LMFDB - Embedded newform 1936.4.a.bs.1.4

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:	$6$
Coefficient field:	$\mathbb{Q}[x]/(x^{6} - \cdots)$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{6} - 3x^{5} - 82x^{4} + 161x^{3} + 1730x^{2} - 2271x - 5931 $ x^6 - 3x^5 - 82x^4 + 161x^3 + 1730x^2 - 2271*x - 5931
Coefficient ring:	$\Z[a_1, \ldots, a_{7}]$
Coefficient ring index:	$ 2^{2}\cdot 11 $
Twist minimal:	no (minimal twist has level 44)
Fricke sign:	$+1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.4
Root			$2.86545$ 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)

$f(q)$	$=$	$q+1.86545 q^{3} -18.4346 q^{5} +29.1909 q^{7} -23.5201 q^{9} -78.0240 q^{13} -34.3888 q^{15} -28.5646 q^{17} -43.5952 q^{19} +54.4541 q^{21} +104.563 q^{23} +214.833 q^{25} -94.2428 q^{27} -42.5332 q^{29} +63.0933 q^{31} -538.121 q^{35} -317.419 q^{37} -145.550 q^{39} +68.3454 q^{41} -227.116 q^{43} +433.583 q^{45} +10.7737 q^{47} +509.106 q^{49} -53.2859 q^{51} -497.037 q^{53} -81.3248 q^{57} +423.387 q^{59} +190.367 q^{61} -686.572 q^{63} +1438.34 q^{65} +386.223 q^{67} +195.057 q^{69} +691.474 q^{71} -318.117 q^{73} +400.761 q^{75} +161.713 q^{79} +459.237 q^{81} +394.114 q^{83} +526.577 q^{85} -79.3436 q^{87} +108.129 q^{89} -2277.59 q^{91} +117.697 q^{93} +803.659 q^{95} -678.738 q^{97} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 6 q - 3 q^{3} - 12 q^{5} + 8 q^{7} + 11 q^{9} + 80 q^{13} + 98 q^{15} + 113 q^{17} - 53 q^{19} + 152 q^{21} + 194 q^{23} + 476 q^{25} - 72 q^{27} + 374 q^{29} - 16 q^{31} - 1044 q^{35} - 456 q^{37} - 592 q^{39}+ \cdots + 683 q^{97}+O(q^{100}) $ 6 * q - 3 * q^3 - 12 * q^5 + 8 * q^7 + 11 * q^9 + 80 * q^13 + 98 * q^15 + 113 * q^17 - 53 * q^19 + 152 * q^21 + 194 * q^23 + 476 * q^25 - 72 * q^27 + 374 * q^29 - 16 * q^31 - 1044 * q^35 - 456 * q^37 - 592 * q^39 + 825 * q^41 - 1165 * q^43 - 670 * q^45 - 1248 * q^47 - 4 * q^49 - 988 * q^51 - 122 * q^53 + 2282 * q^57 + 767 * q^59 + 1010 * q^61 - 254 * q^63 + 1946 * q^65 + 2197 * q^67 + 2648 * q^69 - 598 * q^71 + 1523 * q^73 - 609 * q^75 - 2116 * q^79 - 2106 * q^81 - 1213 * q^83 + 1066 * q^85 - 1518 * q^87 - 477 * q^89 - 2420 * q^91 - 714 * q^93 - 404 * q^95 + 683 * q^97

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
$3$	1.86545	0.359006	0.179503	−	0.983757i	$-0.442551\pi$
$3$	1.86545	0.359006	0.179503	+	0.983757i	$0.442551\pi$
$4$	0	0
$5$	−18.4346	−1.64884	−0.824419	−	0.565980i	$-0.808498\pi$
$5$	−18.4346	−1.64884	−0.824419	+	0.565980i	$0.808498\pi$
$6$	0	0
$7$	29.1909	1.57616	0.788079	−	0.615574i	$-0.211076\pi$
$7$	29.1909	1.57616	0.788079	+	0.615574i	$0.211076\pi$
$8$	0	0
$9$	−23.5201	−0.871114
$10$	0	0
$11$	0	0
$11$	0	0
$12$	0	0
$13$	−78.0240	−1.66461	−0.832307	−	0.554315i	$-0.812980\pi$
$13$	−78.0240	−1.66461	−0.832307	+	0.554315i	$0.812980\pi$
$14$	0	0
$15$	−34.3888	−0.591943
$16$	0	0
$17$	−28.5646	−0.407526	−0.203763	−	0.979020i	$-0.565317\pi$
$17$	−28.5646	−0.407526	−0.203763	+	0.979020i	$0.565317\pi$
$18$	0	0
$19$	−43.5952	−0.526391	−0.263195	−	0.964743i	$-0.584776\pi$
$19$	−43.5952	−0.526391	−0.263195	+	0.964743i	$0.584776\pi$
$20$	0	0
$21$	54.4541	0.565851
$22$	0	0
$23$	104.563	0.947952	0.473976	−	0.880538i	$-0.342818\pi$
$23$	104.563	0.947952	0.473976	+	0.880538i	$0.342818\pi$
$24$	0	0
$25$	214.833	1.71867
$26$	0	0
$27$	−94.2428	−0.671742
$28$	0	0
$29$	−42.5332	−0.272352	−0.136176	−	0.990685i	$-0.543481\pi$
$29$	−42.5332	−0.272352	−0.136176	+	0.990685i	$0.543481\pi$
$30$	0	0
$31$	63.0933	0.365545	0.182772	−	0.983155i	$-0.441493\pi$
$31$	63.0933	0.365545	0.182772	+	0.983155i	$0.441493\pi$
$32$	0	0
$33$	0	0
$34$	0	0
$35$	−538.121	−2.59883
$36$	0	0
$37$	−317.419	−1.41036	−0.705180	−	0.709029i	$-0.749134\pi$
$37$	−317.419	−1.41036	−0.705180	+	0.709029i	$0.749134\pi$
$38$	0	0
$39$	−145.550	−0.597607
$40$	0	0
$41$	68.3454	0.260336	0.130168	−	0.991492i	$-0.458448\pi$
$41$	68.3454	0.260336	0.130168	+	0.991492i	$0.458448\pi$
$42$	0	0
$43$	−227.116	−0.805463	−0.402731	−	0.915318i	$-0.631939\pi$
$43$	−227.116	−0.805463	−0.402731	+	0.915318i	$0.631939\pi$
$44$	0	0
$45$	433.583	1.43633
$46$	0	0
$47$	10.7737	0.0334364	0.0167182	−	0.999860i	$-0.494678\pi$
$47$	10.7737	0.0334364	0.0167182	+	0.999860i	$0.494678\pi$
$48$	0	0
$49$	509.106	1.48427
$50$	0	0
$51$	−53.2859	−0.146304
$52$	0	0
$53$	−497.037	−1.28817	−0.644087	−	0.764952i	$-0.722763\pi$
$53$	−497.037	−1.28817	−0.644087	+	0.764952i	$0.722763\pi$
$54$	0	0
$55$	0	0
$56$	0	0
$57$	−81.3248	−0.188978
$58$	0	0
$59$	423.387	0.934242	0.467121	−	0.884193i	$-0.345291\pi$
$59$	423.387	0.934242	0.467121	+	0.884193i	$0.345291\pi$
$60$	0	0
$61$	190.367	0.399574	0.199787	−	0.979839i	$-0.435975\pi$
$61$	190.367	0.399574	0.199787	+	0.979839i	$0.435975\pi$
$62$	0	0
$63$	−686.572	−1.37301
$64$	0	0
$65$	1438.34	2.74468
$66$	0	0
$67$	386.223	0.704248	0.352124	−	0.935953i	$-0.385459\pi$
$67$	386.223	0.704248	0.352124	+	0.935953i	$0.385459\pi$
$68$	0	0
$69$	195.057	0.340321
$70$	0	0
$71$	691.474	1.15581	0.577907	−	0.816102i	$-0.303869\pi$
$71$	691.474	1.15581	0.577907	+	0.816102i	$0.303869\pi$
$72$	0	0
$73$	−318.117	−0.510038	−0.255019	−	0.966936i	$-0.582082\pi$
$73$	−318.117	−0.510038	−0.255019	+	0.966936i	$0.582082\pi$
$74$	0	0
$75$	400.761	0.617013
$76$	0	0
$77$	0	0
$78$	0	0
$79$	161.713	0.230305	0.115152	−	0.993348i	$-0.463264\pi$
$79$	161.713	0.230305	0.115152	+	0.993348i	$0.463264\pi$
$80$	0	0
$81$	459.237	0.629955
$82$	0	0
$83$	394.114	0.521201	0.260600	−	0.965447i	$-0.416080\pi$
$83$	394.114	0.521201	0.260600	+	0.965447i	$0.416080\pi$
$84$	0	0
$85$	526.577	0.671944
$86$	0	0
$87$	−79.3436	−0.0977762
$88$	0	0
$89$	108.129	0.128782	0.0643912	−	0.997925i	$-0.479489\pi$
$89$	108.129	0.128782	0.0643912	+	0.997925i	$0.479489\pi$
$90$	0	0
$91$	−2277.59	−2.62369
$92$	0	0
$93$	117.697	0.131233
$94$	0	0
$95$	803.659	0.867934
$96$	0	0
$97$	−678.738	−0.710469	−0.355234	−	0.934777i	$-0.615599\pi$
$97$	−678.738	−0.710469	−0.355234	+	0.934777i	$0.615599\pi$
$98$	0	0
$99$	0	0
$100$	0	0
$101$	1128.96	1.11223	0.556117	−	0.831104i	$-0.312291\pi$
$101$	1128.96	1.11223	0.556117	+	0.831104i	$0.312291\pi$
$102$	0	0
$103$	98.3154	0.0940515	0.0470258	−	0.998894i	$-0.485026\pi$
$103$	98.3154	0.0940515	0.0470258	+	0.998894i	$0.485026\pi$
$104$	0	0
$105$	−1003.84	−0.932996
$106$	0	0
$107$	−544.664	−0.492100	−0.246050	−	0.969257i	$-0.579133\pi$
$107$	−544.664	−0.492100	−0.246050	+	0.969257i	$0.579133\pi$
$108$	0	0
$109$	−367.085	−0.322572	−0.161286	−	0.986908i	$-0.551564\pi$
$109$	−367.085	−0.322572	−0.161286	+	0.986908i	$0.551564\pi$
$110$	0	0
$111$	−592.129	−0.506328
$112$	0	0
$113$	−556.708	−0.463457	−0.231729	−	0.972780i	$-0.574438\pi$
$113$	−556.708	−0.463457	−0.231729	+	0.972780i	$0.574438\pi$
$114$	0	0
$115$	−1927.58	−1.56302
$116$	0	0
$117$	1835.13	1.45007
$118$	0	0
$119$	−833.826	−0.642325
$120$	0	0
$121$	0	0
$122$	0	0
$123$	127.495	0.0934621
$124$	0	0
$125$	−1656.04	−1.18497
$126$	0	0
$127$	−1298.70	−0.907409	−0.453704	−	0.891152i	$-0.649898\pi$
$127$	−1298.70	−0.907409	−0.453704	+	0.891152i	$0.649898\pi$
$128$	0	0
$129$	−423.674	−0.289166
$130$	0	0
$131$	−1110.64	−0.740738	−0.370369	−	0.928885i	$-0.620769\pi$
$131$	−1110.64	−0.740738	−0.370369	+	0.928885i	$0.620769\pi$
$132$	0	0
$133$	−1272.58	−0.829675
$134$	0	0
$135$	1737.33	1.10759
$136$	0	0
$137$	−1186.63	−0.740005	−0.370002	−	0.929031i	$-0.620643\pi$
$137$	−1186.63	−0.740005	−0.370002	+	0.929031i	$0.620643\pi$
$138$	0	0
$139$	2898.86	1.76891	0.884455	−	0.466626i	$-0.154531\pi$
$139$	2898.86	1.76891	0.884455	+	0.466626i	$0.154531\pi$
$140$	0	0
$141$	20.0979	0.0120039
$142$	0	0
$143$	0	0
$144$	0	0
$145$	784.081	0.449065
$146$	0	0
$147$	949.713	0.532864
$148$	0	0
$149$	2937.12	1.61489	0.807444	−	0.589944i	$-0.200850\pi$
$149$	2937.12	1.61489	0.807444	+	0.589944i	$0.200850\pi$
$150$	0	0
$151$	3056.43	1.64721	0.823605	−	0.567164i	$-0.191959\pi$
$151$	3056.43	1.64721	0.823605	+	0.567164i	$0.191959\pi$
$152$	0	0
$153$	671.842	0.355002
$154$	0	0
$155$	−1163.10	−0.602724
$156$	0	0
$157$	−1576.42	−0.801351	−0.400676	−	0.916220i	$-0.631225\pi$
$157$	−1576.42	−0.801351	−0.400676	+	0.916220i	$0.631225\pi$
$158$	0	0
$159$	−927.198	−0.462463
$160$	0	0
$161$	3052.28	1.49412
$162$	0	0
$163$	2831.54	1.36064	0.680318	−	0.732917i	$-0.261842\pi$
$163$	2831.54	1.36064	0.680318	+	0.732917i	$0.261842\pi$
$164$	0	0
$165$	0	0
$166$	0	0
$167$	1246.67	0.577666	0.288833	−	0.957380i	$-0.406733\pi$
$167$	1246.67	0.577666	0.288833	+	0.957380i	$0.406733\pi$
$168$	0	0
$169$	3890.75	1.77094
$170$	0	0
$171$	1025.36	0.458547
$172$	0	0
$173$	1959.83	0.861291	0.430645	−	0.902521i	$-0.358286\pi$
$173$	1959.83	0.861291	0.430645	+	0.902521i	$0.358286\pi$
$174$	0	0
$175$	6271.17	2.70889
$176$	0	0
$177$	789.808	0.335399
$178$	0	0
$179$	−638.909	−0.266784	−0.133392	−	0.991063i	$-0.542587\pi$
$179$	−638.909	−0.266784	−0.133392	+	0.991063i	$0.542587\pi$
$180$	0	0
$181$	−3146.74	−1.29224	−0.646120	−	0.763236i	$-0.723609\pi$
$181$	−3146.74	−1.29224	−0.646120	+	0.763236i	$0.723609\pi$
$182$	0	0
$183$	355.121	0.143450
$184$	0	0
$185$	5851.48	2.32545
$186$	0	0
$187$	0	0
$188$	0	0
$189$	−2751.03	−1.05877
$190$	0	0
$191$	−1763.32	−0.668008	−0.334004	−	0.942572i	$-0.608400\pi$
$191$	−1763.32	−0.668008	−0.334004	+	0.942572i	$0.608400\pi$
$192$	0	0
$193$	−4172.22	−1.55608	−0.778038	−	0.628217i	$-0.783785\pi$
$193$	−4172.22	−1.55608	−0.778038	+	0.628217i	$0.783785\pi$
$194$	0	0
$195$	2683.15	0.985357
$196$	0	0
$197$	−2892.24	−1.04601	−0.523005	−	0.852330i	$-0.675189\pi$
$197$	−2892.24	−1.04601	−0.523005	+	0.852330i	$0.675189\pi$
$198$	0	0
$199$	4281.26	1.52508	0.762539	−	0.646943i	$-0.223953\pi$
$199$	4281.26	1.52508	0.762539	+	0.646943i	$0.223953\pi$
$200$	0	0
$201$	720.480	0.252830
$202$	0	0
$203$	−1241.58	−0.429270
$204$	0	0
$205$	−1259.92	−0.429251
$206$	0	0
$207$	−2459.33	−0.825775
$208$	0	0
$209$	0	0
$210$	0	0
$211$	2747.76	0.896511	0.448256	−	0.893905i	$-0.352045\pi$
$211$	2747.76	0.896511	0.448256	+	0.893905i	$0.352045\pi$
$212$	0	0
$213$	1289.91	0.414945
$214$	0	0
$215$	4186.79	1.32808
$216$	0	0
$217$	1841.75	0.576156
$218$	0	0
$219$	−593.432	−0.183107
$220$	0	0
$221$	2228.73	0.678373
$222$	0	0
$223$	869.528	0.261112	0.130556	−	0.991441i	$-0.458324\pi$
$223$	869.528	0.261112	0.130556	+	0.991441i	$0.458324\pi$
$224$	0	0
$225$	−5052.90	−1.49716
$226$	0	0
$227$	−2856.58	−0.835233	−0.417616	−	0.908623i	$-0.637134\pi$
$227$	−2856.58	−0.835233	−0.417616	+	0.908623i	$0.637134\pi$
$228$	0	0
$229$	5277.91	1.52303	0.761516	−	0.648147i	$-0.224456\pi$
$229$	5277.91	1.52303	0.761516	+	0.648147i	$0.224456\pi$
$230$	0	0
$231$	0	0
$232$	0	0
$233$	−915.390	−0.257378	−0.128689	−	0.991685i	$-0.541077\pi$
$233$	−915.390	−0.257378	−0.128689	+	0.991685i	$0.541077\pi$
$234$	0	0
$235$	−198.609	−0.0551312
$236$	0	0
$237$	301.667	0.0826809
$238$	0	0
$239$	6328.24	1.71272	0.856359	−	0.516381i	$-0.172721\pi$
$239$	6328.24	1.71272	0.856359	+	0.516381i	$0.172721\pi$
$240$	0	0
$241$	540.257	0.144403	0.0722013	−	0.997390i	$-0.476998\pi$
$241$	540.257	0.144403	0.0722013	+	0.997390i	$0.476998\pi$
$242$	0	0
$243$	3401.24	0.897900
$244$	0	0
$245$	−9385.15	−2.44733
$246$	0	0
$247$	3401.47	0.876238
$248$	0	0
$249$	735.201	0.187114
$250$	0	0
$251$	5333.42	1.34121	0.670603	−	0.741816i	$-0.266035\pi$
$251$	5333.42	1.34121	0.670603	+	0.741816i	$0.266035\pi$
$252$	0	0
$253$	0	0
$254$	0	0
$255$	982.303	0.241232
$256$	0	0
$257$	1378.09	0.334485	0.167242	−	0.985916i	$-0.446514\pi$
$257$	1378.09	0.334485	0.167242	+	0.985916i	$0.446514\pi$
$258$	0	0
$259$	−9265.72	−2.22295
$260$	0	0
$261$	1000.38	0.237250
$262$	0	0
$263$	7490.14	1.75613	0.878064	−	0.478543i	$-0.158835\pi$
$263$	7490.14	1.75613	0.878064	+	0.478543i	$0.158835\pi$
$264$	0	0
$265$	9162.66	2.12399
$266$	0	0
$267$	201.709	0.0462337
$268$	0	0
$269$	7291.23	1.65262	0.826308	−	0.563218i	$-0.190437\pi$
$269$	7291.23	1.65262	0.826308	+	0.563218i	$0.190437\pi$
$270$	0	0
$271$	−781.878	−0.175261	−0.0876305	−	0.996153i	$-0.527929\pi$
$271$	−781.878	−0.175261	−0.0876305	+	0.996153i	$0.527929\pi$
$272$	0	0
$273$	−4248.73	−0.941923
$274$	0	0
$275$	0	0
$276$	0	0
$277$	575.042	0.124733	0.0623663	−	0.998053i	$-0.480135\pi$
$277$	575.042	0.124733	0.0623663	+	0.998053i	$0.480135\pi$
$278$	0	0
$279$	−1483.96	−0.318431
$280$	0	0
$281$	5602.67	1.18942	0.594711	−	0.803940i	$-0.297267\pi$
$281$	5602.67	1.18942	0.594711	+	0.803940i	$0.297267\pi$
$282$	0	0
$283$	−898.013	−0.188627	−0.0943133	−	0.995543i	$-0.530066\pi$
$283$	−898.013	−0.188627	−0.0943133	+	0.995543i	$0.530066\pi$
$284$	0	0
$285$	1499.19	0.311594
$286$	0	0
$287$	1995.06	0.410330
$288$	0	0
$289$	−4097.06	−0.833923
$290$	0	0
$291$	−1266.15	−0.255063
$292$	0	0
$293$	8040.51	1.60318	0.801590	−	0.597874i	$-0.203988\pi$
$293$	8040.51	1.60318	0.801590	+	0.597874i	$0.203988\pi$
$294$	0	0
$295$	−7804.96	−1.54041
$296$	0	0
$297$	0	0
$298$	0	0
$299$	−8158.43	−1.57797
$300$	0	0
$301$	−6629.72	−1.26954
$302$	0	0
$303$	2106.02	0.399299
$304$	0	0
$305$	−3509.34	−0.658833
$306$	0	0
$307$	−5491.51	−1.02090	−0.510451	−	0.859907i	$-0.670522\pi$
$307$	−5491.51	−1.02090	−0.510451	+	0.859907i	$0.670522\pi$
$308$	0	0
$309$	183.403	0.0337651
$310$	0	0
$311$	−3195.06	−0.582557	−0.291278	−	0.956638i	$-0.594081\pi$
$311$	−3195.06	−0.582557	−0.291278	+	0.956638i	$0.594081\pi$
$312$	0	0
$313$	−3117.41	−0.562961	−0.281480	−	0.959567i	$-0.590825\pi$
$313$	−3117.41	−0.562961	−0.281480	+	0.959567i	$0.590825\pi$
$314$	0	0
$315$	12656.7	2.26388
$316$	0	0
$317$	240.265	0.0425697	0.0212849	−	0.999773i	$-0.493224\pi$
$317$	240.265	0.0425697	0.0212849	+	0.999773i	$0.493224\pi$
$318$	0	0
$319$	0	0
$320$	0	0
$321$	−1016.04	−0.176667
$322$	0	0
$323$	1245.28	0.214518
$324$	0	0
$325$	−16762.2	−2.86092
$326$	0	0
$327$	−684.779	−0.115805
$328$	0	0
$329$	314.494	0.0527010
$330$	0	0
$331$	9405.79	1.56190	0.780950	−	0.624594i	$-0.214735\pi$
$331$	9405.79	1.56190	0.780950	+	0.624594i	$0.214735\pi$
$332$	0	0
$333$	7465.71	1.22858
$334$	0	0
$335$	−7119.85	−1.16119
$336$	0	0
$337$	4690.42	0.758170	0.379085	−	0.925362i	$-0.376239\pi$
$337$	4690.42	0.758170	0.379085	+	0.925362i	$0.376239\pi$
$338$	0	0
$339$	−1038.51	−0.166384
$340$	0	0
$341$	0	0
$342$	0	0
$343$	4848.78	0.763292
$344$	0	0
$345$	−3595.80	−0.561134
$346$	0	0
$347$	−5358.07	−0.828922	−0.414461	−	0.910067i	$-0.636030\pi$
$347$	−5358.07	−0.828922	−0.414461	+	0.910067i	$0.636030\pi$
$348$	0	0
$349$	987.122	0.151402	0.0757012	−	0.997131i	$-0.475880\pi$
$349$	987.122	0.151402	0.0757012	+	0.997131i	$0.475880\pi$
$350$	0	0
$351$	7353.20	1.11819
$352$	0	0
$353$	−130.359	−0.0196552	−0.00982762	−	0.999952i	$-0.503128\pi$
$353$	−130.359	−0.0196552	−0.00982762	+	0.999952i	$0.503128\pi$
$354$	0	0
$355$	−12747.0	−1.90575
$356$	0	0
$357$	−1555.46	−0.230599
$358$	0	0
$359$	9231.03	1.35709	0.678545	−	0.734559i	$-0.262611\pi$
$359$	9231.03	1.35709	0.678545	+	0.734559i	$0.262611\pi$
$360$	0	0
$361$	−4958.46	−0.722913
$362$	0	0
$363$	0	0
$364$	0	0
$365$	5864.35	0.840971
$366$	0	0
$367$	−8497.29	−1.20860	−0.604299	−	0.796758i	$-0.706547\pi$
$367$	−8497.29	−1.20860	−0.604299	+	0.796758i	$0.706547\pi$
$368$	0	0
$369$	−1607.49	−0.226782
$370$	0	0
$371$	−14508.9	−2.03037
$372$	0	0
$373$	12810.7	1.77832	0.889160	−	0.457597i	$-0.151290\pi$
$373$	12810.7	1.77832	0.889160	+	0.457597i	$0.151290\pi$
$374$	0	0
$375$	−3089.27	−0.425411
$376$	0	0
$377$	3318.61	0.453361
$378$	0	0
$379$	6768.45	0.917340	0.458670	−	0.888607i	$-0.348326\pi$
$379$	6768.45	0.917340	0.458670	+	0.888607i	$0.348326\pi$
$380$	0	0
$381$	−2422.66	−0.325766
$382$	0	0
$383$	11600.9	1.54772	0.773859	−	0.633357i	$-0.218324\pi$
$383$	11600.9	1.54772	0.773859	+	0.633357i	$0.218324\pi$
$384$	0	0
$385$	0	0
$386$	0	0
$387$	5341.79	0.701650
$388$	0	0
$389$	3183.50	0.414936	0.207468	−	0.978242i	$-0.433478\pi$
$389$	3183.50	0.414936	0.207468	+	0.978242i	$0.433478\pi$
$390$	0	0
$391$	−2986.80	−0.386315
$392$	0	0
$393$	−2071.84	−0.265930
$394$	0	0
$395$	−2981.10	−0.379736
$396$	0	0
$397$	4412.72	0.557854	0.278927	−	0.960312i	$-0.410021\pi$
$397$	4412.72	0.557854	0.278927	+	0.960312i	$0.410021\pi$
$398$	0	0
$399$	−2373.94	−0.297859
$400$	0	0
$401$	3770.02	0.469490	0.234745	−	0.972057i	$-0.424574\pi$
$401$	3770.02	0.469490	0.234745	+	0.972057i	$0.424574\pi$
$402$	0	0
$403$	−4922.79	−0.608491
$404$	0	0
$405$	−8465.84	−1.03869
$406$	0	0
$407$	0	0
$408$	0	0
$409$	−6371.80	−0.770331	−0.385165	−	0.922848i	$-0.625856\pi$
$409$	−6371.80	−0.770331	−0.385165	+	0.922848i	$0.625856\pi$
$410$	0	0
$411$	−2213.60	−0.265666
$412$	0	0
$413$	12359.0	1.47251
$414$	0	0
$415$	−7265.33	−0.859375
$416$	0	0
$417$	5407.69	0.635050
$418$	0	0
$419$	−2107.98	−0.245779	−0.122890	−	0.992420i	$-0.539216\pi$
$419$	−2107.98	−0.245779	−0.122890	+	0.992420i	$0.539216\pi$
$420$	0	0
$421$	−8247.39	−0.954758	−0.477379	−	0.878697i	$-0.658413\pi$
$421$	−8247.39	−0.954758	−0.477379	+	0.878697i	$0.658413\pi$
$422$	0	0
$423$	−253.399	−0.0291269
$424$	0	0
$425$	−6136.64	−0.700401
$426$	0	0
$427$	5556.98	0.629792
$428$	0	0
$429$	0	0
$430$	0	0
$431$	−12928.4	−1.44487	−0.722435	−	0.691439i	$-0.756977\pi$
$431$	−12928.4	−1.44487	−0.722435	+	0.691439i	$0.756977\pi$
$432$	0	0
$433$	−7810.22	−0.866825	−0.433413	−	0.901196i	$-0.642691\pi$
$433$	−7810.22	−0.866825	−0.433413	+	0.901196i	$0.642691\pi$
$434$	0	0
$435$	1462.67	0.161217
$436$	0	0
$437$	−4558.45	−0.498994
$438$	0	0
$439$	−17491.4	−1.90163	−0.950817	−	0.309752i	$-0.899754\pi$
$439$	−17491.4	−1.90163	−0.950817	+	0.309752i	$0.899754\pi$
$440$	0	0
$441$	−11974.2	−1.29297
$442$	0	0
$443$	−2569.82	−0.275611	−0.137806	−	0.990459i	$-0.544005\pi$
$443$	−2569.82	−0.275611	−0.137806	+	0.990459i	$0.544005\pi$
$444$	0	0
$445$	−1993.31	−0.212342
$446$	0	0
$447$	5479.06	0.579755
$448$	0	0
$449$	5357.81	0.563142	0.281571	−	0.959540i	$-0.409145\pi$
$449$	5357.81	0.563142	0.281571	+	0.959540i	$0.409145\pi$
$450$	0	0
$451$	0	0
$452$	0	0
$453$	5701.62	0.591359
$454$	0	0
$455$	41986.4	4.32605
$456$	0	0
$457$	−4836.50	−0.495059	−0.247529	−	0.968880i	$-0.579619\pi$
$457$	−4836.50	−0.495059	−0.247529	+	0.968880i	$0.579619\pi$
$458$	0	0
$459$	2692.01	0.273752
$460$	0	0
$461$	18173.0	1.83601	0.918007	−	0.396564i	$-0.129797\pi$
$461$	18173.0	1.83601	0.918007	+	0.396564i	$0.129797\pi$
$462$	0	0
$463$	−13451.4	−1.35019	−0.675095	−	0.737731i	$-0.735897\pi$
$463$	−13451.4	−1.35019	−0.675095	+	0.737731i	$0.735897\pi$
$464$	0	0
$465$	−2169.70	−0.216382
$466$	0	0
$467$	−16081.9	−1.59353	−0.796767	−	0.604286i	$-0.793458\pi$
$467$	−16081.9	−1.59353	−0.796767	+	0.604286i	$0.793458\pi$
$468$	0	0
$469$	11274.2	1.11001
$470$	0	0
$471$	−2940.74	−0.287690
$472$	0	0
$473$	0	0
$474$	0	0
$475$	−9365.71	−0.904691
$476$	0	0
$477$	11690.3	1.12215
$478$	0	0
$479$	3277.03	0.312592	0.156296	−	0.987710i	$-0.450045\pi$
$479$	3277.03	0.312592	0.156296	+	0.987710i	$0.450045\pi$
$480$	0	0
$481$	24766.3	2.34770
$482$	0	0
$483$	5693.89	0.536399
$484$	0	0
$485$	12512.3	1.17145
$486$	0	0
$487$	−1611.80	−0.149975	−0.0749875	−	0.997184i	$-0.523892\pi$
$487$	−1611.80	−0.149975	−0.0749875	+	0.997184i	$0.523892\pi$
$488$	0	0
$489$	5282.11	0.488477
$490$	0	0
$491$	−10763.9	−0.989341	−0.494670	−	0.869081i	$-0.664711\pi$
$491$	−10763.9	−0.989341	−0.494670	+	0.869081i	$0.664711\pi$
$492$	0	0
$493$	1214.94	0.110991
$494$	0	0
$495$	0	0
$496$	0	0
$497$	20184.7	1.82175
$498$	0	0
$499$	−13426.9	−1.20455	−0.602277	−	0.798287i	$-0.705740\pi$
$499$	−13426.9	−1.20455	−0.602277	+	0.798287i	$0.705740\pi$
$500$	0	0
$501$	2325.60	0.207386
$502$	0	0
$503$	18901.5	1.67550	0.837751	−	0.546053i	$-0.183870\pi$
$503$	18901.5	1.67550	0.837751	+	0.546053i	$0.183870\pi$
$504$	0	0
$505$	−20811.9	−1.83389
$506$	0	0
$507$	7258.01	0.635778
$508$	0	0
$509$	10527.8	0.916772	0.458386	−	0.888753i	$-0.348428\pi$
$509$	10527.8	0.916772	0.458386	+	0.888753i	$0.348428\pi$
$510$	0	0
$511$	−9286.11	−0.803901
$512$	0	0
$513$	4108.53	0.353599
$514$	0	0
$515$	−1812.40	−0.155076
$516$	0	0
$517$	0	0
$518$	0	0
$519$	3655.97	0.309209
$520$	0	0
$521$	−1856.19	−0.156086	−0.0780432	−	0.996950i	$-0.524867\pi$
$521$	−1856.19	−0.156086	−0.0780432	+	0.996950i	$0.524867\pi$
$522$	0	0
$523$	−1950.64	−0.163089	−0.0815443	−	0.996670i	$-0.525985\pi$
$523$	−1950.64	−0.163089	−0.0815443	+	0.996670i	$0.525985\pi$
$524$	0	0
$525$	11698.6	0.972509
$526$	0	0
$527$	−1802.24	−0.148969
$528$	0	0
$529$	−1233.57	−0.101386
$530$	0	0
$531$	−9958.10	−0.813832
$532$	0	0
$533$	−5332.58	−0.433358
$534$	0	0
$535$	10040.7	0.811393
$536$	0	0
$537$	−1191.85	−0.0957771
$538$	0	0
$539$	0	0
$540$	0	0
$541$	−87.4698	−0.00695124	−0.00347562	−	0.999994i	$-0.501106\pi$
$541$	−87.4698	−0.00695124	−0.00347562	+	0.999994i	$0.501106\pi$
$542$	0	0
$543$	−5870.09	−0.463922
$544$	0	0
$545$	6767.05	0.531869
$546$	0	0
$547$	13499.5	1.05521	0.527604	−	0.849491i	$-0.323091\pi$
$547$	13499.5	1.05521	0.527604	+	0.849491i	$0.323091\pi$
$548$	0	0
$549$	−4477.45	−0.348075
$550$	0	0
$551$	1854.24	0.143364
$552$	0	0
$553$	4720.53	0.362997
$554$	0	0
$555$	10915.6	0.834853
$556$	0	0
$557$	−6930.80	−0.527230	−0.263615	−	0.964628i	$-0.584915\pi$
$557$	−6930.80	−0.527230	−0.263615	+	0.964628i	$0.584915\pi$
$558$	0	0
$559$	17720.5	1.34078
$560$	0	0
$561$	0	0
$562$	0	0
$563$	15183.5	1.13661	0.568303	−	0.822819i	$-0.307600\pi$
$563$	15183.5	1.13661	0.568303	+	0.822819i	$0.307600\pi$
$564$	0	0
$565$	10262.7	0.764166
$566$	0	0
$567$	13405.5	0.992908
$568$	0	0
$569$	8646.27	0.637031	0.318515	−	0.947918i	$-0.396816\pi$
$569$	8646.27	0.637031	0.318515	+	0.947918i	$0.396816\pi$
$570$	0	0
$571$	306.105	0.0224345	0.0112172	−	0.999937i	$-0.496429\pi$
$571$	306.105	0.0224345	0.0112172	+	0.999937i	$0.496429\pi$
$572$	0	0
$573$	−3289.39	−0.239819
$574$	0	0
$575$	22463.6	1.62922
$576$	0	0
$577$	−14855.6	−1.07183	−0.535916	−	0.844271i	$-0.680034\pi$
$577$	−14855.6	−1.07183	−0.535916	+	0.844271i	$0.680034\pi$
$578$	0	0
$579$	−7783.07	−0.558641
$580$	0	0
$581$	11504.5	0.821494
$582$	0	0
$583$	0	0
$584$	0	0
$585$	−33829.9	−2.39093
$586$	0	0
$587$	−1227.33	−0.0862983	−0.0431492	−	0.999069i	$-0.513739\pi$
$587$	−1227.33	−0.0862983	−0.0431492	+	0.999069i	$0.513739\pi$
$588$	0	0
$589$	−2750.57	−0.192419
$590$	0	0
$591$	−5395.34	−0.375524
$592$	0	0
$593$	14255.5	0.987189	0.493594	−	0.869692i	$-0.335683\pi$
$593$	14255.5	0.987189	0.493594	+	0.869692i	$0.335683\pi$
$594$	0	0
$595$	15371.2	1.05909
$596$	0	0
$597$	7986.48	0.547512
$598$	0	0
$599$	15628.0	1.06602	0.533008	−	0.846110i	$-0.321061\pi$
$599$	15628.0	1.06602	0.533008	+	0.846110i	$0.321061\pi$
$600$	0	0
$601$	−25798.8	−1.75100	−0.875502	−	0.483215i	$-0.839469\pi$
$601$	−25798.8	−1.75100	−0.875502	+	0.483215i	$0.839469\pi$
$602$	0	0
$603$	−9084.00	−0.613481
$604$	0	0
$605$	0	0
$606$	0	0
$607$	21867.5	1.46223	0.731116	−	0.682253i	$-0.239000\pi$
$607$	21867.5	1.46223	0.731116	+	0.682253i	$0.239000\pi$
$608$	0	0
$609$	−2316.11	−0.154111
$610$	0	0
$611$	−840.610	−0.0556586
$612$	0	0
$613$	−22794.9	−1.50192	−0.750961	−	0.660346i	$-0.770410\pi$
$613$	−22794.9	−1.50192	−0.750961	+	0.660346i	$0.770410\pi$
$614$	0	0
$615$	−2350.32	−0.154104
$616$	0	0
$617$	−16843.3	−1.09901	−0.549504	−	0.835491i	$-0.685183\pi$
$617$	−16843.3	−1.09901	−0.549504	+	0.835491i	$0.685183\pi$
$618$	0	0
$619$	18192.4	1.18128	0.590642	−	0.806934i	$-0.298875\pi$
$619$	18192.4	1.18128	0.590642	+	0.806934i	$0.298875\pi$
$620$	0	0
$621$	−9854.31	−0.636779
$622$	0	0
$623$	3156.38	0.202982
$624$	0	0
$625$	3674.24	0.235151
$626$	0	0
$627$	0	0
$628$	0	0
$629$	9066.94	0.574758
$630$	0	0
$631$	−16141.5	−1.01836	−0.509179	−	0.860661i	$-0.670051\pi$
$631$	−16141.5	−1.01836	−0.509179	+	0.860661i	$0.670051\pi$
$632$	0	0
$633$	5125.82	0.321853
$634$	0	0
$635$	23941.0	1.49617
$636$	0	0
$637$	−39722.5	−2.47074
$638$	0	0
$639$	−16263.5	−1.00685
$640$	0	0
$641$	−20392.4	−1.25656	−0.628278	−	0.777989i	$-0.716240\pi$
$641$	−20392.4	−1.25656	−0.628278	+	0.777989i	$0.716240\pi$
$642$	0	0
$643$	−11382.8	−0.698123	−0.349061	−	0.937100i	$-0.613500\pi$
$643$	−11382.8	−0.698123	−0.349061	+	0.937100i	$0.613500\pi$
$644$	0	0
$645$	7810.25	0.476788
$646$	0	0
$647$	−13492.6	−0.819857	−0.409928	−	0.912118i	$-0.634446\pi$
$647$	−13492.6	−0.819857	−0.409928	+	0.912118i	$0.634446\pi$
$648$	0	0
$649$	0	0
$650$	0	0
$651$	3435.69	0.206844
$652$	0	0
$653$	−8516.01	−0.510348	−0.255174	−	0.966895i	$-0.582133\pi$
$653$	−8516.01	−0.510348	−0.255174	+	0.966895i	$0.582133\pi$
$654$	0	0
$655$	20474.1	1.22136
$656$	0	0
$657$	7482.14	0.444302
$658$	0	0
$659$	29637.1	1.75189	0.875946	−	0.482409i	$-0.160238\pi$
$659$	29637.1	1.75189	0.875946	+	0.482409i	$0.160238\pi$
$660$	0	0
$661$	7113.07	0.418557	0.209279	−	0.977856i	$-0.432888\pi$
$661$	7113.07	0.418557	0.209279	+	0.977856i	$0.432888\pi$
$662$	0	0
$663$	4157.58	0.243540
$664$	0	0
$665$	23459.5	1.36800
$666$	0	0
$667$	−4447.40	−0.258177
$668$	0	0
$669$	1622.06	0.0937408
$670$	0	0
$671$	0	0
$672$	0	0
$673$	−11824.8	−0.677285	−0.338643	−	0.940915i	$-0.609968\pi$
$673$	−11824.8	−0.677285	−0.338643	+	0.940915i	$0.609968\pi$
$674$	0	0
$675$	−20246.5	−1.15450
$676$	0	0
$677$	24426.8	1.38670	0.693351	−	0.720600i	$-0.256133\pi$
$677$	24426.8	1.38670	0.693351	+	0.720600i	$0.256133\pi$
$678$	0	0
$679$	−19813.0	−1.11981
$680$	0	0
$681$	−5328.81	−0.299854
$682$	0	0
$683$	−21680.4	−1.21461	−0.607304	−	0.794469i	$-0.707749\pi$
$683$	−21680.4	−1.21461	−0.607304	+	0.794469i	$0.707749\pi$
$684$	0	0
$685$	21875.0	1.22015
$686$	0	0
$687$	9845.69	0.546778
$688$	0	0
$689$	38780.8	2.14431
$690$	0	0
$691$	−7054.92	−0.388397	−0.194198	−	0.980962i	$-0.562211\pi$
$691$	−7054.92	−0.388397	−0.194198	+	0.980962i	$0.562211\pi$
$692$	0	0
$693$	0	0
$694$	0	0
$695$	−53439.3	−2.91665
$696$	0	0
$697$	−1952.26	−0.106093
$698$	0	0
$699$	−1707.61	−0.0924005
$700$	0	0
$701$	−31341.1	−1.68864	−0.844321	−	0.535838i	$-0.819996\pi$
$701$	−31341.1	−1.68864	−0.844321	+	0.535838i	$0.819996\pi$
$702$	0	0
$703$	13837.9	0.742400
$704$	0	0
$705$	−370.495	−0.0197924
$706$	0	0
$707$	32955.3	1.75306
$708$	0	0
$709$	4726.48	0.250362	0.125181	−	0.992134i	$-0.460049\pi$
$709$	4726.48	0.250362	0.125181	+	0.992134i	$0.460049\pi$
$710$	0	0
$711$	−3803.49	−0.200622
$712$	0	0
$713$	6597.23	0.346519
$714$	0	0
$715$	0	0
$716$	0	0
$717$	11805.0	0.614877
$718$	0	0
$719$	27189.9	1.41031	0.705155	−	0.709053i	$-0.250877\pi$
$719$	27189.9	1.41031	0.705155	+	0.709053i	$0.250877\pi$
$720$	0	0
$721$	2869.91	0.148240
$722$	0	0
$723$	1007.82	0.0518415
$724$	0	0
$725$	−9137.55	−0.468083
$726$	0	0
$727$	14387.6	0.733986	0.366993	−	0.930224i	$-0.380387\pi$
$727$	14387.6	0.733986	0.366993	+	0.930224i	$0.380387\pi$
$728$	0	0
$729$	−6054.55	−0.307603
$730$	0	0
$731$	6487.49	0.328247
$732$	0	0
$733$	−29792.5	−1.50124	−0.750621	−	0.660733i	$-0.770246\pi$
$733$	−29792.5	−1.50124	−0.750621	+	0.660733i	$0.770246\pi$
$734$	0	0
$735$	−17507.5	−0.878606
$736$	0	0
$737$	0	0
$738$	0	0
$739$	22439.1	1.11696	0.558481	−	0.829517i	$-0.311384\pi$
$739$	22439.1	1.11696	0.558481	+	0.829517i	$0.311384\pi$
$740$	0	0
$741$	6345.29	0.314575
$742$	0	0
$743$	6996.52	0.345461	0.172731	−	0.984969i	$-0.444741\pi$
$743$	6996.52	0.345461	0.172731	+	0.984969i	$0.444741\pi$
$744$	0	0
$745$	−54144.6	−2.66269
$746$	0	0
$747$	−9269.60	−0.454025
$748$	0	0
$749$	−15899.2	−0.775627
$750$	0	0
$751$	−29020.1	−1.41006	−0.705032	−	0.709175i	$-0.749068\pi$
$751$	−29020.1	−1.41006	−0.705032	+	0.709175i	$0.749068\pi$
$752$	0	0
$753$	9949.24	0.481501
$754$	0	0
$755$	−56344.0	−2.71598
$756$	0	0
$757$	14945.0	0.717549	0.358775	−	0.933424i	$-0.383195\pi$
$757$	14945.0	0.717549	0.358775	+	0.933424i	$0.383195\pi$
$758$	0	0
$759$	0	0
$760$	0	0
$761$	15539.6	0.740221	0.370111	−	0.928988i	$-0.379320\pi$
$761$	15539.6	0.740221	0.370111	+	0.928988i	$0.379320\pi$
$762$	0	0
$763$	−10715.5	−0.508425
$764$	0	0
$765$	−12385.1	−0.585340
$766$	0	0
$767$	−33034.4	−1.55515
$768$	0	0
$769$	−1512.44	−0.0709233	−0.0354616	−	0.999371i	$-0.511290\pi$
$769$	−1512.44	−0.0709233	−0.0354616	+	0.999371i	$0.511290\pi$
$770$	0	0
$771$	2570.75	0.120082
$772$	0	0
$773$	−40691.4	−1.89336	−0.946681	−	0.322173i	$-0.895587\pi$
$773$	−40691.4	−1.89336	−0.946681	+	0.322173i	$0.895587\pi$
$774$	0	0
$775$	13554.6	0.628250
$776$	0	0
$777$	−17284.7	−0.798053
$778$	0	0
$779$	−2979.53	−0.137038
$780$	0	0
$781$	0	0
$782$	0	0
$783$	4008.45	0.182950
$784$	0	0
$785$	29060.7	1.32130
$786$	0	0
$787$	−33472.6	−1.51610	−0.758049	−	0.652197i	$-0.773847\pi$
$787$	−33472.6	−1.51610	−0.758049	+	0.652197i	$0.773847\pi$
$788$	0	0
$789$	13972.5	0.630461
$790$	0	0
$791$	−16250.8	−0.730482
$792$	0	0
$793$	−14853.2	−0.665136
$794$	0	0
$795$	17092.5	0.762526
$796$	0	0
$797$	18807.5	0.835879	0.417939	−	0.908475i	$-0.362752\pi$
$797$	18807.5	0.835879	0.417939	+	0.908475i	$0.362752\pi$
$798$	0	0
$799$	−307.747	−0.0136262
$800$	0	0
$801$	−2543.20	−0.112184
$802$	0	0
$803$	0	0
$804$	0	0
$805$	−56267.6	−2.46357
$806$	0	0
$807$	13601.4	0.593300
$808$	0	0
$809$	−29840.4	−1.29683	−0.648414	−	0.761288i	$-0.724567\pi$
$809$	−29840.4	−1.29683	−0.648414	+	0.761288i	$0.724567\pi$
$810$	0	0
$811$	−21226.7	−0.919075	−0.459538	−	0.888158i	$-0.651985\pi$
$811$	−21226.7	−0.919075	−0.459538	+	0.888158i	$0.651985\pi$
$812$	0	0
$813$	−1458.56	−0.0629198
$814$	0	0
$815$	−52198.3	−2.24347
$816$	0	0
$817$	9901.18	0.423988
$818$	0	0
$819$	53569.1	2.28554
$820$	0	0
$821$	−3923.89	−0.166802	−0.0834012	−	0.996516i	$-0.526578\pi$
$821$	−3923.89	−0.166802	−0.0834012	+	0.996516i	$0.526578\pi$
$822$	0	0
$823$	−20157.5	−0.853762	−0.426881	−	0.904308i	$-0.640388\pi$
$823$	−20157.5	−0.853762	−0.426881	+	0.904308i	$0.640388\pi$
$824$	0	0
$825$	0	0
$826$	0	0
$827$	39082.1	1.64331	0.821654	−	0.569986i	$-0.193051\pi$
$827$	39082.1	1.64331	0.821654	+	0.569986i	$0.193051\pi$
$828$	0	0
$829$	−18903.9	−0.791989	−0.395995	−	0.918253i	$-0.629600\pi$
$829$	−18903.9	−0.791989	−0.395995	+	0.918253i	$0.629600\pi$
$830$	0	0
$831$	1072.71	0.0447798
$832$	0	0
$833$	−14542.4	−0.604880
$834$	0	0
$835$	−22981.8	−0.952477
$836$	0	0
$837$	−5946.09	−0.245552
$838$	0	0
$839$	1351.84	0.0556264	0.0278132	−	0.999613i	$-0.491146\pi$
$839$	1351.84	0.0556264	0.0278132	+	0.999613i	$0.491146\pi$
$840$	0	0
$841$	−22579.9	−0.925824
$842$	0	0
$843$	10451.5	0.427010
$844$	0	0
$845$	−71724.3	−2.91999
$846$	0	0
$847$	0	0
$848$	0	0
$849$	−1675.20	−0.0677181
$850$	0	0
$851$	−33190.3	−1.33695
$852$	0	0
$853$	−34061.8	−1.36724	−0.683619	−	0.729839i	$-0.739595\pi$
$853$	−34061.8	−1.36724	−0.683619	+	0.729839i	$0.739595\pi$
$854$	0	0
$855$	−18902.1	−0.756069
$856$	0	0
$857$	−40085.5	−1.59778	−0.798888	−	0.601480i	$-0.794578\pi$
$857$	−40085.5	−1.59778	−0.798888	+	0.601480i	$0.794578\pi$
$858$	0	0
$859$	20827.3	0.827261	0.413631	−	0.910445i	$-0.364261\pi$
$859$	20827.3	0.827261	0.413631	+	0.910445i	$0.364261\pi$
$860$	0	0
$861$	3721.69	0.147311
$862$	0	0
$863$	−40209.3	−1.58603	−0.793013	−	0.609205i	$-0.791489\pi$
$863$	−40209.3	−1.58603	−0.793013	+	0.609205i	$0.791489\pi$
$864$	0	0
$865$	−36128.7	−1.42013
$866$	0	0
$867$	−7642.87	−0.299384
$868$	0	0
$869$	0	0
$870$	0	0
$871$	−30134.7	−1.17230
$872$	0	0
$873$	15964.0	0.618899
$874$	0	0
$875$	−48341.3	−1.86770
$876$	0	0
$877$	−7031.80	−0.270749	−0.135375	−	0.990794i	$-0.543224\pi$
$877$	−7031.80	−0.270749	−0.135375	+	0.990794i	$0.543224\pi$
$878$	0	0
$879$	14999.2	0.575552
$880$	0	0
$881$	30396.7	1.16242	0.581209	−	0.813754i	$-0.302580\pi$
$881$	30396.7	1.16242	0.581209	+	0.813754i	$0.302580\pi$
$882$	0	0
$883$	1884.50	0.0718215	0.0359107	−	0.999355i	$-0.488567\pi$
$883$	1884.50	0.0718215	0.0359107	+	0.999355i	$0.488567\pi$
$884$	0	0
$885$	−14559.8	−0.553018
$886$	0	0
$887$	1433.13	0.0542499	0.0271249	−	0.999632i	$-0.491365\pi$
$887$	1433.13	0.0542499	0.0271249	+	0.999632i	$0.491365\pi$
$888$	0	0
$889$	−37910.1	−1.43022
$890$	0	0
$891$	0	0
$892$	0	0
$893$	−469.683	−0.0176006
$894$	0	0
$895$	11778.0	0.439883
$896$	0	0
$897$	−15219.2	−0.566503
$898$	0	0
$899$	−2683.56	−0.0995570
$900$	0	0
$901$	14197.7	0.524964
$902$	0	0
$903$	−12367.4	−0.455772
$904$	0	0
$905$	58008.8	2.13069
$906$	0	0
$907$	27460.9	1.00532	0.502659	−	0.864485i	$-0.332355\pi$
$907$	27460.9	1.00532	0.502659	+	0.864485i	$0.332355\pi$
$908$	0	0
$909$	−26553.2	−0.968883
$910$	0	0
$911$	−46905.8	−1.70588	−0.852941	−	0.522008i	$-0.825183\pi$
$911$	−46905.8	−1.70588	−0.852941	+	0.522008i	$0.825183\pi$
$912$	0	0
$913$	0	0
$914$	0	0
$915$	−6546.50	−0.236525
$916$	0	0
$917$	−32420.4	−1.16752
$918$	0	0
$919$	33579.8	1.20533	0.602664	−	0.797995i	$-0.294106\pi$
$919$	33579.8	1.20533	0.602664	+	0.797995i	$0.294106\pi$
$920$	0	0
$921$	−10244.1	−0.366510
$922$	0	0
$923$	−53951.6	−1.92399
$924$	0	0
$925$	−68192.1	−2.42394
$926$	0	0
$927$	−2312.39	−0.0819297
$928$	0	0
$929$	19534.3	0.689883	0.344941	−	0.938624i	$-0.387899\pi$
$929$	19534.3	0.689883	0.344941	+	0.938624i	$0.387899\pi$
$930$	0	0
$931$	−22194.6	−0.781308
$932$	0	0
$933$	−5960.23	−0.209142
$934$	0	0
$935$	0	0
$936$	0	0
$937$	52601.5	1.83396	0.916978	−	0.398938i	$-0.130621\pi$
$937$	52601.5	1.83396	0.916978	+	0.398938i	$0.130621\pi$
$938$	0	0
$939$	−5815.38	−0.202106
$940$	0	0
$941$	34929.1	1.21005	0.605025	−	0.796206i	$-0.293163\pi$
$941$	34929.1	1.21005	0.605025	+	0.796206i	$0.293163\pi$
$942$	0	0
$943$	7146.40	0.246786
$944$	0	0
$945$	50714.0	1.74574
$946$	0	0
$947$	−4206.50	−0.144343	−0.0721715	−	0.997392i	$-0.522993\pi$
$947$	−4206.50	−0.144343	−0.0721715	+	0.997392i	$0.522993\pi$
$948$	0	0
$949$	24820.8	0.849017
$950$	0	0
$951$	448.202	0.0152828
$952$	0	0
$953$	732.399	0.0248948	0.0124474	−	0.999923i	$-0.496038\pi$
$953$	732.399	0.0248948	0.0124474	+	0.999923i	$0.496038\pi$
$954$	0	0
$955$	32506.1	1.10144
$956$	0	0
$957$	0	0
$958$	0	0
$959$	−34638.7	−1.16636
$960$	0	0
$961$	−25810.2	−0.866377
$962$	0	0
$963$	12810.6	0.428675
$964$	0	0
$965$	76913.1	2.56572
$966$	0	0
$967$	−6877.86	−0.228725	−0.114363	−	0.993439i	$-0.536483\pi$
$967$	−6877.86	−0.228725	−0.114363	+	0.993439i	$0.536483\pi$
$968$	0	0
$969$	2323.01	0.0770133
$970$	0	0
$971$	−37675.6	−1.24518	−0.622588	−	0.782550i	$-0.713919\pi$
$971$	−37675.6	−1.24518	−0.622588	+	0.782550i	$0.713919\pi$
$972$	0	0
$973$	84620.3	2.78808
$974$	0	0
$975$	−31269.0	−1.02709
$976$	0	0
$977$	31802.3	1.04140	0.520699	−	0.853741i	$-0.325672\pi$
$977$	31802.3	1.04140	0.520699	+	0.853741i	$0.325672\pi$
$978$	0	0
$979$	0	0
$980$	0	0
$981$	8633.87	0.280997
$982$	0	0
$983$	32494.6	1.05434	0.527170	−	0.849760i	$-0.323253\pi$
$983$	32494.6	1.05434	0.527170	+	0.849760i	$0.323253\pi$
$984$	0	0
$985$	53317.3	1.72470
$986$	0	0
$987$	586.674	0.0189200
$988$	0	0
$989$	−23748.0	−0.763540
$990$	0	0
$991$	−3024.29	−0.0969423	−0.0484712	−	0.998825i	$-0.515435\pi$
$991$	−3024.29	−0.0969423	−0.0484712	+	0.998825i	$0.515435\pi$
$992$	0	0
$993$	17546.0	0.560732
$994$	0	0
$995$	−78923.2	−2.51461
$996$	0	0
$997$	10509.7	0.333848	0.166924	−	0.985970i	$-0.446617\pi$
$997$	10509.7	0.333848	0.166924	+	0.985970i	$0.446617\pi$
$998$	0	0
$999$	29914.4	0.947397

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	1936.4.a.bs.1.4		6
4.3	odd	2		484.4.a.h.1.3		6
11.2	odd	10		176.4.m.d.81.2		12
11.6	odd	10		176.4.m.d.113.2		12
11.10	odd	2		1936.4.a.br.1.4		6
44.35	even	10		44.4.e.a.37.2	yes	12
44.39	even	10		44.4.e.a.25.2	✓	12
44.43	even	2		484.4.a.i.1.3		6
132.35	odd	10		396.4.j.d.37.1		12
132.83	odd	10		396.4.j.d.289.1		12

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
44.4.e.a.25.2	✓	12	44.39	even	10
44.4.e.a.37.2	yes	12	44.35	even	10
176.4.m.d.81.2		12	11.2	odd	10
176.4.m.d.113.2		12	11.6	odd	10
396.4.j.d.37.1		12	132.35	odd	10
396.4.j.d.289.1		12	132.83	odd	10
484.4.a.h.1.3		6	4.3	odd	2
484.4.a.i.1.3		6	44.43	even	2
1936.4.a.br.1.4		6	11.10	odd	2
1936.4.a.bs.1.4		6	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)

\(f(q)\)	\(=\)	\(q+1.86545 q^{3} -18.4346 q^{5} +29.1909 q^{7} -23.5201 q^{9} -78.0240 q^{13} -34.3888 q^{15} -28.5646 q^{17} -43.5952 q^{19} +54.4541 q^{21} +104.563 q^{23} +214.833 q^{25} -94.2428 q^{27} -42.5332 q^{29} +63.0933 q^{31} -538.121 q^{35} -317.419 q^{37} -145.550 q^{39} +68.3454 q^{41} -227.116 q^{43} +433.583 q^{45} +10.7737 q^{47} +509.106 q^{49} -53.2859 q^{51} -497.037 q^{53} -81.3248 q^{57} +423.387 q^{59} +190.367 q^{61} -686.572 q^{63} +1438.34 q^{65} +386.223 q^{67} +195.057 q^{69} +691.474 q^{71} -318.117 q^{73} +400.761 q^{75} +161.713 q^{79} +459.237 q^{81} +394.114 q^{83} +526.577 q^{85} -79.3436 q^{87} +108.129 q^{89} -2277.59 q^{91} +117.697 q^{93} +803.659 q^{95} -678.738 q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 6 q - 3 q^{3} - 12 q^{5} + 8 q^{7} + 11 q^{9} + 80 q^{13} + 98 q^{15} + 113 q^{17} - 53 q^{19} + 152 q^{21} + 194 q^{23} + 476 q^{25} - 72 q^{27} + 374 q^{29} - 16 q^{31} - 1044 q^{35} - 456 q^{37} - 592 q^{39}+ \cdots + 683 q^{97}+O(q^{100}) \) 6 * q - 3 * q^3 - 12 * q^5 + 8 * q^7 + 11 * q^9 + 80 * q^13 + 98 * q^15 + 113 * q^17 - 53 * q^19 + 152 * q^21 + 194 * q^23 + 476 * q^25 - 72 * q^27 + 374 * q^29 - 16 * q^31 - 1044 * q^35 - 456 * q^37 - 592 * q^39 + 825 * q^41 - 1165 * q^43 - 670 * q^45 - 1248 * q^47 - 4 * q^49 - 988 * q^51 - 122 * q^53 + 2282 * q^57 + 767 * q^59 + 1010 * q^61 - 254 * q^63 + 1946 * q^65 + 2197 * q^67 + 2648 * q^69 - 598 * q^71 + 1523 * q^73 - 609 * q^75 - 2116 * q^79 - 2106 * q^81 - 1213 * q^83 + 1066 * q^85 - 1518 * q^87 - 477 * q^89 - 2420 * q^91 - 714 * q^93 - 404 * q^95 + 683 * q^97

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