LMFDB - Embedded newform 4012.2.a.h.1.4

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:	$15$
Coefficient field:	$\mathbb{Q}[x]/(x^{15} - \cdots)$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{15} - 5 x^{14} - 16 x^{13} + 123 x^{12} - 76 x^{11} - 630 x^{10} + 937 x^{9} + 1083 x^{8} - 2602 x^{7} + \cdots - 2 $ x^15 - 5x^14 - 16x^13 + 123x^12 - 76x^11 - 630x^10 + 937x^9 + 1083x^8 - 2602x^7 - 249x^6 + 2736x^5 - 801x^4 - 900x^3 + 429x^2 - 36x - 2
Coefficient ring:	$\Z[a_1, \ldots, a_{11}]$
Coefficient ring index:	$ 1 $
Twist minimal:	yes
Fricke sign:	$1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.4
Root			$-1.49960$ 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)

$f(q)$	$=$	$q-2.30933 q^{3} +2.18242 q^{5} -0.264202 q^{7} +2.33302 q^{9} +O(q^{10})$	$q-2.30933 q^{3} +2.18242 q^{5} -0.264202 q^{7} +2.33302 q^{9} -0.777023 q^{11} +3.18388 q^{13} -5.03993 q^{15} +1.00000 q^{17} -2.74517 q^{19} +0.610130 q^{21} -2.84782 q^{23} -0.237061 q^{25} +1.54028 q^{27} -2.79020 q^{29} -7.75921 q^{31} +1.79441 q^{33} -0.576598 q^{35} +1.58289 q^{37} -7.35263 q^{39} +8.80428 q^{41} -6.38083 q^{43} +5.09162 q^{45} +9.16549 q^{47} -6.93020 q^{49} -2.30933 q^{51} -11.2621 q^{53} -1.69579 q^{55} +6.33952 q^{57} -1.00000 q^{59} +5.95759 q^{61} -0.616388 q^{63} +6.94854 q^{65} +0.642681 q^{67} +6.57656 q^{69} -5.12486 q^{71} +5.17658 q^{73} +0.547452 q^{75} +0.205291 q^{77} -6.54432 q^{79} -10.5561 q^{81} -4.95865 q^{83} +2.18242 q^{85} +6.44351 q^{87} -1.86030 q^{89} -0.841186 q^{91} +17.9186 q^{93} -5.99111 q^{95} -3.35216 q^{97} -1.81281 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 15 q - q^{3} + q^{5} - 11 q^{7} + 16 q^{9}+O(q^{10}) $ 15 * q - q^3 + q^5 - 11 * q^7 + 16 * q^9	$ 15 q - q^{3} + q^{5} - 11 q^{7} + 16 q^{9} - 12 q^{11} - 10 q^{13} - 13 q^{15} + 15 q^{17} - 4 q^{21} - 21 q^{23} + 4 q^{25} - 37 q^{27} + 23 q^{29} - 31 q^{31} - 11 q^{33} - 35 q^{35} - 10 q^{37} - 2 q^{39} - 15 q^{41} - 3 q^{43} + 15 q^{45} - 47 q^{47} + 18 q^{49} - q^{51} - 7 q^{53} - 20 q^{55} - 30 q^{57} - 15 q^{59} + q^{61} + 19 q^{63} + 11 q^{65} - 20 q^{67} - 24 q^{69} - 13 q^{71} - 24 q^{73} - 10 q^{75} + 21 q^{77} - 34 q^{79} - 21 q^{81} - 40 q^{83} + q^{85} - 60 q^{87} - 46 q^{89} - 19 q^{91} + 33 q^{93} - 4 q^{95} - 5 q^{97} - 74 q^{99}+O(q^{100}) $ 15 * q - q^3 + q^5 - 11 * q^7 + 16 * q^9 - 12 * q^11 - 10 * q^13 - 13 * q^15 + 15 * q^17 - 4 * q^21 - 21 * q^23 + 4 * q^25 - 37 * q^27 + 23 * q^29 - 31 * q^31 - 11 * q^33 - 35 * q^35 - 10 * q^37 - 2 * q^39 - 15 * q^41 - 3 * q^43 + 15 * q^45 - 47 * q^47 + 18 * q^49 - q^51 - 7 * q^53 - 20 * q^55 - 30 * q^57 - 15 * q^59 + q^61 + 19 * q^63 + 11 * q^65 - 20 * q^67 - 24 * q^69 - 13 * q^71 - 24 * q^73 - 10 * q^75 + 21 * q^77 - 34 * q^79 - 21 * q^81 - 40 * q^83 + q^85 - 60 * q^87 - 46 * q^89 - 19 * q^91 + 33 * q^93 - 4 * q^95 - 5 * q^97 - 74 * 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
$3$	−2.30933	−1.33329	−0.666647	−	0.745374i	$-0.732271\pi$
$3$	−2.30933	−1.33329	−0.666647	+	0.745374i	$0.732271\pi$
$4$	0	0
$5$	2.18242	0.976006	0.488003	−	0.872842i	$-0.337725\pi$
$5$	2.18242	0.976006	0.488003	+	0.872842i	$0.337725\pi$
$6$	0	0
$7$	−0.264202	−0.0998589	−0.0499295	−	0.998753i	$-0.515900\pi$
$7$	−0.264202	−0.0998589	−0.0499295	+	0.998753i	$0.515900\pi$
$8$	0	0
$9$	2.33302	0.777673
$10$	0	0
$11$	−0.777023	−0.234281	−0.117141	−	0.993115i	$-0.537373\pi$
$11$	−0.777023	−0.234281	−0.117141	+	0.993115i	$0.537373\pi$
$12$	0	0
$13$	3.18388	0.883048	0.441524	−	0.897249i	$-0.354438\pi$
$13$	3.18388	0.883048	0.441524	+	0.897249i	$0.354438\pi$
$14$	0	0
$15$	−5.03993	−1.30130
$16$	0	0
$17$	1.00000	0.242536
$17$	1.00000	0.242536
$18$	0	0
$19$	−2.74517	−0.629786	−0.314893	−	0.949127i	$-0.601969\pi$
$19$	−2.74517	−0.629786	−0.314893	+	0.949127i	$0.601969\pi$
$20$	0	0
$21$	0.610130	0.133141
$22$	0	0
$23$	−2.84782	−0.593811	−0.296906	−	0.954907i	$-0.595955\pi$
$23$	−2.84782	−0.593811	−0.296906	+	0.954907i	$0.595955\pi$
$24$	0	0
$25$	−0.237061	−0.0474121
$26$	0	0
$27$	1.54028	0.296427
$28$	0	0
$29$	−2.79020	−0.518128	−0.259064	−	0.965860i	$-0.583414\pi$
$29$	−2.79020	−0.518128	−0.259064	+	0.965860i	$0.583414\pi$
$30$	0	0
$31$	−7.75921	−1.39360	−0.696798	−	0.717268i	$-0.745392\pi$
$31$	−7.75921	−1.39360	−0.696798	+	0.717268i	$0.745392\pi$
$32$	0	0
$33$	1.79441	0.312366
$34$	0	0
$35$	−0.576598	−0.0974629
$36$	0	0
$37$	1.58289	0.260225	0.130113	−	0.991499i	$-0.458466\pi$
$37$	1.58289	0.260225	0.130113	+	0.991499i	$0.458466\pi$
$38$	0	0
$39$	−7.35263	−1.17736
$40$	0	0
$41$	8.80428	1.37500	0.687499	−	0.726186i	$-0.258709\pi$
$41$	8.80428	1.37500	0.687499	+	0.726186i	$0.258709\pi$
$42$	0	0
$43$	−6.38083	−0.973067	−0.486534	−	0.873662i	$-0.661739\pi$
$43$	−6.38083	−0.973067	−0.486534	+	0.873662i	$0.661739\pi$
$44$	0	0
$45$	5.09162	0.759014
$46$	0	0
$47$	9.16549	1.33692	0.668462	−	0.743746i	$-0.266953\pi$
$47$	9.16549	1.33692	0.668462	+	0.743746i	$0.266953\pi$
$48$	0	0
$49$	−6.93020	−0.990028
$50$	0	0
$51$	−2.30933	−0.323371
$52$	0	0
$53$	−11.2621	−1.54697	−0.773483	−	0.633817i	$-0.781487\pi$
$53$	−11.2621	−1.54697	−0.773483	+	0.633817i	$0.781487\pi$
$54$	0	0
$55$	−1.69579	−0.228660
$56$	0	0
$57$	6.33952	0.839689
$58$	0	0
$59$	−1.00000	−0.130189
$59$	−1.00000	−0.130189
$60$	0	0
$61$	5.95759	0.762792	0.381396	−	0.924412i	$-0.375444\pi$
$61$	5.95759	0.762792	0.381396	+	0.924412i	$0.375444\pi$
$62$	0	0
$63$	−0.616388	−0.0776576
$64$	0	0
$65$	6.94854	0.861860
$66$	0	0
$67$	0.642681	0.0785160	0.0392580	−	0.999229i	$-0.487501\pi$
$67$	0.642681	0.0785160	0.0392580	+	0.999229i	$0.487501\pi$
$68$	0	0
$69$	6.57656	0.791725
$70$	0	0
$71$	−5.12486	−0.608209	−0.304104	−	0.952639i	$-0.598357\pi$
$71$	−5.12486	−0.608209	−0.304104	+	0.952639i	$0.598357\pi$
$72$	0	0
$73$	5.17658	0.605873	0.302936	−	0.953011i	$-0.402033\pi$
$73$	5.17658	0.605873	0.302936	+	0.953011i	$0.402033\pi$
$74$	0	0
$75$	0.547452	0.0632143
$76$	0	0
$77$	0.205291	0.0233951
$78$	0	0
$79$	−6.54432	−0.736293	−0.368147	−	0.929768i	$-0.620008\pi$
$79$	−6.54432	−0.736293	−0.368147	+	0.929768i	$0.620008\pi$
$80$	0	0
$81$	−10.5561	−1.17290
$82$	0	0
$83$	−4.95865	−0.544283	−0.272141	−	0.962257i	$-0.587732\pi$
$83$	−4.95865	−0.544283	−0.272141	+	0.962257i	$0.587732\pi$
$84$	0	0
$85$	2.18242	0.236716
$86$	0	0
$87$	6.44351	0.690817
$88$	0	0
$89$	−1.86030	−0.197191	−0.0985955	−	0.995128i	$-0.531435\pi$
$89$	−1.86030	−0.197191	−0.0985955	+	0.995128i	$0.531435\pi$
$90$	0	0
$91$	−0.841186	−0.0881802
$92$	0	0
$93$	17.9186	1.85807
$94$	0	0
$95$	−5.99111	−0.614675
$96$	0	0
$97$	−3.35216	−0.340361	−0.170180	−	0.985413i	$-0.554435\pi$
$97$	−3.35216	−0.340361	−0.170180	+	0.985413i	$0.554435\pi$
$98$	0	0
$99$	−1.81281	−0.182194
$100$	0	0
$101$	13.7591	1.36908	0.684542	−	0.728974i	$-0.260002\pi$
$101$	13.7591	1.36908	0.684542	+	0.728974i	$0.260002\pi$
$102$	0	0
$103$	16.4240	1.61831	0.809155	−	0.587596i	$-0.199926\pi$
$103$	16.4240	1.61831	0.809155	+	0.587596i	$0.199926\pi$
$104$	0	0
$105$	1.33156	0.129947
$106$	0	0
$107$	−10.3632	−1.00185	−0.500923	−	0.865492i	$-0.667006\pi$
$107$	−10.3632	−1.00185	−0.500923	+	0.865492i	$0.667006\pi$
$108$	0	0
$109$	−1.59710	−0.152974	−0.0764872	−	0.997071i	$-0.524370\pi$
$109$	−1.59710	−0.152974	−0.0764872	+	0.997071i	$0.524370\pi$
$110$	0	0
$111$	−3.65542	−0.346957
$112$	0	0
$113$	6.57876	0.618877	0.309439	−	0.950919i	$-0.399859\pi$
$113$	6.57876	0.618877	0.309439	+	0.950919i	$0.399859\pi$
$114$	0	0
$115$	−6.21512	−0.579563
$116$	0	0
$117$	7.42804	0.686723
$118$	0	0
$119$	−0.264202	−0.0242193
$120$	0	0
$121$	−10.3962	−0.945112
$122$	0	0
$123$	−20.3320	−1.83328
$124$	0	0
$125$	−11.4294	−1.02228
$126$	0	0
$127$	−4.93209	−0.437652	−0.218826	−	0.975764i	$-0.570223\pi$
$127$	−4.93209	−0.437652	−0.218826	+	0.975764i	$0.570223\pi$
$128$	0	0
$129$	14.7355	1.29738
$130$	0	0
$131$	−2.76934	−0.241959	−0.120979	−	0.992655i	$-0.538603\pi$
$131$	−2.76934	−0.241959	−0.120979	+	0.992655i	$0.538603\pi$
$132$	0	0
$133$	0.725279	0.0628897
$134$	0	0
$135$	3.36153	0.289314
$136$	0	0
$137$	2.96232	0.253088	0.126544	−	0.991961i	$-0.459612\pi$
$137$	2.96232	0.253088	0.126544	+	0.991961i	$0.459612\pi$
$138$	0	0
$139$	−0.279251	−0.0236858	−0.0118429	−	0.999930i	$-0.503770\pi$
$139$	−0.279251	−0.0236858	−0.0118429	+	0.999930i	$0.503770\pi$
$140$	0	0
$141$	−21.1662	−1.78251
$142$	0	0
$143$	−2.47395	−0.206882
$144$	0	0
$145$	−6.08938	−0.505696
$146$	0	0
$147$	16.0041	1.32000
$148$	0	0
$149$	5.80585	0.475633	0.237817	−	0.971310i	$-0.423568\pi$
$149$	5.80585	0.475633	0.237817	+	0.971310i	$0.423568\pi$
$150$	0	0
$151$	13.9500	1.13524	0.567619	−	0.823291i	$-0.307865\pi$
$151$	13.9500	1.13524	0.567619	+	0.823291i	$0.307865\pi$
$152$	0	0
$153$	2.33302	0.188613
$154$	0	0
$155$	−16.9338	−1.36016
$156$	0	0
$157$	10.6861	0.852847	0.426423	−	0.904524i	$-0.359773\pi$
$157$	10.6861	0.852847	0.426423	+	0.904524i	$0.359773\pi$
$158$	0	0
$159$	26.0079	2.06256
$160$	0	0
$161$	0.752399	0.0592973
$162$	0	0
$163$	−13.9582	−1.09329	−0.546645	−	0.837364i	$-0.684095\pi$
$163$	−13.9582	−1.09329	−0.546645	+	0.837364i	$0.684095\pi$
$164$	0	0
$165$	3.91614	0.304871
$166$	0	0
$167$	−4.19451	−0.324581	−0.162290	−	0.986743i	$-0.551888\pi$
$167$	−4.19451	−0.324581	−0.162290	+	0.986743i	$0.551888\pi$
$168$	0	0
$169$	−2.86294	−0.220226
$170$	0	0
$171$	−6.40454	−0.489767
$172$	0	0
$173$	−18.4430	−1.40220	−0.701099	−	0.713064i	$-0.747307\pi$
$173$	−18.4430	−1.40220	−0.701099	+	0.713064i	$0.747307\pi$
$174$	0	0
$175$	0.0626318	0.00473452
$176$	0	0
$177$	2.30933	0.173580
$178$	0	0
$179$	4.39087	0.328189	0.164094	−	0.986445i	$-0.447530\pi$
$179$	4.39087	0.328189	0.164094	+	0.986445i	$0.447530\pi$
$180$	0	0
$181$	−11.4840	−0.853597	−0.426799	−	0.904347i	$-0.640359\pi$
$181$	−11.4840	−0.853597	−0.426799	+	0.904347i	$0.640359\pi$
$182$	0	0
$183$	−13.7581	−1.01703
$184$	0	0
$185$	3.45452	0.253981
$186$	0	0
$187$	−0.777023	−0.0568216
$188$	0	0
$189$	−0.406945	−0.0296009
$190$	0	0
$191$	−18.7370	−1.35576	−0.677882	−	0.735170i	$-0.737102\pi$
$191$	−18.7370	−1.35576	−0.677882	+	0.735170i	$0.737102\pi$
$192$	0	0
$193$	−22.1903	−1.59729	−0.798647	−	0.601800i	$-0.794450\pi$
$193$	−22.1903	−1.59729	−0.798647	+	0.601800i	$0.794450\pi$
$194$	0	0
$195$	−16.0465	−1.14911
$196$	0	0
$197$	−2.97763	−0.212148	−0.106074	−	0.994358i	$-0.533828\pi$
$197$	−2.97763	−0.212148	−0.106074	+	0.994358i	$0.533828\pi$
$198$	0	0
$199$	−10.9089	−0.773312	−0.386656	−	0.922224i	$-0.626370\pi$
$199$	−10.9089	−0.773312	−0.386656	+	0.922224i	$0.626370\pi$
$200$	0	0
$201$	−1.48417	−0.104685
$202$	0	0
$203$	0.737177	0.0517397
$204$	0	0
$205$	19.2146	1.34201
$206$	0	0
$207$	−6.64401	−0.461791
$208$	0	0
$209$	2.13306	0.147547
$210$	0	0
$211$	13.7326	0.945390	0.472695	−	0.881226i	$-0.343281\pi$
$211$	13.7326	0.945390	0.472695	+	0.881226i	$0.343281\pi$
$212$	0	0
$213$	11.8350	0.810921
$214$	0	0
$215$	−13.9256	−0.949720
$216$	0	0
$217$	2.05000	0.139163
$218$	0	0
$219$	−11.9544	−0.807807
$220$	0	0
$221$	3.18388	0.214171
$222$	0	0
$223$	−14.8205	−0.992451	−0.496226	−	0.868194i	$-0.665281\pi$
$223$	−14.8205	−0.992451	−0.496226	+	0.868194i	$0.665281\pi$
$224$	0	0
$225$	−0.553067	−0.0368711
$226$	0	0
$227$	1.90629	0.126525	0.0632625	−	0.997997i	$-0.479849\pi$
$227$	1.90629	0.126525	0.0632625	+	0.997997i	$0.479849\pi$
$228$	0	0
$229$	8.77220	0.579684	0.289842	−	0.957075i	$-0.406397\pi$
$229$	8.77220	0.579684	0.289842	+	0.957075i	$0.406397\pi$
$230$	0	0
$231$	−0.474085	−0.0311925
$232$	0	0
$233$	−4.02966	−0.263992	−0.131996	−	0.991250i	$-0.542139\pi$
$233$	−4.02966	−0.263992	−0.131996	+	0.991250i	$0.542139\pi$
$234$	0	0
$235$	20.0029	1.30485
$236$	0	0
$237$	15.1130	0.981695
$238$	0	0
$239$	−23.6870	−1.53218	−0.766092	−	0.642731i	$-0.777801\pi$
$239$	−23.6870	−1.53218	−0.766092	+	0.642731i	$0.777801\pi$
$240$	0	0
$241$	1.57210	0.101268	0.0506339	−	0.998717i	$-0.483876\pi$
$241$	1.57210	0.101268	0.0506339	+	0.998717i	$0.483876\pi$
$242$	0	0
$243$	19.7567	1.26739
$244$	0	0
$245$	−15.1246	−0.966274
$246$	0	0
$247$	−8.74028	−0.556131
$248$	0	0
$249$	11.4512	0.725689
$250$	0	0
$251$	−11.1100	−0.701258	−0.350629	−	0.936515i	$-0.614032\pi$
$251$	−11.1100	−0.701258	−0.350629	+	0.936515i	$0.614032\pi$
$252$	0	0
$253$	2.21282	0.139119
$254$	0	0
$255$	−5.03993	−0.315612
$256$	0	0
$257$	0.347379	0.0216689	0.0108345	−	0.999941i	$-0.496551\pi$
$257$	0.347379	0.0216689	0.0108345	+	0.999941i	$0.496551\pi$
$258$	0	0
$259$	−0.418202	−0.0259858
$260$	0	0
$261$	−6.50960	−0.402934
$262$	0	0
$263$	−8.14307	−0.502123	−0.251062	−	0.967971i	$-0.580780\pi$
$263$	−8.14307	−0.502123	−0.251062	+	0.967971i	$0.580780\pi$
$264$	0	0
$265$	−24.5785	−1.50985
$266$	0	0
$267$	4.29604	0.262914
$268$	0	0
$269$	−17.3567	−1.05826	−0.529129	−	0.848541i	$-0.677481\pi$
$269$	−17.3567	−1.05826	−0.529129	+	0.848541i	$0.677481\pi$
$270$	0	0
$271$	2.66276	0.161751	0.0808755	−	0.996724i	$-0.474228\pi$
$271$	2.66276	0.161751	0.0808755	+	0.996724i	$0.474228\pi$
$272$	0	0
$273$	1.94258	0.117570
$274$	0	0
$275$	0.184202	0.0111078
$276$	0	0
$277$	−27.4787	−1.65103	−0.825516	−	0.564378i	$-0.809116\pi$
$277$	−27.4787	−1.65103	−0.825516	+	0.564378i	$0.809116\pi$
$278$	0	0
$279$	−18.1024	−1.08376
$280$	0	0
$281$	6.81776	0.406713	0.203357	−	0.979105i	$-0.434815\pi$
$281$	6.81776	0.406713	0.203357	+	0.979105i	$0.434815\pi$
$282$	0	0
$283$	−21.2860	−1.26532	−0.632659	−	0.774430i	$-0.718037\pi$
$283$	−21.2860	−1.26532	−0.632659	+	0.774430i	$0.718037\pi$
$284$	0	0
$285$	13.8355	0.819542
$286$	0	0
$287$	−2.32611	−0.137306
$288$	0	0
$289$	1.00000	0.0588235
$290$	0	0
$291$	7.74126	0.453801
$292$	0	0
$293$	−11.1902	−0.653737	−0.326868	−	0.945070i	$-0.605993\pi$
$293$	−11.1902	−0.653737	−0.326868	+	0.945070i	$0.605993\pi$
$294$	0	0
$295$	−2.18242	−0.127065
$296$	0	0
$297$	−1.19683	−0.0694473
$298$	0	0
$299$	−9.06710	−0.524364
$300$	0	0
$301$	1.68583	0.0971694
$302$	0	0
$303$	−31.7744	−1.82539
$304$	0	0
$305$	13.0019	0.744489
$306$	0	0
$307$	−15.3625	−0.876781	−0.438391	−	0.898784i	$-0.644451\pi$
$307$	−15.3625	−0.876781	−0.438391	+	0.898784i	$0.644451\pi$
$308$	0	0
$309$	−37.9286	−2.15768
$310$	0	0
$311$	16.3431	0.926734	0.463367	−	0.886167i	$-0.346641\pi$
$311$	16.3431	0.926734	0.463367	+	0.886167i	$0.346641\pi$
$312$	0	0
$313$	−2.72983	−0.154299	−0.0771497	−	0.997020i	$-0.524582\pi$
$313$	−2.72983	−0.154299	−0.0771497	+	0.997020i	$0.524582\pi$
$314$	0	0
$315$	−1.34522	−0.0757943
$316$	0	0
$317$	−17.4598	−0.980639	−0.490319	−	0.871543i	$-0.663120\pi$
$317$	−17.4598	−0.980639	−0.490319	+	0.871543i	$0.663120\pi$
$318$	0	0
$319$	2.16805	0.121388
$320$	0	0
$321$	23.9320	1.33576
$322$	0	0
$323$	−2.74517	−0.152745
$324$	0	0
$325$	−0.754771	−0.0418672
$326$	0	0
$327$	3.68823	0.203960
$328$	0	0
$329$	−2.42154	−0.133504
$330$	0	0
$331$	34.6457	1.90430	0.952149	−	0.305633i	$-0.0988681\pi$
$331$	34.6457	1.90430	0.952149	+	0.305633i	$0.0988681\pi$
$332$	0	0
$333$	3.69291	0.202370
$334$	0	0
$335$	1.40260	0.0766321
$336$	0	0
$337$	−19.7594	−1.07636	−0.538182	−	0.842828i	$-0.680889\pi$
$337$	−19.7594	−1.07636	−0.538182	+	0.842828i	$0.680889\pi$
$338$	0	0
$339$	−15.1925	−0.825146
$340$	0	0
$341$	6.02909	0.326493
$342$	0	0
$343$	3.68038	0.198722
$344$	0	0
$345$	14.3528	0.772728
$346$	0	0
$347$	4.60482	0.247200	0.123600	−	0.992332i	$-0.460556\pi$
$347$	4.60482	0.247200	0.123600	+	0.992332i	$0.460556\pi$
$348$	0	0
$349$	5.09792	0.272886	0.136443	−	0.990648i	$-0.456433\pi$
$349$	5.09792	0.272886	0.136443	+	0.990648i	$0.456433\pi$
$350$	0	0
$351$	4.90406	0.261759
$352$	0	0
$353$	−1.92998	−0.102723	−0.0513613	−	0.998680i	$-0.516356\pi$
$353$	−1.92998	−0.102723	−0.0513613	+	0.998680i	$0.516356\pi$
$354$	0	0
$355$	−11.1846	−0.593615
$356$	0	0
$357$	0.610130	0.0322915
$358$	0	0
$359$	−28.4770	−1.50296	−0.751478	−	0.659758i	$-0.770659\pi$
$359$	−28.4770	−1.50296	−0.751478	+	0.659758i	$0.770659\pi$
$360$	0	0
$361$	−11.4640	−0.603370
$362$	0	0
$363$	24.0084	1.26011
$364$	0	0
$365$	11.2975	0.591336
$366$	0	0
$367$	−31.1371	−1.62534	−0.812671	−	0.582723i	$-0.801987\pi$
$367$	−31.1371	−1.62534	−0.812671	+	0.582723i	$0.801987\pi$
$368$	0	0
$369$	20.5406	1.06930
$370$	0	0
$371$	2.97546	0.154478
$372$	0	0
$373$	8.66006	0.448401	0.224201	−	0.974543i	$-0.428023\pi$
$373$	8.66006	0.448401	0.224201	+	0.974543i	$0.428023\pi$
$374$	0	0
$375$	26.3944	1.36300
$376$	0	0
$377$	−8.88366	−0.457532
$378$	0	0
$379$	10.5851	0.543722	0.271861	−	0.962337i	$-0.412361\pi$
$379$	10.5851	0.543722	0.271861	+	0.962337i	$0.412361\pi$
$380$	0	0
$381$	11.3898	0.583519
$382$	0	0
$383$	30.1958	1.54293	0.771467	−	0.636269i	$-0.219523\pi$
$383$	30.1958	1.54293	0.771467	+	0.636269i	$0.219523\pi$
$384$	0	0
$385$	0.448030	0.0228337
$386$	0	0
$387$	−14.8866	−0.756728
$388$	0	0
$389$	36.5963	1.85551	0.927753	−	0.373194i	$-0.121737\pi$
$389$	36.5963	1.85551	0.927753	+	0.373194i	$0.121737\pi$
$390$	0	0
$391$	−2.84782	−0.144020
$392$	0	0
$393$	6.39534	0.322602
$394$	0	0
$395$	−14.2824	−0.718627
$396$	0	0
$397$	14.9267	0.749150	0.374575	−	0.927197i	$-0.377789\pi$
$397$	14.9267	0.749150	0.374575	+	0.927197i	$0.377789\pi$
$398$	0	0
$399$	−1.67491	−0.0838505
$400$	0	0
$401$	2.57596	0.128637	0.0643185	−	0.997929i	$-0.479513\pi$
$401$	2.57596	0.128637	0.0643185	+	0.997929i	$0.479513\pi$
$402$	0	0
$403$	−24.7044	−1.23061
$404$	0	0
$405$	−23.0378	−1.14476
$406$	0	0
$407$	−1.22994	−0.0609659
$408$	0	0
$409$	−20.2921	−1.00338	−0.501689	−	0.865048i	$-0.667288\pi$
$409$	−20.2921	−1.00338	−0.501689	+	0.865048i	$0.667288\pi$
$410$	0	0
$411$	−6.84098	−0.337440
$412$	0	0
$413$	0.264202	0.0130005
$414$	0	0
$415$	−10.8218	−0.531223
$416$	0	0
$417$	0.644884	0.0315801
$418$	0	0
$419$	−34.9361	−1.70674	−0.853369	−	0.521307i	$-0.825444\pi$
$419$	−34.9361	−1.70674	−0.853369	+	0.521307i	$0.825444\pi$
$420$	0	0
$421$	−31.4747	−1.53398	−0.766991	−	0.641658i	$-0.778247\pi$
$421$	−31.4747	−1.53398	−0.766991	+	0.641658i	$0.778247\pi$
$422$	0	0
$423$	21.3833	1.03969
$424$	0	0
$425$	−0.237061	−0.0114991
$426$	0	0
$427$	−1.57401	−0.0761715
$428$	0	0
$429$	5.71316	0.275834
$430$	0	0
$431$	0.224040	0.0107916	0.00539582	−	0.999985i	$-0.498282\pi$
$431$	0.224040	0.0107916	0.00539582	+	0.999985i	$0.498282\pi$
$432$	0	0
$433$	32.5051	1.56210	0.781048	−	0.624471i	$-0.214685\pi$
$433$	32.5051	1.56210	0.781048	+	0.624471i	$0.214685\pi$
$434$	0	0
$435$	14.0624	0.674241
$436$	0	0
$437$	7.81775	0.373974
$438$	0	0
$439$	27.4972	1.31237	0.656186	−	0.754599i	$-0.272169\pi$
$439$	27.4972	1.31237	0.656186	+	0.754599i	$0.272169\pi$
$440$	0	0
$441$	−16.1683	−0.769918
$442$	0	0
$443$	6.58600	0.312910	0.156455	−	0.987685i	$-0.449993\pi$
$443$	6.58600	0.312910	0.156455	+	0.987685i	$0.449993\pi$
$444$	0	0
$445$	−4.05994	−0.192460
$446$	0	0
$447$	−13.4076	−0.634159
$448$	0	0
$449$	10.0170	0.472731	0.236366	−	0.971664i	$-0.424044\pi$
$449$	10.0170	0.472731	0.236366	+	0.971664i	$0.424044\pi$
$450$	0	0
$451$	−6.84113	−0.322136
$452$	0	0
$453$	−32.2153	−1.51361
$454$	0	0
$455$	−1.83582	−0.0860644
$456$	0	0
$457$	−19.9947	−0.935312	−0.467656	−	0.883910i	$-0.654901\pi$
$457$	−19.9947	−0.935312	−0.467656	+	0.883910i	$0.654901\pi$
$458$	0	0
$459$	1.54028	0.0718941
$460$	0	0
$461$	−29.5509	−1.37632	−0.688161	−	0.725558i	$-0.741582\pi$
$461$	−29.5509	−1.37632	−0.688161	+	0.725558i	$0.741582\pi$
$462$	0	0
$463$	−29.1860	−1.35639	−0.678193	−	0.734884i	$-0.737237\pi$
$463$	−29.1860	−1.35639	−0.678193	+	0.734884i	$0.737237\pi$
$464$	0	0
$465$	39.1058	1.81349
$466$	0	0
$467$	1.77517	0.0821451	0.0410726	−	0.999156i	$-0.486923\pi$
$467$	1.77517	0.0821451	0.0410726	+	0.999156i	$0.486923\pi$
$468$	0	0
$469$	−0.169798	−0.00784052
$470$	0	0
$471$	−24.6779	−1.13710
$472$	0	0
$473$	4.95805	0.227971
$474$	0	0
$475$	0.650772	0.0298595
$476$	0	0
$477$	−26.2747	−1.20303
$478$	0	0
$479$	−14.0812	−0.643388	−0.321694	−	0.946844i	$-0.604252\pi$
$479$	−14.0812	−0.643388	−0.321694	+	0.946844i	$0.604252\pi$
$480$	0	0
$481$	5.03972	0.229791
$482$	0	0
$483$	−1.73754	−0.0790608
$484$	0	0
$485$	−7.31581	−0.332194
$486$	0	0
$487$	−7.60822	−0.344761	−0.172381	−	0.985030i	$-0.555146\pi$
$487$	−7.60822	−0.344761	−0.172381	+	0.985030i	$0.555146\pi$
$488$	0	0
$489$	32.2341	1.45768
$490$	0	0
$491$	32.4670	1.46521	0.732607	−	0.680651i	$-0.238303\pi$
$491$	32.4670	1.46521	0.732607	+	0.680651i	$0.238303\pi$
$492$	0	0
$493$	−2.79020	−0.125664
$494$	0	0
$495$	−3.95631	−0.177823
$496$	0	0
$497$	1.35400	0.0607350
$498$	0	0
$499$	34.4777	1.54343	0.771717	−	0.635966i	$-0.219398\pi$
$499$	34.4777	1.54343	0.771717	+	0.635966i	$0.219398\pi$
$500$	0	0
$501$	9.68652	0.432762
$502$	0	0
$503$	−11.8225	−0.527140	−0.263570	−	0.964640i	$-0.584900\pi$
$503$	−11.8225	−0.527140	−0.263570	+	0.964640i	$0.584900\pi$
$504$	0	0
$505$	30.0281	1.33623
$506$	0	0
$507$	6.61148	0.293626
$508$	0	0
$509$	−4.76632	−0.211263	−0.105632	−	0.994405i	$-0.533686\pi$
$509$	−4.76632	−0.211263	−0.105632	+	0.994405i	$0.533686\pi$
$510$	0	0
$511$	−1.36766	−0.0605018
$512$	0	0
$513$	−4.22833	−0.186685
$514$	0	0
$515$	35.8441	1.57948
$516$	0	0
$517$	−7.12180	−0.313216
$518$	0	0
$519$	42.5911	1.86954
$520$	0	0
$521$	−19.4958	−0.854129	−0.427064	−	0.904221i	$-0.640452\pi$
$521$	−19.4958	−0.854129	−0.427064	+	0.904221i	$0.640452\pi$
$522$	0	0
$523$	−23.0949	−1.00987	−0.504936	−	0.863157i	$-0.668484\pi$
$523$	−23.0949	−1.00987	−0.504936	+	0.863157i	$0.668484\pi$
$524$	0	0
$525$	−0.144638	−0.00631251
$526$	0	0
$527$	−7.75921	−0.337997
$528$	0	0
$529$	−14.8899	−0.647388
$530$	0	0
$531$	−2.33302	−0.101244
$532$	0	0
$533$	28.0317	1.21419
$534$	0	0
$535$	−22.6168	−0.977808
$536$	0	0
$537$	−10.1400	−0.437572
$538$	0	0
$539$	5.38492	0.231945
$540$	0	0
$541$	4.90687	0.210963	0.105481	−	0.994421i	$-0.466362\pi$
$541$	4.90687	0.210963	0.105481	+	0.994421i	$0.466362\pi$
$542$	0	0
$543$	26.5203	1.13810
$544$	0	0
$545$	−3.48554	−0.149304
$546$	0	0
$547$	−28.7580	−1.22960	−0.614802	−	0.788681i	$-0.710764\pi$
$547$	−28.7580	−1.22960	−0.614802	+	0.788681i	$0.710764\pi$
$548$	0	0
$549$	13.8992	0.593203
$550$	0	0
$551$	7.65959	0.326309
$552$	0	0
$553$	1.72902	0.0735254
$554$	0	0
$555$	−7.97764	−0.338632
$556$	0	0
$557$	21.3533	0.904767	0.452383	−	0.891824i	$-0.350574\pi$
$557$	21.3533	0.904767	0.452383	+	0.891824i	$0.350574\pi$
$558$	0	0
$559$	−20.3158	−0.859265
$560$	0	0
$561$	1.79441	0.0757599
$562$	0	0
$563$	−17.0252	−0.717527	−0.358764	−	0.933428i	$-0.616802\pi$
$563$	−17.0252	−0.717527	−0.358764	+	0.933428i	$0.616802\pi$
$564$	0	0
$565$	14.3576	0.604028
$566$	0	0
$567$	2.78893	0.117124
$568$	0	0
$569$	−16.2194	−0.679954	−0.339977	−	0.940434i	$-0.610419\pi$
$569$	−16.2194	−0.679954	−0.339977	+	0.940434i	$0.610419\pi$
$570$	0	0
$571$	1.68408	0.0704767	0.0352383	−	0.999379i	$-0.488781\pi$
$571$	1.68408	0.0704767	0.0352383	+	0.999379i	$0.488781\pi$
$572$	0	0
$573$	43.2701	1.80763
$574$	0	0
$575$	0.675105	0.0281538
$576$	0	0
$577$	8.56885	0.356726	0.178363	−	0.983965i	$-0.442920\pi$
$577$	8.56885	0.356726	0.178363	+	0.983965i	$0.442920\pi$
$578$	0	0
$579$	51.2448	2.12966
$580$	0	0
$581$	1.31009	0.0543515
$582$	0	0
$583$	8.75090	0.362425
$584$	0	0
$585$	16.2111	0.670246
$586$	0	0
$587$	10.6874	0.441117	0.220558	−	0.975374i	$-0.429212\pi$
$587$	10.6874	0.441117	0.220558	+	0.975374i	$0.429212\pi$
$588$	0	0
$589$	21.3004	0.877666
$590$	0	0
$591$	6.87635	0.282855
$592$	0	0
$593$	4.84302	0.198879	0.0994396	−	0.995044i	$-0.468295\pi$
$593$	4.84302	0.198879	0.0994396	+	0.995044i	$0.468295\pi$
$594$	0	0
$595$	−0.576598	−0.0236382
$596$	0	0
$597$	25.1923	1.03105
$598$	0	0
$599$	−36.3646	−1.48582	−0.742908	−	0.669393i	$-0.766554\pi$
$599$	−36.3646	−1.48582	−0.742908	+	0.669393i	$0.766554\pi$
$600$	0	0
$601$	33.0667	1.34882	0.674409	−	0.738358i	$-0.264399\pi$
$601$	33.0667	1.34882	0.674409	+	0.738358i	$0.264399\pi$
$602$	0	0
$603$	1.49939	0.0610598
$604$	0	0
$605$	−22.6889	−0.922435
$606$	0	0
$607$	−18.6487	−0.756928	−0.378464	−	0.925616i	$-0.623548\pi$
$607$	−18.6487	−0.756928	−0.378464	+	0.925616i	$0.623548\pi$
$608$	0	0
$609$	−1.70239	−0.0689842
$610$	0	0
$611$	29.1818	1.18057
$612$	0	0
$613$	34.7797	1.40474	0.702370	−	0.711812i	$-0.252125\pi$
$613$	34.7797	1.40474	0.702370	+	0.711812i	$0.252125\pi$
$614$	0	0
$615$	−44.3729	−1.78929
$616$	0	0
$617$	6.39059	0.257276	0.128638	−	0.991692i	$-0.458940\pi$
$617$	6.39059	0.257276	0.128638	+	0.991692i	$0.458940\pi$
$618$	0	0
$619$	11.0828	0.445457	0.222729	−	0.974881i	$-0.428504\pi$
$619$	11.0828	0.445457	0.222729	+	0.974881i	$0.428504\pi$
$620$	0	0
$621$	−4.38643	−0.176022
$622$	0	0
$623$	0.491494	0.0196913
$624$	0	0
$625$	−23.7585	−0.950340
$626$	0	0
$627$	−4.92595	−0.196724
$628$	0	0
$629$	1.58289	0.0631139
$630$	0	0
$631$	7.27977	0.289803	0.144902	−	0.989446i	$-0.453713\pi$
$631$	7.27977	0.289803	0.144902	+	0.989446i	$0.453713\pi$
$632$	0	0
$633$	−31.7131	−1.26048
$634$	0	0
$635$	−10.7639	−0.427151
$636$	0	0
$637$	−22.0649	−0.874243
$638$	0	0
$639$	−11.9564	−0.472988
$640$	0	0
$641$	−18.8684	−0.745256	−0.372628	−	0.927981i	$-0.621543\pi$
$641$	−18.8684	−0.745256	−0.372628	+	0.927981i	$0.621543\pi$
$642$	0	0
$643$	17.8729	0.704837	0.352419	−	0.935842i	$-0.385359\pi$
$643$	17.8729	0.704837	0.352419	+	0.935842i	$0.385359\pi$
$644$	0	0
$645$	32.1589	1.26626
$646$	0	0
$647$	21.0616	0.828017	0.414009	−	0.910273i	$-0.364128\pi$
$647$	21.0616	0.828017	0.414009	+	0.910273i	$0.364128\pi$
$648$	0	0
$649$	0.777023	0.0305008
$650$	0	0
$651$	−4.73413	−0.185545
$652$	0	0
$653$	34.3073	1.34255	0.671275	−	0.741208i	$-0.265747\pi$
$653$	34.3073	1.34255	0.671275	+	0.741208i	$0.265747\pi$
$654$	0	0
$655$	−6.04386	−0.236153
$656$	0	0
$657$	12.0771	0.471171
$658$	0	0
$659$	25.8539	1.00712	0.503562	−	0.863959i	$-0.332023\pi$
$659$	25.8539	1.00712	0.503562	+	0.863959i	$0.332023\pi$
$660$	0	0
$661$	46.8338	1.82162	0.910811	−	0.412823i	$-0.135457\pi$
$661$	46.8338	1.82162	0.910811	+	0.412823i	$0.135457\pi$
$662$	0	0
$663$	−7.35263	−0.285552
$664$	0	0
$665$	1.58286	0.0613807
$666$	0	0
$667$	7.94599	0.307670
$668$	0	0
$669$	34.2254	1.32323
$670$	0	0
$671$	−4.62919	−0.178708
$672$	0	0
$673$	16.8065	0.647843	0.323922	−	0.946084i	$-0.394999\pi$
$673$	16.8065	0.647843	0.323922	+	0.946084i	$0.394999\pi$
$674$	0	0
$675$	−0.365140	−0.0140542
$676$	0	0
$677$	28.9638	1.11317	0.556585	−	0.830791i	$-0.312111\pi$
$677$	28.9638	1.11317	0.556585	+	0.830791i	$0.312111\pi$
$678$	0	0
$679$	0.885647	0.0339880
$680$	0	0
$681$	−4.40226	−0.168695
$682$	0	0
$683$	−3.59350	−0.137501	−0.0687507	−	0.997634i	$-0.521901\pi$
$683$	−3.59350	−0.137501	−0.0687507	+	0.997634i	$0.521901\pi$
$684$	0	0
$685$	6.46501	0.247015
$686$	0	0
$687$	−20.2579	−0.772889
$688$	0	0
$689$	−35.8571	−1.36605
$690$	0	0
$691$	40.1288	1.52657	0.763287	−	0.646060i	$-0.223584\pi$
$691$	40.1288	1.52657	0.763287	+	0.646060i	$0.223584\pi$
$692$	0	0
$693$	0.478948	0.0181937
$694$	0	0
$695$	−0.609443	−0.0231175
$696$	0	0
$697$	8.80428	0.333486
$698$	0	0
$699$	9.30582	0.351979
$700$	0	0
$701$	8.60719	0.325089	0.162545	−	0.986701i	$-0.448030\pi$
$701$	8.60719	0.325089	0.162545	+	0.986701i	$0.448030\pi$
$702$	0	0
$703$	−4.34530	−0.163886
$704$	0	0
$705$	−46.1934	−1.73974
$706$	0	0
$707$	−3.63518	−0.136715
$708$	0	0
$709$	−8.49763	−0.319135	−0.159568	−	0.987187i	$-0.551010\pi$
$709$	−8.49763	−0.319135	−0.159568	+	0.987187i	$0.551010\pi$
$710$	0	0
$711$	−15.2680	−0.572596
$712$	0	0
$713$	22.0968	0.827532
$714$	0	0
$715$	−5.39918	−0.201918
$716$	0	0
$717$	54.7012	2.04285
$718$	0	0
$719$	27.4844	1.02500	0.512498	−	0.858689i	$-0.328720\pi$
$719$	27.4844	1.02500	0.512498	+	0.858689i	$0.328720\pi$
$720$	0	0
$721$	−4.33926	−0.161603
$722$	0	0
$723$	−3.63050	−0.135020
$724$	0	0
$725$	0.661447	0.0245655
$726$	0	0
$727$	−1.63488	−0.0606342	−0.0303171	−	0.999540i	$-0.509652\pi$
$727$	−1.63488	−0.0606342	−0.0303171	+	0.999540i	$0.509652\pi$
$728$	0	0
$729$	−13.9565	−0.516907
$730$	0	0
$731$	−6.38083	−0.236003
$732$	0	0
$733$	26.3484	0.973202	0.486601	−	0.873624i	$-0.338237\pi$
$733$	26.3484	0.973202	0.486601	+	0.873624i	$0.338237\pi$
$734$	0	0
$735$	34.9277	1.28833
$736$	0	0
$737$	−0.499378	−0.0183948
$738$	0	0
$739$	12.1784	0.447991	0.223996	−	0.974590i	$-0.428090\pi$
$739$	12.1784	0.447991	0.223996	+	0.974590i	$0.428090\pi$
$740$	0	0
$741$	20.1842	0.741486
$742$	0	0
$743$	−33.8121	−1.24045	−0.620223	−	0.784425i	$-0.712958\pi$
$743$	−33.8121	−1.24045	−0.620223	+	0.784425i	$0.712958\pi$
$744$	0	0
$745$	12.6708	0.464221
$746$	0	0
$747$	−11.5686	−0.423274
$748$	0	0
$749$	2.73797	0.100043
$750$	0	0
$751$	4.06577	0.148362	0.0741811	−	0.997245i	$-0.476366\pi$
$751$	4.06577	0.148362	0.0741811	+	0.997245i	$0.476366\pi$
$752$	0	0
$753$	25.6567	0.934983
$754$	0	0
$755$	30.4448	1.10800
$756$	0	0
$757$	22.9137	0.832813	0.416407	−	0.909178i	$-0.363289\pi$
$757$	22.9137	0.832813	0.416407	+	0.909178i	$0.363289\pi$
$758$	0	0
$759$	−5.11014	−0.185486
$760$	0	0
$761$	45.1163	1.63546	0.817732	−	0.575599i	$-0.195231\pi$
$761$	45.1163	1.63546	0.817732	+	0.575599i	$0.195231\pi$
$762$	0	0
$763$	0.421957	0.0152759
$764$	0	0
$765$	5.09162	0.184088
$766$	0	0
$767$	−3.18388	−0.114963
$768$	0	0
$769$	37.4413	1.35017	0.675083	−	0.737741i	$-0.264108\pi$
$769$	37.4413	1.35017	0.675083	+	0.737741i	$0.264108\pi$
$770$	0	0
$771$	−0.802214	−0.0288910
$772$	0	0
$773$	−53.6994	−1.93143	−0.965717	−	0.259597i	$-0.916410\pi$
$773$	−53.6994	−1.93143	−0.965717	+	0.259597i	$0.916410\pi$
$774$	0	0
$775$	1.83940	0.0660733
$776$	0	0
$777$	0.965768	0.0346467
$778$	0	0
$779$	−24.1693	−0.865953
$780$	0	0
$781$	3.98213	0.142492
$782$	0	0
$783$	−4.29769	−0.153587
$784$	0	0
$785$	23.3216	0.832383
$786$	0	0
$787$	34.8428	1.24201	0.621006	−	0.783805i	$-0.286724\pi$
$787$	34.8428	1.24201	0.621006	+	0.783805i	$0.286724\pi$
$788$	0	0
$789$	18.8051	0.669478
$790$	0	0
$791$	−1.73812	−0.0618004
$792$	0	0
$793$	18.9682	0.673582
$794$	0	0
$795$	56.7600	2.01307
$796$	0	0
$797$	−50.8902	−1.80262	−0.901312	−	0.433170i	$-0.857395\pi$
$797$	−50.8902	−1.80262	−0.901312	+	0.433170i	$0.857395\pi$
$798$	0	0
$799$	9.16549	0.324252
$800$	0	0
$801$	−4.34011	−0.153350
$802$	0	0
$803$	−4.02232	−0.141945
$804$	0	0
$805$	1.64205	0.0578745
$806$	0	0
$807$	40.0825	1.41097
$808$	0	0
$809$	−21.1809	−0.744680	−0.372340	−	0.928096i	$-0.621444\pi$
$809$	−21.1809	−0.744680	−0.372340	+	0.928096i	$0.621444\pi$
$810$	0	0
$811$	52.8163	1.85463	0.927316	−	0.374280i	$-0.122110\pi$
$811$	52.8163	1.85463	0.927316	+	0.374280i	$0.122110\pi$
$812$	0	0
$813$	−6.14919	−0.215662
$814$	0	0
$815$	−30.4626	−1.06706
$816$	0	0
$817$	17.5165	0.612824
$818$	0	0
$819$	−1.96250	−0.0685754
$820$	0	0
$821$	−12.0255	−0.419694	−0.209847	−	0.977734i	$-0.567297\pi$
$821$	−12.0255	−0.419694	−0.209847	+	0.977734i	$0.567297\pi$
$822$	0	0
$823$	−21.1453	−0.737079	−0.368539	−	0.929612i	$-0.620142\pi$
$823$	−21.1453	−0.737079	−0.368539	+	0.929612i	$0.620142\pi$
$824$	0	0
$825$	−0.425383	−0.0148099
$826$	0	0
$827$	20.1210	0.699675	0.349838	−	0.936810i	$-0.386237\pi$
$827$	20.1210	0.699675	0.349838	+	0.936810i	$0.386237\pi$
$828$	0	0
$829$	13.0477	0.453165	0.226583	−	0.973992i	$-0.427245\pi$
$829$	13.0477	0.453165	0.226583	+	0.973992i	$0.427245\pi$
$830$	0	0
$831$	63.4574	2.20131
$832$	0	0
$833$	−6.93020	−0.240117
$834$	0	0
$835$	−9.15416	−0.316793
$836$	0	0
$837$	−11.9514	−0.413099
$838$	0	0
$839$	−10.3360	−0.356837	−0.178419	−	0.983955i	$-0.557098\pi$
$839$	−10.3360	−0.356837	−0.178419	+	0.983955i	$0.557098\pi$
$840$	0	0
$841$	−21.2148	−0.731544
$842$	0	0
$843$	−15.7445	−0.542269
$844$	0	0
$845$	−6.24812	−0.214942
$846$	0	0
$847$	2.74670	0.0943779
$848$	0	0
$849$	49.1564	1.68704
$850$	0	0
$851$	−4.50778	−0.154525
$852$	0	0
$853$	−49.0564	−1.67966	−0.839830	−	0.542850i	$-0.817345\pi$
$853$	−49.0564	−1.67966	−0.839830	+	0.542850i	$0.817345\pi$
$854$	0	0
$855$	−13.9774	−0.478016
$856$	0	0
$857$	−13.5236	−0.461957	−0.230978	−	0.972959i	$-0.574193\pi$
$857$	−13.5236	−0.461957	−0.230978	+	0.972959i	$0.574193\pi$
$858$	0	0
$859$	34.1851	1.16638	0.583190	−	0.812336i	$-0.301804\pi$
$859$	34.1851	1.16638	0.583190	+	0.812336i	$0.301804\pi$
$860$	0	0
$861$	5.37175	0.183069
$862$	0	0
$863$	51.3226	1.74704	0.873520	−	0.486788i	$-0.161832\pi$
$863$	51.3226	1.74704	0.873520	+	0.486788i	$0.161832\pi$
$864$	0	0
$865$	−40.2504	−1.36855
$866$	0	0
$867$	−2.30933	−0.0784291
$868$	0	0
$869$	5.08509	0.172500
$870$	0	0
$871$	2.04622	0.0693334
$872$	0	0
$873$	−7.82066	−0.264689
$874$	0	0
$875$	3.01968	0.102084
$876$	0	0
$877$	−15.2530	−0.515058	−0.257529	−	0.966271i	$-0.582908\pi$
$877$	−15.2530	−0.515058	−0.257529	+	0.966271i	$0.582908\pi$
$878$	0	0
$879$	25.8418	0.871623
$880$	0	0
$881$	−20.4253	−0.688147	−0.344074	−	0.938943i	$-0.611807\pi$
$881$	−20.4253	−0.688147	−0.344074	+	0.938943i	$0.611807\pi$
$882$	0	0
$883$	4.25446	0.143174	0.0715870	−	0.997434i	$-0.477194\pi$
$883$	4.25446	0.143174	0.0715870	+	0.997434i	$0.477194\pi$
$884$	0	0
$885$	5.03993	0.169415
$886$	0	0
$887$	−16.1458	−0.542123	−0.271062	−	0.962562i	$-0.587375\pi$
$887$	−16.1458	−0.542123	−0.271062	+	0.962562i	$0.587375\pi$
$888$	0	0
$889$	1.30307	0.0437035
$890$	0	0
$891$	8.20232	0.274788
$892$	0	0
$893$	−25.1608	−0.841976
$894$	0	0
$895$	9.58269	0.320314
$896$	0	0
$897$	20.9389	0.699131
$898$	0	0
$899$	21.6498	0.722060
$900$	0	0
$901$	−11.2621	−0.375194
$902$	0	0
$903$	−3.89314	−0.129555
$904$	0	0
$905$	−25.0628	−0.833116
$906$	0	0
$907$	46.2281	1.53498	0.767490	−	0.641061i	$-0.221505\pi$
$907$	46.2281	1.53498	0.767490	+	0.641061i	$0.221505\pi$
$908$	0	0
$909$	32.1003	1.06470
$910$	0	0
$911$	47.7380	1.58163	0.790815	−	0.612055i	$-0.209657\pi$
$911$	47.7380	1.58163	0.790815	+	0.612055i	$0.209657\pi$
$912$	0	0
$913$	3.85299	0.127515
$914$	0	0
$915$	−30.0258	−0.992623
$916$	0	0
$917$	0.731666	0.0241617
$918$	0	0
$919$	−40.7466	−1.34411	−0.672053	−	0.740503i	$-0.734587\pi$
$919$	−40.7466	−1.34411	−0.672053	+	0.740503i	$0.734587\pi$
$920$	0	0
$921$	35.4770	1.16901
$922$	0	0
$923$	−16.3169	−0.537078
$924$	0	0
$925$	−0.375241	−0.0123378
$926$	0	0
$927$	38.3176	1.25852
$928$	0	0
$929$	19.9328	0.653973	0.326987	−	0.945029i	$-0.393967\pi$
$929$	19.9328	0.653973	0.326987	+	0.945029i	$0.393967\pi$
$930$	0	0
$931$	19.0246	0.623505
$932$	0	0
$933$	−37.7417	−1.23561
$934$	0	0
$935$	−1.69579	−0.0554582
$936$	0	0
$937$	−20.1660	−0.658794	−0.329397	−	0.944192i	$-0.606845\pi$
$937$	−20.1660	−0.658794	−0.329397	+	0.944192i	$0.606845\pi$
$938$	0	0
$939$	6.30410	0.205726
$940$	0	0
$941$	32.9009	1.07254	0.536270	−	0.844046i	$-0.319833\pi$
$941$	32.9009	1.07254	0.536270	+	0.844046i	$0.319833\pi$
$942$	0	0
$943$	−25.0730	−0.816488
$944$	0	0
$945$	−0.888122	−0.0288906
$946$	0	0
$947$	37.6910	1.22479	0.612396	−	0.790551i	$-0.290206\pi$
$947$	37.6910	1.22479	0.612396	+	0.790551i	$0.290206\pi$
$948$	0	0
$949$	16.4816	0.535015
$950$	0	0
$951$	40.3204	1.30748
$952$	0	0
$953$	−25.3987	−0.822745	−0.411372	−	0.911467i	$-0.634950\pi$
$953$	−25.3987	−0.822745	−0.411372	+	0.911467i	$0.634950\pi$
$954$	0	0
$955$	−40.8920	−1.32323
$956$	0	0
$957$	−5.00676	−0.161845
$958$	0	0
$959$	−0.782650	−0.0252731
$960$	0	0
$961$	29.2054	0.942108
$962$	0	0
$963$	−24.1775	−0.779109
$964$	0	0
$965$	−48.4285	−1.55897
$966$	0	0
$967$	−13.4214	−0.431605	−0.215802	−	0.976437i	$-0.569237\pi$
$967$	−13.4214	−0.431605	−0.215802	+	0.976437i	$0.569237\pi$
$968$	0	0
$969$	6.33952	0.203655
$970$	0	0
$971$	2.10676	0.0676092	0.0338046	−	0.999428i	$-0.489238\pi$
$971$	2.10676	0.0676092	0.0338046	+	0.999428i	$0.489238\pi$
$972$	0	0
$973$	0.0737787	0.00236524
$974$	0	0
$975$	1.74302	0.0558213
$976$	0	0
$977$	38.0453	1.21718	0.608589	−	0.793486i	$-0.291736\pi$
$977$	38.0453	1.21718	0.608589	+	0.793486i	$0.291736\pi$
$978$	0	0
$979$	1.44549	0.0461982
$980$	0	0
$981$	−3.72606	−0.118964
$982$	0	0
$983$	−29.8434	−0.951856	−0.475928	−	0.879484i	$-0.657888\pi$
$983$	−29.8434	−0.951856	−0.475928	+	0.879484i	$0.657888\pi$
$984$	0	0
$985$	−6.49843	−0.207057
$986$	0	0
$987$	5.59214	0.178000
$988$	0	0
$989$	18.1714	0.577818
$990$	0	0
$991$	14.2464	0.452553	0.226277	−	0.974063i	$-0.427345\pi$
$991$	14.2464	0.452553	0.226277	+	0.974063i	$0.427345\pi$
$992$	0	0
$993$	−80.0084	−2.53899
$994$	0	0
$995$	−23.8078	−0.754757
$996$	0	0
$997$	−32.4399	−1.02738	−0.513691	−	0.857976i	$-0.671722\pi$
$997$	−32.4399	−1.02738	−0.513691	+	0.857976i	$0.671722\pi$
$998$	0	0
$999$	2.43809	0.0771378

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	4012.2.a.h.1.4	✓	15

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
4012.2.a.h.1.4	✓	15	1.1	even	1	trivial

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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)

\(f(q)\)	\(=\)	\(q-2.30933 q^{3} +2.18242 q^{5} -0.264202 q^{7} +2.33302 q^{9} +O(q^{10})\)	\(q-2.30933 q^{3} +2.18242 q^{5} -0.264202 q^{7} +2.33302 q^{9} -0.777023 q^{11} +3.18388 q^{13} -5.03993 q^{15} +1.00000 q^{17} -2.74517 q^{19} +0.610130 q^{21} -2.84782 q^{23} -0.237061 q^{25} +1.54028 q^{27} -2.79020 q^{29} -7.75921 q^{31} +1.79441 q^{33} -0.576598 q^{35} +1.58289 q^{37} -7.35263 q^{39} +8.80428 q^{41} -6.38083 q^{43} +5.09162 q^{45} +9.16549 q^{47} -6.93020 q^{49} -2.30933 q^{51} -11.2621 q^{53} -1.69579 q^{55} +6.33952 q^{57} -1.00000 q^{59} +5.95759 q^{61} -0.616388 q^{63} +6.94854 q^{65} +0.642681 q^{67} +6.57656 q^{69} -5.12486 q^{71} +5.17658 q^{73} +0.547452 q^{75} +0.205291 q^{77} -6.54432 q^{79} -10.5561 q^{81} -4.95865 q^{83} +2.18242 q^{85} +6.44351 q^{87} -1.86030 q^{89} -0.841186 q^{91} +17.9186 q^{93} -5.99111 q^{95} -3.35216 q^{97} -1.81281 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 15 q - q^{3} + q^{5} - 11 q^{7} + 16 q^{9}+O(q^{10}) \) 15 * q - q^3 + q^5 - 11 * q^7 + 16 * q^9	\( 15 q - q^{3} + q^{5} - 11 q^{7} + 16 q^{9} - 12 q^{11} - 10 q^{13} - 13 q^{15} + 15 q^{17} - 4 q^{21} - 21 q^{23} + 4 q^{25} - 37 q^{27} + 23 q^{29} - 31 q^{31} - 11 q^{33} - 35 q^{35} - 10 q^{37} - 2 q^{39} - 15 q^{41} - 3 q^{43} + 15 q^{45} - 47 q^{47} + 18 q^{49} - q^{51} - 7 q^{53} - 20 q^{55} - 30 q^{57} - 15 q^{59} + q^{61} + 19 q^{63} + 11 q^{65} - 20 q^{67} - 24 q^{69} - 13 q^{71} - 24 q^{73} - 10 q^{75} + 21 q^{77} - 34 q^{79} - 21 q^{81} - 40 q^{83} + q^{85} - 60 q^{87} - 46 q^{89} - 19 q^{91} + 33 q^{93} - 4 q^{95} - 5 q^{97} - 74 q^{99}+O(q^{100}) \) 15 * q - q^3 + q^5 - 11 * q^7 + 16 * q^9 - 12 * q^11 - 10 * q^13 - 13 * q^15 + 15 * q^17 - 4 * q^21 - 21 * q^23 + 4 * q^25 - 37 * q^27 + 23 * q^29 - 31 * q^31 - 11 * q^33 - 35 * q^35 - 10 * q^37 - 2 * q^39 - 15 * q^41 - 3 * q^43 + 15 * q^45 - 47 * q^47 + 18 * q^49 - q^51 - 7 * q^53 - 20 * q^55 - 30 * q^57 - 15 * q^59 + q^61 + 19 * q^63 + 11 * q^65 - 20 * q^67 - 24 * q^69 - 13 * q^71 - 24 * q^73 - 10 * q^75 + 21 * q^77 - 34 * q^79 - 21 * q^81 - 40 * q^83 + q^85 - 60 * q^87 - 46 * q^89 - 19 * q^91 + 33 * q^93 - 4 * q^95 - 5 * q^97 - 74 * q^99

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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