LMFDB - Embedded newform 4012.2.a.h.1.2

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

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

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(4012, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([0, 0, 0]))

N = Newforms(chi, 2, names="a")

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

chi := DirichletCharacter("4012.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 4012 = 2^{2} \cdot 17 \cdot 59 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	4012.a (trivial)

Newform invariants

comment: select newform

sage: f = N[0] # Warning: the index may be different

gp: f = lf[1] \\ Warning: the index may be different

Self dual:	yes
Analytic conductor:	$32.0359812909$
Analytic rank:	$1$
Dimension:	$15$
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			1.2
Root			$2.10632$ 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)

$f(q)$	$=$	$q-2.74294 q^{3} -0.560055 q^{5} +1.03063 q^{7} +4.52374 q^{9} +O(q^{10})$	$q-2.74294 q^{3} -0.560055 q^{5} +1.03063 q^{7} +4.52374 q^{9} +5.17316 q^{11} -4.63284 q^{13} +1.53620 q^{15} +1.00000 q^{17} -7.68154 q^{19} -2.82696 q^{21} +1.46716 q^{23} -4.68634 q^{25} -4.17954 q^{27} +7.77275 q^{29} +2.95059 q^{31} -14.1897 q^{33} -0.577209 q^{35} -1.08993 q^{37} +12.7076 q^{39} -2.08800 q^{41} -3.40274 q^{43} -2.53354 q^{45} -9.02441 q^{47} -5.93780 q^{49} -2.74294 q^{51} +13.6020 q^{53} -2.89725 q^{55} +21.0700 q^{57} -1.00000 q^{59} +1.43003 q^{61} +4.66230 q^{63} +2.59464 q^{65} +13.5158 q^{67} -4.02435 q^{69} -0.0700800 q^{71} +4.70938 q^{73} +12.8544 q^{75} +5.33161 q^{77} -16.5699 q^{79} -2.10698 q^{81} +11.2382 q^{83} -0.560055 q^{85} -21.3202 q^{87} -3.66413 q^{89} -4.77474 q^{91} -8.09330 q^{93} +4.30208 q^{95} +12.1576 q^{97} +23.4020 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 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
$3$	−2.74294	−1.58364	−0.791820	−	0.610755i	$-0.790866\pi$
$3$	−2.74294	−1.58364	−0.791820	+	0.610755i	$0.790866\pi$
$4$	0	0
$5$	−0.560055	−0.250464	−0.125232	−	0.992127i	$-0.539968\pi$
$5$	−0.560055	−0.250464	−0.125232	+	0.992127i	$0.539968\pi$
$6$	0	0
$7$	1.03063	0.389541	0.194771	−	0.980849i	$-0.437604\pi$
$7$	1.03063	0.389541	0.194771	+	0.980849i	$0.437604\pi$
$8$	0	0
$9$	4.52374	1.50791
$10$	0	0
$11$	5.17316	1.55977	0.779883	−	0.625926i	$-0.215279\pi$
$11$	5.17316	1.55977	0.779883	+	0.625926i	$0.215279\pi$
$12$	0	0
$13$	−4.63284	−1.28492	−0.642459	−	0.766320i	$-0.722086\pi$
$13$	−4.63284	−1.28492	−0.642459	+	0.766320i	$0.722086\pi$
$14$	0	0
$15$	1.53620	0.396645
$16$	0	0
$17$	1.00000	0.242536
$17$	1.00000	0.242536
$18$	0	0
$19$	−7.68154	−1.76227	−0.881133	−	0.472868i	$-0.843219\pi$
$19$	−7.68154	−1.76227	−0.881133	+	0.472868i	$0.843219\pi$
$20$	0	0
$21$	−2.82696	−0.616893
$22$	0	0
$23$	1.46716	0.305925	0.152963	−	0.988232i	$-0.451119\pi$
$23$	1.46716	0.305925	0.152963	+	0.988232i	$0.451119\pi$
$24$	0	0
$25$	−4.68634	−0.937268
$26$	0	0
$27$	−4.17954	−0.804354
$28$	0	0
$29$	7.77275	1.44336	0.721682	−	0.692225i	$-0.243370\pi$
$29$	7.77275	1.44336	0.721682	+	0.692225i	$0.243370\pi$
$30$	0	0
$31$	2.95059	0.529941	0.264971	−	0.964256i	$-0.414638\pi$
$31$	2.95059	0.529941	0.264971	+	0.964256i	$0.414638\pi$
$32$	0	0
$33$	−14.1897	−2.47011
$34$	0	0
$35$	−0.577209	−0.0975661
$36$	0	0
$37$	−1.08993	−0.179183	−0.0895917	−	0.995979i	$-0.528556\pi$
$37$	−1.08993	−0.179183	−0.0895917	+	0.995979i	$0.528556\pi$
$38$	0	0
$39$	12.7076	2.03485
$40$	0	0
$41$	−2.08800	−0.326090	−0.163045	−	0.986619i	$-0.552132\pi$
$41$	−2.08800	−0.326090	−0.163045	+	0.986619i	$0.552132\pi$
$42$	0	0
$43$	−3.40274	−0.518913	−0.259456	−	0.965755i	$-0.583543\pi$
$43$	−3.40274	−0.518913	−0.259456	+	0.965755i	$0.583543\pi$
$44$	0	0
$45$	−2.53354	−0.377678
$46$	0	0
$47$	−9.02441	−1.31635	−0.658173	−	0.752867i	$-0.728670\pi$
$47$	−9.02441	−1.31635	−0.658173	+	0.752867i	$0.728670\pi$
$48$	0	0
$49$	−5.93780	−0.848258
$50$	0	0
$51$	−2.74294	−0.384089
$52$	0	0
$53$	13.6020	1.86838	0.934191	−	0.356773i	$-0.116123\pi$
$53$	13.6020	1.86838	0.934191	+	0.356773i	$0.116123\pi$
$54$	0	0
$55$	−2.89725	−0.390665
$56$	0	0
$57$	21.0700	2.79080
$58$	0	0
$59$	−1.00000	−0.130189
$59$	−1.00000	−0.130189
$60$	0	0
$61$	1.43003	0.183097	0.0915485	−	0.995801i	$-0.470818\pi$
$61$	1.43003	0.183097	0.0915485	+	0.995801i	$0.470818\pi$
$62$	0	0
$63$	4.66230	0.587395
$64$	0	0
$65$	2.59464	0.321826
$66$	0	0
$67$	13.5158	1.65122	0.825609	−	0.564243i	$-0.190832\pi$
$67$	13.5158	1.65122	0.825609	+	0.564243i	$0.190832\pi$
$68$	0	0
$69$	−4.02435	−0.484475
$70$	0	0
$71$	−0.0700800	−0.00831697	−0.00415848	−	0.999991i	$-0.501324\pi$
$71$	−0.0700800	−0.00831697	−0.00415848	+	0.999991i	$0.501324\pi$
$72$	0	0
$73$	4.70938	0.551191	0.275596	−	0.961274i	$-0.411125\pi$
$73$	4.70938	0.551191	0.275596	+	0.961274i	$0.411125\pi$
$74$	0	0
$75$	12.8544	1.48429
$76$	0	0
$77$	5.33161	0.607593
$78$	0	0
$79$	−16.5699	−1.86426	−0.932132	−	0.362119i	$-0.882053\pi$
$79$	−16.5699	−1.86426	−0.932132	+	0.362119i	$0.882053\pi$
$80$	0	0
$81$	−2.10698	−0.234108
$82$	0	0
$83$	11.2382	1.23356	0.616779	−	0.787137i	$-0.288437\pi$
$83$	11.2382	1.23356	0.616779	+	0.787137i	$0.288437\pi$
$84$	0	0
$85$	−0.560055	−0.0607465
$86$	0	0
$87$	−21.3202	−2.28577
$88$	0	0
$89$	−3.66413	−0.388397	−0.194199	−	0.980962i	$-0.562211\pi$
$89$	−3.66413	−0.388397	−0.194199	+	0.980962i	$0.562211\pi$
$90$	0	0
$91$	−4.77474	−0.500529
$92$	0	0
$93$	−8.09330	−0.839236
$94$	0	0
$95$	4.30208	0.441384
$96$	0	0
$97$	12.1576	1.23442	0.617208	−	0.786800i	$-0.288264\pi$
$97$	12.1576	1.23442	0.617208	+	0.786800i	$0.288264\pi$
$98$	0	0
$99$	23.4020	2.35199
$100$	0	0
$101$	9.67881	0.963078	0.481539	−	0.876425i	$-0.340078\pi$
$101$	9.67881	0.963078	0.481539	+	0.876425i	$0.340078\pi$
$102$	0	0
$103$	−6.31042	−0.621784	−0.310892	−	0.950445i	$-0.600628\pi$
$103$	−6.31042	−0.621784	−0.310892	+	0.950445i	$0.600628\pi$
$104$	0	0
$105$	1.58325	0.154510
$106$	0	0
$107$	−6.66901	−0.644718	−0.322359	−	0.946618i	$-0.604476\pi$
$107$	−6.66901	−0.644718	−0.322359	+	0.946618i	$0.604476\pi$
$108$	0	0
$109$	8.76973	0.839988	0.419994	−	0.907527i	$-0.362032\pi$
$109$	8.76973	0.839988	0.419994	+	0.907527i	$0.362032\pi$
$110$	0	0
$111$	2.98962	0.283762
$112$	0	0
$113$	−12.5511	−1.18071	−0.590354	−	0.807144i	$-0.701012\pi$
$113$	−12.5511	−1.18071	−0.590354	+	0.807144i	$0.701012\pi$
$114$	0	0
$115$	−0.821692	−0.0766232
$116$	0	0
$117$	−20.9578	−1.93755
$118$	0	0
$119$	1.03063	0.0944777
$120$	0	0
$121$	15.7616	1.43287
$122$	0	0
$123$	5.72726	0.516409
$124$	0	0
$125$	5.42488	0.485216
$126$	0	0
$127$	−22.3605	−1.98418	−0.992088	−	0.125544i	$-0.959932\pi$
$127$	−22.3605	−1.98418	−0.992088	+	0.125544i	$0.959932\pi$
$128$	0	0
$129$	9.33352	0.821770
$130$	0	0
$131$	19.5643	1.70934	0.854671	−	0.519169i	$-0.173759\pi$
$131$	19.5643	1.70934	0.854671	+	0.519169i	$0.173759\pi$
$132$	0	0
$133$	−7.91682	−0.686476
$134$	0	0
$135$	2.34077	0.201462
$136$	0	0
$137$	8.54575	0.730113	0.365056	−	0.930985i	$-0.381050\pi$
$137$	8.54575	0.730113	0.365056	+	0.930985i	$0.381050\pi$
$138$	0	0
$139$	−17.2947	−1.46692	−0.733458	−	0.679735i	$-0.762095\pi$
$139$	−17.2947	−1.46692	−0.733458	+	0.679735i	$0.762095\pi$
$140$	0	0
$141$	24.7535	2.08462
$142$	0	0
$143$	−23.9664	−2.00417
$144$	0	0
$145$	−4.35316	−0.361511
$146$	0	0
$147$	16.2871	1.34333
$148$	0	0
$149$	13.6391	1.11736	0.558678	−	0.829385i	$-0.311309\pi$
$149$	13.6391	1.11736	0.558678	+	0.829385i	$0.311309\pi$
$150$	0	0
$151$	6.05070	0.492399	0.246200	−	0.969219i	$-0.420818\pi$
$151$	6.05070	0.492399	0.246200	+	0.969219i	$0.420818\pi$
$152$	0	0
$153$	4.52374	0.365723
$154$	0	0
$155$	−1.65249	−0.132731
$156$	0	0
$157$	−17.6913	−1.41192	−0.705958	−	0.708253i	$-0.749483\pi$
$157$	−17.6913	−1.41192	−0.705958	+	0.708253i	$0.749483\pi$
$158$	0	0
$159$	−37.3096	−2.95884
$160$	0	0
$161$	1.51210	0.119170
$162$	0	0
$163$	−19.0257	−1.49021	−0.745105	−	0.666947i	$-0.767601\pi$
$163$	−19.0257	−1.49021	−0.745105	+	0.666947i	$0.767601\pi$
$164$	0	0
$165$	7.94700	0.618673
$166$	0	0
$167$	−14.0370	−1.08622	−0.543109	−	0.839662i	$-0.682753\pi$
$167$	−14.0370	−1.08622	−0.543109	+	0.839662i	$0.682753\pi$
$168$	0	0
$169$	8.46318	0.651014
$170$	0	0
$171$	−34.7493	−2.65735
$172$	0	0
$173$	−16.3611	−1.24391	−0.621957	−	0.783051i	$-0.713662\pi$
$173$	−16.3611	−1.24391	−0.621957	+	0.783051i	$0.713662\pi$
$174$	0	0
$175$	−4.82988	−0.365105
$176$	0	0
$177$	2.74294	0.206172
$178$	0	0
$179$	4.57781	0.342162	0.171081	−	0.985257i	$-0.445274\pi$
$179$	4.57781	0.342162	0.171081	+	0.985257i	$0.445274\pi$
$180$	0	0
$181$	−0.629353	−0.0467794	−0.0233897	−	0.999726i	$-0.507446\pi$
$181$	−0.629353	−0.0467794	−0.0233897	+	0.999726i	$0.507446\pi$
$182$	0	0
$183$	−3.92250	−0.289960
$184$	0	0
$185$	0.610420	0.0448790
$186$	0	0
$187$	5.17316	0.378299
$188$	0	0
$189$	−4.30756	−0.313329
$190$	0	0
$191$	−18.0394	−1.30528	−0.652641	−	0.757667i	$-0.726339\pi$
$191$	−18.0394	−1.30528	−0.652641	+	0.757667i	$0.726339\pi$
$192$	0	0
$193$	−15.8263	−1.13921	−0.569603	−	0.821920i	$-0.692903\pi$
$193$	−15.8263	−1.13921	−0.569603	+	0.821920i	$0.692903\pi$
$194$	0	0
$195$	−7.11696	−0.509656
$196$	0	0
$197$	−3.97810	−0.283428	−0.141714	−	0.989908i	$-0.545261\pi$
$197$	−3.97810	−0.283428	−0.141714	+	0.989908i	$0.545261\pi$
$198$	0	0
$199$	0.706120	0.0500555	0.0250278	−	0.999687i	$-0.492033\pi$
$199$	0.706120	0.0500555	0.0250278	+	0.999687i	$0.492033\pi$
$200$	0	0
$201$	−37.0731	−2.61493
$202$	0	0
$203$	8.01082	0.562250
$204$	0	0
$205$	1.16939	0.0816739
$206$	0	0
$207$	6.63708	0.461309
$208$	0	0
$209$	−39.7378	−2.74872
$210$	0	0
$211$	3.81305	0.262501	0.131251	−	0.991349i	$-0.458101\pi$
$211$	3.81305	0.262501	0.131251	+	0.991349i	$0.458101\pi$
$212$	0	0
$213$	0.192226	0.0131711
$214$	0	0
$215$	1.90572	0.129969
$216$	0	0
$217$	3.04096	0.206434
$218$	0	0
$219$	−12.9176	−0.872889
$220$	0	0
$221$	−4.63284	−0.311638
$222$	0	0
$223$	−29.7895	−1.99485	−0.997425	−	0.0717198i	$-0.977151\pi$
$223$	−29.7895	−1.99485	−0.997425	+	0.0717198i	$0.977151\pi$
$224$	0	0
$225$	−21.1998	−1.41332
$226$	0	0
$227$	−23.3939	−1.55271	−0.776354	−	0.630297i	$-0.782933\pi$
$227$	−23.3939	−1.55271	−0.776354	+	0.630297i	$0.782933\pi$
$228$	0	0
$229$	−17.9389	−1.18543	−0.592717	−	0.805410i	$-0.701945\pi$
$229$	−17.9389	−1.18543	−0.592717	+	0.805410i	$0.701945\pi$
$230$	0	0
$231$	−14.6243	−0.962209
$232$	0	0
$233$	−7.47483	−0.489692	−0.244846	−	0.969562i	$-0.578737\pi$
$233$	−7.47483	−0.489692	−0.244846	+	0.969562i	$0.578737\pi$
$234$	0	0
$235$	5.05416	0.329697
$236$	0	0
$237$	45.4504	2.95232
$238$	0	0
$239$	13.8015	0.892742	0.446371	−	0.894848i	$-0.352716\pi$
$239$	13.8015	0.892742	0.446371	+	0.894848i	$0.352716\pi$
$240$	0	0
$241$	29.3075	1.88786	0.943932	−	0.330141i	$-0.107096\pi$
$241$	29.3075	1.88786	0.943932	+	0.330141i	$0.107096\pi$
$242$	0	0
$243$	18.3179	1.17510
$244$	0	0
$245$	3.32549	0.212458
$246$	0	0
$247$	35.5873	2.26437
$248$	0	0
$249$	−30.8259	−1.95351
$250$	0	0
$251$	2.67144	0.168620	0.0843099	−	0.996440i	$-0.473131\pi$
$251$	2.67144	0.168620	0.0843099	+	0.996440i	$0.473131\pi$
$252$	0	0
$253$	7.58987	0.477171
$254$	0	0
$255$	1.53620	0.0962005
$256$	0	0
$257$	−20.6689	−1.28929	−0.644645	−	0.764482i	$-0.722995\pi$
$257$	−20.6689	−1.28929	−0.644645	+	0.764482i	$0.722995\pi$
$258$	0	0
$259$	−1.12331	−0.0697993
$260$	0	0
$261$	35.1619	2.17647
$262$	0	0
$263$	−26.0641	−1.60718	−0.803592	−	0.595181i	$-0.797080\pi$
$263$	−26.0641	−1.60718	−0.803592	+	0.595181i	$0.797080\pi$
$264$	0	0
$265$	−7.61788	−0.467962
$266$	0	0
$267$	10.0505	0.615082
$268$	0	0
$269$	−10.4578	−0.637622	−0.318811	−	0.947818i	$-0.603284\pi$
$269$	−10.4578	−0.637622	−0.318811	+	0.947818i	$0.603284\pi$
$270$	0	0
$271$	−20.8115	−1.26421	−0.632104	−	0.774883i	$-0.717809\pi$
$271$	−20.8115	−1.26421	−0.632104	+	0.774883i	$0.717809\pi$
$272$	0	0
$273$	13.0968	0.792657
$274$	0	0
$275$	−24.2432	−1.46192
$276$	0	0
$277$	−10.3796	−0.623649	−0.311824	−	0.950140i	$-0.600940\pi$
$277$	−10.3796	−0.623649	−0.311824	+	0.950140i	$0.600940\pi$
$278$	0	0
$279$	13.3477	0.799106
$280$	0	0
$281$	−4.52394	−0.269875	−0.134938	−	0.990854i	$-0.543083\pi$
$281$	−4.52394	−0.269875	−0.134938	+	0.990854i	$0.543083\pi$
$282$	0	0
$283$	−11.0440	−0.656499	−0.328250	−	0.944591i	$-0.606459\pi$
$283$	−11.0440	−0.656499	−0.328250	+	0.944591i	$0.606459\pi$
$284$	0	0
$285$	−11.8004	−0.698994
$286$	0	0
$287$	−2.15195	−0.127026
$288$	0	0
$289$	1.00000	0.0588235
$290$	0	0
$291$	−33.3476	−1.95487
$292$	0	0
$293$	18.9859	1.10917	0.554585	−	0.832127i	$-0.312877\pi$
$293$	18.9859	1.10917	0.554585	+	0.832127i	$0.312877\pi$
$294$	0	0
$295$	0.560055	0.0326076
$296$	0	0
$297$	−21.6214	−1.25460
$298$	0	0
$299$	−6.79714	−0.393089
$300$	0	0
$301$	−3.50696	−0.202138
$302$	0	0
$303$	−26.5484	−1.52517
$304$	0	0
$305$	−0.800897	−0.0458592
$306$	0	0
$307$	−27.4129	−1.56454	−0.782268	−	0.622942i	$-0.785937\pi$
$307$	−27.4129	−1.56454	−0.782268	+	0.622942i	$0.785937\pi$
$308$	0	0
$309$	17.3091	0.984682
$310$	0	0
$311$	−0.0768449	−0.00435748	−0.00217874	−	0.999998i	$-0.500694\pi$
$311$	−0.0768449	−0.00435748	−0.00217874	+	0.999998i	$0.500694\pi$
$312$	0	0
$313$	−25.9528	−1.46694	−0.733471	−	0.679721i	$-0.762101\pi$
$313$	−25.9528	−1.46694	−0.733471	+	0.679721i	$0.762101\pi$
$314$	0	0
$315$	−2.61114	−0.147121
$316$	0	0
$317$	8.72878	0.490257	0.245129	−	0.969491i	$-0.421170\pi$
$317$	8.72878	0.490257	0.245129	+	0.969491i	$0.421170\pi$
$318$	0	0
$319$	40.2097	2.25131
$320$	0	0
$321$	18.2927	1.02100
$322$	0	0
$323$	−7.68154	−0.427412
$324$	0	0
$325$	21.7110	1.20431
$326$	0	0
$327$	−24.0549	−1.33024
$328$	0	0
$329$	−9.30083	−0.512771
$330$	0	0
$331$	−5.10057	−0.280352	−0.140176	−	0.990127i	$-0.544767\pi$
$331$	−5.10057	−0.280352	−0.140176	+	0.990127i	$0.544767\pi$
$332$	0	0
$333$	−4.93056	−0.270193
$334$	0	0
$335$	−7.56959	−0.413571
$336$	0	0
$337$	19.3220	1.05254	0.526268	−	0.850319i	$-0.323591\pi$
$337$	19.3220	1.05254	0.526268	+	0.850319i	$0.323591\pi$
$338$	0	0
$339$	34.4270	1.86982
$340$	0	0
$341$	15.2639	0.826584
$342$	0	0
$343$	−13.3341	−0.719973
$344$	0	0
$345$	2.25386	0.121344
$346$	0	0
$347$	−17.7702	−0.953954	−0.476977	−	0.878916i	$-0.658268\pi$
$347$	−17.7702	−0.953954	−0.476977	+	0.878916i	$0.658268\pi$
$348$	0	0
$349$	31.1565	1.66777	0.833884	−	0.551940i	$-0.186112\pi$
$349$	31.1565	1.66777	0.833884	+	0.551940i	$0.186112\pi$
$350$	0	0
$351$	19.3631	1.03353
$352$	0	0
$353$	−5.29069	−0.281595	−0.140798	−	0.990038i	$-0.544967\pi$
$353$	−5.29069	−0.281595	−0.140798	+	0.990038i	$0.544967\pi$
$354$	0	0
$355$	0.0392486	0.00208310
$356$	0	0
$357$	−2.82696	−0.149619
$358$	0	0
$359$	14.0854	0.743398	0.371699	−	0.928353i	$-0.378775\pi$
$359$	14.0854	0.743398	0.371699	+	0.928353i	$0.378775\pi$
$360$	0	0
$361$	40.0061	2.10558
$362$	0	0
$363$	−43.2331	−2.26915
$364$	0	0
$365$	−2.63751	−0.138054
$366$	0	0
$367$	−8.09151	−0.422373	−0.211187	−	0.977446i	$-0.567733\pi$
$367$	−8.09151	−0.422373	−0.211187	+	0.977446i	$0.567733\pi$
$368$	0	0
$369$	−9.44556	−0.491716
$370$	0	0
$371$	14.0187	0.727812
$372$	0	0
$373$	−32.4470	−1.68004	−0.840021	−	0.542553i	$-0.817458\pi$
$373$	−32.4470	−1.68004	−0.840021	+	0.542553i	$0.817458\pi$
$374$	0	0
$375$	−14.8801	−0.768407
$376$	0	0
$377$	−36.0099	−1.85460
$378$	0	0
$379$	21.2667	1.09239	0.546197	−	0.837656i	$-0.316075\pi$
$379$	21.2667	1.09239	0.546197	+	0.837656i	$0.316075\pi$
$380$	0	0
$381$	61.3337	3.14222
$382$	0	0
$383$	27.3781	1.39896	0.699479	−	0.714653i	$-0.253415\pi$
$383$	27.3781	1.39896	0.699479	+	0.714653i	$0.253415\pi$
$384$	0	0
$385$	−2.98599	−0.152180
$386$	0	0
$387$	−15.3931	−0.782476
$388$	0	0
$389$	−27.1858	−1.37837	−0.689187	−	0.724584i	$-0.742032\pi$
$389$	−27.1858	−1.37837	−0.689187	+	0.724584i	$0.742032\pi$
$390$	0	0
$391$	1.46716	0.0741977
$392$	0	0
$393$	−53.6638	−2.70698
$394$	0	0
$395$	9.28007	0.466931
$396$	0	0
$397$	21.7457	1.09139	0.545694	−	0.837985i	$-0.316266\pi$
$397$	21.7457	1.09139	0.545694	+	0.837985i	$0.316266\pi$
$398$	0	0
$399$	21.7154	1.08713
$400$	0	0
$401$	35.8629	1.79091	0.895453	−	0.445156i	$-0.146852\pi$
$401$	35.8629	1.79091	0.895453	+	0.445156i	$0.146852\pi$
$402$	0	0
$403$	−13.6696	−0.680931
$404$	0	0
$405$	1.18002	0.0586357
$406$	0	0
$407$	−5.63838	−0.279484
$408$	0	0
$409$	20.6180	1.01949	0.509746	−	0.860325i	$-0.329739\pi$
$409$	20.6180	1.01949	0.509746	+	0.860325i	$0.329739\pi$
$410$	0	0
$411$	−23.4405	−1.15624
$412$	0	0
$413$	−1.03063	−0.0507140
$414$	0	0
$415$	−6.29403	−0.308962
$416$	0	0
$417$	47.4384	2.32307
$418$	0	0
$419$	−19.1953	−0.937749	−0.468875	−	0.883265i	$-0.655340\pi$
$419$	−19.1953	−0.937749	−0.468875	+	0.883265i	$0.655340\pi$
$420$	0	0
$421$	26.7932	1.30582	0.652910	−	0.757436i	$-0.273548\pi$
$421$	26.7932	1.30582	0.652910	+	0.757436i	$0.273548\pi$
$422$	0	0
$423$	−40.8241	−1.98494
$424$	0	0
$425$	−4.68634	−0.227321
$426$	0	0
$427$	1.47383	0.0713239
$428$	0	0
$429$	65.7385	3.17388
$430$	0	0
$431$	2.22702	0.107272	0.0536359	−	0.998561i	$-0.482919\pi$
$431$	2.22702	0.107272	0.0536359	+	0.998561i	$0.482919\pi$
$432$	0	0
$433$	−21.3534	−1.02618	−0.513090	−	0.858335i	$-0.671499\pi$
$433$	−21.3534	−1.02618	−0.513090	+	0.858335i	$0.671499\pi$
$434$	0	0
$435$	11.9405	0.572502
$436$	0	0
$437$	−11.2701	−0.539121
$438$	0	0
$439$	24.7867	1.18301	0.591503	−	0.806303i	$-0.298535\pi$
$439$	24.7867	1.18301	0.591503	+	0.806303i	$0.298535\pi$
$440$	0	0
$441$	−26.8611	−1.27910
$442$	0	0
$443$	13.8726	0.659108	0.329554	−	0.944137i	$-0.393102\pi$
$443$	13.8726	0.659108	0.329554	+	0.944137i	$0.393102\pi$
$444$	0	0
$445$	2.05212	0.0972796
$446$	0	0
$447$	−37.4112	−1.76949
$448$	0	0
$449$	17.1634	0.809992	0.404996	−	0.914318i	$-0.367273\pi$
$449$	17.1634	0.809992	0.404996	+	0.914318i	$0.367273\pi$
$450$	0	0
$451$	−10.8015	−0.508624
$452$	0	0
$453$	−16.5967	−0.779783
$454$	0	0
$455$	2.67411	0.125364
$456$	0	0
$457$	2.91385	0.136304	0.0681520	−	0.997675i	$-0.478290\pi$
$457$	2.91385	0.136304	0.0681520	+	0.997675i	$0.478290\pi$
$458$	0	0
$459$	−4.17954	−0.195084
$460$	0	0
$461$	−21.4628	−0.999624	−0.499812	−	0.866134i	$-0.666598\pi$
$461$	−21.4628	−0.999624	−0.499812	+	0.866134i	$0.666598\pi$
$462$	0	0
$463$	29.9897	1.39374	0.696869	−	0.717198i	$-0.254576\pi$
$463$	29.9897	1.39374	0.696869	+	0.717198i	$0.254576\pi$
$464$	0	0
$465$	4.53269	0.210198
$466$	0	0
$467$	−20.3030	−0.939512	−0.469756	−	0.882796i	$-0.655658\pi$
$467$	−20.3030	−0.939512	−0.469756	+	0.882796i	$0.655658\pi$
$468$	0	0
$469$	13.9298	0.643218
$470$	0	0
$471$	48.5261	2.23597
$472$	0	0
$473$	−17.6029	−0.809382
$474$	0	0
$475$	35.9983	1.65172
$476$	0	0
$477$	61.5321	2.81736
$478$	0	0
$479$	−16.1851	−0.739518	−0.369759	−	0.929128i	$-0.620560\pi$
$479$	−16.1851	−0.739518	−0.369759	+	0.929128i	$0.620560\pi$
$480$	0	0
$481$	5.04947	0.230236
$482$	0	0
$483$	−4.14762	−0.188723
$484$	0	0
$485$	−6.80891	−0.309177
$486$	0	0
$487$	−2.47282	−0.112054	−0.0560270	−	0.998429i	$-0.517843\pi$
$487$	−2.47282	−0.112054	−0.0560270	+	0.998429i	$0.517843\pi$
$488$	0	0
$489$	52.1865	2.35996
$490$	0	0
$491$	−42.1435	−1.90191	−0.950954	−	0.309332i	$-0.899895\pi$
$491$	−42.1435	−1.90191	−0.950954	+	0.309332i	$0.899895\pi$
$492$	0	0
$493$	7.77275	0.350067
$494$	0	0
$495$	−13.1064	−0.589090
$496$	0	0
$497$	−0.0722265	−0.00323980
$498$	0	0
$499$	−25.3301	−1.13393	−0.566966	−	0.823741i	$-0.691883\pi$
$499$	−25.3301	−1.13393	−0.566966	+	0.823741i	$0.691883\pi$
$500$	0	0
$501$	38.5028	1.72018
$502$	0	0
$503$	−43.6671	−1.94702	−0.973510	−	0.228644i	$-0.926571\pi$
$503$	−43.6671	−1.94702	−0.973510	+	0.228644i	$0.926571\pi$
$504$	0	0
$505$	−5.42066	−0.241216
$506$	0	0
$507$	−23.2140	−1.03097
$508$	0	0
$509$	12.9653	0.574676	0.287338	−	0.957829i	$-0.407230\pi$
$509$	12.9653	0.574676	0.287338	+	0.957829i	$0.407230\pi$
$510$	0	0
$511$	4.85363	0.214712
$512$	0	0
$513$	32.1053	1.41749
$514$	0	0
$515$	3.53418	0.155735
$516$	0	0
$517$	−46.6847	−2.05319
$518$	0	0
$519$	44.8777	1.96991
$520$	0	0
$521$	−19.7874	−0.866902	−0.433451	−	0.901177i	$-0.642704\pi$
$521$	−19.7874	−0.866902	−0.433451	+	0.901177i	$0.642704\pi$
$522$	0	0
$523$	13.9107	0.608272	0.304136	−	0.952629i	$-0.401632\pi$
$523$	13.9107	0.608272	0.304136	+	0.952629i	$0.401632\pi$
$524$	0	0
$525$	13.2481	0.578194
$526$	0	0
$527$	2.95059	0.128530
$528$	0	0
$529$	−20.8474	−0.906410
$530$	0	0
$531$	−4.52374	−0.196314
$532$	0	0
$533$	9.67335	0.418999
$534$	0	0
$535$	3.73501	0.161479
$536$	0	0
$537$	−12.5567	−0.541861
$538$	0	0
$539$	−30.7172	−1.32308
$540$	0	0
$541$	−0.501110	−0.0215444	−0.0107722	−	0.999942i	$-0.503429\pi$
$541$	−0.501110	−0.0215444	−0.0107722	+	0.999942i	$0.503429\pi$
$542$	0	0
$543$	1.72628	0.0740818
$544$	0	0
$545$	−4.91153	−0.210387
$546$	0	0
$547$	30.9283	1.32240	0.661200	−	0.750210i	$-0.270047\pi$
$547$	30.9283	1.32240	0.661200	+	0.750210i	$0.270047\pi$
$548$	0	0
$549$	6.46910	0.276095
$550$	0	0
$551$	−59.7067	−2.54359
$552$	0	0
$553$	−17.0775	−0.726208
$554$	0	0
$555$	−1.67435	−0.0710721
$556$	0	0
$557$	12.8586	0.544838	0.272419	−	0.962179i	$-0.412176\pi$
$557$	12.8586	0.544838	0.272419	+	0.962179i	$0.412176\pi$
$558$	0	0
$559$	15.7643	0.666760
$560$	0	0
$561$	−14.1897	−0.599089
$562$	0	0
$563$	9.72190	0.409729	0.204865	−	0.978790i	$-0.434325\pi$
$563$	9.72190	0.409729	0.204865	+	0.978790i	$0.434325\pi$
$564$	0	0
$565$	7.02930	0.295725
$566$	0	0
$567$	−2.17151	−0.0911949
$568$	0	0
$569$	−40.8753	−1.71358	−0.856792	−	0.515663i	$-0.827546\pi$
$569$	−40.8753	−1.71358	−0.856792	+	0.515663i	$0.827546\pi$
$570$	0	0
$571$	−5.23472	−0.219066	−0.109533	−	0.993983i	$-0.534936\pi$
$571$	−5.23472	−0.219066	−0.109533	+	0.993983i	$0.534936\pi$
$572$	0	0
$573$	49.4810	2.06710
$574$	0	0
$575$	−6.87563	−0.286734
$576$	0	0
$577$	−6.13205	−0.255280	−0.127640	−	0.991821i	$-0.540740\pi$
$577$	−6.13205	−0.255280	−0.127640	+	0.991821i	$0.540740\pi$
$578$	0	0
$579$	43.4108	1.80409
$580$	0	0
$581$	11.5825	0.480522
$582$	0	0
$583$	70.3654	2.91424
$584$	0	0
$585$	11.7375	0.485286
$586$	0	0
$587$	−13.3823	−0.552347	−0.276173	−	0.961108i	$-0.589066\pi$
$587$	−13.3823	−0.552347	−0.276173	+	0.961108i	$0.589066\pi$
$588$	0	0
$589$	−22.6651	−0.933898
$590$	0	0
$591$	10.9117	0.448848
$592$	0	0
$593$	31.9068	1.31026	0.655128	−	0.755518i	$-0.272615\pi$
$593$	31.9068	1.31026	0.655128	+	0.755518i	$0.272615\pi$
$594$	0	0
$595$	−0.577209	−0.0236633
$596$	0	0
$597$	−1.93685	−0.0792699
$598$	0	0
$599$	−22.0947	−0.902765	−0.451383	−	0.892331i	$-0.649069\pi$
$599$	−22.0947	−0.902765	−0.451383	+	0.892331i	$0.649069\pi$
$600$	0	0
$601$	−41.5468	−1.69473	−0.847366	−	0.531010i	$-0.821813\pi$
$601$	−41.5468	−1.69473	−0.847366	+	0.531010i	$0.821813\pi$
$602$	0	0
$603$	61.1420	2.48990
$604$	0	0
$605$	−8.82733	−0.358882
$606$	0	0
$607$	13.2686	0.538556	0.269278	−	0.963063i	$-0.413215\pi$
$607$	13.2686	0.538556	0.269278	+	0.963063i	$0.413215\pi$
$608$	0	0
$609$	−21.9732	−0.890401
$610$	0	0
$611$	41.8086	1.69140
$612$	0	0
$613$	−28.0521	−1.13301	−0.566506	−	0.824057i	$-0.691705\pi$
$613$	−28.0521	−1.13301	−0.566506	+	0.824057i	$0.691705\pi$
$614$	0	0
$615$	−3.20758	−0.129342
$616$	0	0
$617$	20.5791	0.828485	0.414243	−	0.910167i	$-0.364047\pi$
$617$	20.5791	0.828485	0.414243	+	0.910167i	$0.364047\pi$
$618$	0	0
$619$	−8.81346	−0.354243	−0.177121	−	0.984189i	$-0.556679\pi$
$619$	−8.81346	−0.354243	−0.177121	+	0.984189i	$0.556679\pi$
$620$	0	0
$621$	−6.13208	−0.246072
$622$	0	0
$623$	−3.77637	−0.151297
$624$	0	0
$625$	20.3935	0.815739
$626$	0	0
$627$	108.999	4.35299
$628$	0	0
$629$	−1.08993	−0.0434583
$630$	0	0
$631$	14.1613	0.563752	0.281876	−	0.959451i	$-0.409043\pi$
$631$	14.1613	0.563752	0.281876	+	0.959451i	$0.409043\pi$
$632$	0	0
$633$	−10.4590	−0.415707
$634$	0	0
$635$	12.5231	0.496965
$636$	0	0
$637$	27.5089	1.08994
$638$	0	0
$639$	−0.317024	−0.0125413
$640$	0	0
$641$	3.93575	0.155453	0.0777263	−	0.996975i	$-0.475234\pi$
$641$	3.93575	0.155453	0.0777263	+	0.996975i	$0.475234\pi$
$642$	0	0
$643$	−10.9249	−0.430837	−0.215419	−	0.976522i	$-0.569112\pi$
$643$	−10.9249	−0.430837	−0.215419	+	0.976522i	$0.569112\pi$
$644$	0	0
$645$	−5.22728	−0.205824
$646$	0	0
$647$	24.6497	0.969078	0.484539	−	0.874770i	$-0.338987\pi$
$647$	24.6497	0.969078	0.484539	+	0.874770i	$0.338987\pi$
$648$	0	0
$649$	−5.17316	−0.203064
$650$	0	0
$651$	−8.34119	−0.326917
$652$	0	0
$653$	−39.1759	−1.53307	−0.766535	−	0.642203i	$-0.778021\pi$
$653$	−39.1759	−1.53307	−0.766535	+	0.642203i	$0.778021\pi$
$654$	0	0
$655$	−10.9571	−0.428129
$656$	0	0
$657$	21.3040	0.831149
$658$	0	0
$659$	−19.3417	−0.753446	−0.376723	−	0.926326i	$-0.622949\pi$
$659$	−19.3417	−0.753446	−0.376723	+	0.926326i	$0.622949\pi$
$660$	0	0
$661$	−13.4720	−0.523999	−0.261999	−	0.965068i	$-0.584382\pi$
$661$	−13.4720	−0.523999	−0.261999	+	0.965068i	$0.584382\pi$
$662$	0	0
$663$	12.7076	0.493523
$664$	0	0
$665$	4.43385	0.171937
$666$	0	0
$667$	11.4039	0.441561
$668$	0	0
$669$	81.7108	3.15912
$670$	0	0
$671$	7.39779	0.285588
$672$	0	0
$673$	−45.8425	−1.76710	−0.883549	−	0.468339i	$-0.844853\pi$
$673$	−45.8425	−1.76710	−0.883549	+	0.468339i	$0.844853\pi$
$674$	0	0
$675$	19.5868	0.753895
$676$	0	0
$677$	−13.2310	−0.508510	−0.254255	−	0.967137i	$-0.581830\pi$
$677$	−13.2310	−0.508510	−0.254255	+	0.967137i	$0.581830\pi$
$678$	0	0
$679$	12.5300	0.480856
$680$	0	0
$681$	64.1682	2.45893
$682$	0	0
$683$	−14.9161	−0.570750	−0.285375	−	0.958416i	$-0.592118\pi$
$683$	−14.9161	−0.570750	−0.285375	+	0.958416i	$0.592118\pi$
$684$	0	0
$685$	−4.78609	−0.182867
$686$	0	0
$687$	49.2054	1.87730
$688$	0	0
$689$	−63.0160	−2.40072
$690$	0	0
$691$	−5.22718	−0.198851	−0.0994256	−	0.995045i	$-0.531701\pi$
$691$	−5.22718	−0.198851	−0.0994256	+	0.995045i	$0.531701\pi$
$692$	0	0
$693$	24.1188	0.916199
$694$	0	0
$695$	9.68597	0.367410
$696$	0	0
$697$	−2.08800	−0.0790885
$698$	0	0
$699$	20.5030	0.775496
$700$	0	0
$701$	23.7273	0.896168	0.448084	−	0.893991i	$-0.352106\pi$
$701$	23.7273	0.896168	0.448084	+	0.893991i	$0.352106\pi$
$702$	0	0
$703$	8.37234	0.315769
$704$	0	0
$705$	−13.8633	−0.522122
$706$	0	0
$707$	9.97527	0.375159
$708$	0	0
$709$	−10.3049	−0.387008	−0.193504	−	0.981100i	$-0.561985\pi$
$709$	−10.3049	−0.387008	−0.193504	+	0.981100i	$0.561985\pi$
$710$	0	0
$711$	−74.9581	−2.81115
$712$	0	0
$713$	4.32900	0.162122
$714$	0	0
$715$	13.4225	0.501973
$716$	0	0
$717$	−37.8566	−1.41378
$718$	0	0
$719$	−49.2574	−1.83699	−0.918495	−	0.395433i	$-0.870595\pi$
$719$	−49.2574	−1.83699	−0.918495	+	0.395433i	$0.870595\pi$
$720$	0	0
$721$	−6.50371	−0.242211
$722$	0	0
$723$	−80.3889	−2.98970
$724$	0	0
$725$	−36.4257	−1.35282
$726$	0	0
$727$	−30.0358	−1.11396	−0.556982	−	0.830524i	$-0.688041\pi$
$727$	−30.0358	−1.11396	−0.556982	+	0.830524i	$0.688041\pi$
$728$	0	0
$729$	−43.9242	−1.62682
$730$	0	0
$731$	−3.40274	−0.125855
$732$	0	0
$733$	11.2926	0.417100	0.208550	−	0.978012i	$-0.433126\pi$
$733$	11.2926	0.417100	0.208550	+	0.978012i	$0.433126\pi$
$734$	0	0
$735$	−9.12164	−0.336457
$736$	0	0
$737$	69.9194	2.57551
$738$	0	0
$739$	−14.6692	−0.539616	−0.269808	−	0.962914i	$-0.586960\pi$
$739$	−14.6692	−0.539616	−0.269808	+	0.962914i	$0.586960\pi$
$740$	0	0
$741$	−97.6141	−3.58594
$742$	0	0
$743$	7.75273	0.284420	0.142210	−	0.989837i	$-0.454579\pi$
$743$	7.75273	0.284420	0.142210	+	0.989837i	$0.454579\pi$
$744$	0	0
$745$	−7.63862	−0.279857
$746$	0	0
$747$	50.8389	1.86010
$748$	0	0
$749$	−6.87328	−0.251144
$750$	0	0
$751$	22.4696	0.819926	0.409963	−	0.912102i	$-0.365542\pi$
$751$	22.4696	0.819926	0.409963	+	0.912102i	$0.365542\pi$
$752$	0	0
$753$	−7.32761	−0.267033
$754$	0	0
$755$	−3.38872	−0.123328
$756$	0	0
$757$	−0.360873	−0.0131162	−0.00655808	−	0.999978i	$-0.502088\pi$
$757$	−0.360873	−0.0131162	−0.00655808	+	0.999978i	$0.502088\pi$
$758$	0	0
$759$	−20.8186	−0.755667
$760$	0	0
$761$	−21.3247	−0.773020	−0.386510	−	0.922285i	$-0.626320\pi$
$761$	−21.3247	−0.773020	−0.386510	+	0.922285i	$0.626320\pi$
$762$	0	0
$763$	9.03835	0.327210
$764$	0	0
$765$	−2.53354	−0.0916005
$766$	0	0
$767$	4.63284	0.167282
$768$	0	0
$769$	18.1098	0.653056	0.326528	−	0.945188i	$-0.394121\pi$
$769$	18.1098	0.653056	0.326528	+	0.945188i	$0.394121\pi$
$770$	0	0
$771$	56.6936	2.04177
$772$	0	0
$773$	15.3622	0.552540	0.276270	−	0.961080i	$-0.410902\pi$
$773$	15.3622	0.552540	0.276270	+	0.961080i	$0.410902\pi$
$774$	0	0
$775$	−13.8275	−0.496697
$776$	0	0
$777$	3.08119	0.110537
$778$	0	0
$779$	16.0390	0.574658
$780$	0	0
$781$	−0.362535	−0.0129725
$782$	0	0
$783$	−32.4865	−1.16097
$784$	0	0
$785$	9.90807	0.353634
$786$	0	0
$787$	33.6165	1.19830	0.599149	−	0.800638i	$-0.295506\pi$
$787$	33.6165	1.19830	0.599149	+	0.800638i	$0.295506\pi$
$788$	0	0
$789$	71.4925	2.54520
$790$	0	0
$791$	−12.9355	−0.459935
$792$	0	0
$793$	−6.62511	−0.235265
$794$	0	0
$795$	20.8954	0.741084
$796$	0	0
$797$	5.83433	0.206663	0.103331	−	0.994647i	$-0.467050\pi$
$797$	5.83433	0.206663	0.103331	+	0.994647i	$0.467050\pi$
$798$	0	0
$799$	−9.02441	−0.319261
$800$	0	0
$801$	−16.5756	−0.585670
$802$	0	0
$803$	24.3624	0.859729
$804$	0	0
$805$	−0.846861	−0.0298479
$806$	0	0
$807$	28.6851	1.00976
$808$	0	0
$809$	22.2284	0.781509	0.390754	−	0.920495i	$-0.372214\pi$
$809$	22.2284	0.781509	0.390754	+	0.920495i	$0.372214\pi$
$810$	0	0
$811$	42.9505	1.50820	0.754098	−	0.656762i	$-0.228074\pi$
$811$	42.9505	1.50820	0.754098	+	0.656762i	$0.228074\pi$
$812$	0	0
$813$	57.0848	2.00205
$814$	0	0
$815$	10.6554	0.373244
$816$	0	0
$817$	26.1383	0.914462
$818$	0	0
$819$	−21.5997	−0.754754
$820$	0	0
$821$	10.0324	0.350135	0.175067	−	0.984556i	$-0.443986\pi$
$821$	10.0324	0.350135	0.175067	+	0.984556i	$0.443986\pi$
$822$	0	0
$823$	12.8604	0.448284	0.224142	−	0.974556i	$-0.428042\pi$
$823$	12.8604	0.448284	0.224142	+	0.974556i	$0.428042\pi$
$824$	0	0
$825$	66.4977	2.31515
$826$	0	0
$827$	−47.7358	−1.65994	−0.829969	−	0.557810i	$-0.811642\pi$
$827$	−47.7358	−1.65994	−0.829969	+	0.557810i	$0.811642\pi$
$828$	0	0
$829$	−6.72445	−0.233550	−0.116775	−	0.993158i	$-0.537256\pi$
$829$	−6.72445	−0.233550	−0.116775	+	0.993158i	$0.537256\pi$
$830$	0	0
$831$	28.4706	0.987635
$832$	0	0
$833$	−5.93780	−0.205733
$834$	0	0
$835$	7.86151	0.272059
$836$	0	0
$837$	−12.3321	−0.426260
$838$	0	0
$839$	50.1140	1.73013	0.865063	−	0.501663i	$-0.167278\pi$
$839$	50.1140	1.73013	0.865063	+	0.501663i	$0.167278\pi$
$840$	0	0
$841$	31.4156	1.08330
$842$	0	0
$843$	12.4089	0.427385
$844$	0	0
$845$	−4.73984	−0.163056
$846$	0	0
$847$	16.2443	0.558162
$848$	0	0
$849$	30.2931	1.03966
$850$	0	0
$851$	−1.59911	−0.0548167
$852$	0	0
$853$	−38.8061	−1.32869	−0.664347	−	0.747424i	$-0.731290\pi$
$853$	−38.8061	−1.32869	−0.664347	+	0.747424i	$0.731290\pi$
$854$	0	0
$855$	19.4615	0.665570
$856$	0	0
$857$	33.5880	1.14734	0.573672	−	0.819085i	$-0.305518\pi$
$857$	33.5880	1.14734	0.573672	+	0.819085i	$0.305518\pi$
$858$	0	0
$859$	−14.9315	−0.509455	−0.254727	−	0.967013i	$-0.581986\pi$
$859$	−14.9315	−0.509455	−0.254727	+	0.967013i	$0.581986\pi$
$860$	0	0
$861$	5.90268	0.201163
$862$	0	0
$863$	−18.9450	−0.644894	−0.322447	−	0.946588i	$-0.604505\pi$
$863$	−18.9450	−0.644894	−0.322447	+	0.946588i	$0.604505\pi$
$864$	0	0
$865$	9.16313	0.311556
$866$	0	0
$867$	−2.74294	−0.0931553
$868$	0	0
$869$	−85.7189	−2.90781
$870$	0	0
$871$	−62.6165	−2.12168
$872$	0	0
$873$	54.9978	1.86139
$874$	0	0
$875$	5.59104	0.189012
$876$	0	0
$877$	−16.3344	−0.551574	−0.275787	−	0.961219i	$-0.588938\pi$
$877$	−16.3344	−0.551574	−0.275787	+	0.961219i	$0.588938\pi$
$878$	0	0
$879$	−52.0773	−1.75652
$880$	0	0
$881$	5.41565	0.182458	0.0912289	−	0.995830i	$-0.470920\pi$
$881$	5.41565	0.182458	0.0912289	+	0.995830i	$0.470920\pi$
$882$	0	0
$883$	−21.7329	−0.731370	−0.365685	−	0.930739i	$-0.619165\pi$
$883$	−21.7329	−0.731370	−0.365685	+	0.930739i	$0.619165\pi$
$884$	0	0
$885$	−1.53620	−0.0516388
$886$	0	0
$887$	19.1177	0.641909	0.320954	−	0.947095i	$-0.395996\pi$
$887$	19.1177	0.641909	0.320954	+	0.947095i	$0.395996\pi$
$888$	0	0
$889$	−23.0454	−0.772919
$890$	0	0
$891$	−10.8997	−0.365154
$892$	0	0
$893$	69.3214	2.31975
$894$	0	0
$895$	−2.56382	−0.0856992
$896$	0	0
$897$	18.6442	0.622511
$898$	0	0
$899$	22.9342	0.764898
$900$	0	0
$901$	13.6020	0.453149
$902$	0	0
$903$	9.61940	0.320114
$904$	0	0
$905$	0.352472	0.0117166
$906$	0	0
$907$	38.4624	1.27712	0.638561	−	0.769571i	$-0.279530\pi$
$907$	38.4624	1.27712	0.638561	+	0.769571i	$0.279530\pi$
$908$	0	0
$909$	43.7845	1.45224
$910$	0	0
$911$	−27.8912	−0.924078	−0.462039	−	0.886860i	$-0.652882\pi$
$911$	−27.8912	−0.924078	−0.462039	+	0.886860i	$0.652882\pi$
$912$	0	0
$913$	58.1372	1.92406
$914$	0	0
$915$	2.19682	0.0726245
$916$	0	0
$917$	20.1636	0.665860
$918$	0	0
$919$	35.6029	1.17443	0.587215	−	0.809431i	$-0.300224\pi$
$919$	35.6029	1.17443	0.587215	+	0.809431i	$0.300224\pi$
$920$	0	0
$921$	75.1921	2.47766
$922$	0	0
$923$	0.324669	0.0106866
$924$	0	0
$925$	5.10778	0.167943
$926$	0	0
$927$	−28.5467	−0.937597
$928$	0	0
$929$	21.9694	0.720794	0.360397	−	0.932799i	$-0.382641\pi$
$929$	21.9694	0.720794	0.360397	+	0.932799i	$0.382641\pi$
$930$	0	0
$931$	45.6115	1.49486
$932$	0	0
$933$	0.210781	0.00690067
$934$	0	0
$935$	−2.89725	−0.0947502
$936$	0	0
$937$	30.1936	0.986381	0.493191	−	0.869921i	$-0.335831\pi$
$937$	30.1936	0.986381	0.493191	+	0.869921i	$0.335831\pi$
$938$	0	0
$939$	71.1872	2.32311
$940$	0	0
$941$	43.1344	1.40614	0.703071	−	0.711120i	$-0.251812\pi$
$941$	43.1344	1.40614	0.703071	+	0.711120i	$0.251812\pi$
$942$	0	0
$943$	−3.06343	−0.0997592
$944$	0	0
$945$	2.41247	0.0784776
$946$	0	0
$947$	−35.8584	−1.16524	−0.582620	−	0.812744i	$-0.697973\pi$
$947$	−35.8584	−1.16524	−0.582620	+	0.812744i	$0.697973\pi$
$948$	0	0
$949$	−21.8178	−0.708236
$950$	0	0
$951$	−23.9426	−0.776391
$952$	0	0
$953$	−53.8915	−1.74572	−0.872859	−	0.487972i	$-0.837737\pi$
$953$	−53.8915	−1.74572	−0.872859	+	0.487972i	$0.837737\pi$
$954$	0	0
$955$	10.1030	0.326926
$956$	0	0
$957$	−110.293	−3.56526
$958$	0	0
$959$	8.80750	0.284409
$960$	0	0
$961$	−22.2940	−0.719162
$962$	0	0
$963$	−30.1689	−0.972179
$964$	0	0
$965$	8.86362	0.285330
$966$	0	0
$967$	16.0384	0.515760	0.257880	−	0.966177i	$-0.416976\pi$
$967$	16.0384	0.515760	0.257880	+	0.966177i	$0.416976\pi$
$968$	0	0
$969$	21.0700	0.676867
$970$	0	0
$971$	45.4983	1.46011	0.730054	−	0.683389i	$-0.239495\pi$
$971$	45.4983	1.46011	0.730054	+	0.683389i	$0.239495\pi$
$972$	0	0
$973$	−17.8244	−0.571424
$974$	0	0
$975$	−59.5522	−1.90720
$976$	0	0
$977$	47.1604	1.50879	0.754397	−	0.656418i	$-0.227929\pi$
$977$	47.1604	1.50879	0.754397	+	0.656418i	$0.227929\pi$
$978$	0	0
$979$	−18.9551	−0.605809
$980$	0	0
$981$	39.6720	1.26663
$982$	0	0
$983$	3.90408	0.124521	0.0622605	−	0.998060i	$-0.480169\pi$
$983$	3.90408	0.124521	0.0622605	+	0.998060i	$0.480169\pi$
$984$	0	0
$985$	2.22796	0.0709886
$986$	0	0
$987$	25.5116	0.812045
$988$	0	0
$989$	−4.99238	−0.158748
$990$	0	0
$991$	5.53543	0.175839	0.0879194	−	0.996128i	$-0.471978\pi$
$991$	5.53543	0.175839	0.0879194	+	0.996128i	$0.471978\pi$
$992$	0	0
$993$	13.9906	0.443977
$994$	0	0
$995$	−0.395466	−0.0125371
$996$	0	0
$997$	25.9740	0.822606	0.411303	−	0.911499i	$-0.365074\pi$
$997$	25.9740	0.822606	0.411303	+	0.911499i	$0.365074\pi$
$998$	0	0
$999$	4.55541	0.144127

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	4012.2.a.h.1.2	✓	15

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
4012.2.a.h.1.2	✓	15	1.1	even	1	trivial

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Universe	Knowledge

Classical	Maass
Hilbert	Bianchi

Elliptic curves over $\Q$
Elliptic curves over $\Q(\alpha)$
Genus 2 curves over $\Q$
Higher genus families
Abelian varieties over $\F_{q}$

Level:	\( N \)	\(=\)	\( 4012 = 2^{2} \cdot 17 \cdot 59 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	4012.a (trivial)

\(f(q)\)	\(=\)	\(q-2.74294 q^{3} -0.560055 q^{5} +1.03063 q^{7} +4.52374 q^{9} +O(q^{10})\)	\(q-2.74294 q^{3} -0.560055 q^{5} +1.03063 q^{7} +4.52374 q^{9} +5.17316 q^{11} -4.63284 q^{13} +1.53620 q^{15} +1.00000 q^{17} -7.68154 q^{19} -2.82696 q^{21} +1.46716 q^{23} -4.68634 q^{25} -4.17954 q^{27} +7.77275 q^{29} +2.95059 q^{31} -14.1897 q^{33} -0.577209 q^{35} -1.08993 q^{37} +12.7076 q^{39} -2.08800 q^{41} -3.40274 q^{43} -2.53354 q^{45} -9.02441 q^{47} -5.93780 q^{49} -2.74294 q^{51} +13.6020 q^{53} -2.89725 q^{55} +21.0700 q^{57} -1.00000 q^{59} +1.43003 q^{61} +4.66230 q^{63} +2.59464 q^{65} +13.5158 q^{67} -4.02435 q^{69} -0.0700800 q^{71} +4.70938 q^{73} +12.8544 q^{75} +5.33161 q^{77} -16.5699 q^{79} -2.10698 q^{81} +11.2382 q^{83} -0.560055 q^{85} -21.3202 q^{87} -3.66413 q^{89} -4.77474 q^{91} -8.09330 q^{93} +4.30208 q^{95} +12.1576 q^{97} +23.4020 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 15 q - q^{3} + q^{5} - 11 q^{7} + 16 q^{9}+O(q^{10}) \) 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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