LMFDB - Embedded newform 6036.2.a.f.1.6

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [6036,2,Mod(1,6036)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(6036, 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("6036.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 6036 = 2^{2} \cdot 3 \cdot 503 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	6036.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:	$48.1977026600$
Analytic rank:	$1$
Dimension:	$14$
Coefficient field:	$\mathbb{Q}[x]/(x^{14} - \cdots)$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{14} - 6 x^{13} - 12 x^{12} + 112 x^{11} + 7 x^{10} - 710 x^{9} + 281 x^{8} + 1850 x^{7} - 830 x^{6} + \cdots + 1 $ x^14 - 6x^13 - 12x^12 + 112x^11 + 7x^10 - 710x^9 + 281x^8 + 1850x^7 - 830x^6 - 1918x^5 + 623x^4 + 480x^3 - 29x^2 - 17*x + 1
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			1.6
Root			$0.736579$ of defining polynomial
Character	$\chi$	$=$	6036.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-1.00000 q^{3} -0.736579 q^{5} +1.91291 q^{7} +1.00000 q^{9} +O(q^{10})$	$q-1.00000 q^{3} -0.736579 q^{5} +1.91291 q^{7} +1.00000 q^{9} +3.35146 q^{11} +3.35801 q^{13} +0.736579 q^{15} -1.53670 q^{17} -8.41616 q^{19} -1.91291 q^{21} -1.97902 q^{23} -4.45745 q^{25} -1.00000 q^{27} +2.87055 q^{29} +2.41143 q^{31} -3.35146 q^{33} -1.40901 q^{35} -2.73842 q^{37} -3.35801 q^{39} -8.34545 q^{41} +1.91023 q^{43} -0.736579 q^{45} +3.72365 q^{47} -3.34076 q^{49} +1.53670 q^{51} -2.49339 q^{53} -2.46861 q^{55} +8.41616 q^{57} -15.1251 q^{59} +0.175972 q^{61} +1.91291 q^{63} -2.47344 q^{65} -7.66365 q^{67} +1.97902 q^{69} +5.41730 q^{71} +1.32988 q^{73} +4.45745 q^{75} +6.41105 q^{77} +4.15790 q^{79} +1.00000 q^{81} +1.75779 q^{83} +1.13190 q^{85} -2.87055 q^{87} -4.80040 q^{89} +6.42359 q^{91} -2.41143 q^{93} +6.19917 q^{95} -7.00567 q^{97} +3.35146 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 14 q - 14 q^{3} - 6 q^{5} - 7 q^{7} + 14 q^{9}+O(q^{10}) $ 14 * q - 14 * q^3 - 6 * q^5 - 7 * q^7 + 14 * q^9	$ 14 q - 14 q^{3} - 6 q^{5} - 7 q^{7} + 14 q^{9} + q^{11} - q^{13} + 6 q^{15} - 6 q^{17} + q^{19} + 7 q^{21} + 10 q^{23} - 10 q^{25} - 14 q^{27} - 6 q^{29} - 5 q^{31} - q^{33} + 17 q^{35} - 12 q^{37} + q^{39} - 21 q^{41} + 8 q^{43} - 6 q^{45} + 18 q^{47} - 19 q^{49} + 6 q^{51} - q^{53} - q^{57} + 14 q^{59} - 19 q^{61} - 7 q^{63} + 17 q^{65} + 17 q^{67} - 10 q^{69} - 13 q^{71} - 12 q^{73} + 10 q^{75} - 9 q^{77} - 8 q^{79} + 14 q^{81} + 11 q^{83} - 17 q^{85} + 6 q^{87} - 9 q^{89} - 5 q^{91} + 5 q^{93} + 8 q^{95} - 25 q^{97} + q^{99}+O(q^{100}) $ 14 * q - 14 * q^3 - 6 * q^5 - 7 * q^7 + 14 * q^9 + q^11 - q^13 + 6 * q^15 - 6 * q^17 + q^19 + 7 * q^21 + 10 * q^23 - 10 * q^25 - 14 * q^27 - 6 * q^29 - 5 * q^31 - q^33 + 17 * q^35 - 12 * q^37 + q^39 - 21 * q^41 + 8 * q^43 - 6 * q^45 + 18 * q^47 - 19 * q^49 + 6 * q^51 - q^53 - q^57 + 14 * q^59 - 19 * q^61 - 7 * q^63 + 17 * q^65 + 17 * q^67 - 10 * q^69 - 13 * q^71 - 12 * q^73 + 10 * q^75 - 9 * q^77 - 8 * q^79 + 14 * q^81 + 11 * q^83 - 17 * q^85 + 6 * q^87 - 9 * q^89 - 5 * q^91 + 5 * q^93 + 8 * q^95 - 25 * q^97 + 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
$5$	−0.736579	−0.329408	−0.164704	−	0.986343i	$-0.552667\pi$
$5$	−0.736579	−0.329408	−0.164704	+	0.986343i	$0.552667\pi$
$6$	0	0
$7$	1.91291	0.723013	0.361507	−	0.932370i	$-0.382262\pi$
$7$	1.91291	0.723013	0.361507	+	0.932370i	$0.382262\pi$
$8$	0	0
$9$	1.00000	0.333333
$10$	0	0
$11$	3.35146	1.01050	0.505251	−	0.862972i	$-0.331400\pi$
$11$	3.35146	1.01050	0.505251	+	0.862972i	$0.331400\pi$
$12$	0	0
$13$	3.35801	0.931345	0.465673	−	0.884957i	$-0.345812\pi$
$13$	3.35801	0.931345	0.465673	+	0.884957i	$0.345812\pi$
$14$	0	0
$15$	0.736579	0.190184
$16$	0	0
$17$	−1.53670	−0.372704	−0.186352	−	0.982483i	$-0.559666\pi$
$17$	−1.53670	−0.372704	−0.186352	+	0.982483i	$0.559666\pi$
$18$	0	0
$19$	−8.41616	−1.93080	−0.965400	−	0.260773i	$-0.916023\pi$
$19$	−8.41616	−1.93080	−0.965400	+	0.260773i	$0.916023\pi$
$20$	0	0
$21$	−1.91291	−0.417432
$22$	0	0
$23$	−1.97902	−0.412654	−0.206327	−	0.978483i	$-0.566151\pi$
$23$	−1.97902	−0.412654	−0.206327	+	0.978483i	$0.566151\pi$
$24$	0	0
$25$	−4.45745	−0.891490
$26$	0	0
$27$	−1.00000	−0.192450
$28$	0	0
$29$	2.87055	0.533048	0.266524	−	0.963828i	$-0.414125\pi$
$29$	2.87055	0.533048	0.266524	+	0.963828i	$0.414125\pi$
$30$	0	0
$31$	2.41143	0.433106	0.216553	−	0.976271i	$-0.430519\pi$
$31$	2.41143	0.433106	0.216553	+	0.976271i	$0.430519\pi$
$32$	0	0
$33$	−3.35146	−0.583414
$34$	0	0
$35$	−1.40901	−0.238167
$36$	0	0
$37$	−2.73842	−0.450193	−0.225097	−	0.974336i	$-0.572270\pi$
$37$	−2.73842	−0.450193	−0.225097	+	0.974336i	$0.572270\pi$
$38$	0	0
$39$	−3.35801	−0.537712
$40$	0	0
$41$	−8.34545	−1.30334	−0.651670	−	0.758502i	$-0.725931\pi$
$41$	−8.34545	−1.30334	−0.651670	+	0.758502i	$0.725931\pi$
$42$	0	0
$43$	1.91023	0.291308	0.145654	−	0.989336i	$-0.453471\pi$
$43$	1.91023	0.291308	0.145654	+	0.989336i	$0.453471\pi$
$44$	0	0
$45$	−0.736579	−0.109803
$46$	0	0
$47$	3.72365	0.543151	0.271575	−	0.962417i	$-0.412455\pi$
$47$	3.72365	0.543151	0.271575	+	0.962417i	$0.412455\pi$
$48$	0	0
$49$	−3.34076	−0.477252
$50$	0	0
$51$	1.53670	0.215181
$52$	0	0
$53$	−2.49339	−0.342493	−0.171247	−	0.985228i	$-0.554779\pi$
$53$	−2.49339	−0.342493	−0.171247	+	0.985228i	$0.554779\pi$
$54$	0	0
$55$	−2.46861	−0.332868
$56$	0	0
$57$	8.41616	1.11475
$58$	0	0
$59$	−15.1251	−1.96912	−0.984560	−	0.175050i	$-0.943991\pi$
$59$	−15.1251	−1.96912	−0.984560	+	0.175050i	$0.943991\pi$
$60$	0	0
$61$	0.175972	0.0225309	0.0112655	−	0.999937i	$-0.496414\pi$
$61$	0.175972	0.0225309	0.0112655	+	0.999937i	$0.496414\pi$
$62$	0	0
$63$	1.91291	0.241004
$64$	0	0
$65$	−2.47344	−0.306793
$66$	0	0
$67$	−7.66365	−0.936264	−0.468132	−	0.883659i	$-0.655073\pi$
$67$	−7.66365	−0.936264	−0.468132	+	0.883659i	$0.655073\pi$
$68$	0	0
$69$	1.97902	0.238246
$70$	0	0
$71$	5.41730	0.642916	0.321458	−	0.946924i	$-0.395827\pi$
$71$	5.41730	0.642916	0.321458	+	0.946924i	$0.395827\pi$
$72$	0	0
$73$	1.32988	0.155651	0.0778254	−	0.996967i	$-0.475202\pi$
$73$	1.32988	0.155651	0.0778254	+	0.996967i	$0.475202\pi$
$74$	0	0
$75$	4.45745	0.514702
$76$	0	0
$77$	6.41105	0.730607
$78$	0	0
$79$	4.15790	0.467801	0.233900	−	0.972261i	$-0.424851\pi$
$79$	4.15790	0.467801	0.233900	+	0.972261i	$0.424851\pi$
$80$	0	0
$81$	1.00000	0.111111
$82$	0	0
$83$	1.75779	0.192943	0.0964714	−	0.995336i	$-0.469244\pi$
$83$	1.75779	0.192943	0.0964714	+	0.995336i	$0.469244\pi$
$84$	0	0
$85$	1.13190	0.122772
$86$	0	0
$87$	−2.87055	−0.307755
$88$	0	0
$89$	−4.80040	−0.508841	−0.254420	−	0.967094i	$-0.581885\pi$
$89$	−4.80040	−0.508841	−0.254420	+	0.967094i	$0.581885\pi$
$90$	0	0
$91$	6.42359	0.673375
$92$	0	0
$93$	−2.41143	−0.250054
$94$	0	0
$95$	6.19917	0.636021
$96$	0	0
$97$	−7.00567	−0.711318	−0.355659	−	0.934616i	$-0.615743\pi$
$97$	−7.00567	−0.711318	−0.355659	+	0.934616i	$0.615743\pi$
$98$	0	0
$99$	3.35146	0.336834
$100$	0	0
$101$	−4.49325	−0.447095	−0.223547	−	0.974693i	$-0.571764\pi$
$101$	−4.49325	−0.447095	−0.223547	+	0.974693i	$0.571764\pi$
$102$	0	0
$103$	−16.4827	−1.62409	−0.812043	−	0.583598i	$-0.801644\pi$
$103$	−16.4827	−1.62409	−0.812043	+	0.583598i	$0.801644\pi$
$104$	0	0
$105$	1.40901	0.137506
$106$	0	0
$107$	3.57919	0.346013	0.173007	−	0.984921i	$-0.444652\pi$
$107$	3.57919	0.346013	0.173007	+	0.984921i	$0.444652\pi$
$108$	0	0
$109$	11.1697	1.06987	0.534933	−	0.844894i	$-0.320337\pi$
$109$	11.1697	1.06987	0.534933	+	0.844894i	$0.320337\pi$
$110$	0	0
$111$	2.73842	0.259919
$112$	0	0
$113$	6.16831	0.580266	0.290133	−	0.956986i	$-0.406300\pi$
$113$	6.16831	0.580266	0.290133	+	0.956986i	$0.406300\pi$
$114$	0	0
$115$	1.45770	0.135932
$116$	0	0
$117$	3.35801	0.310448
$118$	0	0
$119$	−2.93957	−0.269470
$120$	0	0
$121$	0.232279	0.0211163
$122$	0	0
$123$	8.34545	0.752484
$124$	0	0
$125$	6.96616	0.623072
$126$	0	0
$127$	2.48428	0.220444	0.110222	−	0.993907i	$-0.464844\pi$
$127$	2.48428	0.220444	0.110222	+	0.993907i	$0.464844\pi$
$128$	0	0
$129$	−1.91023	−0.168186
$130$	0	0
$131$	6.33290	0.553308	0.276654	−	0.960970i	$-0.410774\pi$
$131$	6.33290	0.553308	0.276654	+	0.960970i	$0.410774\pi$
$132$	0	0
$133$	−16.0994	−1.39599
$134$	0	0
$135$	0.736579	0.0633946
$136$	0	0
$137$	−4.00768	−0.342399	−0.171200	−	0.985236i	$-0.554764\pi$
$137$	−4.00768	−0.342399	−0.171200	+	0.985236i	$0.554764\pi$
$138$	0	0
$139$	−3.01108	−0.255397	−0.127698	−	0.991813i	$-0.540759\pi$
$139$	−3.01108	−0.255397	−0.127698	+	0.991813i	$0.540759\pi$
$140$	0	0
$141$	−3.72365	−0.313588
$142$	0	0
$143$	11.2542	0.941127
$144$	0	0
$145$	−2.11439	−0.175590
$146$	0	0
$147$	3.34076	0.275541
$148$	0	0
$149$	24.1996	1.98251	0.991256	−	0.131953i	$-0.0421248\pi$
$149$	24.1996	1.98251	0.991256	+	0.131953i	$0.0421248\pi$
$150$	0	0
$151$	11.3865	0.926616	0.463308	−	0.886197i	$-0.346662\pi$
$151$	11.3865	0.926616	0.463308	+	0.886197i	$0.346662\pi$
$152$	0	0
$153$	−1.53670	−0.124235
$154$	0	0
$155$	−1.77621	−0.142669
$156$	0	0
$157$	−8.04660	−0.642188	−0.321094	−	0.947047i	$-0.604051\pi$
$157$	−8.04660	−0.642188	−0.321094	+	0.947047i	$0.604051\pi$
$158$	0	0
$159$	2.49339	0.197739
$160$	0	0
$161$	−3.78569	−0.298354
$162$	0	0
$163$	−13.6678	−1.07054	−0.535272	−	0.844680i	$-0.679791\pi$
$163$	−13.6678	−1.07054	−0.535272	+	0.844680i	$0.679791\pi$
$164$	0	0
$165$	2.46861	0.192181
$166$	0	0
$167$	5.14117	0.397835	0.198918	−	0.980016i	$-0.436257\pi$
$167$	5.14117	0.397835	0.198918	+	0.980016i	$0.436257\pi$
$168$	0	0
$169$	−1.72375	−0.132596
$170$	0	0
$171$	−8.41616	−0.643600
$172$	0	0
$173$	−19.7281	−1.49990	−0.749948	−	0.661497i	$-0.769922\pi$
$173$	−19.7281	−1.49990	−0.749948	+	0.661497i	$0.769922\pi$
$174$	0	0
$175$	−8.52672	−0.644559
$176$	0	0
$177$	15.1251	1.13687
$178$	0	0
$179$	13.7925	1.03090	0.515448	−	0.856921i	$-0.327625\pi$
$179$	13.7925	1.03090	0.515448	+	0.856921i	$0.327625\pi$
$180$	0	0
$181$	22.7976	1.69453	0.847265	−	0.531170i	$-0.178248\pi$
$181$	22.7976	1.69453	0.847265	+	0.531170i	$0.178248\pi$
$182$	0	0
$183$	−0.175972	−0.0130082
$184$	0	0
$185$	2.01706	0.148297
$186$	0	0
$187$	−5.15018	−0.376618
$188$	0	0
$189$	−1.91291	−0.139144
$190$	0	0
$191$	12.6871	0.918004	0.459002	−	0.888435i	$-0.348207\pi$
$191$	12.6871	0.918004	0.459002	+	0.888435i	$0.348207\pi$
$192$	0	0
$193$	−20.1813	−1.45268	−0.726340	−	0.687336i	$-0.758780\pi$
$193$	−20.1813	−1.45268	−0.726340	+	0.687336i	$0.758780\pi$
$194$	0	0
$195$	2.47344	0.177127
$196$	0	0
$197$	6.17319	0.439822	0.219911	−	0.975520i	$-0.429423\pi$
$197$	6.17319	0.439822	0.219911	+	0.975520i	$0.429423\pi$
$198$	0	0
$199$	6.61307	0.468788	0.234394	−	0.972142i	$-0.424689\pi$
$199$	6.61307	0.468788	0.234394	+	0.972142i	$0.424689\pi$
$200$	0	0
$201$	7.66365	0.540552
$202$	0	0
$203$	5.49112	0.385401
$204$	0	0
$205$	6.14709	0.429331
$206$	0	0
$207$	−1.97902	−0.137551
$208$	0	0
$209$	−28.2064	−1.95108
$210$	0	0
$211$	−26.5667	−1.82893	−0.914463	−	0.404669i	$-0.867387\pi$
$211$	−26.5667	−1.82893	−0.914463	+	0.404669i	$0.867387\pi$
$212$	0	0
$213$	−5.41730	−0.371187
$214$	0	0
$215$	−1.40704	−0.0959591
$216$	0	0
$217$	4.61286	0.313141
$218$	0	0
$219$	−1.32988	−0.0898650
$220$	0	0
$221$	−5.16025	−0.347116
$222$	0	0
$223$	1.62309	0.108690	0.0543451	−	0.998522i	$-0.482693\pi$
$223$	1.62309	0.108690	0.0543451	+	0.998522i	$0.482693\pi$
$224$	0	0
$225$	−4.45745	−0.297163
$226$	0	0
$227$	−3.28587	−0.218091	−0.109045	−	0.994037i	$-0.534779\pi$
$227$	−3.28587	−0.218091	−0.109045	+	0.994037i	$0.534779\pi$
$228$	0	0
$229$	−25.9901	−1.71747	−0.858737	−	0.512417i	$-0.828750\pi$
$229$	−25.9901	−1.71747	−0.858737	+	0.512417i	$0.828750\pi$
$230$	0	0
$231$	−6.41105	−0.421816
$232$	0	0
$233$	−18.7420	−1.22783	−0.613915	−	0.789372i	$-0.710406\pi$
$233$	−18.7420	−1.22783	−0.613915	+	0.789372i	$0.710406\pi$
$234$	0	0
$235$	−2.74277	−0.178918
$236$	0	0
$237$	−4.15790	−0.270085
$238$	0	0
$239$	0.259054	0.0167568	0.00837842	−	0.999965i	$-0.497333\pi$
$239$	0.259054	0.0167568	0.00837842	+	0.999965i	$0.497333\pi$
$240$	0	0
$241$	−12.3055	−0.792669	−0.396335	−	0.918106i	$-0.629718\pi$
$241$	−12.3055	−0.792669	−0.396335	+	0.918106i	$0.629718\pi$
$242$	0	0
$243$	−1.00000	−0.0641500
$244$	0	0
$245$	2.46073	0.157211
$246$	0	0
$247$	−28.2616	−1.79824
$248$	0	0
$249$	−1.75779	−0.111396
$250$	0	0
$251$	6.61769	0.417705	0.208853	−	0.977947i	$-0.433027\pi$
$251$	6.61769	0.417705	0.208853	+	0.977947i	$0.433027\pi$
$252$	0	0
$253$	−6.63260	−0.416988
$254$	0	0
$255$	−1.13190	−0.0708823
$256$	0	0
$257$	15.1088	0.942458	0.471229	−	0.882011i	$-0.343810\pi$
$257$	15.1088	0.942458	0.471229	+	0.882011i	$0.343810\pi$
$258$	0	0
$259$	−5.23836	−0.325496
$260$	0	0
$261$	2.87055	0.177683
$262$	0	0
$263$	−7.33151	−0.452080	−0.226040	−	0.974118i	$-0.572578\pi$
$263$	−7.33151	−0.452080	−0.226040	+	0.974118i	$0.572578\pi$
$264$	0	0
$265$	1.83658	0.112820
$266$	0	0
$267$	4.80040	0.293779
$268$	0	0
$269$	4.03612	0.246087	0.123043	−	0.992401i	$-0.460735\pi$
$269$	4.03612	0.246087	0.123043	+	0.992401i	$0.460735\pi$
$270$	0	0
$271$	−17.5487	−1.06601	−0.533005	−	0.846112i	$-0.678937\pi$
$271$	−17.5487	−1.06601	−0.533005	+	0.846112i	$0.678937\pi$
$272$	0	0
$273$	−6.42359	−0.388773
$274$	0	0
$275$	−14.9390	−0.900854
$276$	0	0
$277$	8.35488	0.501996	0.250998	−	0.967988i	$-0.419241\pi$
$277$	8.35488	0.501996	0.250998	+	0.967988i	$0.419241\pi$
$278$	0	0
$279$	2.41143	0.144369
$280$	0	0
$281$	−17.6407	−1.05235	−0.526177	−	0.850375i	$-0.676375\pi$
$281$	−17.6407	−1.05235	−0.526177	+	0.850375i	$0.676375\pi$
$282$	0	0
$283$	−19.8352	−1.17908	−0.589539	−	0.807740i	$-0.700690\pi$
$283$	−19.8352	−1.17908	−0.589539	+	0.807740i	$0.700690\pi$
$284$	0	0
$285$	−6.19917	−0.367207
$286$	0	0
$287$	−15.9641	−0.942333
$288$	0	0
$289$	−14.6386	−0.861092
$290$	0	0
$291$	7.00567	0.410679
$292$	0	0
$293$	3.09348	0.180723	0.0903615	−	0.995909i	$-0.471198\pi$
$293$	3.09348	0.180723	0.0903615	+	0.995909i	$0.471198\pi$
$294$	0	0
$295$	11.1408	0.648644
$296$	0	0
$297$	−3.35146	−0.194471
$298$	0	0
$299$	−6.64557	−0.384323
$300$	0	0
$301$	3.65411	0.210619
$302$	0	0
$303$	4.49325	0.258130
$304$	0	0
$305$	−0.129617	−0.00742187
$306$	0	0
$307$	8.98421	0.512756	0.256378	−	0.966577i	$-0.417471\pi$
$307$	8.98421	0.512756	0.256378	+	0.966577i	$0.417471\pi$
$308$	0	0
$309$	16.4827	0.937666
$310$	0	0
$311$	−14.6120	−0.828568	−0.414284	−	0.910148i	$-0.635968\pi$
$311$	−14.6120	−0.828568	−0.414284	+	0.910148i	$0.635968\pi$
$312$	0	0
$313$	−13.4152	−0.758273	−0.379137	−	0.925341i	$-0.623779\pi$
$313$	−13.4152	−0.758273	−0.379137	+	0.925341i	$0.623779\pi$
$314$	0	0
$315$	−1.40901	−0.0793888
$316$	0	0
$317$	−22.5821	−1.26834	−0.634168	−	0.773196i	$-0.718657\pi$
$317$	−22.5821	−1.26834	−0.634168	+	0.773196i	$0.718657\pi$
$318$	0	0
$319$	9.62053	0.538647
$320$	0	0
$321$	−3.57919	−0.199771
$322$	0	0
$323$	12.9331	0.719616
$324$	0	0
$325$	−14.9682	−0.830285
$326$	0	0
$327$	−11.1697	−0.617687
$328$	0	0
$329$	7.12303	0.392705
$330$	0	0
$331$	−24.6848	−1.35680	−0.678399	−	0.734694i	$-0.737326\pi$
$331$	−24.6848	−1.35680	−0.678399	+	0.734694i	$0.737326\pi$
$332$	0	0
$333$	−2.73842	−0.150064
$334$	0	0
$335$	5.64489	0.308413
$336$	0	0
$337$	−11.6432	−0.634248	−0.317124	−	0.948384i	$-0.602717\pi$
$337$	−11.6432	−0.634248	−0.317124	+	0.948384i	$0.602717\pi$
$338$	0	0
$339$	−6.16831	−0.335017
$340$	0	0
$341$	8.08182	0.437655
$342$	0	0
$343$	−19.7810	−1.06807
$344$	0	0
$345$	−1.45770	−0.0784802
$346$	0	0
$347$	−23.3017	−1.25090	−0.625450	−	0.780264i	$-0.715085\pi$
$347$	−23.3017	−1.25090	−0.625450	+	0.780264i	$0.715085\pi$
$348$	0	0
$349$	−30.6714	−1.64180	−0.820901	−	0.571070i	$-0.806528\pi$
$349$	−30.6714	−1.64180	−0.820901	+	0.571070i	$0.806528\pi$
$350$	0	0
$351$	−3.35801	−0.179237
$352$	0	0
$353$	−8.63120	−0.459392	−0.229696	−	0.973262i	$-0.573773\pi$
$353$	−8.63120	−0.459392	−0.229696	+	0.973262i	$0.573773\pi$
$354$	0	0
$355$	−3.99027	−0.211782
$356$	0	0
$357$	2.93957	0.155578
$358$	0	0
$359$	−5.35782	−0.282775	−0.141388	−	0.989954i	$-0.545156\pi$
$359$	−5.35782	−0.282775	−0.141388	+	0.989954i	$0.545156\pi$
$360$	0	0
$361$	51.8318	2.72799
$362$	0	0
$363$	−0.232279	−0.0121915
$364$	0	0
$365$	−0.979562	−0.0512726
$366$	0	0
$367$	−19.1887	−1.00164	−0.500820	−	0.865551i	$-0.666968\pi$
$367$	−19.1887	−1.00164	−0.500820	+	0.865551i	$0.666968\pi$
$368$	0	0
$369$	−8.34545	−0.434447
$370$	0	0
$371$	−4.76964	−0.247627
$372$	0	0
$373$	12.7469	0.660009	0.330004	−	0.943979i	$-0.392950\pi$
$373$	12.7469	0.660009	0.330004	+	0.943979i	$0.392950\pi$
$374$	0	0
$375$	−6.96616	−0.359731
$376$	0	0
$377$	9.63935	0.496452
$378$	0	0
$379$	21.6529	1.11223	0.556116	−	0.831105i	$-0.312291\pi$
$379$	21.6529	1.11223	0.556116	+	0.831105i	$0.312291\pi$
$380$	0	0
$381$	−2.48428	−0.127273
$382$	0	0
$383$	17.5601	0.897279	0.448639	−	0.893713i	$-0.351909\pi$
$383$	17.5601	0.897279	0.448639	+	0.893713i	$0.351909\pi$
$384$	0	0
$385$	−4.72225	−0.240668
$386$	0	0
$387$	1.91023	0.0971025
$388$	0	0
$389$	−36.1457	−1.83266	−0.916330	−	0.400425i	$-0.868863\pi$
$389$	−36.1457	−1.83266	−0.916330	+	0.400425i	$0.868863\pi$
$390$	0	0
$391$	3.04115	0.153798
$392$	0	0
$393$	−6.33290	−0.319453
$394$	0	0
$395$	−3.06263	−0.154097
$396$	0	0
$397$	19.1567	0.961446	0.480723	−	0.876873i	$-0.340374\pi$
$397$	19.1567	0.961446	0.480723	+	0.876873i	$0.340374\pi$
$398$	0	0
$399$	16.0994	0.805978
$400$	0	0
$401$	−15.6439	−0.781220	−0.390610	−	0.920556i	$-0.627736\pi$
$401$	−15.6439	−0.781220	−0.390610	+	0.920556i	$0.627736\pi$
$402$	0	0
$403$	8.09762	0.403371
$404$	0	0
$405$	−0.736579	−0.0366009
$406$	0	0
$407$	−9.17770	−0.454922
$408$	0	0
$409$	−31.8152	−1.57316	−0.786581	−	0.617487i	$-0.788151\pi$
$409$	−31.8152	−1.57316	−0.786581	+	0.617487i	$0.788151\pi$
$410$	0	0
$411$	4.00768	0.197684
$412$	0	0
$413$	−28.9330	−1.42370
$414$	0	0
$415$	−1.29475	−0.0635569
$416$	0	0
$417$	3.01108	0.147453
$418$	0	0
$419$	1.54331	0.0753957	0.0376979	−	0.999289i	$-0.487998\pi$
$419$	1.54331	0.0753957	0.0376979	+	0.999289i	$0.487998\pi$
$420$	0	0
$421$	34.6662	1.68953	0.844765	−	0.535138i	$-0.179740\pi$
$421$	34.6662	1.68953	0.844765	+	0.535138i	$0.179740\pi$
$422$	0	0
$423$	3.72365	0.181050
$424$	0	0
$425$	6.84975	0.332262
$426$	0	0
$427$	0.336619	0.0162902
$428$	0	0
$429$	−11.2542	−0.543360
$430$	0	0
$431$	19.0601	0.918094	0.459047	−	0.888412i	$-0.348191\pi$
$431$	19.0601	0.918094	0.459047	+	0.888412i	$0.348191\pi$
$432$	0	0
$433$	−12.3902	−0.595433	−0.297717	−	0.954654i	$-0.596225\pi$
$433$	−12.3902	−0.595433	−0.297717	+	0.954654i	$0.596225\pi$
$434$	0	0
$435$	2.11439	0.101377
$436$	0	0
$437$	16.6557	0.796753
$438$	0	0
$439$	−17.1379	−0.817946	−0.408973	−	0.912547i	$-0.634113\pi$
$439$	−17.1379	−0.817946	−0.408973	+	0.912547i	$0.634113\pi$
$440$	0	0
$441$	−3.34076	−0.159084
$442$	0	0
$443$	21.7553	1.03362	0.516812	−	0.856099i	$-0.327119\pi$
$443$	21.7553	1.03362	0.516812	+	0.856099i	$0.327119\pi$
$444$	0	0
$445$	3.53587	0.167616
$446$	0	0
$447$	−24.1996	−1.14460
$448$	0	0
$449$	31.0371	1.46473	0.732367	−	0.680910i	$-0.238416\pi$
$449$	31.0371	1.46473	0.732367	+	0.680910i	$0.238416\pi$
$450$	0	0
$451$	−27.9694	−1.31703
$452$	0	0
$453$	−11.3865	−0.534982
$454$	0	0
$455$	−4.73148	−0.221815
$456$	0	0
$457$	19.5403	0.914055	0.457027	−	0.889453i	$-0.348914\pi$
$457$	19.5403	0.914055	0.457027	+	0.889453i	$0.348914\pi$
$458$	0	0
$459$	1.53670	0.0717269
$460$	0	0
$461$	14.4177	0.671499	0.335749	−	0.941951i	$-0.391010\pi$
$461$	14.4177	0.671499	0.335749	+	0.941951i	$0.391010\pi$
$462$	0	0
$463$	37.9801	1.76508	0.882541	−	0.470235i	$-0.155831\pi$
$463$	37.9801	1.76508	0.882541	+	0.470235i	$0.155831\pi$
$464$	0	0
$465$	1.77621	0.0823698
$466$	0	0
$467$	36.6314	1.69510	0.847549	−	0.530717i	$-0.178077\pi$
$467$	36.6314	1.69510	0.847549	+	0.530717i	$0.178077\pi$
$468$	0	0
$469$	−14.6599	−0.676932
$470$	0	0
$471$	8.04660	0.370768
$472$	0	0
$473$	6.40206	0.294367
$474$	0	0
$475$	37.5146	1.72129
$476$	0	0
$477$	−2.49339	−0.114164
$478$	0	0
$479$	12.7319	0.581735	0.290868	−	0.956763i	$-0.406056\pi$
$479$	12.7319	0.581735	0.290868	+	0.956763i	$0.406056\pi$
$480$	0	0
$481$	−9.19565	−0.419285
$482$	0	0
$483$	3.78569	0.172255
$484$	0	0
$485$	5.16023	0.234314
$486$	0	0
$487$	12.3591	0.560042	0.280021	−	0.959994i	$-0.409659\pi$
$487$	12.3591	0.560042	0.280021	+	0.959994i	$0.409659\pi$
$488$	0	0
$489$	13.6678	0.618079
$490$	0	0
$491$	23.3762	1.05495	0.527477	−	0.849569i	$-0.323138\pi$
$491$	23.3762	1.05495	0.527477	+	0.849569i	$0.323138\pi$
$492$	0	0
$493$	−4.41117	−0.198669
$494$	0	0
$495$	−2.46861	−0.110956
$496$	0	0
$497$	10.3628	0.464837
$498$	0	0
$499$	−2.96917	−0.132918	−0.0664592	−	0.997789i	$-0.521170\pi$
$499$	−2.96917	−0.132918	−0.0664592	+	0.997789i	$0.521170\pi$
$500$	0	0
$501$	−5.14117	−0.229690
$502$	0	0
$503$	1.00000	0.0445878
$503$	1.00000	0.0445878
$504$	0	0
$505$	3.30963	0.147277
$506$	0	0
$507$	1.72375	0.0765545
$508$	0	0
$509$	−17.9403	−0.795191	−0.397595	−	0.917561i	$-0.630155\pi$
$509$	−17.9403	−0.795191	−0.397595	+	0.917561i	$0.630155\pi$
$510$	0	0
$511$	2.54395	0.112538
$512$	0	0
$513$	8.41616	0.371583
$514$	0	0
$515$	12.1408	0.534987
$516$	0	0
$517$	12.4797	0.548856
$518$	0	0
$519$	19.7281	0.865965
$520$	0	0
$521$	−23.6782	−1.03736	−0.518681	−	0.854968i	$-0.673577\pi$
$521$	−23.6782	−1.03736	−0.518681	+	0.854968i	$0.673577\pi$
$522$	0	0
$523$	12.7963	0.559543	0.279771	−	0.960067i	$-0.409741\pi$
$523$	12.7963	0.559543	0.279771	+	0.960067i	$0.409741\pi$
$524$	0	0
$525$	8.52672	0.372137
$526$	0	0
$527$	−3.70564	−0.161420
$528$	0	0
$529$	−19.0835	−0.829717
$530$	0	0
$531$	−15.1251	−0.656373
$532$	0	0
$533$	−28.0241	−1.21386
$534$	0	0
$535$	−2.63636	−0.113980
$536$	0	0
$537$	−13.7925	−0.595189
$538$	0	0
$539$	−11.1964	−0.482264
$540$	0	0
$541$	−13.7919	−0.592959	−0.296480	−	0.955039i	$-0.595813\pi$
$541$	−13.7919	−0.592959	−0.296480	+	0.955039i	$0.595813\pi$
$542$	0	0
$543$	−22.7976	−0.978337
$544$	0	0
$545$	−8.22739	−0.352423
$546$	0	0
$547$	−33.0967	−1.41511	−0.707557	−	0.706656i	$-0.750203\pi$
$547$	−33.0967	−1.41511	−0.707557	+	0.706656i	$0.750203\pi$
$548$	0	0
$549$	0.175972	0.00751031
$550$	0	0
$551$	−24.1590	−1.02921
$552$	0	0
$553$	7.95371	0.338226
$554$	0	0
$555$	−2.01706	−0.0856195
$556$	0	0
$557$	−24.8057	−1.05105	−0.525525	−	0.850778i	$-0.676131\pi$
$557$	−24.8057	−1.05105	−0.525525	+	0.850778i	$0.676131\pi$
$558$	0	0
$559$	6.41458	0.271308
$560$	0	0
$561$	5.15018	0.217441
$562$	0	0
$563$	46.2489	1.94916	0.974580	−	0.224042i	$-0.0719252\pi$
$563$	46.2489	1.94916	0.974580	+	0.224042i	$0.0719252\pi$
$564$	0	0
$565$	−4.54345	−0.191144
$566$	0	0
$567$	1.91291	0.0803348
$568$	0	0
$569$	−23.6516	−0.991525	−0.495762	−	0.868458i	$-0.665111\pi$
$569$	−23.6516	−0.991525	−0.495762	+	0.868458i	$0.665111\pi$
$570$	0	0
$571$	−25.5223	−1.06808	−0.534038	−	0.845460i	$-0.679326\pi$
$571$	−25.5223	−1.06808	−0.534038	+	0.845460i	$0.679326\pi$
$572$	0	0
$573$	−12.6871	−0.530010
$574$	0	0
$575$	8.82138	0.367877
$576$	0	0
$577$	25.2370	1.05063	0.525315	−	0.850908i	$-0.323947\pi$
$577$	25.2370	1.05063	0.525315	+	0.850908i	$0.323947\pi$
$578$	0	0
$579$	20.1813	0.838705
$580$	0	0
$581$	3.36251	0.139500
$582$	0	0
$583$	−8.35649	−0.346090
$584$	0	0
$585$	−2.47344	−0.102264
$586$	0	0
$587$	21.0289	0.867956	0.433978	−	0.900924i	$-0.357110\pi$
$587$	21.0289	0.867956	0.433978	+	0.900924i	$0.357110\pi$
$588$	0	0
$589$	−20.2950	−0.836241
$590$	0	0
$591$	−6.17319	−0.253931
$592$	0	0
$593$	−26.1219	−1.07270	−0.536349	−	0.843996i	$-0.680197\pi$
$593$	−26.1219	−1.07270	−0.536349	+	0.843996i	$0.680197\pi$
$594$	0	0
$595$	2.16522	0.0887656
$596$	0	0
$597$	−6.61307	−0.270655
$598$	0	0
$599$	2.03570	0.0831765	0.0415883	−	0.999135i	$-0.486758\pi$
$599$	2.03570	0.0831765	0.0415883	+	0.999135i	$0.486758\pi$
$600$	0	0
$601$	−3.81334	−0.155549	−0.0777747	−	0.996971i	$-0.524781\pi$
$601$	−3.81334	−0.155549	−0.0777747	+	0.996971i	$0.524781\pi$
$602$	0	0
$603$	−7.66365	−0.312088
$604$	0	0
$605$	−0.171092	−0.00695587
$606$	0	0
$607$	42.2882	1.71642	0.858212	−	0.513296i	$-0.171576\pi$
$607$	42.2882	1.71642	0.858212	+	0.513296i	$0.171576\pi$
$608$	0	0
$609$	−5.49112	−0.222511
$610$	0	0
$611$	12.5041	0.505861
$612$	0	0
$613$	−33.6651	−1.35972	−0.679859	−	0.733342i	$-0.737959\pi$
$613$	−33.6651	−1.35972	−0.679859	+	0.733342i	$0.737959\pi$
$614$	0	0
$615$	−6.14709	−0.247874
$616$	0	0
$617$	−34.7430	−1.39870	−0.699350	−	0.714780i	$-0.746527\pi$
$617$	−34.7430	−1.39870	−0.699350	+	0.714780i	$0.746527\pi$
$618$	0	0
$619$	−33.5891	−1.35006	−0.675029	−	0.737791i	$-0.735869\pi$
$619$	−33.5891	−1.35006	−0.675029	+	0.737791i	$0.735869\pi$
$620$	0	0
$621$	1.97902	0.0794153
$622$	0	0
$623$	−9.18274	−0.367899
$624$	0	0
$625$	17.1561	0.686245
$626$	0	0
$627$	28.2064	1.12646
$628$	0	0
$629$	4.20812	0.167789
$630$	0	0
$631$	−9.27213	−0.369118	−0.184559	−	0.982821i	$-0.559086\pi$
$631$	−9.27213	−0.369118	−0.184559	+	0.982821i	$0.559086\pi$
$632$	0	0
$633$	26.5667	1.05593
$634$	0	0
$635$	−1.82987	−0.0726160
$636$	0	0
$637$	−11.2183	−0.444486
$638$	0	0
$639$	5.41730	0.214305
$640$	0	0
$641$	−28.1920	−1.11352	−0.556759	−	0.830674i	$-0.687955\pi$
$641$	−28.1920	−1.11352	−0.556759	+	0.830674i	$0.687955\pi$
$642$	0	0
$643$	20.6234	0.813309	0.406654	−	0.913582i	$-0.366695\pi$
$643$	20.6234	0.813309	0.406654	+	0.913582i	$0.366695\pi$
$644$	0	0
$645$	1.40704	0.0554020
$646$	0	0
$647$	18.0050	0.707848	0.353924	−	0.935274i	$-0.384847\pi$
$647$	18.0050	0.707848	0.353924	+	0.935274i	$0.384847\pi$
$648$	0	0
$649$	−50.6911	−1.98980
$650$	0	0
$651$	−4.61286	−0.180792
$652$	0	0
$653$	33.8854	1.32604	0.663018	−	0.748603i	$-0.269275\pi$
$653$	33.8854	1.32604	0.663018	+	0.748603i	$0.269275\pi$
$654$	0	0
$655$	−4.66469	−0.182264
$656$	0	0
$657$	1.32988	0.0518836
$658$	0	0
$659$	40.3981	1.57369	0.786844	−	0.617152i	$-0.211714\pi$
$659$	40.3981	1.57369	0.786844	+	0.617152i	$0.211714\pi$
$660$	0	0
$661$	−30.2160	−1.17527	−0.587634	−	0.809127i	$-0.699940\pi$
$661$	−30.2160	−1.17527	−0.587634	+	0.809127i	$0.699940\pi$
$662$	0	0
$663$	5.16025	0.200407
$664$	0	0
$665$	11.8585	0.459852
$666$	0	0
$667$	−5.68088	−0.219964
$668$	0	0
$669$	−1.62309	−0.0627523
$670$	0	0
$671$	0.589763	0.0227676
$672$	0	0
$673$	−31.0505	−1.19691	−0.598454	−	0.801158i	$-0.704218\pi$
$673$	−31.0505	−1.19691	−0.598454	+	0.801158i	$0.704218\pi$
$674$	0	0
$675$	4.45745	0.171567
$676$	0	0
$677$	19.3827	0.744937	0.372468	−	0.928045i	$-0.378511\pi$
$677$	19.3827	0.744937	0.372468	+	0.928045i	$0.378511\pi$
$678$	0	0
$679$	−13.4012	−0.514292
$680$	0	0
$681$	3.28587	0.125915
$682$	0	0
$683$	−5.07756	−0.194288	−0.0971438	−	0.995270i	$-0.530971\pi$
$683$	−5.07756	−0.194288	−0.0971438	+	0.995270i	$0.530971\pi$
$684$	0	0
$685$	2.95197	0.112789
$686$	0	0
$687$	25.9901	0.991584
$688$	0	0
$689$	−8.37283	−0.318979
$690$	0	0
$691$	33.9104	1.29001	0.645005	−	0.764178i	$-0.276855\pi$
$691$	33.9104	1.29001	0.645005	+	0.764178i	$0.276855\pi$
$692$	0	0
$693$	6.41105	0.243536
$694$	0	0
$695$	2.21790	0.0841297
$696$	0	0
$697$	12.8244	0.485760
$698$	0	0
$699$	18.7420	0.708888
$700$	0	0
$701$	−31.2859	−1.18165	−0.590826	−	0.806799i	$-0.701198\pi$
$701$	−31.2859	−1.18165	−0.590826	+	0.806799i	$0.701198\pi$
$702$	0	0
$703$	23.0470	0.869233
$704$	0	0
$705$	2.74277	0.103299
$706$	0	0
$707$	−8.59519	−0.323255
$708$	0	0
$709$	−11.5958	−0.435490	−0.217745	−	0.976006i	$-0.569870\pi$
$709$	−11.5958	−0.435490	−0.217745	+	0.976006i	$0.569870\pi$
$710$	0	0
$711$	4.15790	0.155934
$712$	0	0
$713$	−4.77227	−0.178723
$714$	0	0
$715$	−8.28964	−0.310015
$716$	0	0
$717$	−0.259054	−0.00967456
$718$	0	0
$719$	−28.1217	−1.04876	−0.524381	−	0.851484i	$-0.675703\pi$
$719$	−28.1217	−1.04876	−0.524381	+	0.851484i	$0.675703\pi$
$720$	0	0
$721$	−31.5299	−1.17424
$722$	0	0
$723$	12.3055	0.457648
$724$	0	0
$725$	−12.7953	−0.475207
$726$	0	0
$727$	19.6733	0.729642	0.364821	−	0.931078i	$-0.381130\pi$
$727$	19.6733	0.729642	0.364821	+	0.931078i	$0.381130\pi$
$728$	0	0
$729$	1.00000	0.0370370
$730$	0	0
$731$	−2.93545	−0.108571
$732$	0	0
$733$	−3.55148	−0.131177	−0.0655885	−	0.997847i	$-0.520892\pi$
$733$	−3.55148	−0.131177	−0.0655885	+	0.997847i	$0.520892\pi$
$734$	0	0
$735$	−2.46073	−0.0907656
$736$	0	0
$737$	−25.6844	−0.946098
$738$	0	0
$739$	15.0790	0.554691	0.277346	−	0.960770i	$-0.410545\pi$
$739$	15.0790	0.554691	0.277346	+	0.960770i	$0.410545\pi$
$740$	0	0
$741$	28.2616	1.03822
$742$	0	0
$743$	−31.1197	−1.14167	−0.570835	−	0.821064i	$-0.693381\pi$
$743$	−31.1197	−1.14167	−0.570835	+	0.821064i	$0.693381\pi$
$744$	0	0
$745$	−17.8250	−0.653056
$746$	0	0
$747$	1.75779	0.0643143
$748$	0	0
$749$	6.84668	0.250172
$750$	0	0
$751$	27.2872	0.995723	0.497862	−	0.867257i	$-0.334119\pi$
$751$	27.2872	0.995723	0.497862	+	0.867257i	$0.334119\pi$
$752$	0	0
$753$	−6.61769	−0.241162
$754$	0	0
$755$	−8.38702	−0.305235
$756$	0	0
$757$	−3.28948	−0.119558	−0.0597790	−	0.998212i	$-0.519040\pi$
$757$	−3.28948	−0.119558	−0.0597790	+	0.998212i	$0.519040\pi$
$758$	0	0
$759$	6.63260	0.240748
$760$	0	0
$761$	46.6892	1.69248	0.846240	−	0.532801i	$-0.178861\pi$
$761$	46.6892	1.69248	0.846240	+	0.532801i	$0.178861\pi$
$762$	0	0
$763$	21.3667	0.773528
$764$	0	0
$765$	1.13190	0.0409239
$766$	0	0
$767$	−50.7902	−1.83393
$768$	0	0
$769$	−15.6719	−0.565143	−0.282572	−	0.959246i	$-0.591187\pi$
$769$	−15.6719	−0.565143	−0.282572	+	0.959246i	$0.591187\pi$
$770$	0	0
$771$	−15.1088	−0.544129
$772$	0	0
$773$	23.8158	0.856596	0.428298	−	0.903638i	$-0.359113\pi$
$773$	23.8158	0.856596	0.428298	+	0.903638i	$0.359113\pi$
$774$	0	0
$775$	−10.7488	−0.386110
$776$	0	0
$777$	5.23836	0.187925
$778$	0	0
$779$	70.2367	2.51649
$780$	0	0
$781$	18.1559	0.649668
$782$	0	0
$783$	−2.87055	−0.102585
$784$	0	0
$785$	5.92695	0.211542
$786$	0	0
$787$	−3.75798	−0.133957	−0.0669787	−	0.997754i	$-0.521336\pi$
$787$	−3.75798	−0.133957	−0.0669787	+	0.997754i	$0.521336\pi$
$788$	0	0
$789$	7.33151	0.261009
$790$	0	0
$791$	11.7994	0.419540
$792$	0	0
$793$	0.590917	0.0209841
$794$	0	0
$795$	−1.83658	−0.0651367
$796$	0	0
$797$	51.2367	1.81490	0.907449	−	0.420162i	$-0.138027\pi$
$797$	51.2367	1.81490	0.907449	+	0.420162i	$0.138027\pi$
$798$	0	0
$799$	−5.72213	−0.202434
$800$	0	0
$801$	−4.80040	−0.169614
$802$	0	0
$803$	4.45704	0.157286
$804$	0	0
$805$	2.78846	0.0982804
$806$	0	0
$807$	−4.03612	−0.142078
$808$	0	0
$809$	12.3215	0.433202	0.216601	−	0.976260i	$-0.430503\pi$
$809$	12.3215	0.433202	0.216601	+	0.976260i	$0.430503\pi$
$810$	0	0
$811$	−21.1715	−0.743432	−0.371716	−	0.928346i	$-0.621231\pi$
$811$	−21.1715	−0.743432	−0.371716	+	0.928346i	$0.621231\pi$
$812$	0	0
$813$	17.5487	0.615461
$814$	0	0
$815$	10.0674	0.352646
$816$	0	0
$817$	−16.0768	−0.562457
$818$	0	0
$819$	6.42359	0.224458
$820$	0	0
$821$	46.2761	1.61505	0.807524	−	0.589835i	$-0.200807\pi$
$821$	46.2761	1.61505	0.807524	+	0.589835i	$0.200807\pi$
$822$	0	0
$823$	13.8588	0.483089	0.241544	−	0.970390i	$-0.422346\pi$
$823$	13.8588	0.483089	0.241544	+	0.970390i	$0.422346\pi$
$824$	0	0
$825$	14.9390	0.520108
$826$	0	0
$827$	16.0662	0.558677	0.279339	−	0.960193i	$-0.409885\pi$
$827$	16.0662	0.558677	0.279339	+	0.960193i	$0.409885\pi$
$828$	0	0
$829$	31.1793	1.08290	0.541451	−	0.840732i	$-0.317875\pi$
$829$	31.1793	1.08290	0.541451	+	0.840732i	$0.317875\pi$
$830$	0	0
$831$	−8.35488	−0.289827
$832$	0	0
$833$	5.13374	0.177873
$834$	0	0
$835$	−3.78688	−0.131050
$836$	0	0
$837$	−2.41143	−0.0833513
$838$	0	0
$839$	−5.75074	−0.198538	−0.0992688	−	0.995061i	$-0.531650\pi$
$839$	−5.75074	−0.198538	−0.0992688	+	0.995061i	$0.531650\pi$
$840$	0	0
$841$	−20.7599	−0.715860
$842$	0	0
$843$	17.6407	0.607577
$844$	0	0
$845$	1.26968	0.0436783
$846$	0	0
$847$	0.444330	0.0152674
$848$	0	0
$849$	19.8352	0.680741
$850$	0	0
$851$	5.41938	0.185774
$852$	0	0
$853$	−9.01478	−0.308660	−0.154330	−	0.988019i	$-0.549322\pi$
$853$	−9.01478	−0.308660	−0.154330	+	0.988019i	$0.549322\pi$
$854$	0	0
$855$	6.19917	0.212007
$856$	0	0
$857$	15.6244	0.533718	0.266859	−	0.963736i	$-0.414014\pi$
$857$	15.6244	0.533718	0.266859	+	0.963736i	$0.414014\pi$
$858$	0	0
$859$	−44.2612	−1.51017	−0.755087	−	0.655625i	$-0.772405\pi$
$859$	−44.2612	−1.51017	−0.755087	+	0.655625i	$0.772405\pi$
$860$	0	0
$861$	15.9641	0.544056
$862$	0	0
$863$	−13.3303	−0.453769	−0.226884	−	0.973922i	$-0.572854\pi$
$863$	−13.3303	−0.453769	−0.226884	+	0.973922i	$0.572854\pi$
$864$	0	0
$865$	14.5313	0.494078
$866$	0	0
$867$	14.6386	0.497152
$868$	0	0
$869$	13.9350	0.472714
$870$	0	0
$871$	−25.7346	−0.871985
$872$	0	0
$873$	−7.00567	−0.237106
$874$	0	0
$875$	13.3257	0.450490
$876$	0	0
$877$	13.7519	0.464370	0.232185	−	0.972672i	$-0.425412\pi$
$877$	13.7519	0.464370	0.232185	+	0.972672i	$0.425412\pi$
$878$	0	0
$879$	−3.09348	−0.104341
$880$	0	0
$881$	−24.8600	−0.837554	−0.418777	−	0.908089i	$-0.637541\pi$
$881$	−24.8600	−0.837554	−0.418777	+	0.908089i	$0.637541\pi$
$882$	0	0
$883$	11.0790	0.372837	0.186418	−	0.982470i	$-0.440312\pi$
$883$	11.0790	0.372837	0.186418	+	0.982470i	$0.440312\pi$
$884$	0	0
$885$	−11.1408	−0.374495
$886$	0	0
$887$	−46.8749	−1.57390	−0.786952	−	0.617014i	$-0.788342\pi$
$887$	−46.8749	−1.57390	−0.786952	+	0.617014i	$0.788342\pi$
$888$	0	0
$889$	4.75221	0.159384
$890$	0	0
$891$	3.35146	0.112278
$892$	0	0
$893$	−31.3389	−1.04872
$894$	0	0
$895$	−10.1592	−0.339586
$896$	0	0
$897$	6.64557	0.221889
$898$	0	0
$899$	6.92214	0.230866
$900$	0	0
$901$	3.83158	0.127648
$902$	0	0
$903$	−3.65411	−0.121601
$904$	0	0
$905$	−16.7922	−0.558192
$906$	0	0
$907$	−29.5673	−0.981765	−0.490882	−	0.871226i	$-0.663325\pi$
$907$	−29.5673	−0.981765	−0.490882	+	0.871226i	$0.663325\pi$
$908$	0	0
$909$	−4.49325	−0.149032
$910$	0	0
$911$	5.02746	0.166567	0.0832837	−	0.996526i	$-0.473459\pi$
$911$	5.02746	0.166567	0.0832837	+	0.996526i	$0.473459\pi$
$912$	0	0
$913$	5.89117	0.194969
$914$	0	0
$915$	0.129617	0.00428502
$916$	0	0
$917$	12.1143	0.400049
$918$	0	0
$919$	18.1476	0.598633	0.299317	−	0.954154i	$-0.403241\pi$
$919$	18.1476	0.598633	0.299317	+	0.954154i	$0.403241\pi$
$920$	0	0
$921$	−8.98421	−0.296040
$922$	0	0
$923$	18.1914	0.598776
$924$	0	0
$925$	12.2064	0.401343
$926$	0	0
$927$	−16.4827	−0.541362
$928$	0	0
$929$	5.44986	0.178804	0.0894020	−	0.995996i	$-0.471504\pi$
$929$	5.44986	0.178804	0.0894020	+	0.995996i	$0.471504\pi$
$930$	0	0
$931$	28.1164	0.921477
$932$	0	0
$933$	14.6120	0.478374
$934$	0	0
$935$	3.79351	0.124061
$936$	0	0
$937$	−6.69115	−0.218590	−0.109295	−	0.994009i	$-0.534859\pi$
$937$	−6.69115	−0.218590	−0.109295	+	0.994009i	$0.534859\pi$
$938$	0	0
$939$	13.4152	0.437789
$940$	0	0
$941$	−42.6008	−1.38875	−0.694374	−	0.719615i	$-0.744318\pi$
$941$	−42.6008	−1.38875	−0.694374	+	0.719615i	$0.744318\pi$
$942$	0	0
$943$	16.5158	0.537829
$944$	0	0
$945$	1.40901	0.0458352
$946$	0	0
$947$	57.4540	1.86700	0.933502	−	0.358572i	$-0.116736\pi$
$947$	57.4540	1.86700	0.933502	+	0.358572i	$0.116736\pi$
$948$	0	0
$949$	4.46576	0.144965
$950$	0	0
$951$	22.5821	0.732274
$952$	0	0
$953$	13.4828	0.436752	0.218376	−	0.975865i	$-0.429924\pi$
$953$	13.4828	0.436752	0.218376	+	0.975865i	$0.429924\pi$
$954$	0	0
$955$	−9.34503	−0.302398
$956$	0	0
$957$	−9.62053	−0.310988
$958$	0	0
$959$	−7.66635	−0.247559
$960$	0	0
$961$	−25.1850	−0.812419
$962$	0	0
$963$	3.57919	0.115338
$964$	0	0
$965$	14.8651	0.478525
$966$	0	0
$967$	46.4793	1.49467	0.747337	−	0.664446i	$-0.231332\pi$
$967$	46.4793	1.49467	0.747337	+	0.664446i	$0.231332\pi$
$968$	0	0
$969$	−12.9331	−0.415471
$970$	0	0
$971$	21.5888	0.692818	0.346409	−	0.938084i	$-0.387401\pi$
$971$	21.5888	0.692818	0.346409	+	0.938084i	$0.387401\pi$
$972$	0	0
$973$	−5.75994	−0.184655
$974$	0	0
$975$	14.9682	0.479365
$976$	0	0
$977$	−2.12544	−0.0679987	−0.0339994	−	0.999422i	$-0.510824\pi$
$977$	−2.12544	−0.0679987	−0.0339994	+	0.999422i	$0.510824\pi$
$978$	0	0
$979$	−16.0883	−0.514185
$980$	0	0
$981$	11.1697	0.356622
$982$	0	0
$983$	−9.88457	−0.315269	−0.157634	−	0.987498i	$-0.550387\pi$
$983$	−9.88457	−0.315269	−0.157634	+	0.987498i	$0.550387\pi$
$984$	0	0
$985$	−4.54704	−0.144881
$986$	0	0
$987$	−7.12303	−0.226729
$988$	0	0
$989$	−3.78038	−0.120209
$990$	0	0
$991$	40.3738	1.28252	0.641258	−	0.767326i	$-0.278413\pi$
$991$	40.3738	1.28252	0.641258	+	0.767326i	$0.278413\pi$
$992$	0	0
$993$	24.6848	0.783348
$994$	0	0
$995$	−4.87105	−0.154423
$996$	0	0
$997$	35.8217	1.13449	0.567243	−	0.823551i	$-0.308010\pi$
$997$	35.8217	1.13449	0.567243	+	0.823551i	$0.308010\pi$
$998$	0	0
$999$	2.73842	0.0866397

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	6036.2.a.f.1.6	✓	14

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
6036.2.a.f.1.6	✓	14	1.1	even	1	trivial

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Classical	Maass
Hilbert	Bianchi

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

Level:	\( N \)	\(=\)	\( 6036 = 2^{2} \cdot 3 \cdot 503 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	6036.a (trivial)

\(f(q)\)	\(=\)	\(q-1.00000 q^{3} -0.736579 q^{5} +1.91291 q^{7} +1.00000 q^{9} +O(q^{10})\)	\(q-1.00000 q^{3} -0.736579 q^{5} +1.91291 q^{7} +1.00000 q^{9} +3.35146 q^{11} +3.35801 q^{13} +0.736579 q^{15} -1.53670 q^{17} -8.41616 q^{19} -1.91291 q^{21} -1.97902 q^{23} -4.45745 q^{25} -1.00000 q^{27} +2.87055 q^{29} +2.41143 q^{31} -3.35146 q^{33} -1.40901 q^{35} -2.73842 q^{37} -3.35801 q^{39} -8.34545 q^{41} +1.91023 q^{43} -0.736579 q^{45} +3.72365 q^{47} -3.34076 q^{49} +1.53670 q^{51} -2.49339 q^{53} -2.46861 q^{55} +8.41616 q^{57} -15.1251 q^{59} +0.175972 q^{61} +1.91291 q^{63} -2.47344 q^{65} -7.66365 q^{67} +1.97902 q^{69} +5.41730 q^{71} +1.32988 q^{73} +4.45745 q^{75} +6.41105 q^{77} +4.15790 q^{79} +1.00000 q^{81} +1.75779 q^{83} +1.13190 q^{85} -2.87055 q^{87} -4.80040 q^{89} +6.42359 q^{91} -2.41143 q^{93} +6.19917 q^{95} -7.00567 q^{97} +3.35146 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 14 q - 14 q^{3} - 6 q^{5} - 7 q^{7} + 14 q^{9}+O(q^{10}) \) 14 * q - 14 * q^3 - 6 * q^5 - 7 * q^7 + 14 * q^9	\( 14 q - 14 q^{3} - 6 q^{5} - 7 q^{7} + 14 q^{9} + q^{11} - q^{13} + 6 q^{15} - 6 q^{17} + q^{19} + 7 q^{21} + 10 q^{23} - 10 q^{25} - 14 q^{27} - 6 q^{29} - 5 q^{31} - q^{33} + 17 q^{35} - 12 q^{37} + q^{39} - 21 q^{41} + 8 q^{43} - 6 q^{45} + 18 q^{47} - 19 q^{49} + 6 q^{51} - q^{53} - q^{57} + 14 q^{59} - 19 q^{61} - 7 q^{63} + 17 q^{65} + 17 q^{67} - 10 q^{69} - 13 q^{71} - 12 q^{73} + 10 q^{75} - 9 q^{77} - 8 q^{79} + 14 q^{81} + 11 q^{83} - 17 q^{85} + 6 q^{87} - 9 q^{89} - 5 q^{91} + 5 q^{93} + 8 q^{95} - 25 q^{97} + q^{99}+O(q^{100}) \) 14 * q - 14 * q^3 - 6 * q^5 - 7 * q^7 + 14 * q^9 + q^11 - q^13 + 6 * q^15 - 6 * q^17 + q^19 + 7 * q^21 + 10 * q^23 - 10 * q^25 - 14 * q^27 - 6 * q^29 - 5 * q^31 - q^33 + 17 * q^35 - 12 * q^37 + q^39 - 21 * q^41 + 8 * q^43 - 6 * q^45 + 18 * q^47 - 19 * q^49 + 6 * q^51 - q^53 - q^57 + 14 * q^59 - 19 * q^61 - 7 * q^63 + 17 * q^65 + 17 * q^67 - 10 * q^69 - 13 * q^71 - 12 * q^73 + 10 * q^75 - 9 * q^77 - 8 * q^79 + 14 * q^81 + 11 * q^83 - 17 * q^85 + 6 * q^87 - 9 * q^89 - 5 * q^91 + 5 * q^93 + 8 * q^95 - 25 * q^97 + q^99

Introduction

L-functions

Modular forms

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