LMFDB - Embedded newform 1936.4.a.bm.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:	4.4.978025.2
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{4} - 2x^{3} - 99x^{2} + 100x + 2420 $ x^4 - 2x^3 - 99x^2 + 100*x + 2420
Coefficient ring:	$\Z[a_1, \ldots, a_{7}]$
Coefficient ring index:	$ 11 $
Twist minimal:	no (minimal twist has level 22)
Fricke sign:	$1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.2
Root			$-7.19378$ 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)

$f(q)$	$=$	$q-7.57575 q^{3} +5.40810 q^{5} -22.1498 q^{7} +30.3919 q^{9} +O(q^{10})$	$q-7.57575 q^{3} +5.40810 q^{5} -22.1498 q^{7} +30.3919 q^{9} -76.8269 q^{13} -40.9704 q^{15} -59.2883 q^{17} +95.2626 q^{19} +167.801 q^{21} -142.484 q^{23} -95.7524 q^{25} -25.6965 q^{27} +20.4183 q^{29} -213.304 q^{31} -119.788 q^{35} -145.578 q^{37} +582.021 q^{39} -82.8326 q^{41} -151.373 q^{43} +164.363 q^{45} -90.3643 q^{47} +147.615 q^{49} +449.153 q^{51} -234.849 q^{53} -721.685 q^{57} -302.497 q^{59} +149.279 q^{61} -673.176 q^{63} -415.488 q^{65} -826.236 q^{67} +1079.42 q^{69} +898.965 q^{71} +137.993 q^{73} +725.396 q^{75} -304.459 q^{79} -625.912 q^{81} -764.294 q^{83} -320.637 q^{85} -154.684 q^{87} -313.100 q^{89} +1701.70 q^{91} +1615.94 q^{93} +515.189 q^{95} -582.505 q^{97} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 4 q - 4 q^{3} + 25 q^{5} - 3 q^{7} + 102 q^{9}+O(q^{10}) $ 4 * q - 4 * q^3 + 25 * q^5 - 3 * q^7 + 102 * q^9	$ 4 q - 4 q^{3} + 25 q^{5} - 3 q^{7} + 102 q^{9} - 41 q^{13} - 68 q^{15} + 52 q^{17} - 16 q^{19} + 25 q^{21} - 314 q^{23} - 21 q^{25} - 286 q^{27} + 561 q^{29} - 199 q^{31} + 714 q^{35} + 357 q^{37} + 1038 q^{39} + 32 q^{41} + 721 q^{43} + 1326 q^{45} - 403 q^{47} + 823 q^{49} + 174 q^{51} - 133 q^{53} - 1031 q^{57} - 1016 q^{59} - 919 q^{61} + 1367 q^{63} + 69 q^{65} - 289 q^{67} - 1620 q^{69} + 1205 q^{71} - 1234 q^{73} + 911 q^{75} + 603 q^{79} - 1400 q^{81} - 1514 q^{83} + 717 q^{85} + 1061 q^{87} - 1101 q^{89} + 2306 q^{91} - 2298 q^{93} + 1766 q^{95} + 2116 q^{97}+O(q^{100}) $ 4 * q - 4 * q^3 + 25 * q^5 - 3 * q^7 + 102 * q^9 - 41 * q^13 - 68 * q^15 + 52 * q^17 - 16 * q^19 + 25 * q^21 - 314 * q^23 - 21 * q^25 - 286 * q^27 + 561 * q^29 - 199 * q^31 + 714 * q^35 + 357 * q^37 + 1038 * q^39 + 32 * q^41 + 721 * q^43 + 1326 * q^45 - 403 * q^47 + 823 * q^49 + 174 * q^51 - 133 * q^53 - 1031 * q^57 - 1016 * q^59 - 919 * q^61 + 1367 * q^63 + 69 * q^65 - 289 * q^67 - 1620 * q^69 + 1205 * q^71 - 1234 * q^73 + 911 * q^75 + 603 * q^79 - 1400 * q^81 - 1514 * q^83 + 717 * q^85 + 1061 * q^87 - 1101 * q^89 + 2306 * q^91 - 2298 * q^93 + 1766 * q^95 + 2116 * 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
$3$	−7.57575	−1.45795	−0.728977	−	0.684539i	$-0.760004\pi$
$3$	−7.57575	−1.45795	−0.728977	+	0.684539i	$0.760004\pi$
$4$	0	0
$5$	5.40810	0.483715	0.241858	−	0.970312i	$-0.422243\pi$
$5$	5.40810	0.483715	0.241858	+	0.970312i	$0.422243\pi$
$6$	0	0
$7$	−22.1498	−1.19598	−0.597989	−	0.801504i	$-0.704033\pi$
$7$	−22.1498	−1.19598	−0.597989	+	0.801504i	$0.704033\pi$
$8$	0	0
$9$	30.3919	1.12563
$10$	0	0
$11$	0	0
$11$	0	0
$12$	0	0
$13$	−76.8269	−1.63907	−0.819537	−	0.573027i	$-0.805769\pi$
$13$	−76.8269	−1.63907	−0.819537	+	0.573027i	$0.805769\pi$
$14$	0	0
$15$	−40.9704	−0.705234
$16$	0	0
$17$	−59.2883	−0.845855	−0.422927	−	0.906164i	$-0.638997\pi$
$17$	−59.2883	−0.845855	−0.422927	+	0.906164i	$0.638997\pi$
$18$	0	0
$19$	95.2626	1.15025	0.575124	−	0.818066i	$-0.304954\pi$
$19$	95.2626	1.15025	0.575124	+	0.818066i	$0.304954\pi$
$20$	0	0
$21$	167.801	1.74368
$22$	0	0
$23$	−142.484	−1.29174	−0.645868	−	0.763449i	$-0.723505\pi$
$23$	−142.484	−1.29174	−0.645868	+	0.763449i	$0.723505\pi$
$24$	0	0
$25$	−95.7524	−0.766020
$26$	0	0
$27$	−25.6965	−0.183159
$28$	0	0
$29$	20.4183	0.130744	0.0653721	−	0.997861i	$-0.479177\pi$
$29$	20.4183	0.130744	0.0653721	+	0.997861i	$0.479177\pi$
$30$	0	0
$31$	−213.304	−1.23582	−0.617912	−	0.786247i	$-0.712021\pi$
$31$	−213.304	−1.23582	−0.617912	+	0.786247i	$0.712021\pi$
$32$	0	0
$33$	0	0
$34$	0	0
$35$	−119.788	−0.578513
$36$	0	0
$37$	−145.578	−0.646832	−0.323416	−	0.946257i	$-0.604831\pi$
$37$	−145.578	−0.646832	−0.323416	+	0.946257i	$0.604831\pi$
$38$	0	0
$39$	582.021	2.38969
$40$	0	0
$41$	−82.8326	−0.315519	−0.157759	−	0.987478i	$-0.550427\pi$
$41$	−82.8326	−0.315519	−0.157759	+	0.987478i	$0.550427\pi$
$42$	0	0
$43$	−151.373	−0.536841	−0.268420	−	0.963302i	$-0.586502\pi$
$43$	−151.373	−0.536841	−0.268420	+	0.963302i	$0.586502\pi$
$44$	0	0
$45$	164.363	0.544483
$46$	0	0
$47$	−90.3643	−0.280447	−0.140223	−	0.990120i	$-0.544782\pi$
$47$	−90.3643	−0.280447	−0.140223	+	0.990120i	$0.544782\pi$
$48$	0	0
$49$	147.615	0.430363
$50$	0	0
$51$	449.153	1.23322
$52$	0	0
$53$	−234.849	−0.608660	−0.304330	−	0.952567i	$-0.598433\pi$
$53$	−234.849	−0.608660	−0.304330	+	0.952567i	$0.598433\pi$
$54$	0	0
$55$	0	0
$56$	0	0
$57$	−721.685	−1.67701
$58$	0	0
$59$	−302.497	−0.667488	−0.333744	−	0.942664i	$-0.608312\pi$
$59$	−302.497	−0.667488	−0.333744	+	0.942664i	$0.608312\pi$
$60$	0	0
$61$	149.279	0.313332	0.156666	−	0.987652i	$-0.449925\pi$
$61$	149.279	0.313332	0.156666	+	0.987652i	$0.449925\pi$
$62$	0	0
$63$	−673.176	−1.34623
$64$	0	0
$65$	−415.488	−0.792845
$66$	0	0
$67$	−826.236	−1.50658	−0.753290	−	0.657689i	$-0.771534\pi$
$67$	−826.236	−1.50658	−0.753290	+	0.657689i	$0.771534\pi$
$68$	0	0
$69$	1079.42	1.88329
$70$	0	0
$71$	898.965	1.50264	0.751321	−	0.659937i	$-0.229417\pi$
$71$	898.965	1.50264	0.751321	+	0.659937i	$0.229417\pi$
$72$	0	0
$73$	137.993	0.221245	0.110623	−	0.993862i	$-0.464716\pi$
$73$	137.993	0.221245	0.110623	+	0.993862i	$0.464716\pi$
$74$	0	0
$75$	725.396	1.11682
$76$	0	0
$77$	0	0
$78$	0	0
$79$	−304.459	−0.433598	−0.216799	−	0.976216i	$-0.569562\pi$
$79$	−304.459	−0.433598	−0.216799	+	0.976216i	$0.569562\pi$
$80$	0	0
$81$	−625.912	−0.858590
$82$	0	0
$83$	−764.294	−1.01075	−0.505374	−	0.862900i	$-0.668646\pi$
$83$	−764.294	−1.01075	−0.505374	+	0.862900i	$0.668646\pi$
$84$	0	0
$85$	−320.637	−0.409153
$86$	0	0
$87$	−154.684	−0.190619
$88$	0	0
$89$	−313.100	−0.372905	−0.186452	−	0.982464i	$-0.559699\pi$
$89$	−313.100	−0.372905	−0.186452	+	0.982464i	$0.559699\pi$
$90$	0	0
$91$	1701.70	1.96030
$92$	0	0
$93$	1615.94	1.80177
$94$	0	0
$95$	515.189	0.556393
$96$	0	0
$97$	−582.505	−0.609737	−0.304868	−	0.952394i	$-0.598612\pi$
$97$	−582.505	−0.609737	−0.304868	+	0.952394i	$0.598612\pi$
$98$	0	0
$99$	0	0
$100$	0	0
$101$	141.554	0.139457	0.0697286	−	0.997566i	$-0.477787\pi$
$101$	141.554	0.139457	0.0697286	+	0.997566i	$0.477787\pi$
$102$	0	0
$103$	−841.380	−0.804890	−0.402445	−	0.915444i	$-0.631839\pi$
$103$	−841.380	−0.804890	−0.402445	+	0.915444i	$0.631839\pi$
$104$	0	0
$105$	907.487	0.843444
$106$	0	0
$107$	−64.4319	−0.0582137	−0.0291069	−	0.999576i	$-0.509266\pi$
$107$	−64.4319	−0.0582137	−0.0291069	+	0.999576i	$0.509266\pi$
$108$	0	0
$109$	−1559.04	−1.36999	−0.684995	−	0.728547i	$-0.740196\pi$
$109$	−1559.04	−1.36999	−0.684995	+	0.728547i	$0.740196\pi$
$110$	0	0
$111$	1102.86	0.943052
$112$	0	0
$113$	−2239.14	−1.86408	−0.932039	−	0.362359i	$-0.881971\pi$
$113$	−2239.14	−1.86408	−0.932039	+	0.362359i	$0.881971\pi$
$114$	0	0
$115$	−770.567	−0.624833
$116$	0	0
$117$	−2334.92	−1.84499
$118$	0	0
$119$	1313.23	1.01162
$120$	0	0
$121$	0	0
$122$	0	0
$123$	627.519	0.460012
$124$	0	0
$125$	−1193.85	−0.854251
$126$	0	0
$127$	1093.46	0.764004	0.382002	−	0.924162i	$-0.375235\pi$
$127$	1093.46	0.764004	0.382002	+	0.924162i	$0.375235\pi$
$128$	0	0
$129$	1146.76	0.782688
$130$	0	0
$131$	−2466.16	−1.64481	−0.822403	−	0.568906i	$-0.807367\pi$
$131$	−2466.16	−1.64481	−0.822403	+	0.568906i	$0.807367\pi$
$132$	0	0
$133$	−2110.05	−1.37567
$134$	0	0
$135$	−138.970	−0.0885970
$136$	0	0
$137$	−973.286	−0.606959	−0.303480	−	0.952838i	$-0.598148\pi$
$137$	−973.286	−0.606959	−0.303480	+	0.952838i	$0.598148\pi$
$138$	0	0
$139$	−1263.04	−0.770717	−0.385358	−	0.922767i	$-0.625922\pi$
$139$	−1263.04	−0.770717	−0.385358	+	0.922767i	$0.625922\pi$
$140$	0	0
$141$	684.577	0.408878
$142$	0	0
$143$	0	0
$144$	0	0
$145$	110.424	0.0632430
$146$	0	0
$147$	−1118.29	−0.627449
$148$	0	0
$149$	489.340	0.269049	0.134525	−	0.990910i	$-0.457049\pi$
$149$	489.340	0.269049	0.134525	+	0.990910i	$0.457049\pi$
$150$	0	0
$151$	1707.77	0.920376	0.460188	−	0.887822i	$-0.347782\pi$
$151$	1707.77	0.920376	0.460188	+	0.887822i	$0.347782\pi$
$152$	0	0
$153$	−1801.89	−0.952118
$154$	0	0
$155$	−1153.57	−0.597787
$156$	0	0
$157$	2561.73	1.30222	0.651108	−	0.758985i	$-0.274304\pi$
$157$	2561.73	1.30222	0.651108	+	0.758985i	$0.274304\pi$
$158$	0	0
$159$	1779.16	0.887397
$160$	0	0
$161$	3155.99	1.54489
$162$	0	0
$163$	−1816.09	−0.872681	−0.436340	−	0.899782i	$-0.643726\pi$
$163$	−1816.09	−0.872681	−0.436340	+	0.899782i	$0.643726\pi$
$164$	0	0
$165$	0	0
$166$	0	0
$167$	378.925	0.175581	0.0877907	−	0.996139i	$-0.472019\pi$
$167$	378.925	0.175581	0.0877907	+	0.996139i	$0.472019\pi$
$168$	0	0
$169$	3705.38	1.68656
$170$	0	0
$171$	2895.21	1.29475
$172$	0	0
$173$	−50.7220	−0.0222909	−0.0111454	−	0.999938i	$-0.503548\pi$
$173$	−50.7220	−0.0222909	−0.0111454	+	0.999938i	$0.503548\pi$
$174$	0	0
$175$	2120.90	0.916142
$176$	0	0
$177$	2291.64	0.973167
$178$	0	0
$179$	−20.8678	−0.00871359	−0.00435679	−	0.999991i	$-0.501387\pi$
$179$	−20.8678	−0.00871359	−0.00435679	+	0.999991i	$0.501387\pi$
$180$	0	0
$181$	−176.153	−0.0723391	−0.0361695	−	0.999346i	$-0.511516\pi$
$181$	−176.153	−0.0723391	−0.0361695	+	0.999346i	$0.511516\pi$
$182$	0	0
$183$	−1130.90	−0.456824
$184$	0	0
$185$	−787.298	−0.312883
$186$	0	0
$187$	0	0
$188$	0	0
$189$	569.174	0.219055
$190$	0	0
$191$	1159.22	0.439153	0.219576	−	0.975595i	$-0.429532\pi$
$191$	1159.22	0.439153	0.219576	+	0.975595i	$0.429532\pi$
$192$	0	0
$193$	1284.94	0.479232	0.239616	−	0.970868i	$-0.422978\pi$
$193$	1284.94	0.479232	0.239616	+	0.970868i	$0.422978\pi$
$194$	0	0
$195$	3147.63	1.15593
$196$	0	0
$197$	2685.06	0.971078	0.485539	−	0.874215i	$-0.338623\pi$
$197$	2685.06	0.971078	0.485539	+	0.874215i	$0.338623\pi$
$198$	0	0
$199$	1333.54	0.475036	0.237518	−	0.971383i	$-0.423666\pi$
$199$	1333.54	0.475036	0.237518	+	0.971383i	$0.423666\pi$
$200$	0	0
$201$	6259.36	2.19652
$202$	0	0
$203$	−452.262	−0.156367
$204$	0	0
$205$	−447.967	−0.152621
$206$	0	0
$207$	−4330.36	−1.45401
$208$	0	0
$209$	0	0
$210$	0	0
$211$	−1628.17	−0.531221	−0.265611	−	0.964080i	$-0.585574\pi$
$211$	−1628.17	−0.531221	−0.265611	+	0.964080i	$0.585574\pi$
$212$	0	0
$213$	−6810.33	−2.19078
$214$	0	0
$215$	−818.640	−0.259678
$216$	0	0
$217$	4724.65	1.47802
$218$	0	0
$219$	−1045.40	−0.322565
$220$	0	0
$221$	4554.94	1.38642
$222$	0	0
$223$	2628.19	0.789224	0.394612	−	0.918848i	$-0.370879\pi$
$223$	2628.19	0.789224	0.394612	+	0.918848i	$0.370879\pi$
$224$	0	0
$225$	−2910.10	−0.862253
$226$	0	0
$227$	1702.56	0.497809	0.248904	−	0.968528i	$-0.419930\pi$
$227$	1702.56	0.497809	0.248904	+	0.968528i	$0.419930\pi$
$228$	0	0
$229$	5395.95	1.55709	0.778546	−	0.627587i	$-0.215957\pi$
$229$	5395.95	1.55709	0.778546	+	0.627587i	$0.215957\pi$
$230$	0	0
$231$	0	0
$232$	0	0
$233$	6054.18	1.70224	0.851122	−	0.524969i	$-0.175923\pi$
$233$	6054.18	1.70224	0.851122	+	0.524969i	$0.175923\pi$
$234$	0	0
$235$	−488.699	−0.135656
$236$	0	0
$237$	2306.50	0.632166
$238$	0	0
$239$	3136.60	0.848911	0.424456	−	0.905449i	$-0.360466\pi$
$239$	3136.60	0.848911	0.424456	+	0.905449i	$0.360466\pi$
$240$	0	0
$241$	6499.26	1.73715	0.868577	−	0.495555i	$-0.165035\pi$
$241$	6499.26	1.73715	0.868577	+	0.495555i	$0.165035\pi$
$242$	0	0
$243$	5435.56	1.43494
$244$	0	0
$245$	798.314	0.208173
$246$	0	0
$247$	−7318.73	−1.88534
$248$	0	0
$249$	5790.10	1.47362
$250$	0	0
$251$	−5565.79	−1.39964	−0.699820	−	0.714320i	$-0.746736\pi$
$251$	−5565.79	−1.39964	−0.699820	+	0.714320i	$0.746736\pi$
$252$	0	0
$253$	0	0
$254$	0	0
$255$	2429.07	0.596526
$256$	0	0
$257$	−6036.31	−1.46512	−0.732558	−	0.680705i	$-0.761673\pi$
$257$	−6036.31	−1.46512	−0.732558	+	0.680705i	$0.761673\pi$
$258$	0	0
$259$	3224.52	0.773597
$260$	0	0
$261$	620.552	0.147169
$262$	0	0
$263$	−708.442	−0.166100	−0.0830502	−	0.996545i	$-0.526466\pi$
$263$	−708.442	−0.166100	−0.0830502	+	0.996545i	$0.526466\pi$
$264$	0	0
$265$	−1270.09	−0.294418
$266$	0	0
$267$	2371.96	0.543677
$268$	0	0
$269$	−4809.71	−1.09016	−0.545080	−	0.838384i	$-0.683501\pi$
$269$	−4809.71	−1.09016	−0.545080	+	0.838384i	$0.683501\pi$
$270$	0	0
$271$	−5240.88	−1.17476	−0.587382	−	0.809310i	$-0.699841\pi$
$271$	−5240.88	−1.17476	−0.587382	+	0.809310i	$0.699841\pi$
$272$	0	0
$273$	−12891.7	−2.85802
$274$	0	0
$275$	0	0
$276$	0	0
$277$	5339.75	1.15825	0.579123	−	0.815240i	$-0.303395\pi$
$277$	5339.75	1.15825	0.579123	+	0.815240i	$0.303395\pi$
$278$	0	0
$279$	−6482.73	−1.39108
$280$	0	0
$281$	3279.97	0.696322	0.348161	−	0.937435i	$-0.386806\pi$
$281$	3279.97	0.696322	0.348161	+	0.937435i	$0.386806\pi$
$282$	0	0
$283$	−4895.49	−1.02829	−0.514146	−	0.857703i	$-0.671891\pi$
$283$	−4895.49	−1.02829	−0.514146	+	0.857703i	$0.671891\pi$
$284$	0	0
$285$	−3902.95	−0.811195
$286$	0	0
$287$	1834.73	0.377354
$288$	0	0
$289$	−1397.89	−0.284529
$290$	0	0
$291$	4412.91	0.888968
$292$	0	0
$293$	−6705.80	−1.33705	−0.668527	−	0.743688i	$-0.733075\pi$
$293$	−6705.80	−1.33705	−0.668527	+	0.743688i	$0.733075\pi$
$294$	0	0
$295$	−1635.94	−0.322874
$296$	0	0
$297$	0	0
$298$	0	0
$299$	10946.6	2.11725
$300$	0	0
$301$	3352.88	0.642049
$302$	0	0
$303$	−1072.38	−0.203322
$304$	0	0
$305$	807.318	0.151564
$306$	0	0
$307$	9507.29	1.76746	0.883730	−	0.467998i	$-0.155025\pi$
$307$	9507.29	1.76746	0.883730	+	0.467998i	$0.155025\pi$
$308$	0	0
$309$	6374.08	1.17349
$310$	0	0
$311$	−5768.07	−1.05170	−0.525848	−	0.850579i	$-0.676252\pi$
$311$	−5768.07	−1.05170	−0.525848	+	0.850579i	$0.676252\pi$
$312$	0	0
$313$	5977.75	1.07950	0.539748	−	0.841826i	$-0.318519\pi$
$313$	5977.75	1.07950	0.539748	+	0.841826i	$0.318519\pi$
$314$	0	0
$315$	−3640.60	−0.651190
$316$	0	0
$317$	5231.93	0.926987	0.463493	−	0.886100i	$-0.346596\pi$
$317$	5231.93	0.926987	0.463493	+	0.886100i	$0.346596\pi$
$318$	0	0
$319$	0	0
$320$	0	0
$321$	488.120	0.0848729
$322$	0	0
$323$	−5647.96	−0.972944
$324$	0	0
$325$	7356.37	1.25556
$326$	0	0
$327$	11810.9	1.99738
$328$	0	0
$329$	2001.55	0.335408
$330$	0	0
$331$	3963.72	0.658205	0.329102	−	0.944294i	$-0.393254\pi$
$331$	3963.72	0.658205	0.329102	+	0.944294i	$0.393254\pi$
$332$	0	0
$333$	−4424.39	−0.728093
$334$	0	0
$335$	−4468.37	−0.728756
$336$	0	0
$337$	−5106.60	−0.825443	−0.412721	−	0.910857i	$-0.635422\pi$
$337$	−5106.60	−0.825443	−0.412721	+	0.910857i	$0.635422\pi$
$338$	0	0
$339$	16963.2	2.71774
$340$	0	0
$341$	0	0
$342$	0	0
$343$	4327.75	0.681273
$344$	0	0
$345$	5837.62	0.910977
$346$	0	0
$347$	7545.75	1.16737	0.583685	−	0.811980i	$-0.301610\pi$
$347$	7545.75	1.16737	0.583685	+	0.811980i	$0.301610\pi$
$348$	0	0
$349$	3256.71	0.499507	0.249753	−	0.968309i	$-0.419650\pi$
$349$	3256.71	0.499507	0.249753	+	0.968309i	$0.419650\pi$
$350$	0	0
$351$	1974.19	0.300212
$352$	0	0
$353$	−6506.83	−0.981087	−0.490543	−	0.871417i	$-0.663202\pi$
$353$	−6506.83	−0.981087	−0.490543	+	0.871417i	$0.663202\pi$
$354$	0	0
$355$	4861.69	0.726851
$356$	0	0
$357$	−9948.67	−1.47490
$358$	0	0
$359$	4066.21	0.597789	0.298895	−	0.954286i	$-0.403382\pi$
$359$	4066.21	0.597789	0.298895	+	0.954286i	$0.403382\pi$
$360$	0	0
$361$	2215.95	0.323073
$362$	0	0
$363$	0	0
$364$	0	0
$365$	746.282	0.107020
$366$	0	0
$367$	1392.27	0.198028	0.0990138	−	0.995086i	$-0.468431\pi$
$367$	1392.27	0.198028	0.0990138	+	0.995086i	$0.468431\pi$
$368$	0	0
$369$	−2517.44	−0.355157
$370$	0	0
$371$	5201.86	0.727944
$372$	0	0
$373$	−9440.30	−1.31046	−0.655228	−	0.755431i	$-0.727428\pi$
$373$	−9440.30	−1.31046	−0.655228	+	0.755431i	$0.727428\pi$
$374$	0	0
$375$	9044.32	1.24546
$376$	0	0
$377$	−1568.68	−0.214299
$378$	0	0
$379$	−1290.35	−0.174884	−0.0874418	−	0.996170i	$-0.527869\pi$
$379$	−1290.35	−0.174884	−0.0874418	+	0.996170i	$0.527869\pi$
$380$	0	0
$381$	−8283.74	−1.11388
$382$	0	0
$383$	5652.89	0.754176	0.377088	−	0.926177i	$-0.376925\pi$
$383$	5652.89	0.754176	0.377088	+	0.926177i	$0.376925\pi$
$384$	0	0
$385$	0	0
$386$	0	0
$387$	−4600.52	−0.604283
$388$	0	0
$389$	−12838.0	−1.67330	−0.836648	−	0.547741i	$-0.815488\pi$
$389$	−12838.0	−1.67330	−0.836648	+	0.547741i	$0.815488\pi$
$390$	0	0
$391$	8447.63	1.09262
$392$	0	0
$393$	18683.0	2.39805
$394$	0	0
$395$	−1646.54	−0.209738
$396$	0	0
$397$	7691.26	0.972326	0.486163	−	0.873868i	$-0.338396\pi$
$397$	7691.26	0.972326	0.486163	+	0.873868i	$0.338396\pi$
$398$	0	0
$399$	15985.2	2.00567
$400$	0	0
$401$	7786.17	0.969633	0.484817	−	0.874616i	$-0.338886\pi$
$401$	7786.17	0.969633	0.484817	+	0.874616i	$0.338886\pi$
$402$	0	0
$403$	16387.5	2.02561
$404$	0	0
$405$	−3385.00	−0.415313
$406$	0	0
$407$	0	0
$408$	0	0
$409$	8859.88	1.07113	0.535566	−	0.844493i	$-0.320098\pi$
$409$	8859.88	1.07113	0.535566	+	0.844493i	$0.320098\pi$
$410$	0	0
$411$	7373.37	0.884918
$412$	0	0
$413$	6700.26	0.798301
$414$	0	0
$415$	−4133.38	−0.488914
$416$	0	0
$417$	9568.47	1.12367
$418$	0	0
$419$	−9469.08	−1.10405	−0.552023	−	0.833829i	$-0.686144\pi$
$419$	−9469.08	−1.10405	−0.552023	+	0.833829i	$0.686144\pi$
$420$	0	0
$421$	5177.94	0.599424	0.299712	−	0.954030i	$-0.403109\pi$
$421$	5177.94	0.599424	0.299712	+	0.954030i	$0.403109\pi$
$422$	0	0
$423$	−2746.35	−0.315678
$424$	0	0
$425$	5677.00	0.647941
$426$	0	0
$427$	−3306.51	−0.374739
$428$	0	0
$429$	0	0
$430$	0	0
$431$	10463.6	1.16940	0.584700	−	0.811249i	$-0.301212\pi$
$431$	10463.6	1.16940	0.584700	+	0.811249i	$0.301212\pi$
$432$	0	0
$433$	15838.2	1.75782	0.878911	−	0.476986i	$-0.158271\pi$
$433$	15838.2	1.75782	0.878911	+	0.476986i	$0.158271\pi$
$434$	0	0
$435$	−836.546	−0.0922053
$436$	0	0
$437$	−13573.4	−1.48582
$438$	0	0
$439$	−4824.70	−0.524534	−0.262267	−	0.964995i	$-0.584470\pi$
$439$	−4824.70	−0.524534	−0.262267	+	0.964995i	$0.584470\pi$
$440$	0	0
$441$	4486.29	0.484429
$442$	0	0
$443$	−3583.38	−0.384315	−0.192158	−	0.981364i	$-0.561549\pi$
$443$	−3583.38	−0.384315	−0.192158	+	0.981364i	$0.561549\pi$
$444$	0	0
$445$	−1693.27	−0.180380
$446$	0	0
$447$	−3707.12	−0.392261
$448$	0	0
$449$	−7795.98	−0.819410	−0.409705	−	0.912218i	$-0.634368\pi$
$449$	−7795.98	−0.819410	−0.409705	+	0.912218i	$0.634368\pi$
$450$	0	0
$451$	0	0
$452$	0	0
$453$	−12937.7	−1.34186
$454$	0	0
$455$	9202.98	0.948225
$456$	0	0
$457$	−3348.02	−0.342700	−0.171350	−	0.985210i	$-0.554813\pi$
$457$	−3348.02	−0.342700	−0.171350	+	0.985210i	$0.554813\pi$
$458$	0	0
$459$	1523.51	0.154926
$460$	0	0
$461$	−7171.96	−0.724580	−0.362290	−	0.932065i	$-0.618005\pi$
$461$	−7171.96	−0.724580	−0.362290	+	0.932065i	$0.618005\pi$
$462$	0	0
$463$	−4034.74	−0.404990	−0.202495	−	0.979283i	$-0.564905\pi$
$463$	−4034.74	−0.404990	−0.202495	+	0.979283i	$0.564905\pi$
$464$	0	0
$465$	8739.16	0.871546
$466$	0	0
$467$	11821.8	1.17141	0.585706	−	0.810523i	$-0.300817\pi$
$467$	11821.8	1.17141	0.585706	+	0.810523i	$0.300817\pi$
$468$	0	0
$469$	18301.0	1.80184
$470$	0	0
$471$	−19407.0	−1.89857
$472$	0	0
$473$	0	0
$474$	0	0
$475$	−9121.62	−0.881113
$476$	0	0
$477$	−7137.51	−0.685124
$478$	0	0
$479$	−8659.94	−0.826060	−0.413030	−	0.910717i	$-0.635530\pi$
$479$	−8659.94	−0.826060	−0.413030	+	0.910717i	$0.635530\pi$
$480$	0	0
$481$	11184.3	1.06021
$482$	0	0
$483$	−23909.0	−2.25238
$484$	0	0
$485$	−3150.25	−0.294939
$486$	0	0
$487$	−7427.66	−0.691128	−0.345564	−	0.938395i	$-0.612312\pi$
$487$	−7427.66	−0.691128	−0.345564	+	0.938395i	$0.612312\pi$
$488$	0	0
$489$	13758.2	1.27233
$490$	0	0
$491$	−5543.89	−0.509556	−0.254778	−	0.967000i	$-0.582002\pi$
$491$	−5543.89	−0.509556	−0.254778	+	0.967000i	$0.582002\pi$
$492$	0	0
$493$	−1210.57	−0.110591
$494$	0	0
$495$	0	0
$496$	0	0
$497$	−19911.9	−1.79713
$498$	0	0
$499$	−20412.0	−1.83120	−0.915599	−	0.402093i	$-0.868283\pi$
$499$	−20412.0	−1.83120	−0.915599	+	0.402093i	$0.868283\pi$
$500$	0	0
$501$	−2870.64	−0.255990
$502$	0	0
$503$	−182.742	−0.0161989	−0.00809946	−	0.999967i	$-0.502578\pi$
$503$	−182.742	−0.0161989	−0.00809946	+	0.999967i	$0.502578\pi$
$504$	0	0
$505$	765.540	0.0674576
$506$	0	0
$507$	−28071.0	−2.45893
$508$	0	0
$509$	4755.29	0.414095	0.207048	−	0.978331i	$-0.433614\pi$
$509$	4755.29	0.414095	0.207048	+	0.978331i	$0.433614\pi$
$510$	0	0
$511$	−3056.53	−0.264604
$512$	0	0
$513$	−2447.92	−0.210679
$514$	0	0
$515$	−4550.27	−0.389337
$516$	0	0
$517$	0	0
$518$	0	0
$519$	384.257	0.0324990
$520$	0	0
$521$	−11731.1	−0.986470	−0.493235	−	0.869896i	$-0.664186\pi$
$521$	−11731.1	−0.986470	−0.493235	+	0.869896i	$0.664186\pi$
$522$	0	0
$523$	−343.839	−0.0287477	−0.0143738	−	0.999897i	$-0.504575\pi$
$523$	−343.839	−0.0287477	−0.0143738	+	0.999897i	$0.504575\pi$
$524$	0	0
$525$	−16067.4	−1.33569
$526$	0	0
$527$	12646.4	1.04533
$528$	0	0
$529$	8134.66	0.668584
$530$	0	0
$531$	−9193.49	−0.751343
$532$	0	0
$533$	6363.77	0.517159
$534$	0	0
$535$	−348.454	−0.0281589
$536$	0	0
$537$	158.089	0.0127040
$538$	0	0
$539$	0	0
$540$	0	0
$541$	17243.8	1.37037	0.685183	−	0.728371i	$-0.259722\pi$
$541$	17243.8	1.37037	0.685183	+	0.728371i	$0.259722\pi$
$542$	0	0
$543$	1334.49	0.105467
$544$	0	0
$545$	−8431.45	−0.662685
$546$	0	0
$547$	−7643.21	−0.597441	−0.298720	−	0.954341i	$-0.596560\pi$
$547$	−7643.21	−0.597441	−0.298720	+	0.954341i	$0.596560\pi$
$548$	0	0
$549$	4536.89	0.352696
$550$	0	0
$551$	1945.10	0.150388
$552$	0	0
$553$	6743.70	0.518574
$554$	0	0
$555$	5964.37	0.456168
$556$	0	0
$557$	−26106.2	−1.98591	−0.992957	−	0.118471i	$-0.962201\pi$
$557$	−26106.2	−1.98591	−0.992957	+	0.118471i	$0.962201\pi$
$558$	0	0
$559$	11629.5	0.879921
$560$	0	0
$561$	0	0
$562$	0	0
$563$	−18073.9	−1.35297	−0.676487	−	0.736455i	$-0.736498\pi$
$563$	−18073.9	−1.35297	−0.676487	+	0.736455i	$0.736498\pi$
$564$	0	0
$565$	−12109.5	−0.901682
$566$	0	0
$567$	13863.8	1.02685
$568$	0	0
$569$	−7805.16	−0.575060	−0.287530	−	0.957772i	$-0.592834\pi$
$569$	−7805.16	−0.575060	−0.287530	+	0.957772i	$0.592834\pi$
$570$	0	0
$571$	−13039.6	−0.955672	−0.477836	−	0.878449i	$-0.658579\pi$
$571$	−13039.6	−0.955672	−0.477836	+	0.878449i	$0.658579\pi$
$572$	0	0
$573$	−8781.96	−0.640264
$574$	0	0
$575$	13643.2	0.989496
$576$	0	0
$577$	−9569.43	−0.690434	−0.345217	−	0.938523i	$-0.612195\pi$
$577$	−9569.43	−0.690434	−0.345217	+	0.938523i	$0.612195\pi$
$578$	0	0
$579$	−9734.36	−0.698698
$580$	0	0
$581$	16929.0	1.20883
$582$	0	0
$583$	0	0
$584$	0	0
$585$	−12627.5	−0.892448
$586$	0	0
$587$	−9727.00	−0.683946	−0.341973	−	0.939710i	$-0.611095\pi$
$587$	−9727.00	−0.683946	−0.341973	+	0.939710i	$0.611095\pi$
$588$	0	0
$589$	−20319.9	−1.42151
$590$	0	0
$591$	−20341.3	−1.41579
$592$	0	0
$593$	−6926.77	−0.479677	−0.239838	−	0.970813i	$-0.577094\pi$
$593$	−6926.77	−0.479677	−0.239838	+	0.970813i	$0.577094\pi$
$594$	0	0
$595$	7102.06	0.489338
$596$	0	0
$597$	−10102.6	−0.692580
$598$	0	0
$599$	21984.7	1.49961	0.749807	−	0.661656i	$-0.230146\pi$
$599$	21984.7	1.49961	0.749807	+	0.661656i	$0.230146\pi$
$600$	0	0
$601$	7363.27	0.499757	0.249878	−	0.968277i	$-0.419609\pi$
$601$	7363.27	0.499757	0.249878	+	0.968277i	$0.419609\pi$
$602$	0	0
$603$	−25110.9	−1.69585
$604$	0	0
$605$	0	0
$606$	0	0
$607$	10168.4	0.679938	0.339969	−	0.940437i	$-0.389583\pi$
$607$	10168.4	0.679938	0.339969	+	0.940437i	$0.389583\pi$
$608$	0	0
$609$	3426.22	0.227976
$610$	0	0
$611$	6942.41	0.459673
$612$	0	0
$613$	−25084.5	−1.65278	−0.826389	−	0.563099i	$-0.809609\pi$
$613$	−25084.5	−1.65278	−0.826389	+	0.563099i	$0.809609\pi$
$614$	0	0
$615$	3393.68	0.222515
$616$	0	0
$617$	−26335.5	−1.71836	−0.859178	−	0.511677i	$-0.829024\pi$
$617$	−26335.5	−1.71836	−0.859178	+	0.511677i	$0.829024\pi$
$618$	0	0
$619$	24643.2	1.60015	0.800075	−	0.599901i	$-0.204793\pi$
$619$	24643.2	1.60015	0.800075	+	0.599901i	$0.204793\pi$
$620$	0	0
$621$	3661.34	0.236594
$622$	0	0
$623$	6935.10	0.445986
$624$	0	0
$625$	5512.59	0.352806
$626$	0	0
$627$	0	0
$628$	0	0
$629$	8631.05	0.547126
$630$	0	0
$631$	−7455.32	−0.470351	−0.235176	−	0.971953i	$-0.575567\pi$
$631$	−7455.32	−0.470351	−0.235176	+	0.971953i	$0.575567\pi$
$632$	0	0
$633$	12334.6	0.774496
$634$	0	0
$635$	5913.52	0.369560
$636$	0	0
$637$	−11340.8	−0.705397
$638$	0	0
$639$	27321.3	1.69141
$640$	0	0
$641$	−2778.67	−0.171218	−0.0856090	−	0.996329i	$-0.527284\pi$
$641$	−2778.67	−0.171218	−0.0856090	+	0.996329i	$0.527284\pi$
$642$	0	0
$643$	−7557.71	−0.463526	−0.231763	−	0.972772i	$-0.574449\pi$
$643$	−7557.71	−0.463526	−0.231763	+	0.972772i	$0.574449\pi$
$644$	0	0
$645$	6201.81	0.378598
$646$	0	0
$647$	27430.7	1.66679	0.833394	−	0.552679i	$-0.186395\pi$
$647$	27430.7	1.66679	0.833394	+	0.552679i	$0.186395\pi$
$648$	0	0
$649$	0	0
$650$	0	0
$651$	−35792.7	−2.15488
$652$	0	0
$653$	284.728	0.0170632	0.00853160	−	0.999964i	$-0.497284\pi$
$653$	284.728	0.0170632	0.00853160	+	0.999964i	$0.497284\pi$
$654$	0	0
$655$	−13337.2	−0.795617
$656$	0	0
$657$	4193.89	0.249040
$658$	0	0
$659$	20747.0	1.22639	0.613193	−	0.789933i	$-0.289885\pi$
$659$	20747.0	1.22639	0.613193	+	0.789933i	$0.289885\pi$
$660$	0	0
$661$	−18908.8	−1.11266	−0.556328	−	0.830963i	$-0.687790\pi$
$661$	−18908.8	−1.11266	−0.556328	+	0.830963i	$0.687790\pi$
$662$	0	0
$663$	−34507.1	−2.02133
$664$	0	0
$665$	−11411.4	−0.665434
$666$	0	0
$667$	−2909.28	−0.168887
$668$	0	0
$669$	−19910.5	−1.15065
$670$	0	0
$671$	0	0
$672$	0	0
$673$	−29865.9	−1.71062	−0.855309	−	0.518118i	$-0.826633\pi$
$673$	−29865.9	−1.71062	−0.855309	+	0.518118i	$0.826633\pi$
$674$	0	0
$675$	2460.51	0.140304
$676$	0	0
$677$	17280.7	0.981024	0.490512	−	0.871435i	$-0.336810\pi$
$677$	17280.7	0.981024	0.490512	+	0.871435i	$0.336810\pi$
$678$	0	0
$679$	12902.4	0.729232
$680$	0	0
$681$	−12898.1	−0.725782
$682$	0	0
$683$	−25104.0	−1.40641	−0.703204	−	0.710988i	$-0.748248\pi$
$683$	−25104.0	−1.40641	−0.703204	+	0.710988i	$0.748248\pi$
$684$	0	0
$685$	−5263.63	−0.293595
$686$	0	0
$687$	−40878.3	−2.27017
$688$	0	0
$689$	18042.7	0.997638
$690$	0	0
$691$	26163.8	1.44040	0.720201	−	0.693765i	$-0.244050\pi$
$691$	26163.8	1.44040	0.720201	+	0.693765i	$0.244050\pi$
$692$	0	0
$693$	0	0
$694$	0	0
$695$	−6830.64	−0.372807
$696$	0	0
$697$	4911.00	0.266883
$698$	0	0
$699$	−45865.0	−2.48179
$700$	0	0
$701$	1519.45	0.0818669	0.0409334	−	0.999162i	$-0.486967\pi$
$701$	1519.45	0.0818669	0.0409334	+	0.999162i	$0.486967\pi$
$702$	0	0
$703$	−13868.1	−0.744018
$704$	0	0
$705$	3702.26	0.197781
$706$	0	0
$707$	−3135.40	−0.166788
$708$	0	0
$709$	−21733.9	−1.15125	−0.575623	−	0.817715i	$-0.695240\pi$
$709$	−21733.9	−1.15125	−0.575623	+	0.817715i	$0.695240\pi$
$710$	0	0
$711$	−9253.09	−0.488070
$712$	0	0
$713$	30392.4	1.59636
$714$	0	0
$715$	0	0
$716$	0	0
$717$	−23762.1	−1.23767
$718$	0	0
$719$	14071.8	0.729890	0.364945	−	0.931029i	$-0.381088\pi$
$719$	14071.8	0.729890	0.364945	+	0.931029i	$0.381088\pi$
$720$	0	0
$721$	18636.4	0.962630
$722$	0	0
$723$	−49236.7	−2.53269
$724$	0	0
$725$	−1955.10	−0.100153
$726$	0	0
$727$	−10409.2	−0.531027	−0.265513	−	0.964107i	$-0.585541\pi$
$727$	−10409.2	−0.531027	−0.265513	+	0.964107i	$0.585541\pi$
$728$	0	0
$729$	−24278.8	−1.23349
$730$	0	0
$731$	8974.65	0.454089
$732$	0	0
$733$	19394.9	0.977307	0.488653	−	0.872478i	$-0.337488\pi$
$733$	19394.9	0.977307	0.488653	+	0.872478i	$0.337488\pi$
$734$	0	0
$735$	−6047.83	−0.303507
$736$	0	0
$737$	0	0
$738$	0	0
$739$	−6672.24	−0.332128	−0.166064	−	0.986115i	$-0.553106\pi$
$739$	−6672.24	−0.332128	−0.166064	+	0.986115i	$0.553106\pi$
$740$	0	0
$741$	55444.8	2.74874
$742$	0	0
$743$	6565.18	0.324163	0.162082	−	0.986777i	$-0.448179\pi$
$743$	6565.18	0.324163	0.162082	+	0.986777i	$0.448179\pi$
$744$	0	0
$745$	2646.40	0.130143
$746$	0	0
$747$	−23228.4	−1.13773
$748$	0	0
$749$	1427.16	0.0696223
$750$	0	0
$751$	−18938.7	−0.920216	−0.460108	−	0.887863i	$-0.652189\pi$
$751$	−18938.7	−0.920216	−0.460108	+	0.887863i	$0.652189\pi$
$752$	0	0
$753$	42165.0	2.04061
$754$	0	0
$755$	9235.82	0.445200
$756$	0	0
$757$	−10958.4	−0.526141	−0.263070	−	0.964777i	$-0.584735\pi$
$757$	−10958.4	−0.526141	−0.263070	+	0.964777i	$0.584735\pi$
$758$	0	0
$759$	0	0
$760$	0	0
$761$	35442.2	1.68828	0.844138	−	0.536126i	$-0.180113\pi$
$761$	35442.2	1.68828	0.844138	+	0.536126i	$0.180113\pi$
$762$	0	0
$763$	34532.5	1.63848
$764$	0	0
$765$	−9744.79	−0.460554
$766$	0	0
$767$	23240.0	1.09406
$768$	0	0
$769$	−4444.21	−0.208404	−0.104202	−	0.994556i	$-0.533229\pi$
$769$	−4444.21	−0.208404	−0.104202	+	0.994556i	$0.533229\pi$
$770$	0	0
$771$	45729.5	2.13607
$772$	0	0
$773$	10160.2	0.472750	0.236375	−	0.971662i	$-0.424041\pi$
$773$	10160.2	0.472750	0.236375	+	0.971662i	$0.424041\pi$
$774$	0	0
$775$	20424.4	0.946666
$776$	0	0
$777$	−24428.1	−1.12787
$778$	0	0
$779$	−7890.84	−0.362925
$780$	0	0
$781$	0	0
$782$	0	0
$783$	−524.680	−0.0239470
$784$	0	0
$785$	13854.1	0.629902
$786$	0	0
$787$	−18270.1	−0.827520	−0.413760	−	0.910386i	$-0.635785\pi$
$787$	−18270.1	−0.827520	−0.413760	+	0.910386i	$0.635785\pi$
$788$	0	0
$789$	5366.98	0.242167
$790$	0	0
$791$	49596.6	2.22939
$792$	0	0
$793$	−11468.7	−0.513575
$794$	0	0
$795$	9621.85	0.429248
$796$	0	0
$797$	17984.0	0.799279	0.399639	−	0.916672i	$-0.369135\pi$
$797$	17984.0	0.799279	0.399639	+	0.916672i	$0.369135\pi$
$798$	0	0
$799$	5357.55	0.237217
$800$	0	0
$801$	−9515.71	−0.419752
$802$	0	0
$803$	0	0
$804$	0	0
$805$	17067.9	0.747286
$806$	0	0
$807$	36437.1	1.58940
$808$	0	0
$809$	−5144.24	−0.223562	−0.111781	−	0.993733i	$-0.535656\pi$
$809$	−5144.24	−0.223562	−0.111781	+	0.993733i	$0.535656\pi$
$810$	0	0
$811$	−8255.99	−0.357469	−0.178734	−	0.983897i	$-0.557200\pi$
$811$	−8255.99	−0.357469	−0.178734	+	0.983897i	$0.557200\pi$
$812$	0	0
$813$	39703.6	1.71275
$814$	0	0
$815$	−9821.58	−0.422129
$816$	0	0
$817$	−14420.2	−0.617500
$818$	0	0
$819$	51718.1	2.20656
$820$	0	0
$821$	−247.250	−0.0105105	−0.00525523	−	0.999986i	$-0.501673\pi$
$821$	−247.250	−0.0105105	−0.00525523	+	0.999986i	$0.501673\pi$
$822$	0	0
$823$	−11794.8	−0.499564	−0.249782	−	0.968302i	$-0.580359\pi$
$823$	−11794.8	−0.499564	−0.249782	+	0.968302i	$0.580359\pi$
$824$	0	0
$825$	0	0
$826$	0	0
$827$	−2771.66	−0.116542	−0.0582709	−	0.998301i	$-0.518559\pi$
$827$	−2771.66	−0.116542	−0.0582709	+	0.998301i	$0.518559\pi$
$828$	0	0
$829$	−27298.0	−1.14367	−0.571834	−	0.820370i	$-0.693768\pi$
$829$	−27298.0	−1.14367	−0.571834	+	0.820370i	$0.693768\pi$
$830$	0	0
$831$	−40452.6	−1.68867
$832$	0	0
$833$	−8751.82	−0.364025
$834$	0	0
$835$	2049.26	0.0849314
$836$	0	0
$837$	5481.18	0.226353
$838$	0	0
$839$	−8754.74	−0.360247	−0.180123	−	0.983644i	$-0.557650\pi$
$839$	−8754.74	−0.360247	−0.180123	+	0.983644i	$0.557650\pi$
$840$	0	0
$841$	−23972.1	−0.982906
$842$	0	0
$843$	−24848.2	−1.01521
$844$	0	0
$845$	20039.1	0.815816
$846$	0	0
$847$	0	0
$848$	0	0
$849$	37087.0	1.49920
$850$	0	0
$851$	20742.5	0.835537
$852$	0	0
$853$	−41184.3	−1.65313	−0.826566	−	0.562839i	$-0.809709\pi$
$853$	−41184.3	−1.65313	−0.826566	+	0.562839i	$0.809709\pi$
$854$	0	0
$855$	15657.6	0.626291
$856$	0	0
$857$	18020.0	0.718265	0.359132	−	0.933287i	$-0.383073\pi$
$857$	18020.0	0.718265	0.359132	+	0.933287i	$0.383073\pi$
$858$	0	0
$859$	−6159.11	−0.244640	−0.122320	−	0.992491i	$-0.539033\pi$
$859$	−6159.11	−0.244640	−0.122320	+	0.992491i	$0.539033\pi$
$860$	0	0
$861$	−13899.4	−0.550164
$862$	0	0
$863$	−28932.6	−1.14122	−0.570612	−	0.821220i	$-0.693294\pi$
$863$	−28932.6	−1.14122	−0.570612	+	0.821220i	$0.693294\pi$
$864$	0	0
$865$	−274.309	−0.0107824
$866$	0	0
$867$	10590.1	0.414831
$868$	0	0
$869$	0	0
$870$	0	0
$871$	63477.2	2.46940
$872$	0	0
$873$	−17703.5	−0.686337
$874$	0	0
$875$	26443.6	1.02166
$876$	0	0
$877$	45152.6	1.73854	0.869268	−	0.494341i	$-0.164591\pi$
$877$	45152.6	1.73854	0.869268	+	0.494341i	$0.164591\pi$
$878$	0	0
$879$	50801.4	1.94936
$880$	0	0
$881$	22263.5	0.851391	0.425696	−	0.904866i	$-0.360029\pi$
$881$	22263.5	0.851391	0.425696	+	0.904866i	$0.360029\pi$
$882$	0	0
$883$	−38215.5	−1.45646	−0.728230	−	0.685332i	$-0.759657\pi$
$883$	−38215.5	−1.45646	−0.728230	+	0.685332i	$0.759657\pi$
$884$	0	0
$885$	12393.4	0.470736
$886$	0	0
$887$	48160.6	1.82308	0.911541	−	0.411208i	$-0.134893\pi$
$887$	48160.6	1.82308	0.911541	+	0.411208i	$0.134893\pi$
$888$	0	0
$889$	−24219.8	−0.913731
$890$	0	0
$891$	0	0
$892$	0	0
$893$	−8608.34	−0.322583
$894$	0	0
$895$	−112.855	−0.00421490
$896$	0	0
$897$	−82928.7	−3.08685
$898$	0	0
$899$	−4355.31	−0.161577
$900$	0	0
$901$	13923.8	0.514838
$902$	0	0
$903$	−25400.6	−0.936078
$904$	0	0
$905$	−952.655	−0.0349915
$906$	0	0
$907$	−8509.88	−0.311539	−0.155770	−	0.987793i	$-0.549786\pi$
$907$	−8509.88	−0.311539	−0.155770	+	0.987793i	$0.549786\pi$
$908$	0	0
$909$	4302.11	0.156977
$910$	0	0
$911$	−11283.5	−0.410360	−0.205180	−	0.978724i	$-0.565778\pi$
$911$	−11283.5	−0.410360	−0.205180	+	0.978724i	$0.565778\pi$
$912$	0	0
$913$	0	0
$914$	0	0
$915$	−6116.04	−0.220973
$916$	0	0
$917$	54625.0	1.96715
$918$	0	0
$919$	26556.9	0.953245	0.476622	−	0.879108i	$-0.341861\pi$
$919$	26556.9	0.953245	0.476622	+	0.879108i	$0.341861\pi$
$920$	0	0
$921$	−72024.9	−2.57687
$922$	0	0
$923$	−69064.7	−2.46294
$924$	0	0
$925$	13939.4	0.495486
$926$	0	0
$927$	−25571.2	−0.906006
$928$	0	0
$929$	−47913.1	−1.69212	−0.846059	−	0.533089i	$-0.821031\pi$
$929$	−47913.1	−1.69212	−0.846059	+	0.533089i	$0.821031\pi$
$930$	0	0
$931$	14062.1	0.495025
$932$	0	0
$933$	43697.5	1.53332
$934$	0	0
$935$	0	0
$936$	0	0
$937$	−5718.95	−0.199391	−0.0996957	−	0.995018i	$-0.531787\pi$
$937$	−5718.95	−0.199391	−0.0996957	+	0.995018i	$0.531787\pi$
$938$	0	0
$939$	−45285.9	−1.57386
$940$	0	0
$941$	11105.6	0.384732	0.192366	−	0.981323i	$-0.438384\pi$
$941$	11105.6	0.384732	0.192366	+	0.981323i	$0.438384\pi$
$942$	0	0
$943$	11802.3	0.407567
$944$	0	0
$945$	3078.15	0.105960
$946$	0	0
$947$	−39759.1	−1.36430	−0.682152	−	0.731210i	$-0.738956\pi$
$947$	−39759.1	−1.36430	−0.682152	+	0.731210i	$0.738956\pi$
$948$	0	0
$949$	−10601.6	−0.362637
$950$	0	0
$951$	−39635.8	−1.35150
$952$	0	0
$953$	15932.0	0.541539	0.270770	−	0.962644i	$-0.412722\pi$
$953$	15932.0	0.541539	0.270770	+	0.962644i	$0.412722\pi$
$954$	0	0
$955$	6269.18	0.212425
$956$	0	0
$957$	0	0
$958$	0	0
$959$	21558.1	0.725910
$960$	0	0
$961$	15707.7	0.527262
$962$	0	0
$963$	−1958.21	−0.0655270
$964$	0	0
$965$	6949.07	0.231812
$966$	0	0
$967$	14952.9	0.497264	0.248632	−	0.968598i	$-0.420019\pi$
$967$	14952.9	0.497264	0.248632	+	0.968598i	$0.420019\pi$
$968$	0	0
$969$	42787.5	1.41851
$970$	0	0
$971$	−27937.3	−0.923327	−0.461663	−	0.887055i	$-0.652747\pi$
$971$	−27937.3	−0.923327	−0.461663	+	0.887055i	$0.652747\pi$
$972$	0	0
$973$	27976.1	0.921760
$974$	0	0
$975$	−55730.0	−1.83055
$976$	0	0
$977$	33359.2	1.09238	0.546191	−	0.837661i	$-0.316077\pi$
$977$	33359.2	1.09238	0.546191	+	0.837661i	$0.316077\pi$
$978$	0	0
$979$	0	0
$980$	0	0
$981$	−47382.3	−1.54210
$982$	0	0
$983$	−12350.1	−0.400718	−0.200359	−	0.979723i	$-0.564211\pi$
$983$	−12350.1	−0.400718	−0.200359	+	0.979723i	$0.564211\pi$
$984$	0	0
$985$	14521.1	0.469725
$986$	0	0
$987$	−15163.3	−0.489009
$988$	0	0
$989$	21568.2	0.693457
$990$	0	0
$991$	30154.0	0.966571	0.483286	−	0.875463i	$-0.339443\pi$
$991$	30154.0	0.966571	0.483286	+	0.875463i	$0.339443\pi$
$992$	0	0
$993$	−30028.1	−0.959632
$994$	0	0
$995$	7211.91	0.229782
$996$	0	0
$997$	−2093.09	−0.0664884	−0.0332442	−	0.999447i	$-0.510584\pi$
$997$	−2093.09	−0.0664884	−0.0332442	+	0.999447i	$0.510584\pi$
$998$	0	0
$999$	3740.84	0.118473

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	1936.4.a.bm.1.2		4
4.3	odd	2		242.4.a.o.1.3		4
11.7	odd	10		176.4.m.b.49.2		8
11.8	odd	10		176.4.m.b.97.2		8
11.10	odd	2		1936.4.a.bn.1.2		4
12.11	even	2		2178.4.a.bt.1.3		4
44.3	odd	10		242.4.c.q.9.1		8
44.7	even	10		22.4.c.b.5.1	✓	8
44.15	odd	10		242.4.c.q.27.1		8
44.19	even	10		22.4.c.b.9.1	yes	8
44.27	odd	10		242.4.c.n.3.2		8
44.31	odd	10		242.4.c.n.81.2		8
44.35	even	10		242.4.c.r.81.2		8
44.39	even	10		242.4.c.r.3.2		8
44.43	even	2		242.4.a.n.1.3		4
132.95	odd	10		198.4.f.d.181.2		8
132.107	odd	10		198.4.f.d.163.2		8
132.131	odd	2		2178.4.a.by.1.3		4

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
22.4.c.b.5.1	✓	8	44.7	even	10
22.4.c.b.9.1	yes	8	44.19	even	10
176.4.m.b.49.2		8	11.7	odd	10
176.4.m.b.97.2		8	11.8	odd	10
198.4.f.d.163.2		8	132.107	odd	10
198.4.f.d.181.2		8	132.95	odd	10
242.4.a.n.1.3		4	44.43	even	2
242.4.a.o.1.3		4	4.3	odd	2
242.4.c.n.3.2		8	44.27	odd	10
242.4.c.n.81.2		8	44.31	odd	10
242.4.c.q.9.1		8	44.3	odd	10
242.4.c.q.27.1		8	44.15	odd	10
242.4.c.r.3.2		8	44.39	even	10
242.4.c.r.81.2		8	44.35	even	10
1936.4.a.bm.1.2		4	1.1	even	1	trivial
1936.4.a.bn.1.2		4	11.10	odd	2
2178.4.a.bt.1.3		4	12.11	even	2
2178.4.a.by.1.3		4	132.131	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:	\( N \)	\(=\)	\( 1936 = 2^{4} \cdot 11^{2} \)
Weight:	\( k \)	\(=\)	\( 4 \)
Character orbit:	\([\chi]\)	\(=\)	1936.a (trivial)

\(f(q)\)	\(=\)	\(q-7.57575 q^{3} +5.40810 q^{5} -22.1498 q^{7} +30.3919 q^{9} +O(q^{10})\)	\(q-7.57575 q^{3} +5.40810 q^{5} -22.1498 q^{7} +30.3919 q^{9} -76.8269 q^{13} -40.9704 q^{15} -59.2883 q^{17} +95.2626 q^{19} +167.801 q^{21} -142.484 q^{23} -95.7524 q^{25} -25.6965 q^{27} +20.4183 q^{29} -213.304 q^{31} -119.788 q^{35} -145.578 q^{37} +582.021 q^{39} -82.8326 q^{41} -151.373 q^{43} +164.363 q^{45} -90.3643 q^{47} +147.615 q^{49} +449.153 q^{51} -234.849 q^{53} -721.685 q^{57} -302.497 q^{59} +149.279 q^{61} -673.176 q^{63} -415.488 q^{65} -826.236 q^{67} +1079.42 q^{69} +898.965 q^{71} +137.993 q^{73} +725.396 q^{75} -304.459 q^{79} -625.912 q^{81} -764.294 q^{83} -320.637 q^{85} -154.684 q^{87} -313.100 q^{89} +1701.70 q^{91} +1615.94 q^{93} +515.189 q^{95} -582.505 q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 4 q - 4 q^{3} + 25 q^{5} - 3 q^{7} + 102 q^{9}+O(q^{10}) \) 4 * q - 4 * q^3 + 25 * q^5 - 3 * q^7 + 102 * q^9	\( 4 q - 4 q^{3} + 25 q^{5} - 3 q^{7} + 102 q^{9} - 41 q^{13} - 68 q^{15} + 52 q^{17} - 16 q^{19} + 25 q^{21} - 314 q^{23} - 21 q^{25} - 286 q^{27} + 561 q^{29} - 199 q^{31} + 714 q^{35} + 357 q^{37} + 1038 q^{39} + 32 q^{41} + 721 q^{43} + 1326 q^{45} - 403 q^{47} + 823 q^{49} + 174 q^{51} - 133 q^{53} - 1031 q^{57} - 1016 q^{59} - 919 q^{61} + 1367 q^{63} + 69 q^{65} - 289 q^{67} - 1620 q^{69} + 1205 q^{71} - 1234 q^{73} + 911 q^{75} + 603 q^{79} - 1400 q^{81} - 1514 q^{83} + 717 q^{85} + 1061 q^{87} - 1101 q^{89} + 2306 q^{91} - 2298 q^{93} + 1766 q^{95} + 2116 q^{97}+O(q^{100}) \) 4 * q - 4 * q^3 + 25 * q^5 - 3 * q^7 + 102 * q^9 - 41 * q^13 - 68 * q^15 + 52 * q^17 - 16 * q^19 + 25 * q^21 - 314 * q^23 - 21 * q^25 - 286 * q^27 + 561 * q^29 - 199 * q^31 + 714 * q^35 + 357 * q^37 + 1038 * q^39 + 32 * q^41 + 721 * q^43 + 1326 * q^45 - 403 * q^47 + 823 * q^49 + 174 * q^51 - 133 * q^53 - 1031 * q^57 - 1016 * q^59 - 919 * q^61 + 1367 * q^63 + 69 * q^65 - 289 * q^67 - 1620 * q^69 + 1205 * q^71 - 1234 * q^73 + 911 * q^75 + 603 * q^79 - 1400 * q^81 - 1514 * q^83 + 717 * q^85 + 1061 * q^87 - 1101 * q^89 + 2306 * q^91 - 2298 * q^93 + 1766 * q^95 + 2116 * q^97

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