LMFDB - Embedded newform 4012.2.a.d.1.1

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:	$\Z[a_1, \ldots, a_{5}]$
Coefficient ring index:	$ 1 $
Twist minimal:	yes
Fricke sign:	$1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.1
Root			$1.61803$ 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)

$f(q)$	$=$	$q-3.23607 q^{3} +0.381966 q^{5} -3.00000 q^{7} +7.47214 q^{9} +O(q^{10})$	$q-3.23607 q^{3} +0.381966 q^{5} -3.00000 q^{7} +7.47214 q^{9} +1.61803 q^{11} -3.61803 q^{13} -1.23607 q^{15} -1.00000 q^{17} +2.23607 q^{19} +9.70820 q^{21} +2.38197 q^{23} -4.85410 q^{25} -14.4721 q^{27} +5.85410 q^{29} -9.00000 q^{31} -5.23607 q^{33} -1.14590 q^{35} +3.00000 q^{37} +11.7082 q^{39} +9.00000 q^{41} -1.38197 q^{43} +2.85410 q^{45} +3.85410 q^{47} +2.00000 q^{49} +3.23607 q^{51} +4.70820 q^{53} +0.618034 q^{55} -7.23607 q^{57} +1.00000 q^{59} +7.47214 q^{61} -22.4164 q^{63} -1.38197 q^{65} -10.7082 q^{67} -7.70820 q^{69} +13.0902 q^{71} +9.18034 q^{73} +15.7082 q^{75} -4.85410 q^{77} +0.0901699 q^{79} +24.4164 q^{81} -5.47214 q^{83} -0.381966 q^{85} -18.9443 q^{87} +12.3262 q^{89} +10.8541 q^{91} +29.1246 q^{93} +0.854102 q^{95} -4.90983 q^{97} +12.0902 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 2 q - 2 q^{3} + 3 q^{5} - 6 q^{7} + 6 q^{9}+O(q^{10}) $ 2 * q - 2 * q^3 + 3 * q^5 - 6 * q^7 + 6 * q^9	$ 2 q - 2 q^{3} + 3 q^{5} - 6 q^{7} + 6 q^{9} + q^{11} - 5 q^{13} + 2 q^{15} - 2 q^{17} + 6 q^{21} + 7 q^{23} - 3 q^{25} - 20 q^{27} + 5 q^{29} - 18 q^{31} - 6 q^{33} - 9 q^{35} + 6 q^{37} + 10 q^{39} + 18 q^{41} - 5 q^{43} - q^{45} + q^{47} + 4 q^{49} + 2 q^{51} - 4 q^{53} - q^{55} - 10 q^{57} + 2 q^{59} + 6 q^{61} - 18 q^{63} - 5 q^{65} - 8 q^{67} - 2 q^{69} + 15 q^{71} - 4 q^{73} + 18 q^{75} - 3 q^{77} - 11 q^{79} + 22 q^{81} - 2 q^{83} - 3 q^{85} - 20 q^{87} + 9 q^{89} + 15 q^{91} + 18 q^{93} - 5 q^{95} - 21 q^{97} + 13 q^{99}+O(q^{100}) $ 2 * q - 2 * q^3 + 3 * q^5 - 6 * q^7 + 6 * q^9 + q^11 - 5 * q^13 + 2 * q^15 - 2 * q^17 + 6 * q^21 + 7 * q^23 - 3 * q^25 - 20 * q^27 + 5 * q^29 - 18 * q^31 - 6 * q^33 - 9 * q^35 + 6 * q^37 + 10 * q^39 + 18 * q^41 - 5 * q^43 - q^45 + q^47 + 4 * q^49 + 2 * q^51 - 4 * q^53 - q^55 - 10 * q^57 + 2 * q^59 + 6 * q^61 - 18 * q^63 - 5 * q^65 - 8 * q^67 - 2 * q^69 + 15 * q^71 - 4 * q^73 + 18 * q^75 - 3 * q^77 - 11 * q^79 + 22 * q^81 - 2 * q^83 - 3 * q^85 - 20 * q^87 + 9 * q^89 + 15 * q^91 + 18 * q^93 - 5 * q^95 - 21 * q^97 + 13 * q^99

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
$3$	−3.23607	−1.86834	−0.934172	−	0.356822i	$-0.883860\pi$
$3$	−3.23607	−1.86834	−0.934172	+	0.356822i	$0.883860\pi$
$4$	0	0
$5$	0.381966	0.170820	0.0854102	−	0.996346i	$-0.472780\pi$
$5$	0.381966	0.170820	0.0854102	+	0.996346i	$0.472780\pi$
$6$	0	0
$7$	−3.00000	−1.13389	−0.566947	−	0.823754i	$-0.691875\pi$
$7$	−3.00000	−1.13389	−0.566947	+	0.823754i	$0.691875\pi$
$8$	0	0
$9$	7.47214	2.49071
$10$	0	0
$11$	1.61803	0.487856	0.243928	−	0.969793i	$-0.421564\pi$
$11$	1.61803	0.487856	0.243928	+	0.969793i	$0.421564\pi$
$12$	0	0
$13$	−3.61803	−1.00346	−0.501731	−	0.865024i	$-0.667303\pi$
$13$	−3.61803	−1.00346	−0.501731	+	0.865024i	$0.667303\pi$
$14$	0	0
$15$	−1.23607	−0.319151
$16$	0	0
$17$	−1.00000	−0.242536
$17$	−1.00000	−0.242536
$18$	0	0
$19$	2.23607	0.512989	0.256495	−	0.966546i	$-0.417432\pi$
$19$	2.23607	0.512989	0.256495	+	0.966546i	$0.417432\pi$
$20$	0	0
$21$	9.70820	2.11850
$22$	0	0
$23$	2.38197	0.496674	0.248337	−	0.968674i	$-0.420116\pi$
$23$	2.38197	0.496674	0.248337	+	0.968674i	$0.420116\pi$
$24$	0	0
$25$	−4.85410	−0.970820
$26$	0	0
$27$	−14.4721	−2.78516
$28$	0	0
$29$	5.85410	1.08708	0.543540	−	0.839383i	$-0.317084\pi$
$29$	5.85410	1.08708	0.543540	+	0.839383i	$0.317084\pi$
$30$	0	0
$31$	−9.00000	−1.61645	−0.808224	−	0.588875i	$-0.799571\pi$
$31$	−9.00000	−1.61645	−0.808224	+	0.588875i	$0.799571\pi$
$32$	0	0
$33$	−5.23607	−0.911482
$34$	0	0
$35$	−1.14590	−0.193692
$36$	0	0
$37$	3.00000	0.493197	0.246598	−	0.969118i	$-0.420687\pi$
$37$	3.00000	0.493197	0.246598	+	0.969118i	$0.420687\pi$
$38$	0	0
$39$	11.7082	1.87481
$40$	0	0
$41$	9.00000	1.40556	0.702782	−	0.711405i	$-0.251941\pi$
$41$	9.00000	1.40556	0.702782	+	0.711405i	$0.251941\pi$
$42$	0	0
$43$	−1.38197	−0.210748	−0.105374	−	0.994433i	$-0.533604\pi$
$43$	−1.38197	−0.210748	−0.105374	+	0.994433i	$0.533604\pi$
$44$	0	0
$45$	2.85410	0.425464
$46$	0	0
$47$	3.85410	0.562179	0.281089	−	0.959682i	$-0.409304\pi$
$47$	3.85410	0.562179	0.281089	+	0.959682i	$0.409304\pi$
$48$	0	0
$49$	2.00000	0.285714
$50$	0	0
$51$	3.23607	0.453140
$52$	0	0
$53$	4.70820	0.646722	0.323361	−	0.946276i	$-0.395187\pi$
$53$	4.70820	0.646722	0.323361	+	0.946276i	$0.395187\pi$
$54$	0	0
$55$	0.618034	0.0833357
$56$	0	0
$57$	−7.23607	−0.958441
$58$	0	0
$59$	1.00000	0.130189
$59$	1.00000	0.130189
$60$	0	0
$61$	7.47214	0.956709	0.478354	−	0.878167i	$-0.341233\pi$
$61$	7.47214	0.956709	0.478354	+	0.878167i	$0.341233\pi$
$62$	0	0
$63$	−22.4164	−2.82420
$64$	0	0
$65$	−1.38197	−0.171412
$66$	0	0
$67$	−10.7082	−1.30822	−0.654108	−	0.756401i	$-0.726956\pi$
$67$	−10.7082	−1.30822	−0.654108	+	0.756401i	$0.726956\pi$
$68$	0	0
$69$	−7.70820	−0.927959
$70$	0	0
$71$	13.0902	1.55352	0.776759	−	0.629798i	$-0.216862\pi$
$71$	13.0902	1.55352	0.776759	+	0.629798i	$0.216862\pi$
$72$	0	0
$73$	9.18034	1.07448	0.537239	−	0.843430i	$-0.319467\pi$
$73$	9.18034	1.07448	0.537239	+	0.843430i	$0.319467\pi$
$74$	0	0
$75$	15.7082	1.81383
$76$	0	0
$77$	−4.85410	−0.553176
$78$	0	0
$79$	0.0901699	0.0101449	0.00507246	−	0.999987i	$-0.498385\pi$
$79$	0.0901699	0.0101449	0.00507246	+	0.999987i	$0.498385\pi$
$80$	0	0
$81$	24.4164	2.71293
$82$	0	0
$83$	−5.47214	−0.600645	−0.300322	−	0.953838i	$-0.597094\pi$
$83$	−5.47214	−0.600645	−0.300322	+	0.953838i	$0.597094\pi$
$84$	0	0
$85$	−0.381966	−0.0414300
$86$	0	0
$87$	−18.9443	−2.03104
$88$	0	0
$89$	12.3262	1.30658	0.653289	−	0.757108i	$-0.273389\pi$
$89$	12.3262	1.30658	0.653289	+	0.757108i	$0.273389\pi$
$90$	0	0
$91$	10.8541	1.13782
$92$	0	0
$93$	29.1246	3.02008
$94$	0	0
$95$	0.854102	0.0876290
$96$	0	0
$97$	−4.90983	−0.498518	−0.249259	−	0.968437i	$-0.580187\pi$
$97$	−4.90983	−0.498518	−0.249259	+	0.968437i	$0.580187\pi$
$98$	0	0
$99$	12.0902	1.21511
$100$	0	0
$101$	−7.94427	−0.790485	−0.395242	−	0.918577i	$-0.629339\pi$
$101$	−7.94427	−0.790485	−0.395242	+	0.918577i	$0.629339\pi$
$102$	0	0
$103$	−9.52786	−0.938808	−0.469404	−	0.882983i	$-0.655531\pi$
$103$	−9.52786	−0.938808	−0.469404	+	0.882983i	$0.655531\pi$
$104$	0	0
$105$	3.70820	0.361884
$106$	0	0
$107$	2.56231	0.247707	0.123854	−	0.992300i	$-0.460475\pi$
$107$	2.56231	0.247707	0.123854	+	0.992300i	$0.460475\pi$
$108$	0	0
$109$	−5.70820	−0.546747	−0.273373	−	0.961908i	$-0.588139\pi$
$109$	−5.70820	−0.546747	−0.273373	+	0.961908i	$0.588139\pi$
$110$	0	0
$111$	−9.70820	−0.921462
$112$	0	0
$113$	16.7082	1.57178	0.785888	−	0.618369i	$-0.212206\pi$
$113$	16.7082	1.57178	0.785888	+	0.618369i	$0.212206\pi$
$114$	0	0
$115$	0.909830	0.0848421
$116$	0	0
$117$	−27.0344	−2.49934
$118$	0	0
$119$	3.00000	0.275010
$120$	0	0
$121$	−8.38197	−0.761997
$122$	0	0
$123$	−29.1246	−2.62608
$124$	0	0
$125$	−3.76393	−0.336656
$126$	0	0
$127$	−9.47214	−0.840516	−0.420258	−	0.907405i	$-0.638060\pi$
$127$	−9.47214	−0.840516	−0.420258	+	0.907405i	$0.638060\pi$
$128$	0	0
$129$	4.47214	0.393750
$130$	0	0
$131$	−21.7082	−1.89665	−0.948327	−	0.317294i	$-0.897226\pi$
$131$	−21.7082	−1.89665	−0.948327	+	0.317294i	$0.897226\pi$
$132$	0	0
$133$	−6.70820	−0.581675
$134$	0	0
$135$	−5.52786	−0.475763
$136$	0	0
$137$	−3.18034	−0.271715	−0.135857	−	0.990728i	$-0.543379\pi$
$137$	−3.18034	−0.271715	−0.135857	+	0.990728i	$0.543379\pi$
$138$	0	0
$139$	11.0000	0.933008	0.466504	−	0.884519i	$-0.345513\pi$
$139$	11.0000	0.933008	0.466504	+	0.884519i	$0.345513\pi$
$140$	0	0
$141$	−12.4721	−1.05034
$142$	0	0
$143$	−5.85410	−0.489545
$144$	0	0
$145$	2.23607	0.185695
$146$	0	0
$147$	−6.47214	−0.533813
$148$	0	0
$149$	14.7082	1.20494	0.602472	−	0.798140i	$-0.294183\pi$
$149$	14.7082	1.20494	0.602472	+	0.798140i	$0.294183\pi$
$150$	0	0
$151$	−9.38197	−0.763494	−0.381747	−	0.924267i	$-0.624677\pi$
$151$	−9.38197	−0.763494	−0.381747	+	0.924267i	$0.624677\pi$
$152$	0	0
$153$	−7.47214	−0.604086
$154$	0	0
$155$	−3.43769	−0.276122
$156$	0	0
$157$	−8.85410	−0.706634	−0.353317	−	0.935504i	$-0.614946\pi$
$157$	−8.85410	−0.706634	−0.353317	+	0.935504i	$0.614946\pi$
$158$	0	0
$159$	−15.2361	−1.20830
$160$	0	0
$161$	−7.14590	−0.563176
$162$	0	0
$163$	19.5623	1.53224	0.766119	−	0.642699i	$-0.222185\pi$
$163$	19.5623	1.53224	0.766119	+	0.642699i	$0.222185\pi$
$164$	0	0
$165$	−2.00000	−0.155700
$166$	0	0
$167$	4.85410	0.375622	0.187811	−	0.982205i	$-0.439861\pi$
$167$	4.85410	0.375622	0.187811	+	0.982205i	$0.439861\pi$
$168$	0	0
$169$	0.0901699	0.00693615
$170$	0	0
$171$	16.7082	1.27771
$172$	0	0
$173$	5.29180	0.402328	0.201164	−	0.979558i	$-0.435528\pi$
$173$	5.29180	0.402328	0.201164	+	0.979558i	$0.435528\pi$
$174$	0	0
$175$	14.5623	1.10081
$176$	0	0
$177$	−3.23607	−0.243238
$178$	0	0
$179$	−17.5066	−1.30850	−0.654252	−	0.756277i	$-0.727016\pi$
$179$	−17.5066	−1.30850	−0.654252	+	0.756277i	$0.727016\pi$
$180$	0	0
$181$	6.47214	0.481070	0.240535	−	0.970640i	$-0.422677\pi$
$181$	6.47214	0.481070	0.240535	+	0.970640i	$0.422677\pi$
$182$	0	0
$183$	−24.1803	−1.78746
$184$	0	0
$185$	1.14590	0.0842481
$186$	0	0
$187$	−1.61803	−0.118322
$188$	0	0
$189$	43.4164	3.15808
$190$	0	0
$191$	9.52786	0.689412	0.344706	−	0.938711i	$-0.387979\pi$
$191$	9.52786	0.689412	0.344706	+	0.938711i	$0.387979\pi$
$192$	0	0
$193$	−22.1246	−1.59256	−0.796282	−	0.604925i	$-0.793203\pi$
$193$	−22.1246	−1.59256	−0.796282	+	0.604925i	$0.793203\pi$
$194$	0	0
$195$	4.47214	0.320256
$196$	0	0
$197$	2.38197	0.169708	0.0848540	−	0.996393i	$-0.472958\pi$
$197$	2.38197	0.169708	0.0848540	+	0.996393i	$0.472958\pi$
$198$	0	0
$199$	−20.9443	−1.48470	−0.742350	−	0.670012i	$-0.766289\pi$
$199$	−20.9443	−1.48470	−0.742350	+	0.670012i	$0.766289\pi$
$200$	0	0
$201$	34.6525	2.44420
$202$	0	0
$203$	−17.5623	−1.23263
$204$	0	0
$205$	3.43769	0.240099
$206$	0	0
$207$	17.7984	1.23707
$208$	0	0
$209$	3.61803	0.250265
$210$	0	0
$211$	8.41641	0.579409	0.289705	−	0.957116i	$-0.406443\pi$
$211$	8.41641	0.579409	0.289705	+	0.957116i	$0.406443\pi$
$212$	0	0
$213$	−42.3607	−2.90251
$214$	0	0
$215$	−0.527864	−0.0360000
$216$	0	0
$217$	27.0000	1.83288
$218$	0	0
$219$	−29.7082	−2.00749
$220$	0	0
$221$	3.61803	0.243375
$222$	0	0
$223$	−21.8541	−1.46346	−0.731729	−	0.681595i	$-0.761287\pi$
$223$	−21.8541	−1.46346	−0.731729	+	0.681595i	$0.761287\pi$
$224$	0	0
$225$	−36.2705	−2.41803
$226$	0	0
$227$	−1.67376	−0.111091	−0.0555457	−	0.998456i	$-0.517690\pi$
$227$	−1.67376	−0.111091	−0.0555457	+	0.998456i	$0.517690\pi$
$228$	0	0
$229$	2.32624	0.153722	0.0768611	−	0.997042i	$-0.475510\pi$
$229$	2.32624	0.153722	0.0768611	+	0.997042i	$0.475510\pi$
$230$	0	0
$231$	15.7082	1.03352
$232$	0	0
$233$	−13.1246	−0.859822	−0.429911	−	0.902871i	$-0.641455\pi$
$233$	−13.1246	−0.859822	−0.429911	+	0.902871i	$0.641455\pi$
$234$	0	0
$235$	1.47214	0.0960316
$236$	0	0
$237$	−0.291796	−0.0189542
$238$	0	0
$239$	−18.7426	−1.21236	−0.606180	−	0.795327i	$-0.707299\pi$
$239$	−18.7426	−1.21236	−0.606180	+	0.795327i	$0.707299\pi$
$240$	0	0
$241$	−20.7426	−1.33615	−0.668076	−	0.744093i	$-0.732882\pi$
$241$	−20.7426	−1.33615	−0.668076	+	0.744093i	$0.732882\pi$
$242$	0	0
$243$	−35.5967	−2.28353
$244$	0	0
$245$	0.763932	0.0488058
$246$	0	0
$247$	−8.09017	−0.514765
$248$	0	0
$249$	17.7082	1.12221
$250$	0	0
$251$	8.27051	0.522030	0.261015	−	0.965335i	$-0.415943\pi$
$251$	8.27051	0.522030	0.261015	+	0.965335i	$0.415943\pi$
$252$	0	0
$253$	3.85410	0.242305
$254$	0	0
$255$	1.23607	0.0774056
$256$	0	0
$257$	−22.5967	−1.40955	−0.704773	−	0.709433i	$-0.748951\pi$
$257$	−22.5967	−1.40955	−0.704773	+	0.709433i	$0.748951\pi$
$258$	0	0
$259$	−9.00000	−0.559233
$260$	0	0
$261$	43.7426	2.70760
$262$	0	0
$263$	10.9098	0.672729	0.336364	−	0.941732i	$-0.390803\pi$
$263$	10.9098	0.672729	0.336364	+	0.941732i	$0.390803\pi$
$264$	0	0
$265$	1.79837	0.110473
$266$	0	0
$267$	−39.8885	−2.44114
$268$	0	0
$269$	−5.38197	−0.328144	−0.164072	−	0.986448i	$-0.552463\pi$
$269$	−5.38197	−0.328144	−0.164072	+	0.986448i	$0.552463\pi$
$270$	0	0
$271$	−21.6525	−1.31529	−0.657647	−	0.753326i	$-0.728448\pi$
$271$	−21.6525	−1.31529	−0.657647	+	0.753326i	$0.728448\pi$
$272$	0	0
$273$	−35.1246	−2.12584
$274$	0	0
$275$	−7.85410	−0.473620
$276$	0	0
$277$	−1.20163	−0.0721987	−0.0360994	−	0.999348i	$-0.511493\pi$
$277$	−1.20163	−0.0721987	−0.0360994	+	0.999348i	$0.511493\pi$
$278$	0	0
$279$	−67.2492	−4.02611
$280$	0	0
$281$	−9.85410	−0.587846	−0.293923	−	0.955829i	$-0.594961\pi$
$281$	−9.85410	−0.587846	−0.293923	+	0.955829i	$0.594961\pi$
$282$	0	0
$283$	−22.0344	−1.30981	−0.654906	−	0.755711i	$-0.727292\pi$
$283$	−22.0344	−1.30981	−0.654906	+	0.755711i	$0.727292\pi$
$284$	0	0
$285$	−2.76393	−0.163721
$286$	0	0
$287$	−27.0000	−1.59376
$288$	0	0
$289$	1.00000	0.0588235
$290$	0	0
$291$	15.8885	0.931403
$292$	0	0
$293$	−13.0344	−0.761480	−0.380740	−	0.924682i	$-0.624331\pi$
$293$	−13.0344	−0.761480	−0.380740	+	0.924682i	$0.624331\pi$
$294$	0	0
$295$	0.381966	0.0222389
$296$	0	0
$297$	−23.4164	−1.35876
$298$	0	0
$299$	−8.61803	−0.498394
$300$	0	0
$301$	4.14590	0.238966
$302$	0	0
$303$	25.7082	1.47690
$304$	0	0
$305$	2.85410	0.163425
$306$	0	0
$307$	27.6525	1.57821	0.789105	−	0.614258i	$-0.210545\pi$
$307$	27.6525	1.57821	0.789105	+	0.614258i	$0.210545\pi$
$308$	0	0
$309$	30.8328	1.75402
$310$	0	0
$311$	−29.0000	−1.64444	−0.822220	−	0.569170i	$-0.807264\pi$
$311$	−29.0000	−1.64444	−0.822220	+	0.569170i	$0.807264\pi$
$312$	0	0
$313$	−30.2705	−1.71099	−0.855495	−	0.517811i	$-0.826747\pi$
$313$	−30.2705	−1.71099	−0.855495	+	0.517811i	$0.826747\pi$
$314$	0	0
$315$	−8.56231	−0.482431
$316$	0	0
$317$	29.3262	1.64713	0.823563	−	0.567225i	$-0.191983\pi$
$317$	29.3262	1.64713	0.823563	+	0.567225i	$0.191983\pi$
$318$	0	0
$319$	9.47214	0.530338
$320$	0	0
$321$	−8.29180	−0.462803
$322$	0	0
$323$	−2.23607	−0.124418
$324$	0	0
$325$	17.5623	0.974181
$326$	0	0
$327$	18.4721	1.02151
$328$	0	0
$329$	−11.5623	−0.637451
$330$	0	0
$331$	2.90983	0.159939	0.0799694	−	0.996797i	$-0.474518\pi$
$331$	2.90983	0.159939	0.0799694	+	0.996797i	$0.474518\pi$
$332$	0	0
$333$	22.4164	1.22841
$334$	0	0
$335$	−4.09017	−0.223470
$336$	0	0
$337$	2.67376	0.145649	0.0728246	−	0.997345i	$-0.476799\pi$
$337$	2.67376	0.145649	0.0728246	+	0.997345i	$0.476799\pi$
$338$	0	0
$339$	−54.0689	−2.93662
$340$	0	0
$341$	−14.5623	−0.788593
$342$	0	0
$343$	15.0000	0.809924
$344$	0	0
$345$	−2.94427	−0.158514
$346$	0	0
$347$	−24.7639	−1.32940	−0.664699	−	0.747111i	$-0.731440\pi$
$347$	−24.7639	−1.32940	−0.664699	+	0.747111i	$0.731440\pi$
$348$	0	0
$349$	2.47214	0.132330	0.0661652	−	0.997809i	$-0.478924\pi$
$349$	2.47214	0.132330	0.0661652	+	0.997809i	$0.478924\pi$
$350$	0	0
$351$	52.3607	2.79481
$352$	0	0
$353$	−1.61803	−0.0861193	−0.0430596	−	0.999073i	$-0.513711\pi$
$353$	−1.61803	−0.0861193	−0.0430596	+	0.999073i	$0.513711\pi$
$354$	0	0
$355$	5.00000	0.265372
$356$	0	0
$357$	−9.70820	−0.513813
$358$	0	0
$359$	6.76393	0.356987	0.178493	−	0.983941i	$-0.442878\pi$
$359$	6.76393	0.356987	0.178493	+	0.983941i	$0.442878\pi$
$360$	0	0
$361$	−14.0000	−0.736842
$362$	0	0
$363$	27.1246	1.42367
$364$	0	0
$365$	3.50658	0.183543
$366$	0	0
$367$	−23.5066	−1.22703	−0.613517	−	0.789682i	$-0.710246\pi$
$367$	−23.5066	−1.22703	−0.613517	+	0.789682i	$0.710246\pi$
$368$	0	0
$369$	67.2492	3.50085
$370$	0	0
$371$	−14.1246	−0.733313
$372$	0	0
$373$	−2.23607	−0.115779	−0.0578896	−	0.998323i	$-0.518437\pi$
$373$	−2.23607	−0.115779	−0.0578896	+	0.998323i	$0.518437\pi$
$374$	0	0
$375$	12.1803	0.628990
$376$	0	0
$377$	−21.1803	−1.09084
$378$	0	0
$379$	−15.7984	−0.811508	−0.405754	−	0.913982i	$-0.632991\pi$
$379$	−15.7984	−0.811508	−0.405754	+	0.913982i	$0.632991\pi$
$380$	0	0
$381$	30.6525	1.57037
$382$	0	0
$383$	13.4164	0.685546	0.342773	−	0.939418i	$-0.388634\pi$
$383$	13.4164	0.685546	0.342773	+	0.939418i	$0.388634\pi$
$384$	0	0
$385$	−1.85410	−0.0944938
$386$	0	0
$387$	−10.3262	−0.524912
$388$	0	0
$389$	−10.1803	−0.516164	−0.258082	−	0.966123i	$-0.583090\pi$
$389$	−10.1803	−0.516164	−0.258082	+	0.966123i	$0.583090\pi$
$390$	0	0
$391$	−2.38197	−0.120461
$392$	0	0
$393$	70.2492	3.54360
$394$	0	0
$395$	0.0344419	0.00173296
$396$	0	0
$397$	−5.14590	−0.258265	−0.129133	−	0.991627i	$-0.541219\pi$
$397$	−5.14590	−0.258265	−0.129133	+	0.991627i	$0.541219\pi$
$398$	0	0
$399$	21.7082	1.08677
$400$	0	0
$401$	−5.18034	−0.258694	−0.129347	−	0.991599i	$-0.541288\pi$
$401$	−5.18034	−0.258694	−0.129347	+	0.991599i	$0.541288\pi$
$402$	0	0
$403$	32.5623	1.62204
$404$	0	0
$405$	9.32624	0.463424
$406$	0	0
$407$	4.85410	0.240609
$408$	0	0
$409$	−31.1246	−1.53901	−0.769507	−	0.638639i	$-0.779498\pi$
$409$	−31.1246	−1.53901	−0.769507	+	0.638639i	$0.779498\pi$
$410$	0	0
$411$	10.2918	0.507657
$412$	0	0
$413$	−3.00000	−0.147620
$414$	0	0
$415$	−2.09017	−0.102602
$416$	0	0
$417$	−35.5967	−1.74318
$418$	0	0
$419$	37.5967	1.83672	0.918361	−	0.395744i	$-0.129513\pi$
$419$	37.5967	1.83672	0.918361	+	0.395744i	$0.129513\pi$
$420$	0	0
$421$	13.4164	0.653876	0.326938	−	0.945046i	$-0.393983\pi$
$421$	13.4164	0.653876	0.326938	+	0.945046i	$0.393983\pi$
$422$	0	0
$423$	28.7984	1.40022
$424$	0	0
$425$	4.85410	0.235459
$426$	0	0
$427$	−22.4164	−1.08481
$428$	0	0
$429$	18.9443	0.914638
$430$	0	0
$431$	36.1246	1.74006	0.870031	−	0.492998i	$-0.164099\pi$
$431$	36.1246	1.74006	0.870031	+	0.492998i	$0.164099\pi$
$432$	0	0
$433$	−22.1459	−1.06426	−0.532132	−	0.846661i	$-0.678609\pi$
$433$	−22.1459	−1.06426	−0.532132	+	0.846661i	$0.678609\pi$
$434$	0	0
$435$	−7.23607	−0.346943
$436$	0	0
$437$	5.32624	0.254789
$438$	0	0
$439$	−1.61803	−0.0772245	−0.0386123	−	0.999254i	$-0.512294\pi$
$439$	−1.61803	−0.0772245	−0.0386123	+	0.999254i	$0.512294\pi$
$440$	0	0
$441$	14.9443	0.711632
$442$	0	0
$443$	−25.8328	−1.22735	−0.613677	−	0.789557i	$-0.710310\pi$
$443$	−25.8328	−1.22735	−0.613677	+	0.789557i	$0.710310\pi$
$444$	0	0
$445$	4.70820	0.223190
$446$	0	0
$447$	−47.5967	−2.25125
$448$	0	0
$449$	−17.4721	−0.824561	−0.412281	−	0.911057i	$-0.635268\pi$
$449$	−17.4721	−0.824561	−0.412281	+	0.911057i	$0.635268\pi$
$450$	0	0
$451$	14.5623	0.685712
$452$	0	0
$453$	30.3607	1.42647
$454$	0	0
$455$	4.14590	0.194363
$456$	0	0
$457$	12.1246	0.567165	0.283583	−	0.958948i	$-0.408477\pi$
$457$	12.1246	0.567165	0.283583	+	0.958948i	$0.408477\pi$
$458$	0	0
$459$	14.4721	0.675501
$460$	0	0
$461$	38.3050	1.78404	0.892020	−	0.451996i	$-0.149288\pi$
$461$	38.3050	1.78404	0.892020	+	0.451996i	$0.149288\pi$
$462$	0	0
$463$	27.6525	1.28512	0.642560	−	0.766236i	$-0.277872\pi$
$463$	27.6525	1.28512	0.642560	+	0.766236i	$0.277872\pi$
$464$	0	0
$465$	11.1246	0.515892
$466$	0	0
$467$	34.9230	1.61604	0.808022	−	0.589153i	$-0.200538\pi$
$467$	34.9230	1.61604	0.808022	+	0.589153i	$0.200538\pi$
$468$	0	0
$469$	32.1246	1.48338
$470$	0	0
$471$	28.6525	1.32024
$472$	0	0
$473$	−2.23607	−0.102815
$474$	0	0
$475$	−10.8541	−0.498020
$476$	0	0
$477$	35.1803	1.61080
$478$	0	0
$479$	−15.3607	−0.701847	−0.350924	−	0.936404i	$-0.614132\pi$
$479$	−15.3607	−0.701847	−0.350924	+	0.936404i	$0.614132\pi$
$480$	0	0
$481$	−10.8541	−0.494904
$482$	0	0
$483$	23.1246	1.05221
$484$	0	0
$485$	−1.87539	−0.0851570
$486$	0	0
$487$	10.4164	0.472012	0.236006	−	0.971752i	$-0.424161\pi$
$487$	10.4164	0.472012	0.236006	+	0.971752i	$0.424161\pi$
$488$	0	0
$489$	−63.3050	−2.86275
$490$	0	0
$491$	20.8328	0.940172	0.470086	−	0.882621i	$-0.344223\pi$
$491$	20.8328	0.940172	0.470086	+	0.882621i	$0.344223\pi$
$492$	0	0
$493$	−5.85410	−0.263655
$494$	0	0
$495$	4.61803	0.207565
$496$	0	0
$497$	−39.2705	−1.76152
$498$	0	0
$499$	−11.2361	−0.502995	−0.251498	−	0.967858i	$-0.580923\pi$
$499$	−11.2361	−0.502995	−0.251498	+	0.967858i	$0.580923\pi$
$500$	0	0
$501$	−15.7082	−0.701791
$502$	0	0
$503$	10.6180	0.473435	0.236717	−	0.971579i	$-0.423928\pi$
$503$	10.6180	0.473435	0.236717	+	0.971579i	$0.423928\pi$
$504$	0	0
$505$	−3.03444	−0.135031
$506$	0	0
$507$	−0.291796	−0.0129591
$508$	0	0
$509$	−8.47214	−0.375521	−0.187760	−	0.982215i	$-0.560123\pi$
$509$	−8.47214	−0.375521	−0.187760	+	0.982215i	$0.560123\pi$
$510$	0	0
$511$	−27.5410	−1.21834
$512$	0	0
$513$	−32.3607	−1.42876
$514$	0	0
$515$	−3.63932	−0.160368
$516$	0	0
$517$	6.23607	0.274262
$518$	0	0
$519$	−17.1246	−0.751687
$520$	0	0
$521$	−13.0557	−0.571982	−0.285991	−	0.958232i	$-0.592323\pi$
$521$	−13.0557	−0.571982	−0.285991	+	0.958232i	$0.592323\pi$
$522$	0	0
$523$	−2.85410	−0.124801	−0.0624006	−	0.998051i	$-0.519876\pi$
$523$	−2.85410	−0.124801	−0.0624006	+	0.998051i	$0.519876\pi$
$524$	0	0
$525$	−47.1246	−2.05669
$526$	0	0
$527$	9.00000	0.392046
$528$	0	0
$529$	−17.3262	−0.753315
$530$	0	0
$531$	7.47214	0.324263
$532$	0	0
$533$	−32.5623	−1.41043
$534$	0	0
$535$	0.978714	0.0423135
$536$	0	0
$537$	56.6525	2.44473
$538$	0	0
$539$	3.23607	0.139387
$540$	0	0
$541$	−11.2705	−0.484557	−0.242279	−	0.970207i	$-0.577895\pi$
$541$	−11.2705	−0.484557	−0.242279	+	0.970207i	$0.577895\pi$
$542$	0	0
$543$	−20.9443	−0.898805
$544$	0	0
$545$	−2.18034	−0.0933955
$546$	0	0
$547$	5.18034	0.221495	0.110748	−	0.993849i	$-0.464675\pi$
$547$	5.18034	0.221495	0.110748	+	0.993849i	$0.464675\pi$
$548$	0	0
$549$	55.8328	2.38289
$550$	0	0
$551$	13.0902	0.557660
$552$	0	0
$553$	−0.270510	−0.0115032
$554$	0	0
$555$	−3.70820	−0.157404
$556$	0	0
$557$	−45.3951	−1.92345	−0.961727	−	0.274011i	$-0.911649\pi$
$557$	−45.3951	−1.92345	−0.961727	+	0.274011i	$0.911649\pi$
$558$	0	0
$559$	5.00000	0.211477
$560$	0	0
$561$	5.23607	0.221067
$562$	0	0
$563$	17.0000	0.716465	0.358232	−	0.933632i	$-0.383380\pi$
$563$	17.0000	0.716465	0.358232	+	0.933632i	$0.383380\pi$
$564$	0	0
$565$	6.38197	0.268491
$566$	0	0
$567$	−73.2492	−3.07618
$568$	0	0
$569$	−20.5967	−0.863461	−0.431730	−	0.902003i	$-0.642097\pi$
$569$	−20.5967	−0.863461	−0.431730	+	0.902003i	$0.642097\pi$
$570$	0	0
$571$	−10.8541	−0.454230	−0.227115	−	0.973868i	$-0.572929\pi$
$571$	−10.8541	−0.454230	−0.227115	+	0.973868i	$0.572929\pi$
$572$	0	0
$573$	−30.8328	−1.28806
$574$	0	0
$575$	−11.5623	−0.482181
$576$	0	0
$577$	−32.0000	−1.33218	−0.666089	−	0.745873i	$-0.732033\pi$
$577$	−32.0000	−1.33218	−0.666089	+	0.745873i	$0.732033\pi$
$578$	0	0
$579$	71.5967	2.97546
$580$	0	0
$581$	16.4164	0.681067
$582$	0	0
$583$	7.61803	0.315507
$584$	0	0
$585$	−10.3262	−0.426937
$586$	0	0
$587$	−9.00000	−0.371470	−0.185735	−	0.982600i	$-0.559467\pi$
$587$	−9.00000	−0.371470	−0.185735	+	0.982600i	$0.559467\pi$
$588$	0	0
$589$	−20.1246	−0.829220
$590$	0	0
$591$	−7.70820	−0.317073
$592$	0	0
$593$	−38.3607	−1.57528	−0.787642	−	0.616133i	$-0.788698\pi$
$593$	−38.3607	−1.57528	−0.787642	+	0.616133i	$0.788698\pi$
$594$	0	0
$595$	1.14590	0.0469772
$596$	0	0
$597$	67.7771	2.77393
$598$	0	0
$599$	20.0557	0.819455	0.409727	−	0.912208i	$-0.365624\pi$
$599$	20.0557	0.819455	0.409727	+	0.912208i	$0.365624\pi$
$600$	0	0
$601$	8.32624	0.339634	0.169817	−	0.985476i	$-0.445682\pi$
$601$	8.32624	0.339634	0.169817	+	0.985476i	$0.445682\pi$
$602$	0	0
$603$	−80.0132	−3.25839
$604$	0	0
$605$	−3.20163	−0.130165
$606$	0	0
$607$	−1.96556	−0.0797795	−0.0398898	−	0.999204i	$-0.512701\pi$
$607$	−1.96556	−0.0797795	−0.0398898	+	0.999204i	$0.512701\pi$
$608$	0	0
$609$	56.8328	2.30298
$610$	0	0
$611$	−13.9443	−0.564125
$612$	0	0
$613$	17.6738	0.713837	0.356918	−	0.934136i	$-0.383827\pi$
$613$	17.6738	0.713837	0.356918	+	0.934136i	$0.383827\pi$
$614$	0	0
$615$	−11.1246	−0.448588
$616$	0	0
$617$	28.0000	1.12724	0.563619	−	0.826035i	$-0.309409\pi$
$617$	28.0000	1.12724	0.563619	+	0.826035i	$0.309409\pi$
$618$	0	0
$619$	20.5066	0.824229	0.412114	−	0.911132i	$-0.364790\pi$
$619$	20.5066	0.824229	0.412114	+	0.911132i	$0.364790\pi$
$620$	0	0
$621$	−34.4721	−1.38332
$622$	0	0
$623$	−36.9787	−1.48152
$624$	0	0
$625$	22.8328	0.913313
$626$	0	0
$627$	−11.7082	−0.467581
$628$	0	0
$629$	−3.00000	−0.119618
$630$	0	0
$631$	35.4508	1.41128	0.705638	−	0.708572i	$-0.250660\pi$
$631$	35.4508	1.41128	0.705638	+	0.708572i	$0.250660\pi$
$632$	0	0
$633$	−27.2361	−1.08254
$634$	0	0
$635$	−3.61803	−0.143577
$636$	0	0
$637$	−7.23607	−0.286703
$638$	0	0
$639$	97.8115	3.86936
$640$	0	0
$641$	4.67376	0.184603	0.0923013	−	0.995731i	$-0.470578\pi$
$641$	4.67376	0.184603	0.0923013	+	0.995731i	$0.470578\pi$
$642$	0	0
$643$	−15.4721	−0.610161	−0.305081	−	0.952326i	$-0.598683\pi$
$643$	−15.4721	−0.610161	−0.305081	+	0.952326i	$0.598683\pi$
$644$	0	0
$645$	1.70820	0.0672605
$646$	0	0
$647$	42.7984	1.68258	0.841289	−	0.540586i	$-0.181797\pi$
$647$	42.7984	1.68258	0.841289	+	0.540586i	$0.181797\pi$
$648$	0	0
$649$	1.61803	0.0635134
$650$	0	0
$651$	−87.3738	−3.42445
$652$	0	0
$653$	9.83282	0.384788	0.192394	−	0.981318i	$-0.438375\pi$
$653$	9.83282	0.384788	0.192394	+	0.981318i	$0.438375\pi$
$654$	0	0
$655$	−8.29180	−0.323987
$656$	0	0
$657$	68.5967	2.67621
$658$	0	0
$659$	11.3475	0.442037	0.221018	−	0.975270i	$-0.429062\pi$
$659$	11.3475	0.442037	0.221018	+	0.975270i	$0.429062\pi$
$660$	0	0
$661$	−28.0132	−1.08959	−0.544793	−	0.838571i	$-0.683392\pi$
$661$	−28.0132	−1.08959	−0.544793	+	0.838571i	$0.683392\pi$
$662$	0	0
$663$	−11.7082	−0.454709
$664$	0	0
$665$	−2.56231	−0.0993620
$666$	0	0
$667$	13.9443	0.539924
$668$	0	0
$669$	70.7214	2.73425
$670$	0	0
$671$	12.0902	0.466736
$672$	0	0
$673$	−3.47214	−0.133841	−0.0669205	−	0.997758i	$-0.521317\pi$
$673$	−3.47214	−0.133841	−0.0669205	+	0.997758i	$0.521317\pi$
$674$	0	0
$675$	70.2492	2.70389
$676$	0	0
$677$	−22.4508	−0.862856	−0.431428	−	0.902147i	$-0.641990\pi$
$677$	−22.4508	−0.862856	−0.431428	+	0.902147i	$0.641990\pi$
$678$	0	0
$679$	14.7295	0.565266
$680$	0	0
$681$	5.41641	0.207557
$682$	0	0
$683$	21.3607	0.817344	0.408672	−	0.912681i	$-0.365992\pi$
$683$	21.3607	0.817344	0.408672	+	0.912681i	$0.365992\pi$
$684$	0	0
$685$	−1.21478	−0.0464144
$686$	0	0
$687$	−7.52786	−0.287206
$688$	0	0
$689$	−17.0344	−0.648961
$690$	0	0
$691$	36.3050	1.38111	0.690553	−	0.723282i	$-0.257367\pi$
$691$	36.3050	1.38111	0.690553	+	0.723282i	$0.257367\pi$
$692$	0	0
$693$	−36.2705	−1.37780
$694$	0	0
$695$	4.20163	0.159377
$696$	0	0
$697$	−9.00000	−0.340899
$698$	0	0
$699$	42.4721	1.60644
$700$	0	0
$701$	−15.5967	−0.589081	−0.294541	−	0.955639i	$-0.595167\pi$
$701$	−15.5967	−0.589081	−0.294541	+	0.955639i	$0.595167\pi$
$702$	0	0
$703$	6.70820	0.253005
$704$	0	0
$705$	−4.76393	−0.179420
$706$	0	0
$707$	23.8328	0.896325
$708$	0	0
$709$	−7.03444	−0.264184	−0.132092	−	0.991237i	$-0.542169\pi$
$709$	−7.03444	−0.264184	−0.132092	+	0.991237i	$0.542169\pi$
$710$	0	0
$711$	0.673762	0.0252681
$712$	0	0
$713$	−21.4377	−0.802848
$714$	0	0
$715$	−2.23607	−0.0836242
$716$	0	0
$717$	60.6525	2.26511
$718$	0	0
$719$	−18.4721	−0.688894	−0.344447	−	0.938806i	$-0.611934\pi$
$719$	−18.4721	−0.688894	−0.344447	+	0.938806i	$0.611934\pi$
$720$	0	0
$721$	28.5836	1.06451
$722$	0	0
$723$	67.1246	2.49639
$724$	0	0
$725$	−28.4164	−1.05536
$726$	0	0
$727$	−26.9230	−0.998518	−0.499259	−	0.866453i	$-0.666394\pi$
$727$	−26.9230	−0.998518	−0.499259	+	0.866453i	$0.666394\pi$
$728$	0	0
$729$	41.9443	1.55349
$730$	0	0
$731$	1.38197	0.0511139
$732$	0	0
$733$	−8.96556	−0.331150	−0.165575	−	0.986197i	$-0.552948\pi$
$733$	−8.96556	−0.331150	−0.165575	+	0.986197i	$0.552948\pi$
$734$	0	0
$735$	−2.47214	−0.0911861
$736$	0	0
$737$	−17.3262	−0.638220
$738$	0	0
$739$	43.4853	1.59963	0.799816	−	0.600245i	$-0.204930\pi$
$739$	43.4853	1.59963	0.799816	+	0.600245i	$0.204930\pi$
$740$	0	0
$741$	26.1803	0.961759
$742$	0	0
$743$	−42.2361	−1.54949	−0.774746	−	0.632273i	$-0.782122\pi$
$743$	−42.2361	−1.54949	−0.774746	+	0.632273i	$0.782122\pi$
$744$	0	0
$745$	5.61803	0.205829
$746$	0	0
$747$	−40.8885	−1.49603
$748$	0	0
$749$	−7.68692	−0.280874
$750$	0	0
$751$	3.41641	0.124666	0.0623332	−	0.998055i	$-0.480146\pi$
$751$	3.41641	0.124666	0.0623332	+	0.998055i	$0.480146\pi$
$752$	0	0
$753$	−26.7639	−0.975332
$754$	0	0
$755$	−3.58359	−0.130420
$756$	0	0
$757$	33.4164	1.21454	0.607270	−	0.794496i	$-0.292265\pi$
$757$	33.4164	1.21454	0.607270	+	0.794496i	$0.292265\pi$
$758$	0	0
$759$	−12.4721	−0.452710
$760$	0	0
$761$	4.36068	0.158075	0.0790373	−	0.996872i	$-0.474815\pi$
$761$	4.36068	0.158075	0.0790373	+	0.996872i	$0.474815\pi$
$762$	0	0
$763$	17.1246	0.619953
$764$	0	0
$765$	−2.85410	−0.103190
$766$	0	0
$767$	−3.61803	−0.130640
$768$	0	0
$769$	−47.8673	−1.72614	−0.863069	−	0.505086i	$-0.831461\pi$
$769$	−47.8673	−1.72614	−0.863069	+	0.505086i	$0.831461\pi$
$770$	0	0
$771$	73.1246	2.63352
$772$	0	0
$773$	31.5623	1.13522	0.567609	−	0.823299i	$-0.307869\pi$
$773$	31.5623	1.13522	0.567609	+	0.823299i	$0.307869\pi$
$774$	0	0
$775$	43.6869	1.56928
$776$	0	0
$777$	29.1246	1.04484
$778$	0	0
$779$	20.1246	0.721039
$780$	0	0
$781$	21.1803	0.757892
$782$	0	0
$783$	−84.7214	−3.02769
$784$	0	0
$785$	−3.38197	−0.120708
$786$	0	0
$787$	−34.6869	−1.23646	−0.618228	−	0.785999i	$-0.712149\pi$
$787$	−34.6869	−1.23646	−0.618228	+	0.785999i	$0.712149\pi$
$788$	0	0
$789$	−35.3050	−1.25689
$790$	0	0
$791$	−50.1246	−1.78223
$792$	0	0
$793$	−27.0344	−0.960021
$794$	0	0
$795$	−5.81966	−0.206402
$796$	0	0
$797$	9.47214	0.335520	0.167760	−	0.985828i	$-0.446347\pi$
$797$	9.47214	0.335520	0.167760	+	0.985828i	$0.446347\pi$
$798$	0	0
$799$	−3.85410	−0.136348
$800$	0	0
$801$	92.1033	3.25431
$802$	0	0
$803$	14.8541	0.524190
$804$	0	0
$805$	−2.72949	−0.0962019
$806$	0	0
$807$	17.4164	0.613087
$808$	0	0
$809$	−16.2918	−0.572789	−0.286395	−	0.958112i	$-0.592457\pi$
$809$	−16.2918	−0.572789	−0.286395	+	0.958112i	$0.592457\pi$
$810$	0	0
$811$	−8.50658	−0.298706	−0.149353	−	0.988784i	$-0.547719\pi$
$811$	−8.50658	−0.298706	−0.149353	+	0.988784i	$0.547719\pi$
$812$	0	0
$813$	70.0689	2.45742
$814$	0	0
$815$	7.47214	0.261738
$816$	0	0
$817$	−3.09017	−0.108111
$818$	0	0
$819$	81.1033	2.83398
$820$	0	0
$821$	13.6738	0.477218	0.238609	−	0.971116i	$-0.423309\pi$
$821$	13.6738	0.477218	0.238609	+	0.971116i	$0.423309\pi$
$822$	0	0
$823$	−15.8328	−0.551897	−0.275949	−	0.961172i	$-0.588992\pi$
$823$	−15.8328	−0.551897	−0.275949	+	0.961172i	$0.588992\pi$
$824$	0	0
$825$	25.4164	0.884886
$826$	0	0
$827$	−56.4164	−1.96179	−0.980895	−	0.194536i	$-0.937680\pi$
$827$	−56.4164	−1.96179	−0.980895	+	0.194536i	$0.937680\pi$
$828$	0	0
$829$	0	0		−	1.00000i	$-0.5\pi$
$829$	0	0			1.00000i	$0.5\pi$
$830$	0	0
$831$	3.88854	0.134892
$832$	0	0
$833$	−2.00000	−0.0692959
$834$	0	0
$835$	1.85410	0.0641638
$836$	0	0
$837$	130.249	4.50207
$838$	0	0
$839$	−51.1246	−1.76502	−0.882509	−	0.470296i	$-0.844147\pi$
$839$	−51.1246	−1.76502	−0.882509	+	0.470296i	$0.844147\pi$
$840$	0	0
$841$	5.27051	0.181742
$842$	0	0
$843$	31.8885	1.09830
$844$	0	0
$845$	0.0344419	0.00118484
$846$	0	0
$847$	25.1459	0.864023
$848$	0	0
$849$	71.3050	2.44718
$850$	0	0
$851$	7.14590	0.244958
$852$	0	0
$853$	22.3475	0.765165	0.382582	−	0.923921i	$-0.375035\pi$
$853$	22.3475	0.765165	0.382582	+	0.923921i	$0.375035\pi$
$854$	0	0
$855$	6.38197	0.218259
$856$	0	0
$857$	26.6180	0.909255	0.454627	−	0.890682i	$-0.349772\pi$
$857$	26.6180	0.909255	0.454627	+	0.890682i	$0.349772\pi$
$858$	0	0
$859$	32.7082	1.11599	0.557995	−	0.829844i	$-0.311571\pi$
$859$	32.7082	1.11599	0.557995	+	0.829844i	$0.311571\pi$
$860$	0	0
$861$	87.3738	2.97769
$862$	0	0
$863$	−30.4508	−1.03656	−0.518279	−	0.855211i	$-0.673427\pi$
$863$	−30.4508	−1.03656	−0.518279	+	0.855211i	$0.673427\pi$
$864$	0	0
$865$	2.02129	0.0687258
$866$	0	0
$867$	−3.23607	−0.109903
$868$	0	0
$869$	0.145898	0.00494925
$870$	0	0
$871$	38.7426	1.31274
$872$	0	0
$873$	−36.6869	−1.24166
$874$	0	0
$875$	11.2918	0.381732
$876$	0	0
$877$	22.0344	0.744050	0.372025	−	0.928223i	$-0.378664\pi$
$877$	22.0344	0.744050	0.372025	+	0.928223i	$0.378664\pi$
$878$	0	0
$879$	42.1803	1.42271
$880$	0	0
$881$	0.978714	0.0329737	0.0164869	−	0.999864i	$-0.494752\pi$
$881$	0.978714	0.0329737	0.0164869	+	0.999864i	$0.494752\pi$
$882$	0	0
$883$	−17.9098	−0.602714	−0.301357	−	0.953511i	$-0.597440\pi$
$883$	−17.9098	−0.602714	−0.301357	+	0.953511i	$0.597440\pi$
$884$	0	0
$885$	−1.23607	−0.0415500
$886$	0	0
$887$	51.7214	1.73663	0.868317	−	0.496010i	$-0.165202\pi$
$887$	51.7214	1.73663	0.868317	+	0.496010i	$0.165202\pi$
$888$	0	0
$889$	28.4164	0.953056
$890$	0	0
$891$	39.5066	1.32352
$892$	0	0
$893$	8.61803	0.288392
$894$	0	0
$895$	−6.68692	−0.223519
$896$	0	0
$897$	27.8885	0.931171
$898$	0	0
$899$	−52.6869	−1.75721
$900$	0	0
$901$	−4.70820	−0.156853
$902$	0	0
$903$	−13.4164	−0.446470
$904$	0	0
$905$	2.47214	0.0821766
$906$	0	0
$907$	7.50658	0.249252	0.124626	−	0.992204i	$-0.460227\pi$
$907$	7.50658	0.249252	0.124626	+	0.992204i	$0.460227\pi$
$908$	0	0
$909$	−59.3607	−1.96887
$910$	0	0
$911$	−9.56231	−0.316813	−0.158407	−	0.987374i	$-0.550636\pi$
$911$	−9.56231	−0.316813	−0.158407	+	0.987374i	$0.550636\pi$
$912$	0	0
$913$	−8.85410	−0.293028
$914$	0	0
$915$	−9.23607	−0.305335
$916$	0	0
$917$	65.1246	2.15060
$918$	0	0
$919$	−41.8541	−1.38064	−0.690320	−	0.723504i	$-0.742530\pi$
$919$	−41.8541	−1.38064	−0.690320	+	0.723504i	$0.742530\pi$
$920$	0	0
$921$	−89.4853	−2.94864
$922$	0	0
$923$	−47.3607	−1.55890
$924$	0	0
$925$	−14.5623	−0.478806
$926$	0	0
$927$	−71.1935	−2.33830
$928$	0	0
$929$	28.3607	0.930484	0.465242	−	0.885184i	$-0.345967\pi$
$929$	28.3607	0.930484	0.465242	+	0.885184i	$0.345967\pi$
$930$	0	0
$931$	4.47214	0.146568
$932$	0	0
$933$	93.8460	3.07238
$934$	0	0
$935$	−0.618034	−0.0202119
$936$	0	0
$937$	−31.3050	−1.02269	−0.511344	−	0.859376i	$-0.670852\pi$
$937$	−31.3050	−1.02269	−0.511344	+	0.859376i	$0.670852\pi$
$938$	0	0
$939$	97.9574	3.19672
$940$	0	0
$941$	5.14590	0.167751	0.0838757	−	0.996476i	$-0.473270\pi$
$941$	5.14590	0.167751	0.0838757	+	0.996476i	$0.473270\pi$
$942$	0	0
$943$	21.4377	0.698107
$944$	0	0
$945$	16.5836	0.539464
$946$	0	0
$947$	−45.7771	−1.48756	−0.743778	−	0.668427i	$-0.766968\pi$
$947$	−45.7771	−1.48756	−0.743778	+	0.668427i	$0.766968\pi$
$948$	0	0
$949$	−33.2148	−1.07820
$950$	0	0
$951$	−94.9017	−3.07740
$952$	0	0
$953$	1.32624	0.0429611	0.0214805	−	0.999769i	$-0.493162\pi$
$953$	1.32624	0.0429611	0.0214805	+	0.999769i	$0.493162\pi$
$954$	0	0
$955$	3.63932	0.117766
$956$	0	0
$957$	−30.6525	−0.990854
$958$	0	0
$959$	9.54102	0.308096
$960$	0	0
$961$	50.0000	1.61290
$962$	0	0
$963$	19.1459	0.616968
$964$	0	0
$965$	−8.45085	−0.272042
$966$	0	0
$967$	50.8673	1.63578	0.817890	−	0.575374i	$-0.195144\pi$
$967$	50.8673	1.63578	0.817890	+	0.575374i	$0.195144\pi$
$968$	0	0
$969$	7.23607	0.232456
$970$	0	0
$971$	−37.2705	−1.19607	−0.598034	−	0.801471i	$-0.704051\pi$
$971$	−37.2705	−1.19607	−0.598034	+	0.801471i	$0.704051\pi$
$972$	0	0
$973$	−33.0000	−1.05793
$974$	0	0
$975$	−56.8328	−1.82011
$976$	0	0
$977$	46.5755	1.49008	0.745041	−	0.667019i	$-0.232430\pi$
$977$	46.5755	1.49008	0.745041	+	0.667019i	$0.232430\pi$
$978$	0	0
$979$	19.9443	0.637422
$980$	0	0
$981$	−42.6525	−1.36179
$982$	0	0
$983$	−19.5623	−0.623941	−0.311970	−	0.950092i	$-0.600989\pi$
$983$	−19.5623	−0.623941	−0.311970	+	0.950092i	$0.600989\pi$
$984$	0	0
$985$	0.909830	0.0289896
$986$	0	0
$987$	37.4164	1.19098
$988$	0	0
$989$	−3.29180	−0.104673
$990$	0	0
$991$	33.2918	1.05755	0.528774	−	0.848762i	$-0.322652\pi$
$991$	33.2918	1.05755	0.528774	+	0.848762i	$0.322652\pi$
$992$	0	0
$993$	−9.41641	−0.298821
$994$	0	0
$995$	−8.00000	−0.253617
$996$	0	0
$997$	−35.8541	−1.13551	−0.567755	−	0.823197i	$-0.692188\pi$
$997$	−35.8541	−1.13551	−0.567755	+	0.823197i	$0.692188\pi$
$998$	0	0
$999$	−43.4164	−1.37363

Twists

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

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
4012.2.a.d.1.1	✓	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)

\(f(q)\)	\(=\)	\(q-3.23607 q^{3} +0.381966 q^{5} -3.00000 q^{7} +7.47214 q^{9} +O(q^{10})\)	\(q-3.23607 q^{3} +0.381966 q^{5} -3.00000 q^{7} +7.47214 q^{9} +1.61803 q^{11} -3.61803 q^{13} -1.23607 q^{15} -1.00000 q^{17} +2.23607 q^{19} +9.70820 q^{21} +2.38197 q^{23} -4.85410 q^{25} -14.4721 q^{27} +5.85410 q^{29} -9.00000 q^{31} -5.23607 q^{33} -1.14590 q^{35} +3.00000 q^{37} +11.7082 q^{39} +9.00000 q^{41} -1.38197 q^{43} +2.85410 q^{45} +3.85410 q^{47} +2.00000 q^{49} +3.23607 q^{51} +4.70820 q^{53} +0.618034 q^{55} -7.23607 q^{57} +1.00000 q^{59} +7.47214 q^{61} -22.4164 q^{63} -1.38197 q^{65} -10.7082 q^{67} -7.70820 q^{69} +13.0902 q^{71} +9.18034 q^{73} +15.7082 q^{75} -4.85410 q^{77} +0.0901699 q^{79} +24.4164 q^{81} -5.47214 q^{83} -0.381966 q^{85} -18.9443 q^{87} +12.3262 q^{89} +10.8541 q^{91} +29.1246 q^{93} +0.854102 q^{95} -4.90983 q^{97} +12.0902 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 2 q - 2 q^{3} + 3 q^{5} - 6 q^{7} + 6 q^{9}+O(q^{10}) \) 2 * q - 2 * q^3 + 3 * q^5 - 6 * q^7 + 6 * q^9	\( 2 q - 2 q^{3} + 3 q^{5} - 6 q^{7} + 6 q^{9} + q^{11} - 5 q^{13} + 2 q^{15} - 2 q^{17} + 6 q^{21} + 7 q^{23} - 3 q^{25} - 20 q^{27} + 5 q^{29} - 18 q^{31} - 6 q^{33} - 9 q^{35} + 6 q^{37} + 10 q^{39} + 18 q^{41} - 5 q^{43} - q^{45} + q^{47} + 4 q^{49} + 2 q^{51} - 4 q^{53} - q^{55} - 10 q^{57} + 2 q^{59} + 6 q^{61} - 18 q^{63} - 5 q^{65} - 8 q^{67} - 2 q^{69} + 15 q^{71} - 4 q^{73} + 18 q^{75} - 3 q^{77} - 11 q^{79} + 22 q^{81} - 2 q^{83} - 3 q^{85} - 20 q^{87} + 9 q^{89} + 15 q^{91} + 18 q^{93} - 5 q^{95} - 21 q^{97} + 13 q^{99}+O(q^{100}) \) 2 * q - 2 * q^3 + 3 * q^5 - 6 * q^7 + 6 * q^9 + q^11 - 5 * q^13 + 2 * q^15 - 2 * q^17 + 6 * q^21 + 7 * q^23 - 3 * q^25 - 20 * q^27 + 5 * q^29 - 18 * q^31 - 6 * q^33 - 9 * q^35 + 6 * q^37 + 10 * q^39 + 18 * q^41 - 5 * q^43 - q^45 + q^47 + 4 * q^49 + 2 * q^51 - 4 * q^53 - q^55 - 10 * q^57 + 2 * q^59 + 6 * q^61 - 18 * q^63 - 5 * q^65 - 8 * q^67 - 2 * q^69 + 15 * q^71 - 4 * q^73 + 18 * q^75 - 3 * q^77 - 11 * q^79 + 22 * q^81 - 2 * q^83 - 3 * q^85 - 20 * q^87 + 9 * q^89 + 15 * q^91 + 18 * q^93 - 5 * q^95 - 21 * q^97 + 13 * q^99

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

Related objects

Downloads

Learn more