LMFDB - Embedded newform 6936.2.a.bj.1.1

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [6936,2,Mod(1,6936)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(6936, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([0, 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("6936.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 6936 = 2^{3} \cdot 3 \cdot 17^{2} $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	6936.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:	$55.3842388420$
Analytic rank:	$0$
Dimension:	$6$
Coefficient field:	6.6.50874368.1
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{6} - 2x^{5} - 13x^{4} + 12x^{3} + 56x^{2} + 8x - 34 $ x^6 - 2x^5 - 13x^4 + 12x^3 + 56x^2 + 8*x - 34
Coefficient ring:	$\Z[a_1, \ldots, a_{7}]$
Coefficient ring index:	$ 2^{2} $
Twist minimal:	no (minimal twist has level 408)
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.1
Root			$3.11578$ of defining polynomial
Character	$\chi$	$=$	6936.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+1.00000 q^{3} -3.36112 q^{5} -1.25969 q^{7} +1.00000 q^{9} +O(q^{10})$	$q+1.00000 q^{3} -3.36112 q^{5} -1.25969 q^{7} +1.00000 q^{9} +5.29711 q^{11} +3.51564 q^{13} -3.36112 q^{15} -3.51564 q^{19} -1.25969 q^{21} +2.02199 q^{23} +6.29711 q^{25} +1.00000 q^{27} +7.57068 q^{29} +6.23156 q^{31} +5.29711 q^{33} +4.23396 q^{35} -1.12639 q^{37} +3.51564 q^{39} -9.46466 q^{41} -5.29711 q^{43} -3.36112 q^{45} +7.43832 q^{47} -5.41318 q^{49} -10.2446 q^{53} -17.8042 q^{55} -3.51564 q^{57} -12.3040 q^{59} +4.83951 q^{61} -1.25969 q^{63} -11.8165 q^{65} -5.78147 q^{67} +2.02199 q^{69} -7.45135 q^{71} +3.16458 q^{73} +6.29711 q^{75} -6.67271 q^{77} +7.73508 q^{79} +1.00000 q^{81} +7.87539 q^{83} +7.57068 q^{87} -2.05942 q^{89} -4.42861 q^{91} +6.23156 q^{93} +11.8165 q^{95} +1.94902 q^{97} +5.29711 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 6 q + 6 q^{3} + 2 q^{5} + 4 q^{7} + 6 q^{9}+O(q^{10}) $ 6 * q + 6 * q^3 + 2 * q^5 + 4 * q^7 + 6 * q^9	$ 6 q + 6 q^{3} + 2 q^{5} + 4 q^{7} + 6 q^{9} + 6 q^{11} + 2 q^{13} + 2 q^{15} - 2 q^{19} + 4 q^{21} + 10 q^{23} + 12 q^{25} + 6 q^{27} - 8 q^{29} + 4 q^{31} + 6 q^{33} + 20 q^{35} + 12 q^{37} + 2 q^{39} + 6 q^{41} - 6 q^{43} + 2 q^{45} + 4 q^{47} + 18 q^{49} + 12 q^{53} - 10 q^{55} - 2 q^{57} + 16 q^{59} + 20 q^{61} + 4 q^{63} - 22 q^{65} - 28 q^{67} + 10 q^{69} + 36 q^{71} - 12 q^{73} + 12 q^{75} + 20 q^{77} + 16 q^{79} + 6 q^{81} + 20 q^{83} - 8 q^{87} + 4 q^{89} + 36 q^{91} + 4 q^{93} + 22 q^{95} - 32 q^{97} + 6 q^{99}+O(q^{100}) $ 6 * q + 6 * q^3 + 2 * q^5 + 4 * q^7 + 6 * q^9 + 6 * q^11 + 2 * q^13 + 2 * q^15 - 2 * q^19 + 4 * q^21 + 10 * q^23 + 12 * q^25 + 6 * q^27 - 8 * q^29 + 4 * q^31 + 6 * q^33 + 20 * q^35 + 12 * q^37 + 2 * q^39 + 6 * q^41 - 6 * q^43 + 2 * q^45 + 4 * q^47 + 18 * q^49 + 12 * q^53 - 10 * q^55 - 2 * q^57 + 16 * q^59 + 20 * q^61 + 4 * q^63 - 22 * q^65 - 28 * q^67 + 10 * q^69 + 36 * q^71 - 12 * q^73 + 12 * q^75 + 20 * q^77 + 16 * q^79 + 6 * q^81 + 20 * q^83 - 8 * q^87 + 4 * q^89 + 36 * q^91 + 4 * q^93 + 22 * q^95 - 32 * q^97 + 6 * 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
$3$	1.00000	0.577350
$3$	1.00000	0.577350
$4$	0	0
$5$	−3.36112	−1.50314	−0.751569	−	0.659655i	$-0.770702\pi$
$5$	−3.36112	−1.50314	−0.751569	+	0.659655i	$0.770702\pi$
$6$	0	0
$7$	−1.25969	−0.476117	−0.238059	−	0.971251i	$-0.576511\pi$
$7$	−1.25969	−0.476117	−0.238059	+	0.971251i	$0.576511\pi$
$8$	0	0
$9$	1.00000	0.333333
$10$	0	0
$11$	5.29711	1.59714	0.798570	−	0.601903i	$-0.205590\pi$
$11$	5.29711	1.59714	0.798570	+	0.601903i	$0.205590\pi$
$12$	0	0
$13$	3.51564	0.975064	0.487532	−	0.873105i	$-0.337897\pi$
$13$	3.51564	0.975064	0.487532	+	0.873105i	$0.337897\pi$
$14$	0	0
$15$	−3.36112	−0.867837
$16$	0	0
$17$	0	0
$17$	0	0
$18$	0	0
$19$	−3.51564	−0.806544	−0.403272	−	0.915080i	$-0.632127\pi$
$19$	−3.51564	−0.806544	−0.403272	+	0.915080i	$0.632127\pi$
$20$	0	0
$21$	−1.25969	−0.274887
$22$	0	0
$23$	2.02199	0.421615	0.210807	−	0.977528i	$-0.432391\pi$
$23$	2.02199	0.421615	0.210807	+	0.977528i	$0.432391\pi$
$24$	0	0
$25$	6.29711	1.25942
$26$	0	0
$27$	1.00000	0.192450
$28$	0	0
$29$	7.57068	1.40584	0.702920	−	0.711269i	$-0.251879\pi$
$29$	7.57068	1.40584	0.702920	+	0.711269i	$0.251879\pi$
$30$	0	0
$31$	6.23156	1.11922	0.559610	−	0.828756i	$-0.310951\pi$
$31$	6.23156	1.11922	0.559610	+	0.828756i	$0.310951\pi$
$32$	0	0
$33$	5.29711	0.922109
$34$	0	0
$35$	4.23396	0.715670
$36$	0	0
$37$	−1.12639	−0.185178	−0.0925891	−	0.995704i	$-0.529514\pi$
$37$	−1.12639	−0.185178	−0.0925891	+	0.995704i	$0.529514\pi$
$38$	0	0
$39$	3.51564	0.562953
$40$	0	0
$41$	−9.46466	−1.47813	−0.739066	−	0.673633i	$-0.764733\pi$
$41$	−9.46466	−1.47813	−0.739066	+	0.673633i	$0.764733\pi$
$42$	0	0
$43$	−5.29711	−0.807802	−0.403901	−	0.914803i	$-0.632346\pi$
$43$	−5.29711	−0.807802	−0.403901	+	0.914803i	$0.632346\pi$
$44$	0	0
$45$	−3.36112	−0.501046
$46$	0	0
$47$	7.43832	1.08499	0.542495	−	0.840059i	$-0.317480\pi$
$47$	7.43832	1.08499	0.542495	+	0.840059i	$0.317480\pi$
$48$	0	0
$49$	−5.41318	−0.773312
$50$	0	0
$51$	0	0
$52$	0	0
$53$	−10.2446	−1.40720	−0.703601	−	0.710595i	$-0.748426\pi$
$53$	−10.2446	−1.40720	−0.703601	+	0.710595i	$0.748426\pi$
$54$	0	0
$55$	−17.8042	−2.40072
$56$	0	0
$57$	−3.51564	−0.465658
$58$	0	0
$59$	−12.3040	−1.60184	−0.800922	−	0.598768i	$-0.795657\pi$
$59$	−12.3040	−1.60184	−0.800922	+	0.598768i	$0.795657\pi$
$60$	0	0
$61$	4.83951	0.619636	0.309818	−	0.950796i	$-0.399732\pi$
$61$	4.83951	0.619636	0.309818	+	0.950796i	$0.399732\pi$
$62$	0	0
$63$	−1.25969	−0.158706
$64$	0	0
$65$	−11.8165	−1.46565
$66$	0	0
$67$	−5.78147	−0.706319	−0.353159	−	0.935563i	$-0.614893\pi$
$67$	−5.78147	−0.706319	−0.353159	+	0.935563i	$0.614893\pi$
$68$	0	0
$69$	2.02199	0.243419
$70$	0	0
$71$	−7.45135	−0.884312	−0.442156	−	0.896938i	$-0.645786\pi$
$71$	−7.45135	−0.884312	−0.442156	+	0.896938i	$0.645786\pi$
$72$	0	0
$73$	3.16458	0.370386	0.185193	−	0.982702i	$-0.440709\pi$
$73$	3.16458	0.370386	0.185193	+	0.982702i	$0.440709\pi$
$74$	0	0
$75$	6.29711	0.727128
$76$	0	0
$77$	−6.67271	−0.760426
$78$	0	0
$79$	7.73508	0.870264	0.435132	−	0.900367i	$-0.356702\pi$
$79$	7.73508	0.870264	0.435132	+	0.900367i	$0.356702\pi$
$80$	0	0
$81$	1.00000	0.111111
$82$	0	0
$83$	7.87539	0.864436	0.432218	−	0.901769i	$-0.357731\pi$
$83$	7.87539	0.864436	0.432218	+	0.901769i	$0.357731\pi$
$84$	0	0
$85$	0	0
$86$	0	0
$87$	7.57068	0.811662
$88$	0	0
$89$	−2.05942	−0.218298	−0.109149	−	0.994025i	$-0.534813\pi$
$89$	−2.05942	−0.218298	−0.109149	+	0.994025i	$0.534813\pi$
$90$	0	0
$91$	−4.42861	−0.464245
$92$	0	0
$93$	6.23156	0.646182
$94$	0	0
$95$	11.8165	1.21235
$96$	0	0
$97$	1.94902	0.197893	0.0989465	−	0.995093i	$-0.468453\pi$
$97$	1.94902	0.197893	0.0989465	+	0.995093i	$0.468453\pi$
$98$	0	0
$99$	5.29711	0.532380
$100$	0	0
$101$	11.7692	1.17108	0.585539	−	0.810644i	$-0.300883\pi$
$101$	11.7692	1.17108	0.585539	+	0.810644i	$0.300883\pi$
$102$	0	0
$103$	−13.0223	−1.28313	−0.641564	−	0.767070i	$-0.721714\pi$
$103$	−13.0223	−1.28313	−0.641564	+	0.767070i	$0.721714\pi$
$104$	0	0
$105$	4.23396	0.413192
$106$	0	0
$107$	12.8229	1.23963	0.619816	−	0.784747i	$-0.287207\pi$
$107$	12.8229	1.23963	0.619816	+	0.784747i	$0.287207\pi$
$108$	0	0
$109$	18.5677	1.77846	0.889232	−	0.457456i	$-0.151239\pi$
$109$	18.5677	1.77846	0.889232	+	0.457456i	$0.151239\pi$
$110$	0	0
$111$	−1.12639	−0.106913
$112$	0	0
$113$	20.6240	1.94014	0.970070	−	0.242827i	$-0.0780749\pi$
$113$	20.6240	1.94014	0.970070	+	0.242827i	$0.0780749\pi$
$114$	0	0
$115$	−6.79616	−0.633745
$116$	0	0
$117$	3.51564	0.325021
$118$	0	0
$119$	0	0
$120$	0	0
$121$	17.0594	1.55085
$122$	0	0
$123$	−9.46466	−0.853400
$124$	0	0
$125$	−4.35974	−0.389947
$126$	0	0
$127$	5.17828	0.459498	0.229749	−	0.973250i	$-0.426209\pi$
$127$	5.17828	0.459498	0.229749	+	0.973250i	$0.426209\pi$
$128$	0	0
$129$	−5.29711	−0.466385
$130$	0	0
$131$	−9.97261	−0.871311	−0.435656	−	0.900113i	$-0.643483\pi$
$131$	−9.97261	−0.871311	−0.435656	+	0.900113i	$0.643483\pi$
$132$	0	0
$133$	4.42861	0.384010
$134$	0	0
$135$	−3.36112	−0.289279
$136$	0	0
$137$	14.1917	1.21248	0.606238	−	0.795284i	$-0.292678\pi$
$137$	14.1917	1.21248	0.606238	+	0.795284i	$0.292678\pi$
$138$	0	0
$139$	3.89065	0.330001	0.165000	−	0.986293i	$-0.447237\pi$
$139$	3.89065	0.330001	0.165000	+	0.986293i	$0.447237\pi$
$140$	0	0
$141$	7.43832	0.626420
$142$	0	0
$143$	18.6227	1.55731
$144$	0	0
$145$	−25.4460	−2.11317
$146$	0	0
$147$	−5.41318	−0.446472
$148$	0	0
$149$	16.5225	1.35358	0.676789	−	0.736177i	$-0.263371\pi$
$149$	16.5225	1.35358	0.676789	+	0.736177i	$0.263371\pi$
$150$	0	0
$151$	21.2898	1.73254	0.866271	−	0.499575i	$-0.166510\pi$
$151$	21.2898	1.73254	0.866271	+	0.499575i	$0.166510\pi$
$152$	0	0
$153$	0	0
$154$	0	0
$155$	−20.9450	−1.68234
$156$	0	0
$157$	−3.79638	−0.302984	−0.151492	−	0.988458i	$-0.548408\pi$
$157$	−3.79638	−0.302984	−0.151492	+	0.988458i	$0.548408\pi$
$158$	0	0
$159$	−10.2446	−0.812448
$160$	0	0
$161$	−2.54708	−0.200738
$162$	0	0
$163$	−3.45675	−0.270753	−0.135377	−	0.990794i	$-0.543224\pi$
$163$	−3.45675	−0.270753	−0.135377	+	0.990794i	$0.543224\pi$
$164$	0	0
$165$	−17.8042	−1.38606
$166$	0	0
$167$	1.69112	0.130863	0.0654315	−	0.997857i	$-0.479158\pi$
$167$	1.69112	0.130863	0.0654315	+	0.997857i	$0.479158\pi$
$168$	0	0
$169$	−0.640257	−0.0492505
$170$	0	0
$171$	−3.51564	−0.268848
$172$	0	0
$173$	10.2957	0.782770	0.391385	−	0.920227i	$-0.371996\pi$
$173$	10.2957	0.782770	0.391385	+	0.920227i	$0.371996\pi$
$174$	0	0
$175$	−7.93240	−0.599633
$176$	0	0
$177$	−12.3040	−0.924825
$178$	0	0
$179$	18.8811	1.41124	0.705620	−	0.708590i	$-0.250669\pi$
$179$	18.8811	1.41124	0.705620	+	0.708590i	$0.250669\pi$
$180$	0	0
$181$	−17.2273	−1.28049	−0.640247	−	0.768169i	$-0.721168\pi$
$181$	−17.2273	−1.28049	−0.640247	+	0.768169i	$0.721168\pi$
$182$	0	0
$183$	4.83951	0.357747
$184$	0	0
$185$	3.78594	0.278348
$186$	0	0
$187$	0	0
$188$	0	0
$189$	−1.25969	−0.0916289
$190$	0	0
$191$	1.42735	0.103280	0.0516399	−	0.998666i	$-0.483555\pi$
$191$	1.42735	0.103280	0.0516399	+	0.998666i	$0.483555\pi$
$192$	0	0
$193$	8.98031	0.646417	0.323208	−	0.946328i	$-0.395239\pi$
$193$	8.98031	0.646417	0.323208	+	0.946328i	$0.395239\pi$
$194$	0	0
$195$	−11.8165	−0.846196
$196$	0	0
$197$	11.6920	0.833020	0.416510	−	0.909131i	$-0.363253\pi$
$197$	11.6920	0.833020	0.416510	+	0.909131i	$0.363253\pi$
$198$	0	0
$199$	−22.0940	−1.56620	−0.783102	−	0.621894i	$-0.786364\pi$
$199$	−22.0940	−1.56620	−0.783102	+	0.621894i	$0.786364\pi$
$200$	0	0
$201$	−5.78147	−0.407793
$202$	0	0
$203$	−9.53670	−0.669345
$204$	0	0
$205$	31.8118	2.22184
$206$	0	0
$207$	2.02199	0.140538
$208$	0	0
$209$	−18.6227	−1.28816
$210$	0	0
$211$	2.68777	0.185034	0.0925170	−	0.995711i	$-0.470509\pi$
$211$	2.68777	0.185034	0.0925170	+	0.995711i	$0.470509\pi$
$212$	0	0
$213$	−7.45135	−0.510558
$214$	0	0
$215$	17.8042	1.21424
$216$	0	0
$217$	−7.84982	−0.532881
$218$	0	0
$219$	3.16458	0.213842
$220$	0	0
$221$	0	0
$222$	0	0
$223$	−10.9974	−0.736441	−0.368220	−	0.929739i	$-0.620033\pi$
$223$	−10.9974	−0.736441	−0.368220	+	0.929739i	$0.620033\pi$
$224$	0	0
$225$	6.29711	0.419807
$226$	0	0
$227$	28.2796	1.87698	0.938492	−	0.345302i	$-0.112224\pi$
$227$	28.2796	1.87698	0.938492	+	0.345302i	$0.112224\pi$
$228$	0	0
$229$	2.65048	0.175149	0.0875744	−	0.996158i	$-0.472088\pi$
$229$	2.65048	0.175149	0.0875744	+	0.996158i	$0.472088\pi$
$230$	0	0
$231$	−6.67271	−0.439032
$232$	0	0
$233$	−6.62360	−0.433926	−0.216963	−	0.976180i	$-0.569615\pi$
$233$	−6.62360	−0.433926	−0.216963	+	0.976180i	$0.569615\pi$
$234$	0	0
$235$	−25.0011	−1.63089
$236$	0	0
$237$	7.73508	0.502447
$238$	0	0
$239$	4.07582	0.263643	0.131822	−	0.991273i	$-0.457917\pi$
$239$	4.07582	0.263643	0.131822	+	0.991273i	$0.457917\pi$
$240$	0	0
$241$	25.1955	1.62298	0.811492	−	0.584363i	$-0.198656\pi$
$241$	25.1955	1.62298	0.811492	+	0.584363i	$0.198656\pi$
$242$	0	0
$243$	1.00000	0.0641500
$244$	0	0
$245$	18.1944	1.16239
$246$	0	0
$247$	−12.3597	−0.786432
$248$	0	0
$249$	7.87539	0.499082
$250$	0	0
$251$	8.39177	0.529684	0.264842	−	0.964292i	$-0.414680\pi$
$251$	8.39177	0.529684	0.264842	+	0.964292i	$0.414680\pi$
$252$	0	0
$253$	10.7107	0.673377
$254$	0	0
$255$	0	0
$256$	0	0
$257$	−30.4927	−1.90208	−0.951040	−	0.309067i	$-0.899983\pi$
$257$	−30.4927	−1.90208	−0.951040	+	0.309067i	$0.899983\pi$
$258$	0	0
$259$	1.41891	0.0881665
$260$	0	0
$261$	7.57068	0.468613
$262$	0	0
$263$	19.2976	1.18994	0.594971	−	0.803747i	$-0.297163\pi$
$263$	19.2976	1.18994	0.594971	+	0.803747i	$0.297163\pi$
$264$	0	0
$265$	34.4333	2.11522
$266$	0	0
$267$	−2.05942	−0.126034
$268$	0	0
$269$	0.462077	0.0281734	0.0140867	−	0.999901i	$-0.495516\pi$
$269$	0.462077	0.0281734	0.0140867	+	0.999901i	$0.495516\pi$
$270$	0	0
$271$	22.4044	1.36097	0.680485	−	0.732762i	$-0.261769\pi$
$271$	22.4044	1.36097	0.680485	+	0.732762i	$0.261769\pi$
$272$	0	0
$273$	−4.42861	−0.268032
$274$	0	0
$275$	33.3565	2.01147
$276$	0	0
$277$	−4.04868	−0.243261	−0.121631	−	0.992575i	$-0.538812\pi$
$277$	−4.04868	−0.243261	−0.121631	+	0.992575i	$0.538812\pi$
$278$	0	0
$279$	6.23156	0.373074
$280$	0	0
$281$	−5.58551	−0.333203	−0.166602	−	0.986024i	$-0.553279\pi$
$281$	−5.58551	−0.333203	−0.166602	+	0.986024i	$0.553279\pi$
$282$	0	0
$283$	−15.2926	−0.909052	−0.454526	−	0.890733i	$-0.650191\pi$
$283$	−15.2926	−0.909052	−0.454526	+	0.890733i	$0.650191\pi$
$284$	0	0
$285$	11.8165	0.699948
$286$	0	0
$287$	11.9225	0.703764
$288$	0	0
$289$	0	0
$290$	0	0
$291$	1.94902	0.114254
$292$	0	0
$293$	19.8030	1.15691	0.578453	−	0.815716i	$-0.303657\pi$
$293$	19.8030	1.15691	0.578453	+	0.815716i	$0.303657\pi$
$294$	0	0
$295$	41.3552	2.40779
$296$	0	0
$297$	5.29711	0.307370
$298$	0	0
$299$	7.10861	0.411101
$300$	0	0
$301$	6.67271	0.384609
$302$	0	0
$303$	11.7692	0.676123
$304$	0	0
$305$	−16.2662	−0.931398
$306$	0	0
$307$	−31.2649	−1.78438	−0.892192	−	0.451657i	$-0.850833\pi$
$307$	−31.2649	−1.78438	−0.892192	+	0.451657i	$0.850833\pi$
$308$	0	0
$309$	−13.0223	−0.740814
$310$	0	0
$311$	2.02585	0.114875	0.0574377	−	0.998349i	$-0.481707\pi$
$311$	2.02585	0.114875	0.0574377	+	0.998349i	$0.481707\pi$
$312$	0	0
$313$	0.0758942	0.00428980	0.00214490	−	0.999998i	$-0.499317\pi$
$313$	0.0758942	0.00428980	0.00214490	+	0.999998i	$0.499317\pi$
$314$	0	0
$315$	4.23396	0.238557
$316$	0	0
$317$	−21.2536	−1.19372	−0.596860	−	0.802345i	$-0.703585\pi$
$317$	−21.2536	−1.19372	−0.596860	+	0.802345i	$0.703585\pi$
$318$	0	0
$319$	40.1027	2.24532
$320$	0	0
$321$	12.8229	0.715702
$322$	0	0
$323$	0	0
$324$	0	0
$325$	22.1384	1.22802
$326$	0	0
$327$	18.5677	1.02680
$328$	0	0
$329$	−9.36997	−0.516583
$330$	0	0
$331$	−6.98531	−0.383947	−0.191974	−	0.981400i	$-0.561489\pi$
$331$	−6.98531	−0.383947	−0.191974	+	0.981400i	$0.561489\pi$
$332$	0	0
$333$	−1.12639	−0.0617260
$334$	0	0
$335$	19.4322	1.06169
$336$	0	0
$337$	−22.2486	−1.21196	−0.605979	−	0.795481i	$-0.707218\pi$
$337$	−22.2486	−1.21196	−0.605979	+	0.795481i	$0.707218\pi$
$338$	0	0
$339$	20.6240	1.12014
$340$	0	0
$341$	33.0093	1.78755
$342$	0	0
$343$	15.6367	0.844305
$344$	0	0
$345$	−6.79616	−0.365893
$346$	0	0
$347$	21.2324	1.13982	0.569908	−	0.821708i	$-0.306979\pi$
$347$	21.2324	1.13982	0.569908	+	0.821708i	$0.306979\pi$
$348$	0	0
$349$	−25.0044	−1.33846	−0.669229	−	0.743057i	$-0.733375\pi$
$349$	−25.0044	−1.33846	−0.669229	+	0.743057i	$0.733375\pi$
$350$	0	0
$351$	3.51564	0.187651
$352$	0	0
$353$	15.8608	0.844185	0.422092	−	0.906553i	$-0.361296\pi$
$353$	15.8608	0.844185	0.422092	+	0.906553i	$0.361296\pi$
$354$	0	0
$355$	25.0449	1.32924
$356$	0	0
$357$	0	0
$358$	0	0
$359$	23.5921	1.24515	0.622573	−	0.782562i	$-0.286087\pi$
$359$	23.5921	1.24515	0.622573	+	0.782562i	$0.286087\pi$
$360$	0	0
$361$	−6.64026	−0.349487
$362$	0	0
$363$	17.0594	0.895386
$364$	0	0
$365$	−10.6365	−0.556741
$366$	0	0
$367$	13.3785	0.698353	0.349177	−	0.937057i	$-0.386461\pi$
$367$	13.3785	0.698353	0.349177	+	0.937057i	$0.386461\pi$
$368$	0	0
$369$	−9.46466	−0.492711
$370$	0	0
$371$	12.9050	0.669993
$372$	0	0
$373$	−14.3508	−0.743055	−0.371528	−	0.928422i	$-0.621166\pi$
$373$	−14.3508	−0.743055	−0.371528	+	0.928422i	$0.621166\pi$
$374$	0	0
$375$	−4.35974	−0.225136
$376$	0	0
$377$	26.6158	1.37078
$378$	0	0
$379$	−19.9186	−1.02315	−0.511574	−	0.859239i	$-0.670937\pi$
$379$	−19.9186	−1.02315	−0.511574	+	0.859239i	$0.670937\pi$
$380$	0	0
$381$	5.17828	0.265291
$382$	0	0
$383$	18.2886	0.934502	0.467251	−	0.884125i	$-0.345244\pi$
$383$	18.2886	0.934502	0.467251	+	0.884125i	$0.345244\pi$
$384$	0	0
$385$	22.4278	1.14302
$386$	0	0
$387$	−5.29711	−0.269267
$388$	0	0
$389$	7.92188	0.401655	0.200828	−	0.979627i	$-0.435637\pi$
$389$	7.92188	0.401655	0.200828	+	0.979627i	$0.435637\pi$
$390$	0	0
$391$	0	0
$392$	0	0
$393$	−9.97261	−0.503052
$394$	0	0
$395$	−25.9985	−1.30813
$396$	0	0
$397$	−20.0170	−1.00462	−0.502312	−	0.864687i	$-0.667517\pi$
$397$	−20.0170	−1.00462	−0.502312	+	0.864687i	$0.667517\pi$
$398$	0	0
$399$	4.42861	0.221708
$400$	0	0
$401$	−19.3094	−0.964263	−0.482131	−	0.876099i	$-0.660137\pi$
$401$	−19.3094	−0.964263	−0.482131	+	0.876099i	$0.660137\pi$
$402$	0	0
$403$	21.9079	1.09131
$404$	0	0
$405$	−3.36112	−0.167015
$406$	0	0
$407$	−5.96664	−0.295755
$408$	0	0
$409$	−27.0359	−1.33684	−0.668420	−	0.743784i	$-0.733029\pi$
$409$	−27.0359	−1.33684	−0.668420	+	0.743784i	$0.733029\pi$
$410$	0	0
$411$	14.1917	0.700023
$412$	0	0
$413$	15.4992	0.762666
$414$	0	0
$415$	−26.4701	−1.29937
$416$	0	0
$417$	3.89065	0.190526
$418$	0	0
$419$	−3.68623	−0.180084	−0.0900421	−	0.995938i	$-0.528700\pi$
$419$	−3.68623	−0.180084	−0.0900421	+	0.995938i	$0.528700\pi$
$420$	0	0
$421$	−7.95271	−0.387591	−0.193796	−	0.981042i	$-0.562080\pi$
$421$	−7.95271	−0.387591	−0.193796	+	0.981042i	$0.562080\pi$
$422$	0	0
$423$	7.43832	0.361664
$424$	0	0
$425$	0	0
$426$	0	0
$427$	−6.09627	−0.295019
$428$	0	0
$429$	18.6227	0.899115
$430$	0	0
$431$	25.7945	1.24248	0.621239	−	0.783622i	$-0.286630\pi$
$431$	25.7945	1.24248	0.621239	+	0.783622i	$0.286630\pi$
$432$	0	0
$433$	23.3482	1.12204	0.561021	−	0.827802i	$-0.310409\pi$
$433$	23.3482	1.12204	0.561021	+	0.827802i	$0.310409\pi$
$434$	0	0
$435$	−25.4460	−1.22004
$436$	0	0
$437$	−7.10861	−0.340051
$438$	0	0
$439$	11.0617	0.527946	0.263973	−	0.964530i	$-0.414967\pi$
$439$	11.0617	0.527946	0.263973	+	0.964530i	$0.414967\pi$
$440$	0	0
$441$	−5.41318	−0.257771
$442$	0	0
$443$	9.66132	0.459023	0.229511	−	0.973306i	$-0.426287\pi$
$443$	9.66132	0.459023	0.229511	+	0.973306i	$0.426287\pi$
$444$	0	0
$445$	6.92194	0.328131
$446$	0	0
$447$	16.5225	0.781489
$448$	0	0
$449$	32.5200	1.53471	0.767357	−	0.641220i	$-0.221571\pi$
$449$	32.5200	1.53471	0.767357	+	0.641220i	$0.221571\pi$
$450$	0	0
$451$	−50.1354	−2.36078
$452$	0	0
$453$	21.2898	1.00028
$454$	0	0
$455$	14.8851	0.697824
$456$	0	0
$457$	−20.4726	−0.957667	−0.478833	−	0.877906i	$-0.658940\pi$
$457$	−20.4726	−0.957667	−0.478833	+	0.877906i	$0.658940\pi$
$458$	0	0
$459$	0	0
$460$	0	0
$461$	8.36140	0.389429	0.194715	−	0.980860i	$-0.437622\pi$
$461$	8.36140	0.389429	0.194715	+	0.980860i	$0.437622\pi$
$462$	0	0
$463$	−3.21364	−0.149351	−0.0746753	−	0.997208i	$-0.523792\pi$
$463$	−3.21364	−0.149351	−0.0746753	+	0.997208i	$0.523792\pi$
$464$	0	0
$465$	−20.9450	−0.971301
$466$	0	0
$467$	−16.4242	−0.760019	−0.380010	−	0.924983i	$-0.624079\pi$
$467$	−16.4242	−0.760019	−0.380010	+	0.924983i	$0.624079\pi$
$468$	0	0
$469$	7.28285	0.336291
$470$	0	0
$471$	−3.79638	−0.174928
$472$	0	0
$473$	−28.0594	−1.29017
$474$	0	0
$475$	−22.1384	−1.01578
$476$	0	0
$477$	−10.2446	−0.469067
$478$	0	0
$479$	17.4787	0.798624	0.399312	−	0.916815i	$-0.369249\pi$
$479$	17.4787	0.798624	0.399312	+	0.916815i	$0.369249\pi$
$480$	0	0
$481$	−3.96000	−0.180560
$482$	0	0
$483$	−2.54708	−0.115896
$484$	0	0
$485$	−6.55089	−0.297460
$486$	0	0
$487$	−33.2876	−1.50840	−0.754202	−	0.656642i	$-0.771976\pi$
$487$	−33.2876	−1.50840	−0.754202	+	0.656642i	$0.771976\pi$
$488$	0	0
$489$	−3.45675	−0.156319
$490$	0	0
$491$	−17.8636	−0.806173	−0.403086	−	0.915162i	$-0.632063\pi$
$491$	−17.8636	−0.806173	−0.403086	+	0.915162i	$0.632063\pi$
$492$	0	0
$493$	0	0
$494$	0	0
$495$	−17.8042	−0.800240
$496$	0	0
$497$	9.38638	0.421037
$498$	0	0
$499$	0.849109	0.0380113	0.0190057	−	0.999819i	$-0.493950\pi$
$499$	0.849109	0.0380113	0.0190057	+	0.999819i	$0.493950\pi$
$500$	0	0
$501$	1.69112	0.0755538
$502$	0	0
$503$	−9.60917	−0.428452	−0.214226	−	0.976784i	$-0.568723\pi$
$503$	−9.60917	−0.428452	−0.214226	+	0.976784i	$0.568723\pi$
$504$	0	0
$505$	−39.5576	−1.76029
$506$	0	0
$507$	−0.640257	−0.0284348
$508$	0	0
$509$	23.2619	1.03107	0.515533	−	0.856870i	$-0.327594\pi$
$509$	23.2619	1.03107	0.515533	+	0.856870i	$0.327594\pi$
$510$	0	0
$511$	−3.98638	−0.176347
$512$	0	0
$513$	−3.51564	−0.155219
$514$	0	0
$515$	43.7695	1.92872
$516$	0	0
$517$	39.4016	1.73288
$518$	0	0
$519$	10.2957	0.451933
$520$	0	0
$521$	28.9788	1.26958	0.634792	−	0.772683i	$-0.281086\pi$
$521$	28.9788	1.26958	0.634792	+	0.772683i	$0.281086\pi$
$522$	0	0
$523$	−3.85560	−0.168594	−0.0842969	−	0.996441i	$-0.526864\pi$
$523$	−3.85560	−0.168594	−0.0842969	+	0.996441i	$0.526864\pi$
$524$	0	0
$525$	−7.93240	−0.346198
$526$	0	0
$527$	0	0
$528$	0	0
$529$	−18.9115	−0.822241
$530$	0	0
$531$	−12.3040	−0.533948
$532$	0	0
$533$	−33.2744	−1.44127
$534$	0	0
$535$	−43.0991	−1.86334
$536$	0	0
$537$	18.8811	0.814780
$538$	0	0
$539$	−28.6742	−1.23509
$540$	0	0
$541$	−2.21546	−0.0952503	−0.0476251	−	0.998865i	$-0.515165\pi$
$541$	−2.21546	−0.0952503	−0.0476251	+	0.998865i	$0.515165\pi$
$542$	0	0
$543$	−17.2273	−0.739294
$544$	0	0
$545$	−62.4083	−2.67328
$546$	0	0
$547$	20.1417	0.861198	0.430599	−	0.902543i	$-0.358302\pi$
$547$	20.1417	0.861198	0.430599	+	0.902543i	$0.358302\pi$
$548$	0	0
$549$	4.83951	0.206545
$550$	0	0
$551$	−26.6158	−1.13387
$552$	0	0
$553$	−9.74379	−0.414348
$554$	0	0
$555$	3.78594	0.160704
$556$	0	0
$557$	−26.1303	−1.10718	−0.553588	−	0.832791i	$-0.686742\pi$
$557$	−26.1303	−1.10718	−0.553588	+	0.832791i	$0.686742\pi$
$558$	0	0
$559$	−18.6227	−0.787658
$560$	0	0
$561$	0	0
$562$	0	0
$563$	20.8262	0.877718	0.438859	−	0.898556i	$-0.355383\pi$
$563$	20.8262	0.877718	0.438859	+	0.898556i	$0.355383\pi$
$564$	0	0
$565$	−69.3196	−2.91630
$566$	0	0
$567$	−1.25969	−0.0529019
$568$	0	0
$569$	−23.6182	−0.990126	−0.495063	−	0.868857i	$-0.664855\pi$
$569$	−23.6182	−0.990126	−0.495063	+	0.868857i	$0.664855\pi$
$570$	0	0
$571$	33.3032	1.39370	0.696849	−	0.717218i	$-0.254585\pi$
$571$	33.3032	1.39370	0.696849	+	0.717218i	$0.254585\pi$
$572$	0	0
$573$	1.42735	0.0596286
$574$	0	0
$575$	12.7327	0.530991
$576$	0	0
$577$	29.7818	1.23983	0.619917	−	0.784668i	$-0.287166\pi$
$577$	29.7818	1.23983	0.619917	+	0.784668i	$0.287166\pi$
$578$	0	0
$579$	8.98031	0.373209
$580$	0	0
$581$	−9.92053	−0.411573
$582$	0	0
$583$	−54.2667	−2.24750
$584$	0	0
$585$	−11.8165	−0.488552
$586$	0	0
$587$	−13.3509	−0.551051	−0.275526	−	0.961294i	$-0.588852\pi$
$587$	−13.3509	−0.551051	−0.275526	+	0.961294i	$0.588852\pi$
$588$	0	0
$589$	−21.9079	−0.902701
$590$	0	0
$591$	11.6920	0.480944
$592$	0	0
$593$	2.23097	0.0916149	0.0458074	−	0.998950i	$-0.485414\pi$
$593$	2.23097	0.0916149	0.0458074	+	0.998950i	$0.485414\pi$
$594$	0	0
$595$	0	0
$596$	0	0
$597$	−22.0940	−0.904248
$598$	0	0
$599$	−42.1932	−1.72397	−0.861983	−	0.506937i	$-0.830778\pi$
$599$	−42.1932	−1.72397	−0.861983	+	0.506937i	$0.830778\pi$
$600$	0	0
$601$	2.92591	0.119350	0.0596752	−	0.998218i	$-0.480993\pi$
$601$	2.92591	0.119350	0.0596752	+	0.998218i	$0.480993\pi$
$602$	0	0
$603$	−5.78147	−0.235440
$604$	0	0
$605$	−57.3386	−2.33115
$606$	0	0
$607$	23.2101	0.942068	0.471034	−	0.882115i	$-0.343881\pi$
$607$	23.2101	0.942068	0.471034	+	0.882115i	$0.343881\pi$
$608$	0	0
$609$	−9.53670	−0.386447
$610$	0	0
$611$	26.1505	1.05794
$612$	0	0
$613$	45.8752	1.85288	0.926442	−	0.376438i	$-0.122851\pi$
$613$	45.8752	1.85288	0.926442	+	0.376438i	$0.122851\pi$
$614$	0	0
$615$	31.8118	1.28278
$616$	0	0
$617$	−29.3961	−1.18344	−0.591721	−	0.806143i	$-0.701551\pi$
$617$	−29.3961	−1.18344	−0.591721	+	0.806143i	$0.701551\pi$
$618$	0	0
$619$	25.9424	1.04271	0.521357	−	0.853339i	$-0.325426\pi$
$619$	25.9424	1.04271	0.521357	+	0.853339i	$0.325426\pi$
$620$	0	0
$621$	2.02199	0.0811398
$622$	0	0
$623$	2.59422	0.103935
$624$	0	0
$625$	−16.8319	−0.673278
$626$	0	0
$627$	−18.6227	−0.743721
$628$	0	0
$629$	0	0
$630$	0	0
$631$	−0.483870	−0.0192626	−0.00963128	−	0.999954i	$-0.503066\pi$
$631$	−0.483870	−0.0192626	−0.00963128	+	0.999954i	$0.503066\pi$
$632$	0	0
$633$	2.68777	0.106829
$634$	0	0
$635$	−17.4048	−0.690689
$636$	0	0
$637$	−19.0308	−0.754029
$638$	0	0
$639$	−7.45135	−0.294771
$640$	0	0
$641$	−26.7987	−1.05849	−0.529243	−	0.848471i	$-0.677524\pi$
$641$	−26.7987	−1.05849	−0.529243	+	0.848471i	$0.677524\pi$
$642$	0	0
$643$	38.8010	1.53016	0.765080	−	0.643935i	$-0.222699\pi$
$643$	38.8010	1.53016	0.765080	+	0.643935i	$0.222699\pi$
$644$	0	0
$645$	17.8042	0.701040
$646$	0	0
$647$	6.88192	0.270556	0.135278	−	0.990808i	$-0.456807\pi$
$647$	6.88192	0.270556	0.135278	+	0.990808i	$0.456807\pi$
$648$	0	0
$649$	−65.1757	−2.55837
$650$	0	0
$651$	−7.84982	−0.307659
$652$	0	0
$653$	32.1249	1.25714	0.628572	−	0.777752i	$-0.283640\pi$
$653$	32.1249	1.25714	0.628572	+	0.777752i	$0.283640\pi$
$654$	0	0
$655$	33.5191	1.30970
$656$	0	0
$657$	3.16458	0.123462
$658$	0	0
$659$	25.4302	0.990621	0.495311	−	0.868716i	$-0.335054\pi$
$659$	25.4302	0.990621	0.495311	+	0.868716i	$0.335054\pi$
$660$	0	0
$661$	27.1665	1.05665	0.528327	−	0.849041i	$-0.322819\pi$
$661$	27.1665	1.05665	0.528327	+	0.849041i	$0.322819\pi$
$662$	0	0
$663$	0	0
$664$	0	0
$665$	−14.8851	−0.577219
$666$	0	0
$667$	15.3079	0.592723
$668$	0	0
$669$	−10.9974	−0.425184
$670$	0	0
$671$	25.6354	0.989644
$672$	0	0
$673$	15.3432	0.591438	0.295719	−	0.955275i	$-0.404441\pi$
$673$	15.3432	0.591438	0.295719	+	0.955275i	$0.404441\pi$
$674$	0	0
$675$	6.29711	0.242376
$676$	0	0
$677$	10.6261	0.408395	0.204197	−	0.978930i	$-0.434542\pi$
$677$	10.6261	0.408395	0.204197	+	0.978930i	$0.434542\pi$
$678$	0	0
$679$	−2.45516	−0.0942203
$680$	0	0
$681$	28.2796	1.08368
$682$	0	0
$683$	−36.1231	−1.38221	−0.691106	−	0.722753i	$-0.742876\pi$
$683$	−36.1231	−1.38221	−0.691106	+	0.722753i	$0.742876\pi$
$684$	0	0
$685$	−47.6998	−1.82252
$686$	0	0
$687$	2.65048	0.101122
$688$	0	0
$689$	−36.0163	−1.37211
$690$	0	0
$691$	42.7074	1.62467	0.812333	−	0.583193i	$-0.198197\pi$
$691$	42.7074	1.62467	0.812333	+	0.583193i	$0.198197\pi$
$692$	0	0
$693$	−6.67271	−0.253475
$694$	0	0
$695$	−13.0769	−0.496037
$696$	0	0
$697$	0	0
$698$	0	0
$699$	−6.62360	−0.250527
$700$	0	0
$701$	26.3392	0.994817	0.497408	−	0.867516i	$-0.334285\pi$
$701$	26.3392	0.994817	0.497408	+	0.867516i	$0.334285\pi$
$702$	0	0
$703$	3.96000	0.149354
$704$	0	0
$705$	−25.0011	−0.941595
$706$	0	0
$707$	−14.8255	−0.557571
$708$	0	0
$709$	42.6594	1.60211	0.801054	−	0.598592i	$-0.204273\pi$
$709$	42.6594	1.60211	0.801054	+	0.598592i	$0.204273\pi$
$710$	0	0
$711$	7.73508	0.290088
$712$	0	0
$713$	12.6002	0.471880
$714$	0	0
$715$	−62.5933	−2.34085
$716$	0	0
$717$	4.07582	0.152214
$718$	0	0
$719$	45.3554	1.69147	0.845736	−	0.533602i	$-0.179162\pi$
$719$	45.3554	1.69147	0.845736	+	0.533602i	$0.179162\pi$
$720$	0	0
$721$	16.4041	0.610919
$722$	0	0
$723$	25.1955	0.937031
$724$	0	0
$725$	47.6734	1.77055
$726$	0	0
$727$	1.38784	0.0514723	0.0257362	−	0.999669i	$-0.491807\pi$
$727$	1.38784	0.0514723	0.0257362	+	0.999669i	$0.491807\pi$
$728$	0	0
$729$	1.00000	0.0370370
$730$	0	0
$731$	0	0
$732$	0	0
$733$	−3.38058	−0.124865	−0.0624323	−	0.998049i	$-0.519886\pi$
$733$	−3.38058	−0.124865	−0.0624323	+	0.998049i	$0.519886\pi$
$734$	0	0
$735$	18.1944	0.671109
$736$	0	0
$737$	−30.6251	−1.12809
$738$	0	0
$739$	17.1232	0.629888	0.314944	−	0.949110i	$-0.398014\pi$
$739$	17.1232	0.629888	0.314944	+	0.949110i	$0.398014\pi$
$740$	0	0
$741$	−12.3597	−0.454047
$742$	0	0
$743$	39.9129	1.46426	0.732132	−	0.681163i	$-0.238525\pi$
$743$	39.9129	1.46426	0.732132	+	0.681163i	$0.238525\pi$
$744$	0	0
$745$	−55.5342	−2.03461
$746$	0	0
$747$	7.87539	0.288145
$748$	0	0
$749$	−16.1528	−0.590211
$750$	0	0
$751$	−15.4882	−0.565172	−0.282586	−	0.959242i	$-0.591192\pi$
$751$	−15.4882	−0.565172	−0.282586	+	0.959242i	$0.591192\pi$
$752$	0	0
$753$	8.39177	0.305813
$754$	0	0
$755$	−71.5576	−2.60425
$756$	0	0
$757$	−34.4118	−1.25072	−0.625360	−	0.780336i	$-0.715048\pi$
$757$	−34.4118	−1.25072	−0.625360	+	0.780336i	$0.715048\pi$
$758$	0	0
$759$	10.7107	0.388775
$760$	0	0
$761$	−24.7518	−0.897254	−0.448627	−	0.893719i	$-0.648087\pi$
$761$	−24.7518	−0.897254	−0.448627	+	0.893719i	$0.648087\pi$
$762$	0	0
$763$	−23.3895	−0.846758
$764$	0	0
$765$	0	0
$766$	0	0
$767$	−43.2565	−1.56190
$768$	0	0
$769$	−48.0717	−1.73351	−0.866755	−	0.498735i	$-0.833798\pi$
$769$	−48.0717	−1.73351	−0.866755	+	0.498735i	$0.833798\pi$
$770$	0	0
$771$	−30.4927	−1.09817
$772$	0	0
$773$	8.96309	0.322380	0.161190	−	0.986923i	$-0.448467\pi$
$773$	8.96309	0.322380	0.161190	+	0.986923i	$0.448467\pi$
$774$	0	0
$775$	39.2408	1.40957
$776$	0	0
$777$	1.41891	0.0509030
$778$	0	0
$779$	33.2744	1.19218
$780$	0	0
$781$	−39.4706	−1.41237
$782$	0	0
$783$	7.57068	0.270554
$784$	0	0
$785$	12.7601	0.455427
$786$	0	0
$787$	12.1933	0.434645	0.217323	−	0.976100i	$-0.430268\pi$
$787$	12.1933	0.434645	0.217323	+	0.976100i	$0.430268\pi$
$788$	0	0
$789$	19.2976	0.687014
$790$	0	0
$791$	−25.9798	−0.923734
$792$	0	0
$793$	17.0140	0.604184
$794$	0	0
$795$	34.4333	1.22122
$796$	0	0
$797$	−5.75047	−0.203692	−0.101846	−	0.994800i	$-0.532475\pi$
$797$	−5.75047	−0.203692	−0.101846	+	0.994800i	$0.532475\pi$
$798$	0	0
$799$	0	0
$800$	0	0
$801$	−2.05942	−0.0727659
$802$	0	0
$803$	16.7631	0.591558
$804$	0	0
$805$	8.56104	0.301737
$806$	0	0
$807$	0.462077	0.0162659
$808$	0	0
$809$	26.8339	0.943431	0.471716	−	0.881751i	$-0.343635\pi$
$809$	26.8339	0.943431	0.471716	+	0.881751i	$0.343635\pi$
$810$	0	0
$811$	47.1210	1.65464	0.827322	−	0.561728i	$-0.189863\pi$
$811$	47.1210	1.65464	0.827322	+	0.561728i	$0.189863\pi$
$812$	0	0
$813$	22.4044	0.785756
$814$	0	0
$815$	11.6185	0.406979
$816$	0	0
$817$	18.6227	0.651528
$818$	0	0
$819$	−4.42861	−0.154748
$820$	0	0
$821$	42.2962	1.47615	0.738074	−	0.674720i	$-0.235736\pi$
$821$	42.2962	1.47615	0.738074	+	0.674720i	$0.235736\pi$
$822$	0	0
$823$	−22.7732	−0.793823	−0.396911	−	0.917857i	$-0.629918\pi$
$823$	−22.7732	−0.793823	−0.396911	+	0.917857i	$0.629918\pi$
$824$	0	0
$825$	33.3565	1.16132
$826$	0	0
$827$	−25.9111	−0.901017	−0.450509	−	0.892772i	$-0.648757\pi$
$827$	−25.9111	−0.901017	−0.450509	+	0.892772i	$0.648757\pi$
$828$	0	0
$829$	−35.7645	−1.24215	−0.621077	−	0.783749i	$-0.713305\pi$
$829$	−35.7645	−1.24215	−0.621077	+	0.783749i	$0.713305\pi$
$830$	0	0
$831$	−4.04868	−0.140447
$832$	0	0
$833$	0	0
$834$	0	0
$835$	−5.68406	−0.196705
$836$	0	0
$837$	6.23156	0.215394
$838$	0	0
$839$	−25.8312	−0.891792	−0.445896	−	0.895085i	$-0.647115\pi$
$839$	−25.8312	−0.891792	−0.445896	+	0.895085i	$0.647115\pi$
$840$	0	0
$841$	28.3152	0.976387
$842$	0	0
$843$	−5.58551	−0.192375
$844$	0	0
$845$	2.15198	0.0740303
$846$	0	0
$847$	−21.4895	−0.738388
$848$	0	0
$849$	−15.2926	−0.524841
$850$	0	0
$851$	−2.27756	−0.0780738
$852$	0	0
$853$	−13.0292	−0.446110	−0.223055	−	0.974806i	$-0.571603\pi$
$853$	−13.0292	−0.446110	−0.223055	+	0.974806i	$0.571603\pi$
$854$	0	0
$855$	11.8165	0.404115
$856$	0	0
$857$	−48.5245	−1.65756	−0.828782	−	0.559571i	$-0.810966\pi$
$857$	−48.5245	−1.65756	−0.828782	+	0.559571i	$0.810966\pi$
$858$	0	0
$859$	16.8468	0.574805	0.287402	−	0.957810i	$-0.407208\pi$
$859$	16.8468	0.574805	0.287402	+	0.957810i	$0.407208\pi$
$860$	0	0
$861$	11.9225	0.406319
$862$	0	0
$863$	29.9924	1.02095	0.510477	−	0.859892i	$-0.329469\pi$
$863$	29.9924	1.02095	0.510477	+	0.859892i	$0.329469\pi$
$864$	0	0
$865$	−34.6052	−1.17661
$866$	0	0
$867$	0	0
$868$	0	0
$869$	40.9736	1.38993
$870$	0	0
$871$	−20.3256	−0.688706
$872$	0	0
$873$	1.94902	0.0659643
$874$	0	0
$875$	5.49192	0.185661
$876$	0	0
$877$	−40.3794	−1.36352	−0.681759	−	0.731577i	$-0.738785\pi$
$877$	−40.3794	−1.36352	−0.681759	+	0.731577i	$0.738785\pi$
$878$	0	0
$879$	19.8030	0.667940
$880$	0	0
$881$	11.4388	0.385383	0.192691	−	0.981259i	$-0.438278\pi$
$881$	11.4388	0.385383	0.192691	+	0.981259i	$0.438278\pi$
$882$	0	0
$883$	−8.32819	−0.280266	−0.140133	−	0.990133i	$-0.544753\pi$
$883$	−8.32819	−0.280266	−0.140133	+	0.990133i	$0.544753\pi$
$884$	0	0
$885$	41.3552	1.39014
$886$	0	0
$887$	−3.67919	−0.123535	−0.0617676	−	0.998091i	$-0.519674\pi$
$887$	−3.67919	−0.123535	−0.0617676	+	0.998091i	$0.519674\pi$
$888$	0	0
$889$	−6.52302	−0.218775
$890$	0	0
$891$	5.29711	0.177460
$892$	0	0
$893$	−26.1505	−0.875093
$894$	0	0
$895$	−63.4616	−2.12129
$896$	0	0
$897$	7.10861	0.237349
$898$	0	0
$899$	47.1771	1.57345
$900$	0	0
$901$	0	0
$902$	0	0
$903$	6.67271	0.222054
$904$	0	0
$905$	57.9030	1.92476
$906$	0	0
$907$	53.7166	1.78363	0.891815	−	0.452401i	$-0.149432\pi$
$907$	53.7166	1.78363	0.891815	+	0.452401i	$0.149432\pi$
$908$	0	0
$909$	11.7692	0.390360
$910$	0	0
$911$	−15.1047	−0.500440	−0.250220	−	0.968189i	$-0.580503\pi$
$911$	−15.1047	−0.500440	−0.250220	+	0.968189i	$0.580503\pi$
$912$	0	0
$913$	41.7168	1.38062
$914$	0	0
$915$	−16.2662	−0.537743
$916$	0	0
$917$	12.5624	0.414846
$918$	0	0
$919$	40.6982	1.34251	0.671256	−	0.741226i	$-0.265755\pi$
$919$	40.6982	1.34251	0.671256	+	0.741226i	$0.265755\pi$
$920$	0	0
$921$	−31.2649	−1.03021
$922$	0	0
$923$	−26.1963	−0.862261
$924$	0	0
$925$	−7.09303	−0.233217
$926$	0	0
$927$	−13.0223	−0.427709
$928$	0	0
$929$	−51.3009	−1.68313	−0.841563	−	0.540158i	$-0.818364\pi$
$929$	−51.3009	−1.68313	−0.841563	+	0.540158i	$0.818364\pi$
$930$	0	0
$931$	19.0308	0.623710
$932$	0	0
$933$	2.02585	0.0663233
$934$	0	0
$935$	0	0
$936$	0	0
$937$	−18.0890	−0.590941	−0.295470	−	0.955352i	$-0.595476\pi$
$937$	−18.0890	−0.590941	−0.295470	+	0.955352i	$0.595476\pi$
$938$	0	0
$939$	0.0758942	0.00247672
$940$	0	0
$941$	7.43107	0.242246	0.121123	−	0.992638i	$-0.461350\pi$
$941$	7.43107	0.242246	0.121123	+	0.992638i	$0.461350\pi$
$942$	0	0
$943$	−19.1375	−0.623202
$944$	0	0
$945$	4.23396	0.137731
$946$	0	0
$947$	−40.8531	−1.32755	−0.663774	−	0.747934i	$-0.731046\pi$
$947$	−40.8531	−1.32755	−0.663774	+	0.747934i	$0.731046\pi$
$948$	0	0
$949$	11.1255	0.361150
$950$	0	0
$951$	−21.2536	−0.689195
$952$	0	0
$953$	−9.17477	−0.297200	−0.148600	−	0.988897i	$-0.547477\pi$
$953$	−9.17477	−0.297200	−0.148600	+	0.988897i	$0.547477\pi$
$954$	0	0
$955$	−4.79751	−0.155244
$956$	0	0
$957$	40.1027	1.29634
$958$	0	0
$959$	−17.8771	−0.577281
$960$	0	0
$961$	7.83231	0.252655
$962$	0	0
$963$	12.8229	0.413211
$964$	0	0
$965$	−30.1839	−0.971653
$966$	0	0
$967$	−18.1721	−0.584374	−0.292187	−	0.956361i	$-0.594383\pi$
$967$	−18.1721	−0.584374	−0.292187	+	0.956361i	$0.594383\pi$
$968$	0	0
$969$	0	0
$970$	0	0
$971$	39.7526	1.27572	0.637861	−	0.770151i	$-0.279819\pi$
$971$	39.7526	1.27572	0.637861	+	0.770151i	$0.279819\pi$
$972$	0	0
$973$	−4.90101	−0.157119
$974$	0	0
$975$	22.1384	0.708996
$976$	0	0
$977$	−17.1433	−0.548463	−0.274231	−	0.961664i	$-0.588423\pi$
$977$	−17.1433	−0.548463	−0.274231	+	0.961664i	$0.588423\pi$
$978$	0	0
$979$	−10.9090	−0.348652
$980$	0	0
$981$	18.5677	0.592821
$982$	0	0
$983$	−7.64269	−0.243764	−0.121882	−	0.992545i	$-0.538893\pi$
$983$	−7.64269	−0.243764	−0.121882	+	0.992545i	$0.538893\pi$
$984$	0	0
$985$	−39.2981	−1.25214
$986$	0	0
$987$	−9.36997	−0.298249
$988$	0	0
$989$	−10.7107	−0.340581
$990$	0	0
$991$	−10.9018	−0.346308	−0.173154	−	0.984895i	$-0.555396\pi$
$991$	−10.9018	−0.346308	−0.173154	+	0.984895i	$0.555396\pi$
$992$	0	0
$993$	−6.98531	−0.221672
$994$	0	0
$995$	74.2606	2.35422
$996$	0	0
$997$	−39.5572	−1.25279	−0.626395	−	0.779506i	$-0.715470\pi$
$997$	−39.5572	−1.25279	−0.626395	+	0.779506i	$0.715470\pi$
$998$	0	0
$999$	−1.12639	−0.0356375

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	6936.2.a.bj.1.1		6
17.2	even	8		408.2.v.c.361.6	yes	12
17.9	even	8		408.2.v.c.217.6	✓	12
17.16	even	2		6936.2.a.bg.1.6		6
51.2	odd	8		1224.2.w.k.361.2		12
51.26	odd	8		1224.2.w.k.217.2		12
68.19	odd	8		816.2.bd.f.769.3		12
68.43	odd	8		816.2.bd.f.625.3		12
204.155	even	8		2448.2.be.y.1585.2		12
204.179	even	8		2448.2.be.y.1441.2		12

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
408.2.v.c.217.6	✓	12	17.9	even	8
408.2.v.c.361.6	yes	12	17.2	even	8
816.2.bd.f.625.3		12	68.43	odd	8
816.2.bd.f.769.3		12	68.19	odd	8
1224.2.w.k.217.2		12	51.26	odd	8
1224.2.w.k.361.2		12	51.2	odd	8
2448.2.be.y.1441.2		12	204.179	even	8
2448.2.be.y.1585.2		12	204.155	even	8
6936.2.a.bg.1.6		6	17.16	even	2
6936.2.a.bj.1.1		6	1.1	even	1	trivial

Overview	Random
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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:	\( N \)	\(=\)	\( 6936 = 2^{3} \cdot 3 \cdot 17^{2} \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	6936.a (trivial)

\(f(q)\)	\(=\)	\(q+1.00000 q^{3} -3.36112 q^{5} -1.25969 q^{7} +1.00000 q^{9} +O(q^{10})\)	\(q+1.00000 q^{3} -3.36112 q^{5} -1.25969 q^{7} +1.00000 q^{9} +5.29711 q^{11} +3.51564 q^{13} -3.36112 q^{15} -3.51564 q^{19} -1.25969 q^{21} +2.02199 q^{23} +6.29711 q^{25} +1.00000 q^{27} +7.57068 q^{29} +6.23156 q^{31} +5.29711 q^{33} +4.23396 q^{35} -1.12639 q^{37} +3.51564 q^{39} -9.46466 q^{41} -5.29711 q^{43} -3.36112 q^{45} +7.43832 q^{47} -5.41318 q^{49} -10.2446 q^{53} -17.8042 q^{55} -3.51564 q^{57} -12.3040 q^{59} +4.83951 q^{61} -1.25969 q^{63} -11.8165 q^{65} -5.78147 q^{67} +2.02199 q^{69} -7.45135 q^{71} +3.16458 q^{73} +6.29711 q^{75} -6.67271 q^{77} +7.73508 q^{79} +1.00000 q^{81} +7.87539 q^{83} +7.57068 q^{87} -2.05942 q^{89} -4.42861 q^{91} +6.23156 q^{93} +11.8165 q^{95} +1.94902 q^{97} +5.29711 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 6 q + 6 q^{3} + 2 q^{5} + 4 q^{7} + 6 q^{9}+O(q^{10}) \) 6 * q + 6 * q^3 + 2 * q^5 + 4 * q^7 + 6 * q^9	\( 6 q + 6 q^{3} + 2 q^{5} + 4 q^{7} + 6 q^{9} + 6 q^{11} + 2 q^{13} + 2 q^{15} - 2 q^{19} + 4 q^{21} + 10 q^{23} + 12 q^{25} + 6 q^{27} - 8 q^{29} + 4 q^{31} + 6 q^{33} + 20 q^{35} + 12 q^{37} + 2 q^{39} + 6 q^{41} - 6 q^{43} + 2 q^{45} + 4 q^{47} + 18 q^{49} + 12 q^{53} - 10 q^{55} - 2 q^{57} + 16 q^{59} + 20 q^{61} + 4 q^{63} - 22 q^{65} - 28 q^{67} + 10 q^{69} + 36 q^{71} - 12 q^{73} + 12 q^{75} + 20 q^{77} + 16 q^{79} + 6 q^{81} + 20 q^{83} - 8 q^{87} + 4 q^{89} + 36 q^{91} + 4 q^{93} + 22 q^{95} - 32 q^{97} + 6 q^{99}+O(q^{100}) \) 6 * q + 6 * q^3 + 2 * q^5 + 4 * q^7 + 6 * q^9 + 6 * q^11 + 2 * q^13 + 2 * q^15 - 2 * q^19 + 4 * q^21 + 10 * q^23 + 12 * q^25 + 6 * q^27 - 8 * q^29 + 4 * q^31 + 6 * q^33 + 20 * q^35 + 12 * q^37 + 2 * q^39 + 6 * q^41 - 6 * q^43 + 2 * q^45 + 4 * q^47 + 18 * q^49 + 12 * q^53 - 10 * q^55 - 2 * q^57 + 16 * q^59 + 20 * q^61 + 4 * q^63 - 22 * q^65 - 28 * q^67 + 10 * q^69 + 36 * q^71 - 12 * q^73 + 12 * q^75 + 20 * q^77 + 16 * q^79 + 6 * q^81 + 20 * q^83 - 8 * q^87 + 4 * q^89 + 36 * q^91 + 4 * q^93 + 22 * q^95 - 32 * q^97 + 6 * q^99

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