LMFDB - Embedded newform 1936.4.a.bp.1.2

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [1936,4,Mod(1,1936)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(1936, 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("1936.1");

S:= CuspForms(chi, 4);

N := Newforms(S);

Level:	$ N $	$=$	$ 1936 = 2^{4} \cdot 11^{2} $
Weight:	$ k $	$=$	$ 4 $
Character orbit:	$[\chi]$	$=$	1936.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:	$114.227697771$
Analytic rank:	$0$
Dimension:	$4$
Coefficient field:	4.4.4166757.2
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{4} - 2x^{3} - 42x^{2} + 43x + 64 $ x^4 - 2x^3 - 42x^2 + 43*x + 64
Coefficient ring:	$\Z[a_1, \ldots, a_{7}]$
Coefficient ring index:	$ 2^{3} $
Twist minimal:	no (minimal twist has level 968)
Fricke sign:	$+1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.2
Root			$-5.95803$ of defining polynomial
Character	$\chi$	$=$	1936.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-0.556674 q^{3} +16.8608 q^{5} -23.8321 q^{7} -26.6901 q^{9} +O(q^{10})$	$q-0.556674 q^{3} +16.8608 q^{5} -23.8321 q^{7} -26.6901 q^{9} +6.17353 q^{13} -9.38598 q^{15} -124.260 q^{17} -99.5077 q^{19} +13.2667 q^{21} -170.384 q^{23} +159.287 q^{25} +29.8879 q^{27} +259.911 q^{29} +86.9640 q^{31} -401.829 q^{35} +96.4817 q^{37} -3.43664 q^{39} +89.7487 q^{41} -330.694 q^{43} -450.017 q^{45} +450.605 q^{47} +224.971 q^{49} +69.1721 q^{51} +339.389 q^{53} +55.3934 q^{57} +506.209 q^{59} +405.053 q^{61} +636.082 q^{63} +104.091 q^{65} -83.9133 q^{67} +94.8486 q^{69} +839.049 q^{71} +516.166 q^{73} -88.6709 q^{75} +407.382 q^{79} +703.995 q^{81} -1073.80 q^{83} -2095.12 q^{85} -144.686 q^{87} -25.4661 q^{89} -147.128 q^{91} -48.4106 q^{93} -1677.78 q^{95} +597.209 q^{97} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 4 q + 3 q^{3} + 9 q^{5} + 8 q^{7} + 31 q^{9}+O(q^{10}) $ 4 * q + 3 * q^3 + 9 * q^5 + 8 * q^7 + 31 * q^9	$ 4 q + 3 q^{3} + 9 q^{5} + 8 q^{7} + 31 q^{9} + 80 q^{13} + 47 q^{15} - 16 q^{17} + 8 q^{19} + 192 q^{21} + 71 q^{23} + 11 q^{25} + 81 q^{27} + 240 q^{29} + 115 q^{31} - 168 q^{35} + 315 q^{37} + 232 q^{39} - 592 q^{41} - 624 q^{43} + 108 q^{45} + 304 q^{47} + 36 q^{49} - 1256 q^{51} + 184 q^{53} + 592 q^{57} + 805 q^{59} + 240 q^{61} + 2288 q^{63} + 1424 q^{65} - 119 q^{67} + 1499 q^{69} - 723 q^{71} + 1936 q^{73} + 300 q^{75} - 1520 q^{79} + 700 q^{81} - 2016 q^{83} - 3040 q^{85} - 3152 q^{87} + 367 q^{89} + 2848 q^{91} - 1625 q^{93} - 3400 q^{95} + 881 q^{97}+O(q^{100}) $ 4 * q + 3 * q^3 + 9 * q^5 + 8 * q^7 + 31 * q^9 + 80 * q^13 + 47 * q^15 - 16 * q^17 + 8 * q^19 + 192 * q^21 + 71 * q^23 + 11 * q^25 + 81 * q^27 + 240 * q^29 + 115 * q^31 - 168 * q^35 + 315 * q^37 + 232 * q^39 - 592 * q^41 - 624 * q^43 + 108 * q^45 + 304 * q^47 + 36 * q^49 - 1256 * q^51 + 184 * q^53 + 592 * q^57 + 805 * q^59 + 240 * q^61 + 2288 * q^63 + 1424 * q^65 - 119 * q^67 + 1499 * q^69 - 723 * q^71 + 1936 * q^73 + 300 * q^75 - 1520 * q^79 + 700 * q^81 - 2016 * q^83 - 3040 * q^85 - 3152 * q^87 + 367 * q^89 + 2848 * q^91 - 1625 * q^93 - 3400 * q^95 + 881 * q^97

Display 100 coefficients 10 coefficients

Coefficient data

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

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

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

$2$	0	0
$2$	0	0
$3$	−0.556674	−0.107132	−0.0535660	−	0.998564i	$-0.517059\pi$
$3$	−0.556674	−0.107132	−0.0535660	+	0.998564i	$0.517059\pi$
$4$	0	0
$5$	16.8608	1.50808	0.754038	−	0.656830i	$-0.228103\pi$
$5$	16.8608	1.50808	0.754038	+	0.656830i	$0.228103\pi$
$6$	0	0
$7$	−23.8321	−1.28681	−0.643407	−	0.765524i	$-0.722480\pi$
$7$	−23.8321	−1.28681	−0.643407	+	0.765524i	$0.722480\pi$
$8$	0	0
$9$	−26.6901	−0.988523
$10$	0	0
$11$	0	0
$11$	0	0
$12$	0	0
$13$	6.17353	0.131710	0.0658549	−	0.997829i	$-0.479023\pi$
$13$	6.17353	0.131710	0.0658549	+	0.997829i	$0.479023\pi$
$14$	0	0
$15$	−9.38598	−0.161563
$16$	0	0
$17$	−124.260	−1.77279	−0.886393	−	0.462933i	$-0.846797\pi$
$17$	−124.260	−1.77279	−0.886393	+	0.462933i	$0.846797\pi$
$18$	0	0
$19$	−99.5077	−1.20151	−0.600754	−	0.799434i	$-0.705133\pi$
$19$	−99.5077	−1.20151	−0.600754	+	0.799434i	$0.705133\pi$
$20$	0	0
$21$	13.2667	0.137859
$22$	0	0
$23$	−170.384	−1.54468	−0.772339	−	0.635210i	$-0.780914\pi$
$23$	−170.384	−1.54468	−0.772339	+	0.635210i	$0.780914\pi$
$24$	0	0
$25$	159.287	1.27430
$26$	0	0
$27$	29.8879	0.213034
$28$	0	0
$29$	259.911	1.66428	0.832142	−	0.554562i	$-0.187114\pi$
$29$	259.911	1.66428	0.832142	+	0.554562i	$0.187114\pi$
$30$	0	0
$31$	86.9640	0.503845	0.251923	−	0.967747i	$-0.418937\pi$
$31$	86.9640	0.503845	0.251923	+	0.967747i	$0.418937\pi$
$32$	0	0
$33$	0	0
$34$	0	0
$35$	−401.829	−1.94061
$36$	0	0
$37$	96.4817	0.428689	0.214345	−	0.976758i	$-0.431238\pi$
$37$	96.4817	0.428689	0.214345	+	0.976758i	$0.431238\pi$
$38$	0	0
$39$	−3.43664	−0.0141103
$40$	0	0
$41$	89.7487	0.341863	0.170932	−	0.985283i	$-0.445322\pi$
$41$	89.7487	0.341863	0.170932	+	0.985283i	$0.445322\pi$
$42$	0	0
$43$	−330.694	−1.17280	−0.586400	−	0.810022i	$-0.699455\pi$
$43$	−330.694	−1.17280	−0.586400	+	0.810022i	$0.699455\pi$
$44$	0	0
$45$	−450.017	−1.49077
$46$	0	0
$47$	450.605	1.39846	0.699229	−	0.714898i	$-0.253527\pi$
$47$	450.605	1.39846	0.699229	+	0.714898i	$0.253527\pi$
$48$	0	0
$49$	224.971	0.655891
$50$	0	0
$51$	69.1721	0.189922
$52$	0	0
$53$	339.389	0.879597	0.439798	−	0.898097i	$-0.355050\pi$
$53$	339.389	0.879597	0.439798	+	0.898097i	$0.355050\pi$
$54$	0	0
$55$	0	0
$56$	0	0
$57$	55.3934	0.128720
$58$	0	0
$59$	506.209	1.11700	0.558498	−	0.829506i	$-0.311378\pi$
$59$	506.209	1.11700	0.558498	+	0.829506i	$0.311378\pi$
$60$	0	0
$61$	405.053	0.850192	0.425096	−	0.905148i	$-0.360240\pi$
$61$	405.053	0.850192	0.425096	+	0.905148i	$0.360240\pi$
$62$	0	0
$63$	636.082	1.27205
$64$	0	0
$65$	104.091	0.198629
$66$	0	0
$67$	−83.9133	−0.153009	−0.0765047	−	0.997069i	$-0.524376\pi$
$67$	−83.9133	−0.153009	−0.0765047	+	0.997069i	$0.524376\pi$
$68$	0	0
$69$	94.8486	0.165485
$70$	0	0
$71$	839.049	1.40249	0.701245	−	0.712920i	$-0.252628\pi$
$71$	839.049	1.40249	0.701245	+	0.712920i	$0.252628\pi$
$72$	0	0
$73$	516.166	0.827570	0.413785	−	0.910375i	$-0.364206\pi$
$73$	516.166	0.827570	0.413785	+	0.910375i	$0.364206\pi$
$74$	0	0
$75$	−88.6709	−0.136518
$76$	0	0
$77$	0	0
$78$	0	0
$79$	407.382	0.580178	0.290089	−	0.957000i	$-0.406315\pi$
$79$	407.382	0.580178	0.290089	+	0.957000i	$0.406315\pi$
$80$	0	0
$81$	703.995	0.965700
$82$	0	0
$83$	−1073.80	−1.42006	−0.710029	−	0.704172i	$-0.751318\pi$
$83$	−1073.80	−1.42006	−0.710029	+	0.704172i	$0.751318\pi$
$84$	0	0
$85$	−2095.12	−2.67350
$86$	0	0
$87$	−144.686	−0.178298
$88$	0	0
$89$	−25.4661	−0.0303304	−0.0151652	−	0.999885i	$-0.504827\pi$
$89$	−25.4661	−0.0303304	−0.0151652	+	0.999885i	$0.504827\pi$
$90$	0	0
$91$	−147.128	−0.169486
$92$	0	0
$93$	−48.4106	−0.0539780
$94$	0	0
$95$	−1677.78	−1.81197
$96$	0	0
$97$	597.209	0.625127	0.312564	−	0.949897i	$-0.398812\pi$
$97$	597.209	0.625127	0.312564	+	0.949897i	$0.398812\pi$
$98$	0	0
$99$	0	0
$100$	0	0
$101$	3.99320	0.00393404	0.00196702	−	0.999998i	$-0.499374\pi$
$101$	3.99320	0.00393404	0.00196702	+	0.999998i	$0.499374\pi$
$102$	0	0
$103$	476.702	0.456027	0.228014	−	0.973658i	$-0.426777\pi$
$103$	476.702	0.456027	0.228014	+	0.973658i	$0.426777\pi$
$104$	0	0
$105$	223.688	0.207902
$106$	0	0
$107$	1803.08	1.62907	0.814536	−	0.580113i	$-0.196992\pi$
$107$	1803.08	1.62907	0.814536	+	0.580113i	$0.196992\pi$
$108$	0	0
$109$	1363.21	1.19791	0.598955	−	0.800783i	$-0.295583\pi$
$109$	1363.21	1.19791	0.598955	+	0.800783i	$0.295583\pi$
$110$	0	0
$111$	−53.7089	−0.0459263
$112$	0	0
$113$	608.692	0.506734	0.253367	−	0.967370i	$-0.418462\pi$
$113$	608.692	0.506734	0.253367	+	0.967370i	$0.418462\pi$
$114$	0	0
$115$	−2872.82	−2.32949
$116$	0	0
$117$	−164.772	−0.130198
$118$	0	0
$119$	2961.37	2.28125
$120$	0	0
$121$	0	0
$122$	0	0
$123$	−49.9608	−0.0366245
$124$	0	0
$125$	578.105	0.413658
$126$	0	0
$127$	902.462	0.630556	0.315278	−	0.948999i	$-0.397902\pi$
$127$	902.462	0.630556	0.315278	+	0.948999i	$0.397902\pi$
$128$	0	0
$129$	184.089	0.125644
$130$	0	0
$131$	−734.925	−0.490158	−0.245079	−	0.969503i	$-0.578814\pi$
$131$	−734.925	−0.490158	−0.245079	+	0.969503i	$0.578814\pi$
$132$	0	0
$133$	2371.48	1.54612
$134$	0	0
$135$	503.934	0.321272
$136$	0	0
$137$	−2197.03	−1.37011	−0.685056	−	0.728490i	$-0.740222\pi$
$137$	−2197.03	−1.37011	−0.685056	+	0.728490i	$0.740222\pi$
$138$	0	0
$139$	−2277.60	−1.38981	−0.694906	−	0.719100i	$-0.744554\pi$
$139$	−2277.60	−1.38981	−0.694906	+	0.719100i	$0.744554\pi$
$140$	0	0
$141$	−250.840	−0.149820
$142$	0	0
$143$	0	0
$144$	0	0
$145$	4382.31	2.50987
$146$	0	0
$147$	−125.235	−0.0702670
$148$	0	0
$149$	2534.86	1.39372	0.696858	−	0.717209i	$-0.254581\pi$
$149$	2534.86	1.39372	0.696858	+	0.717209i	$0.254581\pi$
$150$	0	0
$151$	665.753	0.358796	0.179398	−	0.983777i	$-0.442585\pi$
$151$	665.753	0.358796	0.179398	+	0.983777i	$0.442585\pi$
$152$	0	0
$153$	3316.50	1.75244
$154$	0	0
$155$	1466.28	0.759837
$156$	0	0
$157$	1485.07	0.754913	0.377456	−	0.926027i	$-0.376799\pi$
$157$	1485.07	0.754913	0.377456	+	0.926027i	$0.376799\pi$
$158$	0	0
$159$	−188.929	−0.0942330
$160$	0	0
$161$	4060.63	1.98772
$162$	0	0
$163$	−3846.90	−1.84855	−0.924273	−	0.381733i	$-0.875327\pi$
$163$	−3846.90	−1.84855	−0.924273	+	0.381733i	$0.875327\pi$
$164$	0	0
$165$	0	0
$166$	0	0
$167$	−3654.90	−1.69356	−0.846780	−	0.531944i	$-0.821462\pi$
$167$	−3654.90	−1.69356	−0.846780	+	0.531944i	$0.821462\pi$
$168$	0	0
$169$	−2158.89	−0.982653
$170$	0	0
$171$	2655.87	1.18772
$172$	0	0
$173$	−144.568	−0.0635335	−0.0317667	−	0.999495i	$-0.510113\pi$
$173$	−144.568	−0.0635335	−0.0317667	+	0.999495i	$0.510113\pi$
$174$	0	0
$175$	−3796.15	−1.63978
$176$	0	0
$177$	−281.793	−0.119666
$178$	0	0
$179$	1647.27	0.687838	0.343919	−	0.938999i	$-0.388245\pi$
$179$	1647.27	0.687838	0.343919	+	0.938999i	$0.388245\pi$
$180$	0	0
$181$	−3518.17	−1.44477	−0.722385	−	0.691491i	$-0.756954\pi$
$181$	−3518.17	−1.44477	−0.722385	+	0.691491i	$0.756954\pi$
$182$	0	0
$183$	−225.483	−0.0910828
$184$	0	0
$185$	1626.76	0.646496
$186$	0	0
$187$	0	0
$188$	0	0
$189$	−712.293	−0.274136
$190$	0	0
$191$	−3495.82	−1.32434	−0.662169	−	0.749354i	$-0.730364\pi$
$191$	−3495.82	−1.32434	−0.662169	+	0.749354i	$0.730364\pi$
$192$	0	0
$193$	−4653.87	−1.73571	−0.867857	−	0.496814i	$-0.834503\pi$
$193$	−4653.87	−1.73571	−0.867857	+	0.496814i	$0.834503\pi$
$194$	0	0
$195$	−57.9446	−0.0212795
$196$	0	0
$197$	−2401.27	−0.868442	−0.434221	−	0.900806i	$-0.642976\pi$
$197$	−2401.27	−0.868442	−0.434221	+	0.900806i	$0.642976\pi$
$198$	0	0
$199$	3715.90	1.32368	0.661842	−	0.749644i	$-0.269775\pi$
$199$	3715.90	1.32368	0.661842	+	0.749644i	$0.269775\pi$
$200$	0	0
$201$	46.7123	0.0163922
$202$	0	0
$203$	−6194.23	−2.14163
$204$	0	0
$205$	1513.24	0.515556
$206$	0	0
$207$	4547.58	1.52695
$208$	0	0
$209$	0	0
$210$	0	0
$211$	−36.1850	−0.0118061	−0.00590303	−	0.999983i	$-0.501879\pi$
$211$	−36.1850	−0.0118061	−0.00590303	+	0.999983i	$0.501879\pi$
$212$	0	0
$213$	−467.077	−0.150252
$214$	0	0
$215$	−5575.77	−1.76867
$216$	0	0
$217$	−2072.54	−0.648355
$218$	0	0
$219$	−287.336	−0.0886593
$220$	0	0
$221$	−767.120	−0.233493
$222$	0	0
$223$	5644.43	1.69497	0.847486	−	0.530818i	$-0.178115\pi$
$223$	5644.43	1.69497	0.847486	+	0.530818i	$0.178115\pi$
$224$	0	0
$225$	−4251.38	−1.25967
$226$	0	0
$227$	−3926.52	−1.14807	−0.574036	−	0.818830i	$-0.694623\pi$
$227$	−3926.52	−1.14807	−0.574036	+	0.818830i	$0.694623\pi$
$228$	0	0
$229$	3510.09	1.01290	0.506448	−	0.862270i	$-0.330958\pi$
$229$	3510.09	1.01290	0.506448	+	0.862270i	$0.330958\pi$
$230$	0	0
$231$	0	0
$232$	0	0
$233$	−2621.90	−0.737194	−0.368597	−	0.929589i	$-0.620162\pi$
$233$	−2621.90	−0.737194	−0.368597	+	0.929589i	$0.620162\pi$
$234$	0	0
$235$	7597.57	2.10898
$236$	0	0
$237$	−226.779	−0.0621557
$238$	0	0
$239$	1445.38	0.391188	0.195594	−	0.980685i	$-0.437336\pi$
$239$	1445.38	0.391188	0.195594	+	0.980685i	$0.437336\pi$
$240$	0	0
$241$	960.689	0.256778	0.128389	−	0.991724i	$-0.459019\pi$
$241$	960.689	0.256778	0.128389	+	0.991724i	$0.459019\pi$
$242$	0	0
$243$	−1198.87	−0.316492
$244$	0	0
$245$	3793.19	0.989134
$246$	0	0
$247$	−614.314	−0.158250
$248$	0	0
$249$	597.757	0.152134
$250$	0	0
$251$	450.517	0.113292	0.0566462	−	0.998394i	$-0.481959\pi$
$251$	450.517	0.113292	0.0566462	+	0.998394i	$0.481959\pi$
$252$	0	0
$253$	0	0
$254$	0	0
$255$	1166.30	0.286417
$256$	0	0
$257$	5518.27	1.33938	0.669690	−	0.742641i	$-0.266427\pi$
$257$	5518.27	1.33938	0.669690	+	0.742641i	$0.266427\pi$
$258$	0	0
$259$	−2299.37	−0.551643
$260$	0	0
$261$	−6937.05	−1.64518
$262$	0	0
$263$	3727.46	0.873936	0.436968	−	0.899477i	$-0.356052\pi$
$263$	3727.46	0.873936	0.436968	+	0.899477i	$0.356052\pi$
$264$	0	0
$265$	5722.37	1.32650
$266$	0	0
$267$	14.1763	0.00324935
$268$	0	0
$269$	−1349.68	−0.305917	−0.152958	−	0.988233i	$-0.548880\pi$
$269$	−1349.68	−0.305917	−0.152958	+	0.988233i	$0.548880\pi$
$270$	0	0
$271$	6064.42	1.35936	0.679682	−	0.733507i	$-0.262118\pi$
$271$	6064.42	1.35936	0.679682	+	0.733507i	$0.262118\pi$
$272$	0	0
$273$	81.9026	0.0181574
$274$	0	0
$275$	0	0
$276$	0	0
$277$	−4460.37	−0.967501	−0.483751	−	0.875206i	$-0.660726\pi$
$277$	−4460.37	−0.967501	−0.483751	+	0.875206i	$0.660726\pi$
$278$	0	0
$279$	−2321.08	−0.498063
$280$	0	0
$281$	5972.61	1.26796	0.633979	−	0.773350i	$-0.281420\pi$
$281$	5972.61	1.26796	0.633979	+	0.773350i	$0.281420\pi$
$282$	0	0
$283$	1484.03	0.311719	0.155860	−	0.987779i	$-0.450185\pi$
$283$	1484.03	0.311719	0.155860	+	0.987779i	$0.450185\pi$
$284$	0	0
$285$	933.977	0.194119
$286$	0	0
$287$	−2138.90	−0.439915
$288$	0	0
$289$	10527.4	2.14277
$290$	0	0
$291$	−332.451	−0.0669711
$292$	0	0
$293$	−3419.84	−0.681874	−0.340937	−	0.940086i	$-0.610744\pi$
$293$	−3419.84	−0.681874	−0.340937	+	0.940086i	$0.610744\pi$
$294$	0	0
$295$	8535.09	1.68452
$296$	0	0
$297$	0	0
$298$	0	0
$299$	−1051.87	−0.203449
$300$	0	0
$301$	7881.15	1.50918
$302$	0	0
$303$	−2.22291	−0.000421462 0
$304$	0	0
$305$	6829.52	1.28216
$306$	0	0
$307$	−214.895	−0.0399501	−0.0199751	−	0.999800i	$-0.506359\pi$
$307$	−214.895	−0.0399501	−0.0199751	+	0.999800i	$0.506359\pi$
$308$	0	0
$309$	−265.367	−0.0488551
$310$	0	0
$311$	438.080	0.0798754	0.0399377	−	0.999202i	$-0.487284\pi$
$311$	438.080	0.0798754	0.0399377	+	0.999202i	$0.487284\pi$
$312$	0	0
$313$	7472.96	1.34951	0.674755	−	0.738042i	$-0.264249\pi$
$313$	7472.96	1.34951	0.674755	+	0.738042i	$0.264249\pi$
$314$	0	0
$315$	10724.9	1.91834
$316$	0	0
$317$	3406.09	0.603487	0.301743	−	0.953389i	$-0.402431\pi$
$317$	3406.09	0.603487	0.301743	+	0.953389i	$0.402431\pi$
$318$	0	0
$319$	0	0
$320$	0	0
$321$	−1003.73	−0.174526
$322$	0	0
$323$	12364.8	2.13002
$324$	0	0
$325$	983.362	0.167837
$326$	0	0
$327$	−758.866	−0.128335
$328$	0	0
$329$	−10738.9	−1.79956
$330$	0	0
$331$	34.6019	0.00574590	0.00287295	−	0.999996i	$-0.499086\pi$
$331$	34.6019	0.00574590	0.00287295	+	0.999996i	$0.499086\pi$
$332$	0	0
$333$	−2575.11	−0.423769
$334$	0	0
$335$	−1414.85	−0.230750
$336$	0	0
$337$	−4857.82	−0.785230	−0.392615	−	0.919703i	$-0.628430\pi$
$337$	−4857.82	−0.785230	−0.392615	+	0.919703i	$0.628430\pi$
$338$	0	0
$339$	−338.843	−0.0542874
$340$	0	0
$341$	0	0
$342$	0	0
$343$	2812.89	0.442804
$344$	0	0
$345$	1599.22	0.249563
$346$	0	0
$347$	680.824	0.105327	0.0526636	−	0.998612i	$-0.483229\pi$
$347$	680.824	0.105327	0.0526636	+	0.998612i	$0.483229\pi$
$348$	0	0
$349$	9750.68	1.49554	0.747768	−	0.663961i	$-0.231126\pi$
$349$	9750.68	1.49554	0.747768	+	0.663961i	$0.231126\pi$
$350$	0	0
$351$	184.514	0.0280587
$352$	0	0
$353$	−6441.92	−0.971299	−0.485650	−	0.874153i	$-0.661417\pi$
$353$	−6441.92	−0.971299	−0.485650	+	0.874153i	$0.661417\pi$
$354$	0	0
$355$	14147.0	2.11506
$356$	0	0
$357$	−1648.52	−0.244395
$358$	0	0
$359$	−8325.12	−1.22391	−0.611954	−	0.790893i	$-0.709616\pi$
$359$	−8325.12	−1.22391	−0.611954	+	0.790893i	$0.709616\pi$
$360$	0	0
$361$	3042.79	0.443620
$362$	0	0
$363$	0	0
$364$	0	0
$365$	8702.97	1.24804
$366$	0	0
$367$	1119.03	0.159163	0.0795815	−	0.996828i	$-0.474642\pi$
$367$	1119.03	0.159163	0.0795815	+	0.996828i	$0.474642\pi$
$368$	0	0
$369$	−2395.40	−0.337940
$370$	0	0
$371$	−8088.36	−1.13188
$372$	0	0
$373$	3647.68	0.506353	0.253177	−	0.967420i	$-0.418525\pi$
$373$	3647.68	0.506353	0.253177	+	0.967420i	$0.418525\pi$
$374$	0	0
$375$	−321.816	−0.0443160
$376$	0	0
$377$	1604.57	0.219203
$378$	0	0
$379$	7137.08	0.967301	0.483651	−	0.875261i	$-0.339311\pi$
$379$	7137.08	0.967301	0.483651	+	0.875261i	$0.339311\pi$
$380$	0	0
$381$	−502.378	−0.0675527
$382$	0	0
$383$	8150.46	1.08739	0.543693	−	0.839284i	$-0.317026\pi$
$383$	8150.46	1.08739	0.543693	+	0.839284i	$0.317026\pi$
$384$	0	0
$385$	0	0
$386$	0	0
$387$	8826.27	1.15934
$388$	0	0
$389$	1465.98	0.191074	0.0955372	−	0.995426i	$-0.469543\pi$
$389$	1465.98	0.191074	0.0955372	+	0.995426i	$0.469543\pi$
$390$	0	0
$391$	21171.9	2.73839
$392$	0	0
$393$	409.114	0.0525116
$394$	0	0
$395$	6868.79	0.874953
$396$	0	0
$397$	−1287.93	−0.162819	−0.0814095	−	0.996681i	$-0.525942\pi$
$397$	−1287.93	−0.162819	−0.0814095	+	0.996681i	$0.525942\pi$
$398$	0	0
$399$	−1320.14	−0.165639
$400$	0	0
$401$	−13458.1	−1.67597	−0.837986	−	0.545692i	$-0.816267\pi$
$401$	−13458.1	−1.67597	−0.837986	+	0.545692i	$0.816267\pi$
$402$	0	0
$403$	536.875	0.0663614
$404$	0	0
$405$	11869.9	1.45635
$406$	0	0
$407$	0	0
$408$	0	0
$409$	−1237.41	−0.149599	−0.0747994	−	0.997199i	$-0.523832\pi$
$409$	−1237.41	−0.149599	−0.0747994	+	0.997199i	$0.523832\pi$
$410$	0	0
$411$	1223.03	0.146783
$412$	0	0
$413$	−12064.0	−1.43737
$414$	0	0
$415$	−18105.1	−2.14156
$416$	0	0
$417$	1267.88	0.148893
$418$	0	0
$419$	−933.173	−0.108803	−0.0544016	−	0.998519i	$-0.517325\pi$
$419$	−933.173	−0.108803	−0.0544016	+	0.998519i	$0.517325\pi$
$420$	0	0
$421$	14303.9	1.65589	0.827947	−	0.560806i	$-0.189509\pi$
$421$	14303.9	1.65589	0.827947	+	0.560806i	$0.189509\pi$
$422$	0	0
$423$	−12026.7	−1.38241
$424$	0	0
$425$	−19792.9	−2.25905
$426$	0	0
$427$	−9653.28	−1.09404
$428$	0	0
$429$	0	0
$430$	0	0
$431$	10582.7	1.18272	0.591359	−	0.806408i	$-0.298592\pi$
$431$	10582.7	1.18272	0.591359	+	0.806408i	$0.298592\pi$
$432$	0	0
$433$	2853.68	0.316719	0.158359	−	0.987382i	$-0.449380\pi$
$433$	2853.68	0.316719	0.158359	+	0.987382i	$0.449380\pi$
$434$	0	0
$435$	−2439.52	−0.268887
$436$	0	0
$437$	16954.6	1.85594
$438$	0	0
$439$	16020.6	1.74174	0.870870	−	0.491514i	$-0.163556\pi$
$439$	16020.6	1.74174	0.870870	+	0.491514i	$0.163556\pi$
$440$	0	0
$441$	−6004.49	−0.648363
$442$	0	0
$443$	13885.6	1.48922	0.744608	−	0.667502i	$-0.232637\pi$
$443$	13885.6	1.48922	0.744608	+	0.667502i	$0.232637\pi$
$444$	0	0
$445$	−429.379	−0.0457405
$446$	0	0
$447$	−1411.09	−0.149312
$448$	0	0
$449$	6009.69	0.631658	0.315829	−	0.948816i	$-0.397717\pi$
$449$	6009.69	0.631658	0.315829	+	0.948816i	$0.397717\pi$
$450$	0	0
$451$	0	0
$452$	0	0
$453$	−370.607	−0.0384385
$454$	0	0
$455$	−2480.70	−0.255598
$456$	0	0
$457$	−4428.35	−0.453281	−0.226641	−	0.973978i	$-0.572774\pi$
$457$	−4428.35	−0.453281	−0.226641	+	0.973978i	$0.572774\pi$
$458$	0	0
$459$	−3713.86	−0.377665
$460$	0	0
$461$	−8342.07	−0.842796	−0.421398	−	0.906876i	$-0.638460\pi$
$461$	−8342.07	−0.842796	−0.421398	+	0.906876i	$0.638460\pi$
$462$	0	0
$463$	7839.47	0.786892	0.393446	−	0.919348i	$-0.371283\pi$
$463$	7839.47	0.786892	0.393446	+	0.919348i	$0.371283\pi$
$464$	0	0
$465$	−816.243	−0.0814029
$466$	0	0
$467$	394.019	0.0390429	0.0195214	−	0.999809i	$-0.493786\pi$
$467$	394.019	0.0390429	0.0195214	+	0.999809i	$0.493786\pi$
$468$	0	0
$469$	1999.83	0.196895
$470$	0	0
$471$	−826.699	−0.0808753
$472$	0	0
$473$	0	0
$474$	0	0
$475$	−15850.3	−1.53107
$476$	0	0
$477$	−9058.33	−0.869501
$478$	0	0
$479$	−8038.90	−0.766820	−0.383410	−	0.923578i	$-0.625250\pi$
$479$	−8038.90	−0.766820	−0.383410	+	0.923578i	$0.625250\pi$
$480$	0	0
$481$	595.633	0.0564626
$482$	0	0
$483$	−2260.45	−0.212948
$484$	0	0
$485$	10069.4	0.942740
$486$	0	0
$487$	−6556.67	−0.610085	−0.305042	−	0.952339i	$-0.598671\pi$
$487$	−6556.67	−0.610085	−0.305042	+	0.952339i	$0.598671\pi$
$488$	0	0
$489$	2141.47	0.198038
$490$	0	0
$491$	13008.7	1.19567	0.597835	−	0.801619i	$-0.296028\pi$
$491$	13008.7	1.19567	0.597835	+	0.801619i	$0.296028\pi$
$492$	0	0
$493$	−32296.4	−2.95042
$494$	0	0
$495$	0	0
$496$	0	0
$497$	−19996.3	−1.80474
$498$	0	0
$499$	−13264.2	−1.18995	−0.594976	−	0.803743i	$-0.702839\pi$
$499$	−13264.2	−1.18995	−0.594976	+	0.803743i	$0.702839\pi$
$500$	0	0
$501$	2034.59	0.181434
$502$	0	0
$503$	14241.0	1.26238	0.631189	−	0.775629i	$-0.282567\pi$
$503$	14241.0	1.26238	0.631189	+	0.775629i	$0.282567\pi$
$504$	0	0
$505$	67.3286	0.00593284
$506$	0	0
$507$	1201.80	0.105274
$508$	0	0
$509$	−8676.03	−0.755517	−0.377758	−	0.925904i	$-0.623305\pi$
$509$	−8676.03	−0.755517	−0.377758	+	0.925904i	$0.623305\pi$
$510$	0	0
$511$	−12301.3	−1.06493
$512$	0	0
$513$	−2974.08	−0.255962
$514$	0	0
$515$	8037.57	0.687724
$516$	0	0
$517$	0	0
$518$	0	0
$519$	80.4772	0.00680647
$520$	0	0
$521$	−14259.3	−1.19906	−0.599531	−	0.800351i	$-0.704646\pi$
$521$	−14259.3	−1.19906	−0.599531	+	0.800351i	$0.704646\pi$
$522$	0	0
$523$	4972.54	0.415744	0.207872	−	0.978156i	$-0.433346\pi$
$523$	4972.54	0.415744	0.207872	+	0.978156i	$0.433346\pi$
$524$	0	0
$525$	2113.22	0.175673
$526$	0	0
$527$	−10806.1	−0.893210
$528$	0	0
$529$	16863.9	1.38603
$530$	0	0
$531$	−13510.8	−1.10418
$532$	0	0
$533$	554.066	0.0450268
$534$	0	0
$535$	30401.4	2.45676
$536$	0	0
$537$	−916.995	−0.0736895
$538$	0	0
$539$	0	0
$540$	0	0
$541$	306.943	0.0243928	0.0121964	−	0.999926i	$-0.496118\pi$
$541$	306.943	0.0243928	0.0121964	+	0.999926i	$0.496118\pi$
$542$	0	0
$543$	1958.47	0.154781
$544$	0	0
$545$	22984.9	1.80654
$546$	0	0
$547$	−8696.33	−0.679759	−0.339880	−	0.940469i	$-0.610386\pi$
$547$	−8696.33	−0.679759	−0.339880	+	0.940469i	$0.610386\pi$
$548$	0	0
$549$	−10810.9	−0.840434
$550$	0	0
$551$	−25863.1	−1.99965
$552$	0	0
$553$	−9708.79	−0.746582
$554$	0	0
$555$	−905.576	−0.0692604
$556$	0	0
$557$	−11477.0	−0.873065	−0.436533	−	0.899688i	$-0.643794\pi$
$557$	−11477.0	−0.873065	−0.436533	+	0.899688i	$0.643794\pi$
$558$	0	0
$559$	−2041.55	−0.154469
$560$	0	0
$561$	0	0
$562$	0	0
$563$	17947.0	1.34348	0.671739	−	0.740788i	$-0.265548\pi$
$563$	17947.0	1.34348	0.671739	+	0.740788i	$0.265548\pi$
$564$	0	0
$565$	10263.0	0.764193
$566$	0	0
$567$	−16777.7	−1.24268
$568$	0	0
$569$	−20841.5	−1.53554	−0.767768	−	0.640728i	$-0.778633\pi$
$569$	−20841.5	−1.53554	−0.767768	+	0.640728i	$0.778633\pi$
$570$	0	0
$571$	10334.4	0.757414	0.378707	−	0.925517i	$-0.376369\pi$
$571$	10334.4	0.757414	0.378707	+	0.925517i	$0.376369\pi$
$572$	0	0
$573$	1946.03	0.141879
$574$	0	0
$575$	−27140.0	−1.96838
$576$	0	0
$577$	−19615.8	−1.41528	−0.707639	−	0.706574i	$-0.750240\pi$
$577$	−19615.8	−1.41528	−0.707639	+	0.706574i	$0.750240\pi$
$578$	0	0
$579$	2590.69	0.185951
$580$	0	0
$581$	25591.0	1.82735
$582$	0	0
$583$	0	0
$584$	0	0
$585$	−2778.19	−0.196349
$586$	0	0
$587$	−10513.5	−0.739251	−0.369626	−	0.929181i	$-0.620514\pi$
$587$	−10513.5	−0.739251	−0.369626	+	0.929181i	$0.620514\pi$
$588$	0	0
$589$	−8653.59	−0.605374
$590$	0	0
$591$	1336.72	0.0930379
$592$	0	0
$593$	−11093.6	−0.768229	−0.384114	−	0.923285i	$-0.625493\pi$
$593$	−11093.6	−0.768229	−0.384114	+	0.923285i	$0.625493\pi$
$594$	0	0
$595$	49931.1	3.44030
$596$	0	0
$597$	−2068.54	−0.141809
$598$	0	0
$599$	−13389.5	−0.913323	−0.456661	−	0.889641i	$-0.650955\pi$
$599$	−13389.5	−0.913323	−0.456661	+	0.889641i	$0.650955\pi$
$600$	0	0
$601$	21203.8	1.43914	0.719569	−	0.694421i	$-0.244339\pi$
$601$	21203.8	1.43914	0.719569	+	0.694421i	$0.244339\pi$
$602$	0	0
$603$	2239.65	0.151253
$604$	0	0
$605$	0	0
$606$	0	0
$607$	−911.644	−0.0609596	−0.0304798	−	0.999535i	$-0.509704\pi$
$607$	−911.644	−0.0609596	−0.0304798	+	0.999535i	$0.509704\pi$
$608$	0	0
$609$	3448.17	0.229437
$610$	0	0
$611$	2781.82	0.184191
$612$	0	0
$613$	11430.3	0.753125	0.376562	−	0.926391i	$-0.377106\pi$
$613$	11430.3	0.753125	0.376562	+	0.926391i	$0.377106\pi$
$614$	0	0
$615$	−842.380	−0.0552326
$616$	0	0
$617$	−6982.96	−0.455630	−0.227815	−	0.973704i	$-0.573158\pi$
$617$	−6982.96	−0.455630	−0.227815	+	0.973704i	$0.573158\pi$
$618$	0	0
$619$	13697.0	0.889382	0.444691	−	0.895684i	$-0.353314\pi$
$619$	13697.0	0.889382	0.444691	+	0.895684i	$0.353314\pi$
$620$	0	0
$621$	−5092.43	−0.329070
$622$	0	0
$623$	606.912	0.0390296
$624$	0	0
$625$	−10163.5	−0.650467
$626$	0	0
$627$	0	0
$628$	0	0
$629$	−11988.8	−0.759974
$630$	0	0
$631$	3700.84	0.233483	0.116742	−	0.993162i	$-0.462755\pi$
$631$	3700.84	0.233483	0.116742	+	0.993162i	$0.462755\pi$
$632$	0	0
$633$	20.1433	0.00126481
$634$	0	0
$635$	15216.2	0.950926
$636$	0	0
$637$	1388.86	0.0863873
$638$	0	0
$639$	−22394.3	−1.38639
$640$	0	0
$641$	1431.92	0.0882330	0.0441165	−	0.999026i	$-0.485953\pi$
$641$	1431.92	0.0882330	0.0441165	+	0.999026i	$0.485953\pi$
$642$	0	0
$643$	27881.7	1.71003	0.855014	−	0.518606i	$-0.173549\pi$
$643$	27881.7	1.71003	0.855014	+	0.518606i	$0.173549\pi$
$644$	0	0
$645$	3103.89	0.189481
$646$	0	0
$647$	−10377.1	−0.630550	−0.315275	−	0.949000i	$-0.602097\pi$
$647$	−10377.1	−0.630550	−0.315275	+	0.949000i	$0.602097\pi$
$648$	0	0
$649$	0	0
$650$	0	0
$651$	1153.73	0.0694596
$652$	0	0
$653$	−11905.5	−0.713472	−0.356736	−	0.934205i	$-0.616110\pi$
$653$	−11905.5	−0.713472	−0.356736	+	0.934205i	$0.616110\pi$
$654$	0	0
$655$	−12391.4	−0.739196
$656$	0	0
$657$	−13776.5	−0.818072
$658$	0	0
$659$	−14253.0	−0.842513	−0.421257	−	0.906941i	$-0.638411\pi$
$659$	−14253.0	−0.842513	−0.421257	+	0.906941i	$0.638411\pi$
$660$	0	0
$661$	3356.44	0.197504	0.0987522	−	0.995112i	$-0.468515\pi$
$661$	3356.44	0.197504	0.0987522	+	0.995112i	$0.468515\pi$
$662$	0	0
$663$	427.036	0.0250146
$664$	0	0
$665$	39985.1	2.33166
$666$	0	0
$667$	−44284.8	−2.57078
$668$	0	0
$669$	−3142.11	−0.181586
$670$	0	0
$671$	0	0
$672$	0	0
$673$	12592.7	0.721267	0.360633	−	0.932708i	$-0.382561\pi$
$673$	12592.7	0.721267	0.360633	+	0.932708i	$0.382561\pi$
$674$	0	0
$675$	4760.75	0.271469
$676$	0	0
$677$	−30588.2	−1.73648	−0.868242	−	0.496141i	$-0.834750\pi$
$677$	−30588.2	−1.73648	−0.868242	+	0.496141i	$0.834750\pi$
$678$	0	0
$679$	−14232.8	−0.804423
$680$	0	0
$681$	2185.79	0.122995
$682$	0	0
$683$	26902.5	1.50717	0.753585	−	0.657351i	$-0.228323\pi$
$683$	26902.5	1.50717	0.753585	+	0.657351i	$0.228323\pi$
$684$	0	0
$685$	−37043.8	−2.06623
$686$	0	0
$687$	−1953.98	−0.108514
$688$	0	0
$689$	2095.23	0.115852
$690$	0	0
$691$	−20905.7	−1.15093	−0.575463	−	0.817828i	$-0.695178\pi$
$691$	−20905.7	−1.15093	−0.575463	+	0.817828i	$0.695178\pi$
$692$	0	0
$693$	0	0
$694$	0	0
$695$	−38402.3	−2.09594
$696$	0	0
$697$	−11152.1	−0.606051
$698$	0	0
$699$	1459.54	0.0789770
$700$	0	0
$701$	−19993.1	−1.07722	−0.538609	−	0.842556i	$-0.681050\pi$
$701$	−19993.1	−1.07722	−0.538609	+	0.842556i	$0.681050\pi$
$702$	0	0
$703$	−9600.68	−0.515073
$704$	0	0
$705$	−4229.37	−0.225939
$706$	0	0
$707$	−95.1665	−0.00506238
$708$	0	0
$709$	857.539	0.0454239	0.0227120	−	0.999742i	$-0.492770\pi$
$709$	857.539	0.0454239	0.0227120	+	0.999742i	$0.492770\pi$
$710$	0	0
$711$	−10873.1	−0.573519
$712$	0	0
$713$	−14817.3	−0.778279
$714$	0	0
$715$	0	0
$716$	0	0
$717$	−804.607	−0.0419088
$718$	0	0
$719$	14156.1	0.734259	0.367129	−	0.930170i	$-0.380341\pi$
$719$	14156.1	0.734259	0.367129	+	0.930170i	$0.380341\pi$
$720$	0	0
$721$	−11360.8	−0.586822
$722$	0	0
$723$	−534.791	−0.0275091
$724$	0	0
$725$	41400.4	2.12079
$726$	0	0
$727$	−33330.0	−1.70033	−0.850166	−	0.526514i	$-0.823499\pi$
$727$	−33330.0	−1.70033	−0.850166	+	0.526514i	$0.823499\pi$
$728$	0	0
$729$	−18340.5	−0.931794
$730$	0	0
$731$	41091.9	2.07912
$732$	0	0
$733$	19705.1	0.992938	0.496469	−	0.868055i	$-0.334630\pi$
$733$	19705.1	0.992938	0.496469	+	0.868055i	$0.334630\pi$
$734$	0	0
$735$	−2111.57	−0.105968
$736$	0	0
$737$	0	0
$738$	0	0
$739$	−6447.71	−0.320951	−0.160476	−	0.987040i	$-0.551303\pi$
$739$	−6447.71	−0.320951	−0.160476	+	0.987040i	$0.551303\pi$
$740$	0	0
$741$	341.973	0.0169537
$742$	0	0
$743$	27926.2	1.37888	0.689442	−	0.724340i	$-0.257856\pi$
$743$	27926.2	1.37888	0.689442	+	0.724340i	$0.257856\pi$
$744$	0	0
$745$	42739.8	2.10183
$746$	0	0
$747$	28659.9	1.40376
$748$	0	0
$749$	−42971.3	−2.09631
$750$	0	0
$751$	−26183.2	−1.27222	−0.636112	−	0.771597i	$-0.719458\pi$
$751$	−26183.2	−1.27222	−0.636112	+	0.771597i	$0.719458\pi$
$752$	0	0
$753$	−250.791	−0.0121372
$754$	0	0
$755$	11225.1	0.541092
$756$	0	0
$757$	−1997.56	−0.0959080	−0.0479540	−	0.998850i	$-0.515270\pi$
$757$	−1997.56	−0.0959080	−0.0479540	+	0.998850i	$0.515270\pi$
$758$	0	0
$759$	0	0
$760$	0	0
$761$	26391.5	1.25715	0.628576	−	0.777748i	$-0.283638\pi$
$761$	26391.5	1.25715	0.628576	+	0.777748i	$0.283638\pi$
$762$	0	0
$763$	−32488.3	−1.54149
$764$	0	0
$765$	55918.9	2.64281
$766$	0	0
$767$	3125.09	0.147119
$768$	0	0
$769$	19903.1	0.933322	0.466661	−	0.884436i	$-0.345457\pi$
$769$	19903.1	0.933322	0.466661	+	0.884436i	$0.345457\pi$
$770$	0	0
$771$	−3071.88	−0.143490
$772$	0	0
$773$	19476.3	0.906228	0.453114	−	0.891453i	$-0.350313\pi$
$773$	19476.3	0.906228	0.453114	+	0.891453i	$0.350313\pi$
$774$	0	0
$775$	13852.2	0.642048
$776$	0	0
$777$	1280.00	0.0590987
$778$	0	0
$779$	−8930.69	−0.410751
$780$	0	0
$781$	0	0
$782$	0	0
$783$	7768.19	0.354550
$784$	0	0
$785$	25039.4	1.13847
$786$	0	0
$787$	4899.05	0.221896	0.110948	−	0.993826i	$-0.464611\pi$
$787$	4899.05	0.221896	0.110948	+	0.993826i	$0.464611\pi$
$788$	0	0
$789$	−2074.98	−0.0936265
$790$	0	0
$791$	−14506.4	−0.652072
$792$	0	0
$793$	2500.61	0.111979
$794$	0	0
$795$	−3185.50	−0.142111
$796$	0	0
$797$	41253.5	1.83347	0.916733	−	0.399501i	$-0.130817\pi$
$797$	41253.5	1.83347	0.916733	+	0.399501i	$0.130817\pi$
$798$	0	0
$799$	−55992.0	−2.47917
$800$	0	0
$801$	679.694	0.0299823
$802$	0	0
$803$	0	0
$804$	0	0
$805$	68465.4	2.99763
$806$	0	0
$807$	751.333	0.0327735
$808$	0	0
$809$	−4088.66	−0.177688	−0.0888441	−	0.996046i	$-0.528317\pi$
$809$	−4088.66	−0.177688	−0.0888441	+	0.996046i	$0.528317\pi$
$810$	0	0
$811$	21807.4	0.944219	0.472109	−	0.881540i	$-0.343493\pi$
$811$	21807.4	0.944219	0.472109	+	0.881540i	$0.343493\pi$
$812$	0	0
$813$	−3375.91	−0.145631
$814$	0	0
$815$	−64861.9	−2.78775
$816$	0	0
$817$	32906.6	1.40913
$818$	0	0
$819$	3926.87	0.167541
$820$	0	0
$821$	16327.1	0.694054	0.347027	−	0.937855i	$-0.387191\pi$
$821$	16327.1	0.694054	0.347027	+	0.937855i	$0.387191\pi$
$822$	0	0
$823$	6533.33	0.276716	0.138358	−	0.990382i	$-0.455818\pi$
$823$	6533.33	0.276716	0.138358	+	0.990382i	$0.455818\pi$
$824$	0	0
$825$	0	0
$826$	0	0
$827$	3638.29	0.152982	0.0764908	−	0.997070i	$-0.475628\pi$
$827$	3638.29	0.152982	0.0764908	+	0.997070i	$0.475628\pi$
$828$	0	0
$829$	−21220.7	−0.889053	−0.444526	−	0.895766i	$-0.646628\pi$
$829$	−21220.7	−0.889053	−0.444526	+	0.895766i	$0.646628\pi$
$830$	0	0
$831$	2482.98	0.103650
$832$	0	0
$833$	−27954.8	−1.16276
$834$	0	0
$835$	−61624.5	−2.55402
$836$	0	0
$837$	2599.17	0.107336
$838$	0	0
$839$	−18766.3	−0.772212	−0.386106	−	0.922455i	$-0.626180\pi$
$839$	−18766.3	−0.772212	−0.386106	+	0.922455i	$0.626180\pi$
$840$	0	0
$841$	43164.7	1.76984
$842$	0	0
$843$	−3324.80	−0.135839
$844$	0	0
$845$	−36400.6	−1.48192
$846$	0	0
$847$	0	0
$848$	0	0
$849$	−826.122	−0.0333951
$850$	0	0
$851$	−16439.0	−0.662187
$852$	0	0
$853$	18039.1	0.724086	0.362043	−	0.932161i	$-0.382079\pi$
$853$	18039.1	0.724086	0.362043	+	0.932161i	$0.382079\pi$
$854$	0	0
$855$	44780.2	1.79117
$856$	0	0
$857$	30631.9	1.22096	0.610482	−	0.792030i	$-0.290976\pi$
$857$	30631.9	1.22096	0.610482	+	0.792030i	$0.290976\pi$
$858$	0	0
$859$	3325.07	0.132072	0.0660361	−	0.997817i	$-0.478965\pi$
$859$	3325.07	0.132072	0.0660361	+	0.997817i	$0.478965\pi$
$860$	0	0
$861$	1190.67	0.0471289
$862$	0	0
$863$	11549.4	0.455559	0.227779	−	0.973713i	$-0.426853\pi$
$863$	11549.4	0.455559	0.227779	+	0.973713i	$0.426853\pi$
$864$	0	0
$865$	−2437.53	−0.0958133
$866$	0	0
$867$	−5860.35	−0.229559
$868$	0	0
$869$	0	0
$870$	0	0
$871$	−518.041	−0.0201529
$872$	0	0
$873$	−15939.6	−0.617952
$874$	0	0
$875$	−13777.5	−0.532301
$876$	0	0
$877$	−37872.4	−1.45822	−0.729110	−	0.684397i	$-0.760066\pi$
$877$	−37872.4	−1.45822	−0.729110	+	0.684397i	$0.760066\pi$
$878$	0	0
$879$	1903.74	0.0730505
$880$	0	0
$881$	−22422.0	−0.857454	−0.428727	−	0.903434i	$-0.641038\pi$
$881$	−22422.0	−0.857454	−0.428727	+	0.903434i	$0.641038\pi$
$882$	0	0
$883$	30138.1	1.14862	0.574309	−	0.818639i	$-0.305271\pi$
$883$	30138.1	1.14862	0.574309	+	0.818639i	$0.305271\pi$
$884$	0	0
$885$	−4751.26	−0.180466
$886$	0	0
$887$	10319.5	0.390635	0.195318	−	0.980740i	$-0.437426\pi$
$887$	10319.5	0.390635	0.195318	+	0.980740i	$0.437426\pi$
$888$	0	0
$889$	−21507.6	−0.811408
$890$	0	0
$891$	0	0
$892$	0	0
$893$	−44838.7	−1.68026
$894$	0	0
$895$	27774.4	1.03731
$896$	0	0
$897$	585.551	0.0217959
$898$	0	0
$899$	22602.9	0.838542
$900$	0	0
$901$	−42172.3	−1.55934
$902$	0	0
$903$	−4387.23	−0.161681
$904$	0	0
$905$	−59319.1	−2.17882
$906$	0	0
$907$	17929.5	0.656381	0.328191	−	0.944612i	$-0.393561\pi$
$907$	17929.5	0.656381	0.328191	+	0.944612i	$0.393561\pi$
$908$	0	0
$909$	−106.579	−0.00388889
$910$	0	0
$911$	14692.8	0.534351	0.267176	−	0.963648i	$-0.413910\pi$
$911$	14692.8	0.534351	0.267176	+	0.963648i	$0.413910\pi$
$912$	0	0
$913$	0	0
$914$	0	0
$915$	−3801.82	−0.137360
$916$	0	0
$917$	17514.8	0.630742
$918$	0	0
$919$	−1503.29	−0.0539595	−0.0269798	−	0.999636i	$-0.508589\pi$
$919$	−1503.29	−0.0539595	−0.0269798	+	0.999636i	$0.508589\pi$
$920$	0	0
$921$	119.626	0.00427994
$922$	0	0
$923$	5179.89	0.184722
$924$	0	0
$925$	15368.3	0.546277
$926$	0	0
$927$	−12723.2	−0.450793
$928$	0	0
$929$	−12257.4	−0.432888	−0.216444	−	0.976295i	$-0.569446\pi$
$929$	−12257.4	−0.432888	−0.216444	+	0.976295i	$0.569446\pi$
$930$	0	0
$931$	−22386.3	−0.788058
$932$	0	0
$933$	−243.868	−0.00855721
$934$	0	0
$935$	0	0
$936$	0	0
$937$	1385.10	0.0482917	0.0241459	−	0.999708i	$-0.492313\pi$
$937$	1385.10	0.0482917	0.0241459	+	0.999708i	$0.492313\pi$
$938$	0	0
$939$	−4160.00	−0.144576
$940$	0	0
$941$	19152.4	0.663498	0.331749	−	0.943368i	$-0.392361\pi$
$941$	19152.4	0.663498	0.331749	+	0.943368i	$0.392361\pi$
$942$	0	0
$943$	−15291.8	−0.528069
$944$	0	0
$945$	−12009.8	−0.413418
$946$	0	0
$947$	24452.2	0.839061	0.419531	−	0.907741i	$-0.362195\pi$
$947$	24452.2	0.839061	0.419531	+	0.907741i	$0.362195\pi$
$948$	0	0
$949$	3186.56	0.108999
$950$	0	0
$951$	−1896.09	−0.0646528
$952$	0	0
$953$	38595.4	1.31189	0.655943	−	0.754810i	$-0.272271\pi$
$953$	38595.4	1.31189	0.655943	+	0.754810i	$0.272271\pi$
$954$	0	0
$955$	−58942.3	−1.99720
$956$	0	0
$957$	0	0
$958$	0	0
$959$	52360.0	1.76308
$960$	0	0
$961$	−22228.3	−0.746140
$962$	0	0
$963$	−48124.5	−1.61037
$964$	0	0
$965$	−78468.0	−2.61759
$966$	0	0
$967$	17975.2	0.597769	0.298885	−	0.954289i	$-0.403385\pi$
$967$	17975.2	0.597769	0.298885	+	0.954289i	$0.403385\pi$
$968$	0	0
$969$	−6883.16	−0.228193
$970$	0	0
$971$	6149.28	0.203233	0.101617	−	0.994824i	$-0.467598\pi$
$971$	6149.28	0.203233	0.101617	+	0.994824i	$0.467598\pi$
$972$	0	0
$973$	54280.2	1.78843
$974$	0	0
$975$	−547.412	−0.0179807
$976$	0	0
$977$	16675.9	0.546068	0.273034	−	0.962004i	$-0.411973\pi$
$977$	16675.9	0.546068	0.273034	+	0.962004i	$0.411973\pi$
$978$	0	0
$979$	0	0
$980$	0	0
$981$	−36384.3	−1.18416
$982$	0	0
$983$	−22937.4	−0.744240	−0.372120	−	0.928185i	$-0.621369\pi$
$983$	−22937.4	−0.744240	−0.372120	+	0.928185i	$0.621369\pi$
$984$	0	0
$985$	−40487.3	−1.30968
$986$	0	0
$987$	5978.06	0.192790
$988$	0	0
$989$	56345.2	1.81160
$990$	0	0
$991$	−6407.71	−0.205396	−0.102698	−	0.994713i	$-0.532748\pi$
$991$	−6407.71	−0.205396	−0.102698	+	0.994713i	$0.532748\pi$
$992$	0	0
$993$	−19.2620	−0.000615570 0
$994$	0	0
$995$	62653.0	1.99622
$996$	0	0
$997$	3016.40	0.0958176	0.0479088	−	0.998852i	$-0.484744\pi$
$997$	3016.40	0.0958176	0.0479088	+	0.998852i	$0.484744\pi$
$998$	0	0
$999$	2883.64	0.0913256

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	1936.4.a.bp.1.2		4
4.3	odd	2		968.4.a.l.1.3	✓	4
11.10	odd	2		1936.4.a.bo.1.2		4
44.43	even	2		968.4.a.m.1.3	yes	4

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
968.4.a.l.1.3	✓	4	4.3	odd	2
968.4.a.m.1.3	yes	4	44.43	even	2
1936.4.a.bo.1.2		4	11.10	odd	2
1936.4.a.bp.1.2		4	1.1	even	1	trivial

Overview	Random
Universe	Knowledge

Classical	Maass
Hilbert	Bianchi

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

Level:	\( N \)	\(=\)	\( 1936 = 2^{4} \cdot 11^{2} \)
Weight:	\( k \)	\(=\)	\( 4 \)
Character orbit:	\([\chi]\)	\(=\)	1936.a (trivial)

\(f(q)\)	\(=\)	\(q-0.556674 q^{3} +16.8608 q^{5} -23.8321 q^{7} -26.6901 q^{9} +O(q^{10})\)	\(q-0.556674 q^{3} +16.8608 q^{5} -23.8321 q^{7} -26.6901 q^{9} +6.17353 q^{13} -9.38598 q^{15} -124.260 q^{17} -99.5077 q^{19} +13.2667 q^{21} -170.384 q^{23} +159.287 q^{25} +29.8879 q^{27} +259.911 q^{29} +86.9640 q^{31} -401.829 q^{35} +96.4817 q^{37} -3.43664 q^{39} +89.7487 q^{41} -330.694 q^{43} -450.017 q^{45} +450.605 q^{47} +224.971 q^{49} +69.1721 q^{51} +339.389 q^{53} +55.3934 q^{57} +506.209 q^{59} +405.053 q^{61} +636.082 q^{63} +104.091 q^{65} -83.9133 q^{67} +94.8486 q^{69} +839.049 q^{71} +516.166 q^{73} -88.6709 q^{75} +407.382 q^{79} +703.995 q^{81} -1073.80 q^{83} -2095.12 q^{85} -144.686 q^{87} -25.4661 q^{89} -147.128 q^{91} -48.4106 q^{93} -1677.78 q^{95} +597.209 q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 4 q + 3 q^{3} + 9 q^{5} + 8 q^{7} + 31 q^{9}+O(q^{10}) \) 4 * q + 3 * q^3 + 9 * q^5 + 8 * q^7 + 31 * q^9	\( 4 q + 3 q^{3} + 9 q^{5} + 8 q^{7} + 31 q^{9} + 80 q^{13} + 47 q^{15} - 16 q^{17} + 8 q^{19} + 192 q^{21} + 71 q^{23} + 11 q^{25} + 81 q^{27} + 240 q^{29} + 115 q^{31} - 168 q^{35} + 315 q^{37} + 232 q^{39} - 592 q^{41} - 624 q^{43} + 108 q^{45} + 304 q^{47} + 36 q^{49} - 1256 q^{51} + 184 q^{53} + 592 q^{57} + 805 q^{59} + 240 q^{61} + 2288 q^{63} + 1424 q^{65} - 119 q^{67} + 1499 q^{69} - 723 q^{71} + 1936 q^{73} + 300 q^{75} - 1520 q^{79} + 700 q^{81} - 2016 q^{83} - 3040 q^{85} - 3152 q^{87} + 367 q^{89} + 2848 q^{91} - 1625 q^{93} - 3400 q^{95} + 881 q^{97}+O(q^{100}) \) 4 * q + 3 * q^3 + 9 * q^5 + 8 * q^7 + 31 * q^9 + 80 * q^13 + 47 * q^15 - 16 * q^17 + 8 * q^19 + 192 * q^21 + 71 * q^23 + 11 * q^25 + 81 * q^27 + 240 * q^29 + 115 * q^31 - 168 * q^35 + 315 * q^37 + 232 * q^39 - 592 * q^41 - 624 * q^43 + 108 * q^45 + 304 * q^47 + 36 * q^49 - 1256 * q^51 + 184 * q^53 + 592 * q^57 + 805 * q^59 + 240 * q^61 + 2288 * q^63 + 1424 * q^65 - 119 * q^67 + 1499 * q^69 - 723 * q^71 + 1936 * q^73 + 300 * q^75 - 1520 * q^79 + 700 * q^81 - 2016 * q^83 - 3040 * q^85 - 3152 * q^87 + 367 * q^89 + 2848 * q^91 - 1625 * q^93 - 3400 * q^95 + 881 * q^97

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