LMFDB - Embedded newform 4012.2.a.e.1.2

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [4012,2,Mod(1,4012)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(4012, 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("4012.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 4012 = 2^{2} \cdot 17 \cdot 59 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	4012.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:	$32.0359812909$
Analytic rank:	$1$
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:	$ 2 $
Twist minimal:	yes
Fricke sign:	$1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.2
Root			$-0.618034$ of defining polynomial
Character	$\chi$	$=$	4012.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+2.23607 q^{3} -1.00000 q^{5} -2.23607 q^{7} +2.00000 q^{9} +O(q^{10})$	$q+2.23607 q^{3} -1.00000 q^{5} -2.23607 q^{7} +2.00000 q^{9} +4.47214 q^{11} -3.23607 q^{13} -2.23607 q^{15} -1.00000 q^{17} -6.70820 q^{19} -5.00000 q^{21} +4.47214 q^{23} -4.00000 q^{25} -2.23607 q^{27} +1.00000 q^{29} +6.00000 q^{31} +10.0000 q^{33} +2.23607 q^{35} -11.7082 q^{37} -7.23607 q^{39} -7.47214 q^{41} +0.472136 q^{43} -2.00000 q^{45} -2.00000 q^{47} -2.00000 q^{49} -2.23607 q^{51} +9.47214 q^{53} -4.47214 q^{55} -15.0000 q^{57} +1.00000 q^{59} +0.291796 q^{61} -4.47214 q^{63} +3.23607 q^{65} -14.9443 q^{67} +10.0000 q^{69} +2.47214 q^{71} -13.4164 q^{73} -8.94427 q^{75} -10.0000 q^{77} +4.70820 q^{79} -11.0000 q^{81} +1.23607 q^{83} +1.00000 q^{85} +2.23607 q^{87} +6.47214 q^{89} +7.23607 q^{91} +13.4164 q^{93} +6.70820 q^{95} -7.70820 q^{97} +8.94427 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 2 q - 2 q^{5} + 4 q^{9}+O(q^{10}) $ 2 * q - 2 * q^5 + 4 * q^9	$ 2 q - 2 q^{5} + 4 q^{9} - 2 q^{13} - 2 q^{17} - 10 q^{21} - 8 q^{25} + 2 q^{29} + 12 q^{31} + 20 q^{33} - 10 q^{37} - 10 q^{39} - 6 q^{41} - 8 q^{43} - 4 q^{45} - 4 q^{47} - 4 q^{49} + 10 q^{53} - 30 q^{57} + 2 q^{59} + 14 q^{61} + 2 q^{65} - 12 q^{67} + 20 q^{69} - 4 q^{71} - 20 q^{77} - 4 q^{79} - 22 q^{81} - 2 q^{83} + 2 q^{85} + 4 q^{89} + 10 q^{91} - 2 q^{97}+O(q^{100}) $ 2 * q - 2 * q^5 + 4 * q^9 - 2 * q^13 - 2 * q^17 - 10 * q^21 - 8 * q^25 + 2 * q^29 + 12 * q^31 + 20 * q^33 - 10 * q^37 - 10 * q^39 - 6 * q^41 - 8 * q^43 - 4 * q^45 - 4 * q^47 - 4 * q^49 + 10 * q^53 - 30 * q^57 + 2 * q^59 + 14 * q^61 + 2 * q^65 - 12 * q^67 + 20 * q^69 - 4 * q^71 - 20 * q^77 - 4 * q^79 - 22 * q^81 - 2 * q^83 + 2 * q^85 + 4 * q^89 + 10 * q^91 - 2 * 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
$3$	2.23607	1.29099	0.645497	−	0.763763i	$-0.276650\pi$
$3$	2.23607	1.29099	0.645497	+	0.763763i	$0.276650\pi$
$4$	0	0
$5$	−1.00000	−0.447214	−0.223607	−	0.974679i	$-0.571783\pi$
$5$	−1.00000	−0.447214	−0.223607	+	0.974679i	$0.571783\pi$
$6$	0	0
$7$	−2.23607	−0.845154	−0.422577	−	0.906327i	$-0.638874\pi$
$7$	−2.23607	−0.845154	−0.422577	+	0.906327i	$0.638874\pi$
$8$	0	0
$9$	2.00000	0.666667
$10$	0	0
$11$	4.47214	1.34840	0.674200	−	0.738549i	$-0.264489\pi$
$11$	4.47214	1.34840	0.674200	+	0.738549i	$0.264489\pi$
$12$	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
$15$	−2.23607	−0.577350
$16$	0	0
$17$	−1.00000	−0.242536
$17$	−1.00000	−0.242536
$18$	0	0
$19$	−6.70820	−1.53897	−0.769484	−	0.638666i	$-0.779486\pi$
$19$	−6.70820	−1.53897	−0.769484	+	0.638666i	$0.779486\pi$
$20$	0	0
$21$	−5.00000	−1.09109
$22$	0	0
$23$	4.47214	0.932505	0.466252	−	0.884652i	$-0.345604\pi$
$23$	4.47214	0.932505	0.466252	+	0.884652i	$0.345604\pi$
$24$	0	0
$25$	−4.00000	−0.800000
$26$	0	0
$27$	−2.23607	−0.430331
$28$	0	0
$29$	1.00000	0.185695	0.0928477	−	0.995680i	$-0.470403\pi$
$29$	1.00000	0.185695	0.0928477	+	0.995680i	$0.470403\pi$
$30$	0	0
$31$	6.00000	1.07763	0.538816	−	0.842424i	$-0.318872\pi$
$31$	6.00000	1.07763	0.538816	+	0.842424i	$0.318872\pi$
$32$	0	0
$33$	10.0000	1.74078
$34$	0	0
$35$	2.23607	0.377964
$36$	0	0
$37$	−11.7082	−1.92482	−0.962408	−	0.271606i	$-0.912445\pi$
$37$	−11.7082	−1.92482	−0.962408	+	0.271606i	$0.912445\pi$
$38$	0	0
$39$	−7.23607	−1.15870
$40$	0	0
$41$	−7.47214	−1.16695	−0.583476	−	0.812131i	$-0.698308\pi$
$41$	−7.47214	−1.16695	−0.583476	+	0.812131i	$0.698308\pi$
$42$	0	0
$43$	0.472136	0.0720001	0.0360000	−	0.999352i	$-0.488538\pi$
$43$	0.472136	0.0720001	0.0360000	+	0.999352i	$0.488538\pi$
$44$	0	0
$45$	−2.00000	−0.298142
$46$	0	0
$47$	−2.00000	−0.291730	−0.145865	−	0.989305i	$-0.546597\pi$
$47$	−2.00000	−0.291730	−0.145865	+	0.989305i	$0.546597\pi$
$48$	0	0
$49$	−2.00000	−0.285714
$50$	0	0
$51$	−2.23607	−0.313112
$52$	0	0
$53$	9.47214	1.30110	0.650549	−	0.759464i	$-0.274539\pi$
$53$	9.47214	1.30110	0.650549	+	0.759464i	$0.274539\pi$
$54$	0	0
$55$	−4.47214	−0.603023
$56$	0	0
$57$	−15.0000	−1.98680
$58$	0	0
$59$	1.00000	0.130189
$59$	1.00000	0.130189
$60$	0	0
$61$	0.291796	0.0373607	0.0186803	−	0.999826i	$-0.494054\pi$
$61$	0.291796	0.0373607	0.0186803	+	0.999826i	$0.494054\pi$
$62$	0	0
$63$	−4.47214	−0.563436
$64$	0	0
$65$	3.23607	0.401385
$66$	0	0
$67$	−14.9443	−1.82573	−0.912867	−	0.408258i	$-0.866136\pi$
$67$	−14.9443	−1.82573	−0.912867	+	0.408258i	$0.866136\pi$
$68$	0	0
$69$	10.0000	1.20386
$70$	0	0
$71$	2.47214	0.293389	0.146694	−	0.989182i	$-0.453137\pi$
$71$	2.47214	0.293389	0.146694	+	0.989182i	$0.453137\pi$
$72$	0	0
$73$	−13.4164	−1.57027	−0.785136	−	0.619324i	$-0.787407\pi$
$73$	−13.4164	−1.57027	−0.785136	+	0.619324i	$0.787407\pi$
$74$	0	0
$75$	−8.94427	−1.03280
$76$	0	0
$77$	−10.0000	−1.13961
$78$	0	0
$79$	4.70820	0.529714	0.264857	−	0.964288i	$-0.414675\pi$
$79$	4.70820	0.529714	0.264857	+	0.964288i	$0.414675\pi$
$80$	0	0
$81$	−11.0000	−1.22222
$82$	0	0
$83$	1.23607	0.135676	0.0678380	−	0.997696i	$-0.478390\pi$
$83$	1.23607	0.135676	0.0678380	+	0.997696i	$0.478390\pi$
$84$	0	0
$85$	1.00000	0.108465
$86$	0	0
$87$	2.23607	0.239732
$88$	0	0
$89$	6.47214	0.686045	0.343023	−	0.939327i	$-0.388549\pi$
$89$	6.47214	0.686045	0.343023	+	0.939327i	$0.388549\pi$
$90$	0	0
$91$	7.23607	0.758546
$92$	0	0
$93$	13.4164	1.39122
$94$	0	0
$95$	6.70820	0.688247
$96$	0	0
$97$	−7.70820	−0.782650	−0.391325	−	0.920253i	$-0.627983\pi$
$97$	−7.70820	−0.782650	−0.391325	+	0.920253i	$0.627983\pi$
$98$	0	0
$99$	8.94427	0.898933
$100$	0	0
$101$	−1.70820	−0.169973	−0.0849863	−	0.996382i	$-0.527085\pi$
$101$	−1.70820	−0.169973	−0.0849863	+	0.996382i	$0.527085\pi$
$102$	0	0
$103$	1.70820	0.168314	0.0841572	−	0.996452i	$-0.473180\pi$
$103$	1.70820	0.168314	0.0841572	+	0.996452i	$0.473180\pi$
$104$	0	0
$105$	5.00000	0.487950
$106$	0	0
$107$	10.2361	0.989558	0.494779	−	0.869019i	$-0.335249\pi$
$107$	10.2361	0.989558	0.494779	+	0.869019i	$0.335249\pi$
$108$	0	0
$109$	−8.47214	−0.811483	−0.405742	−	0.913988i	$-0.632987\pi$
$109$	−8.47214	−0.811483	−0.405742	+	0.913988i	$0.632987\pi$
$110$	0	0
$111$	−26.1803	−2.48493
$112$	0	0
$113$	−7.23607	−0.680712	−0.340356	−	0.940297i	$-0.610548\pi$
$113$	−7.23607	−0.680712	−0.340356	+	0.940297i	$0.610548\pi$
$114$	0	0
$115$	−4.47214	−0.417029
$116$	0	0
$117$	−6.47214	−0.598349
$118$	0	0
$119$	2.23607	0.204980
$120$	0	0
$121$	9.00000	0.818182
$122$	0	0
$123$	−16.7082	−1.50653
$124$	0	0
$125$	9.00000	0.804984
$126$	0	0
$127$	10.2361	0.908304	0.454152	−	0.890924i	$-0.349942\pi$
$127$	10.2361	0.908304	0.454152	+	0.890924i	$0.349942\pi$
$128$	0	0
$129$	1.05573	0.0929517
$130$	0	0
$131$	−8.94427	−0.781465	−0.390732	−	0.920504i	$-0.627778\pi$
$131$	−8.94427	−0.781465	−0.390732	+	0.920504i	$0.627778\pi$
$132$	0	0
$133$	15.0000	1.30066
$134$	0	0
$135$	2.23607	0.192450
$136$	0	0
$137$	−17.0000	−1.45241	−0.726204	−	0.687479i	$-0.758717\pi$
$137$	−17.0000	−1.45241	−0.726204	+	0.687479i	$0.758717\pi$
$138$	0	0
$139$	16.9443	1.43719	0.718597	−	0.695427i	$-0.244785\pi$
$139$	16.9443	1.43719	0.718597	+	0.695427i	$0.244785\pi$
$140$	0	0
$141$	−4.47214	−0.376622
$142$	0	0
$143$	−14.4721	−1.21022
$144$	0	0
$145$	−1.00000	−0.0830455
$146$	0	0
$147$	−4.47214	−0.368856
$148$	0	0
$149$	16.6525	1.36422	0.682112	−	0.731248i	$-0.261062\pi$
$149$	16.6525	1.36422	0.682112	+	0.731248i	$0.261062\pi$
$150$	0	0
$151$	−4.76393	−0.387683	−0.193842	−	0.981033i	$-0.562095\pi$
$151$	−4.76393	−0.387683	−0.193842	+	0.981033i	$0.562095\pi$
$152$	0	0
$153$	−2.00000	−0.161690
$154$	0	0
$155$	−6.00000	−0.481932
$156$	0	0
$157$	0.180340	0.0143927	0.00719634	−	0.999974i	$-0.497709\pi$
$157$	0.180340	0.0143927	0.00719634	+	0.999974i	$0.497709\pi$
$158$	0	0
$159$	21.1803	1.67971
$160$	0	0
$161$	−10.0000	−0.788110
$162$	0	0
$163$	−20.9443	−1.64048	−0.820241	−	0.572018i	$-0.806161\pi$
$163$	−20.9443	−1.64048	−0.820241	+	0.572018i	$0.806161\pi$
$164$	0	0
$165$	−10.0000	−0.778499
$166$	0	0
$167$	−9.29180	−0.719021	−0.359510	−	0.933141i	$-0.617056\pi$
$167$	−9.29180	−0.719021	−0.359510	+	0.933141i	$0.617056\pi$
$168$	0	0
$169$	−2.52786	−0.194451
$170$	0	0
$171$	−13.4164	−1.02598
$172$	0	0
$173$	18.1803	1.38223	0.691113	−	0.722747i	$-0.257121\pi$
$173$	18.1803	1.38223	0.691113	+	0.722747i	$0.257121\pi$
$174$	0	0
$175$	8.94427	0.676123
$176$	0	0
$177$	2.23607	0.168073
$178$	0	0
$179$	4.18034	0.312453	0.156227	−	0.987721i	$-0.450067\pi$
$179$	4.18034	0.312453	0.156227	+	0.987721i	$0.450067\pi$
$180$	0	0
$181$	−20.8885	−1.55263	−0.776317	−	0.630343i	$-0.782914\pi$
$181$	−20.8885	−1.55263	−0.776317	+	0.630343i	$0.782914\pi$
$182$	0	0
$183$	0.652476	0.0482324
$184$	0	0
$185$	11.7082	0.860804
$186$	0	0
$187$	−4.47214	−0.327035
$188$	0	0
$189$	5.00000	0.363696
$190$	0	0
$191$	−12.9443	−0.936615	−0.468307	−	0.883566i	$-0.655136\pi$
$191$	−12.9443	−0.936615	−0.468307	+	0.883566i	$0.655136\pi$
$192$	0	0
$193$	2.05573	0.147975	0.0739873	−	0.997259i	$-0.476428\pi$
$193$	2.05573	0.147975	0.0739873	+	0.997259i	$0.476428\pi$
$194$	0	0
$195$	7.23607	0.518186
$196$	0	0
$197$	−15.8885	−1.13201	−0.566006	−	0.824401i	$-0.691512\pi$
$197$	−15.8885	−1.13201	−0.566006	+	0.824401i	$0.691512\pi$
$198$	0	0
$199$	−22.2361	−1.57627	−0.788137	−	0.615500i	$-0.788954\pi$
$199$	−22.2361	−1.57627	−0.788137	+	0.615500i	$0.788954\pi$
$200$	0	0
$201$	−33.4164	−2.35701
$202$	0	0
$203$	−2.23607	−0.156941
$204$	0	0
$205$	7.47214	0.521877
$206$	0	0
$207$	8.94427	0.621670
$208$	0	0
$209$	−30.0000	−2.07514
$210$	0	0
$211$	−17.5279	−1.20667	−0.603334	−	0.797489i	$-0.706161\pi$
$211$	−17.5279	−1.20667	−0.603334	+	0.797489i	$0.706161\pi$
$212$	0	0
$213$	5.52786	0.378763
$214$	0	0
$215$	−0.472136	−0.0321994
$216$	0	0
$217$	−13.4164	−0.910765
$218$	0	0
$219$	−30.0000	−2.02721
$220$	0	0
$221$	3.23607	0.217681
$222$	0	0
$223$	6.47214	0.433406	0.216703	−	0.976238i	$-0.430470\pi$
$223$	6.47214	0.433406	0.216703	+	0.976238i	$0.430470\pi$
$224$	0	0
$225$	−8.00000	−0.533333
$226$	0	0
$227$	10.6525	0.707030	0.353515	−	0.935429i	$-0.384986\pi$
$227$	10.6525	0.707030	0.353515	+	0.935429i	$0.384986\pi$
$228$	0	0
$229$	−9.70820	−0.641536	−0.320768	−	0.947158i	$-0.603941\pi$
$229$	−9.70820	−0.641536	−0.320768	+	0.947158i	$0.603941\pi$
$230$	0	0
$231$	−22.3607	−1.47122
$232$	0	0
$233$	1.23607	0.0809775	0.0404888	−	0.999180i	$-0.487109\pi$
$233$	1.23607	0.0809775	0.0404888	+	0.999180i	$0.487109\pi$
$234$	0	0
$235$	2.00000	0.130466
$236$	0	0
$237$	10.5279	0.683858
$238$	0	0
$239$	18.1246	1.17238	0.586192	−	0.810172i	$-0.300626\pi$
$239$	18.1246	1.17238	0.586192	+	0.810172i	$0.300626\pi$
$240$	0	0
$241$	−11.9443	−0.769398	−0.384699	−	0.923042i	$-0.625695\pi$
$241$	−11.9443	−0.769398	−0.384699	+	0.923042i	$0.625695\pi$
$242$	0	0
$243$	−17.8885	−1.14755
$244$	0	0
$245$	2.00000	0.127775
$246$	0	0
$247$	21.7082	1.38126
$248$	0	0
$249$	2.76393	0.175157
$250$	0	0
$251$	23.6525	1.49293	0.746466	−	0.665424i	$-0.231749\pi$
$251$	23.6525	1.49293	0.746466	+	0.665424i	$0.231749\pi$
$252$	0	0
$253$	20.0000	1.25739
$254$	0	0
$255$	2.23607	0.140028
$256$	0	0
$257$	1.00000	0.0623783	0.0311891	−	0.999514i	$-0.490071\pi$
$257$	1.00000	0.0623783	0.0311891	+	0.999514i	$0.490071\pi$
$258$	0	0
$259$	26.1803	1.62677
$260$	0	0
$261$	2.00000	0.123797
$262$	0	0
$263$	6.70820	0.413646	0.206823	−	0.978378i	$-0.433688\pi$
$263$	6.70820	0.413646	0.206823	+	0.978378i	$0.433688\pi$
$264$	0	0
$265$	−9.47214	−0.581869
$266$	0	0
$267$	14.4721	0.885680
$268$	0	0
$269$	6.18034	0.376822	0.188411	−	0.982090i	$-0.439666\pi$
$269$	6.18034	0.376822	0.188411	+	0.982090i	$0.439666\pi$
$270$	0	0
$271$	15.1803	0.922140	0.461070	−	0.887364i	$-0.347466\pi$
$271$	15.1803	0.922140	0.461070	+	0.887364i	$0.347466\pi$
$272$	0	0
$273$	16.1803	0.979279
$274$	0	0
$275$	−17.8885	−1.07872
$276$	0	0
$277$	−1.00000	−0.0600842	−0.0300421	−	0.999549i	$-0.509564\pi$
$277$	−1.00000	−0.0600842	−0.0300421	+	0.999549i	$0.509564\pi$
$278$	0	0
$279$	12.0000	0.718421
$280$	0	0
$281$	19.3607	1.15496	0.577481	−	0.816404i	$-0.304036\pi$
$281$	19.3607	1.15496	0.577481	+	0.816404i	$0.304036\pi$
$282$	0	0
$283$	14.6525	0.870999	0.435500	−	0.900189i	$-0.356572\pi$
$283$	14.6525	0.870999	0.435500	+	0.900189i	$0.356572\pi$
$284$	0	0
$285$	15.0000	0.888523
$286$	0	0
$287$	16.7082	0.986254
$288$	0	0
$289$	1.00000	0.0588235
$290$	0	0
$291$	−17.2361	−1.01040
$292$	0	0
$293$	12.4164	0.725374	0.362687	−	0.931911i	$-0.381859\pi$
$293$	12.4164	0.725374	0.362687	+	0.931911i	$0.381859\pi$
$294$	0	0
$295$	−1.00000	−0.0582223
$296$	0	0
$297$	−10.0000	−0.580259
$298$	0	0
$299$	−14.4721	−0.836945
$300$	0	0
$301$	−1.05573	−0.0608512
$302$	0	0
$303$	−3.81966	−0.219434
$304$	0	0
$305$	−0.291796	−0.0167082
$306$	0	0
$307$	11.6525	0.665042	0.332521	−	0.943096i	$-0.392101\pi$
$307$	11.6525	0.665042	0.332521	+	0.943096i	$0.392101\pi$
$308$	0	0
$309$	3.81966	0.217293
$310$	0	0
$311$	18.7082	1.06084	0.530422	−	0.847733i	$-0.322033\pi$
$311$	18.7082	1.06084	0.530422	+	0.847733i	$0.322033\pi$
$312$	0	0
$313$	−26.1803	−1.47980	−0.739900	−	0.672717i	$-0.765127\pi$
$313$	−26.1803	−1.47980	−0.739900	+	0.672717i	$0.765127\pi$
$314$	0	0
$315$	4.47214	0.251976
$316$	0	0
$317$	−12.4721	−0.700505	−0.350252	−	0.936655i	$-0.613904\pi$
$317$	−12.4721	−0.700505	−0.350252	+	0.936655i	$0.613904\pi$
$318$	0	0
$319$	4.47214	0.250392
$320$	0	0
$321$	22.8885	1.27751
$322$	0	0
$323$	6.70820	0.373254
$324$	0	0
$325$	12.9443	0.718019
$326$	0	0
$327$	−18.9443	−1.04762
$328$	0	0
$329$	4.47214	0.246557
$330$	0	0
$331$	−3.65248	−0.200758	−0.100379	−	0.994949i	$-0.532006\pi$
$331$	−3.65248	−0.200758	−0.100379	+	0.994949i	$0.532006\pi$
$332$	0	0
$333$	−23.4164	−1.28321
$334$	0	0
$335$	14.9443	0.816493
$336$	0	0
$337$	29.3050	1.59634	0.798171	−	0.602431i	$-0.205801\pi$
$337$	29.3050	1.59634	0.798171	+	0.602431i	$0.205801\pi$
$338$	0	0
$339$	−16.1803	−0.878795
$340$	0	0
$341$	26.8328	1.45308
$342$	0	0
$343$	20.1246	1.08663
$344$	0	0
$345$	−10.0000	−0.538382
$346$	0	0
$347$	−17.5967	−0.944643	−0.472321	−	0.881426i	$-0.656584\pi$
$347$	−17.5967	−0.944643	−0.472321	+	0.881426i	$0.656584\pi$
$348$	0	0
$349$	3.70820	0.198496	0.0992478	−	0.995063i	$-0.468356\pi$
$349$	3.70820	0.198496	0.0992478	+	0.995063i	$0.468356\pi$
$350$	0	0
$351$	7.23607	0.386233
$352$	0	0
$353$	−5.23607	−0.278688	−0.139344	−	0.990244i	$-0.544499\pi$
$353$	−5.23607	−0.278688	−0.139344	+	0.990244i	$0.544499\pi$
$354$	0	0
$355$	−2.47214	−0.131207
$356$	0	0
$357$	5.00000	0.264628
$358$	0	0
$359$	−18.7082	−0.987381	−0.493691	−	0.869638i	$-0.664352\pi$
$359$	−18.7082	−0.987381	−0.493691	+	0.869638i	$0.664352\pi$
$360$	0	0
$361$	26.0000	1.36842
$362$	0	0
$363$	20.1246	1.05627
$364$	0	0
$365$	13.4164	0.702247
$366$	0	0
$367$	24.6525	1.28685	0.643424	−	0.765510i	$-0.277513\pi$
$367$	24.6525	1.28685	0.643424	+	0.765510i	$0.277513\pi$
$368$	0	0
$369$	−14.9443	−0.777968
$370$	0	0
$371$	−21.1803	−1.09963
$372$	0	0
$373$	0.472136	0.0244463	0.0122231	−	0.999925i	$-0.496109\pi$
$373$	0.472136	0.0244463	0.0122231	+	0.999925i	$0.496109\pi$
$374$	0	0
$375$	20.1246	1.03923
$376$	0	0
$377$	−3.23607	−0.166666
$378$	0	0
$379$	6.70820	0.344577	0.172289	−	0.985047i	$-0.444884\pi$
$379$	6.70820	0.344577	0.172289	+	0.985047i	$0.444884\pi$
$380$	0	0
$381$	22.8885	1.17262
$382$	0	0
$383$	−16.0000	−0.817562	−0.408781	−	0.912633i	$-0.634046\pi$
$383$	−16.0000	−0.817562	−0.408781	+	0.912633i	$0.634046\pi$
$384$	0	0
$385$	10.0000	0.509647
$386$	0	0
$387$	0.944272	0.0480000
$388$	0	0
$389$	−8.47214	−0.429554	−0.214777	−	0.976663i	$-0.568903\pi$
$389$	−8.47214	−0.429554	−0.214777	+	0.976663i	$0.568903\pi$
$390$	0	0
$391$	−4.47214	−0.226166
$392$	0	0
$393$	−20.0000	−1.00887
$394$	0	0
$395$	−4.70820	−0.236895
$396$	0	0
$397$	−6.76393	−0.339472	−0.169736	−	0.985490i	$-0.554292\pi$
$397$	−6.76393	−0.339472	−0.169736	+	0.985490i	$0.554292\pi$
$398$	0	0
$399$	33.5410	1.67915
$400$	0	0
$401$	−1.12461	−0.0561604	−0.0280802	−	0.999606i	$-0.508939\pi$
$401$	−1.12461	−0.0561604	−0.0280802	+	0.999606i	$0.508939\pi$
$402$	0	0
$403$	−19.4164	−0.967200
$404$	0	0
$405$	11.0000	0.546594
$406$	0	0
$407$	−52.3607	−2.59542
$408$	0	0
$409$	30.6525	1.51567	0.757834	−	0.652448i	$-0.226258\pi$
$409$	30.6525	1.51567	0.757834	+	0.652448i	$0.226258\pi$
$410$	0	0
$411$	−38.0132	−1.87505
$412$	0	0
$413$	−2.23607	−0.110030
$414$	0	0
$415$	−1.23607	−0.0606762
$416$	0	0
$417$	37.8885	1.85541
$418$	0	0
$419$	12.1803	0.595049	0.297524	−	0.954714i	$-0.403839\pi$
$419$	12.1803	0.595049	0.297524	+	0.954714i	$0.403839\pi$
$420$	0	0
$421$	6.00000	0.292422	0.146211	−	0.989253i	$-0.453292\pi$
$421$	6.00000	0.292422	0.146211	+	0.989253i	$0.453292\pi$
$422$	0	0
$423$	−4.00000	−0.194487
$424$	0	0
$425$	4.00000	0.194029
$426$	0	0
$427$	−0.652476	−0.0315755
$428$	0	0
$429$	−32.3607	−1.56239
$430$	0	0
$431$	8.94427	0.430830	0.215415	−	0.976523i	$-0.430890\pi$
$431$	8.94427	0.430830	0.215415	+	0.976523i	$0.430890\pi$
$432$	0	0
$433$	−23.4721	−1.12800	−0.563999	−	0.825775i	$-0.690738\pi$
$433$	−23.4721	−1.12800	−0.563999	+	0.825775i	$0.690738\pi$
$434$	0	0
$435$	−2.23607	−0.107211
$436$	0	0
$437$	−30.0000	−1.43509
$438$	0	0
$439$	26.8328	1.28066	0.640330	−	0.768100i	$-0.278798\pi$
$439$	26.8328	1.28066	0.640330	+	0.768100i	$0.278798\pi$
$440$	0	0
$441$	−4.00000	−0.190476
$442$	0	0
$443$	3.52786	0.167614	0.0838069	−	0.996482i	$-0.473292\pi$
$443$	3.52786	0.167614	0.0838069	+	0.996482i	$0.473292\pi$
$444$	0	0
$445$	−6.47214	−0.306809
$446$	0	0
$447$	37.2361	1.76121
$448$	0	0
$449$	13.3607	0.630529	0.315265	−	0.949004i	$-0.397907\pi$
$449$	13.3607	0.630529	0.315265	+	0.949004i	$0.397907\pi$
$450$	0	0
$451$	−33.4164	−1.57352
$452$	0	0
$453$	−10.6525	−0.500497
$454$	0	0
$455$	−7.23607	−0.339232
$456$	0	0
$457$	12.2918	0.574986	0.287493	−	0.957783i	$-0.407178\pi$
$457$	12.2918	0.574986	0.287493	+	0.957783i	$0.407178\pi$
$458$	0	0
$459$	2.23607	0.104371
$460$	0	0
$461$	9.05573	0.421767	0.210884	−	0.977511i	$-0.432366\pi$
$461$	9.05573	0.421767	0.210884	+	0.977511i	$0.432366\pi$
$462$	0	0
$463$	−9.05573	−0.420855	−0.210428	−	0.977609i	$-0.567486\pi$
$463$	−9.05573	−0.420855	−0.210428	+	0.977609i	$0.567486\pi$
$464$	0	0
$465$	−13.4164	−0.622171
$466$	0	0
$467$	−34.3607	−1.59002	−0.795011	−	0.606595i	$-0.792535\pi$
$467$	−34.3607	−1.59002	−0.795011	+	0.606595i	$0.792535\pi$
$468$	0	0
$469$	33.4164	1.54303
$470$	0	0
$471$	0.403252	0.0185809
$472$	0	0
$473$	2.11146	0.0970849
$474$	0	0
$475$	26.8328	1.23117
$476$	0	0
$477$	18.9443	0.867399
$478$	0	0
$479$	−14.8328	−0.677729	−0.338864	−	0.940835i	$-0.610043\pi$
$479$	−14.8328	−0.677729	−0.338864	+	0.940835i	$0.610043\pi$
$480$	0	0
$481$	37.8885	1.72757
$482$	0	0
$483$	−22.3607	−1.01745
$484$	0	0
$485$	7.70820	0.350012
$486$	0	0
$487$	13.7639	0.623703	0.311852	−	0.950131i	$-0.399051\pi$
$487$	13.7639	0.623703	0.311852	+	0.950131i	$0.399051\pi$
$488$	0	0
$489$	−46.8328	−2.11785
$490$	0	0
$491$	33.7639	1.52374	0.761872	−	0.647727i	$-0.224280\pi$
$491$	33.7639	1.52374	0.761872	+	0.647727i	$0.224280\pi$
$492$	0	0
$493$	−1.00000	−0.0450377
$494$	0	0
$495$	−8.94427	−0.402015
$496$	0	0
$497$	−5.52786	−0.247959
$498$	0	0
$499$	39.5410	1.77010	0.885050	−	0.465497i	$-0.154124\pi$
$499$	39.5410	1.77010	0.885050	+	0.465497i	$0.154124\pi$
$500$	0	0
$501$	−20.7771	−0.928252
$502$	0	0
$503$	−5.70820	−0.254516	−0.127258	−	0.991870i	$-0.540618\pi$
$503$	−5.70820	−0.254516	−0.127258	+	0.991870i	$0.540618\pi$
$504$	0	0
$505$	1.70820	0.0760141
$506$	0	0
$507$	−5.65248	−0.251035
$508$	0	0
$509$	−28.7639	−1.27494	−0.637469	−	0.770476i	$-0.720019\pi$
$509$	−28.7639	−1.27494	−0.637469	+	0.770476i	$0.720019\pi$
$510$	0	0
$511$	30.0000	1.32712
$512$	0	0
$513$	15.0000	0.662266
$514$	0	0
$515$	−1.70820	−0.0752725
$516$	0	0
$517$	−8.94427	−0.393369
$518$	0	0
$519$	40.6525	1.78445
$520$	0	0
$521$	13.0557	0.571982	0.285991	−	0.958232i	$-0.407677\pi$
$521$	13.0557	0.571982	0.285991	+	0.958232i	$0.407677\pi$
$522$	0	0
$523$	17.7639	0.776763	0.388381	−	0.921499i	$-0.373034\pi$
$523$	17.7639	0.776763	0.388381	+	0.921499i	$0.373034\pi$
$524$	0	0
$525$	20.0000	0.872872
$526$	0	0
$527$	−6.00000	−0.261364
$528$	0	0
$529$	−3.00000	−0.130435
$530$	0	0
$531$	2.00000	0.0867926
$532$	0	0
$533$	24.1803	1.04737
$534$	0	0
$535$	−10.2361	−0.442544
$536$	0	0
$537$	9.34752	0.403375
$538$	0	0
$539$	−8.94427	−0.385257
$540$	0	0
$541$	−18.3607	−0.789387	−0.394694	−	0.918813i	$-0.629149\pi$
$541$	−18.3607	−0.789387	−0.394694	+	0.918813i	$0.629149\pi$
$542$	0	0
$543$	−46.7082	−2.00444
$544$	0	0
$545$	8.47214	0.362906
$546$	0	0
$547$	−32.9443	−1.40860	−0.704298	−	0.709905i	$-0.748738\pi$
$547$	−32.9443	−1.40860	−0.704298	+	0.709905i	$0.748738\pi$
$548$	0	0
$549$	0.583592	0.0249071
$550$	0	0
$551$	−6.70820	−0.285779
$552$	0	0
$553$	−10.5279	−0.447690
$554$	0	0
$555$	26.1803	1.11129
$556$	0	0
$557$	20.8885	0.885076	0.442538	−	0.896750i	$-0.354078\pi$
$557$	20.8885	0.885076	0.442538	+	0.896750i	$0.354078\pi$
$558$	0	0
$559$	−1.52786	−0.0646218
$560$	0	0
$561$	−10.0000	−0.422200
$562$	0	0
$563$	−42.8328	−1.80519	−0.902594	−	0.430493i	$-0.858340\pi$
$563$	−42.8328	−1.80519	−0.902594	+	0.430493i	$0.858340\pi$
$564$	0	0
$565$	7.23607	0.304424
$566$	0	0
$567$	24.5967	1.03297
$568$	0	0
$569$	−24.0689	−1.00902	−0.504510	−	0.863406i	$-0.668327\pi$
$569$	−24.0689	−1.00902	−0.504510	+	0.863406i	$0.668327\pi$
$570$	0	0
$571$	−21.7082	−0.908460	−0.454230	−	0.890884i	$-0.650086\pi$
$571$	−21.7082	−0.908460	−0.454230	+	0.890884i	$0.650086\pi$
$572$	0	0
$573$	−28.9443	−1.20916
$574$	0	0
$575$	−17.8885	−0.746004
$576$	0	0
$577$	−11.0000	−0.457936	−0.228968	−	0.973434i	$-0.573535\pi$
$577$	−11.0000	−0.457936	−0.228968	+	0.973434i	$0.573535\pi$
$578$	0	0
$579$	4.59675	0.191034
$580$	0	0
$581$	−2.76393	−0.114667
$582$	0	0
$583$	42.3607	1.75440
$584$	0	0
$585$	6.47214	0.267590
$586$	0	0
$587$	−2.47214	−0.102036	−0.0510180	−	0.998698i	$-0.516247\pi$
$587$	−2.47214	−0.102036	−0.0510180	+	0.998698i	$0.516247\pi$
$588$	0	0
$589$	−40.2492	−1.65844
$590$	0	0
$591$	−35.5279	−1.46142
$592$	0	0
$593$	−24.3050	−0.998085	−0.499042	−	0.866578i	$-0.666315\pi$
$593$	−24.3050	−0.998085	−0.499042	+	0.866578i	$0.666315\pi$
$594$	0	0
$595$	−2.23607	−0.0916698
$596$	0	0
$597$	−49.7214	−2.03496
$598$	0	0
$599$	−1.76393	−0.0720723	−0.0360362	−	0.999350i	$-0.511473\pi$
$599$	−1.76393	−0.0720723	−0.0360362	+	0.999350i	$0.511473\pi$
$600$	0	0
$601$	−13.5279	−0.551813	−0.275907	−	0.961184i	$-0.588978\pi$
$601$	−13.5279	−0.551813	−0.275907	+	0.961184i	$0.588978\pi$
$602$	0	0
$603$	−29.8885	−1.21716
$604$	0	0
$605$	−9.00000	−0.365902
$606$	0	0
$607$	−3.76393	−0.152773	−0.0763866	−	0.997078i	$-0.524338\pi$
$607$	−3.76393	−0.152773	−0.0763866	+	0.997078i	$0.524338\pi$
$608$	0	0
$609$	−5.00000	−0.202610
$610$	0	0
$611$	6.47214	0.261835
$612$	0	0
$613$	18.4721	0.746083	0.373041	−	0.927815i	$-0.378315\pi$
$613$	18.4721	0.746083	0.373041	+	0.927815i	$0.378315\pi$
$614$	0	0
$615$	16.7082	0.673740
$616$	0	0
$617$	45.3607	1.82615	0.913076	−	0.407789i	$-0.133700\pi$
$617$	45.3607	1.82615	0.913076	+	0.407789i	$0.133700\pi$
$618$	0	0
$619$	0.819660	0.0329449	0.0164725	−	0.999864i	$-0.494756\pi$
$619$	0.819660	0.0329449	0.0164725	+	0.999864i	$0.494756\pi$
$620$	0	0
$621$	−10.0000	−0.401286
$622$	0	0
$623$	−14.4721	−0.579814
$624$	0	0
$625$	11.0000	0.440000
$626$	0	0
$627$	−67.0820	−2.67900
$628$	0	0
$629$	11.7082	0.466837
$630$	0	0
$631$	−0.583592	−0.0232324	−0.0116162	−	0.999933i	$-0.503698\pi$
$631$	−0.583592	−0.0232324	−0.0116162	+	0.999933i	$0.503698\pi$
$632$	0	0
$633$	−39.1935	−1.55780
$634$	0	0
$635$	−10.2361	−0.406206
$636$	0	0
$637$	6.47214	0.256435
$638$	0	0
$639$	4.94427	0.195592
$640$	0	0
$641$	14.0000	0.552967	0.276483	−	0.961019i	$-0.410831\pi$
$641$	14.0000	0.552967	0.276483	+	0.961019i	$0.410831\pi$
$642$	0	0
$643$	−21.7639	−0.858286	−0.429143	−	0.903237i	$-0.641184\pi$
$643$	−21.7639	−0.858286	−0.429143	+	0.903237i	$0.641184\pi$
$644$	0	0
$645$	−1.05573	−0.0415693
$646$	0	0
$647$	−0.347524	−0.0136626	−0.00683129	−	0.999977i	$-0.502174\pi$
$647$	−0.347524	−0.0136626	−0.00683129	+	0.999977i	$0.502174\pi$
$648$	0	0
$649$	4.47214	0.175547
$650$	0	0
$651$	−30.0000	−1.17579
$652$	0	0
$653$	15.3607	0.601110	0.300555	−	0.953765i	$-0.402828\pi$
$653$	15.3607	0.601110	0.300555	+	0.953765i	$0.402828\pi$
$654$	0	0
$655$	8.94427	0.349482
$656$	0	0
$657$	−26.8328	−1.04685
$658$	0	0
$659$	42.0689	1.63877	0.819386	−	0.573243i	$-0.194315\pi$
$659$	42.0689	1.63877	0.819386	+	0.573243i	$0.194315\pi$
$660$	0	0
$661$	−14.8885	−0.579097	−0.289549	−	0.957163i	$-0.593505\pi$
$661$	−14.8885	−0.579097	−0.289549	+	0.957163i	$0.593505\pi$
$662$	0	0
$663$	7.23607	0.281026
$664$	0	0
$665$	−15.0000	−0.581675
$666$	0	0
$667$	4.47214	0.173162
$668$	0	0
$669$	14.4721	0.559525
$670$	0	0
$671$	1.30495	0.0503771
$672$	0	0
$673$	47.4164	1.82777	0.913884	−	0.405975i	$-0.133068\pi$
$673$	47.4164	1.82777	0.913884	+	0.405975i	$0.133068\pi$
$674$	0	0
$675$	8.94427	0.344265
$676$	0	0
$677$	26.0000	0.999261	0.499631	−	0.866239i	$-0.333469\pi$
$677$	26.0000	0.999261	0.499631	+	0.866239i	$0.333469\pi$
$678$	0	0
$679$	17.2361	0.661460
$680$	0	0
$681$	23.8197	0.912771
$682$	0	0
$683$	−48.6525	−1.86164	−0.930818	−	0.365484i	$-0.880903\pi$
$683$	−48.6525	−1.86164	−0.930818	+	0.365484i	$0.880903\pi$
$684$	0	0
$685$	17.0000	0.649537
$686$	0	0
$687$	−21.7082	−0.828220
$688$	0	0
$689$	−30.6525	−1.16777
$690$	0	0
$691$	−9.41641	−0.358217	−0.179109	−	0.983829i	$-0.557321\pi$
$691$	−9.41641	−0.358217	−0.179109	+	0.983829i	$0.557321\pi$
$692$	0	0
$693$	−20.0000	−0.759737
$694$	0	0
$695$	−16.9443	−0.642733
$696$	0	0
$697$	7.47214	0.283027
$698$	0	0
$699$	2.76393	0.104542
$700$	0	0
$701$	11.5279	0.435401	0.217701	−	0.976016i	$-0.430144\pi$
$701$	11.5279	0.435401	0.217701	+	0.976016i	$0.430144\pi$
$702$	0	0
$703$	78.5410	2.96223
$704$	0	0
$705$	4.47214	0.168430
$706$	0	0
$707$	3.81966	0.143653
$708$	0	0
$709$	−4.88854	−0.183593	−0.0917966	−	0.995778i	$-0.529261\pi$
$709$	−4.88854	−0.183593	−0.0917966	+	0.995778i	$0.529261\pi$
$710$	0	0
$711$	9.41641	0.353143
$712$	0	0
$713$	26.8328	1.00490
$714$	0	0
$715$	14.4721	0.541227
$716$	0	0
$717$	40.5279	1.51354
$718$	0	0
$719$	−42.1803	−1.57306	−0.786531	−	0.617551i	$-0.788125\pi$
$719$	−42.1803	−1.57306	−0.786531	+	0.617551i	$0.788125\pi$
$720$	0	0
$721$	−3.81966	−0.142252
$722$	0	0
$723$	−26.7082	−0.993289
$724$	0	0
$725$	−4.00000	−0.148556
$726$	0	0
$727$	−43.7771	−1.62360	−0.811801	−	0.583934i	$-0.801513\pi$
$727$	−43.7771	−1.62360	−0.811801	+	0.583934i	$0.801513\pi$
$728$	0	0
$729$	−7.00000	−0.259259
$730$	0	0
$731$	−0.472136	−0.0174626
$732$	0	0
$733$	23.8885	0.882343	0.441172	−	0.897423i	$-0.354563\pi$
$733$	23.8885	0.882343	0.441172	+	0.897423i	$0.354563\pi$
$734$	0	0
$735$	4.47214	0.164957
$736$	0	0
$737$	−66.8328	−2.46182
$738$	0	0
$739$	−10.0689	−0.370390	−0.185195	−	0.982702i	$-0.559292\pi$
$739$	−10.0689	−0.370390	−0.185195	+	0.982702i	$0.559292\pi$
$740$	0	0
$741$	48.5410	1.78320
$742$	0	0
$743$	33.8885	1.24325	0.621625	−	0.783315i	$-0.286473\pi$
$743$	33.8885	1.24325	0.621625	+	0.783315i	$0.286473\pi$
$744$	0	0
$745$	−16.6525	−0.610100
$746$	0	0
$747$	2.47214	0.0904507
$748$	0	0
$749$	−22.8885	−0.836329
$750$	0	0
$751$	5.41641	0.197648	0.0988238	−	0.995105i	$-0.468492\pi$
$751$	5.41641	0.197648	0.0988238	+	0.995105i	$0.468492\pi$
$752$	0	0
$753$	52.8885	1.92737
$754$	0	0
$755$	4.76393	0.173377
$756$	0	0
$757$	37.0000	1.34479	0.672394	−	0.740193i	$-0.265266\pi$
$757$	37.0000	1.34479	0.672394	+	0.740193i	$0.265266\pi$
$758$	0	0
$759$	44.7214	1.62328
$760$	0	0
$761$	24.4164	0.885094	0.442547	−	0.896745i	$-0.354075\pi$
$761$	24.4164	0.885094	0.442547	+	0.896745i	$0.354075\pi$
$762$	0	0
$763$	18.9443	0.685829
$764$	0	0
$765$	2.00000	0.0723102
$766$	0	0
$767$	−3.23607	−0.116848
$768$	0	0
$769$	21.2361	0.765792	0.382896	−	0.923791i	$-0.374927\pi$
$769$	21.2361	0.765792	0.382896	+	0.923791i	$0.374927\pi$
$770$	0	0
$771$	2.23607	0.0805300
$772$	0	0
$773$	−31.0132	−1.11547	−0.557733	−	0.830021i	$-0.688329\pi$
$773$	−31.0132	−1.11547	−0.557733	+	0.830021i	$0.688329\pi$
$774$	0	0
$775$	−24.0000	−0.862105
$776$	0	0
$777$	58.5410	2.10015
$778$	0	0
$779$	50.1246	1.79590
$780$	0	0
$781$	11.0557	0.395605
$782$	0	0
$783$	−2.23607	−0.0799106
$784$	0	0
$785$	−0.180340	−0.00643661
$786$	0	0
$787$	10.4721	0.373291	0.186646	−	0.982427i	$-0.440238\pi$
$787$	10.4721	0.373291	0.186646	+	0.982427i	$0.440238\pi$
$788$	0	0
$789$	15.0000	0.534014
$790$	0	0
$791$	16.1803	0.575307
$792$	0	0
$793$	−0.944272	−0.0335321
$794$	0	0
$795$	−21.1803	−0.751189
$796$	0	0
$797$	−8.87539	−0.314382	−0.157191	−	0.987568i	$-0.550244\pi$
$797$	−8.87539	−0.314382	−0.157191	+	0.987568i	$0.550244\pi$
$798$	0	0
$799$	2.00000	0.0707549
$800$	0	0
$801$	12.9443	0.457363
$802$	0	0
$803$	−60.0000	−2.11735
$804$	0	0
$805$	10.0000	0.352454
$806$	0	0
$807$	13.8197	0.486475
$808$	0	0
$809$	−2.76393	−0.0971747	−0.0485873	−	0.998819i	$-0.515472\pi$
$809$	−2.76393	−0.0971747	−0.0485873	+	0.998819i	$0.515472\pi$
$810$	0	0
$811$	−22.5410	−0.791522	−0.395761	−	0.918353i	$-0.629519\pi$
$811$	−22.5410	−0.791522	−0.395761	+	0.918353i	$0.629519\pi$
$812$	0	0
$813$	33.9443	1.19048
$814$	0	0
$815$	20.9443	0.733646
$816$	0	0
$817$	−3.16718	−0.110806
$818$	0	0
$819$	14.4721	0.505697
$820$	0	0
$821$	34.6525	1.20938	0.604690	−	0.796461i	$-0.293297\pi$
$821$	34.6525	1.20938	0.604690	+	0.796461i	$0.293297\pi$
$822$	0	0
$823$	10.1803	0.354864	0.177432	−	0.984133i	$-0.443221\pi$
$823$	10.1803	0.354864	0.177432	+	0.984133i	$0.443221\pi$
$824$	0	0
$825$	−40.0000	−1.39262
$826$	0	0
$827$	−35.4164	−1.23155	−0.615775	−	0.787922i	$-0.711157\pi$
$827$	−35.4164	−1.23155	−0.615775	+	0.787922i	$0.711157\pi$
$828$	0	0
$829$	−3.00000	−0.104194	−0.0520972	−	0.998642i	$-0.516591\pi$
$829$	−3.00000	−0.104194	−0.0520972	+	0.998642i	$0.516591\pi$
$830$	0	0
$831$	−2.23607	−0.0775683
$832$	0	0
$833$	2.00000	0.0692959
$834$	0	0
$835$	9.29180	0.321556
$836$	0	0
$837$	−13.4164	−0.463739
$838$	0	0
$839$	−25.7082	−0.887546	−0.443773	−	0.896139i	$-0.646360\pi$
$839$	−25.7082	−0.887546	−0.443773	+	0.896139i	$0.646360\pi$
$840$	0	0
$841$	−28.0000	−0.965517
$842$	0	0
$843$	43.2918	1.49105
$844$	0	0
$845$	2.52786	0.0869612
$846$	0	0
$847$	−20.1246	−0.691490
$848$	0	0
$849$	32.7639	1.12446
$850$	0	0
$851$	−52.3607	−1.79490
$852$	0	0
$853$	−40.5279	−1.38765	−0.693824	−	0.720145i	$-0.744075\pi$
$853$	−40.5279	−1.38765	−0.693824	+	0.720145i	$0.744075\pi$
$854$	0	0
$855$	13.4164	0.458831
$856$	0	0
$857$	−52.0000	−1.77629	−0.888143	−	0.459567i	$-0.848005\pi$
$857$	−52.0000	−1.77629	−0.888143	+	0.459567i	$0.848005\pi$
$858$	0	0
$859$	8.47214	0.289066	0.144533	−	0.989500i	$-0.453832\pi$
$859$	8.47214	0.289066	0.144533	+	0.989500i	$0.453832\pi$
$860$	0	0
$861$	37.3607	1.27325
$862$	0	0
$863$	7.52786	0.256251	0.128126	−	0.991758i	$-0.459104\pi$
$863$	7.52786	0.256251	0.128126	+	0.991758i	$0.459104\pi$
$864$	0	0
$865$	−18.1803	−0.618150
$866$	0	0
$867$	2.23607	0.0759408
$868$	0	0
$869$	21.0557	0.714267
$870$	0	0
$871$	48.3607	1.63864
$872$	0	0
$873$	−15.4164	−0.521766
$874$	0	0
$875$	−20.1246	−0.680336
$876$	0	0
$877$	−27.0000	−0.911725	−0.455863	−	0.890050i	$-0.650669\pi$
$877$	−27.0000	−0.911725	−0.455863	+	0.890050i	$0.650669\pi$
$878$	0	0
$879$	27.7639	0.936454
$880$	0	0
$881$	52.6525	1.77391	0.886953	−	0.461859i	$-0.152817\pi$
$881$	52.6525	1.77391	0.886953	+	0.461859i	$0.152817\pi$
$882$	0	0
$883$	−14.8197	−0.498721	−0.249361	−	0.968411i	$-0.580220\pi$
$883$	−14.8197	−0.498721	−0.249361	+	0.968411i	$0.580220\pi$
$884$	0	0
$885$	−2.23607	−0.0751646
$886$	0	0
$887$	17.7771	0.596896	0.298448	−	0.954426i	$-0.403531\pi$
$887$	17.7771	0.596896	0.298448	+	0.954426i	$0.403531\pi$
$888$	0	0
$889$	−22.8885	−0.767657
$890$	0	0
$891$	−49.1935	−1.64804
$892$	0	0
$893$	13.4164	0.448963
$894$	0	0
$895$	−4.18034	−0.139733
$896$	0	0
$897$	−32.3607	−1.08049
$898$	0	0
$899$	6.00000	0.200111
$900$	0	0
$901$	−9.47214	−0.315563
$902$	0	0
$903$	−2.36068	−0.0785585
$904$	0	0
$905$	20.8885	0.694359
$906$	0	0
$907$	4.70820	0.156333	0.0781667	−	0.996940i	$-0.475093\pi$
$907$	4.70820	0.156333	0.0781667	+	0.996940i	$0.475093\pi$
$908$	0	0
$909$	−3.41641	−0.113315
$910$	0	0
$911$	−1.87539	−0.0621344	−0.0310672	−	0.999517i	$-0.509891\pi$
$911$	−1.87539	−0.0621344	−0.0310672	+	0.999517i	$0.509891\pi$
$912$	0	0
$913$	5.52786	0.182946
$914$	0	0
$915$	−0.652476	−0.0215702
$916$	0	0
$917$	20.0000	0.660458
$918$	0	0
$919$	40.0689	1.32175	0.660875	−	0.750496i	$-0.270185\pi$
$919$	40.0689	1.32175	0.660875	+	0.750496i	$0.270185\pi$
$920$	0	0
$921$	26.0557	0.858565
$922$	0	0
$923$	−8.00000	−0.263323
$924$	0	0
$925$	46.8328	1.53985
$926$	0	0
$927$	3.41641	0.112210
$928$	0	0
$929$	29.0557	0.953288	0.476644	−	0.879097i	$-0.341853\pi$
$929$	29.0557	0.953288	0.476644	+	0.879097i	$0.341853\pi$
$930$	0	0
$931$	13.4164	0.439705
$932$	0	0
$933$	41.8328	1.36954
$934$	0	0
$935$	4.47214	0.146254
$936$	0	0
$937$	−36.9443	−1.20692	−0.603458	−	0.797394i	$-0.706211\pi$
$937$	−36.9443	−1.20692	−0.603458	+	0.797394i	$0.706211\pi$
$938$	0	0
$939$	−58.5410	−1.91041
$940$	0	0
$941$	19.8197	0.646102	0.323051	−	0.946381i	$-0.395291\pi$
$941$	19.8197	0.646102	0.323051	+	0.946381i	$0.395291\pi$
$942$	0	0
$943$	−33.4164	−1.08819
$944$	0	0
$945$	−5.00000	−0.162650
$946$	0	0
$947$	−0.819660	−0.0266354	−0.0133177	−	0.999911i	$-0.504239\pi$
$947$	−0.819660	−0.0266354	−0.0133177	+	0.999911i	$0.504239\pi$
$948$	0	0
$949$	43.4164	1.40936
$950$	0	0
$951$	−27.8885	−0.904348
$952$	0	0
$953$	24.2492	0.785509	0.392755	−	0.919643i	$-0.371522\pi$
$953$	24.2492	0.785509	0.392755	+	0.919643i	$0.371522\pi$
$954$	0	0
$955$	12.9443	0.418867
$956$	0	0
$957$	10.0000	0.323254
$958$	0	0
$959$	38.0132	1.22751
$960$	0	0
$961$	5.00000	0.161290
$962$	0	0
$963$	20.4721	0.659705
$964$	0	0
$965$	−2.05573	−0.0661762
$966$	0	0
$967$	41.2361	1.32606	0.663031	−	0.748592i	$-0.269270\pi$
$967$	41.2361	1.32606	0.663031	+	0.748592i	$0.269270\pi$
$968$	0	0
$969$	15.0000	0.481869
$970$	0	0
$971$	−52.1246	−1.67276	−0.836379	−	0.548151i	$-0.815332\pi$
$971$	−52.1246	−1.67276	−0.836379	+	0.548151i	$0.815332\pi$
$972$	0	0
$973$	−37.8885	−1.21465
$974$	0	0
$975$	28.9443	0.926959
$976$	0	0
$977$	−35.4164	−1.13307	−0.566536	−	0.824037i	$-0.691717\pi$
$977$	−35.4164	−1.13307	−0.566536	+	0.824037i	$0.691717\pi$
$978$	0	0
$979$	28.9443	0.925063
$980$	0	0
$981$	−16.9443	−0.540989
$982$	0	0
$983$	−29.7771	−0.949742	−0.474871	−	0.880056i	$-0.657505\pi$
$983$	−29.7771	−0.949742	−0.474871	+	0.880056i	$0.657505\pi$
$984$	0	0
$985$	15.8885	0.506251
$986$	0	0
$987$	10.0000	0.318304
$988$	0	0
$989$	2.11146	0.0671404
$990$	0	0
$991$	−59.5967	−1.89315	−0.946577	−	0.322479i	$-0.895484\pi$
$991$	−59.5967	−1.89315	−0.946577	+	0.322479i	$0.895484\pi$
$992$	0	0
$993$	−8.16718	−0.259178
$994$	0	0
$995$	22.2361	0.704931
$996$	0	0
$997$	4.05573	0.128446	0.0642231	−	0.997936i	$-0.479543\pi$
$997$	4.05573	0.128446	0.0642231	+	0.997936i	$0.479543\pi$
$998$	0	0
$999$	26.1803	0.828309

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	4012.2.a.e.1.2	✓	2

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
4012.2.a.e.1.2	✓	2	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:	\( N \)	\(=\)	\( 4012 = 2^{2} \cdot 17 \cdot 59 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	4012.a (trivial)

\(f(q)\)	\(=\)	\(q+2.23607 q^{3} -1.00000 q^{5} -2.23607 q^{7} +2.00000 q^{9} +O(q^{10})\)	\(q+2.23607 q^{3} -1.00000 q^{5} -2.23607 q^{7} +2.00000 q^{9} +4.47214 q^{11} -3.23607 q^{13} -2.23607 q^{15} -1.00000 q^{17} -6.70820 q^{19} -5.00000 q^{21} +4.47214 q^{23} -4.00000 q^{25} -2.23607 q^{27} +1.00000 q^{29} +6.00000 q^{31} +10.0000 q^{33} +2.23607 q^{35} -11.7082 q^{37} -7.23607 q^{39} -7.47214 q^{41} +0.472136 q^{43} -2.00000 q^{45} -2.00000 q^{47} -2.00000 q^{49} -2.23607 q^{51} +9.47214 q^{53} -4.47214 q^{55} -15.0000 q^{57} +1.00000 q^{59} +0.291796 q^{61} -4.47214 q^{63} +3.23607 q^{65} -14.9443 q^{67} +10.0000 q^{69} +2.47214 q^{71} -13.4164 q^{73} -8.94427 q^{75} -10.0000 q^{77} +4.70820 q^{79} -11.0000 q^{81} +1.23607 q^{83} +1.00000 q^{85} +2.23607 q^{87} +6.47214 q^{89} +7.23607 q^{91} +13.4164 q^{93} +6.70820 q^{95} -7.70820 q^{97} +8.94427 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 2 q - 2 q^{5} + 4 q^{9}+O(q^{10}) \) 2 * q - 2 * q^5 + 4 * q^9	\( 2 q - 2 q^{5} + 4 q^{9} - 2 q^{13} - 2 q^{17} - 10 q^{21} - 8 q^{25} + 2 q^{29} + 12 q^{31} + 20 q^{33} - 10 q^{37} - 10 q^{39} - 6 q^{41} - 8 q^{43} - 4 q^{45} - 4 q^{47} - 4 q^{49} + 10 q^{53} - 30 q^{57} + 2 q^{59} + 14 q^{61} + 2 q^{65} - 12 q^{67} + 20 q^{69} - 4 q^{71} - 20 q^{77} - 4 q^{79} - 22 q^{81} - 2 q^{83} + 2 q^{85} + 4 q^{89} + 10 q^{91} - 2 q^{97}+O(q^{100}) \) 2 * q - 2 * q^5 + 4 * q^9 - 2 * q^13 - 2 * q^17 - 10 * q^21 - 8 * q^25 + 2 * q^29 + 12 * q^31 + 20 * q^33 - 10 * q^37 - 10 * q^39 - 6 * q^41 - 8 * q^43 - 4 * q^45 - 4 * q^47 - 4 * q^49 + 10 * q^53 - 30 * q^57 + 2 * q^59 + 14 * q^61 + 2 * q^65 - 12 * q^67 + 20 * q^69 - 4 * q^71 - 20 * q^77 - 4 * q^79 - 22 * q^81 - 2 * q^83 + 2 * q^85 + 4 * q^89 + 10 * q^91 - 2 * q^97

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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