LMFDB - Embedded newform 4012.2.a.g.1.5

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:	$12$
Coefficient field:	$\mathbb{Q}[x]/(x^{12} - \cdots)$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{12} - 4 x^{11} - 13 x^{10} + 60 x^{9} + 48 x^{8} - 289 x^{7} - 89 x^{6} + 602 x^{5} + 161 x^{4} + \cdots + 82 $ x^12 - 4x^11 - 13x^10 + 60x^9 + 48x^8 - 289x^7 - 89x^6 + 602x^5 + 161x^4 - 555x^3 - 197x^2 + 181*x + 82
Coefficient ring:	$\Z[a_1, \ldots, a_{7}]$
Coefficient ring index:	$ 1 $
Twist minimal:	yes
Fricke sign:	$1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.5
Root			$1.18762$ 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)

$f(q)$	$=$	$q-1.18762 q^{3} +1.03337 q^{5} +2.52681 q^{7} -1.58957 q^{9} +O(q^{10})$	$q-1.18762 q^{3} +1.03337 q^{5} +2.52681 q^{7} -1.58957 q^{9} +4.63016 q^{11} -5.51519 q^{13} -1.22725 q^{15} -1.00000 q^{17} +4.35203 q^{19} -3.00088 q^{21} -4.68684 q^{23} -3.93214 q^{25} +5.45065 q^{27} -8.15402 q^{29} -0.288042 q^{31} -5.49885 q^{33} +2.61113 q^{35} -8.28324 q^{37} +6.54993 q^{39} +5.92083 q^{41} -4.48394 q^{43} -1.64261 q^{45} -5.50830 q^{47} -0.615243 q^{49} +1.18762 q^{51} +3.79017 q^{53} +4.78467 q^{55} -5.16854 q^{57} +1.00000 q^{59} +0.623587 q^{61} -4.01653 q^{63} -5.69924 q^{65} -10.7757 q^{67} +5.56616 q^{69} +1.81909 q^{71} +0.894382 q^{73} +4.66988 q^{75} +11.6995 q^{77} +0.647603 q^{79} -1.70457 q^{81} +5.53842 q^{83} -1.03337 q^{85} +9.68384 q^{87} -1.18899 q^{89} -13.9358 q^{91} +0.342084 q^{93} +4.49726 q^{95} -9.08803 q^{97} -7.35995 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 12 q - 4 q^{3} + 3 q^{5} + 2 q^{7} + 6 q^{9}+O(q^{10}) $ 12 * q - 4 * q^3 + 3 * q^5 + 2 * q^7 + 6 * q^9	$ 12 q - 4 q^{3} + 3 q^{5} + 2 q^{7} + 6 q^{9} - 5 q^{11} + 9 q^{13} - 16 q^{15} - 12 q^{17} - 15 q^{19} + 5 q^{21} - 22 q^{23} + 5 q^{25} - 16 q^{27} - 17 q^{29} + 5 q^{31} - 33 q^{33} - 9 q^{35} - 2 q^{37} - 24 q^{39} - 18 q^{41} - 16 q^{43} + 30 q^{45} - 60 q^{47} + 10 q^{49} + 4 q^{51} - 24 q^{53} - q^{55} + 41 q^{57} + 12 q^{59} - 17 q^{61} - 23 q^{63} + 2 q^{65} - 22 q^{67} + 6 q^{69} - 10 q^{71} + 4 q^{73} - 62 q^{75} - 6 q^{77} + 48 q^{79} + 4 q^{81} - 20 q^{83} - 3 q^{85} + 13 q^{87} - 29 q^{89} - 28 q^{91} - 31 q^{93} - 48 q^{95} - 26 q^{97} - 3 q^{99}+O(q^{100}) $ 12 * q - 4 * q^3 + 3 * q^5 + 2 * q^7 + 6 * q^9 - 5 * q^11 + 9 * q^13 - 16 * q^15 - 12 * q^17 - 15 * q^19 + 5 * q^21 - 22 * q^23 + 5 * q^25 - 16 * q^27 - 17 * q^29 + 5 * q^31 - 33 * q^33 - 9 * q^35 - 2 * q^37 - 24 * q^39 - 18 * q^41 - 16 * q^43 + 30 * q^45 - 60 * q^47 + 10 * q^49 + 4 * q^51 - 24 * q^53 - q^55 + 41 * q^57 + 12 * q^59 - 17 * q^61 - 23 * q^63 + 2 * q^65 - 22 * q^67 + 6 * q^69 - 10 * q^71 + 4 * q^73 - 62 * q^75 - 6 * q^77 + 48 * q^79 + 4 * q^81 - 20 * q^83 - 3 * q^85 + 13 * q^87 - 29 * q^89 - 28 * q^91 - 31 * q^93 - 48 * q^95 - 26 * q^97 - 3 * 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
$3$	−1.18762	−0.685671	−0.342835	−	0.939396i	$-0.611387\pi$
$3$	−1.18762	−0.685671	−0.342835	+	0.939396i	$0.611387\pi$
$4$	0	0
$5$	1.03337	0.462138	0.231069	−	0.972937i	$-0.425778\pi$
$5$	1.03337	0.462138	0.231069	+	0.972937i	$0.425778\pi$
$6$	0	0
$7$	2.52681	0.955044	0.477522	−	0.878620i	$-0.341535\pi$
$7$	2.52681	0.955044	0.477522	+	0.878620i	$0.341535\pi$
$8$	0	0
$9$	−1.58957	−0.529856
$10$	0	0
$11$	4.63016	1.39605	0.698023	−	0.716076i	$-0.254063\pi$
$11$	4.63016	1.39605	0.698023	+	0.716076i	$0.254063\pi$
$12$	0	0
$13$	−5.51519	−1.52964	−0.764820	−	0.644245i	$-0.777172\pi$
$13$	−5.51519	−1.52964	−0.764820	+	0.644245i	$0.777172\pi$
$14$	0	0
$15$	−1.22725	−0.316874
$16$	0	0
$17$	−1.00000	−0.242536
$17$	−1.00000	−0.242536
$18$	0	0
$19$	4.35203	0.998424	0.499212	−	0.866480i	$-0.333623\pi$
$19$	4.35203	0.998424	0.499212	+	0.866480i	$0.333623\pi$
$20$	0	0
$21$	−3.00088	−0.654845
$22$	0	0
$23$	−4.68684	−0.977273	−0.488636	−	0.872488i	$-0.662506\pi$
$23$	−4.68684	−0.977273	−0.488636	+	0.872488i	$0.662506\pi$
$24$	0	0
$25$	−3.93214	−0.786429
$26$	0	0
$27$	5.45065	1.04898
$28$	0	0
$29$	−8.15402	−1.51416	−0.757081	−	0.653321i	$-0.773375\pi$
$29$	−8.15402	−1.51416	−0.757081	+	0.653321i	$0.773375\pi$
$30$	0	0
$31$	−0.288042	−0.0517339	−0.0258670	−	0.999665i	$-0.508235\pi$
$31$	−0.288042	−0.0517339	−0.0258670	+	0.999665i	$0.508235\pi$
$32$	0	0
$33$	−5.49885	−0.957227
$34$	0	0
$35$	2.61113	0.441362
$36$	0	0
$37$	−8.28324	−1.36176	−0.680878	−	0.732397i	$-0.738402\pi$
$37$	−8.28324	−1.36176	−0.680878	+	0.732397i	$0.738402\pi$
$38$	0	0
$39$	6.54993	1.04883
$40$	0	0
$41$	5.92083	0.924679	0.462339	−	0.886703i	$-0.347010\pi$
$41$	5.92083	0.924679	0.462339	+	0.886703i	$0.347010\pi$
$42$	0	0
$43$	−4.48394	−0.683794	−0.341897	−	0.939737i	$-0.611069\pi$
$43$	−4.48394	−0.683794	−0.341897	+	0.939737i	$0.611069\pi$
$44$	0	0
$45$	−1.64261	−0.244866
$46$	0	0
$47$	−5.50830	−0.803467	−0.401734	−	0.915757i	$-0.631592\pi$
$47$	−5.50830	−0.803467	−0.401734	+	0.915757i	$0.631592\pi$
$48$	0	0
$49$	−0.615243	−0.0878919
$50$	0	0
$51$	1.18762	0.166300
$52$	0	0
$53$	3.79017	0.520620	0.260310	−	0.965525i	$-0.416175\pi$
$53$	3.79017	0.520620	0.260310	+	0.965525i	$0.416175\pi$
$54$	0	0
$55$	4.78467	0.645165
$56$	0	0
$57$	−5.16854	−0.684590
$58$	0	0
$59$	1.00000	0.130189
$59$	1.00000	0.130189
$60$	0	0
$61$	0.623587	0.0798421	0.0399210	−	0.999203i	$-0.487289\pi$
$61$	0.623587	0.0798421	0.0399210	+	0.999203i	$0.487289\pi$
$62$	0	0
$63$	−4.01653	−0.506035
$64$	0	0
$65$	−5.69924	−0.706904
$66$	0	0
$67$	−10.7757	−1.31646	−0.658229	−	0.752818i	$-0.728694\pi$
$67$	−10.7757	−1.31646	−0.658229	+	0.752818i	$0.728694\pi$
$68$	0	0
$69$	5.56616	0.670087
$70$	0	0
$71$	1.81909	0.215886	0.107943	−	0.994157i	$-0.465574\pi$
$71$	1.81909	0.215886	0.107943	+	0.994157i	$0.465574\pi$
$72$	0	0
$73$	0.894382	0.104680	0.0523398	−	0.998629i	$-0.483332\pi$
$73$	0.894382	0.104680	0.0523398	+	0.998629i	$0.483332\pi$
$74$	0	0
$75$	4.66988	0.539231
$76$	0	0
$77$	11.6995	1.33328
$78$	0	0
$79$	0.647603	0.0728610	0.0364305	−	0.999336i	$-0.488401\pi$
$79$	0.647603	0.0728610	0.0364305	+	0.999336i	$0.488401\pi$
$80$	0	0
$81$	−1.70457	−0.189397
$82$	0	0
$83$	5.53842	0.607920	0.303960	−	0.952685i	$-0.401691\pi$
$83$	5.53842	0.607920	0.303960	+	0.952685i	$0.401691\pi$
$84$	0	0
$85$	−1.03337	−0.112085
$86$	0	0
$87$	9.68384	1.03822
$88$	0	0
$89$	−1.18899	−0.126033	−0.0630166	−	0.998012i	$-0.520072\pi$
$89$	−1.18899	−0.126033	−0.0630166	+	0.998012i	$0.520072\pi$
$90$	0	0
$91$	−13.9358	−1.46087
$92$	0	0
$93$	0.342084	0.0354724
$94$	0	0
$95$	4.49726	0.461409
$96$	0	0
$97$	−9.08803	−0.922749	−0.461375	−	0.887205i	$-0.652644\pi$
$97$	−9.08803	−0.922749	−0.461375	+	0.887205i	$0.652644\pi$
$98$	0	0
$99$	−7.35995	−0.739703
$100$	0	0
$101$	−2.33733	−0.232573	−0.116286	−	0.993216i	$-0.537099\pi$
$101$	−2.33733	−0.232573	−0.116286	+	0.993216i	$0.537099\pi$
$102$	0	0
$103$	6.01022	0.592204	0.296102	−	0.955156i	$-0.404313\pi$
$103$	6.01022	0.592204	0.296102	+	0.955156i	$0.404313\pi$
$104$	0	0
$105$	−3.10102	−0.302629
$106$	0	0
$107$	−19.5086	−1.88597	−0.942984	−	0.332838i	$-0.891994\pi$
$107$	−19.5086	−1.88597	−0.942984	+	0.332838i	$0.891994\pi$
$108$	0	0
$109$	19.1263	1.83197	0.915983	−	0.401217i	$-0.131413\pi$
$109$	19.1263	1.83197	0.915983	+	0.401217i	$0.131413\pi$
$110$	0	0
$111$	9.83731	0.933716
$112$	0	0
$113$	−5.29162	−0.497794	−0.248897	−	0.968530i	$-0.580068\pi$
$113$	−5.29162	−0.497794	−0.248897	+	0.968530i	$0.580068\pi$
$114$	0	0
$115$	−4.84324	−0.451634
$116$	0	0
$117$	8.76677	0.810488
$118$	0	0
$119$	−2.52681	−0.231632
$120$	0	0
$121$	10.4384	0.948943
$122$	0	0
$123$	−7.03168	−0.634025
$124$	0	0
$125$	−9.23022	−0.825576
$126$	0	0
$127$	14.5684	1.29274	0.646370	−	0.763024i	$-0.276286\pi$
$127$	14.5684	1.29274	0.646370	+	0.763024i	$0.276286\pi$
$128$	0	0
$129$	5.32520	0.468858
$130$	0	0
$131$	−1.64696	−0.143896	−0.0719479	−	0.997408i	$-0.522922\pi$
$131$	−1.64696	−0.143896	−0.0719479	+	0.997408i	$0.522922\pi$
$132$	0	0
$133$	10.9967	0.953538
$134$	0	0
$135$	5.63254	0.484772
$136$	0	0
$137$	1.99828	0.170725	0.0853625	−	0.996350i	$-0.472795\pi$
$137$	1.99828	0.170725	0.0853625	+	0.996350i	$0.472795\pi$
$138$	0	0
$139$	−16.6631	−1.41335	−0.706673	−	0.707540i	$-0.749805\pi$
$139$	−16.6631	−1.41335	−0.706673	+	0.707540i	$0.749805\pi$
$140$	0	0
$141$	6.54174	0.550914
$142$	0	0
$143$	−25.5362	−2.13545
$144$	0	0
$145$	−8.42613	−0.699752
$146$	0	0
$147$	0.730673	0.0602649
$148$	0	0
$149$	−13.2853	−1.08837	−0.544187	−	0.838964i	$-0.683162\pi$
$149$	−13.2853	−1.08837	−0.544187	+	0.838964i	$0.683162\pi$
$150$	0	0
$151$	−9.45570	−0.769494	−0.384747	−	0.923022i	$-0.625711\pi$
$151$	−9.45570	−0.769494	−0.384747	+	0.923022i	$0.625711\pi$
$152$	0	0
$153$	1.58957	0.128509
$154$	0	0
$155$	−0.297655	−0.0239082
$156$	0	0
$157$	12.1686	0.971160	0.485580	−	0.874192i	$-0.338608\pi$
$157$	12.1686	0.971160	0.485580	+	0.874192i	$0.338608\pi$
$158$	0	0
$159$	−4.50127	−0.356974
$160$	0	0
$161$	−11.8427	−0.933338
$162$	0	0
$163$	−23.6966	−1.85606	−0.928030	−	0.372506i	$-0.878499\pi$
$163$	−23.6966	−1.85606	−0.928030	+	0.372506i	$0.878499\pi$
$164$	0	0
$165$	−5.68236	−0.442371
$166$	0	0
$167$	15.1640	1.17343	0.586713	−	0.809795i	$-0.300422\pi$
$167$	15.1640	1.17343	0.586713	+	0.809795i	$0.300422\pi$
$168$	0	0
$169$	17.4173	1.33980
$170$	0	0
$171$	−6.91784	−0.529021
$172$	0	0
$173$	13.1395	0.998978	0.499489	−	0.866320i	$-0.333521\pi$
$173$	13.1395	0.998978	0.499489	+	0.866320i	$0.333521\pi$
$174$	0	0
$175$	−9.93577	−0.751074
$176$	0	0
$177$	−1.18762	−0.0892667
$178$	0	0
$179$	13.9694	1.04412	0.522059	−	0.852909i	$-0.325164\pi$
$179$	13.9694	1.04412	0.522059	+	0.852909i	$0.325164\pi$
$180$	0	0
$181$	6.23709	0.463599	0.231799	−	0.972764i	$-0.425539\pi$
$181$	6.23709	0.463599	0.231799	+	0.972764i	$0.425539\pi$
$182$	0	0
$183$	−0.740581	−0.0547454
$184$	0	0
$185$	−8.55966	−0.629319
$186$	0	0
$187$	−4.63016	−0.338591
$188$	0	0
$189$	13.7727	1.00182
$190$	0	0
$191$	6.28159	0.454520	0.227260	−	0.973834i	$-0.427023\pi$
$191$	6.28159	0.454520	0.227260	+	0.973834i	$0.427023\pi$
$192$	0	0
$193$	−18.1330	−1.30524	−0.652621	−	0.757684i	$-0.726331\pi$
$193$	−18.1330	−1.30524	−0.652621	+	0.757684i	$0.726331\pi$
$194$	0	0
$195$	6.76851	0.484703
$196$	0	0
$197$	−12.3431	−0.879411	−0.439706	−	0.898142i	$-0.644917\pi$
$197$	−12.3431	−0.879411	−0.439706	+	0.898142i	$0.644917\pi$
$198$	0	0
$199$	−7.76295	−0.550301	−0.275151	−	0.961401i	$-0.588728\pi$
$199$	−7.76295	−0.550301	−0.275151	+	0.961401i	$0.588728\pi$
$200$	0	0
$201$	12.7974	0.902657
$202$	0	0
$203$	−20.6036	−1.44609
$204$	0	0
$205$	6.11842	0.427329
$206$	0	0
$207$	7.45004	0.517814
$208$	0	0
$209$	20.1506	1.39385
$210$	0	0
$211$	12.9000	0.888074	0.444037	−	0.896009i	$-0.353546\pi$
$211$	12.9000	0.888074	0.444037	+	0.896009i	$0.353546\pi$
$212$	0	0
$213$	−2.16038	−0.148027
$214$	0	0
$215$	−4.63357	−0.316007
$216$	0	0
$217$	−0.727828	−0.0494082
$218$	0	0
$219$	−1.06218	−0.0717757
$220$	0	0
$221$	5.51519	0.370992
$222$	0	0
$223$	−14.9830	−1.00333	−0.501667	−	0.865061i	$-0.667280\pi$
$223$	−14.9830	−1.00333	−0.501667	+	0.865061i	$0.667280\pi$
$224$	0	0
$225$	6.25041	0.416694
$226$	0	0
$227$	−3.65735	−0.242747	−0.121374	−	0.992607i	$-0.538730\pi$
$227$	−3.65735	−0.242747	−0.121374	+	0.992607i	$0.538730\pi$
$228$	0	0
$229$	10.3092	0.681249	0.340624	−	0.940199i	$-0.389362\pi$
$229$	10.3092	0.681249	0.340624	+	0.940199i	$0.389362\pi$
$230$	0	0
$231$	−13.8945	−0.914194
$232$	0	0
$233$	−24.6674	−1.61602	−0.808009	−	0.589171i	$-0.799455\pi$
$233$	−24.6674	−1.61602	−0.808009	+	0.589171i	$0.799455\pi$
$234$	0	0
$235$	−5.69211	−0.371313
$236$	0	0
$237$	−0.769103	−0.0499586
$238$	0	0
$239$	−5.11018	−0.330550	−0.165275	−	0.986248i	$-0.552851\pi$
$239$	−5.11018	−0.330550	−0.165275	+	0.986248i	$0.552851\pi$
$240$	0	0
$241$	−18.4630	−1.18930	−0.594652	−	0.803983i	$-0.702710\pi$
$241$	−18.4630	−1.18930	−0.594652	+	0.803983i	$0.702710\pi$
$242$	0	0
$243$	−14.3276	−0.919113
$244$	0	0
$245$	−0.635774	−0.0406181
$246$	0	0
$247$	−24.0023	−1.52723
$248$	0	0
$249$	−6.57751	−0.416833
$250$	0	0
$251$	7.44336	0.469821	0.234910	−	0.972017i	$-0.424520\pi$
$251$	7.44336	0.469821	0.234910	+	0.972017i	$0.424520\pi$
$252$	0	0
$253$	−21.7008	−1.36432
$254$	0	0
$255$	1.22725	0.0768533
$256$	0	0
$257$	−5.58926	−0.348649	−0.174324	−	0.984688i	$-0.555774\pi$
$257$	−5.58926	−0.348649	−0.174324	+	0.984688i	$0.555774\pi$
$258$	0	0
$259$	−20.9302	−1.30054
$260$	0	0
$261$	12.9614	0.802288
$262$	0	0
$263$	−8.41046	−0.518611	−0.259305	−	0.965795i	$-0.583494\pi$
$263$	−8.41046	−0.518611	−0.259305	+	0.965795i	$0.583494\pi$
$264$	0	0
$265$	3.91666	0.240598
$266$	0	0
$267$	1.41207	0.0864172
$268$	0	0
$269$	−6.36903	−0.388327	−0.194163	−	0.980969i	$-0.562199\pi$
$269$	−6.36903	−0.388327	−0.194163	+	0.980969i	$0.562199\pi$
$270$	0	0
$271$	21.6443	1.31480	0.657398	−	0.753544i	$-0.271657\pi$
$271$	21.6443	1.31480	0.657398	+	0.753544i	$0.271657\pi$
$272$	0	0
$273$	16.5504	1.00168
$274$	0	0
$275$	−18.2065	−1.09789
$276$	0	0
$277$	−24.4894	−1.47142	−0.735712	−	0.677295i	$-0.763152\pi$
$277$	−24.4894	−1.47142	−0.735712	+	0.677295i	$0.763152\pi$
$278$	0	0
$279$	0.457863	0.0274115
$280$	0	0
$281$	−0.662298	−0.0395094	−0.0197547	−	0.999805i	$-0.506289\pi$
$281$	−0.662298	−0.0395094	−0.0197547	+	0.999805i	$0.506289\pi$
$282$	0	0
$283$	1.52952	0.0909208	0.0454604	−	0.998966i	$-0.485525\pi$
$283$	1.52952	0.0909208	0.0454604	+	0.998966i	$0.485525\pi$
$284$	0	0
$285$	−5.34102	−0.316375
$286$	0	0
$287$	14.9608	0.883108
$288$	0	0
$289$	1.00000	0.0588235
$290$	0	0
$291$	10.7931	0.632702
$292$	0	0
$293$	−13.2926	−0.776564	−0.388282	−	0.921541i	$-0.626931\pi$
$293$	−13.2926	−0.776564	−0.388282	+	0.921541i	$0.626931\pi$
$294$	0	0
$295$	1.03337	0.0601652
$296$	0	0
$297$	25.2374	1.46442
$298$	0	0
$299$	25.8488	1.49487
$300$	0	0
$301$	−11.3301	−0.653053
$302$	0	0
$303$	2.77585	0.159468
$304$	0	0
$305$	0.644396	0.0368980
$306$	0	0
$307$	−14.3656	−0.819887	−0.409944	−	0.912111i	$-0.634452\pi$
$307$	−14.3656	−0.819887	−0.409944	+	0.912111i	$0.634452\pi$
$308$	0	0
$309$	−7.13783	−0.406057
$310$	0	0
$311$	−17.6746	−1.00223	−0.501117	−	0.865380i	$-0.667077\pi$
$311$	−17.6746	−1.00223	−0.501117	+	0.865380i	$0.667077\pi$
$312$	0	0
$313$	−23.2229	−1.31264	−0.656318	−	0.754484i	$-0.727887\pi$
$313$	−23.2229	−1.31264	−0.656318	+	0.754484i	$0.727887\pi$
$314$	0	0
$315$	−4.15057	−0.233858
$316$	0	0
$317$	8.36278	0.469700	0.234850	−	0.972032i	$-0.424540\pi$
$317$	8.36278	0.469700	0.234850	+	0.972032i	$0.424540\pi$
$318$	0	0
$319$	−37.7544	−2.11384
$320$	0	0
$321$	23.1687	1.29315
$322$	0	0
$323$	−4.35203	−0.242153
$324$	0	0
$325$	21.6865	1.20295
$326$	0	0
$327$	−22.7147	−1.25612
$328$	0	0
$329$	−13.9184	−0.767346
$330$	0	0
$331$	−22.2097	−1.22076	−0.610379	−	0.792110i	$-0.708983\pi$
$331$	−22.2097	−1.22076	−0.610379	+	0.792110i	$0.708983\pi$
$332$	0	0
$333$	13.1668	0.721535
$334$	0	0
$335$	−11.1353	−0.608385
$336$	0	0
$337$	24.1815	1.31725	0.658625	−	0.752471i	$-0.271138\pi$
$337$	24.1815	1.31725	0.658625	+	0.752471i	$0.271138\pi$
$338$	0	0
$339$	6.28442	0.341323
$340$	0	0
$341$	−1.33368	−0.0722229
$342$	0	0
$343$	−19.2423	−1.03898
$344$	0	0
$345$	5.75191	0.309672
$346$	0	0
$347$	21.7712	1.16874	0.584370	−	0.811487i	$-0.301342\pi$
$347$	21.7712	1.16874	0.584370	+	0.811487i	$0.301342\pi$
$348$	0	0
$349$	−0.197978	−0.0105975	−0.00529877	−	0.999986i	$-0.501687\pi$
$349$	−0.197978	−0.0105975	−0.00529877	+	0.999986i	$0.501687\pi$
$350$	0	0
$351$	−30.0614	−1.60456
$352$	0	0
$353$	−27.6504	−1.47168	−0.735841	−	0.677154i	$-0.763213\pi$
$353$	−27.6504	−1.47168	−0.735841	+	0.677154i	$0.763213\pi$
$354$	0	0
$355$	1.87980	0.0997692
$356$	0	0
$357$	3.00088	0.158823
$358$	0	0
$359$	−14.5218	−0.766429	−0.383214	−	0.923659i	$-0.625183\pi$
$359$	−14.5218	−0.766429	−0.383214	+	0.923659i	$0.625183\pi$
$360$	0	0
$361$	−0.0598483	−0.00314991
$362$	0	0
$363$	−12.3968	−0.650662
$364$	0	0
$365$	0.924229	0.0483763
$366$	0	0
$367$	24.0002	1.25280	0.626401	−	0.779501i	$-0.284527\pi$
$367$	24.0002	1.25280	0.626401	+	0.779501i	$0.284527\pi$
$368$	0	0
$369$	−9.41157	−0.489947
$370$	0	0
$371$	9.57704	0.497215
$372$	0	0
$373$	−10.6393	−0.550881	−0.275441	−	0.961318i	$-0.588824\pi$
$373$	−10.6393	−0.550881	−0.275441	+	0.961318i	$0.588824\pi$
$374$	0	0
$375$	10.9620	0.566073
$376$	0	0
$377$	44.9710	2.31612
$378$	0	0
$379$	3.06460	0.157418	0.0787089	−	0.996898i	$-0.474920\pi$
$379$	3.06460	0.157418	0.0787089	+	0.996898i	$0.474920\pi$
$380$	0	0
$381$	−17.3017	−0.886394
$382$	0	0
$383$	−29.1353	−1.48874	−0.744372	−	0.667765i	$-0.767251\pi$
$383$	−29.1353	−1.48874	−0.744372	+	0.667765i	$0.767251\pi$
$384$	0	0
$385$	12.0899	0.616161
$386$	0	0
$387$	7.12753	0.362313
$388$	0	0
$389$	17.8129	0.903150	0.451575	−	0.892233i	$-0.350862\pi$
$389$	17.8129	0.903150	0.451575	+	0.892233i	$0.350862\pi$
$390$	0	0
$391$	4.68684	0.237023
$392$	0	0
$393$	1.95596	0.0986651
$394$	0	0
$395$	0.669214	0.0336718
$396$	0	0
$397$	35.9205	1.80280	0.901400	−	0.432988i	$-0.142541\pi$
$397$	35.9205	1.80280	0.901400	+	0.432988i	$0.142541\pi$
$398$	0	0
$399$	−13.0599	−0.653813
$400$	0	0
$401$	−32.0902	−1.60251	−0.801255	−	0.598323i	$-0.795834\pi$
$401$	−32.0902	−1.60251	−0.801255	+	0.598323i	$0.795834\pi$
$402$	0	0
$403$	1.58861	0.0791342
$404$	0	0
$405$	−1.76145	−0.0875274
$406$	0	0
$407$	−38.3527	−1.90107
$408$	0	0
$409$	−32.6614	−1.61500	−0.807500	−	0.589867i	$-0.799180\pi$
$409$	−32.6614	−1.61500	−0.807500	+	0.589867i	$0.799180\pi$
$410$	0	0
$411$	−2.37320	−0.117061
$412$	0	0
$413$	2.52681	0.124336
$414$	0	0
$415$	5.72324	0.280943
$416$	0	0
$417$	19.7894	0.969090
$418$	0	0
$419$	−16.6891	−0.815314	−0.407657	−	0.913135i	$-0.633654\pi$
$419$	−16.6891	−0.815314	−0.407657	+	0.913135i	$0.633654\pi$
$420$	0	0
$421$	12.8125	0.624441	0.312221	−	0.950010i	$-0.398927\pi$
$421$	12.8125	0.624441	0.312221	+	0.950010i	$0.398927\pi$
$422$	0	0
$423$	8.75581	0.425722
$424$	0	0
$425$	3.93214	0.190737
$426$	0	0
$427$	1.57568	0.0762527
$428$	0	0
$429$	30.3272	1.46421
$430$	0	0
$431$	13.5966	0.654927	0.327463	−	0.944864i	$-0.393806\pi$
$431$	13.5966	0.654927	0.327463	+	0.944864i	$0.393806\pi$
$432$	0	0
$433$	28.9831	1.39284	0.696420	−	0.717634i	$-0.254775\pi$
$433$	28.9831	1.39284	0.696420	+	0.717634i	$0.254775\pi$
$434$	0	0
$435$	10.0070	0.479799
$436$	0	0
$437$	−20.3972	−0.975732
$438$	0	0
$439$	−14.8551	−0.708997	−0.354498	−	0.935057i	$-0.615348\pi$
$439$	−14.8551	−0.708997	−0.354498	+	0.935057i	$0.615348\pi$
$440$	0	0
$441$	0.977970	0.0465700
$442$	0	0
$443$	−4.99505	−0.237322	−0.118661	−	0.992935i	$-0.537860\pi$
$443$	−4.99505	−0.237322	−0.118661	+	0.992935i	$0.537860\pi$
$444$	0	0
$445$	−1.22867	−0.0582447
$446$	0	0
$447$	15.7778	0.746266
$448$	0	0
$449$	32.6933	1.54289	0.771445	−	0.636296i	$-0.219534\pi$
$449$	32.6933	1.54289	0.771445	+	0.636296i	$0.219534\pi$
$450$	0	0
$451$	27.4144	1.29089
$452$	0	0
$453$	11.2297	0.527619
$454$	0	0
$455$	−14.4009	−0.675124
$456$	0	0
$457$	29.8505	1.39635	0.698174	−	0.715928i	$-0.253996\pi$
$457$	29.8505	1.39635	0.698174	+	0.715928i	$0.253996\pi$
$458$	0	0
$459$	−5.45065	−0.254414
$460$	0	0
$461$	−23.9016	−1.11321	−0.556604	−	0.830778i	$-0.687896\pi$
$461$	−23.9016	−1.11321	−0.556604	+	0.830778i	$0.687896\pi$
$462$	0	0
$463$	33.1565	1.54091	0.770457	−	0.637492i	$-0.220028\pi$
$463$	33.1565	1.54091	0.770457	+	0.637492i	$0.220028\pi$
$464$	0	0
$465$	0.353499	0.0163931
$466$	0	0
$467$	25.0742	1.16030	0.580149	−	0.814511i	$-0.302994\pi$
$467$	25.0742	1.16030	0.580149	+	0.814511i	$0.302994\pi$
$468$	0	0
$469$	−27.2281	−1.25728
$470$	0	0
$471$	−14.4516	−0.665896
$472$	0	0
$473$	−20.7614	−0.954608
$474$	0	0
$475$	−17.1128	−0.785189
$476$	0	0
$477$	−6.02474	−0.275854
$478$	0	0
$479$	6.64989	0.303841	0.151921	−	0.988393i	$-0.451454\pi$
$479$	6.64989	0.303841	0.151921	+	0.988393i	$0.451454\pi$
$480$	0	0
$481$	45.6837	2.08300
$482$	0	0
$483$	14.0646	0.639962
$484$	0	0
$485$	−9.39131	−0.426437
$486$	0	0
$487$	20.5561	0.931487	0.465744	−	0.884920i	$-0.345787\pi$
$487$	20.5561	0.931487	0.465744	+	0.884920i	$0.345787\pi$
$488$	0	0
$489$	28.1424	1.27265
$490$	0	0
$491$	−21.1923	−0.956393	−0.478197	−	0.878253i	$-0.658709\pi$
$491$	−21.1923	−0.956393	−0.478197	+	0.878253i	$0.658709\pi$
$492$	0	0
$493$	8.15402	0.367238
$494$	0	0
$495$	−7.60556	−0.341845
$496$	0	0
$497$	4.59649	0.206181
$498$	0	0
$499$	−19.2403	−0.861313	−0.430656	−	0.902516i	$-0.641718\pi$
$499$	−19.2403	−0.861313	−0.430656	+	0.902516i	$0.641718\pi$
$500$	0	0
$501$	−18.0090	−0.804583
$502$	0	0
$503$	−13.3508	−0.595285	−0.297642	−	0.954677i	$-0.596200\pi$
$503$	−13.3508	−0.595285	−0.297642	+	0.954677i	$0.596200\pi$
$504$	0	0
$505$	−2.41533	−0.107481
$506$	0	0
$507$	−20.6851	−0.918659
$508$	0	0
$509$	37.7144	1.67166	0.835832	−	0.548986i	$-0.184986\pi$
$509$	37.7144	1.67166	0.835832	+	0.548986i	$0.184986\pi$
$510$	0	0
$511$	2.25993	0.0999735
$512$	0	0
$513$	23.7214	1.04732
$514$	0	0
$515$	6.21078	0.273680
$516$	0	0
$517$	−25.5043	−1.12168
$518$	0	0
$519$	−15.6047	−0.684970
$520$	0	0
$521$	5.26252	0.230555	0.115277	−	0.993333i	$-0.463224\pi$
$521$	5.26252	0.230555	0.115277	+	0.993333i	$0.463224\pi$
$522$	0	0
$523$	−8.41943	−0.368156	−0.184078	−	0.982912i	$-0.558930\pi$
$523$	−8.41943	−0.368156	−0.184078	+	0.982912i	$0.558930\pi$
$524$	0	0
$525$	11.7999	0.514989
$526$	0	0
$527$	0.288042	0.0125473
$528$	0	0
$529$	−1.03358	−0.0449381
$530$	0	0
$531$	−1.58957	−0.0689814
$532$	0	0
$533$	−32.6545	−1.41442
$534$	0	0
$535$	−20.1596	−0.871577
$536$	0	0
$537$	−16.5902	−0.715921
$538$	0	0
$539$	−2.84867	−0.122701
$540$	0	0
$541$	20.0821	0.863398	0.431699	−	0.902018i	$-0.357914\pi$
$541$	20.0821	0.863398	0.431699	+	0.902018i	$0.357914\pi$
$542$	0	0
$543$	−7.40726	−0.317876
$544$	0	0
$545$	19.7646	0.846620
$546$	0	0
$547$	2.01168	0.0860131	0.0430066	−	0.999075i	$-0.486306\pi$
$547$	2.01168	0.0860131	0.0430066	+	0.999075i	$0.486306\pi$
$548$	0	0
$549$	−0.991233	−0.0423048
$550$	0	0
$551$	−35.4865	−1.51178
$552$	0	0
$553$	1.63637	0.0695854
$554$	0	0
$555$	10.1656	0.431505
$556$	0	0
$557$	21.1857	0.897668	0.448834	−	0.893615i	$-0.351840\pi$
$557$	21.1857	0.897668	0.448834	+	0.893615i	$0.351840\pi$
$558$	0	0
$559$	24.7298	1.04596
$560$	0	0
$561$	5.49885	0.232162
$562$	0	0
$563$	−13.3745	−0.563667	−0.281834	−	0.959463i	$-0.590943\pi$
$563$	−13.3745	−0.563667	−0.281834	+	0.959463i	$0.590943\pi$
$564$	0	0
$565$	−5.46821	−0.230049
$566$	0	0
$567$	−4.30712	−0.180882
$568$	0	0
$569$	22.8533	0.958060	0.479030	−	0.877799i	$-0.340988\pi$
$569$	22.8533	0.958060	0.479030	+	0.877799i	$0.340988\pi$
$570$	0	0
$571$	11.3829	0.476359	0.238179	−	0.971221i	$-0.423449\pi$
$571$	11.3829	0.476359	0.238179	+	0.971221i	$0.423449\pi$
$572$	0	0
$573$	−7.46011	−0.311651
$574$	0	0
$575$	18.4293	0.768555
$576$	0	0
$577$	−20.1452	−0.838655	−0.419328	−	0.907835i	$-0.637734\pi$
$577$	−20.1452	−0.838655	−0.419328	+	0.907835i	$0.637734\pi$
$578$	0	0
$579$	21.5351	0.894967
$580$	0	0
$581$	13.9945	0.580590
$582$	0	0
$583$	17.5491	0.726810
$584$	0	0
$585$	9.05933	0.374557
$586$	0	0
$587$	11.3827	0.469816	0.234908	−	0.972018i	$-0.424521\pi$
$587$	11.3827	0.469816	0.234908	+	0.972018i	$0.424521\pi$
$588$	0	0
$589$	−1.25357	−0.0516524
$590$	0	0
$591$	14.6589	0.602986
$592$	0	0
$593$	−42.7338	−1.75487	−0.877433	−	0.479699i	$-0.840746\pi$
$593$	−42.7338	−1.75487	−0.877433	+	0.479699i	$0.840746\pi$
$594$	0	0
$595$	−2.61113	−0.107046
$596$	0	0
$597$	9.21941	0.377325
$598$	0	0
$599$	30.7320	1.25568	0.627839	−	0.778344i	$-0.283940\pi$
$599$	30.7320	1.25568	0.627839	+	0.778344i	$0.283940\pi$
$600$	0	0
$601$	−30.1861	−1.23132	−0.615658	−	0.788014i	$-0.711110\pi$
$601$	−30.1861	−1.23132	−0.615658	+	0.788014i	$0.711110\pi$
$602$	0	0
$603$	17.1287	0.697533
$604$	0	0
$605$	10.7867	0.438542
$606$	0	0
$607$	21.0474	0.854288	0.427144	−	0.904184i	$-0.359520\pi$
$607$	21.0474	0.854288	0.427144	+	0.904184i	$0.359520\pi$
$608$	0	0
$609$	24.4692	0.991542
$610$	0	0
$611$	30.3793	1.22902
$612$	0	0
$613$	22.4809	0.907995	0.453998	−	0.891003i	$-0.349997\pi$
$613$	22.4809	0.907995	0.453998	+	0.891003i	$0.349997\pi$
$614$	0	0
$615$	−7.26633	−0.293007
$616$	0	0
$617$	−47.7816	−1.92361	−0.961806	−	0.273730i	$-0.911742\pi$
$617$	−47.7816	−1.92361	−0.961806	+	0.273730i	$0.911742\pi$
$618$	0	0
$619$	−1.12745	−0.0453161	−0.0226580	−	0.999743i	$-0.507213\pi$
$619$	−1.12745	−0.0453161	−0.0226580	+	0.999743i	$0.507213\pi$
$620$	0	0
$621$	−25.5463	−1.02514
$622$	0	0
$623$	−3.00436	−0.120367
$624$	0	0
$625$	10.1225	0.404899
$626$	0	0
$627$	−23.9312	−0.955719
$628$	0	0
$629$	8.28324	0.330274
$630$	0	0
$631$	−10.8411	−0.431576	−0.215788	−	0.976440i	$-0.569232\pi$
$631$	−10.8411	−0.431576	−0.215788	+	0.976440i	$0.569232\pi$
$632$	0	0
$633$	−15.3203	−0.608926
$634$	0	0
$635$	15.0546	0.597424
$636$	0	0
$637$	3.39318	0.134443
$638$	0	0
$639$	−2.89157	−0.114389
$640$	0	0
$641$	17.4969	0.691086	0.345543	−	0.938403i	$-0.387695\pi$
$641$	17.4969	0.691086	0.345543	+	0.938403i	$0.387695\pi$
$642$	0	0
$643$	39.7035	1.56575	0.782877	−	0.622176i	$-0.213751\pi$
$643$	39.7035	1.56575	0.782877	+	0.622176i	$0.213751\pi$
$644$	0	0
$645$	5.50291	0.216677
$646$	0	0
$647$	−17.0242	−0.669290	−0.334645	−	0.942344i	$-0.608616\pi$
$647$	−17.0242	−0.669290	−0.334645	+	0.942344i	$0.608616\pi$
$648$	0	0
$649$	4.63016	0.181750
$650$	0	0
$651$	0.864380	0.0338777
$652$	0	0
$653$	40.7887	1.59618	0.798092	−	0.602536i	$-0.205843\pi$
$653$	40.7887	1.59618	0.798092	+	0.602536i	$0.205843\pi$
$654$	0	0
$655$	−1.70192	−0.0664996
$656$	0	0
$657$	−1.42168	−0.0554651
$658$	0	0
$659$	−7.20821	−0.280792	−0.140396	−	0.990095i	$-0.544838\pi$
$659$	−7.20821	−0.280792	−0.140396	+	0.990095i	$0.544838\pi$
$660$	0	0
$661$	34.9067	1.35771	0.678857	−	0.734270i	$-0.262475\pi$
$661$	34.9067	1.35771	0.678857	+	0.734270i	$0.262475\pi$
$662$	0	0
$663$	−6.54993	−0.254378
$664$	0	0
$665$	11.3637	0.440666
$666$	0	0
$667$	38.2165	1.47975
$668$	0	0
$669$	17.7940	0.687957
$670$	0	0
$671$	2.88730	0.111463
$672$	0	0
$673$	−6.96846	−0.268614	−0.134307	−	0.990940i	$-0.542881\pi$
$673$	−6.96846	−0.268614	−0.134307	+	0.990940i	$0.542881\pi$
$674$	0	0
$675$	−21.4327	−0.824946
$676$	0	0
$677$	28.8553	1.10900	0.554500	−	0.832184i	$-0.312910\pi$
$677$	28.8553	1.10900	0.554500	+	0.832184i	$0.312910\pi$
$678$	0	0
$679$	−22.9637	−0.881266
$680$	0	0
$681$	4.34353	0.166445
$682$	0	0
$683$	40.0525	1.53257	0.766283	−	0.642503i	$-0.222104\pi$
$683$	40.0525	1.53257	0.766283	+	0.642503i	$0.222104\pi$
$684$	0	0
$685$	2.06497	0.0788984
$686$	0	0
$687$	−12.2433	−0.467112
$688$	0	0
$689$	−20.9035	−0.796361
$690$	0	0
$691$	−10.5665	−0.401969	−0.200985	−	0.979594i	$-0.564414\pi$
$691$	−10.5665	−0.401969	−0.200985	+	0.979594i	$0.564414\pi$
$692$	0	0
$693$	−18.5972	−0.706449
$694$	0	0
$695$	−17.2192	−0.653161
$696$	0	0
$697$	−5.92083	−0.224268
$698$	0	0
$699$	29.2954	1.10806
$700$	0	0
$701$	−35.6541	−1.34664	−0.673319	−	0.739352i	$-0.735132\pi$
$701$	−35.6541	−1.34664	−0.673319	+	0.739352i	$0.735132\pi$
$702$	0	0
$703$	−36.0489	−1.35961
$704$	0	0
$705$	6.76005	0.254598
$706$	0	0
$707$	−5.90598	−0.222117
$708$	0	0
$709$	−13.2597	−0.497979	−0.248990	−	0.968506i	$-0.580098\pi$
$709$	−13.2597	−0.497979	−0.248990	+	0.968506i	$0.580098\pi$
$710$	0	0
$711$	−1.02941	−0.0386058
$712$	0	0
$713$	1.35001	0.0505582
$714$	0	0
$715$	−26.3884	−0.986870
$716$	0	0
$717$	6.06893	0.226648
$718$	0	0
$719$	−35.7261	−1.33236	−0.666180	−	0.745791i	$-0.732072\pi$
$719$	−35.7261	−1.33236	−0.666180	+	0.745791i	$0.732072\pi$
$720$	0	0
$721$	15.1867	0.565581
$722$	0	0
$723$	21.9269	0.815471
$724$	0	0
$725$	32.0628	1.19078
$726$	0	0
$727$	45.6976	1.69483	0.847415	−	0.530932i	$-0.178158\pi$
$727$	45.6976	1.69483	0.847415	+	0.530932i	$0.178158\pi$
$728$	0	0
$729$	22.1294	0.819606
$730$	0	0
$731$	4.48394	0.165845
$732$	0	0
$733$	−25.7661	−0.951691	−0.475846	−	0.879529i	$-0.657858\pi$
$733$	−25.7661	−0.951691	−0.475846	+	0.879529i	$0.657858\pi$
$734$	0	0
$735$	0.755056	0.0278507
$736$	0	0
$737$	−49.8931	−1.83784
$738$	0	0
$739$	25.5492	0.939843	0.469922	−	0.882708i	$-0.344282\pi$
$739$	25.5492	0.939843	0.469922	+	0.882708i	$0.344282\pi$
$740$	0	0
$741$	28.5055	1.04718
$742$	0	0
$743$	44.7258	1.64083	0.820415	−	0.571769i	$-0.193743\pi$
$743$	44.7258	1.64083	0.820415	+	0.571769i	$0.193743\pi$
$744$	0	0
$745$	−13.7286	−0.502978
$746$	0	0
$747$	−8.80369	−0.322110
$748$	0	0
$749$	−49.2945	−1.80118
$750$	0	0
$751$	27.4166	1.00045	0.500224	−	0.865896i	$-0.333251\pi$
$751$	27.4166	1.00045	0.500224	+	0.865896i	$0.333251\pi$
$752$	0	0
$753$	−8.83985	−0.322142
$754$	0	0
$755$	−9.77124	−0.355612
$756$	0	0
$757$	26.8335	0.975280	0.487640	−	0.873045i	$-0.337858\pi$
$757$	26.8335	0.975280	0.487640	+	0.873045i	$0.337858\pi$
$758$	0	0
$759$	25.7722	0.935472
$760$	0	0
$761$	−37.6103	−1.36337	−0.681686	−	0.731645i	$-0.738753\pi$
$761$	−37.6103	−1.36337	−0.681686	+	0.731645i	$0.738753\pi$
$762$	0	0
$763$	48.3284	1.74961
$764$	0	0
$765$	1.64261	0.0593888
$766$	0	0
$767$	−5.51519	−0.199142
$768$	0	0
$769$	33.0477	1.19173	0.595866	−	0.803084i	$-0.296809\pi$
$769$	33.0477	1.19173	0.595866	+	0.803084i	$0.296809\pi$
$770$	0	0
$771$	6.63790	0.239058
$772$	0	0
$773$	−28.1772	−1.01347	−0.506733	−	0.862103i	$-0.669147\pi$
$773$	−28.1772	−1.01347	−0.506733	+	0.862103i	$0.669147\pi$
$774$	0	0
$775$	1.13262	0.0406851
$776$	0	0
$777$	24.8570	0.891740
$778$	0	0
$779$	25.7676	0.923221
$780$	0	0
$781$	8.42268	0.301387
$782$	0	0
$783$	−44.4446	−1.58832
$784$	0	0
$785$	12.5747	0.448810
$786$	0	0
$787$	14.5459	0.518507	0.259253	−	0.965809i	$-0.416524\pi$
$787$	14.5459	0.518507	0.259253	+	0.965809i	$0.416524\pi$
$788$	0	0
$789$	9.98839	0.355596
$790$	0	0
$791$	−13.3709	−0.475415
$792$	0	0
$793$	−3.43920	−0.122130
$794$	0	0
$795$	−4.65148	−0.164971
$796$	0	0
$797$	22.5698	0.799462	0.399731	−	0.916633i	$-0.369104\pi$
$797$	22.5698	0.799462	0.399731	+	0.916633i	$0.369104\pi$
$798$	0	0
$799$	5.50830	0.194869
$800$	0	0
$801$	1.88999	0.0667794
$802$	0	0
$803$	4.14113	0.146137
$804$	0	0
$805$	−12.2379	−0.431331
$806$	0	0
$807$	7.56396	0.266264
$808$	0	0
$809$	3.61148	0.126973	0.0634865	−	0.997983i	$-0.479778\pi$
$809$	3.61148	0.126973	0.0634865	+	0.997983i	$0.479778\pi$
$810$	0	0
$811$	33.7605	1.18549	0.592746	−	0.805390i	$-0.298044\pi$
$811$	33.7605	1.18549	0.592746	+	0.805390i	$0.298044\pi$
$812$	0	0
$813$	−25.7051	−0.901517
$814$	0	0
$815$	−24.4874	−0.857755
$816$	0	0
$817$	−19.5142	−0.682717
$818$	0	0
$819$	22.1519	0.774052
$820$	0	0
$821$	−7.93314	−0.276869	−0.138434	−	0.990372i	$-0.544207\pi$
$821$	−7.93314	−0.276869	−0.138434	+	0.990372i	$0.544207\pi$
$822$	0	0
$823$	−12.6073	−0.439463	−0.219731	−	0.975560i	$-0.570518\pi$
$823$	−12.6073	−0.439463	−0.219731	+	0.975560i	$0.570518\pi$
$824$	0	0
$825$	21.6223	0.752791
$826$	0	0
$827$	44.0830	1.53292	0.766458	−	0.642294i	$-0.222017\pi$
$827$	44.0830	1.53292	0.766458	+	0.642294i	$0.222017\pi$
$828$	0	0
$829$	−17.4568	−0.606300	−0.303150	−	0.952943i	$-0.598038\pi$
$829$	−17.4568	−0.606300	−0.303150	+	0.952943i	$0.598038\pi$
$830$	0	0
$831$	29.0840	1.00891
$832$	0	0
$833$	0.615243	0.0213169
$834$	0	0
$835$	15.6700	0.542284
$836$	0	0
$837$	−1.57002	−0.0542677
$838$	0	0
$839$	−52.3730	−1.80812	−0.904058	−	0.427410i	$-0.859426\pi$
$839$	−52.3730	−1.80812	−0.904058	+	0.427410i	$0.859426\pi$
$840$	0	0
$841$	37.4880	1.29269
$842$	0	0
$843$	0.786555	0.0270904
$844$	0	0
$845$	17.9986	0.619170
$846$	0	0
$847$	26.3758	0.906282
$848$	0	0
$849$	−1.81649	−0.0623417
$850$	0	0
$851$	38.8222	1.33081
$852$	0	0
$853$	47.1584	1.61467	0.807336	−	0.590092i	$-0.200909\pi$
$853$	47.1584	1.61467	0.807336	+	0.590092i	$0.200909\pi$
$854$	0	0
$855$	−7.14870	−0.244480
$856$	0	0
$857$	−9.84517	−0.336305	−0.168152	−	0.985761i	$-0.553780\pi$
$857$	−9.84517	−0.336305	−0.168152	+	0.985761i	$0.553780\pi$
$858$	0	0
$859$	16.2690	0.555091	0.277546	−	0.960712i	$-0.410479\pi$
$859$	16.2690	0.555091	0.277546	+	0.960712i	$0.410479\pi$
$860$	0	0
$861$	−17.7677	−0.605521
$862$	0	0
$863$	18.7237	0.637362	0.318681	−	0.947862i	$-0.396760\pi$
$863$	18.7237	0.637362	0.318681	+	0.947862i	$0.396760\pi$
$864$	0	0
$865$	13.5780	0.461665
$866$	0	0
$867$	−1.18762	−0.0403336
$868$	0	0
$869$	2.99850	0.101717
$870$	0	0
$871$	59.4299	2.01371
$872$	0	0
$873$	14.4460	0.488924
$874$	0	0
$875$	−23.3230	−0.788461
$876$	0	0
$877$	47.9591	1.61946	0.809732	−	0.586799i	$-0.199612\pi$
$877$	47.9591	1.61946	0.809732	+	0.586799i	$0.199612\pi$
$878$	0	0
$879$	15.7865	0.532467
$880$	0	0
$881$	−36.3116	−1.22337	−0.611685	−	0.791102i	$-0.709508\pi$
$881$	−36.3116	−1.22337	−0.611685	+	0.791102i	$0.709508\pi$
$882$	0	0
$883$	14.7832	0.497493	0.248746	−	0.968569i	$-0.419981\pi$
$883$	14.7832	0.497493	0.248746	+	0.968569i	$0.419981\pi$
$884$	0	0
$885$	−1.22725	−0.0412535
$886$	0	0
$887$	−17.5647	−0.589765	−0.294882	−	0.955534i	$-0.595281\pi$
$887$	−17.5647	−0.589765	−0.294882	+	0.955534i	$0.595281\pi$
$888$	0	0
$889$	36.8117	1.23462
$890$	0	0
$891$	−7.89244	−0.264407
$892$	0	0
$893$	−23.9723	−0.802201
$894$	0	0
$895$	14.4355	0.482526
$896$	0	0
$897$	−30.6985	−1.02499
$898$	0	0
$899$	2.34870	0.0783336
$900$	0	0
$901$	−3.79017	−0.126269
$902$	0	0
$903$	13.4558	0.447779
$904$	0	0
$905$	6.44522	0.214247
$906$	0	0
$907$	−26.9853	−0.896034	−0.448017	−	0.894025i	$-0.647870\pi$
$907$	−26.9853	−0.896034	−0.448017	+	0.894025i	$0.647870\pi$
$908$	0	0
$909$	3.71534	0.123230
$910$	0	0
$911$	15.0678	0.499217	0.249609	−	0.968347i	$-0.419698\pi$
$911$	15.0678	0.499217	0.249609	+	0.968347i	$0.419698\pi$
$912$	0	0
$913$	25.6437	0.848684
$914$	0	0
$915$	−0.765296	−0.0252999
$916$	0	0
$917$	−4.16156	−0.137427
$918$	0	0
$919$	46.0500	1.51905	0.759524	−	0.650479i	$-0.225432\pi$
$919$	46.0500	1.51905	0.759524	+	0.650479i	$0.225432\pi$
$920$	0	0
$921$	17.0608	0.562172
$922$	0	0
$923$	−10.0326	−0.330228
$924$	0	0
$925$	32.5709	1.07092
$926$	0	0
$927$	−9.55365	−0.313783
$928$	0	0
$929$	−18.6977	−0.613451	−0.306725	−	0.951798i	$-0.599233\pi$
$929$	−18.6977	−0.613451	−0.306725	+	0.951798i	$0.599233\pi$
$930$	0	0
$931$	−2.67755	−0.0877533
$932$	0	0
$933$	20.9906	0.687202
$934$	0	0
$935$	−4.78467	−0.156476
$936$	0	0
$937$	−3.18701	−0.104115	−0.0520576	−	0.998644i	$-0.516578\pi$
$937$	−3.18701	−0.104115	−0.0520576	+	0.998644i	$0.516578\pi$
$938$	0	0
$939$	27.5799	0.900036
$940$	0	0
$941$	13.3743	0.435989	0.217994	−	0.975950i	$-0.430049\pi$
$941$	13.3743	0.435989	0.217994	+	0.975950i	$0.430049\pi$
$942$	0	0
$943$	−27.7500	−0.903663
$944$	0	0
$945$	14.2323	0.462978
$946$	0	0
$947$	26.1771	0.850641	0.425321	−	0.905043i	$-0.360161\pi$
$947$	26.1771	0.850641	0.425321	+	0.905043i	$0.360161\pi$
$948$	0	0
$949$	−4.93269	−0.160122
$950$	0	0
$951$	−9.93177	−0.322060
$952$	0	0
$953$	−44.8044	−1.45136	−0.725679	−	0.688034i	$-0.758474\pi$
$953$	−44.8044	−1.45136	−0.725679	+	0.688034i	$0.758474\pi$
$954$	0	0
$955$	6.49121	0.210051
$956$	0	0
$957$	44.8377	1.44940
$958$	0	0
$959$	5.04928	0.163050
$960$	0	0
$961$	−30.9170	−0.997324
$962$	0	0
$963$	31.0103	0.999291
$964$	0	0
$965$	−18.7381	−0.603202
$966$	0	0
$967$	14.2867	0.459430	0.229715	−	0.973258i	$-0.426221\pi$
$967$	14.2867	0.459430	0.229715	+	0.973258i	$0.426221\pi$
$968$	0	0
$969$	5.16854	0.166037
$970$	0	0
$971$	33.2314	1.06645	0.533224	−	0.845974i	$-0.320980\pi$
$971$	33.2314	1.06645	0.533224	+	0.845974i	$0.320980\pi$
$972$	0	0
$973$	−42.1045	−1.34981
$974$	0	0
$975$	−25.7553	−0.824829
$976$	0	0
$977$	−7.86530	−0.251633	−0.125817	−	0.992054i	$-0.540155\pi$
$977$	−7.86530	−0.251633	−0.125817	+	0.992054i	$0.540155\pi$
$978$	0	0
$979$	−5.50523	−0.175948
$980$	0	0
$981$	−30.4025	−0.970678
$982$	0	0
$983$	−60.8191	−1.93983	−0.969914	−	0.243447i	$-0.921722\pi$
$983$	−60.8191	−1.93983	−0.969914	+	0.243447i	$0.921722\pi$
$984$	0	0
$985$	−12.7550	−0.406409
$986$	0	0
$987$	16.5297	0.526147
$988$	0	0
$989$	21.0155	0.668254
$990$	0	0
$991$	−12.1698	−0.386588	−0.193294	−	0.981141i	$-0.561917\pi$
$991$	−12.1698	−0.386588	−0.193294	+	0.981141i	$0.561917\pi$
$992$	0	0
$993$	26.3766	0.837037
$994$	0	0
$995$	−8.02201	−0.254315
$996$	0	0
$997$	−50.0742	−1.58587	−0.792933	−	0.609308i	$-0.791447\pi$
$997$	−50.0742	−1.58587	−0.792933	+	0.609308i	$0.791447\pi$
$998$	0	0
$999$	−45.1490	−1.42845

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	4012.2.a.g.1.5	✓	12

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
4012.2.a.g.1.5	✓	12	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)

\(f(q)\)	\(=\)	\(q-1.18762 q^{3} +1.03337 q^{5} +2.52681 q^{7} -1.58957 q^{9} +O(q^{10})\)	\(q-1.18762 q^{3} +1.03337 q^{5} +2.52681 q^{7} -1.58957 q^{9} +4.63016 q^{11} -5.51519 q^{13} -1.22725 q^{15} -1.00000 q^{17} +4.35203 q^{19} -3.00088 q^{21} -4.68684 q^{23} -3.93214 q^{25} +5.45065 q^{27} -8.15402 q^{29} -0.288042 q^{31} -5.49885 q^{33} +2.61113 q^{35} -8.28324 q^{37} +6.54993 q^{39} +5.92083 q^{41} -4.48394 q^{43} -1.64261 q^{45} -5.50830 q^{47} -0.615243 q^{49} +1.18762 q^{51} +3.79017 q^{53} +4.78467 q^{55} -5.16854 q^{57} +1.00000 q^{59} +0.623587 q^{61} -4.01653 q^{63} -5.69924 q^{65} -10.7757 q^{67} +5.56616 q^{69} +1.81909 q^{71} +0.894382 q^{73} +4.66988 q^{75} +11.6995 q^{77} +0.647603 q^{79} -1.70457 q^{81} +5.53842 q^{83} -1.03337 q^{85} +9.68384 q^{87} -1.18899 q^{89} -13.9358 q^{91} +0.342084 q^{93} +4.49726 q^{95} -9.08803 q^{97} -7.35995 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 12 q - 4 q^{3} + 3 q^{5} + 2 q^{7} + 6 q^{9}+O(q^{10}) \) 12 * q - 4 * q^3 + 3 * q^5 + 2 * q^7 + 6 * q^9	\( 12 q - 4 q^{3} + 3 q^{5} + 2 q^{7} + 6 q^{9} - 5 q^{11} + 9 q^{13} - 16 q^{15} - 12 q^{17} - 15 q^{19} + 5 q^{21} - 22 q^{23} + 5 q^{25} - 16 q^{27} - 17 q^{29} + 5 q^{31} - 33 q^{33} - 9 q^{35} - 2 q^{37} - 24 q^{39} - 18 q^{41} - 16 q^{43} + 30 q^{45} - 60 q^{47} + 10 q^{49} + 4 q^{51} - 24 q^{53} - q^{55} + 41 q^{57} + 12 q^{59} - 17 q^{61} - 23 q^{63} + 2 q^{65} - 22 q^{67} + 6 q^{69} - 10 q^{71} + 4 q^{73} - 62 q^{75} - 6 q^{77} + 48 q^{79} + 4 q^{81} - 20 q^{83} - 3 q^{85} + 13 q^{87} - 29 q^{89} - 28 q^{91} - 31 q^{93} - 48 q^{95} - 26 q^{97} - 3 q^{99}+O(q^{100}) \) 12 * q - 4 * q^3 + 3 * q^5 + 2 * q^7 + 6 * q^9 - 5 * q^11 + 9 * q^13 - 16 * q^15 - 12 * q^17 - 15 * q^19 + 5 * q^21 - 22 * q^23 + 5 * q^25 - 16 * q^27 - 17 * q^29 + 5 * q^31 - 33 * q^33 - 9 * q^35 - 2 * q^37 - 24 * q^39 - 18 * q^41 - 16 * q^43 + 30 * q^45 - 60 * q^47 + 10 * q^49 + 4 * q^51 - 24 * q^53 - q^55 + 41 * q^57 + 12 * q^59 - 17 * q^61 - 23 * q^63 + 2 * q^65 - 22 * q^67 + 6 * q^69 - 10 * q^71 + 4 * q^73 - 62 * q^75 - 6 * q^77 + 48 * q^79 + 4 * q^81 - 20 * q^83 - 3 * q^85 + 13 * q^87 - 29 * q^89 - 28 * q^91 - 31 * q^93 - 48 * q^95 - 26 * q^97 - 3 * q^99

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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