LMFDB - Embedded newform 1936.2.a.o.1.1

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [1936,2,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, 2, names="a")

//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code

chi := DirichletCharacter("1936.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 1936 = 2^{4} \cdot 11^{2} $
Weight:	$ k $	$=$	$ 2 $
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:	$15.4590378313$
Analytic rank:	$0$
Dimension:	$2$
Coefficient field:	$\Q(\zeta_{10})^+$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{2} - x - 1 $ x^2 - x - 1
Coefficient ring:	$\Z[a_1, a_2, a_3]$
Coefficient ring index:	$ 1 $
Twist minimal:	no (minimal twist has level 22)
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.1
Root			$1.61803$ 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)

$f(q)$	$=$	$q-2.61803 q^{3} -1.23607 q^{5} +2.00000 q^{7} +3.85410 q^{9} -3.23607 q^{13} +3.23607 q^{15} +1.61803 q^{17} -0.854102 q^{19} -5.23607 q^{21} +3.23607 q^{23} -3.47214 q^{25} -2.23607 q^{27} -4.47214 q^{29} -2.00000 q^{31} -2.47214 q^{35} +9.70820 q^{37} +8.47214 q^{39} +3.38197 q^{41} -11.5623 q^{43} -4.76393 q^{45} -2.47214 q^{47} -3.00000 q^{49} -4.23607 q^{51} -10.4721 q^{53} +2.23607 q^{57} +6.38197 q^{59} +6.47214 q^{61} +7.70820 q^{63} +4.00000 q^{65} +0.0901699 q^{67} -8.47214 q^{69} +0.763932 q^{71} +12.6180 q^{73} +9.09017 q^{75} +13.4164 q^{79} -5.70820 q^{81} +6.32624 q^{83} -2.00000 q^{85} +11.7082 q^{87} +3.09017 q^{89} -6.47214 q^{91} +5.23607 q^{93} +1.05573 q^{95} +13.8541 q^{97} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 2 q - 3 q^{3} + 2 q^{5} + 4 q^{7} + q^{9} - 2 q^{13} + 2 q^{15} + q^{17} + 5 q^{19} - 6 q^{21} + 2 q^{23} + 2 q^{25} - 4 q^{31} + 4 q^{35} + 6 q^{37} + 8 q^{39} + 9 q^{41} - 3 q^{43} - 14 q^{45} + 4 q^{47}+ \cdots + 21 q^{97}+O(q^{100}) $ 2 * q - 3 * q^3 + 2 * q^5 + 4 * q^7 + q^9 - 2 * q^13 + 2 * q^15 + q^17 + 5 * q^19 - 6 * q^21 + 2 * q^23 + 2 * q^25 - 4 * q^31 + 4 * q^35 + 6 * q^37 + 8 * q^39 + 9 * q^41 - 3 * q^43 - 14 * q^45 + 4 * q^47 - 6 * q^49 - 4 * q^51 - 12 * q^53 + 15 * q^59 + 4 * q^61 + 2 * q^63 + 8 * q^65 - 11 * q^67 - 8 * q^69 + 6 * q^71 + 23 * q^73 + 7 * q^75 + 2 * q^81 - 3 * q^83 - 4 * q^85 + 10 * q^87 - 5 * q^89 - 4 * q^91 + 6 * q^93 + 20 * q^95 + 21 * 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
$3$	−2.61803	−1.51152	−0.755761	−	0.654847i	$-0.772733\pi$
$3$	−2.61803	−1.51152	−0.755761	+	0.654847i	$0.772733\pi$
$4$	0	0
$5$	−1.23607	−0.552786	−0.276393	−	0.961045i	$-0.589139\pi$
$5$	−1.23607	−0.552786	−0.276393	+	0.961045i	$0.589139\pi$
$6$	0	0
$7$	2.00000	0.755929	0.377964	−	0.925820i	$-0.376624\pi$
$7$	2.00000	0.755929	0.377964	+	0.925820i	$0.376624\pi$
$8$	0	0
$9$	3.85410	1.28470
$10$	0	0
$11$	0	0
$11$	0	0
$12$	0	0
$13$	−3.23607	−0.897524	−0.448762	−	0.893651i	$-0.648135\pi$
$13$	−3.23607	−0.897524	−0.448762	+	0.893651i	$0.648135\pi$
$14$	0	0
$15$	3.23607	0.835549
$16$	0	0
$17$	1.61803	0.392431	0.196215	−	0.980561i	$-0.437135\pi$
$17$	1.61803	0.392431	0.196215	+	0.980561i	$0.437135\pi$
$18$	0	0
$19$	−0.854102	−0.195944	−0.0979722	−	0.995189i	$-0.531236\pi$
$19$	−0.854102	−0.195944	−0.0979722	+	0.995189i	$0.531236\pi$
$20$	0	0
$21$	−5.23607	−1.14260
$22$	0	0
$23$	3.23607	0.674767	0.337383	−	0.941367i	$-0.390458\pi$
$23$	3.23607	0.674767	0.337383	+	0.941367i	$0.390458\pi$
$24$	0	0
$25$	−3.47214	−0.694427
$26$	0	0
$27$	−2.23607	−0.430331
$28$	0	0
$29$	−4.47214	−0.830455	−0.415227	−	0.909718i	$-0.636298\pi$
$29$	−4.47214	−0.830455	−0.415227	+	0.909718i	$0.636298\pi$
$30$	0	0
$31$	−2.00000	−0.359211	−0.179605	−	0.983739i	$-0.557482\pi$
$31$	−2.00000	−0.359211	−0.179605	+	0.983739i	$0.557482\pi$
$32$	0	0
$33$	0	0
$34$	0	0
$35$	−2.47214	−0.417867
$36$	0	0
$37$	9.70820	1.59602	0.798009	−	0.602645i	$-0.205886\pi$
$37$	9.70820	1.59602	0.798009	+	0.602645i	$0.205886\pi$
$38$	0	0
$39$	8.47214	1.35663
$40$	0	0
$41$	3.38197	0.528174	0.264087	−	0.964499i	$-0.414929\pi$
$41$	3.38197	0.528174	0.264087	+	0.964499i	$0.414929\pi$
$42$	0	0
$43$	−11.5623	−1.76324	−0.881618	−	0.471964i	$-0.843545\pi$
$43$	−11.5623	−1.76324	−0.881618	+	0.471964i	$0.843545\pi$
$44$	0	0
$45$	−4.76393	−0.710165
$46$	0	0
$47$	−2.47214	−0.360598	−0.180299	−	0.983612i	$-0.557707\pi$
$47$	−2.47214	−0.360598	−0.180299	+	0.983612i	$0.557707\pi$
$48$	0	0
$49$	−3.00000	−0.428571
$50$	0	0
$51$	−4.23607	−0.593168
$52$	0	0
$53$	−10.4721	−1.43846	−0.719229	−	0.694773i	$-0.755505\pi$
$53$	−10.4721	−1.43846	−0.719229	+	0.694773i	$0.755505\pi$
$54$	0	0
$55$	0	0
$56$	0	0
$57$	2.23607	0.296174
$58$	0	0
$59$	6.38197	0.830861	0.415431	−	0.909625i	$-0.363631\pi$
$59$	6.38197	0.830861	0.415431	+	0.909625i	$0.363631\pi$
$60$	0	0
$61$	6.47214	0.828672	0.414336	−	0.910124i	$-0.364014\pi$
$61$	6.47214	0.828672	0.414336	+	0.910124i	$0.364014\pi$
$62$	0	0
$63$	7.70820	0.971142
$64$	0	0
$65$	4.00000	0.496139
$66$	0	0
$67$	0.0901699	0.0110160	0.00550801	−	0.999985i	$-0.498247\pi$
$67$	0.0901699	0.0110160	0.00550801	+	0.999985i	$0.498247\pi$
$68$	0	0
$69$	−8.47214	−1.01993
$70$	0	0
$71$	0.763932	0.0906621	0.0453310	−	0.998972i	$-0.485566\pi$
$71$	0.763932	0.0906621	0.0453310	+	0.998972i	$0.485566\pi$
$72$	0	0
$73$	12.6180	1.47683	0.738415	−	0.674347i	$-0.235575\pi$
$73$	12.6180	1.47683	0.738415	+	0.674347i	$0.235575\pi$
$74$	0	0
$75$	9.09017	1.04964
$76$	0	0
$77$	0	0
$78$	0	0
$79$	13.4164	1.50946	0.754732	−	0.656033i	$-0.227767\pi$
$79$	13.4164	1.50946	0.754732	+	0.656033i	$0.227767\pi$
$80$	0	0
$81$	−5.70820	−0.634245
$82$	0	0
$83$	6.32624	0.694395	0.347197	−	0.937792i	$-0.387133\pi$
$83$	6.32624	0.694395	0.347197	+	0.937792i	$0.387133\pi$
$84$	0	0
$85$	−2.00000	−0.216930
$86$	0	0
$87$	11.7082	1.25525
$88$	0	0
$89$	3.09017	0.327557	0.163779	−	0.986497i	$-0.447632\pi$
$89$	3.09017	0.327557	0.163779	+	0.986497i	$0.447632\pi$
$90$	0	0
$91$	−6.47214	−0.678464
$92$	0	0
$93$	5.23607	0.542955
$94$	0	0
$95$	1.05573	0.108315
$96$	0	0
$97$	13.8541	1.40667	0.703335	−	0.710858i	$-0.251693\pi$
$97$	13.8541	1.40667	0.703335	+	0.710858i	$0.251693\pi$
$98$	0	0
$99$	0	0
$100$	0	0
$101$	18.1803	1.80901	0.904506	−	0.426461i	$-0.140240\pi$
$101$	18.1803	1.80901	0.904506	+	0.426461i	$0.140240\pi$
$102$	0	0
$103$	−2.29180	−0.225817	−0.112909	−	0.993605i	$-0.536017\pi$
$103$	−2.29180	−0.225817	−0.112909	+	0.993605i	$0.536017\pi$
$104$	0	0
$105$	6.47214	0.631616
$106$	0	0
$107$	7.85410	0.759285	0.379642	−	0.925133i	$-0.376047\pi$
$107$	7.85410	0.759285	0.379642	+	0.925133i	$0.376047\pi$
$108$	0	0
$109$	1.05573	0.101120	0.0505602	−	0.998721i	$-0.483899\pi$
$109$	1.05573	0.101120	0.0505602	+	0.998721i	$0.483899\pi$
$110$	0	0
$111$	−25.4164	−2.41242
$112$	0	0
$113$	4.85410	0.456636	0.228318	−	0.973587i	$-0.426677\pi$
$113$	4.85410	0.456636	0.228318	+	0.973587i	$0.426677\pi$
$114$	0	0
$115$	−4.00000	−0.373002
$116$	0	0
$117$	−12.4721	−1.15305
$118$	0	0
$119$	3.23607	0.296650
$120$	0	0
$121$	0	0
$122$	0	0
$123$	−8.85410	−0.798347
$124$	0	0
$125$	10.4721	0.936656
$126$	0	0
$127$	−6.94427	−0.616204	−0.308102	−	0.951353i	$-0.599694\pi$
$127$	−6.94427	−0.616204	−0.308102	+	0.951353i	$0.599694\pi$
$128$	0	0
$129$	30.2705	2.66517
$130$	0	0
$131$	17.7984	1.55505	0.777526	−	0.628851i	$-0.216475\pi$
$131$	17.7984	1.55505	0.777526	+	0.628851i	$0.216475\pi$
$132$	0	0
$133$	−1.70820	−0.148120
$134$	0	0
$135$	2.76393	0.237881
$136$	0	0
$137$	4.90983	0.419475	0.209738	−	0.977758i	$-0.432739\pi$
$137$	4.90983	0.419475	0.209738	+	0.977758i	$0.432739\pi$
$138$	0	0
$139$	0	0		−	1.00000i	$-0.5\pi$
$139$	0	0			1.00000i	$0.5\pi$
$140$	0	0
$141$	6.47214	0.545052
$142$	0	0
$143$	0	0
$144$	0	0
$145$	5.52786	0.459064
$146$	0	0
$147$	7.85410	0.647795
$148$	0	0
$149$	16.1803	1.32555	0.662773	−	0.748821i	$-0.269380\pi$
$149$	16.1803	1.32555	0.662773	+	0.748821i	$0.269380\pi$
$150$	0	0
$151$	8.00000	0.651031	0.325515	−	0.945537i	$-0.394462\pi$
$151$	8.00000	0.651031	0.325515	+	0.945537i	$0.394462\pi$
$152$	0	0
$153$	6.23607	0.504156
$154$	0	0
$155$	2.47214	0.198567
$156$	0	0
$157$	−3.70820	−0.295947	−0.147973	−	0.988991i	$-0.547275\pi$
$157$	−3.70820	−0.295947	−0.147973	+	0.988991i	$0.547275\pi$
$158$	0	0
$159$	27.4164	2.17426
$160$	0	0
$161$	6.47214	0.510076
$162$	0	0
$163$	−12.0902	−0.946975	−0.473488	−	0.880800i	$-0.657005\pi$
$163$	−12.0902	−0.946975	−0.473488	+	0.880800i	$0.657005\pi$
$164$	0	0
$165$	0	0
$166$	0	0
$167$	19.2361	1.48853	0.744266	−	0.667884i	$-0.232800\pi$
$167$	19.2361	1.48853	0.744266	+	0.667884i	$0.232800\pi$
$168$	0	0
$169$	−2.52786	−0.194451
$170$	0	0
$171$	−3.29180	−0.251730
$172$	0	0
$173$	−13.8885	−1.05593	−0.527963	−	0.849267i	$-0.677044\pi$
$173$	−13.8885	−1.05593	−0.527963	+	0.849267i	$0.677044\pi$
$174$	0	0
$175$	−6.94427	−0.524938
$176$	0	0
$177$	−16.7082	−1.25587
$178$	0	0
$179$	6.38197	0.477011	0.238505	−	0.971141i	$-0.423343\pi$
$179$	6.38197	0.477011	0.238505	+	0.971141i	$0.423343\pi$
$180$	0	0
$181$	18.1803	1.35133	0.675667	−	0.737207i	$-0.263856\pi$
$181$	18.1803	1.35133	0.675667	+	0.737207i	$0.263856\pi$
$182$	0	0
$183$	−16.9443	−1.25256
$184$	0	0
$185$	−12.0000	−0.882258
$186$	0	0
$187$	0	0
$188$	0	0
$189$	−4.47214	−0.325300
$190$	0	0
$191$	−9.23607	−0.668298	−0.334149	−	0.942520i	$-0.608449\pi$
$191$	−9.23607	−0.668298	−0.334149	+	0.942520i	$0.608449\pi$
$192$	0	0
$193$	−9.41641	−0.677808	−0.338904	−	0.940821i	$-0.610056\pi$
$193$	−9.41641	−0.677808	−0.338904	+	0.940821i	$0.610056\pi$
$194$	0	0
$195$	−10.4721	−0.749925
$196$	0	0
$197$	−3.05573	−0.217712	−0.108856	−	0.994058i	$-0.534719\pi$
$197$	−3.05573	−0.217712	−0.108856	+	0.994058i	$0.534719\pi$
$198$	0	0
$199$	1.05573	0.0748386	0.0374193	−	0.999300i	$-0.488086\pi$
$199$	1.05573	0.0748386	0.0374193	+	0.999300i	$0.488086\pi$
$200$	0	0
$201$	−0.236068	−0.0166510
$202$	0	0
$203$	−8.94427	−0.627765
$204$	0	0
$205$	−4.18034	−0.291968
$206$	0	0
$207$	12.4721	0.866873
$208$	0	0
$209$	0	0
$210$	0	0
$211$	−5.09017	−0.350422	−0.175211	−	0.984531i	$-0.556061\pi$
$211$	−5.09017	−0.350422	−0.175211	+	0.984531i	$0.556061\pi$
$212$	0	0
$213$	−2.00000	−0.137038
$214$	0	0
$215$	14.2918	0.974692
$216$	0	0
$217$	−4.00000	−0.271538
$218$	0	0
$219$	−33.0344	−2.23226
$220$	0	0
$221$	−5.23607	−0.352216
$222$	0	0
$223$	−12.2918	−0.823120	−0.411560	−	0.911383i	$-0.635016\pi$
$223$	−12.2918	−0.823120	−0.411560	+	0.911383i	$0.635016\pi$
$224$	0	0
$225$	−13.3820	−0.892131
$226$	0	0
$227$	23.5066	1.56019	0.780093	−	0.625663i	$-0.215172\pi$
$227$	23.5066	1.56019	0.780093	+	0.625663i	$0.215172\pi$
$228$	0	0
$229$	−11.7082	−0.773700	−0.386850	−	0.922143i	$-0.626437\pi$
$229$	−11.7082	−0.773700	−0.386850	+	0.922143i	$0.626437\pi$
$230$	0	0
$231$	0	0
$232$	0	0
$233$	−7.38197	−0.483609	−0.241804	−	0.970325i	$-0.577739\pi$
$233$	−7.38197	−0.483609	−0.241804	+	0.970325i	$0.577739\pi$
$234$	0	0
$235$	3.05573	0.199334
$236$	0	0
$237$	−35.1246	−2.28159
$238$	0	0
$239$	8.29180	0.536352	0.268176	−	0.963370i	$-0.413579\pi$
$239$	8.29180	0.536352	0.268176	+	0.963370i	$0.413579\pi$
$240$	0	0
$241$	0.0901699	0.00580836	0.00290418	−	0.999996i	$-0.499076\pi$
$241$	0.0901699	0.00580836	0.00290418	+	0.999996i	$0.499076\pi$
$242$	0	0
$243$	21.6525	1.38901
$244$	0	0
$245$	3.70820	0.236908
$246$	0	0
$247$	2.76393	0.175865
$248$	0	0
$249$	−16.5623	−1.04959
$250$	0	0
$251$	−3.05573	−0.192876	−0.0964379	−	0.995339i	$-0.530745\pi$
$251$	−3.05573	−0.192876	−0.0964379	+	0.995339i	$0.530745\pi$
$252$	0	0
$253$	0	0
$254$	0	0
$255$	5.23607	0.327895
$256$	0	0
$257$	−3.38197	−0.210961	−0.105481	−	0.994421i	$-0.533638\pi$
$257$	−3.38197	−0.210961	−0.105481	+	0.994421i	$0.533638\pi$
$258$	0	0
$259$	19.4164	1.20648
$260$	0	0
$261$	−17.2361	−1.06689
$262$	0	0
$263$	18.7639	1.15703	0.578517	−	0.815670i	$-0.303632\pi$
$263$	18.7639	1.15703	0.578517	+	0.815670i	$0.303632\pi$
$264$	0	0
$265$	12.9443	0.795160
$266$	0	0
$267$	−8.09017	−0.495110
$268$	0	0
$269$	−16.1803	−0.986533	−0.493266	−	0.869878i	$-0.664197\pi$
$269$	−16.1803	−0.986533	−0.493266	+	0.869878i	$0.664197\pi$
$270$	0	0
$271$	−2.00000	−0.121491	−0.0607457	−	0.998153i	$-0.519348\pi$
$271$	−2.00000	−0.121491	−0.0607457	+	0.998153i	$0.519348\pi$
$272$	0	0
$273$	16.9443	1.02551
$274$	0	0
$275$	0	0
$276$	0	0
$277$	−10.2918	−0.618374	−0.309187	−	0.951001i	$-0.600057\pi$
$277$	−10.2918	−0.618374	−0.309187	+	0.951001i	$0.600057\pi$
$278$	0	0
$279$	−7.70820	−0.461478
$280$	0	0
$281$	−4.90983	−0.292896	−0.146448	−	0.989218i	$-0.546784\pi$
$281$	−4.90983	−0.292896	−0.146448	+	0.989218i	$0.546784\pi$
$282$	0	0
$283$	−24.0000	−1.42665	−0.713326	−	0.700832i	$-0.752812\pi$
$283$	−24.0000	−1.42665	−0.713326	+	0.700832i	$0.752812\pi$
$284$	0	0
$285$	−2.76393	−0.163721
$286$	0	0
$287$	6.76393	0.399262
$288$	0	0
$289$	−14.3820	−0.845998
$290$	0	0
$291$	−36.2705	−2.12621
$292$	0	0
$293$	16.3607	0.955801	0.477901	−	0.878414i	$-0.341398\pi$
$293$	16.3607	0.955801	0.477901	+	0.878414i	$0.341398\pi$
$294$	0	0
$295$	−7.88854	−0.459289
$296$	0	0
$297$	0	0
$298$	0	0
$299$	−10.4721	−0.605619
$300$	0	0
$301$	−23.1246	−1.33288
$302$	0	0
$303$	−47.5967	−2.73436
$304$	0	0
$305$	−8.00000	−0.458079
$306$	0	0
$307$	−3.20163	−0.182726	−0.0913632	−	0.995818i	$-0.529122\pi$
$307$	−3.20163	−0.182726	−0.0913632	+	0.995818i	$0.529122\pi$
$308$	0	0
$309$	6.00000	0.341328
$310$	0	0
$311$	32.0689	1.81846	0.909230	−	0.416295i	$-0.136672\pi$
$311$	32.0689	1.81846	0.909230	+	0.416295i	$0.136672\pi$
$312$	0	0
$313$	−11.3262	−0.640197	−0.320098	−	0.947384i	$-0.603716\pi$
$313$	−11.3262	−0.640197	−0.320098	+	0.947384i	$0.603716\pi$
$314$	0	0
$315$	−9.52786	−0.536834
$316$	0	0
$317$	9.70820	0.545267	0.272634	−	0.962118i	$-0.412105\pi$
$317$	9.70820	0.545267	0.272634	+	0.962118i	$0.412105\pi$
$318$	0	0
$319$	0	0
$320$	0	0
$321$	−20.5623	−1.14768
$322$	0	0
$323$	−1.38197	−0.0768946
$324$	0	0
$325$	11.2361	0.623265
$326$	0	0
$327$	−2.76393	−0.152846
$328$	0	0
$329$	−4.94427	−0.272587
$330$	0	0
$331$	27.2705	1.49892	0.749461	−	0.662048i	$-0.230313\pi$
$331$	27.2705	1.49892	0.749461	+	0.662048i	$0.230313\pi$
$332$	0	0
$333$	37.4164	2.05041
$334$	0	0
$335$	−0.111456	−0.00608950
$336$	0	0
$337$	−2.20163	−0.119930	−0.0599651	−	0.998200i	$-0.519099\pi$
$337$	−2.20163	−0.119930	−0.0599651	+	0.998200i	$0.519099\pi$
$338$	0	0
$339$	−12.7082	−0.690215
$340$	0	0
$341$	0	0
$342$	0	0
$343$	−20.0000	−1.07990
$344$	0	0
$345$	10.4721	0.563801
$346$	0	0
$347$	−14.3820	−0.772064	−0.386032	−	0.922485i	$-0.626155\pi$
$347$	−14.3820	−0.772064	−0.386032	+	0.922485i	$0.626155\pi$
$348$	0	0
$349$	29.5967	1.58428	0.792139	−	0.610341i	$-0.208968\pi$
$349$	29.5967	1.58428	0.792139	+	0.610341i	$0.208968\pi$
$350$	0	0
$351$	7.23607	0.386233
$352$	0	0
$353$	30.3820	1.61707	0.808534	−	0.588449i	$-0.200261\pi$
$353$	30.3820	1.61707	0.808534	+	0.588449i	$0.200261\pi$
$354$	0	0
$355$	−0.944272	−0.0501167
$356$	0	0
$357$	−8.47214	−0.448393
$358$	0	0
$359$	35.1246	1.85381	0.926903	−	0.375301i	$-0.122461\pi$
$359$	35.1246	1.85381	0.926903	+	0.375301i	$0.122461\pi$
$360$	0	0
$361$	−18.2705	−0.961606
$362$	0	0
$363$	0	0
$364$	0	0
$365$	−15.5967	−0.816371
$366$	0	0
$367$	−6.29180	−0.328429	−0.164215	−	0.986425i	$-0.552509\pi$
$367$	−6.29180	−0.328429	−0.164215	+	0.986425i	$0.552509\pi$
$368$	0	0
$369$	13.0344	0.678546
$370$	0	0
$371$	−20.9443	−1.08737
$372$	0	0
$373$	−17.7082	−0.916896	−0.458448	−	0.888721i	$-0.651594\pi$
$373$	−17.7082	−0.916896	−0.458448	+	0.888721i	$0.651594\pi$
$374$	0	0
$375$	−27.4164	−1.41578
$376$	0	0
$377$	14.4721	0.745353
$378$	0	0
$379$	19.2705	0.989860	0.494930	−	0.868933i	$-0.335194\pi$
$379$	19.2705	0.989860	0.494930	+	0.868933i	$0.335194\pi$
$380$	0	0
$381$	18.1803	0.931407
$382$	0	0
$383$	−16.3607	−0.835992	−0.417996	−	0.908449i	$-0.637267\pi$
$383$	−16.3607	−0.835992	−0.417996	+	0.908449i	$0.637267\pi$
$384$	0	0
$385$	0	0
$386$	0	0
$387$	−44.5623	−2.26523
$388$	0	0
$389$	20.6525	1.04712	0.523561	−	0.851988i	$-0.324603\pi$
$389$	20.6525	1.04712	0.523561	+	0.851988i	$0.324603\pi$
$390$	0	0
$391$	5.23607	0.264799
$392$	0	0
$393$	−46.5967	−2.35049
$394$	0	0
$395$	−16.5836	−0.834411
$396$	0	0
$397$	23.1246	1.16059	0.580295	−	0.814406i	$-0.302937\pi$
$397$	23.1246	1.16059	0.580295	+	0.814406i	$0.302937\pi$
$398$	0	0
$399$	4.47214	0.223887
$400$	0	0
$401$	6.79837	0.339495	0.169747	−	0.985488i	$-0.445705\pi$
$401$	6.79837	0.339495	0.169747	+	0.985488i	$0.445705\pi$
$402$	0	0
$403$	6.47214	0.322400
$404$	0	0
$405$	7.05573	0.350602
$406$	0	0
$407$	0	0
$408$	0	0
$409$	13.4164	0.663399	0.331699	−	0.943385i	$-0.392378\pi$
$409$	13.4164	0.663399	0.331699	+	0.943385i	$0.392378\pi$
$410$	0	0
$411$	−12.8541	−0.634046
$412$	0	0
$413$	12.7639	0.628072
$414$	0	0
$415$	−7.81966	−0.383852
$416$	0	0
$417$	0	0
$418$	0	0
$419$	4.14590	0.202540	0.101270	−	0.994859i	$-0.467709\pi$
$419$	4.14590	0.202540	0.101270	+	0.994859i	$0.467709\pi$
$420$	0	0
$421$	−4.58359	−0.223391	−0.111695	−	0.993743i	$-0.535628\pi$
$421$	−4.58359	−0.223391	−0.111695	+	0.993743i	$0.535628\pi$
$422$	0	0
$423$	−9.52786	−0.463261
$424$	0	0
$425$	−5.61803	−0.272515
$426$	0	0
$427$	12.9443	0.626417
$428$	0	0
$429$	0	0
$430$	0	0
$431$	−0.944272	−0.0454840	−0.0227420	−	0.999741i	$-0.507240\pi$
$431$	−0.944272	−0.0454840	−0.0227420	+	0.999741i	$0.507240\pi$
$432$	0	0
$433$	−14.7426	−0.708486	−0.354243	−	0.935153i	$-0.615261\pi$
$433$	−14.7426	−0.708486	−0.354243	+	0.935153i	$0.615261\pi$
$434$	0	0
$435$	−14.4721	−0.693886
$436$	0	0
$437$	−2.76393	−0.132217
$438$	0	0
$439$	33.4164	1.59488	0.797439	−	0.603399i	$-0.206188\pi$
$439$	33.4164	1.59488	0.797439	+	0.603399i	$0.206188\pi$
$440$	0	0
$441$	−11.5623	−0.550586
$442$	0	0
$443$	−11.5623	−0.549342	−0.274671	−	0.961538i	$-0.588569\pi$
$443$	−11.5623	−0.549342	−0.274671	+	0.961538i	$0.588569\pi$
$444$	0	0
$445$	−3.81966	−0.181069
$446$	0	0
$447$	−42.3607	−2.00359
$448$	0	0
$449$	−24.2705	−1.14540	−0.572698	−	0.819766i	$-0.694103\pi$
$449$	−24.2705	−1.14540	−0.572698	+	0.819766i	$0.694103\pi$
$450$	0	0
$451$	0	0
$452$	0	0
$453$	−20.9443	−0.984048
$454$	0	0
$455$	8.00000	0.375046
$456$	0	0
$457$	−29.5623	−1.38287	−0.691433	−	0.722440i	$-0.743020\pi$
$457$	−29.5623	−1.38287	−0.691433	+	0.722440i	$0.743020\pi$
$458$	0	0
$459$	−3.61803	−0.168875
$460$	0	0
$461$	32.6525	1.52078	0.760389	−	0.649468i	$-0.225008\pi$
$461$	32.6525	1.52078	0.760389	+	0.649468i	$0.225008\pi$
$462$	0	0
$463$	−7.41641	−0.344670	−0.172335	−	0.985038i	$-0.555131\pi$
$463$	−7.41641	−0.344670	−0.172335	+	0.985038i	$0.555131\pi$
$464$	0	0
$465$	−6.47214	−0.300138
$466$	0	0
$467$	−29.3050	−1.35607	−0.678036	−	0.735029i	$-0.737169\pi$
$467$	−29.3050	−1.35607	−0.678036	+	0.735029i	$0.737169\pi$
$468$	0	0
$469$	0.180340	0.00832732
$470$	0	0
$471$	9.70820	0.447330
$472$	0	0
$473$	0	0
$474$	0	0
$475$	2.96556	0.136069
$476$	0	0
$477$	−40.3607	−1.84799
$478$	0	0
$479$	25.5279	1.16640	0.583199	−	0.812329i	$-0.301801\pi$
$479$	25.5279	1.16640	0.583199	+	0.812329i	$0.301801\pi$
$480$	0	0
$481$	−31.4164	−1.43246
$482$	0	0
$483$	−16.9443	−0.770991
$484$	0	0
$485$	−17.1246	−0.777589
$486$	0	0
$487$	32.6525	1.47962	0.739812	−	0.672813i	$-0.234914\pi$
$487$	32.6525	1.47962	0.739812	+	0.672813i	$0.234914\pi$
$488$	0	0
$489$	31.6525	1.43137
$490$	0	0
$491$	−16.1459	−0.728654	−0.364327	−	0.931271i	$-0.618701\pi$
$491$	−16.1459	−0.728654	−0.364327	+	0.931271i	$0.618701\pi$
$492$	0	0
$493$	−7.23607	−0.325896
$494$	0	0
$495$	0	0
$496$	0	0
$497$	1.52786	0.0685341
$498$	0	0
$499$	24.1459	1.08092	0.540459	−	0.841370i	$-0.318250\pi$
$499$	24.1459	1.08092	0.540459	+	0.841370i	$0.318250\pi$
$500$	0	0
$501$	−50.3607	−2.24995
$502$	0	0
$503$	−1.23607	−0.0551135	−0.0275568	−	0.999620i	$-0.508773\pi$
$503$	−1.23607	−0.0551135	−0.0275568	+	0.999620i	$0.508773\pi$
$504$	0	0
$505$	−22.4721	−0.999997
$506$	0	0
$507$	6.61803	0.293917
$508$	0	0
$509$	−21.7082	−0.962199	−0.481100	−	0.876666i	$-0.659762\pi$
$509$	−21.7082	−0.962199	−0.481100	+	0.876666i	$0.659762\pi$
$510$	0	0
$511$	25.2361	1.11638
$512$	0	0
$513$	1.90983	0.0843211
$514$	0	0
$515$	2.83282	0.124829
$516$	0	0
$517$	0	0
$518$	0	0
$519$	36.3607	1.59606
$520$	0	0
$521$	9.03444	0.395806	0.197903	−	0.980222i	$-0.436587\pi$
$521$	9.03444	0.395806	0.197903	+	0.980222i	$0.436587\pi$
$522$	0	0
$523$	−33.7984	−1.47790	−0.738950	−	0.673760i	$-0.764678\pi$
$523$	−33.7984	−1.47790	−0.738950	+	0.673760i	$0.764678\pi$
$524$	0	0
$525$	18.1803	0.793455
$526$	0	0
$527$	−3.23607	−0.140965
$528$	0	0
$529$	−12.5279	−0.544690
$530$	0	0
$531$	24.5967	1.06741
$532$	0	0
$533$	−10.9443	−0.474049
$534$	0	0
$535$	−9.70820	−0.419722
$536$	0	0
$537$	−16.7082	−0.721012
$538$	0	0
$539$	0	0
$540$	0	0
$541$	37.1246	1.59611	0.798056	−	0.602583i	$-0.205862\pi$
$541$	37.1246	1.59611	0.798056	+	0.602583i	$0.205862\pi$
$542$	0	0
$543$	−47.5967	−2.04257
$544$	0	0
$545$	−1.30495	−0.0558980
$546$	0	0
$547$	7.32624	0.313247	0.156624	−	0.987658i	$-0.449939\pi$
$547$	7.32624	0.313247	0.156624	+	0.987658i	$0.449939\pi$
$548$	0	0
$549$	24.9443	1.06460
$550$	0	0
$551$	3.81966	0.162723
$552$	0	0
$553$	26.8328	1.14105
$554$	0	0
$555$	31.4164	1.33355
$556$	0	0
$557$	30.7639	1.30351	0.651755	−	0.758430i	$-0.274033\pi$
$557$	30.7639	1.30351	0.651755	+	0.758430i	$0.274033\pi$
$558$	0	0
$559$	37.4164	1.58255
$560$	0	0
$561$	0	0
$562$	0	0
$563$	−2.61803	−0.110337	−0.0551685	−	0.998477i	$-0.517570\pi$
$563$	−2.61803	−0.110337	−0.0551685	+	0.998477i	$0.517570\pi$
$564$	0	0
$565$	−6.00000	−0.252422
$566$	0	0
$567$	−11.4164	−0.479444
$568$	0	0
$569$	−13.2148	−0.553992	−0.276996	−	0.960871i	$-0.589339\pi$
$569$	−13.2148	−0.553992	−0.276996	+	0.960871i	$0.589339\pi$
$570$	0	0
$571$	−6.47214	−0.270850	−0.135425	−	0.990788i	$-0.543240\pi$
$571$	−6.47214	−0.270850	−0.135425	+	0.990788i	$0.543240\pi$
$572$	0	0
$573$	24.1803	1.01015
$574$	0	0
$575$	−11.2361	−0.468576
$576$	0	0
$577$	17.7984	0.740956	0.370478	−	0.928841i	$-0.379194\pi$
$577$	17.7984	0.740956	0.370478	+	0.928841i	$0.379194\pi$
$578$	0	0
$579$	24.6525	1.02452
$580$	0	0
$581$	12.6525	0.524913
$582$	0	0
$583$	0	0
$584$	0	0
$585$	15.4164	0.637390
$586$	0	0
$587$	−28.8541	−1.19094	−0.595468	−	0.803379i	$-0.703033\pi$
$587$	−28.8541	−1.19094	−0.595468	+	0.803379i	$0.703033\pi$
$588$	0	0
$589$	1.70820	0.0703853
$590$	0	0
$591$	8.00000	0.329076
$592$	0	0
$593$	31.6869	1.30123	0.650613	−	0.759410i	$-0.274512\pi$
$593$	31.6869	1.30123	0.650613	+	0.759410i	$0.274512\pi$
$594$	0	0
$595$	−4.00000	−0.163984
$596$	0	0
$597$	−2.76393	−0.113120
$598$	0	0
$599$	−13.4164	−0.548180	−0.274090	−	0.961704i	$-0.588377\pi$
$599$	−13.4164	−0.548180	−0.274090	+	0.961704i	$0.588377\pi$
$600$	0	0
$601$	−33.8541	−1.38094	−0.690469	−	0.723362i	$-0.742596\pi$
$601$	−33.8541	−1.38094	−0.690469	+	0.723362i	$0.742596\pi$
$602$	0	0
$603$	0.347524	0.0141523
$604$	0	0
$605$	0	0
$606$	0	0
$607$	17.1246	0.695067	0.347533	−	0.937668i	$-0.387019\pi$
$607$	17.1246	0.695067	0.347533	+	0.937668i	$0.387019\pi$
$608$	0	0
$609$	23.4164	0.948881
$610$	0	0
$611$	8.00000	0.323645
$612$	0	0
$613$	22.2918	0.900357	0.450179	−	0.892939i	$-0.351360\pi$
$613$	22.2918	0.900357	0.450179	+	0.892939i	$0.351360\pi$
$614$	0	0
$615$	10.9443	0.441316
$616$	0	0
$617$	−32.4508	−1.30642	−0.653211	−	0.757176i	$-0.726579\pi$
$617$	−32.4508	−1.30642	−0.653211	+	0.757176i	$0.726579\pi$
$618$	0	0
$619$	−29.7984	−1.19770	−0.598849	−	0.800862i	$-0.704375\pi$
$619$	−29.7984	−1.19770	−0.598849	+	0.800862i	$0.704375\pi$
$620$	0	0
$621$	−7.23607	−0.290373
$622$	0	0
$623$	6.18034	0.247610
$624$	0	0
$625$	4.41641	0.176656
$626$	0	0
$627$	0	0
$628$	0	0
$629$	15.7082	0.626327
$630$	0	0
$631$	−10.2918	−0.409710	−0.204855	−	0.978792i	$-0.565672\pi$
$631$	−10.2918	−0.409710	−0.204855	+	0.978792i	$0.565672\pi$
$632$	0	0
$633$	13.3262	0.529670
$634$	0	0
$635$	8.58359	0.340629
$636$	0	0
$637$	9.70820	0.384653
$638$	0	0
$639$	2.94427	0.116474
$640$	0	0
$641$	32.3262	1.27681	0.638405	−	0.769701i	$-0.279595\pi$
$641$	32.3262	1.27681	0.638405	+	0.769701i	$0.279595\pi$
$642$	0	0
$643$	6.97871	0.275214	0.137607	−	0.990487i	$-0.456059\pi$
$643$	6.97871	0.275214	0.137607	+	0.990487i	$0.456059\pi$
$644$	0	0
$645$	−37.4164	−1.47327
$646$	0	0
$647$	−46.9443	−1.84557	−0.922785	−	0.385316i	$-0.874093\pi$
$647$	−46.9443	−1.84557	−0.922785	+	0.385316i	$0.874093\pi$
$648$	0	0
$649$	0	0
$650$	0	0
$651$	10.4721	0.410435
$652$	0	0
$653$	−46.6525	−1.82565	−0.912826	−	0.408348i	$-0.866105\pi$
$653$	−46.6525	−1.82565	−0.912826	+	0.408348i	$0.866105\pi$
$654$	0	0
$655$	−22.0000	−0.859611
$656$	0	0
$657$	48.6312	1.89728
$658$	0	0
$659$	28.0902	1.09424	0.547119	−	0.837055i	$-0.315725\pi$
$659$	28.0902	1.09424	0.547119	+	0.837055i	$0.315725\pi$
$660$	0	0
$661$	−12.4721	−0.485110	−0.242555	−	0.970138i	$-0.577985\pi$
$661$	−12.4721	−0.485110	−0.242555	+	0.970138i	$0.577985\pi$
$662$	0	0
$663$	13.7082	0.532383
$664$	0	0
$665$	2.11146	0.0818788
$666$	0	0
$667$	−14.4721	−0.560363
$668$	0	0
$669$	32.1803	1.24416
$670$	0	0
$671$	0	0
$672$	0	0
$673$	3.14590	0.121265	0.0606327	−	0.998160i	$-0.480688\pi$
$673$	3.14590	0.121265	0.0606327	+	0.998160i	$0.480688\pi$
$674$	0	0
$675$	7.76393	0.298834
$676$	0	0
$677$	30.3607	1.16686	0.583428	−	0.812165i	$-0.301711\pi$
$677$	30.3607	1.16686	0.583428	+	0.812165i	$0.301711\pi$
$678$	0	0
$679$	27.7082	1.06334
$680$	0	0
$681$	−61.5410	−2.35826
$682$	0	0
$683$	−9.52786	−0.364574	−0.182287	−	0.983245i	$-0.558350\pi$
$683$	−9.52786	−0.364574	−0.182287	+	0.983245i	$0.558350\pi$
$684$	0	0
$685$	−6.06888	−0.231880
$686$	0	0
$687$	30.6525	1.16946
$688$	0	0
$689$	33.8885	1.29105
$690$	0	0
$691$	−25.2148	−0.959216	−0.479608	−	0.877483i	$-0.659221\pi$
$691$	−25.2148	−0.959216	−0.479608	+	0.877483i	$0.659221\pi$
$692$	0	0
$693$	0	0
$694$	0	0
$695$	0	0
$696$	0	0
$697$	5.47214	0.207272
$698$	0	0
$699$	19.3262	0.730985
$700$	0	0
$701$	38.8328	1.46670	0.733348	−	0.679854i	$-0.237957\pi$
$701$	38.8328	1.46670	0.733348	+	0.679854i	$0.237957\pi$
$702$	0	0
$703$	−8.29180	−0.312731
$704$	0	0
$705$	−8.00000	−0.301297
$706$	0	0
$707$	36.3607	1.36748
$708$	0	0
$709$	15.5279	0.583161	0.291581	−	0.956546i	$-0.405819\pi$
$709$	15.5279	0.583161	0.291581	+	0.956546i	$0.405819\pi$
$710$	0	0
$711$	51.7082	1.93921
$712$	0	0
$713$	−6.47214	−0.242383
$714$	0	0
$715$	0	0
$716$	0	0
$717$	−21.7082	−0.810708
$718$	0	0
$719$	26.8328	1.00070	0.500348	−	0.865825i	$-0.333206\pi$
$719$	26.8328	1.00070	0.500348	+	0.865825i	$0.333206\pi$
$720$	0	0
$721$	−4.58359	−0.170702
$722$	0	0
$723$	−0.236068	−0.00877946
$724$	0	0
$725$	15.5279	0.576690
$726$	0	0
$727$	−43.1246	−1.59940	−0.799702	−	0.600398i	$-0.795009\pi$
$727$	−43.1246	−1.59940	−0.799702	+	0.600398i	$0.795009\pi$
$728$	0	0
$729$	−39.5623	−1.46527
$730$	0	0
$731$	−18.7082	−0.691948
$732$	0	0
$733$	17.4164	0.643290	0.321645	−	0.946860i	$-0.395764\pi$
$733$	17.4164	0.643290	0.321645	+	0.946860i	$0.395764\pi$
$734$	0	0
$735$	−9.70820	−0.358092
$736$	0	0
$737$	0	0
$738$	0	0
$739$	17.4377	0.641456	0.320728	−	0.947171i	$-0.396072\pi$
$739$	17.4377	0.641456	0.320728	+	0.947171i	$0.396072\pi$
$740$	0	0
$741$	−7.23607	−0.265824
$742$	0	0
$743$	13.2361	0.485584	0.242792	−	0.970078i	$-0.421937\pi$
$743$	13.2361	0.485584	0.242792	+	0.970078i	$0.421937\pi$
$744$	0	0
$745$	−20.0000	−0.732743
$746$	0	0
$747$	24.3820	0.892089
$748$	0	0
$749$	15.7082	0.573965
$750$	0	0
$751$	−37.7771	−1.37851	−0.689253	−	0.724521i	$-0.742061\pi$
$751$	−37.7771	−1.37851	−0.689253	+	0.724521i	$0.742061\pi$
$752$	0	0
$753$	8.00000	0.291536
$754$	0	0
$755$	−9.88854	−0.359881
$756$	0	0
$757$	28.6525	1.04139	0.520696	−	0.853742i	$-0.325673\pi$
$757$	28.6525	1.04139	0.520696	+	0.853742i	$0.325673\pi$
$758$	0	0
$759$	0	0
$760$	0	0
$761$	−46.7426	−1.69442	−0.847210	−	0.531258i	$-0.821719\pi$
$761$	−46.7426	−1.69442	−0.847210	+	0.531258i	$0.821719\pi$
$762$	0	0
$763$	2.11146	0.0764398
$764$	0	0
$765$	−7.70820	−0.278691
$766$	0	0
$767$	−20.6525	−0.745718
$768$	0	0
$769$	−13.4164	−0.483808	−0.241904	−	0.970300i	$-0.577772\pi$
$769$	−13.4164	−0.483808	−0.241904	+	0.970300i	$0.577772\pi$
$770$	0	0
$771$	8.85410	0.318873
$772$	0	0
$773$	31.8885	1.14695	0.573476	−	0.819223i	$-0.305595\pi$
$773$	31.8885	1.14695	0.573476	+	0.819223i	$0.305595\pi$
$774$	0	0
$775$	6.94427	0.249446
$776$	0	0
$777$	−50.8328	−1.82362
$778$	0	0
$779$	−2.88854	−0.103493
$780$	0	0
$781$	0	0
$782$	0	0
$783$	10.0000	0.357371
$784$	0	0
$785$	4.58359	0.163595
$786$	0	0
$787$	25.2148	0.898810	0.449405	−	0.893328i	$-0.351636\pi$
$787$	25.2148	0.898810	0.449405	+	0.893328i	$0.351636\pi$
$788$	0	0
$789$	−49.1246	−1.74888
$790$	0	0
$791$	9.70820	0.345184
$792$	0	0
$793$	−20.9443	−0.743753
$794$	0	0
$795$	−33.8885	−1.20190
$796$	0	0
$797$	−29.2361	−1.03559	−0.517797	−	0.855503i	$-0.673248\pi$
$797$	−29.2361	−1.03559	−0.517797	+	0.855503i	$0.673248\pi$
$798$	0	0
$799$	−4.00000	−0.141510
$800$	0	0
$801$	11.9098	0.420813
$802$	0	0
$803$	0	0
$804$	0	0
$805$	−8.00000	−0.281963
$806$	0	0
$807$	42.3607	1.49117
$808$	0	0
$809$	28.0902	0.987598	0.493799	−	0.869576i	$-0.335608\pi$
$809$	28.0902	0.987598	0.493799	+	0.869576i	$0.335608\pi$
$810$	0	0
$811$	−10.6180	−0.372850	−0.186425	−	0.982469i	$-0.559690\pi$
$811$	−10.6180	−0.372850	−0.186425	+	0.982469i	$0.559690\pi$
$812$	0	0
$813$	5.23607	0.183637
$814$	0	0
$815$	14.9443	0.523475
$816$	0	0
$817$	9.87539	0.345496
$818$	0	0
$819$	−24.9443	−0.871623
$820$	0	0
$821$	−36.5410	−1.27529	−0.637645	−	0.770330i	$-0.720091\pi$
$821$	−36.5410	−1.27529	−0.637645	+	0.770330i	$0.720091\pi$
$822$	0	0
$823$	25.5967	0.892247	0.446123	−	0.894972i	$-0.352804\pi$
$823$	25.5967	0.892247	0.446123	+	0.894972i	$0.352804\pi$
$824$	0	0
$825$	0	0
$826$	0	0
$827$	−41.6180	−1.44720	−0.723600	−	0.690219i	$-0.757514\pi$
$827$	−41.6180	−1.44720	−0.723600	+	0.690219i	$0.757514\pi$
$828$	0	0
$829$	48.9443	1.69990	0.849952	−	0.526859i	$-0.176631\pi$
$829$	48.9443	1.69990	0.849952	+	0.526859i	$0.176631\pi$
$830$	0	0
$831$	26.9443	0.934686
$832$	0	0
$833$	−4.85410	−0.168185
$834$	0	0
$835$	−23.7771	−0.822840
$836$	0	0
$837$	4.47214	0.154580
$838$	0	0
$839$	−47.8885	−1.65330	−0.826648	−	0.562719i	$-0.809755\pi$
$839$	−47.8885	−1.65330	−0.826648	+	0.562719i	$0.809755\pi$
$840$	0	0
$841$	−9.00000	−0.310345
$842$	0	0
$843$	12.8541	0.442719
$844$	0	0
$845$	3.12461	0.107490
$846$	0	0
$847$	0	0
$848$	0	0
$849$	62.8328	2.15642
$850$	0	0
$851$	31.4164	1.07694
$852$	0	0
$853$	−21.5279	−0.737100	−0.368550	−	0.929608i	$-0.620146\pi$
$853$	−21.5279	−0.737100	−0.368550	+	0.929608i	$0.620146\pi$
$854$	0	0
$855$	4.06888	0.139153
$856$	0	0
$857$	31.7426	1.08431	0.542154	−	0.840279i	$-0.317609\pi$
$857$	31.7426	1.08431	0.542154	+	0.840279i	$0.317609\pi$
$858$	0	0
$859$	−8.49342	−0.289792	−0.144896	−	0.989447i	$-0.546285\pi$
$859$	−8.49342	−0.289792	−0.144896	+	0.989447i	$0.546285\pi$
$860$	0	0
$861$	−17.7082	−0.603494
$862$	0	0
$863$	28.7639	0.979136	0.489568	−	0.871965i	$-0.337155\pi$
$863$	28.7639	0.979136	0.489568	+	0.871965i	$0.337155\pi$
$864$	0	0
$865$	17.1672	0.583702
$866$	0	0
$867$	37.6525	1.27875
$868$	0	0
$869$	0	0
$870$	0	0
$871$	−0.291796	−0.00988713
$872$	0	0
$873$	53.3951	1.80715
$874$	0	0
$875$	20.9443	0.708046
$876$	0	0
$877$	−5.81966	−0.196516	−0.0982580	−	0.995161i	$-0.531327\pi$
$877$	−5.81966	−0.196516	−0.0982580	+	0.995161i	$0.531327\pi$
$878$	0	0
$879$	−42.8328	−1.44472
$880$	0	0
$881$	45.3394	1.52752	0.763761	−	0.645499i	$-0.223350\pi$
$881$	45.3394	1.52752	0.763761	+	0.645499i	$0.223350\pi$
$882$	0	0
$883$	23.5623	0.792935	0.396467	−	0.918049i	$-0.370236\pi$
$883$	23.5623	0.792935	0.396467	+	0.918049i	$0.370236\pi$
$884$	0	0
$885$	20.6525	0.694225
$886$	0	0
$887$	−33.7771	−1.13412	−0.567062	−	0.823675i	$-0.691920\pi$
$887$	−33.7771	−1.13412	−0.567062	+	0.823675i	$0.691920\pi$
$888$	0	0
$889$	−13.8885	−0.465807
$890$	0	0
$891$	0	0
$892$	0	0
$893$	2.11146	0.0706572
$894$	0	0
$895$	−7.88854	−0.263685
$896$	0	0
$897$	27.4164	0.915407
$898$	0	0
$899$	8.94427	0.298308
$900$	0	0
$901$	−16.9443	−0.564496
$902$	0	0
$903$	60.5410	2.01468
$904$	0	0
$905$	−22.4721	−0.746999
$906$	0	0
$907$	−30.5623	−1.01480	−0.507402	−	0.861709i	$-0.669394\pi$
$907$	−30.5623	−1.01480	−0.507402	+	0.861709i	$0.669394\pi$
$908$	0	0
$909$	70.0689	2.32404
$910$	0	0
$911$	4.18034	0.138501	0.0692504	−	0.997599i	$-0.477939\pi$
$911$	4.18034	0.138501	0.0692504	+	0.997599i	$0.477939\pi$
$912$	0	0
$913$	0	0
$914$	0	0
$915$	20.9443	0.692396
$916$	0	0
$917$	35.5967	1.17551
$918$	0	0
$919$	−16.5836	−0.547042	−0.273521	−	0.961866i	$-0.588188\pi$
$919$	−16.5836	−0.547042	−0.273521	+	0.961866i	$0.588188\pi$
$920$	0	0
$921$	8.38197	0.276195
$922$	0	0
$923$	−2.47214	−0.0813713
$924$	0	0
$925$	−33.7082	−1.10832
$926$	0	0
$927$	−8.83282	−0.290108
$928$	0	0
$929$	−58.7426	−1.92728	−0.963642	−	0.267197i	$-0.913902\pi$
$929$	−58.7426	−1.92728	−0.963642	+	0.267197i	$0.913902\pi$
$930$	0	0
$931$	2.56231	0.0839762
$932$	0	0
$933$	−83.9574	−2.74864
$934$	0	0
$935$	0	0
$936$	0	0
$937$	−31.1459	−1.01749	−0.508746	−	0.860917i	$-0.669891\pi$
$937$	−31.1459	−1.01749	−0.508746	+	0.860917i	$0.669891\pi$
$938$	0	0
$939$	29.6525	0.967672
$940$	0	0
$941$	25.8197	0.841697	0.420848	−	0.907131i	$-0.361732\pi$
$941$	25.8197	0.841697	0.420848	+	0.907131i	$0.361732\pi$
$942$	0	0
$943$	10.9443	0.356395
$944$	0	0
$945$	5.52786	0.179821
$946$	0	0
$947$	−21.2148	−0.689388	−0.344694	−	0.938715i	$-0.612017\pi$
$947$	−21.2148	−0.689388	−0.344694	+	0.938715i	$0.612017\pi$
$948$	0	0
$949$	−40.8328	−1.32549
$950$	0	0
$951$	−25.4164	−0.824183
$952$	0	0
$953$	4.32624	0.140141	0.0700703	−	0.997542i	$-0.477678\pi$
$953$	4.32624	0.140141	0.0700703	+	0.997542i	$0.477678\pi$
$954$	0	0
$955$	11.4164	0.369426
$956$	0	0
$957$	0	0
$958$	0	0
$959$	9.81966	0.317093
$960$	0	0
$961$	−27.0000	−0.870968
$962$	0	0
$963$	30.2705	0.975454
$964$	0	0
$965$	11.6393	0.374683
$966$	0	0
$967$	−38.0000	−1.22200	−0.610999	−	0.791632i	$-0.709232\pi$
$967$	−38.0000	−1.22200	−0.610999	+	0.791632i	$0.709232\pi$
$968$	0	0
$969$	3.61803	0.116228
$970$	0	0
$971$	45.8885	1.47263	0.736317	−	0.676637i	$-0.236563\pi$
$971$	45.8885	1.47263	0.736317	+	0.676637i	$0.236563\pi$
$972$	0	0
$973$	0	0
$974$	0	0
$975$	−29.4164	−0.942079
$976$	0	0
$977$	19.3050	0.617620	0.308810	−	0.951124i	$-0.400069\pi$
$977$	19.3050	0.617620	0.308810	+	0.951124i	$0.400069\pi$
$978$	0	0
$979$	0	0
$980$	0	0
$981$	4.06888	0.129909
$982$	0	0
$983$	6.00000	0.191370	0.0956851	−	0.995412i	$-0.469496\pi$
$983$	6.00000	0.191370	0.0956851	+	0.995412i	$0.469496\pi$
$984$	0	0
$985$	3.77709	0.120348
$986$	0	0
$987$	12.9443	0.412021
$988$	0	0
$989$	−37.4164	−1.18977
$990$	0	0
$991$	36.5410	1.16076	0.580382	−	0.814344i	$-0.302903\pi$
$991$	36.5410	1.16076	0.580382	+	0.814344i	$0.302903\pi$
$992$	0	0
$993$	−71.3951	−2.26566
$994$	0	0
$995$	−1.30495	−0.0413697
$996$	0	0
$997$	−22.6525	−0.717411	−0.358706	−	0.933451i	$-0.616782\pi$
$997$	−22.6525	−0.717411	−0.358706	+	0.933451i	$0.616782\pi$
$998$	0	0
$999$	−21.7082	−0.686817

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	1936.2.a.o.1.1		2
4.3	odd	2		242.2.a.f.1.2		2
8.3	odd	2		7744.2.a.bm.1.1		2
8.5	even	2		7744.2.a.cz.1.2		2
11.3	even	5		176.2.m.c.97.1		4
11.4	even	5		176.2.m.c.49.1		4
11.10	odd	2		1936.2.a.n.1.1		2
12.11	even	2		2178.2.a.p.1.2		2
20.19	odd	2		6050.2.a.bs.1.1		2
44.3	odd	10		22.2.c.a.9.1	yes	4
44.7	even	10		242.2.c.c.27.1		4
44.15	odd	10		22.2.c.a.5.1	✓	4
44.19	even	10		242.2.c.c.9.1		4
44.27	odd	10		242.2.c.a.3.1		4
44.31	odd	10		242.2.c.a.81.1		4
44.35	even	10		242.2.c.d.81.1		4
44.39	even	10		242.2.c.d.3.1		4
44.43	even	2		242.2.a.d.1.2		2
88.3	odd	10		704.2.m.h.449.1		4
88.21	odd	2		7744.2.a.cy.1.2		2
88.37	even	10		704.2.m.a.577.1		4
88.43	even	2		7744.2.a.bn.1.1		2
88.59	odd	10		704.2.m.h.577.1		4
88.69	even	10		704.2.m.a.449.1		4
132.47	even	10		198.2.f.e.163.1		4
132.59	even	10		198.2.f.e.181.1		4
132.131	odd	2		2178.2.a.x.1.2		2
220.3	even	20		550.2.ba.c.449.1		8
220.47	even	20		550.2.ba.c.449.2		8
220.59	odd	10		550.2.h.h.401.1		4
220.103	even	20		550.2.ba.c.49.2		8
220.147	even	20		550.2.ba.c.49.1		8
220.179	odd	10		550.2.h.h.251.1		4
220.219	even	2		6050.2.a.ci.1.1		2

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
22.2.c.a.5.1	✓	4	44.15	odd	10
22.2.c.a.9.1	yes	4	44.3	odd	10
176.2.m.c.49.1		4	11.4	even	5
176.2.m.c.97.1		4	11.3	even	5
198.2.f.e.163.1		4	132.47	even	10
198.2.f.e.181.1		4	132.59	even	10
242.2.a.d.1.2		2	44.43	even	2
242.2.a.f.1.2		2	4.3	odd	2
242.2.c.a.3.1		4	44.27	odd	10
242.2.c.a.81.1		4	44.31	odd	10
242.2.c.c.9.1		4	44.19	even	10
242.2.c.c.27.1		4	44.7	even	10
242.2.c.d.3.1		4	44.39	even	10
242.2.c.d.81.1		4	44.35	even	10
550.2.h.h.251.1		4	220.179	odd	10
550.2.h.h.401.1		4	220.59	odd	10
550.2.ba.c.49.1		8	220.147	even	20
550.2.ba.c.49.2		8	220.103	even	20
550.2.ba.c.449.1		8	220.3	even	20
550.2.ba.c.449.2		8	220.47	even	20
704.2.m.a.449.1		4	88.69	even	10
704.2.m.a.577.1		4	88.37	even	10
704.2.m.h.449.1		4	88.3	odd	10
704.2.m.h.577.1		4	88.59	odd	10
1936.2.a.n.1.1		2	11.10	odd	2
1936.2.a.o.1.1		2	1.1	even	1	trivial
2178.2.a.p.1.2		2	12.11	even	2
2178.2.a.x.1.2		2	132.131	odd	2
6050.2.a.bs.1.1		2	20.19	odd	2
6050.2.a.ci.1.1		2	220.219	even	2
7744.2.a.bm.1.1		2	8.3	odd	2
7744.2.a.bn.1.1		2	88.43	even	2
7744.2.a.cy.1.2		2	88.21	odd	2
7744.2.a.cz.1.2		2	8.5	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:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	1936.a (trivial)

\(f(q)\)	\(=\)	\(q-2.61803 q^{3} -1.23607 q^{5} +2.00000 q^{7} +3.85410 q^{9} -3.23607 q^{13} +3.23607 q^{15} +1.61803 q^{17} -0.854102 q^{19} -5.23607 q^{21} +3.23607 q^{23} -3.47214 q^{25} -2.23607 q^{27} -4.47214 q^{29} -2.00000 q^{31} -2.47214 q^{35} +9.70820 q^{37} +8.47214 q^{39} +3.38197 q^{41} -11.5623 q^{43} -4.76393 q^{45} -2.47214 q^{47} -3.00000 q^{49} -4.23607 q^{51} -10.4721 q^{53} +2.23607 q^{57} +6.38197 q^{59} +6.47214 q^{61} +7.70820 q^{63} +4.00000 q^{65} +0.0901699 q^{67} -8.47214 q^{69} +0.763932 q^{71} +12.6180 q^{73} +9.09017 q^{75} +13.4164 q^{79} -5.70820 q^{81} +6.32624 q^{83} -2.00000 q^{85} +11.7082 q^{87} +3.09017 q^{89} -6.47214 q^{91} +5.23607 q^{93} +1.05573 q^{95} +13.8541 q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 2 q - 3 q^{3} + 2 q^{5} + 4 q^{7} + q^{9} - 2 q^{13} + 2 q^{15} + q^{17} + 5 q^{19} - 6 q^{21} + 2 q^{23} + 2 q^{25} - 4 q^{31} + 4 q^{35} + 6 q^{37} + 8 q^{39} + 9 q^{41} - 3 q^{43} - 14 q^{45} + 4 q^{47}+ \cdots + 21 q^{97}+O(q^{100}) \) 2 * q - 3 * q^3 + 2 * q^5 + 4 * q^7 + q^9 - 2 * q^13 + 2 * q^15 + q^17 + 5 * q^19 - 6 * q^21 + 2 * q^23 + 2 * q^25 - 4 * q^31 + 4 * q^35 + 6 * q^37 + 8 * q^39 + 9 * q^41 - 3 * q^43 - 14 * q^45 + 4 * q^47 - 6 * q^49 - 4 * q^51 - 12 * q^53 + 15 * q^59 + 4 * q^61 + 2 * q^63 + 8 * q^65 - 11 * q^67 - 8 * q^69 + 6 * q^71 + 23 * q^73 + 7 * q^75 + 2 * q^81 - 3 * q^83 - 4 * q^85 + 10 * q^87 - 5 * q^89 - 4 * q^91 + 6 * q^93 + 20 * q^95 + 21 * q^97

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