LMFDB - Embedded newform 1856.4.a.bj.1.8

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [1856,4,Mod(1,1856)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(1856, base_ring=CyclotomicField(2))

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

N = Newforms(chi, 4, names="a")

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

chi := DirichletCharacter("1856.1");

S:= CuspForms(chi, 4);

N := Newforms(S);

Level:	$ N $	$=$	$ 1856 = 2^{6} \cdot 29 $
Weight:	$ k $	$=$	$ 4 $
Character orbit:	$[\chi]$	$=$	1856.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:	$109.507544971$
Analytic rank:	$0$
Dimension:	$12$
Coefficient field:	$\mathbb{Q}[x]/(x^{12} - \cdots)$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{12} - 2 x^{11} - 217 x^{10} + 520 x^{9} + 16022 x^{8} - 37368 x^{7} - 509640 x^{6} + 989168 x^{5} + \cdots + 14751072 $ x^12 - 2x^11 - 217x^10 + 520x^9 + 16022x^8 - 37368x^7 - 509640x^6 + 989168x^5 + 7106592x^4 - 7979328x^3 - 38912400x^2 + 1083456*x + 14751072
Coefficient ring:	$\Z[a_1, \ldots, a_{11}]$
Coefficient ring index:	$ 2^{17} $
Twist minimal:	no (minimal twist has level 928)
Fricke sign:	$1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.8
Root			$-2.71183$ of defining polynomial
Character	$\chi$	$=$	1856.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+1.71183 q^{3} -8.28780 q^{5} -9.40926 q^{7} -24.0696 q^{9} +O(q^{10})$	$q+1.71183 q^{3} -8.28780 q^{5} -9.40926 q^{7} -24.0696 q^{9} +70.9884 q^{11} +55.2478 q^{13} -14.1873 q^{15} -113.376 q^{17} -130.825 q^{19} -16.1070 q^{21} +119.948 q^{23} -56.3124 q^{25} -87.4225 q^{27} -29.0000 q^{29} +152.679 q^{31} +121.520 q^{33} +77.9821 q^{35} -270.093 q^{37} +94.5747 q^{39} +368.077 q^{41} +79.8832 q^{43} +199.484 q^{45} +556.119 q^{47} -254.466 q^{49} -194.080 q^{51} -221.368 q^{53} -588.338 q^{55} -223.951 q^{57} -18.7179 q^{59} +298.553 q^{61} +226.478 q^{63} -457.882 q^{65} -516.187 q^{67} +205.330 q^{69} -568.221 q^{71} -660.868 q^{73} -96.3971 q^{75} -667.949 q^{77} +35.3870 q^{79} +500.228 q^{81} +83.5548 q^{83} +939.637 q^{85} -49.6430 q^{87} +1027.44 q^{89} -519.841 q^{91} +261.361 q^{93} +1084.25 q^{95} -621.640 q^{97} -1708.67 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 12 q - 14 q^{3} + 10 q^{5} + 44 q^{7} + 130 q^{9}+O(q^{10}) $ 12 * q - 14 * q^3 + 10 * q^5 + 44 * q^7 + 130 * q^9	$ 12 q - 14 q^{3} + 10 q^{5} + 44 q^{7} + 130 q^{9} - 46 q^{11} + 34 q^{13} - 50 q^{15} + 36 q^{17} - 148 q^{19} + 92 q^{21} + 328 q^{23} + 486 q^{25} - 326 q^{27} - 348 q^{29} + 374 q^{31} + 710 q^{33} - 204 q^{35} + 340 q^{37} - 122 q^{39} + 32 q^{41} - 462 q^{43} + 1132 q^{45} + 434 q^{47} + 1508 q^{49} - 440 q^{51} - 610 q^{53} + 46 q^{55} - 932 q^{57} - 1240 q^{59} + 1228 q^{61} + 4240 q^{63} + 730 q^{65} - 1672 q^{67} + 528 q^{69} + 3220 q^{71} + 564 q^{73} - 6032 q^{75} - 644 q^{77} + 1862 q^{79} + 3040 q^{81} - 3736 q^{83} + 808 q^{85} + 406 q^{87} + 584 q^{89} - 4844 q^{91} + 3226 q^{93} + 2844 q^{95} + 904 q^{97} - 6832 q^{99}+O(q^{100}) $ 12 * q - 14 * q^3 + 10 * q^5 + 44 * q^7 + 130 * q^9 - 46 * q^11 + 34 * q^13 - 50 * q^15 + 36 * q^17 - 148 * q^19 + 92 * q^21 + 328 * q^23 + 486 * q^25 - 326 * q^27 - 348 * q^29 + 374 * q^31 + 710 * q^33 - 204 * q^35 + 340 * q^37 - 122 * q^39 + 32 * q^41 - 462 * q^43 + 1132 * q^45 + 434 * q^47 + 1508 * q^49 - 440 * q^51 - 610 * q^53 + 46 * q^55 - 932 * q^57 - 1240 * q^59 + 1228 * q^61 + 4240 * q^63 + 730 * q^65 - 1672 * q^67 + 528 * q^69 + 3220 * q^71 + 564 * q^73 - 6032 * q^75 - 644 * q^77 + 1862 * q^79 + 3040 * q^81 - 3736 * q^83 + 808 * q^85 + 406 * q^87 + 584 * q^89 - 4844 * q^91 + 3226 * q^93 + 2844 * q^95 + 904 * q^97 - 6832 * 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
$3$	1.71183	0.329441	0.164721	−	0.986340i	$-0.447328\pi$
$3$	1.71183	0.329441	0.164721	+	0.986340i	$0.447328\pi$
$4$	0	0
$5$	−8.28780	−0.741283	−0.370642	−	0.928776i	$-0.620862\pi$
$5$	−8.28780	−0.741283	−0.370642	+	0.928776i	$0.620862\pi$
$6$	0	0
$7$	−9.40926	−0.508052	−0.254026	−	0.967197i	$-0.581755\pi$
$7$	−9.40926	−0.508052	−0.254026	+	0.967197i	$0.581755\pi$
$8$	0	0
$9$	−24.0696	−0.891468
$10$	0	0
$11$	70.9884	1.94580	0.972901	−	0.231224i	$-0.0742730\pi$
$11$	70.9884	1.94580	0.972901	+	0.231224i	$0.0742730\pi$
$12$	0	0
$13$	55.2478	1.17869	0.589345	−	0.807881i	$-0.299386\pi$
$13$	55.2478	1.17869	0.589345	+	0.807881i	$0.299386\pi$
$14$	0	0
$15$	−14.1873	−0.244209
$16$	0	0
$17$	−113.376	−1.61751	−0.808756	−	0.588144i	$-0.799859\pi$
$17$	−113.376	−1.61751	−0.808756	+	0.588144i	$0.799859\pi$
$18$	0	0
$19$	−130.825	−1.57965	−0.789826	−	0.613331i	$-0.789829\pi$
$19$	−130.825	−1.57965	−0.789826	+	0.613331i	$0.789829\pi$
$20$	0	0
$21$	−16.1070	−0.167374
$22$	0	0
$23$	119.948	1.08743	0.543714	−	0.839270i	$-0.317017\pi$
$23$	119.948	1.08743	0.543714	+	0.839270i	$0.317017\pi$
$24$	0	0
$25$	−56.3124	−0.450499
$26$	0	0
$27$	−87.4225	−0.623128
$28$	0	0
$29$	−29.0000	−0.185695
$29$	−29.0000	−0.185695
$30$	0	0
$31$	152.679	0.884581	0.442291	−	0.896872i	$-0.354166\pi$
$31$	152.679	0.884581	0.442291	+	0.896872i	$0.354166\pi$
$32$	0	0
$33$	121.520	0.641028
$34$	0	0
$35$	77.9821	0.376611
$36$	0	0
$37$	−270.093	−1.20008	−0.600041	−	0.799969i	$-0.704849\pi$
$37$	−270.093	−1.20008	−0.600041	+	0.799969i	$0.704849\pi$
$38$	0	0
$39$	94.5747	0.388310
$40$	0	0
$41$	368.077	1.40205	0.701024	−	0.713137i	$-0.252726\pi$
$41$	368.077	1.40205	0.701024	+	0.713137i	$0.252726\pi$
$42$	0	0
$43$	79.8832	0.283304	0.141652	−	0.989917i	$-0.454759\pi$
$43$	79.8832	0.283304	0.141652	+	0.989917i	$0.454759\pi$
$44$	0	0
$45$	199.484	0.660831
$46$	0	0
$47$	556.119	1.72592	0.862960	−	0.505272i	$-0.168608\pi$
$47$	556.119	1.72592	0.862960	+	0.505272i	$0.168608\pi$
$48$	0	0
$49$	−254.466	−0.741883
$50$	0	0
$51$	−194.080	−0.532876
$52$	0	0
$53$	−221.368	−0.573722	−0.286861	−	0.957972i	$-0.592612\pi$
$53$	−221.368	−0.573722	−0.286861	+	0.957972i	$0.592612\pi$
$54$	0	0
$55$	−588.338	−1.44239
$56$	0	0
$57$	−223.951	−0.520403
$58$	0	0
$59$	−18.7179	−0.0413027	−0.0206514	−	0.999787i	$-0.506574\pi$
$59$	−18.7179	−0.0413027	−0.0206514	+	0.999787i	$0.506574\pi$
$60$	0	0
$61$	298.553	0.626652	0.313326	−	0.949646i	$-0.398557\pi$
$61$	298.553	0.626652	0.313326	+	0.949646i	$0.398557\pi$
$62$	0	0
$63$	226.478	0.452913
$64$	0	0
$65$	−457.882	−0.873743
$66$	0	0
$67$	−516.187	−0.941228	−0.470614	−	0.882339i	$-0.655968\pi$
$67$	−516.187	−0.941228	−0.470614	+	0.882339i	$0.655968\pi$
$68$	0	0
$69$	205.330	0.358244
$70$	0	0
$71$	−568.221	−0.949795	−0.474897	−	0.880041i	$-0.657515\pi$
$71$	−568.221	−0.949795	−0.474897	+	0.880041i	$0.657515\pi$
$72$	0	0
$73$	−660.868	−1.05957	−0.529786	−	0.848131i	$-0.677728\pi$
$73$	−660.868	−1.05957	−0.529786	+	0.848131i	$0.677728\pi$
$74$	0	0
$75$	−96.3971	−0.148413
$76$	0	0
$77$	−667.949	−0.988569
$78$	0	0
$79$	35.3870	0.0503969	0.0251984	−	0.999682i	$-0.491978\pi$
$79$	35.3870	0.0503969	0.0251984	+	0.999682i	$0.491978\pi$
$80$	0	0
$81$	500.228	0.686184
$82$	0	0
$83$	83.5548	0.110498	0.0552490	−	0.998473i	$-0.482405\pi$
$83$	83.5548	0.110498	0.0552490	+	0.998473i	$0.482405\pi$
$84$	0	0
$85$	939.637	1.19904
$86$	0	0
$87$	−49.6430	−0.0611758
$88$	0	0
$89$	1027.44	1.22369	0.611846	−	0.790977i	$-0.290427\pi$
$89$	1027.44	1.22369	0.611846	+	0.790977i	$0.290427\pi$
$90$	0	0
$91$	−519.841	−0.598836
$92$	0	0
$93$	261.361	0.291418
$94$	0	0
$95$	1084.25	1.17097
$96$	0	0
$97$	−621.640	−0.650701	−0.325350	−	0.945594i	$-0.605482\pi$
$97$	−621.640	−0.650701	−0.325350	+	0.945594i	$0.605482\pi$
$98$	0	0
$99$	−1708.67	−1.73462
$100$	0	0
$101$	−480.106	−0.472993	−0.236497	−	0.971632i	$-0.575999\pi$
$101$	−480.106	−0.472993	−0.236497	+	0.971632i	$0.575999\pi$
$102$	0	0
$103$	385.075	0.368375	0.184187	−	0.982891i	$-0.441035\pi$
$103$	385.075	0.368375	0.184187	+	0.982891i	$0.441035\pi$
$104$	0	0
$105$	133.492	0.124071
$106$	0	0
$107$	1765.83	1.59541	0.797706	−	0.603046i	$-0.206047\pi$
$107$	1765.83	1.59541	0.797706	+	0.603046i	$0.206047\pi$
$108$	0	0
$109$	634.006	0.557127	0.278563	−	0.960418i	$-0.410142\pi$
$109$	634.006	0.557127	0.278563	+	0.960418i	$0.410142\pi$
$110$	0	0
$111$	−462.353	−0.395357
$112$	0	0
$113$	717.864	0.597620	0.298810	−	0.954313i	$-0.403410\pi$
$113$	717.864	0.597620	0.298810	+	0.954313i	$0.403410\pi$
$114$	0	0
$115$	−994.104	−0.806093
$116$	0	0
$117$	−1329.79	−1.05077
$118$	0	0
$119$	1066.78	0.821781
$120$	0	0
$121$	3708.35	2.78614
$122$	0	0
$123$	630.085	0.461893
$124$	0	0
$125$	1502.68	1.07523
$126$	0	0
$127$	641.367	0.448127	0.224063	−	0.974575i	$-0.428068\pi$
$127$	641.367	0.448127	0.224063	+	0.974575i	$0.428068\pi$
$128$	0	0
$129$	136.746	0.0933321
$130$	0	0
$131$	63.1476	0.0421163	0.0210581	−	0.999778i	$-0.493296\pi$
$131$	63.1476	0.0421163	0.0210581	+	0.999778i	$0.493296\pi$
$132$	0	0
$133$	1230.97	0.802546
$134$	0	0
$135$	724.540	0.461914
$136$	0	0
$137$	−1329.38	−0.829029	−0.414514	−	0.910043i	$-0.636049\pi$
$137$	−1329.38	−0.829029	−0.414514	+	0.910043i	$0.636049\pi$
$138$	0	0
$139$	28.8252	0.0175894	0.00879469	−	0.999961i	$-0.497201\pi$
$139$	28.8252	0.0175894	0.00879469	+	0.999961i	$0.497201\pi$
$140$	0	0
$141$	951.979	0.568590
$142$	0	0
$143$	3921.95	2.29350
$144$	0	0
$145$	240.346	0.137653
$146$	0	0
$147$	−435.602	−0.244407
$148$	0	0
$149$	623.189	0.342642	0.171321	−	0.985215i	$-0.445197\pi$
$149$	623.189	0.342642	0.171321	+	0.985215i	$0.445197\pi$
$150$	0	0
$151$	2726.75	1.46953	0.734766	−	0.678320i	$-0.237292\pi$
$151$	2726.75	1.46953	0.734766	+	0.678320i	$0.237292\pi$
$152$	0	0
$153$	2728.92	1.44196
$154$	0	0
$155$	−1265.38	−0.655725
$156$	0	0
$157$	2576.81	1.30989	0.654943	−	0.755679i	$-0.272693\pi$
$157$	2576.81	1.30989	0.654943	+	0.755679i	$0.272693\pi$
$158$	0	0
$159$	−378.945	−0.189008
$160$	0	0
$161$	−1128.62	−0.552471
$162$	0	0
$163$	−1996.69	−0.959467	−0.479733	−	0.877414i	$-0.659267\pi$
$163$	−1996.69	−0.959467	−0.479733	+	0.877414i	$0.659267\pi$
$164$	0	0
$165$	−1007.13	−0.475183
$166$	0	0
$167$	2963.72	1.37329	0.686644	−	0.726993i	$-0.259083\pi$
$167$	2963.72	1.37329	0.686644	+	0.726993i	$0.259083\pi$
$168$	0	0
$169$	855.316	0.389311
$170$	0	0
$171$	3148.92	1.40821
$172$	0	0
$173$	−2261.02	−0.993656	−0.496828	−	0.867849i	$-0.665502\pi$
$173$	−2261.02	−0.993656	−0.496828	+	0.867849i	$0.665502\pi$
$174$	0	0
$175$	529.858	0.228877
$176$	0	0
$177$	−32.0418	−0.0136068
$178$	0	0
$179$	1946.14	0.812633	0.406316	−	0.913733i	$-0.366813\pi$
$179$	1946.14	0.812633	0.406316	+	0.913733i	$0.366813\pi$
$180$	0	0
$181$	3260.73	1.33905	0.669526	−	0.742788i	$-0.266497\pi$
$181$	3260.73	1.33905	0.669526	+	0.742788i	$0.266497\pi$
$182$	0	0
$183$	511.072	0.206445
$184$	0	0
$185$	2238.48	0.889601
$186$	0	0
$187$	−8048.38	−3.14736
$188$	0	0
$189$	822.581	0.316582
$190$	0	0
$191$	−3088.23	−1.16993	−0.584966	−	0.811058i	$-0.698892\pi$
$191$	−3088.23	−1.16993	−0.584966	+	0.811058i	$0.698892\pi$
$192$	0	0
$193$	4378.98	1.63319	0.816595	−	0.577210i	$-0.195859\pi$
$193$	4378.98	1.63319	0.816595	+	0.577210i	$0.195859\pi$
$194$	0	0
$195$	−783.816	−0.287847
$196$	0	0
$197$	3148.61	1.13873	0.569364	−	0.822086i	$-0.307190\pi$
$197$	3148.61	1.13873	0.569364	+	0.822086i	$0.307190\pi$
$198$	0	0
$199$	1801.03	0.641567	0.320784	−	0.947152i	$-0.396054\pi$
$199$	1801.03	0.641567	0.320784	+	0.947152i	$0.396054\pi$
$200$	0	0
$201$	−883.623	−0.310080
$202$	0	0
$203$	272.869	0.0943430
$204$	0	0
$205$	−3050.55	−1.03932
$206$	0	0
$207$	−2887.10	−0.969408
$208$	0	0
$209$	−9287.09	−3.07369
$210$	0	0
$211$	789.717	0.257660	0.128830	−	0.991667i	$-0.458878\pi$
$211$	789.717	0.257660	0.128830	+	0.991667i	$0.458878\pi$
$212$	0	0
$213$	−972.697	−0.312902
$214$	0	0
$215$	−662.056	−0.210008
$216$	0	0
$217$	−1436.60	−0.449414
$218$	0	0
$219$	−1131.29	−0.349067
$220$	0	0
$221$	−6263.77	−1.90655
$222$	0	0
$223$	−4287.13	−1.28739	−0.643693	−	0.765283i	$-0.722599\pi$
$223$	−4287.13	−1.28739	−0.643693	+	0.765283i	$0.722599\pi$
$224$	0	0
$225$	1355.42	0.401606
$226$	0	0
$227$	−1991.26	−0.582221	−0.291111	−	0.956689i	$-0.594025\pi$
$227$	−1991.26	−0.582221	−0.291111	+	0.956689i	$0.594025\pi$
$228$	0	0
$229$	3349.42	0.966532	0.483266	−	0.875473i	$-0.339450\pi$
$229$	3349.42	0.966532	0.483266	+	0.875473i	$0.339450\pi$
$230$	0	0
$231$	−1143.41	−0.325676
$232$	0	0
$233$	6311.65	1.77463	0.887317	−	0.461159i	$-0.152566\pi$
$233$	6311.65	1.77463	0.887317	+	0.461159i	$0.152566\pi$
$234$	0	0
$235$	−4609.00	−1.27940
$236$	0	0
$237$	60.5766	0.0166028
$238$	0	0
$239$	−2352.90	−0.636805	−0.318403	−	0.947956i	$-0.603146\pi$
$239$	−2352.90	−0.636805	−0.318403	+	0.947956i	$0.603146\pi$
$240$	0	0
$241$	5729.54	1.53142	0.765709	−	0.643187i	$-0.222388\pi$
$241$	5729.54	1.53142	0.765709	+	0.643187i	$0.222388\pi$
$242$	0	0
$243$	3216.71	0.849186
$244$	0	0
$245$	2108.96	0.549945
$246$	0	0
$247$	−7227.81	−1.86192
$248$	0	0
$249$	143.031	0.0364026
$250$	0	0
$251$	−710.136	−0.178579	−0.0892896	−	0.996006i	$-0.528460\pi$
$251$	−710.136	−0.178579	−0.0892896	+	0.996006i	$0.528460\pi$
$252$	0	0
$253$	8514.91	2.11592
$254$	0	0
$255$	1608.50	0.395012
$256$	0	0
$257$	−3482.49	−0.845260	−0.422630	−	0.906302i	$-0.638893\pi$
$257$	−3482.49	−0.845260	−0.422630	+	0.906302i	$0.638893\pi$
$258$	0	0
$259$	2541.38	0.609704
$260$	0	0
$261$	698.020	0.165542
$262$	0	0
$263$	8117.07	1.90312	0.951559	−	0.307466i	$-0.0994810\pi$
$263$	8117.07	1.90312	0.951559	+	0.307466i	$0.0994810\pi$
$264$	0	0
$265$	1834.66	0.425291
$266$	0	0
$267$	1758.80	0.403135
$268$	0	0
$269$	109.393	0.0247949	0.0123975	−	0.999923i	$-0.496054\pi$
$269$	109.393	0.0247949	0.0123975	+	0.999923i	$0.496054\pi$
$270$	0	0
$271$	−3.75076	−0.000840748 0	−0.000420374	−	1.00000i	$-0.500134\pi$
$271$	−3.75076	−0.000840748 0	−0.000420374		1.00000i	$0.500134\pi$
$272$	0	0
$273$	−889.878	−0.197282
$274$	0	0
$275$	−3997.53	−0.876582
$276$	0	0
$277$	368.156	0.0798568	0.0399284	−	0.999203i	$-0.487287\pi$
$277$	368.156	0.0798568	0.0399284	+	0.999203i	$0.487287\pi$
$278$	0	0
$279$	−3674.94	−0.788576
$280$	0	0
$281$	7284.38	1.54644	0.773220	−	0.634138i	$-0.218645\pi$
$281$	7284.38	1.54644	0.773220	+	0.634138i	$0.218645\pi$
$282$	0	0
$283$	−2747.05	−0.577014	−0.288507	−	0.957478i	$-0.593159\pi$
$283$	−2747.05	−0.577014	−0.288507	+	0.957478i	$0.593159\pi$
$284$	0	0
$285$	1856.06	0.385766
$286$	0	0
$287$	−3463.33	−0.712314
$288$	0	0
$289$	7941.11	1.61635
$290$	0	0
$291$	−1064.14	−0.214368
$292$	0	0
$293$	3312.98	0.660568	0.330284	−	0.943882i	$-0.392856\pi$
$293$	3312.98	0.660568	0.330284	+	0.943882i	$0.392856\pi$
$294$	0	0
$295$	155.130	0.0306170
$296$	0	0
$297$	−6205.98	−1.21248
$298$	0	0
$299$	6626.85	1.28174
$300$	0	0
$301$	−751.642	−0.143933
$302$	0	0
$303$	−821.859	−0.155824
$304$	0	0
$305$	−2474.35	−0.464527
$306$	0	0
$307$	117.629	0.0218679	0.0109339	−	0.999940i	$-0.496520\pi$
$307$	117.629	0.0218679	0.0109339	+	0.999940i	$0.496520\pi$
$308$	0	0
$309$	659.183	0.121358
$310$	0	0
$311$	−755.020	−0.137663	−0.0688316	−	0.997628i	$-0.521927\pi$
$311$	−755.020	−0.137663	−0.0688316	+	0.997628i	$0.521927\pi$
$312$	0	0
$313$	−1478.29	−0.266959	−0.133479	−	0.991052i	$-0.542615\pi$
$313$	−1478.29	−0.266959	−0.133479	+	0.991052i	$0.542615\pi$
$314$	0	0
$315$	−1877.00	−0.335737
$316$	0	0
$317$	8904.99	1.57777	0.788887	−	0.614538i	$-0.210658\pi$
$317$	8904.99	1.57777	0.788887	+	0.614538i	$0.210658\pi$
$318$	0	0
$319$	−2058.66	−0.361326
$320$	0	0
$321$	3022.80	0.525595
$322$	0	0
$323$	14832.5	2.55511
$324$	0	0
$325$	−3111.13	−0.530999
$326$	0	0
$327$	1085.31	0.183541
$328$	0	0
$329$	−5232.66	−0.876858
$330$	0	0
$331$	4878.38	0.810090	0.405045	−	0.914297i	$-0.367256\pi$
$331$	4878.38	0.810090	0.405045	+	0.914297i	$0.367256\pi$
$332$	0	0
$333$	6501.04	1.06984
$334$	0	0
$335$	4278.05	0.697716
$336$	0	0
$337$	2398.08	0.387631	0.193815	−	0.981038i	$-0.437914\pi$
$337$	2398.08	0.387631	0.193815	+	0.981038i	$0.437914\pi$
$338$	0	0
$339$	1228.86	0.196881
$340$	0	0
$341$	10838.5	1.72122
$342$	0	0
$343$	5621.71	0.884968
$344$	0	0
$345$	−1701.74	−0.265560
$346$	0	0
$347$	−6793.41	−1.05098	−0.525489	−	0.850801i	$-0.676117\pi$
$347$	−6793.41	−1.05098	−0.525489	+	0.850801i	$0.676117\pi$
$348$	0	0
$349$	−4272.74	−0.655342	−0.327671	−	0.944792i	$-0.606264\pi$
$349$	−4272.74	−0.655342	−0.327671	+	0.944792i	$0.606264\pi$
$350$	0	0
$351$	−4829.90	−0.734475
$352$	0	0
$353$	9012.18	1.35884	0.679419	−	0.733750i	$-0.262232\pi$
$353$	9012.18	1.35884	0.679419	+	0.733750i	$0.262232\pi$
$354$	0	0
$355$	4709.30	0.704067
$356$	0	0
$357$	1826.15	0.270729
$358$	0	0
$359$	4766.83	0.700790	0.350395	−	0.936602i	$-0.386047\pi$
$359$	4766.83	0.700790	0.350395	+	0.936602i	$0.386047\pi$
$360$	0	0
$361$	10256.3	1.49530
$362$	0	0
$363$	6348.07	0.917871
$364$	0	0
$365$	5477.14	0.785443
$366$	0	0
$367$	2367.41	0.336724	0.168362	−	0.985725i	$-0.446152\pi$
$367$	2367.41	0.336724	0.168362	+	0.985725i	$0.446152\pi$
$368$	0	0
$369$	−8859.49	−1.24988
$370$	0	0
$371$	2082.91	0.291481
$372$	0	0
$373$	−8820.15	−1.22437	−0.612185	−	0.790715i	$-0.709709\pi$
$373$	−8820.15	−1.22437	−0.612185	+	0.790715i	$0.709709\pi$
$374$	0	0
$375$	2572.33	0.354226
$376$	0	0
$377$	−1602.19	−0.218877
$378$	0	0
$379$	−1473.35	−0.199686	−0.0998430	−	0.995003i	$-0.531834\pi$
$379$	−1473.35	−0.199686	−0.0998430	+	0.995003i	$0.531834\pi$
$380$	0	0
$381$	1097.91	0.147632
$382$	0	0
$383$	4220.98	0.563138	0.281569	−	0.959541i	$-0.409145\pi$
$383$	4220.98	0.563138	0.281569	+	0.959541i	$0.409145\pi$
$384$	0	0
$385$	5535.82	0.732810
$386$	0	0
$387$	−1922.76	−0.252556
$388$	0	0
$389$	−13830.8	−1.80270	−0.901349	−	0.433094i	$-0.857422\pi$
$389$	−13830.8	−1.80270	−0.901349	+	0.433094i	$0.857422\pi$
$390$	0	0
$391$	−13599.2	−1.75893
$392$	0	0
$393$	108.098	0.0138748
$394$	0	0
$395$	−293.281	−0.0373584
$396$	0	0
$397$	−7744.90	−0.979107	−0.489554	−	0.871973i	$-0.662840\pi$
$397$	−7744.90	−0.979107	−0.489554	+	0.871973i	$0.662840\pi$
$398$	0	0
$399$	2107.21	0.264392
$400$	0	0
$401$	5490.40	0.683734	0.341867	−	0.939748i	$-0.388941\pi$
$401$	5490.40	0.683734	0.341867	+	0.939748i	$0.388941\pi$
$402$	0	0
$403$	8435.19	1.04265
$404$	0	0
$405$	−4145.79	−0.508657
$406$	0	0
$407$	−19173.5	−2.33512
$408$	0	0
$409$	−11356.6	−1.37297	−0.686487	−	0.727142i	$-0.740848\pi$
$409$	−11356.6	−1.37297	−0.686487	+	0.727142i	$0.740848\pi$
$410$	0	0
$411$	−2275.68	−0.273116
$412$	0	0
$413$	176.122	0.0209840
$414$	0	0
$415$	−692.485	−0.0819102
$416$	0	0
$417$	49.3439	0.00579467
$418$	0	0
$419$	−16478.5	−1.92131	−0.960653	−	0.277751i	$-0.910411\pi$
$419$	−16478.5	−1.92131	−0.960653	+	0.277751i	$0.910411\pi$
$420$	0	0
$421$	−2653.53	−0.307186	−0.153593	−	0.988134i	$-0.549084\pi$
$421$	−2653.53	−0.307186	−0.153593	+	0.988134i	$0.549084\pi$
$422$	0	0
$423$	−13385.6	−1.53860
$424$	0	0
$425$	6384.47	0.728688
$426$	0	0
$427$	−2809.16	−0.318372
$428$	0	0
$429$	6713.71	0.755573
$430$	0	0
$431$	−1761.29	−0.196841	−0.0984203	−	0.995145i	$-0.531379\pi$
$431$	−1761.29	−0.196841	−0.0984203	+	0.995145i	$0.531379\pi$
$432$	0	0
$433$	−10336.6	−1.14722	−0.573610	−	0.819129i	$-0.694457\pi$
$433$	−10336.6	−1.14722	−0.573610	+	0.819129i	$0.694457\pi$
$434$	0	0
$435$	411.431	0.0453486
$436$	0	0
$437$	−15692.2	−1.71776
$438$	0	0
$439$	13521.8	1.47007	0.735036	−	0.678028i	$-0.237165\pi$
$439$	13521.8	1.47007	0.735036	+	0.678028i	$0.237165\pi$
$440$	0	0
$441$	6124.90	0.661365
$442$	0	0
$443$	9509.20	1.01986	0.509928	−	0.860217i	$-0.329672\pi$
$443$	9509.20	1.01986	0.509928	+	0.860217i	$0.329672\pi$
$444$	0	0
$445$	−8515.23	−0.907103
$446$	0	0
$447$	1066.79	0.112880
$448$	0	0
$449$	−1571.72	−0.165198	−0.0825992	−	0.996583i	$-0.526322\pi$
$449$	−1571.72	−0.165198	−0.0825992	+	0.996583i	$0.526322\pi$
$450$	0	0
$451$	26129.2	2.72811
$452$	0	0
$453$	4667.72	0.484125
$454$	0	0
$455$	4308.34	0.443907
$456$	0	0
$457$	5101.82	0.522217	0.261108	−	0.965309i	$-0.415912\pi$
$457$	5101.82	0.522217	0.261108	+	0.965309i	$0.415912\pi$
$458$	0	0
$459$	9911.61	1.00792
$460$	0	0
$461$	−10183.2	−1.02881	−0.514404	−	0.857548i	$-0.671987\pi$
$461$	−10183.2	−1.02881	−0.514404	+	0.857548i	$0.671987\pi$
$462$	0	0
$463$	−17263.5	−1.73283	−0.866417	−	0.499322i	$-0.833582\pi$
$463$	−17263.5	−1.73283	−0.866417	+	0.499322i	$0.833582\pi$
$464$	0	0
$465$	−2166.11	−0.216023
$466$	0	0
$467$	−5511.73	−0.546151	−0.273075	−	0.961993i	$-0.588041\pi$
$467$	−5511.73	−0.546151	−0.273075	+	0.961993i	$0.588041\pi$
$468$	0	0
$469$	4856.94	0.478193
$470$	0	0
$471$	4411.06	0.431531
$472$	0	0
$473$	5670.78	0.551253
$474$	0	0
$475$	7367.09	0.711632
$476$	0	0
$477$	5328.26	0.511455
$478$	0	0
$479$	−17573.4	−1.67630	−0.838152	−	0.545437i	$-0.816364\pi$
$479$	−17573.4	−1.67630	−0.838152	+	0.545437i	$0.816364\pi$
$480$	0	0
$481$	−14922.0	−1.41453
$482$	0	0
$483$	−1932.01	−0.182007
$484$	0	0
$485$	5152.03	0.482353
$486$	0	0
$487$	−3865.76	−0.359700	−0.179850	−	0.983694i	$-0.557561\pi$
$487$	−3865.76	−0.359700	−0.179850	+	0.983694i	$0.557561\pi$
$488$	0	0
$489$	−3418.00	−0.316088
$490$	0	0
$491$	−12294.6	−1.13004	−0.565019	−	0.825078i	$-0.691131\pi$
$491$	−12294.6	−1.13004	−0.565019	+	0.825078i	$0.691131\pi$
$492$	0	0
$493$	3287.90	0.300365
$494$	0	0
$495$	14161.1	1.28584
$496$	0	0
$497$	5346.54	0.482545
$498$	0	0
$499$	8629.37	0.774155	0.387078	−	0.922047i	$-0.373485\pi$
$499$	8629.37	0.774155	0.387078	+	0.922047i	$0.373485\pi$
$500$	0	0
$501$	5073.37	0.452418
$502$	0	0
$503$	15188.2	1.34634	0.673168	−	0.739490i	$-0.264933\pi$
$503$	15188.2	1.34634	0.673168	+	0.739490i	$0.264933\pi$
$504$	0	0
$505$	3979.02	0.350622
$506$	0	0
$507$	1464.15	0.128255
$508$	0	0
$509$	7150.74	0.622694	0.311347	−	0.950296i	$-0.399220\pi$
$509$	7150.74	0.622694	0.311347	+	0.950296i	$0.399220\pi$
$510$	0	0
$511$	6218.28	0.538318
$512$	0	0
$513$	11437.1	0.984326
$514$	0	0
$515$	−3191.43	−0.273070
$516$	0	0
$517$	39478.0	3.35830
$518$	0	0
$519$	−3870.49	−0.327352
$520$	0	0
$521$	−655.463	−0.0551178	−0.0275589	−	0.999620i	$-0.508773\pi$
$521$	−655.463	−0.0551178	−0.0275589	+	0.999620i	$0.508773\pi$
$522$	0	0
$523$	10838.3	0.906168	0.453084	−	0.891468i	$-0.350324\pi$
$523$	10838.3	0.906168	0.453084	+	0.891468i	$0.350324\pi$
$524$	0	0
$525$	907.026	0.0754016
$526$	0	0
$527$	−17310.2	−1.43082
$528$	0	0
$529$	2220.50	0.182502
$530$	0	0
$531$	450.533	0.0368201
$532$	0	0
$533$	20335.4	1.65258
$534$	0	0
$535$	−14634.8	−1.18265
$536$	0	0
$537$	3331.45	0.267715
$538$	0	0
$539$	−18064.1	−1.44356
$540$	0	0
$541$	22416.8	1.78147	0.890733	−	0.454526i	$-0.150191\pi$
$541$	22416.8	1.78147	0.890733	+	0.454526i	$0.150191\pi$
$542$	0	0
$543$	5581.82	0.441139
$544$	0	0
$545$	−5254.52	−0.412989
$546$	0	0
$547$	−20452.2	−1.59867	−0.799334	−	0.600886i	$-0.794814\pi$
$547$	−20452.2	−1.59867	−0.799334	+	0.600886i	$0.794814\pi$
$548$	0	0
$549$	−7186.07	−0.558641
$550$	0	0
$551$	3793.94	0.293334
$552$	0	0
$553$	−332.966	−0.0256043
$554$	0	0
$555$	3831.89	0.293071
$556$	0	0
$557$	−16769.9	−1.27570	−0.637848	−	0.770162i	$-0.720175\pi$
$557$	−16769.9	−1.27570	−0.637848	+	0.770162i	$0.720175\pi$
$558$	0	0
$559$	4413.37	0.333928
$560$	0	0
$561$	−13777.4	−1.03687
$562$	0	0
$563$	1365.59	0.102225	0.0511125	−	0.998693i	$-0.483723\pi$
$563$	1365.59	0.102225	0.0511125	+	0.998693i	$0.483723\pi$
$564$	0	0
$565$	−5949.52	−0.443005
$566$	0	0
$567$	−4706.78	−0.348617
$568$	0	0
$569$	20141.2	1.48394	0.741970	−	0.670433i	$-0.233892\pi$
$569$	20141.2	1.48394	0.741970	+	0.670433i	$0.233892\pi$
$570$	0	0
$571$	−16801.5	−1.23139	−0.615693	−	0.787986i	$-0.711124\pi$
$571$	−16801.5	−1.23139	−0.615693	+	0.787986i	$0.711124\pi$
$572$	0	0
$573$	−5286.53	−0.385424
$574$	0	0
$575$	−6754.55	−0.489886
$576$	0	0
$577$	11696.0	0.843869	0.421935	−	0.906626i	$-0.361351\pi$
$577$	11696.0	0.843869	0.421935	+	0.906626i	$0.361351\pi$
$578$	0	0
$579$	7496.06	0.538041
$580$	0	0
$581$	−786.189	−0.0561387
$582$	0	0
$583$	−15714.6	−1.11635
$584$	0	0
$585$	11021.1	0.778915
$586$	0	0
$587$	5954.62	0.418694	0.209347	−	0.977841i	$-0.432866\pi$
$587$	5954.62	0.418694	0.209347	+	0.977841i	$0.432866\pi$
$588$	0	0
$589$	−19974.3	−1.39733
$590$	0	0
$591$	5389.88	0.375144
$592$	0	0
$593$	−15990.8	−1.10736	−0.553680	−	0.832729i	$-0.686777\pi$
$593$	−15990.8	−1.10736	−0.553680	+	0.832729i	$0.686777\pi$
$594$	0	0
$595$	−8841.29	−0.609173
$596$	0	0
$597$	3083.06	0.211359
$598$	0	0
$599$	12031.8	0.820710	0.410355	−	0.911926i	$-0.365405\pi$
$599$	12031.8	0.820710	0.410355	+	0.911926i	$0.365405\pi$
$600$	0	0
$601$	9585.65	0.650593	0.325297	−	0.945612i	$-0.394536\pi$
$601$	9585.65	0.650593	0.325297	+	0.945612i	$0.394536\pi$
$602$	0	0
$603$	12424.4	0.839075
$604$	0	0
$605$	−30734.1	−2.06532
$606$	0	0
$607$	355.950	0.0238016	0.0119008	−	0.999929i	$-0.496212\pi$
$607$	355.950	0.0238016	0.0119008	+	0.999929i	$0.496212\pi$
$608$	0	0
$609$	467.104	0.0310805
$610$	0	0
$611$	30724.3	2.03432
$612$	0	0
$613$	−4428.45	−0.291783	−0.145892	−	0.989301i	$-0.546605\pi$
$613$	−4428.45	−0.291783	−0.145892	+	0.989301i	$0.546605\pi$
$614$	0	0
$615$	−5222.02	−0.342394
$616$	0	0
$617$	−127.787	−0.00833794	−0.00416897	−	0.999991i	$-0.501327\pi$
$617$	−127.787	−0.00833794	−0.00416897	+	0.999991i	$0.501327\pi$
$618$	0	0
$619$	27816.0	1.80617	0.903086	−	0.429459i	$-0.141296\pi$
$619$	27816.0	1.80617	0.903086	+	0.429459i	$0.141296\pi$
$620$	0	0
$621$	−10486.1	−0.677608
$622$	0	0
$623$	−9667.47	−0.621700
$624$	0	0
$625$	−5414.86	−0.346551
$626$	0	0
$627$	−15897.9	−1.01260
$628$	0	0
$629$	30622.1	1.94115
$630$	0	0
$631$	−631.928	−0.0398679	−0.0199340	−	0.999801i	$-0.506346\pi$
$631$	−631.928	−0.0398679	−0.0199340	+	0.999801i	$0.506346\pi$
$632$	0	0
$633$	1351.86	0.0848840
$634$	0	0
$635$	−5315.52	−0.332189
$636$	0	0
$637$	−14058.7	−0.874450
$638$	0	0
$639$	13676.9	0.846712
$640$	0	0
$641$	−18761.8	−1.15608	−0.578040	−	0.816008i	$-0.696182\pi$
$641$	−18761.8	−1.15608	−0.578040	+	0.816008i	$0.696182\pi$
$642$	0	0
$643$	4443.65	0.272536	0.136268	−	0.990672i	$-0.456489\pi$
$643$	4443.65	0.272536	0.136268	+	0.990672i	$0.456489\pi$
$644$	0	0
$645$	−1133.33	−0.0691855
$646$	0	0
$647$	7906.38	0.480420	0.240210	−	0.970721i	$-0.422784\pi$
$647$	7906.38	0.480420	0.240210	+	0.970721i	$0.422784\pi$
$648$	0	0
$649$	−1328.75	−0.0803669
$650$	0	0
$651$	−2459.21	−0.148055
$652$	0	0
$653$	−25497.5	−1.52802	−0.764008	−	0.645207i	$-0.776771\pi$
$653$	−25497.5	−1.52802	−0.764008	+	0.645207i	$0.776771\pi$
$654$	0	0
$655$	−523.354	−0.0312201
$656$	0	0
$657$	15906.9	0.944575
$658$	0	0
$659$	−16034.5	−0.947826	−0.473913	−	0.880572i	$-0.657159\pi$
$659$	−16034.5	−0.947826	−0.473913	+	0.880572i	$0.657159\pi$
$660$	0	0
$661$	17019.8	1.00150	0.500752	−	0.865591i	$-0.333057\pi$
$661$	17019.8	1.00150	0.500752	+	0.865591i	$0.333057\pi$
$662$	0	0
$663$	−10722.5	−0.628096
$664$	0	0
$665$	−10202.0	−0.594914
$666$	0	0
$667$	−3478.49	−0.201930
$668$	0	0
$669$	−7338.83	−0.424119
$670$	0	0
$671$	21193.8	1.21934
$672$	0	0
$673$	−14092.3	−0.807161	−0.403580	−	0.914944i	$-0.632234\pi$
$673$	−14092.3	−0.807161	−0.403580	+	0.914944i	$0.632234\pi$
$674$	0	0
$675$	4922.97	0.280719
$676$	0	0
$677$	−26148.5	−1.48444	−0.742221	−	0.670155i	$-0.766228\pi$
$677$	−26148.5	−1.48444	−0.742221	+	0.670155i	$0.766228\pi$
$678$	0	0
$679$	5849.17	0.330590
$680$	0	0
$681$	−3408.69	−0.191808
$682$	0	0
$683$	11340.0	0.635303	0.317651	−	0.948208i	$-0.397106\pi$
$683$	11340.0	0.635303	0.317651	+	0.948208i	$0.397106\pi$
$684$	0	0
$685$	11017.7	0.614545
$686$	0	0
$687$	5733.63	0.318416
$688$	0	0
$689$	−12230.1	−0.676241
$690$	0	0
$691$	−9692.86	−0.533623	−0.266812	−	0.963749i	$-0.585970\pi$
$691$	−9692.86	−0.533623	−0.266812	+	0.963749i	$0.585970\pi$
$692$	0	0
$693$	16077.3	0.881278
$694$	0	0
$695$	−238.898	−0.0130387
$696$	0	0
$697$	−41731.1	−2.26783
$698$	0	0
$699$	10804.5	0.584638
$700$	0	0
$701$	10537.1	0.567731	0.283866	−	0.958864i	$-0.408383\pi$
$701$	10537.1	0.567731	0.283866	+	0.958864i	$0.408383\pi$
$702$	0	0
$703$	35335.0	1.89571
$704$	0	0
$705$	−7889.81	−0.421486
$706$	0	0
$707$	4517.44	0.240305
$708$	0	0
$709$	−34974.1	−1.85258	−0.926291	−	0.376809i	$-0.877021\pi$
$709$	−34974.1	−1.85258	−0.926291	+	0.376809i	$0.877021\pi$
$710$	0	0
$711$	−851.754	−0.0449272
$712$	0	0
$713$	18313.6	0.961919
$714$	0	0
$715$	−32504.3	−1.70013
$716$	0	0
$717$	−4027.76	−0.209790
$718$	0	0
$719$	31466.9	1.63215	0.816075	−	0.577946i	$-0.196146\pi$
$719$	31466.9	1.63215	0.816075	+	0.577946i	$0.196146\pi$
$720$	0	0
$721$	−3623.27	−0.187154
$722$	0	0
$723$	9807.98	0.504513
$724$	0	0
$725$	1633.06	0.0836556
$726$	0	0
$727$	18510.7	0.944325	0.472162	−	0.881512i	$-0.343474\pi$
$727$	18510.7	0.944325	0.472162	+	0.881512i	$0.343474\pi$
$728$	0	0
$729$	−7999.70	−0.406427
$730$	0	0
$731$	−9056.83	−0.458248
$732$	0	0
$733$	20105.0	1.01309	0.506546	−	0.862213i	$-0.330922\pi$
$733$	20105.0	1.01309	0.506546	+	0.862213i	$0.330922\pi$
$734$	0	0
$735$	3610.18	0.181175
$736$	0	0
$737$	−36643.3	−1.83144
$738$	0	0
$739$	−9123.58	−0.454149	−0.227075	−	0.973877i	$-0.572916\pi$
$739$	−9123.58	−0.454149	−0.227075	+	0.973877i	$0.572916\pi$
$740$	0	0
$741$	−12372.8	−0.613394
$742$	0	0
$743$	19900.9	0.982631	0.491316	−	0.870982i	$-0.336516\pi$
$743$	19900.9	0.982631	0.491316	+	0.870982i	$0.336516\pi$
$744$	0	0
$745$	−5164.86	−0.253994
$746$	0	0
$747$	−2011.13	−0.0985054
$748$	0	0
$749$	−16615.1	−0.810553
$750$	0	0
$751$	33574.3	1.63135	0.815674	−	0.578512i	$-0.196366\pi$
$751$	33574.3	1.63135	0.815674	+	0.578512i	$0.196366\pi$
$752$	0	0
$753$	−1215.63	−0.0588314
$754$	0	0
$755$	−22598.7	−1.08934
$756$	0	0
$757$	11310.4	0.543045	0.271523	−	0.962432i	$-0.412473\pi$
$757$	11310.4	0.543045	0.271523	+	0.962432i	$0.412473\pi$
$758$	0	0
$759$	14576.1	0.697072
$760$	0	0
$761$	−21252.7	−1.01236	−0.506182	−	0.862426i	$-0.668944\pi$
$761$	−21252.7	−1.01236	−0.506182	+	0.862426i	$0.668944\pi$
$762$	0	0
$763$	−5965.53	−0.283050
$764$	0	0
$765$	−22616.7	−1.06890
$766$	0	0
$767$	−1034.12	−0.0486831
$768$	0	0
$769$	3825.74	0.179401	0.0897006	−	0.995969i	$-0.471409\pi$
$769$	3825.74	0.179401	0.0897006	+	0.995969i	$0.471409\pi$
$770$	0	0
$771$	−5961.43	−0.278464
$772$	0	0
$773$	22840.1	1.06275	0.531373	−	0.847138i	$-0.321676\pi$
$773$	22840.1	1.06275	0.531373	+	0.847138i	$0.321676\pi$
$774$	0	0
$775$	−8597.74	−0.398503
$776$	0	0
$777$	4350.40	0.200862
$778$	0	0
$779$	−48153.8	−2.21475
$780$	0	0
$781$	−40337.1	−1.84811
$782$	0	0
$783$	2535.25	0.115712
$784$	0	0
$785$	−21356.1	−0.970996
$786$	0	0
$787$	19668.2	0.890845	0.445422	−	0.895321i	$-0.353054\pi$
$787$	19668.2	0.890845	0.445422	+	0.895321i	$0.353054\pi$
$788$	0	0
$789$	13895.0	0.626966
$790$	0	0
$791$	−6754.57	−0.303622
$792$	0	0
$793$	16494.4	0.738629
$794$	0	0
$795$	3140.62	0.140108
$796$	0	0
$797$	−38295.1	−1.70199	−0.850993	−	0.525176i	$-0.823999\pi$
$797$	−38295.1	−1.70199	−0.850993	+	0.525176i	$0.823999\pi$
$798$	0	0
$799$	−63050.5	−2.79170
$800$	0	0
$801$	−24730.2	−1.09088
$802$	0	0
$803$	−46914.0	−2.06172
$804$	0	0
$805$	9353.79	0.409537
$806$	0	0
$807$	187.263	0.00816848
$808$	0	0
$809$	425.248	0.0184808	0.00924038	−	0.999957i	$-0.497059\pi$
$809$	425.248	0.0184808	0.00924038	+	0.999957i	$0.497059\pi$
$810$	0	0
$811$	−13172.8	−0.570359	−0.285179	−	0.958474i	$-0.592053\pi$
$811$	−13172.8	−0.570359	−0.285179	+	0.958474i	$0.592053\pi$
$812$	0	0
$813$	−6.42066	−0.000276977 0
$814$	0	0
$815$	16548.2	0.711236
$816$	0	0
$817$	−10450.7	−0.447522
$818$	0	0
$819$	12512.4	0.533844
$820$	0	0
$821$	686.231	0.0291713	0.0145856	−	0.999894i	$-0.495357\pi$
$821$	686.231	0.0291713	0.0145856	+	0.999894i	$0.495357\pi$
$822$	0	0
$823$	−26137.8	−1.10706	−0.553528	−	0.832831i	$-0.686719\pi$
$823$	−26137.8	−1.10706	−0.553528	+	0.832831i	$0.686719\pi$
$824$	0	0
$825$	−6843.08	−0.288782
$826$	0	0
$827$	−11065.6	−0.465282	−0.232641	−	0.972563i	$-0.574737\pi$
$827$	−11065.6	−0.465282	−0.232641	+	0.972563i	$0.574737\pi$
$828$	0	0
$829$	−11377.1	−0.476649	−0.238325	−	0.971186i	$-0.576598\pi$
$829$	−11377.1	−0.476649	−0.238325	+	0.971186i	$0.576598\pi$
$830$	0	0
$831$	630.220	0.0263081
$832$	0	0
$833$	28850.3	1.20000
$834$	0	0
$835$	−24562.7	−1.01800
$836$	0	0
$837$	−13347.6	−0.551207
$838$	0	0
$839$	36828.9	1.51546	0.757732	−	0.652565i	$-0.226307\pi$
$839$	36828.9	1.51546	0.757732	+	0.652565i	$0.226307\pi$
$840$	0	0
$841$	841.000	0.0344828
$842$	0	0
$843$	12469.6	0.509461
$844$	0	0
$845$	−7088.69	−0.288590
$846$	0	0
$847$	−34892.9	−1.41551
$848$	0	0
$849$	−4702.47	−0.190092
$850$	0	0
$851$	−32397.1	−1.30500
$852$	0	0
$853$	21309.6	0.855364	0.427682	−	0.903929i	$-0.359330\pi$
$853$	21309.6	0.855364	0.427682	+	0.903929i	$0.359330\pi$
$854$	0	0
$855$	−26097.6	−1.04388
$856$	0	0
$857$	31171.7	1.24248	0.621241	−	0.783620i	$-0.286629\pi$
$857$	31171.7	1.24248	0.621241	+	0.783620i	$0.286629\pi$
$858$	0	0
$859$	36032.4	1.43121	0.715605	−	0.698505i	$-0.246151\pi$
$859$	36032.4	1.43121	0.715605	+	0.698505i	$0.246151\pi$
$860$	0	0
$861$	−5928.63	−0.234666
$862$	0	0
$863$	−10952.3	−0.432007	−0.216004	−	0.976393i	$-0.569302\pi$
$863$	−10952.3	−0.432007	−0.216004	+	0.976393i	$0.569302\pi$
$864$	0	0
$865$	18738.9	0.736581
$866$	0	0
$867$	13593.8	0.532492
$868$	0	0
$869$	2512.07	0.0980623
$870$	0	0
$871$	−28518.2	−1.10942
$872$	0	0
$873$	14962.6	0.580079
$874$	0	0
$875$	−14139.1	−0.546274
$876$	0	0
$877$	−11601.3	−0.446693	−0.223346	−	0.974739i	$-0.571698\pi$
$877$	−11601.3	−0.446693	−0.223346	+	0.974739i	$0.571698\pi$
$878$	0	0
$879$	5671.25	0.217618
$880$	0	0
$881$	−36417.2	−1.39265	−0.696326	−	0.717726i	$-0.745183\pi$
$881$	−36417.2	−1.39265	−0.696326	+	0.717726i	$0.745183\pi$
$882$	0	0
$883$	−42545.3	−1.62147	−0.810737	−	0.585410i	$-0.800933\pi$
$883$	−42545.3	−1.62147	−0.810737	+	0.585410i	$0.800933\pi$
$884$	0	0
$885$	265.556	0.0100865
$886$	0	0
$887$	−30330.6	−1.14814	−0.574071	−	0.818805i	$-0.694637\pi$
$887$	−30330.6	−1.14814	−0.574071	+	0.818805i	$0.694637\pi$
$888$	0	0
$889$	−6034.79	−0.227672
$890$	0	0
$891$	35510.4	1.33518
$892$	0	0
$893$	−72754.4	−2.72635
$894$	0	0
$895$	−16129.2	−0.602391
$896$	0	0
$897$	11344.0	0.422259
$898$	0	0
$899$	−4427.70	−0.164263
$900$	0	0
$901$	25097.9	0.928003
$902$	0	0
$903$	−1286.68	−0.0474176
$904$	0	0
$905$	−27024.3	−0.992617
$906$	0	0
$907$	27338.0	1.00082	0.500410	−	0.865789i	$-0.333183\pi$
$907$	27338.0	1.00082	0.500410	+	0.865789i	$0.333183\pi$
$908$	0	0
$909$	11556.0	0.421658
$910$	0	0
$911$	−43560.9	−1.58423	−0.792116	−	0.610370i	$-0.791021\pi$
$911$	−43560.9	−1.58423	−0.792116	+	0.610370i	$0.791021\pi$
$912$	0	0
$913$	5931.42	0.215007
$914$	0	0
$915$	−4235.66	−0.153034
$916$	0	0
$917$	−594.172	−0.0213973
$918$	0	0
$919$	45457.6	1.63167	0.815836	−	0.578283i	$-0.196277\pi$
$919$	45457.6	1.63167	0.815836	+	0.578283i	$0.196277\pi$
$920$	0	0
$921$	201.360	0.00720418
$922$	0	0
$923$	−31392.9	−1.11951
$924$	0	0
$925$	15209.6	0.540636
$926$	0	0
$927$	−9268.63	−0.328394
$928$	0	0
$929$	6823.03	0.240965	0.120482	−	0.992715i	$-0.461556\pi$
$929$	6823.03	0.240965	0.120482	+	0.992715i	$0.461556\pi$
$930$	0	0
$931$	33290.6	1.17192
$932$	0	0
$933$	−1292.46	−0.0453520
$934$	0	0
$935$	66703.4	2.33308
$936$	0	0
$937$	9758.66	0.340236	0.170118	−	0.985424i	$-0.445585\pi$
$937$	9758.66	0.340236	0.170118	+	0.985424i	$0.445585\pi$
$938$	0	0
$939$	−2530.58	−0.0879473
$940$	0	0
$941$	40689.7	1.40961	0.704806	−	0.709400i	$-0.251034\pi$
$941$	40689.7	1.40961	0.704806	+	0.709400i	$0.251034\pi$
$942$	0	0
$943$	44150.1	1.52463
$944$	0	0
$945$	−6817.38	−0.234677
$946$	0	0
$947$	11970.4	0.410757	0.205379	−	0.978683i	$-0.434157\pi$
$947$	11970.4	0.410757	0.205379	+	0.978683i	$0.434157\pi$
$948$	0	0
$949$	−36511.5	−1.24891
$950$	0	0
$951$	15243.8	0.519784
$952$	0	0
$953$	−19048.1	−0.647459	−0.323729	−	0.946150i	$-0.604937\pi$
$953$	−19048.1	−0.647459	−0.323729	+	0.946150i	$0.604937\pi$
$954$	0	0
$955$	25594.7	0.867250
$956$	0	0
$957$	−3524.08	−0.119036
$958$	0	0
$959$	12508.5	0.421190
$960$	0	0
$961$	−6480.03	−0.217516
$962$	0	0
$963$	−42502.9	−1.42226
$964$	0	0
$965$	−36292.1	−1.21066
$966$	0	0
$967$	27038.2	0.899162	0.449581	−	0.893240i	$-0.351573\pi$
$967$	27038.2	0.899162	0.449581	+	0.893240i	$0.351573\pi$
$968$	0	0
$969$	25390.6	0.841759
$970$	0	0
$971$	−9345.94	−0.308883	−0.154442	−	0.988002i	$-0.549358\pi$
$971$	−9345.94	−0.308883	−0.154442	+	0.988002i	$0.549358\pi$
$972$	0	0
$973$	−271.224	−0.00893633
$974$	0	0
$975$	−5325.73	−0.174933
$976$	0	0
$977$	4695.57	0.153761	0.0768806	−	0.997040i	$-0.475504\pi$
$977$	4695.57	0.153761	0.0768806	+	0.997040i	$0.475504\pi$
$978$	0	0
$979$	72936.5	2.38106
$980$	0	0
$981$	−15260.3	−0.496661
$982$	0	0
$983$	20018.2	0.649522	0.324761	−	0.945796i	$-0.394716\pi$
$983$	20018.2	0.649522	0.324761	+	0.945796i	$0.394716\pi$
$984$	0	0
$985$	−26095.1	−0.844119
$986$	0	0
$987$	−8957.42	−0.288873
$988$	0	0
$989$	9581.82	0.308073
$990$	0	0
$991$	−12166.3	−0.389984	−0.194992	−	0.980805i	$-0.562468\pi$
$991$	−12166.3	−0.389984	−0.194992	+	0.980805i	$0.562468\pi$
$992$	0	0
$993$	8350.94	0.266877
$994$	0	0
$995$	−14926.6	−0.475583
$996$	0	0
$997$	−33451.4	−1.06260	−0.531302	−	0.847183i	$-0.678297\pi$
$997$	−33451.4	−1.06260	−0.531302	+	0.847183i	$0.678297\pi$
$998$	0	0
$999$	23612.2	0.747805

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	1856.4.a.bj.1.8		12
4.3	odd	2		1856.4.a.bl.1.5		12
8.3	odd	2		928.4.a.h.1.8	✓	12
8.5	even	2		928.4.a.j.1.5	yes	12

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
928.4.a.h.1.8	✓	12	8.3	odd	2
928.4.a.j.1.5	yes	12	8.5	even	2
1856.4.a.bj.1.8		12	1.1	even	1	trivial
1856.4.a.bl.1.5		12	4.3	odd	2

Overview	Random
Universe	Knowledge

Classical	Maass
Hilbert	Bianchi

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

Level:	\( N \)	\(=\)	\( 1856 = 2^{6} \cdot 29 \)
Weight:	\( k \)	\(=\)	\( 4 \)
Character orbit:	\([\chi]\)	\(=\)	1856.a (trivial)

\(f(q)\)	\(=\)	\(q+1.71183 q^{3} -8.28780 q^{5} -9.40926 q^{7} -24.0696 q^{9} +O(q^{10})\)	\(q+1.71183 q^{3} -8.28780 q^{5} -9.40926 q^{7} -24.0696 q^{9} +70.9884 q^{11} +55.2478 q^{13} -14.1873 q^{15} -113.376 q^{17} -130.825 q^{19} -16.1070 q^{21} +119.948 q^{23} -56.3124 q^{25} -87.4225 q^{27} -29.0000 q^{29} +152.679 q^{31} +121.520 q^{33} +77.9821 q^{35} -270.093 q^{37} +94.5747 q^{39} +368.077 q^{41} +79.8832 q^{43} +199.484 q^{45} +556.119 q^{47} -254.466 q^{49} -194.080 q^{51} -221.368 q^{53} -588.338 q^{55} -223.951 q^{57} -18.7179 q^{59} +298.553 q^{61} +226.478 q^{63} -457.882 q^{65} -516.187 q^{67} +205.330 q^{69} -568.221 q^{71} -660.868 q^{73} -96.3971 q^{75} -667.949 q^{77} +35.3870 q^{79} +500.228 q^{81} +83.5548 q^{83} +939.637 q^{85} -49.6430 q^{87} +1027.44 q^{89} -519.841 q^{91} +261.361 q^{93} +1084.25 q^{95} -621.640 q^{97} -1708.67 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 12 q - 14 q^{3} + 10 q^{5} + 44 q^{7} + 130 q^{9}+O(q^{10}) \) 12 * q - 14 * q^3 + 10 * q^5 + 44 * q^7 + 130 * q^9	\( 12 q - 14 q^{3} + 10 q^{5} + 44 q^{7} + 130 q^{9} - 46 q^{11} + 34 q^{13} - 50 q^{15} + 36 q^{17} - 148 q^{19} + 92 q^{21} + 328 q^{23} + 486 q^{25} - 326 q^{27} - 348 q^{29} + 374 q^{31} + 710 q^{33} - 204 q^{35} + 340 q^{37} - 122 q^{39} + 32 q^{41} - 462 q^{43} + 1132 q^{45} + 434 q^{47} + 1508 q^{49} - 440 q^{51} - 610 q^{53} + 46 q^{55} - 932 q^{57} - 1240 q^{59} + 1228 q^{61} + 4240 q^{63} + 730 q^{65} - 1672 q^{67} + 528 q^{69} + 3220 q^{71} + 564 q^{73} - 6032 q^{75} - 644 q^{77} + 1862 q^{79} + 3040 q^{81} - 3736 q^{83} + 808 q^{85} + 406 q^{87} + 584 q^{89} - 4844 q^{91} + 3226 q^{93} + 2844 q^{95} + 904 q^{97} - 6832 q^{99}+O(q^{100}) \) 12 * q - 14 * q^3 + 10 * q^5 + 44 * q^7 + 130 * q^9 - 46 * q^11 + 34 * q^13 - 50 * q^15 + 36 * q^17 - 148 * q^19 + 92 * q^21 + 328 * q^23 + 486 * q^25 - 326 * q^27 - 348 * q^29 + 374 * q^31 + 710 * q^33 - 204 * q^35 + 340 * q^37 - 122 * q^39 + 32 * q^41 - 462 * q^43 + 1132 * q^45 + 434 * q^47 + 1508 * q^49 - 440 * q^51 - 610 * q^53 + 46 * q^55 - 932 * q^57 - 1240 * q^59 + 1228 * q^61 + 4240 * q^63 + 730 * q^65 - 1672 * q^67 + 528 * q^69 + 3220 * q^71 + 564 * q^73 - 6032 * q^75 - 644 * q^77 + 1862 * q^79 + 3040 * q^81 - 3736 * q^83 + 808 * q^85 + 406 * q^87 + 584 * q^89 - 4844 * q^91 + 3226 * q^93 + 2844 * q^95 + 904 * q^97 - 6832 * q^99

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