LMFDB - Embedded newform 1936.4.a.bn.1.1

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:	$1$
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			1.1
Root			$-5.92695$ 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)

$f(q)$	$=$	$q-8.54499 q^{3} +12.7359 q^{5} -23.4611 q^{7} +46.0168 q^{9} +O(q^{10})$	$q-8.54499 q^{3} +12.7359 q^{5} -23.4611 q^{7} +46.0168 q^{9} +11.4654 q^{13} -108.828 q^{15} -65.5022 q^{17} +7.25013 q^{19} +200.475 q^{21} +104.072 q^{23} +37.2034 q^{25} -162.498 q^{27} -127.351 q^{29} +288.811 q^{31} -298.799 q^{35} -85.4023 q^{37} -97.9721 q^{39} -135.444 q^{41} -353.691 q^{43} +586.066 q^{45} +134.604 q^{47} +207.424 q^{49} +559.715 q^{51} +501.431 q^{53} -61.9522 q^{57} -651.658 q^{59} +365.787 q^{61} -1079.61 q^{63} +146.023 q^{65} +294.576 q^{67} -889.297 q^{69} -132.446 q^{71} +469.489 q^{73} -317.903 q^{75} -408.033 q^{79} +146.093 q^{81} +1359.54 q^{83} -834.230 q^{85} +1088.21 q^{87} -260.255 q^{89} -268.992 q^{91} -2467.89 q^{93} +92.3370 q^{95} +1414.53 q^{97} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 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
$3$	−8.54499	−1.64448	−0.822242	−	0.569138i	$-0.807277\pi$
$3$	−8.54499	−1.64448	−0.822242	+	0.569138i	$0.807277\pi$
$4$	0	0
$5$	12.7359	1.13913	0.569567	−	0.821945i	$-0.307111\pi$
$5$	12.7359	1.13913	0.569567	+	0.821945i	$0.307111\pi$
$6$	0	0
$7$	−23.4611	−1.26678	−0.633391	−	0.773832i	$-0.718337\pi$
$7$	−23.4611	−1.26678	−0.633391	+	0.773832i	$0.718337\pi$
$8$	0	0
$9$	46.0168	1.70433
$10$	0	0
$11$	0	0
$11$	0	0
$12$	0	0
$13$	11.4654	0.244611	0.122305	−	0.992493i	$-0.460971\pi$
$13$	11.4654	0.244611	0.122305	+	0.992493i	$0.460971\pi$
$14$	0	0
$15$	−108.828	−1.87329
$16$	0	0
$17$	−65.5022	−0.934507	−0.467253	−	0.884124i	$-0.654756\pi$
$17$	−65.5022	−0.934507	−0.467253	+	0.884124i	$0.654756\pi$
$18$	0	0
$19$	7.25013	0.0875417	0.0437709	−	0.999042i	$-0.486063\pi$
$19$	7.25013	0.0875417	0.0437709	+	0.999042i	$0.486063\pi$
$20$	0	0
$21$	200.475	2.08320
$22$	0	0
$23$	104.072	0.943504	0.471752	−	0.881731i	$-0.343622\pi$
$23$	104.072	0.943504	0.471752	+	0.881731i	$0.343622\pi$
$24$	0	0
$25$	37.2034	0.297627
$26$	0	0
$27$	−162.498	−1.15825
$28$	0	0
$29$	−127.351	−0.815466	−0.407733	−	0.913101i	$-0.633681\pi$
$29$	−127.351	−0.815466	−0.407733	+	0.913101i	$0.633681\pi$
$30$	0	0
$31$	288.811	1.67329	0.836646	−	0.547744i	$-0.184513\pi$
$31$	288.811	1.67329	0.836646	+	0.547744i	$0.184513\pi$
$32$	0	0
$33$	0	0
$34$	0	0
$35$	−298.799	−1.44303
$36$	0	0
$37$	−85.4023	−0.379461	−0.189731	−	0.981836i	$-0.560761\pi$
$37$	−85.4023	−0.379461	−0.189731	+	0.981836i	$0.560761\pi$
$38$	0	0
$39$	−97.9721	−0.402259
$40$	0	0
$41$	−135.444	−0.515921	−0.257960	−	0.966156i	$-0.583050\pi$
$41$	−135.444	−0.515921	−0.257960	+	0.966156i	$0.583050\pi$
$42$	0	0
$43$	−353.691	−1.25436	−0.627178	−	0.778876i	$-0.715790\pi$
$43$	−353.691	−1.25436	−0.627178	+	0.778876i	$0.715790\pi$
$44$	0	0
$45$	586.066	1.94146
$46$	0	0
$47$	134.604	0.417744	0.208872	−	0.977943i	$-0.433021\pi$
$47$	134.604	0.417744	0.208872	+	0.977943i	$0.433021\pi$
$48$	0	0
$49$	207.424	0.604735
$50$	0	0
$51$	559.715	1.53678
$52$	0	0
$53$	501.431	1.29956	0.649782	−	0.760121i	$-0.274860\pi$
$53$	501.431	1.29956	0.649782	+	0.760121i	$0.274860\pi$
$54$	0	0
$55$	0	0
$56$	0	0
$57$	−61.9522	−0.143961
$58$	0	0
$59$	−651.658	−1.43794	−0.718971	−	0.695040i	$-0.755387\pi$
$59$	−651.658	−1.43794	−0.718971	+	0.695040i	$0.755387\pi$
$60$	0	0
$61$	365.787	0.767775	0.383887	−	0.923380i	$-0.374585\pi$
$61$	365.787	0.767775	0.383887	+	0.923380i	$0.374585\pi$
$62$	0	0
$63$	−1079.61	−2.15901
$64$	0	0
$65$	146.023	0.278645
$66$	0	0
$67$	294.576	0.537136	0.268568	−	0.963261i	$-0.413450\pi$
$67$	294.576	0.537136	0.268568	+	0.963261i	$0.413450\pi$
$68$	0	0
$69$	−889.297	−1.55158
$70$	0	0
$71$	−132.446	−0.221387	−0.110694	−	0.993855i	$-0.535307\pi$
$71$	−132.446	−0.221387	−0.110694	+	0.993855i	$0.535307\pi$
$72$	0	0
$73$	469.489	0.752733	0.376367	−	0.926471i	$-0.377173\pi$
$73$	469.489	0.752733	0.376367	+	0.926471i	$0.377173\pi$
$74$	0	0
$75$	−317.903	−0.489443
$76$	0	0
$77$	0	0
$78$	0	0
$79$	−408.033	−0.581105	−0.290552	−	0.956859i	$-0.593839\pi$
$79$	−408.033	−0.581105	−0.290552	+	0.956859i	$0.593839\pi$
$80$	0	0
$81$	146.093	0.200402
$82$	0	0
$83$	1359.54	1.79794	0.898971	−	0.438009i	$-0.144316\pi$
$83$	1359.54	1.79794	0.898971	+	0.438009i	$0.144316\pi$
$84$	0	0
$85$	−834.230	−1.06453
$86$	0	0
$87$	1088.21	1.34102
$88$	0	0
$89$	−260.255	−0.309966	−0.154983	−	0.987917i	$-0.549532\pi$
$89$	−260.255	−0.309966	−0.154983	+	0.987917i	$0.549532\pi$
$90$	0	0
$91$	−268.992	−0.309869
$92$	0	0
$93$	−2467.89	−2.75170
$94$	0	0
$95$	92.3370	0.0997218
$96$	0	0
$97$	1414.53	1.48066	0.740329	−	0.672245i	$-0.234670\pi$
$97$	1414.53	1.48066	0.740329	+	0.672245i	$0.234670\pi$
$98$	0	0
$99$	0	0
$100$	0	0
$101$	−186.617	−0.183852	−0.0919261	−	0.995766i	$-0.529302\pi$
$101$	−186.617	−0.183852	−0.0919261	+	0.995766i	$0.529302\pi$
$102$	0	0
$103$	1172.91	1.12205	0.561023	−	0.827800i	$-0.310408\pi$
$103$	1172.91	1.12205	0.561023	+	0.827800i	$0.310408\pi$
$104$	0	0
$105$	2553.23	2.37305
$106$	0	0
$107$	712.714	0.643931	0.321966	−	0.946751i	$-0.395656\pi$
$107$	712.714	0.643931	0.321966	+	0.946751i	$0.395656\pi$
$108$	0	0
$109$	1247.22	1.09598	0.547989	−	0.836486i	$-0.315394\pi$
$109$	1247.22	1.09598	0.547989	+	0.836486i	$0.315394\pi$
$110$	0	0
$111$	729.762	0.624017
$112$	0	0
$113$	−982.002	−0.817513	−0.408756	−	0.912644i	$-0.634037\pi$
$113$	−982.002	−0.817513	−0.408756	+	0.912644i	$0.634037\pi$
$114$	0	0
$115$	1325.46	1.07478
$116$	0	0
$117$	527.603	0.416897
$118$	0	0
$119$	1536.75	1.18382
$120$	0	0
$121$	0	0
$122$	0	0
$123$	1157.36	0.848423
$124$	0	0
$125$	−1118.17	−0.800097
$126$	0	0
$127$	1550.67	1.08346	0.541730	−	0.840552i	$-0.317769\pi$
$127$	1550.67	1.08346	0.541730	+	0.840552i	$0.317769\pi$
$128$	0	0
$129$	3022.28	2.06277
$130$	0	0
$131$	−742.114	−0.494953	−0.247476	−	0.968894i	$-0.579601\pi$
$131$	−742.114	−0.494953	−0.247476	+	0.968894i	$0.579601\pi$
$132$	0	0
$133$	−170.096	−0.110896
$134$	0	0
$135$	−2069.56	−1.31941
$136$	0	0
$137$	515.678	0.321586	0.160793	−	0.986988i	$-0.448595\pi$
$137$	515.678	0.321586	0.160793	+	0.986988i	$0.448595\pi$
$138$	0	0
$139$	−2463.17	−1.50304	−0.751522	−	0.659709i	$-0.770680\pi$
$139$	−2463.17	−1.50304	−0.751522	+	0.659709i	$0.770680\pi$
$140$	0	0
$141$	−1150.19	−0.686973
$142$	0	0
$143$	0	0
$144$	0	0
$145$	−1621.93	−0.928925
$146$	0	0
$147$	−1772.44	−0.994476
$148$	0	0
$149$	−434.757	−0.239038	−0.119519	−	0.992832i	$-0.538135\pi$
$149$	−434.757	−0.239038	−0.119519	+	0.992832i	$0.538135\pi$
$150$	0	0
$151$	780.992	0.420902	0.210451	−	0.977604i	$-0.432507\pi$
$151$	780.992	0.420902	0.210451	+	0.977604i	$0.432507\pi$
$152$	0	0
$153$	−3014.20	−1.59270
$154$	0	0
$155$	3678.27	1.90610
$156$	0	0
$157$	522.382	0.265545	0.132773	−	0.991147i	$-0.457612\pi$
$157$	522.382	0.265545	0.132773	+	0.991147i	$0.457612\pi$
$158$	0	0
$159$	−4284.72	−2.13711
$160$	0	0
$161$	−2441.65	−1.19521
$162$	0	0
$163$	−2543.29	−1.22212	−0.611060	−	0.791584i	$-0.709257\pi$
$163$	−2543.29	−1.22212	−0.611060	+	0.791584i	$0.709257\pi$
$164$	0	0
$165$	0	0
$166$	0	0
$167$	−1385.44	−0.641968	−0.320984	−	0.947085i	$-0.604014\pi$
$167$	−1385.44	−0.641968	−0.320984	+	0.947085i	$0.604014\pi$
$168$	0	0
$169$	−2065.54	−0.940165
$170$	0	0
$171$	333.628	0.149200
$172$	0	0
$173$	−443.511	−0.194911	−0.0974553	−	0.995240i	$-0.531070\pi$
$173$	−443.511	−0.194911	−0.0974553	+	0.995240i	$0.531070\pi$
$174$	0	0
$175$	−872.833	−0.377029
$176$	0	0
$177$	5568.41	2.36467
$178$	0	0
$179$	−376.639	−0.157270	−0.0786349	−	0.996903i	$-0.525056\pi$
$179$	−376.639	−0.157270	−0.0786349	+	0.996903i	$0.525056\pi$
$180$	0	0
$181$	3293.76	1.35261	0.676307	−	0.736620i	$-0.263579\pi$
$181$	3293.76	1.35261	0.676307	+	0.736620i	$0.263579\pi$
$182$	0	0
$183$	−3125.65	−1.26259
$184$	0	0
$185$	−1087.68	−0.432257
$186$	0	0
$187$	0	0
$188$	0	0
$189$	3812.39	1.46725
$190$	0	0
$191$	−2290.26	−0.867631	−0.433816	−	0.901002i	$-0.642833\pi$
$191$	−2290.26	−0.867631	−0.433816	+	0.901002i	$0.642833\pi$
$192$	0	0
$193$	−2303.40	−0.859079	−0.429540	−	0.903048i	$-0.641324\pi$
$193$	−2303.40	−0.859079	−0.429540	+	0.903048i	$0.641324\pi$
$194$	0	0
$195$	−1247.76	−0.458227
$196$	0	0
$197$	−1041.86	−0.376801	−0.188400	−	0.982092i	$-0.560330\pi$
$197$	−1041.86	−0.376801	−0.188400	+	0.982092i	$0.560330\pi$
$198$	0	0
$199$	−3463.83	−1.23389	−0.616946	−	0.787005i	$-0.711630\pi$
$199$	−3463.83	−1.23389	−0.616946	+	0.787005i	$0.711630\pi$
$200$	0	0
$201$	−2517.14	−0.883312
$202$	0	0
$203$	2987.80	1.03302
$204$	0	0
$205$	−1725.00	−0.587703
$206$	0	0
$207$	4789.08	1.60804
$208$	0	0
$209$	0	0
$210$	0	0
$211$	−2091.98	−0.682548	−0.341274	−	0.939964i	$-0.610858\pi$
$211$	−2091.98	−0.682548	−0.341274	+	0.939964i	$0.610858\pi$
$212$	0	0
$213$	1131.75	0.364067
$214$	0	0
$215$	−4504.57	−1.42888
$216$	0	0
$217$	−6775.84	−2.11969
$218$	0	0
$219$	−4011.78	−1.23786
$220$	0	0
$221$	−751.012	−0.228591
$222$	0	0
$223$	−3703.53	−1.11214	−0.556068	−	0.831137i	$-0.687691\pi$
$223$	−3703.53	−1.11214	−0.556068	+	0.831137i	$0.687691\pi$
$224$	0	0
$225$	1711.98	0.507254
$226$	0	0
$227$	4853.50	1.41911	0.709555	−	0.704650i	$-0.248896\pi$
$227$	4853.50	1.41911	0.709555	+	0.704650i	$0.248896\pi$
$228$	0	0
$229$	−1336.85	−0.385770	−0.192885	−	0.981221i	$-0.561784\pi$
$229$	−1336.85	−0.385770	−0.192885	+	0.981221i	$0.561784\pi$
$230$	0	0
$231$	0	0
$232$	0	0
$233$	−5903.91	−1.65999	−0.829995	−	0.557770i	$-0.811657\pi$
$233$	−5903.91	−1.65999	−0.829995	+	0.557770i	$0.811657\pi$
$234$	0	0
$235$	1714.30	0.475866
$236$	0	0
$237$	3486.63	0.955617
$238$	0	0
$239$	3319.79	0.898490	0.449245	−	0.893409i	$-0.351693\pi$
$239$	3319.79	0.898490	0.449245	+	0.893409i	$0.351693\pi$
$240$	0	0
$241$	−5275.95	−1.41018	−0.705091	−	0.709116i	$-0.749094\pi$
$241$	−5275.95	−1.41018	−0.705091	+	0.709116i	$0.749094\pi$
$242$	0	0
$243$	3139.10	0.828696
$244$	0	0
$245$	2641.73	0.688874
$246$	0	0
$247$	83.1259	0.0214137
$248$	0	0
$249$	−11617.3	−2.95669
$250$	0	0
$251$	1961.85	0.493350	0.246675	−	0.969098i	$-0.420662\pi$
$251$	1961.85	0.493350	0.246675	+	0.969098i	$0.420662\pi$
$252$	0	0
$253$	0	0
$254$	0	0
$255$	7128.48	1.75060
$256$	0	0
$257$	5941.97	1.44222	0.721108	−	0.692822i	$-0.243633\pi$
$257$	5941.97	1.44222	0.721108	+	0.692822i	$0.243633\pi$
$258$	0	0
$259$	2003.63	0.480694
$260$	0	0
$261$	−5860.29	−1.38982
$262$	0	0
$263$	1704.11	0.399544	0.199772	−	0.979842i	$-0.435980\pi$
$263$	1704.11	0.399544	0.199772	+	0.979842i	$0.435980\pi$
$264$	0	0
$265$	6386.18	1.48038
$266$	0	0
$267$	2223.87	0.509734
$268$	0	0
$269$	−5845.33	−1.32489	−0.662446	−	0.749109i	$-0.730482\pi$
$269$	−5845.33	−1.32489	−0.662446	+	0.749109i	$0.730482\pi$
$270$	0	0
$271$	−5905.84	−1.32382	−0.661908	−	0.749585i	$-0.730253\pi$
$271$	−5905.84	−1.32382	−0.661908	+	0.749585i	$0.730253\pi$
$272$	0	0
$273$	2298.53	0.509574
$274$	0	0
$275$	0	0
$276$	0	0
$277$	8882.38	1.92668	0.963341	−	0.268281i	$-0.0864556\pi$
$277$	8882.38	1.92668	0.963341	+	0.268281i	$0.0864556\pi$
$278$	0	0
$279$	13290.2	2.85183
$280$	0	0
$281$	−8151.77	−1.73058	−0.865291	−	0.501269i	$-0.832867\pi$
$281$	−8151.77	−1.73058	−0.865291	+	0.501269i	$0.832867\pi$
$282$	0	0
$283$	−5863.96	−1.23172	−0.615859	−	0.787856i	$-0.711191\pi$
$283$	−5863.96	−1.23172	−0.615859	+	0.787856i	$0.711191\pi$
$284$	0	0
$285$	−789.018	−0.163991
$286$	0	0
$287$	3177.66	0.653559
$288$	0	0
$289$	−622.465	−0.126698
$290$	0	0
$291$	−12087.1	−2.43492
$292$	0	0
$293$	7616.57	1.51865	0.759326	−	0.650711i	$-0.225529\pi$
$293$	7616.57	1.51865	0.759326	+	0.650711i	$0.225529\pi$
$294$	0	0
$295$	−8299.45	−1.63801
$296$	0	0
$297$	0	0
$298$	0	0
$299$	1193.24	0.230791
$300$	0	0
$301$	8297.98	1.58899
$302$	0	0
$303$	1594.64	0.302342
$304$	0	0
$305$	4658.63	0.874598
$306$	0	0
$307$	−4100.68	−0.762339	−0.381170	−	0.924505i	$-0.624479\pi$
$307$	−4100.68	−0.762339	−0.381170	+	0.924505i	$0.624479\pi$
$308$	0	0
$309$	−10022.5	−1.84519
$310$	0	0
$311$	−1456.13	−0.265497	−0.132749	−	0.991150i	$-0.542380\pi$
$311$	−1456.13	−0.265497	−0.132749	+	0.991150i	$0.542380\pi$
$312$	0	0
$313$	−3461.68	−0.625129	−0.312565	−	0.949896i	$-0.601188\pi$
$313$	−3461.68	−0.625129	−0.312565	+	0.949896i	$0.601188\pi$
$314$	0	0
$315$	−13749.8	−2.45940
$316$	0	0
$317$	4992.31	0.884530	0.442265	−	0.896884i	$-0.354175\pi$
$317$	4992.31	0.884530	0.442265	+	0.896884i	$0.354175\pi$
$318$	0	0
$319$	0	0
$320$	0	0
$321$	−6090.13	−1.05893
$322$	0	0
$323$	−474.899	−0.0818083
$324$	0	0
$325$	426.554	0.0728029
$326$	0	0
$327$	−10657.4	−1.80232
$328$	0	0
$329$	−3157.95	−0.529190
$330$	0	0
$331$	−10199.3	−1.69366	−0.846832	−	0.531861i	$-0.821493\pi$
$331$	−10199.3	−1.69366	−0.846832	+	0.531861i	$0.821493\pi$
$332$	0	0
$333$	−3929.94	−0.646725
$334$	0	0
$335$	3751.69	0.611870
$336$	0	0
$337$	−1680.74	−0.271678	−0.135839	−	0.990731i	$-0.543373\pi$
$337$	−1680.74	−0.271678	−0.135839	+	0.990731i	$0.543373\pi$
$338$	0	0
$339$	8391.19	1.34439
$340$	0	0
$341$	0	0
$342$	0	0
$343$	3180.76	0.500715
$344$	0	0
$345$	−11326.0	−1.76745
$346$	0	0
$347$	−10526.3	−1.62847	−0.814236	−	0.580534i	$-0.802844\pi$
$347$	−10526.3	−1.62847	−0.814236	+	0.580534i	$0.802844\pi$
$348$	0	0
$349$	−5300.36	−0.812956	−0.406478	−	0.913660i	$-0.633243\pi$
$349$	−5300.36	−0.812956	−0.406478	+	0.913660i	$0.633243\pi$
$350$	0	0
$351$	−1863.12	−0.283321
$352$	0	0
$353$	−7438.40	−1.12155	−0.560773	−	0.827969i	$-0.689496\pi$
$353$	−7438.40	−1.12155	−0.560773	+	0.827969i	$0.689496\pi$
$354$	0	0
$355$	−1686.82	−0.252190
$356$	0	0
$357$	−13131.5	−1.94676
$358$	0	0
$359$	−10151.3	−1.49238	−0.746191	−	0.665731i	$-0.768120\pi$
$359$	−10151.3	−1.49238	−0.746191	+	0.665731i	$0.768120\pi$
$360$	0	0
$361$	−6806.44	−0.992336
$362$	0	0
$363$	0	0
$364$	0	0
$365$	5979.37	0.857464
$366$	0	0
$367$	−476.803	−0.0678172	−0.0339086	−	0.999425i	$-0.510796\pi$
$367$	−476.803	−0.0678172	−0.0339086	+	0.999425i	$0.510796\pi$
$368$	0	0
$369$	−6232.69	−0.879297
$370$	0	0
$371$	−11764.1	−1.64626
$372$	0	0
$373$	−12738.5	−1.76829	−0.884146	−	0.467210i	$-0.845259\pi$
$373$	−12738.5	−1.76829	−0.884146	+	0.467210i	$0.845259\pi$
$374$	0	0
$375$	9554.74	1.31575
$376$	0	0
$377$	−1460.14	−0.199472
$378$	0	0
$379$	1298.69	0.176013	0.0880067	−	0.996120i	$-0.471950\pi$
$379$	1298.69	0.176013	0.0880067	+	0.996120i	$0.471950\pi$
$380$	0	0
$381$	−13250.4	−1.78173
$382$	0	0
$383$	670.568	0.0894632	0.0447316	−	0.998999i	$-0.485757\pi$
$383$	670.568	0.0894632	0.0447316	+	0.998999i	$0.485757\pi$
$384$	0	0
$385$	0	0
$386$	0	0
$387$	−16275.7	−2.13783
$388$	0	0
$389$	5235.90	0.682444	0.341222	−	0.939983i	$-0.389159\pi$
$389$	5235.90	0.682444	0.341222	+	0.939983i	$0.389159\pi$
$390$	0	0
$391$	−6816.97	−0.881711
$392$	0	0
$393$	6341.36	0.813942
$394$	0	0
$395$	−5196.67	−0.661956
$396$	0	0
$397$	1751.64	0.221442	0.110721	−	0.993852i	$-0.464684\pi$
$397$	1751.64	0.221442	0.110721	+	0.993852i	$0.464684\pi$
$398$	0	0
$399$	1453.47	0.182367
$400$	0	0
$401$	3612.78	0.449909	0.224954	−	0.974369i	$-0.427777\pi$
$401$	3612.78	0.449909	0.224954	+	0.974369i	$0.427777\pi$
$402$	0	0
$403$	3311.35	0.409305
$404$	0	0
$405$	1860.62	0.228284
$406$	0	0
$407$	0	0
$408$	0	0
$409$	−11295.0	−1.36553	−0.682764	−	0.730639i	$-0.739222\pi$
$409$	−11295.0	−1.36553	−0.682764	+	0.730639i	$0.739222\pi$
$410$	0	0
$411$	−4406.46	−0.528844
$412$	0	0
$413$	15288.6	1.82156
$414$	0	0
$415$	17315.0	2.04810
$416$	0	0
$417$	21047.7	2.47173
$418$	0	0
$419$	3680.45	0.429121	0.214560	−	0.976711i	$-0.431168\pi$
$419$	3680.45	0.429121	0.214560	+	0.976711i	$0.431168\pi$
$420$	0	0
$421$	9256.17	1.07154	0.535770	−	0.844364i	$-0.320021\pi$
$421$	9256.17	1.07154	0.535770	+	0.844364i	$0.320021\pi$
$422$	0	0
$423$	6194.03	0.711972
$424$	0	0
$425$	−2436.90	−0.278135
$426$	0	0
$427$	−8581.77	−0.972602
$428$	0	0
$429$	0	0
$430$	0	0
$431$	−3584.04	−0.400550	−0.200275	−	0.979740i	$-0.564184\pi$
$431$	−3584.04	−0.400550	−0.200275	+	0.979740i	$0.564184\pi$
$432$	0	0
$433$	−8306.80	−0.921939	−0.460969	−	0.887416i	$-0.652498\pi$
$433$	−8306.80	−0.921939	−0.460969	+	0.887416i	$0.652498\pi$
$434$	0	0
$435$	13859.4	1.52760
$436$	0	0
$437$	754.538	0.0825960
$438$	0	0
$439$	−11932.4	−1.29727	−0.648634	−	0.761101i	$-0.724659\pi$
$439$	−11932.4	−1.29727	−0.648634	+	0.761101i	$0.724659\pi$
$440$	0	0
$441$	9544.99	1.03067
$442$	0	0
$443$	−1078.82	−0.115703	−0.0578514	−	0.998325i	$-0.518425\pi$
$443$	−1078.82	−0.115703	−0.0578514	+	0.998325i	$0.518425\pi$
$444$	0	0
$445$	−3314.58	−0.353093
$446$	0	0
$447$	3715.00	0.393095
$448$	0	0
$449$	12765.9	1.34179	0.670893	−	0.741554i	$-0.265911\pi$
$449$	12765.9	1.34179	0.670893	+	0.741554i	$0.265911\pi$
$450$	0	0
$451$	0	0
$452$	0	0
$453$	−6673.57	−0.692167
$454$	0	0
$455$	−3425.86	−0.352982
$456$	0	0
$457$	−11226.8	−1.14917	−0.574583	−	0.818446i	$-0.694836\pi$
$457$	−11226.8	−1.14917	−0.574583	+	0.818446i	$0.694836\pi$
$458$	0	0
$459$	10644.0	1.08239
$460$	0	0
$461$	−2160.58	−0.218283	−0.109141	−	0.994026i	$-0.534810\pi$
$461$	−2160.58	−0.218283	−0.109141	+	0.994026i	$0.534810\pi$
$462$	0	0
$463$	−11469.5	−1.15125	−0.575627	−	0.817712i	$-0.695242\pi$
$463$	−11469.5	−1.15125	−0.575627	+	0.817712i	$0.695242\pi$
$464$	0	0
$465$	−31430.8	−3.13456
$466$	0	0
$467$	−2018.39	−0.200000	−0.100000	−	0.994987i	$-0.531884\pi$
$467$	−2018.39	−0.200000	−0.100000	+	0.994987i	$0.531884\pi$
$468$	0	0
$469$	−6911.07	−0.680434
$470$	0	0
$471$	−4463.75	−0.436685
$472$	0	0
$473$	0	0
$474$	0	0
$475$	269.729	0.0260548
$476$	0	0
$477$	23074.3	2.21488
$478$	0	0
$479$	7327.00	0.698913	0.349457	−	0.936953i	$-0.386366\pi$
$479$	7327.00	0.698913	0.349457	+	0.936953i	$0.386366\pi$
$480$	0	0
$481$	−979.176	−0.0928203
$482$	0	0
$483$	20863.9	1.96551
$484$	0	0
$485$	18015.3	1.68667
$486$	0	0
$487$	6189.80	0.575948	0.287974	−	0.957638i	$-0.407018\pi$
$487$	6189.80	0.575948	0.287974	+	0.957638i	$0.407018\pi$
$488$	0	0
$489$	21732.3	2.00976
$490$	0	0
$491$	10120.9	0.930246	0.465123	−	0.885246i	$-0.346010\pi$
$491$	10120.9	0.930246	0.465123	+	0.885246i	$0.346010\pi$
$492$	0	0
$493$	8341.77	0.762058
$494$	0	0
$495$	0	0
$496$	0	0
$497$	3107.34	0.280449
$498$	0	0
$499$	−2562.81	−0.229914	−0.114957	−	0.993370i	$-0.536673\pi$
$499$	−2562.81	−0.229914	−0.114957	+	0.993370i	$0.536673\pi$
$500$	0	0
$501$	11838.6	1.05571
$502$	0	0
$503$	−8601.51	−0.762470	−0.381235	−	0.924478i	$-0.624501\pi$
$503$	−8601.51	−0.762470	−0.381235	+	0.924478i	$0.624501\pi$
$504$	0	0
$505$	−2376.74	−0.209432
$506$	0	0
$507$	17650.0	1.54609
$508$	0	0
$509$	−13945.1	−1.21435	−0.607176	−	0.794567i	$-0.707698\pi$
$509$	−13945.1	−1.21435	−0.607176	+	0.794567i	$0.707698\pi$
$510$	0	0
$511$	−11014.7	−0.953548
$512$	0	0
$513$	−1178.13	−0.101395
$514$	0	0
$515$	14938.1	1.27816
$516$	0	0
$517$	0	0
$518$	0	0
$519$	3789.80	0.320527
$520$	0	0
$521$	−12092.8	−1.01688	−0.508442	−	0.861096i	$-0.669778\pi$
$521$	−12092.8	−1.01688	−0.508442	+	0.861096i	$0.669778\pi$
$522$	0	0
$523$	16691.8	1.39557	0.697784	−	0.716308i	$-0.254169\pi$
$523$	16691.8	1.39557	0.697784	+	0.716308i	$0.254169\pi$
$524$	0	0
$525$	7458.35	0.620017
$526$	0	0
$527$	−18917.8	−1.56370
$528$	0	0
$529$	−1335.94	−0.109800
$530$	0	0
$531$	−29987.2	−2.45072
$532$	0	0
$533$	−1552.92	−0.126200
$534$	0	0
$535$	9077.06	0.733524
$536$	0	0
$537$	3218.37	0.258628
$538$	0	0
$539$	0	0
$540$	0	0
$541$	20673.3	1.64291	0.821456	−	0.570272i	$-0.193162\pi$
$541$	20673.3	1.64291	0.821456	+	0.570272i	$0.193162\pi$
$542$	0	0
$543$	−28145.1	−2.22435
$544$	0	0
$545$	15884.4	1.24847
$546$	0	0
$547$	−9169.80	−0.716768	−0.358384	−	0.933574i	$-0.616672\pi$
$547$	−9169.80	−0.716768	−0.358384	+	0.933574i	$0.616672\pi$
$548$	0	0
$549$	16832.4	1.30854
$550$	0	0
$551$	−923.311	−0.0713873
$552$	0	0
$553$	9572.90	0.736132
$554$	0	0
$555$	9294.18	0.710840
$556$	0	0
$557$	20310.8	1.54506	0.772528	−	0.634981i	$-0.218992\pi$
$557$	20310.8	1.54506	0.772528	+	0.634981i	$0.218992\pi$
$558$	0	0
$559$	−4055.22	−0.306829
$560$	0	0
$561$	0	0
$562$	0	0
$563$	−5563.15	−0.416445	−0.208223	−	0.978081i	$-0.566768\pi$
$563$	−5563.15	−0.416445	−0.208223	+	0.978081i	$0.566768\pi$
$564$	0	0
$565$	−12506.7	−0.931257
$566$	0	0
$567$	−3427.50	−0.253865
$568$	0	0
$569$	−22785.5	−1.67876	−0.839381	−	0.543543i	$-0.817083\pi$
$569$	−22785.5	−1.67876	−0.839381	+	0.543543i	$0.817083\pi$
$570$	0	0
$571$	11157.6	0.817743	0.408871	−	0.912592i	$-0.365922\pi$
$571$	11157.6	0.817743	0.408871	+	0.912592i	$0.365922\pi$
$572$	0	0
$573$	19570.3	1.42681
$574$	0	0
$575$	3871.85	0.280812
$576$	0	0
$577$	12429.1	0.896759	0.448380	−	0.893843i	$-0.352001\pi$
$577$	12429.1	0.896759	0.448380	+	0.893843i	$0.352001\pi$
$578$	0	0
$579$	19682.5	1.41274
$580$	0	0
$581$	−31896.4	−2.27760
$582$	0	0
$583$	0	0
$584$	0	0
$585$	6719.51	0.474902
$586$	0	0
$587$	−6114.94	−0.429967	−0.214984	−	0.976618i	$-0.568970\pi$
$587$	−6114.94	−0.429967	−0.214984	+	0.976618i	$0.568970\pi$
$588$	0	0
$589$	2093.92	0.146483
$590$	0	0
$591$	8902.72	0.619643
$592$	0	0
$593$	7188.32	0.497789	0.248894	−	0.968531i	$-0.419933\pi$
$593$	7188.32	0.497789	0.248894	+	0.968531i	$0.419933\pi$
$594$	0	0
$595$	19572.0	1.34852
$596$	0	0
$597$	29598.4	2.02912
$598$	0	0
$599$	1012.18	0.0690429	0.0345215	−	0.999404i	$-0.489009\pi$
$599$	1012.18	0.0690429	0.0345215	+	0.999404i	$0.489009\pi$
$600$	0	0
$601$	13380.9	0.908184	0.454092	−	0.890955i	$-0.349964\pi$
$601$	13380.9	0.908184	0.454092	+	0.890955i	$0.349964\pi$
$602$	0	0
$603$	13555.4	0.915455
$604$	0	0
$605$	0	0
$606$	0	0
$607$	−546.951	−0.0365734	−0.0182867	−	0.999833i	$-0.505821\pi$
$607$	−546.951	−0.0365734	−0.0182867	+	0.999833i	$0.505821\pi$
$608$	0	0
$609$	−25530.7	−1.69878
$610$	0	0
$611$	1543.29	0.102185
$612$	0	0
$613$	13745.6	0.905676	0.452838	−	0.891593i	$-0.350412\pi$
$613$	13745.6	0.905676	0.452838	+	0.891593i	$0.350412\pi$
$614$	0	0
$615$	14740.1	0.966468
$616$	0	0
$617$	−3323.39	−0.216847	−0.108423	−	0.994105i	$-0.534580\pi$
$617$	−3323.39	−0.216847	−0.108423	+	0.994105i	$0.534580\pi$
$618$	0	0
$619$	21679.1	1.40768	0.703842	−	0.710357i	$-0.251467\pi$
$619$	21679.1	1.40768	0.703842	+	0.710357i	$0.251467\pi$
$620$	0	0
$621$	−16911.6	−1.09282
$622$	0	0
$623$	6105.87	0.392659
$624$	0	0
$625$	−18891.3	−1.20905
$626$	0	0
$627$	0	0
$628$	0	0
$629$	5594.04	0.354609
$630$	0	0
$631$	29888.8	1.88566	0.942831	−	0.333271i	$-0.108153\pi$
$631$	29888.8	1.88566	0.942831	+	0.333271i	$0.108153\pi$
$632$	0	0
$633$	17875.9	1.12244
$634$	0	0
$635$	19749.2	1.23421
$636$	0	0
$637$	2378.21	0.147925
$638$	0	0
$639$	−6094.75	−0.377316
$640$	0	0
$641$	27178.0	1.67467	0.837336	−	0.546689i	$-0.184112\pi$
$641$	27178.0	1.67467	0.837336	+	0.546689i	$0.184112\pi$
$642$	0	0
$643$	−16374.5	−1.00427	−0.502136	−	0.864789i	$-0.667452\pi$
$643$	−16374.5	−1.00427	−0.502136	+	0.864789i	$0.667452\pi$
$644$	0	0
$645$	38491.5	2.34977
$646$	0	0
$647$	−2287.67	−0.139007	−0.0695034	−	0.997582i	$-0.522141\pi$
$647$	−2287.67	−0.139007	−0.0695034	+	0.997582i	$0.522141\pi$
$648$	0	0
$649$	0	0
$650$	0	0
$651$	57899.4	3.48580
$652$	0	0
$653$	−1146.92	−0.0687329	−0.0343665	−	0.999409i	$-0.510941\pi$
$653$	−1146.92	−0.0687329	−0.0343665	+	0.999409i	$0.510941\pi$
$654$	0	0
$655$	−9451.50	−0.563818
$656$	0	0
$657$	21604.4	1.28290
$658$	0	0
$659$	377.923	0.0223396	0.0111698	−	0.999938i	$-0.496444\pi$
$659$	377.923	0.0223396	0.0111698	+	0.999938i	$0.496444\pi$
$660$	0	0
$661$	−17500.4	−1.02978	−0.514892	−	0.857255i	$-0.672168\pi$
$661$	−17500.4	−1.02978	−0.514892	+	0.857255i	$0.672168\pi$
$662$	0	0
$663$	6417.38	0.375913
$664$	0	0
$665$	−2166.33	−0.126326
$666$	0	0
$667$	−13253.7	−0.769395
$668$	0	0
$669$	31646.6	1.82889
$670$	0	0
$671$	0	0
$672$	0	0
$673$	−14510.5	−0.831114	−0.415557	−	0.909567i	$-0.636413\pi$
$673$	−14510.5	−0.831114	−0.415557	+	0.909567i	$0.636413\pi$
$674$	0	0
$675$	−6045.49	−0.344728
$676$	0	0
$677$	469.109	0.0266312	0.0133156	−	0.999911i	$-0.495761\pi$
$677$	469.109	0.0266312	0.0133156	+	0.999911i	$0.495761\pi$
$678$	0	0
$679$	−33186.5	−1.87567
$680$	0	0
$681$	−41473.1	−2.33370
$682$	0	0
$683$	15892.3	0.890342	0.445171	−	0.895446i	$-0.353143\pi$
$683$	15892.3	0.890342	0.445171	+	0.895446i	$0.353143\pi$
$684$	0	0
$685$	6567.63	0.366330
$686$	0	0
$687$	11423.3	0.634392
$688$	0	0
$689$	5749.13	0.317887
$690$	0	0
$691$	6205.64	0.341641	0.170820	−	0.985302i	$-0.445358\pi$
$691$	6205.64	0.341641	0.170820	+	0.985302i	$0.445358\pi$
$692$	0	0
$693$	0	0
$694$	0	0
$695$	−31370.7	−1.71217
$696$	0	0
$697$	8871.85	0.482131
$698$	0	0
$699$	50448.8	2.72983
$700$	0	0
$701$	1872.54	0.100891	0.0504456	−	0.998727i	$-0.483936\pi$
$701$	1872.54	0.100891	0.0504456	+	0.998727i	$0.483936\pi$
$702$	0	0
$703$	−619.178	−0.0332187
$704$	0	0
$705$	−14648.7	−0.782554
$706$	0	0
$707$	4378.24	0.232901
$708$	0	0
$709$	5236.01	0.277352	0.138676	−	0.990338i	$-0.455715\pi$
$709$	5236.01	0.277352	0.138676	+	0.990338i	$0.455715\pi$
$710$	0	0
$711$	−18776.4	−0.990392
$712$	0	0
$713$	30057.3	1.57876
$714$	0	0
$715$	0	0
$716$	0	0
$717$	−28367.5	−1.47755
$718$	0	0
$719$	17274.8	0.896023	0.448011	−	0.894028i	$-0.352132\pi$
$719$	17274.8	0.896023	0.448011	+	0.894028i	$0.352132\pi$
$720$	0	0
$721$	−27517.9	−1.42139
$722$	0	0
$723$	45083.0	2.31902
$724$	0	0
$725$	−4737.89	−0.242705
$726$	0	0
$727$	−17292.7	−0.882190	−0.441095	−	0.897461i	$-0.645410\pi$
$727$	−17292.7	−0.882190	−0.441095	+	0.897461i	$0.645410\pi$
$728$	0	0
$729$	−30768.0	−1.56318
$730$	0	0
$731$	23167.5	1.17220
$732$	0	0
$733$	−13881.3	−0.699480	−0.349740	−	0.936847i	$-0.613730\pi$
$733$	−13881.3	−0.699480	−0.349740	+	0.936847i	$0.613730\pi$
$734$	0	0
$735$	−22573.6	−1.13284
$736$	0	0
$737$	0	0
$738$	0	0
$739$	−6379.26	−0.317544	−0.158772	−	0.987315i	$-0.550753\pi$
$739$	−6379.26	−0.317544	−0.158772	+	0.987315i	$0.550753\pi$
$740$	0	0
$741$	−710.310	−0.0352144
$742$	0	0
$743$	−23042.7	−1.13776	−0.568880	−	0.822421i	$-0.692623\pi$
$743$	−23042.7	−1.13776	−0.568880	+	0.822421i	$0.692623\pi$
$744$	0	0
$745$	−5537.03	−0.272297
$746$	0	0
$747$	62561.8	3.06428
$748$	0	0
$749$	−16721.1	−0.815720
$750$	0	0
$751$	20806.1	1.01095	0.505476	−	0.862841i	$-0.331317\pi$
$751$	20806.1	1.01095	0.505476	+	0.862841i	$0.331317\pi$
$752$	0	0
$753$	−16764.0	−0.811306
$754$	0	0
$755$	9946.65	0.479464
$756$	0	0
$757$	−21837.7	−1.04849	−0.524245	−	0.851568i	$-0.675652\pi$
$757$	−21837.7	−1.04849	−0.524245	+	0.851568i	$0.675652\pi$
$758$	0	0
$759$	0	0
$760$	0	0
$761$	−7139.46	−0.340086	−0.170043	−	0.985437i	$-0.554391\pi$
$761$	−7139.46	−0.340086	−0.170043	+	0.985437i	$0.554391\pi$
$762$	0	0
$763$	−29261.1	−1.38836
$764$	0	0
$765$	−38388.6	−1.81430
$766$	0	0
$767$	−7471.55	−0.351737
$768$	0	0
$769$	15473.8	0.725618	0.362809	−	0.931864i	$-0.381818\pi$
$769$	15473.8	0.725618	0.362809	+	0.931864i	$0.381818\pi$
$770$	0	0
$771$	−50774.0	−2.37170
$772$	0	0
$773$	15240.9	0.709155	0.354577	−	0.935027i	$-0.384625\pi$
$773$	15240.9	0.709155	0.354577	+	0.935027i	$0.384625\pi$
$774$	0	0
$775$	10744.8	0.498017
$776$	0	0
$777$	−17121.0	−0.790494
$778$	0	0
$779$	−981.984	−0.0451646
$780$	0	0
$781$	0	0
$782$	0	0
$783$	20694.3	0.944515
$784$	0	0
$785$	6653.01	0.302492
$786$	0	0
$787$	−36081.4	−1.63426	−0.817131	−	0.576452i	$-0.804437\pi$
$787$	−36081.4	−1.63426	−0.817131	+	0.576452i	$0.804437\pi$
$788$	0	0
$789$	−14561.6	−0.657043
$790$	0	0
$791$	23038.9	1.03561
$792$	0	0
$793$	4193.91	0.187806
$794$	0	0
$795$	−54569.8	−2.43446
$796$	0	0
$797$	3383.49	0.150375	0.0751877	−	0.997169i	$-0.476044\pi$
$797$	3383.49	0.150375	0.0751877	+	0.997169i	$0.476044\pi$
$798$	0	0
$799$	−8816.83	−0.390384
$800$	0	0
$801$	−11976.1	−0.528283
$802$	0	0
$803$	0	0
$804$	0	0
$805$	−31096.7	−1.36151
$806$	0	0
$807$	49948.3	2.17876
$808$	0	0
$809$	13480.3	0.585835	0.292917	−	0.956138i	$-0.405374\pi$
$809$	13480.3	0.585835	0.292917	+	0.956138i	$0.405374\pi$
$810$	0	0
$811$	21423.1	0.927579	0.463790	−	0.885945i	$-0.346489\pi$
$811$	21423.1	0.927579	0.463790	+	0.885945i	$0.346489\pi$
$812$	0	0
$813$	50465.3	2.17699
$814$	0	0
$815$	−32391.1	−1.39216
$816$	0	0
$817$	−2564.30	−0.109809
$818$	0	0
$819$	−12378.2	−0.528117
$820$	0	0
$821$	10664.8	0.453355	0.226677	−	0.973970i	$-0.427214\pi$
$821$	10664.8	0.453355	0.226677	+	0.973970i	$0.427214\pi$
$822$	0	0
$823$	11996.9	0.508123	0.254062	−	0.967188i	$-0.418233\pi$
$823$	11996.9	0.508123	0.254062	+	0.967188i	$0.418233\pi$
$824$	0	0
$825$	0	0
$826$	0	0
$827$	−40880.7	−1.71894	−0.859469	−	0.511188i	$-0.829206\pi$
$827$	−40880.7	−1.71894	−0.859469	+	0.511188i	$0.829206\pi$
$828$	0	0
$829$	−33237.0	−1.39248	−0.696241	−	0.717808i	$-0.745145\pi$
$829$	−33237.0	−1.39248	−0.696241	+	0.717808i	$0.745145\pi$
$830$	0	0
$831$	−75899.9	−3.16840
$832$	0	0
$833$	−13586.7	−0.565128
$834$	0	0
$835$	−17644.9	−0.731288
$836$	0	0
$837$	−46931.4	−1.93809
$838$	0	0
$839$	6368.57	0.262059	0.131029	−	0.991378i	$-0.458172\pi$
$839$	6368.57	0.262059	0.131029	+	0.991378i	$0.458172\pi$
$840$	0	0
$841$	−8170.70	−0.335016
$842$	0	0
$843$	69656.8	2.84591
$844$	0	0
$845$	−26306.6	−1.07097
$846$	0	0
$847$	0	0
$848$	0	0
$849$	50107.5	2.02554
$850$	0	0
$851$	−8888.02	−0.358023
$852$	0	0
$853$	3036.41	0.121881	0.0609406	−	0.998141i	$-0.480590\pi$
$853$	3036.41	0.121881	0.0609406	+	0.998141i	$0.480590\pi$
$854$	0	0
$855$	4249.05	0.169958
$856$	0	0
$857$	−10184.8	−0.405959	−0.202979	−	0.979183i	$-0.565062\pi$
$857$	−10184.8	−0.405959	−0.202979	+	0.979183i	$0.565062\pi$
$858$	0	0
$859$	34929.8	1.38742	0.693708	−	0.720256i	$-0.255976\pi$
$859$	34929.8	1.38742	0.693708	+	0.720256i	$0.255976\pi$
$860$	0	0
$861$	−27153.1	−1.07477
$862$	0	0
$863$	12283.3	0.484504	0.242252	−	0.970213i	$-0.422114\pi$
$863$	12283.3	0.484504	0.242252	+	0.970213i	$0.422114\pi$
$864$	0	0
$865$	−5648.52	−0.222029
$866$	0	0
$867$	5318.96	0.208352
$868$	0	0
$869$	0	0
$870$	0	0
$871$	3377.44	0.131389
$872$	0	0
$873$	65092.2	2.52352
$874$	0	0
$875$	26233.5	1.01355
$876$	0	0
$877$	19137.9	0.736877	0.368438	−	0.929652i	$-0.379893\pi$
$877$	19137.9	0.736877	0.368438	+	0.929652i	$0.379893\pi$
$878$	0	0
$879$	−65083.5	−2.49740
$880$	0	0
$881$	−16737.0	−0.640049	−0.320024	−	0.947409i	$-0.603691\pi$
$881$	−16737.0	−0.640049	−0.320024	+	0.947409i	$0.603691\pi$
$882$	0	0
$883$	18604.2	0.709039	0.354520	−	0.935049i	$-0.384644\pi$
$883$	18604.2	0.709039	0.354520	+	0.935049i	$0.384644\pi$
$884$	0	0
$885$	70918.7	2.69368
$886$	0	0
$887$	−15912.8	−0.602367	−0.301184	−	0.953566i	$-0.597382\pi$
$887$	−15912.8	−0.602367	−0.301184	+	0.953566i	$0.597382\pi$
$888$	0	0
$889$	−36380.4	−1.37251
$890$	0	0
$891$	0	0
$892$	0	0
$893$	975.894	0.0365700
$894$	0	0
$895$	−4796.84	−0.179152
$896$	0	0
$897$	−10196.2	−0.379533
$898$	0	0
$899$	−36780.4	−1.36451
$900$	0	0
$901$	−32844.8	−1.21445
$902$	0	0
$903$	−70906.1	−2.61308
$904$	0	0
$905$	41949.0	1.54081
$906$	0	0
$907$	22555.3	0.825729	0.412864	−	0.910793i	$-0.364528\pi$
$907$	22555.3	0.825729	0.412864	+	0.910793i	$0.364528\pi$
$908$	0	0
$909$	−8587.51	−0.313344
$910$	0	0
$911$	−48359.6	−1.75876	−0.879378	−	0.476125i	$-0.842041\pi$
$911$	−48359.6	−1.75876	−0.879378	+	0.476125i	$0.842041\pi$
$912$	0	0
$913$	0	0
$914$	0	0
$915$	−39808.0	−1.43826
$916$	0	0
$917$	17410.8	0.626997
$918$	0	0
$919$	−21014.3	−0.754295	−0.377148	−	0.926153i	$-0.623095\pi$
$919$	−21014.3	−0.754295	−0.377148	+	0.926153i	$0.623095\pi$
$920$	0	0
$921$	35040.3	1.25365
$922$	0	0
$923$	−1518.56	−0.0541537
$924$	0	0
$925$	−3177.26	−0.112938
$926$	0	0
$927$	53973.8	1.91233
$928$	0	0
$929$	4919.67	0.173745	0.0868725	−	0.996219i	$-0.472313\pi$
$929$	4919.67	0.173745	0.0868725	+	0.996219i	$0.472313\pi$
$930$	0	0
$931$	1503.85	0.0529395
$932$	0	0
$933$	12442.6	0.436606
$934$	0	0
$935$	0	0
$936$	0	0
$937$	−17180.3	−0.598991	−0.299496	−	0.954098i	$-0.596818\pi$
$937$	−17180.3	−0.598991	−0.299496	+	0.954098i	$0.596818\pi$
$938$	0	0
$939$	29580.0	1.02802
$940$	0	0
$941$	−51486.9	−1.78366	−0.891830	−	0.452370i	$-0.850579\pi$
$941$	−51486.9	−1.78366	−0.891830	+	0.452370i	$0.850579\pi$
$942$	0	0
$943$	−14095.9	−0.486773
$944$	0	0
$945$	48554.3	1.67140
$946$	0	0
$947$	−45107.8	−1.54784	−0.773920	−	0.633283i	$-0.781707\pi$
$947$	−45107.8	−1.54784	−0.773920	+	0.633283i	$0.781707\pi$
$948$	0	0
$949$	5382.90	0.184127
$950$	0	0
$951$	−42659.2	−1.45460
$952$	0	0
$953$	25385.8	0.862881	0.431440	−	0.902141i	$-0.358006\pi$
$953$	25385.8	0.862881	0.431440	+	0.902141i	$0.358006\pi$
$954$	0	0
$955$	−29168.6	−0.988349
$956$	0	0
$957$	0	0
$958$	0	0
$959$	−12098.4	−0.407380
$960$	0	0
$961$	53621.0	1.79990
$962$	0	0
$963$	32796.8	1.09747
$964$	0	0
$965$	−29335.9	−0.978607
$966$	0	0
$967$	−26837.3	−0.892482	−0.446241	−	0.894913i	$-0.647238\pi$
$967$	−26837.3	−0.892482	−0.446241	+	0.894913i	$0.647238\pi$
$968$	0	0
$969$	4058.01	0.134532
$970$	0	0
$971$	5451.84	0.180183	0.0900916	−	0.995933i	$-0.471284\pi$
$971$	5451.84	0.180183	0.0900916	+	0.995933i	$0.471284\pi$
$972$	0	0
$973$	57788.6	1.90403
$974$	0	0
$975$	−3644.89	−0.119723
$976$	0	0
$977$	−3608.24	−0.118155	−0.0590776	−	0.998253i	$-0.518816\pi$
$977$	−3608.24	−0.118155	−0.0590776	+	0.998253i	$0.518816\pi$
$978$	0	0
$979$	0	0
$980$	0	0
$981$	57392.9	1.86790
$982$	0	0
$983$	−58045.5	−1.88338	−0.941691	−	0.336478i	$-0.890764\pi$
$983$	−58045.5	−1.88338	−0.941691	+	0.336478i	$0.890764\pi$
$984$	0	0
$985$	−13269.1	−0.429227
$986$	0	0
$987$	26984.7	0.870244
$988$	0	0
$989$	−36809.4	−1.18349
$990$	0	0
$991$	18977.5	0.608315	0.304157	−	0.952622i	$-0.401625\pi$
$991$	18977.5	0.608315	0.304157	+	0.952622i	$0.401625\pi$
$992$	0	0
$993$	87152.7	2.78520
$994$	0	0
$995$	−44115.1	−1.40557
$996$	0	0
$997$	−24260.2	−0.770641	−0.385321	−	0.922783i	$-0.625909\pi$
$997$	−24260.2	−0.770641	−0.385321	+	0.922783i	$0.625909\pi$
$998$	0	0
$999$	13877.7	0.439512

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	1936.4.a.bn.1.1		4
4.3	odd	2		242.4.a.n.1.4		4
11.5	even	5		176.4.m.b.113.1		8
11.9	even	5		176.4.m.b.81.1		8
11.10	odd	2		1936.4.a.bm.1.1		4
12.11	even	2		2178.4.a.by.1.2		4
44.3	odd	10		242.4.c.r.9.1		8
44.7	even	10		242.4.c.n.27.1		8
44.15	odd	10		242.4.c.r.27.1		8
44.19	even	10		242.4.c.n.9.1		8
44.27	odd	10		22.4.c.b.3.2	✓	8
44.31	odd	10		22.4.c.b.15.2	yes	8
44.35	even	10		242.4.c.q.81.2		8
44.39	even	10		242.4.c.q.3.2		8
44.43	even	2		242.4.a.o.1.4		4
132.71	even	10		198.4.f.d.91.2		8
132.119	even	10		198.4.f.d.37.2		8
132.131	odd	2		2178.4.a.bt.1.2		4

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
22.4.c.b.3.2	✓	8	44.27	odd	10
22.4.c.b.15.2	yes	8	44.31	odd	10
176.4.m.b.81.1		8	11.9	even	5
176.4.m.b.113.1		8	11.5	even	5
198.4.f.d.37.2		8	132.119	even	10
198.4.f.d.91.2		8	132.71	even	10
242.4.a.n.1.4		4	4.3	odd	2
242.4.a.o.1.4		4	44.43	even	2
242.4.c.n.9.1		8	44.19	even	10
242.4.c.n.27.1		8	44.7	even	10
242.4.c.q.3.2		8	44.39	even	10
242.4.c.q.81.2		8	44.35	even	10
242.4.c.r.9.1		8	44.3	odd	10
242.4.c.r.27.1		8	44.15	odd	10
1936.4.a.bm.1.1		4	11.10	odd	2
1936.4.a.bn.1.1		4	1.1	even	1	trivial
2178.4.a.bt.1.2		4	132.131	odd	2
2178.4.a.by.1.2		4	12.11	even	2

Overview	Random
Universe	Knowledge

Classical	Maass
Hilbert	Bianchi

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

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

\(f(q)\)	\(=\)	\(q-8.54499 q^{3} +12.7359 q^{5} -23.4611 q^{7} +46.0168 q^{9} +O(q^{10})\)	\(q-8.54499 q^{3} +12.7359 q^{5} -23.4611 q^{7} +46.0168 q^{9} +11.4654 q^{13} -108.828 q^{15} -65.5022 q^{17} +7.25013 q^{19} +200.475 q^{21} +104.072 q^{23} +37.2034 q^{25} -162.498 q^{27} -127.351 q^{29} +288.811 q^{31} -298.799 q^{35} -85.4023 q^{37} -97.9721 q^{39} -135.444 q^{41} -353.691 q^{43} +586.066 q^{45} +134.604 q^{47} +207.424 q^{49} +559.715 q^{51} +501.431 q^{53} -61.9522 q^{57} -651.658 q^{59} +365.787 q^{61} -1079.61 q^{63} +146.023 q^{65} +294.576 q^{67} -889.297 q^{69} -132.446 q^{71} +469.489 q^{73} -317.903 q^{75} -408.033 q^{79} +146.093 q^{81} +1359.54 q^{83} -834.230 q^{85} +1088.21 q^{87} -260.255 q^{89} -268.992 q^{91} -2467.89 q^{93} +92.3370 q^{95} +1414.53 q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 4 q - 4 q^{3} + 25 q^{5} + 3 q^{7} + 102 q^{9}+O(q^{10}) \) 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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