1.35819	0.679095	+	0.734050i	$0.262372\pi$
$488$	0	0
$489$	11249.4	1.04031
$490$	0	0
$491$	−10795.2	−0.992221	−0.496110	−	0.868260i	$-0.665239\pi$
$491$	−10795.2	−0.992221	−0.496110	+	0.868260i	$0.665239\pi$
$492$	0	0
$493$	2732.67	0.249642
$494$	0	0
$495$	−1084.03	−0.0984315
$496$	0	0
$497$	722.319	0.0651920
$498$	0	0
$499$	4488.29	0.402652	0.201326	−	0.979524i	$-0.435475\pi$
$499$	4488.29	0.402652	0.201326	+	0.979524i	$0.435475\pi$
$500$	0	0
$501$	14614.7	1.30326
$502$	0	0
$503$	7038.93	0.623957	0.311979	−	0.950089i	$-0.399008\pi$
$503$	7038.93	0.623957	0.311979	+	0.950089i	$0.399008\pi$
$504$	0	0
$505$	6681.52	0.588760
$506$	0	0
$507$	11297.0	0.989581
$508$	0	0
$509$	−13014.6	−1.13333	−0.566663	−	0.823950i	$-0.691766\pi$
$509$	−13014.6	−1.13333	−0.566663	+	0.823950i	$0.691766\pi$
$510$	0	0
$511$	76.4486	0.00661817
$512$	0	0
$513$	16338.7	1.40618
$514$	0	0
$515$	14188.8	1.21405
$516$	0	0
$517$	−15031.6	−1.27870
$518$	0	0
$519$	20490.3	1.73299
$520$	0	0
$521$	5330.99	0.448282	0.224141	−	0.974557i	$-0.428042\pi$
$521$	5330.99	0.448282	0.224141	+	0.974557i	$0.428042\pi$
$522$	0	0
$523$	−3742.70	−0.312919	−0.156460	−	0.987684i	$-0.550008\pi$
$523$	−3742.70	−0.312919	−0.156460	+	0.987684i	$0.550008\pi$
$524$	0	0
$525$	465.334	0.0386835
$526$	0	0
$527$	208.021	0.0171945
$528$	0	0
$529$	−9831.34	−0.808033
$530$	0	0
$531$	1085.63	0.0887240
$532$	0	0
$533$	1181.15	0.0959873
$534$	0	0
$535$	−12958.2	−1.04716
$536$	0	0
$537$	−22423.6	−1.80195
$538$	0	0
$539$	−17553.1	−1.40272
$540$	0	0
$541$	−21570.3	−1.71420	−0.857099	−	0.515152i	$-0.827736\pi$
$541$	−21570.3	−1.71420	−0.857099	+	0.515152i	$0.827736\pi$
$542$	0	0
$543$	17880.5	1.41312
$544$	0	0
$545$	−15397.5	−1.21019
$546$	0	0
$547$	6410.92	0.501117	0.250558	−	0.968101i	$-0.419386\pi$
$547$	6410.92	0.501117	0.250558	+	0.968101i	$0.419386\pi$
$548$	0	0
$549$	−696.631	−0.0541558
$550$	0	0
$551$	19626.0	1.51742
$552$	0	0
$553$	2747.86	0.211304
$554$	0	0
$555$	−5153.97	−0.394187
$556$	0	0
$557$	−13560.5	−1.03156	−0.515779	−	0.856722i	$-0.672497\pi$
$557$	−13560.5	−1.03156	−0.515779	+	0.856722i	$0.672497\pi$
$558$	0	0
$559$	793.372	0.0600287
$560$	0	0
$561$	−4771.55	−0.359100
$562$	0	0
$563$	17476.4	1.30824	0.654121	−	0.756390i	$-0.273039\pi$
$563$	17476.4	1.30824	0.654121	+	0.756390i	$0.273039\pi$
$564$	0	0
$565$	−15324.8	−1.14109
$566$	0	0
$567$	−1703.49	−0.126173
$568$	0	0
$569$	9653.23	0.711221	0.355610	−	0.934634i	$-0.384273\pi$
$569$	9653.23	0.711221	0.355610	+	0.934634i	$0.384273\pi$
$570$	0	0
$571$	−4717.78	−0.345767	−0.172884	−	0.984942i	$-0.555308\pi$
$571$	−4717.78	−0.345767	−0.172884	+	0.984942i	$0.555308\pi$
$572$	0	0
$573$	13198.8	0.962281
$574$	0	0
$575$	−1915.66	−0.138937
$576$	0	0
$577$	−2393.69	−0.172705	−0.0863523	−	0.996265i	$-0.527521\pi$
$577$	−2393.69	−0.172705	−0.0863523	+	0.996265i	$0.527521\pi$
$578$	0	0
$579$	11538.6	0.828197
$580$	0	0
$581$	−1023.13	−0.0730575
$582$	0	0
$583$	23222.3	1.64969
$584$	0	0
$585$	−217.702	−0.0153861
$586$	0	0
$587$	−13084.7	−0.920037	−0.460018	−	0.887909i	$-0.652157\pi$
$587$	−13084.7	−0.920037	−0.460018	+	0.887909i	$0.652157\pi$
$588$	0	0
$589$	1494.00	0.104515
$590$	0	0
$591$	15332.1	1.06714
$592$	0	0
$593$	−8243.95	−0.570891	−0.285445	−	0.958395i	$-0.592142\pi$
$593$	−8243.95	−0.570891	−0.285445	+	0.958395i	$0.592142\pi$
$594$	0	0
$595$	−340.867	−0.0234860
$596$	0	0
$597$	−2210.85	−0.151565
$598$	0	0
$599$	−2036.48	−0.138912	−0.0694559	−	0.997585i	$-0.522126\pi$
$599$	−2036.48	−0.138912	−0.0694559	+	0.997585i	$0.522126\pi$
$600$	0	0
$601$	16389.8	1.11240	0.556201	−	0.831048i	$-0.312258\pi$
$601$	16389.8	1.11240	0.556201	+	0.831048i	$0.312258\pi$
$602$	0	0
$603$	−127.546	−0.00861369
$604$	0	0
$605$	−12577.5	−0.845204
$606$	0	0
$607$	−19636.7	−1.31306	−0.656530	−	0.754300i	$-0.727976\pi$
$607$	−19636.7	−1.31306	−0.656530	+	0.754300i	$0.727976\pi$
$608$	0	0
$609$	−1887.08	−0.125564
$610$	0	0
$611$	−3018.74	−0.199878
$612$	0	0
$613$	19749.2	1.30125	0.650623	−	0.759401i	$-0.274508\pi$
$613$	19749.2	1.30125	0.650623	+	0.759401i	$0.274508\pi$
$614$	0	0
$615$	5664.97	0.371437
$616$	0	0
$617$	12500.4	0.815638	0.407819	−	0.913063i	$-0.366289\pi$
$617$	12500.4	0.815638	0.407819	+	0.913063i	$0.366289\pi$
$618$	0	0
$619$	24897.0	1.61663	0.808315	−	0.588750i	$-0.200380\pi$
$619$	24897.0	1.61663	0.808315	+	0.588750i	$0.200380\pi$
$620$	0	0
$621$	6467.39	0.417918
$622$	0	0
$623$	1213.98	0.0780691
$624$	0	0
$625$	−9099.03	−0.582338
$626$	0	0
$627$	−34269.2	−2.18274
$628$	0	0
$629$	−1753.12	−0.111131
$630$	0	0
$631$	−9456.81	−0.596624	−0.298312	−	0.954468i	$-0.596424\pi$
$631$	−9456.81	−0.596624	−0.298312	+	0.954468i	$0.596424\pi$
$632$	0	0
$633$	−8060.81	−0.506143
$634$	0	0
$635$	−1139.35	−0.0712030
$636$	0	0
$637$	−3525.12	−0.219263
$638$	0	0
$639$	752.612	0.0465929
$640$	0	0
$641$	−15145.7	−0.933258	−0.466629	−	0.884453i	$-0.654532\pi$
$641$	−15145.7	−0.933258	−0.466629	+	0.884453i	$0.654532\pi$
$642$	0	0
$643$	−8980.82	−0.550807	−0.275404	−	0.961329i	$-0.588811\pi$
$643$	−8980.82	−0.550807	−0.275404	+	0.961329i	$0.588811\pi$
$644$	0	0
$645$	3805.13	0.232290
$646$	0	0
$647$	9379.56	0.569936	0.284968	−	0.958537i	$-0.408017\pi$
$647$	9379.56	0.569936	0.284968	+	0.958537i	$0.408017\pi$
$648$	0	0
$649$	24911.6	1.50672
$650$	0	0
$651$	−143.651	−0.00864841
$652$	0	0
$653$	−27950.4	−1.67501	−0.837506	−	0.546429i	$-0.815987\pi$
$653$	−27950.4	−1.67501	−0.837506	+	0.546429i	$0.815987\pi$
$654$	0	0
$655$	−2397.87	−0.143042
$656$	0	0
$657$	79.6547	0.00473003
$658$	0	0
$659$	−33455.6	−1.97761	−0.988805	−	0.149216i	$-0.952325\pi$
$659$	−33455.6	−1.97761	−0.988805	+	0.149216i	$0.952325\pi$
$660$	0	0
$661$	24836.2	1.46144	0.730722	−	0.682675i	$-0.239183\pi$
$661$	24836.2	1.46144	0.730722	+	0.682675i	$0.239183\pi$
$662$	0	0
$663$	−958.253	−0.0561319
$664$	0	0
$665$	−2448.10	−0.142757
$666$	0	0
$667$	7768.61	0.450977
$668$	0	0
$669$	18250.0	1.05468
$670$	0	0
$671$	−15985.3	−0.919681
$672$	0	0
$673$	−27067.2	−1.55032	−0.775158	−	0.631767i	$-0.782330\pi$
$673$	−27067.2	−1.55032	−0.775158	+	0.631767i	$0.782330\pi$
$674$	0	0
$675$	−5304.43	−0.302470
$676$	0	0
$677$	−15455.5	−0.877406	−0.438703	−	0.898632i	$-0.644562\pi$
$677$	−15455.5	−0.877406	−0.438703	+	0.898632i	$0.644562\pi$
$678$	0	0
$679$	1886.77	0.106639
$680$	0	0
$681$	−1239.10	−0.0697243
$682$	0	0
$683$	−26737.7	−1.49793	−0.748966	−	0.662608i	$-0.769449\pi$
$683$	−26737.7	−1.49793	−0.748966	+	0.662608i	$0.769449\pi$
$684$	0	0
$685$	−27994.4	−1.56148
$686$	0	0
$687$	471.721	0.0261969
$688$	0	0
$689$	4663.64	0.257867
$690$	0	0
$691$	26627.1	1.46591	0.732953	−	0.680279i	$-0.238142\pi$
$691$	26627.1	1.46591	0.732953	+	0.680279i	$0.238142\pi$
$692$	0	0
$693$	254.633	0.0139577
$694$	0	0
$695$	−13225.8	−0.721846
$696$	0	0
$697$	1926.94	0.104718
$698$	0	0
$699$	−17829.9	−0.964791
$700$	0	0
$701$	8597.61	0.463234	0.231617	−	0.972807i	$-0.425598\pi$
$701$	8597.61	0.463234	0.231617	+	0.972807i	$0.425598\pi$
$702$	0	0
$703$	−12590.9	−0.675497
$704$	0	0
$705$	−14478.4	−0.773456
$706$	0	0
$707$	−1569.45	−0.0834870
$708$	0	0
$709$	30214.9	1.60049	0.800244	−	0.599675i	$-0.204704\pi$
$709$	30214.9	1.60049	0.800244	+	0.599675i	$0.204704\pi$
$710$	0	0
$711$	2863.10	0.151019
$712$	0	0
$713$	591.373	0.0310619
$714$	0	0
$715$	−4995.52	−0.261289
$716$	0	0
$717$	−10061.4	−0.524058
$718$	0	0
$719$	10500.2	0.544631	0.272316	−	0.962208i	$-0.412211\pi$
$719$	10500.2	0.544631	0.272316	+	0.962208i	$0.412211\pi$
$720$	0	0
$721$	−3332.87	−0.172153
$722$	0	0
$723$	20333.5	1.04594
$724$	0	0
$725$	−6371.67	−0.326397
$726$	0	0
$727$	−31702.3	−1.61729	−0.808647	−	0.588294i	$-0.799800\pi$
$727$	−31702.3	−1.61729	−0.808647	+	0.588294i	$0.799800\pi$
$728$	0	0
$729$	17770.0	0.902810
$730$	0	0
$731$	1294.32	0.0654885
$732$	0	0
$733$	35987.6	1.81341	0.906706	−	0.421763i	$-0.138589\pi$
$733$	35987.6	1.81341	0.906706	+	0.421763i	$0.138589\pi$
$734$	0	0
$735$	−16907.0	−0.848470
$736$	0	0
$737$	−2926.73	−0.146279
$738$	0	0
$739$	−16160.2	−0.804414	−0.402207	−	0.915549i	$-0.631757\pi$
$739$	−16160.2	−0.804414	−0.402207	+	0.915549i	$0.631757\pi$
$740$	0	0
$741$	−6882.15	−0.341191
$742$	0	0
$743$	25595.1	1.26379	0.631894	−	0.775055i	$-0.282278\pi$
$743$	25595.1	1.26379	0.631894	+	0.775055i	$0.282278\pi$
$744$	0	0
$745$	−8164.75	−0.401521
$746$	0	0
$747$	−1066.03	−0.0522144
$748$	0	0
$749$	3043.81	0.148489
$750$	0	0
$751$	−12606.2	−0.612526	−0.306263	−	0.951947i	$-0.599079\pi$
$751$	−12606.2	−0.612526	−0.306263	+	0.951947i	$0.599079\pi$
$752$	0	0
$753$	−35042.3	−1.69590
$754$	0	0
$755$	15730.9	0.758286
$756$	0	0
$757$	25396.6	1.21936	0.609679	−	0.792648i	$-0.291298\pi$
$757$	25396.6	1.21936	0.609679	+	0.792648i	$0.291298\pi$
$758$	0	0
$759$	−13564.8	−0.648712
$760$	0	0
$761$	−20050.4	−0.955093	−0.477546	−	0.878606i	$-0.658474\pi$
$761$	−20050.4	−0.955093	−0.477546	+	0.878606i	$0.658474\pi$
$762$	0	0
$763$	3616.78	0.171607
$764$	0	0
$765$	−355.162	−0.0167855
$766$	0	0
$767$	5002.90	0.235521
$768$	0	0
$769$	5669.82	0.265876	0.132938	−	0.991124i	$-0.457559\pi$
$769$	5669.82	0.265876	0.132938	+	0.991124i	$0.457559\pi$
$770$	0	0
$771$	37517.1	1.75246
$772$	0	0
$773$	18925.2	0.880587	0.440293	−	0.897854i	$-0.354874\pi$
$773$	18925.2	0.880587	0.440293	+	0.897854i	$0.354874\pi$
$774$	0	0
$775$	−485.033	−0.0224812
$776$	0	0
$777$	1210.64	0.0558962
$778$	0	0
$779$	13839.3	0.636512
$780$	0	0
$781$	17269.9	0.791248
$782$	0	0
$783$	21511.1	0.981795
$784$	0	0
$785$	34095.4	1.55021
$786$	0	0
$787$	1740.38	0.0788283	0.0394141	−	0.999223i	$-0.487451\pi$
$787$	1740.38	0.0788283	0.0394141	+	0.999223i	$0.487451\pi$
$788$	0	0
$789$	−1798.77	−0.0811633
$790$	0	0
$791$	3599.70	0.161809
$792$	0	0
$793$	−3210.27	−0.143758
$794$	0	0
$795$	22367.6	0.997856
$796$	0	0
$797$	22835.0	1.01488	0.507438	−	0.861688i	$-0.330593\pi$
$797$	22835.0	1.01488	0.507438	+	0.861688i	$0.330593\pi$
$798$	0	0
$799$	−4924.82	−0.218057
$800$	0	0
$801$	1264.89	0.0557962
$802$	0	0
$803$	1827.80	0.0803260
$804$	0	0
$805$	−969.037	−0.0424274
$806$	0	0
$807$	−22808.4	−0.994910
$808$	0	0
$809$	28228.2	1.22676	0.613381	−	0.789787i	$-0.289809\pi$
$809$	28228.2	1.22676	0.613381	+	0.789787i	$0.289809\pi$
$810$	0	0
$811$	−28563.2	−1.23673	−0.618366	−	0.785890i	$-0.712205\pi$
$811$	−28563.2	−1.23673	−0.618366	+	0.785890i	$0.712205\pi$
$812$	0	0
$813$	−15792.3	−0.681254
$814$	0	0
$815$	19213.8	0.825804
$816$	0	0
$817$	9295.76	0.398063
$818$	0	0
$819$	51.1370	0.00218177
$820$	0	0
$821$	−33261.4	−1.41393	−0.706963	−	0.707251i	$-0.749935\pi$
$821$	−33261.4	−1.41393	−0.706963	+	0.707251i	$0.749935\pi$
$822$	0	0
$823$	427.352	0.0181003	0.00905016	−	0.999959i	$-0.497119\pi$
$823$	427.352	0.0181003	0.00905016	+	0.999959i	$0.497119\pi$
$824$	0	0
$825$	11125.6	0.469509
$826$	0	0
$827$	6501.35	0.273367	0.136683	−	0.990615i	$-0.456356\pi$
$827$	6501.35	0.273367	0.136683	+	0.990615i	$0.456356\pi$
$828$	0	0
$829$	23991.9	1.00515	0.502577	−	0.864532i	$-0.332385\pi$
$829$	23991.9	1.00515	0.502577	+	0.864532i	$0.332385\pi$
$830$	0	0
$831$	−19272.2	−0.804509
$832$	0	0
$833$	−5750.93	−0.239205
$834$	0	0
$835$	24961.7	1.03453
$836$	0	0
$837$	1637.50	0.0676228
$838$	0	0
$839$	−6156.54	−0.253334	−0.126667	−	0.991945i	$-0.540428\pi$
$839$	−6156.54	−0.253334	−0.126667	+	0.991945i	$0.540428\pi$
$840$	0	0
$841$	1450.14	0.0594587
$842$	0	0
$843$	44537.3	1.81963
$844$	0	0
$845$	19295.2	0.785531
$846$	0	0
$847$	2954.39	0.119851
$848$	0	0
$849$	−36129.9	−1.46051
$850$	0	0
$851$	−4983.89	−0.200758
$852$	0	0
$853$	−27808.9	−1.11625	−0.558123	−	0.829758i	$-0.688478\pi$
$853$	−27808.9	−1.11625	−0.558123	+	0.829758i	$0.688478\pi$
$854$	0	0
$855$	−2550.77	−0.102029
$856$	0	0
$857$	21462.0	0.855460	0.427730	−	0.903907i	$-0.359313\pi$
$857$	21462.0	0.855460	0.427730	+	0.903907i	$0.359313\pi$
$858$	0	0
$859$	−2478.41	−0.0984427	−0.0492214	−	0.998788i	$-0.515674\pi$
$859$	−2478.41	−0.0984427	−0.0492214	+	0.998788i	$0.515674\pi$
$860$	0	0
$861$	−1330.67	−0.0526703
$862$	0	0
$863$	−32289.4	−1.27363	−0.636816	−	0.771016i	$-0.719749\pi$
$863$	−32289.4	−1.27363	−0.636816	+	0.771016i	$0.719749\pi$
$864$	0	0
$865$	34997.2	1.37565
$866$	0	0
$867$	−1563.31	−0.0612372
$868$	0	0
$869$	65698.4	2.56463
$870$	0	0
$871$	−587.765	−0.0228653
$872$	0	0
$873$	1965.90	0.0762149
$874$	0	0
$875$	3301.16	0.127542
$876$	0	0
$877$	−14557.9	−0.560531	−0.280265	−	0.959923i	$-0.590422\pi$
$877$	−14557.9	−0.560531	−0.280265	+	0.959923i	$0.590422\pi$
$878$	0	0
$879$	−32025.9	−1.22890
$880$	0	0
$881$	597.936	0.0228660	0.0114330	−	0.999935i	$-0.496361\pi$
$881$	597.936	0.0228660	0.0114330	+	0.999935i	$0.496361\pi$
$882$	0	0
$883$	39538.1	1.50686	0.753432	−	0.657525i	$-0.228397\pi$
$883$	39538.1	1.50686	0.753432	+	0.657525i	$0.228397\pi$
$884$	0	0
$885$	23994.7	0.911381
$886$	0	0
$887$	−6885.65	−0.260651	−0.130325	−	0.991471i	$-0.541602\pi$
$887$	−6885.65	−0.260651	−0.130325	+	0.991471i	$0.541602\pi$
$888$	0	0
$889$	267.628	0.0100967
$890$	0	0
$891$	−40728.7	−1.53138
$892$	0	0
$893$	−35370.0	−1.32543
$894$	0	0
$895$	−38299.2	−1.43039
$896$	0	0
$897$	−2724.18	−0.101402
$898$	0	0
$899$	1966.96	0.0729721
$900$	0	0
$901$	7608.33	0.281321
$902$	0	0
$903$	−893.804	−0.0329390
$904$	0	0
$905$	30539.6	1.12174
$906$	0	0
$907$	24863.6	0.910233	0.455116	−	0.890432i	$-0.349598\pi$
$907$	24863.6	0.910233	0.455116	+	0.890432i	$0.349598\pi$
$908$	0	0
$909$	−1635.27	−0.0596683
$910$	0	0
$911$	46604.6	1.69493	0.847464	−	0.530852i	$-0.178128\pi$
$911$	46604.6	1.69493	0.847464	+	0.530852i	$0.178128\pi$
$912$	0	0
$913$	−24461.8	−0.886713
$914$	0	0
$915$	−15397.0	−0.556293
$916$	0	0
$917$	563.246	0.0202836
$918$	0	0
$919$	−47914.7	−1.71987	−0.859936	−	0.510403i	$-0.829496\pi$
$919$	−47914.7	−1.71987	−0.859936	+	0.510403i	$0.829496\pi$
$920$	0	0
$921$	45188.7	1.61674
$922$	0	0
$923$	3468.25	0.123682
$924$	0	0
$925$	4087.69	0.145300
$926$	0	0
$927$	−3472.65	−0.123038
$928$	0	0
$929$	−19170.2	−0.677023	−0.338511	−	0.940962i	$-0.609923\pi$
$929$	−19170.2	−0.677023	−0.338511	+	0.940962i	$0.609923\pi$
$930$	0	0
$931$	−41303.1	−1.45398
$932$	0	0
$933$	−53325.4	−1.87116
$934$	0	0
$935$	−8149.76	−0.285054
$936$	0	0
$937$	−31908.3	−1.11248	−0.556242	−	0.831020i	$-0.687757\pi$
$937$	−31908.3	−1.11248	−0.556242	+	0.831020i	$0.687757\pi$
$938$	0	0
$939$	12142.6	0.422001
$940$	0	0
$941$	−34499.4	−1.19516	−0.597582	−	0.801808i	$-0.703872\pi$
$941$	−34499.4	−1.19516	−0.597582	+	0.801808i	$0.703872\pi$
$942$	0	0
$943$	5478.03	0.189172
$944$	0	0
$945$	−2683.24	−0.0923661
$946$	0	0
$947$	−7533.06	−0.258491	−0.129246	−	0.991613i	$-0.541256\pi$
$947$	−7533.06	−0.258491	−0.129246	+	0.991613i	$0.541256\pi$
$948$	0	0
$949$	367.071	0.0125560
$950$	0	0
$951$	34458.0	1.17495
$952$	0	0
$953$	−16653.9	−0.566080	−0.283040	−	0.959108i	$-0.591343\pi$
$953$	−16653.9	−0.566080	−0.283040	+	0.959108i	$0.591343\pi$
$954$	0	0
$955$	22543.4	0.763860
$956$	0	0
$957$	−45118.0	−1.52399
$958$	0	0
$959$	6575.72	0.221419
$960$	0	0
$961$	−29641.3	−0.994974
$962$	0	0
$963$	3171.46	0.106125
$964$	0	0
$965$	19707.8	0.657425
$966$	0	0
$967$	−12695.3	−0.422185	−0.211092	−	0.977466i	$-0.567702\pi$
$967$	−12695.3	−0.422185	−0.211092	+	0.977466i	$0.567702\pi$
$968$	0	0
$969$	−11227.6	−0.372223
$970$	0	0
$971$	544.395	0.0179923	0.00899613	−	0.999960i	$-0.497136\pi$
$971$	544.395	0.0179923	0.00899613	+	0.999960i	$0.497136\pi$
$972$	0	0
$973$	3106.67	0.102359
$974$	0	0
$975$	2234.32	0.0733903
$976$	0	0
$977$	16946.6	0.554934	0.277467	−	0.960735i	$-0.410505\pi$
$977$	16946.6	0.554934	0.277467	+	0.960735i	$0.410505\pi$
$978$	0	0
$979$	29024.9	0.947539
$980$	0	0
$981$	3768.46	0.122648
$982$	0	0
$983$	−11378.9	−0.369206	−0.184603	−	0.982813i	$-0.559100\pi$
$983$	−11378.9	−0.369206	−0.184603	+	0.982813i	$0.559100\pi$
$984$	0	0
$985$	26187.1	0.847097
$986$	0	0
$987$	3400.88	0.109677
$988$	0	0
$989$	3679.56	0.118305
$990$	0	0
$991$	−50637.8	−1.62317	−0.811585	−	0.584234i	$-0.801395\pi$
$991$	−50637.8	−1.62317	−0.811585	+	0.584234i	$0.801395\pi$
$992$	0	0
$993$	−60038.5	−1.91869
$994$	0	0
$995$	−3776.12	−0.120313
$996$	0	0
$997$	−18584.2	−0.590339	−0.295169	−	0.955445i	$-0.595376\pi$
$997$	−18584.2	−0.590339	−0.295169	+	0.955445i	$0.595376\pi$
$998$	0	0
$999$	−13800.3	−0.437059

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	136.4.a.c.1.1	✓	3
3.2	odd	2		1224.4.a.f.1.3		3
4.3	odd	2		272.4.a.g.1.3		3
8.3	odd	2		1088.4.a.z.1.1		3
8.5	even	2		1088.4.a.s.1.3		3
12.11	even	2		2448.4.a.bf.1.3		3
17.16	even	2		2312.4.a.c.1.3		3

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
136.4.a.c.1.1	✓	3	1.1	even	1	trivial
272.4.a.g.1.3		3	4.3	odd	2
1088.4.a.s.1.3		3	8.5	even	2
1088.4.a.z.1.1		3	8.3	odd	2
1224.4.a.f.1.3		3	3.2	odd	2
2312.4.a.c.1.3		3	17.16	even	2
2448.4.a.bf.1.3		3	12.11	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:	\( N \)	\(=\)	\( 136 = 2^{3} \cdot 17 \)
Weight:	\( k \)	\(=\)	\( 4 \)
Character orbit:	\([\chi]\)	\(=\)	136.a (trivial)

\(f(q)\)	\(=\)	\(q-5.40937 q^{3} -9.23914 q^{5} +2.17022 q^{7} +2.26124 q^{9} +O(q^{10})\)	\(q-5.40937 q^{3} -9.23914 q^{5} +2.17022 q^{7} +2.26124 q^{9} +51.8877 q^{11} +10.4204 q^{13} +49.9779 q^{15} +17.0000 q^{17} +122.094 q^{19} -11.7395 q^{21} +48.3286 q^{23} -39.6382 q^{25} +133.821 q^{27} +160.746 q^{29} +12.2365 q^{31} -280.679 q^{33} -20.0510 q^{35} -103.125 q^{37} -56.3678 q^{39} +113.350 q^{41} +76.1363 q^{43} -20.8919 q^{45} -289.695 q^{47} -338.290 q^{49} -91.9592 q^{51} +447.549 q^{53} -479.397 q^{55} -660.449 q^{57} +480.106 q^{59} -308.075 q^{61} +4.90739 q^{63} -96.2757 q^{65} -56.4052 q^{67} -261.427 q^{69} +332.832 q^{71} +35.2262 q^{73} +214.418 q^{75} +112.608 q^{77} +1266.17 q^{79} -784.940 q^{81} -471.438 q^{83} -157.065 q^{85} -869.532 q^{87} +559.380 q^{89} +22.6146 q^{91} -66.1917 q^{93} -1128.04 q^{95} +869.391 q^{97} +117.330 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 3 q + 8 q^{3} + 2 q^{5} + 12 q^{7} + 63 q^{9}+O(q^{10}) \) 3 * q + 8 * q^3 + 2 * q^5 + 12 * q^7 + 63 * q^9	\( 3 q + 8 q^{3} + 2 q^{5} + 12 q^{7} + 63 q^{9} + 72 q^{11} + 50 q^{13} + 64 q^{15} + 51 q^{17} + 124 q^{19} - 32 q^{21} + 60 q^{23} - 75 q^{25} + 512 q^{27} - 54 q^{29} + 300 q^{31} - 48 q^{33} + 248 q^{35} - 542 q^{37} + 320 q^{39} + 30 q^{41} + 52 q^{43} - 502 q^{45} + 16 q^{47} - 677 q^{49} + 136 q^{51} - 518 q^{53} - 608 q^{55} - 896 q^{57} + 132 q^{59} - 614 q^{61} - 852 q^{63} - 148 q^{65} - 332 q^{67} - 1280 q^{69} - 268 q^{71} - 578 q^{73} - 712 q^{75} - 128 q^{77} - 668 q^{79} + 2427 q^{81} + 876 q^{83} + 34 q^{85} - 2400 q^{87} + 1790 q^{89} - 168 q^{91} + 2592 q^{93} - 504 q^{95} + 838 q^{97} + 2040 q^{99}+O(q^{100}) \) 3 * q + 8 * q^3 + 2 * q^5 + 12 * q^7 + 63 * q^9 + 72 * q^11 + 50 * q^13 + 64 * q^15 + 51 * q^17 + 124 * q^19 - 32 * q^21 + 60 * q^23 - 75 * q^25 + 512 * q^27 - 54 * q^29 + 300 * q^31 - 48 * q^33 + 248 * q^35 - 542 * q^37 + 320 * q^39 + 30 * q^41 + 52 * q^43 - 502 * q^45 + 16 * q^47 - 677 * q^49 + 136 * q^51 - 518 * q^53 - 608 * q^55 - 896 * q^57 + 132 * q^59 - 614 * q^61 - 852 * q^63 - 148 * q^65 - 332 * q^67 - 1280 * q^69 - 268 * q^71 - 578 * q^73 - 712 * q^75 - 128 * q^77 - 668 * q^79 + 2427 * q^81 + 876 * q^83 + 34 * q^85 - 2400 * q^87 + 1790 * q^89 - 168 * q^91 + 2592 * q^93 - 504 * q^95 + 838 * q^97 + 2040 * q^99

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