LMFDB - Embedded newform 1104.6.a.g.1.1

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [1104,6,Mod(1,1104)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(1104, base_ring=CyclotomicField(2))

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

N = Newforms(chi, 6, names="a")

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

chi := DirichletCharacter("1104.1");

S:= CuspForms(chi, 6);

N := Newforms(S);

Level:	$ N $	$=$	$ 1104 = 2^{4} \cdot 3 \cdot 23 $
Weight:	$ k $	$=$	$ 6 $
Character orbit:	$[\chi]$	$=$	1104.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:	$177.063737074$
Analytic rank:	$0$
Dimension:	$2$
Coefficient field:	$\Q(\sqrt{514}) $
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{2} - 514 $ x^2 - 514
Coefficient ring:	$\Z[a_1, \ldots, a_{7}]$
Coefficient ring index:	$ 1 $
Twist minimal:	no (minimal twist has level 138)
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.1
Root			$-22.6716$ of defining polynomial
Character	$\chi$	$=$	1104.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+9.00000 q^{3} -48.0147 q^{5} +163.358 q^{7} +81.0000 q^{9} +O(q^{10})$	$q+9.00000 q^{3} -48.0147 q^{5} +163.358 q^{7} +81.0000 q^{9} -261.373 q^{11} +860.118 q^{13} -432.132 q^{15} +676.015 q^{17} -1938.10 q^{19} +1470.22 q^{21} +529.000 q^{23} -819.588 q^{25} +729.000 q^{27} +4962.50 q^{29} +2870.23 q^{31} -2352.35 q^{33} -7843.58 q^{35} +6989.15 q^{37} +7741.06 q^{39} +1864.04 q^{41} -9177.94 q^{43} -3889.19 q^{45} -6600.34 q^{47} +9878.78 q^{49} +6084.13 q^{51} +26953.5 q^{53} +12549.7 q^{55} -17442.9 q^{57} +28436.4 q^{59} -1819.11 q^{61} +13232.0 q^{63} -41298.3 q^{65} -17814.8 q^{67} +4761.00 q^{69} +26660.0 q^{71} -41099.9 q^{73} -7376.29 q^{75} -42697.3 q^{77} -39544.2 q^{79} +6561.00 q^{81} -63245.7 q^{83} -32458.6 q^{85} +44662.5 q^{87} +138695. q^{89} +140507. q^{91} +25832.0 q^{93} +93057.4 q^{95} -173657. q^{97} -21171.2 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 2 q + 18 q^{3} + 40 q^{5} + 100 q^{7} + 162 q^{9}+O(q^{10}) $ 2 * q + 18 * q^3 + 40 * q^5 + 100 * q^7 + 162 * q^9	$ 2 q + 18 q^{3} + 40 q^{5} + 100 q^{7} + 162 q^{9} - 160 q^{11} + 632 q^{13} + 360 q^{15} + 1216 q^{17} - 2924 q^{19} + 900 q^{21} + 1058 q^{23} + 3802 q^{25} + 1458 q^{27} + 5300 q^{29} - 2512 q^{31} - 1440 q^{33} - 13420 q^{35} + 21868 q^{37} + 5688 q^{39} + 9532 q^{41} - 6612 q^{43} + 3240 q^{45} - 3860 q^{47} - 2914 q^{49} + 10944 q^{51} + 27744 q^{53} + 21472 q^{55} - 26316 q^{57} + 59140 q^{59} - 5724 q^{61} + 8100 q^{63} - 61376 q^{65} + 46124 q^{67} + 9522 q^{69} + 16320 q^{71} + 9756 q^{73} + 34218 q^{75} - 49120 q^{77} + 3028 q^{79} + 13122 q^{81} - 61560 q^{83} + 15068 q^{85} + 47700 q^{87} + 65592 q^{89} + 154960 q^{91} - 22608 q^{93} + 6284 q^{95} - 106724 q^{97} - 12960 q^{99}+O(q^{100}) $ 2 * q + 18 * q^3 + 40 * q^5 + 100 * q^7 + 162 * q^9 - 160 * q^11 + 632 * q^13 + 360 * q^15 + 1216 * q^17 - 2924 * q^19 + 900 * q^21 + 1058 * q^23 + 3802 * q^25 + 1458 * q^27 + 5300 * q^29 - 2512 * q^31 - 1440 * q^33 - 13420 * q^35 + 21868 * q^37 + 5688 * q^39 + 9532 * q^41 - 6612 * q^43 + 3240 * q^45 - 3860 * q^47 - 2914 * q^49 + 10944 * q^51 + 27744 * q^53 + 21472 * q^55 - 26316 * q^57 + 59140 * q^59 - 5724 * q^61 + 8100 * q^63 - 61376 * q^65 + 46124 * q^67 + 9522 * q^69 + 16320 * q^71 + 9756 * q^73 + 34218 * q^75 - 49120 * q^77 + 3028 * q^79 + 13122 * q^81 - 61560 * q^83 + 15068 * q^85 + 47700 * q^87 + 65592 * q^89 + 154960 * q^91 - 22608 * q^93 + 6284 * q^95 - 106724 * q^97 - 12960 * 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
$3$	9.00000	0.577350
$3$	9.00000	0.577350
$4$	0	0
$5$	−48.0147	−0.858913	−0.429457	−	0.903088i	$-0.641295\pi$
$5$	−48.0147	−0.858913	−0.429457	+	0.903088i	$0.641295\pi$
$6$	0	0
$7$	163.358	1.26007	0.630035	−	0.776566i	$-0.283040\pi$
$7$	163.358	1.26007	0.630035	+	0.776566i	$0.283040\pi$
$8$	0	0
$9$	81.0000	0.333333
$10$	0	0
$11$	−261.373	−0.651296	−0.325648	−	0.945491i	$-0.605582\pi$
$11$	−261.373	−0.651296	−0.325648	+	0.945491i	$0.605582\pi$
$12$	0	0
$13$	860.118	1.41156	0.705780	−	0.708431i	$-0.250597\pi$
$13$	860.118	1.41156	0.705780	+	0.708431i	$0.250597\pi$
$14$	0	0
$15$	−432.132	−0.495894
$16$	0	0
$17$	676.015	0.567328	0.283664	−	0.958924i	$-0.408450\pi$
$17$	676.015	0.567328	0.283664	+	0.958924i	$0.408450\pi$
$18$	0	0
$19$	−1938.10	−1.23167	−0.615833	−	0.787877i	$-0.711180\pi$
$19$	−1938.10	−1.23167	−0.615833	+	0.787877i	$0.711180\pi$
$20$	0	0
$21$	1470.22	0.727502
$22$	0	0
$23$	529.000	0.208514
$23$	529.000	0.208514
$24$	0	0
$25$	−819.588	−0.262268
$26$	0	0
$27$	729.000	0.192450
$28$	0	0
$29$	4962.50	1.09573	0.547867	−	0.836565i	$-0.315440\pi$
$29$	4962.50	1.09573	0.547867	+	0.836565i	$0.315440\pi$
$30$	0	0
$31$	2870.23	0.536428	0.268214	−	0.963359i	$-0.413567\pi$
$31$	2870.23	0.536428	0.268214	+	0.963359i	$0.413567\pi$
$32$	0	0
$33$	−2352.35	−0.376026
$34$	0	0
$35$	−7843.58	−1.08229
$36$	0	0
$37$	6989.15	0.839305	0.419653	−	0.907685i	$-0.362152\pi$
$37$	6989.15	0.839305	0.419653	+	0.907685i	$0.362152\pi$
$38$	0	0
$39$	7741.06	0.814965
$40$	0	0
$41$	1864.04	0.173179	0.0865895	−	0.996244i	$-0.472403\pi$
$41$	1864.04	0.173179	0.0865895	+	0.996244i	$0.472403\pi$
$42$	0	0
$43$	−9177.94	−0.756962	−0.378481	−	0.925609i	$-0.623553\pi$
$43$	−9177.94	−0.756962	−0.378481	+	0.925609i	$0.623553\pi$
$44$	0	0
$45$	−3889.19	−0.286304
$46$	0	0
$47$	−6600.34	−0.435835	−0.217917	−	0.975967i	$-0.569926\pi$
$47$	−6600.34	−0.435835	−0.217917	+	0.975967i	$0.569926\pi$
$48$	0	0
$49$	9878.78	0.587778
$50$	0	0
$51$	6084.13	0.327547
$52$	0	0
$53$	26953.5	1.31803	0.659015	−	0.752130i	$-0.270973\pi$
$53$	26953.5	1.31803	0.659015	+	0.752130i	$0.270973\pi$
$54$	0	0
$55$	12549.7	0.559406
$56$	0	0
$57$	−17442.9	−0.711103
$58$	0	0
$59$	28436.4	1.06352	0.531759	−	0.846896i	$-0.321531\pi$
$59$	28436.4	1.06352	0.531759	+	0.846896i	$0.321531\pi$
$60$	0	0
$61$	−1819.11	−0.0625942	−0.0312971	−	0.999510i	$-0.509964\pi$
$61$	−1819.11	−0.0625942	−0.0312971	+	0.999510i	$0.509964\pi$
$62$	0	0
$63$	13232.0	0.420024
$64$	0	0
$65$	−41298.3	−1.21241
$66$	0	0
$67$	−17814.8	−0.484836	−0.242418	−	0.970172i	$-0.577941\pi$
$67$	−17814.8	−0.484836	−0.242418	+	0.970172i	$0.577941\pi$
$68$	0	0
$69$	4761.00	0.120386
$70$	0	0
$71$	26660.0	0.627645	0.313823	−	0.949482i	$-0.398390\pi$
$71$	26660.0	0.627645	0.313823	+	0.949482i	$0.398390\pi$
$72$	0	0
$73$	−41099.9	−0.902680	−0.451340	−	0.892352i	$-0.649054\pi$
$73$	−41099.9	−0.902680	−0.451340	+	0.892352i	$0.649054\pi$
$74$	0	0
$75$	−7376.29	−0.151421
$76$	0	0
$77$	−42697.3	−0.820679
$78$	0	0
$79$	−39544.2	−0.712878	−0.356439	−	0.934319i	$-0.616009\pi$
$79$	−39544.2	−0.712878	−0.356439	+	0.934319i	$0.616009\pi$
$80$	0	0
$81$	6561.00	0.111111
$82$	0	0
$83$	−63245.7	−1.00771	−0.503855	−	0.863788i	$-0.668085\pi$
$83$	−63245.7	−1.00771	−0.503855	+	0.863788i	$0.668085\pi$
$84$	0	0
$85$	−32458.6	−0.487285
$86$	0	0
$87$	44662.5	0.632623
$88$	0	0
$89$	138695.	1.85603	0.928016	−	0.372540i	$-0.121513\pi$
$89$	138695.	1.85603	0.928016	+	0.372540i	$0.121513\pi$
$90$	0	0
$91$	140507.	1.77867
$92$	0	0
$93$	25832.0	0.309707
$94$	0	0
$95$	93057.4	1.05789
$96$	0	0
$97$	−173657.	−1.87398	−0.936988	−	0.349363i	$-0.886398\pi$
$97$	−173657.	−1.87398	−0.936988	+	0.349363i	$0.886398\pi$
$98$	0	0
$99$	−21171.2	−0.217099
$100$	0	0
$101$	54598.1	0.532567	0.266283	−	0.963895i	$-0.414204\pi$
$101$	54598.1	0.532567	0.266283	+	0.963895i	$0.414204\pi$
$102$	0	0
$103$	−104026.	−0.966158	−0.483079	−	0.875577i	$-0.660482\pi$
$103$	−104026.	−0.966158	−0.483079	+	0.875577i	$0.660482\pi$
$104$	0	0
$105$	−70592.2	−0.624861
$106$	0	0
$107$	24524.7	0.207083	0.103541	−	0.994625i	$-0.466983\pi$
$107$	24524.7	0.207083	0.103541	+	0.994625i	$0.466983\pi$
$108$	0	0
$109$	−122620.	−0.988545	−0.494273	−	0.869307i	$-0.664566\pi$
$109$	−122620.	−0.988545	−0.494273	+	0.869307i	$0.664566\pi$
$110$	0	0
$111$	62902.3	0.484573
$112$	0	0
$113$	116837.	0.860766	0.430383	−	0.902646i	$-0.358378\pi$
$113$	116837.	0.860766	0.430383	+	0.902646i	$0.358378\pi$
$114$	0	0
$115$	−25399.8	−0.179096
$116$	0	0
$117$	69669.5	0.470520
$118$	0	0
$119$	110432.	0.714873
$120$	0	0
$121$	−92735.4	−0.575814
$122$	0	0
$123$	16776.4	0.0999850
$124$	0	0
$125$	189398.	1.08418
$126$	0	0
$127$	92407.7	0.508392	0.254196	−	0.967153i	$-0.418189\pi$
$127$	92407.7	0.508392	0.254196	+	0.967153i	$0.418189\pi$
$128$	0	0
$129$	−82601.4	−0.437032
$130$	0	0
$131$	267815.	1.36350	0.681751	−	0.731584i	$-0.261219\pi$
$131$	267815.	1.36350	0.681751	+	0.731584i	$0.261219\pi$
$132$	0	0
$133$	−316604.	−1.55199
$134$	0	0
$135$	−35002.7	−0.165298
$136$	0	0
$137$	26189.3	0.119213	0.0596064	−	0.998222i	$-0.481015\pi$
$137$	26189.3	0.119213	0.0596064	+	0.998222i	$0.481015\pi$
$138$	0	0
$139$	313436.	1.37598	0.687990	−	0.725720i	$-0.258493\pi$
$139$	313436.	1.37598	0.687990	+	0.725720i	$0.258493\pi$
$140$	0	0
$141$	−59403.1	−0.251629
$142$	0	0
$143$	−224811.	−0.919343
$144$	0	0
$145$	−238273.	−0.941141
$146$	0	0
$147$	88909.1	0.339354
$148$	0	0
$149$	361347.	1.33339	0.666697	−	0.745329i	$-0.267708\pi$
$149$	361347.	1.33339	0.666697	+	0.745329i	$0.267708\pi$
$150$	0	0
$151$	−208786.	−0.745175	−0.372587	−	0.927997i	$-0.621529\pi$
$151$	−208786.	−0.745175	−0.372587	+	0.927997i	$0.621529\pi$
$152$	0	0
$153$	54757.2	0.189109
$154$	0	0
$155$	−137813.	−0.460745
$156$	0	0
$157$	381523.	1.23530	0.617649	−	0.786454i	$-0.288085\pi$
$157$	381523.	1.23530	0.617649	+	0.786454i	$0.288085\pi$
$158$	0	0
$159$	242581.	0.760965
$160$	0	0
$161$	86416.3	0.262743
$162$	0	0
$163$	−124532.	−0.367122	−0.183561	−	0.983008i	$-0.558763\pi$
$163$	−124532.	−0.367122	−0.183561	+	0.983008i	$0.558763\pi$
$164$	0	0
$165$	112948.	0.322973
$166$	0	0
$167$	−229735.	−0.637435	−0.318717	−	0.947850i	$-0.603252\pi$
$167$	−229735.	−0.637435	−0.318717	+	0.947850i	$0.603252\pi$
$168$	0	0
$169$	368509.	0.992503
$170$	0	0
$171$	−156986.	−0.410555
$172$	0	0
$173$	747402.	1.89862	0.949312	−	0.314335i	$-0.101782\pi$
$173$	747402.	1.89862	0.949312	+	0.314335i	$0.101782\pi$
$174$	0	0
$175$	−133886.	−0.330476
$176$	0	0
$177$	255928.	0.614023
$178$	0	0
$179$	−85323.4	−0.199038	−0.0995189	−	0.995036i	$-0.531730\pi$
$179$	−85323.4	−0.199038	−0.0995189	+	0.995036i	$0.531730\pi$
$180$	0	0
$181$	110195.	0.250014	0.125007	−	0.992156i	$-0.460105\pi$
$181$	110195.	0.250014	0.125007	+	0.992156i	$0.460105\pi$
$182$	0	0
$183$	−16372.0	−0.0361388
$184$	0	0
$185$	−335582.	−0.720890
$186$	0	0
$187$	−176692.	−0.369498
$188$	0	0
$189$	119088.	0.242501
$190$	0	0
$191$	−354991.	−0.704099	−0.352049	−	0.935981i	$-0.614515\pi$
$191$	−354991.	−0.704099	−0.352049	+	0.935981i	$0.614515\pi$
$192$	0	0
$193$	375860.	0.726327	0.363164	−	0.931725i	$-0.381697\pi$
$193$	375860.	0.726327	0.363164	+	0.931725i	$0.381697\pi$
$194$	0	0
$195$	−371685.	−0.699984
$196$	0	0
$197$	693249.	1.27269	0.636347	−	0.771403i	$-0.280445\pi$
$197$	693249.	1.27269	0.636347	+	0.771403i	$0.280445\pi$
$198$	0	0
$199$	958551.	1.71586	0.857931	−	0.513765i	$-0.171750\pi$
$199$	958551.	1.71586	0.857931	+	0.513765i	$0.171750\pi$
$200$	0	0
$201$	−160334.	−0.279920
$202$	0	0
$203$	810663.	1.38070
$204$	0	0
$205$	−89501.3	−0.148746
$206$	0	0
$207$	42849.0	0.0695048
$208$	0	0
$209$	506567.	0.802179
$210$	0	0
$211$	1.02066e6	1.57824	0.789121	−	0.614238i	$-0.210536\pi$
$211$	1.02066e6	1.57824	0.789121	+	0.614238i	$0.210536\pi$
$212$	0	0
$213$	239940.	0.362371
$214$	0	0
$215$	440676.	0.650164
$216$	0	0
$217$	468874.	0.675938
$218$	0	0
$219$	−369899.	−0.521163
$220$	0	0
$221$	581452.	0.800817
$222$	0	0
$223$	779297.	1.04940	0.524700	−	0.851287i	$-0.324177\pi$
$223$	779297.	1.04940	0.524700	+	0.851287i	$0.324177\pi$
$224$	0	0
$225$	−66386.6	−0.0874227
$226$	0	0
$227$	−601887.	−0.775265	−0.387633	−	0.921814i	$-0.626707\pi$
$227$	−601887.	−0.775265	−0.387633	+	0.921814i	$0.626707\pi$
$228$	0	0
$229$	−1.12190e6	−1.41372	−0.706861	−	0.707353i	$-0.749889\pi$
$229$	−1.12190e6	−1.41372	−0.706861	+	0.707353i	$0.749889\pi$
$230$	0	0
$231$	−384275.	−0.473819
$232$	0	0
$233$	728389.	0.878969	0.439485	−	0.898250i	$-0.355161\pi$
$233$	728389.	0.878969	0.439485	+	0.898250i	$0.355161\pi$
$234$	0	0
$235$	316914.	0.374344
$236$	0	0
$237$	−355898.	−0.411580
$238$	0	0
$239$	−1.18897e6	−1.34641	−0.673203	−	0.739457i	$-0.735082\pi$
$239$	−1.18897e6	−1.34641	−0.673203	+	0.739457i	$0.735082\pi$
$240$	0	0
$241$	1.22533e6	1.35898	0.679488	−	0.733687i	$-0.262202\pi$
$241$	1.22533e6	1.35898	0.679488	+	0.733687i	$0.262202\pi$
$242$	0	0
$243$	59049.0	0.0641500
$244$	0	0
$245$	−474327.	−0.504850
$246$	0	0
$247$	−1.66700e6	−1.73857
$248$	0	0
$249$	−569211.	−0.581802
$250$	0	0
$251$	1.62441e6	1.62746	0.813731	−	0.581241i	$-0.197433\pi$
$251$	1.62441e6	1.62746	0.813731	+	0.581241i	$0.197433\pi$
$252$	0	0
$253$	−138266.	−0.135805
$254$	0	0
$255$	−292128.	−0.281334
$256$	0	0
$257$	−1.69969e6	−1.60523	−0.802614	−	0.596499i	$-0.796558\pi$
$257$	−1.69969e6	−1.60523	−0.802614	+	0.596499i	$0.796558\pi$
$258$	0	0
$259$	1.14173e6	1.05758
$260$	0	0
$261$	401962.	0.365245
$262$	0	0
$263$	1.10279e6	0.983111	0.491555	−	0.870846i	$-0.336429\pi$
$263$	1.10279e6	0.983111	0.491555	+	0.870846i	$0.336429\pi$
$264$	0	0
$265$	−1.29416e6	−1.13207
$266$	0	0
$267$	1.24825e6	1.07158
$268$	0	0
$269$	573995.	0.483645	0.241823	−	0.970320i	$-0.422255\pi$
$269$	573995.	0.483645	0.241823	+	0.970320i	$0.422255\pi$
$270$	0	0
$271$	−1.69774e6	−1.40426	−0.702132	−	0.712047i	$-0.747768\pi$
$271$	−1.69774e6	−1.40426	−0.702132	+	0.712047i	$0.747768\pi$
$272$	0	0
$273$	1.26456e6	1.02691
$274$	0	0
$275$	214218.	0.170814
$276$	0	0
$277$	1.67908e6	1.31483	0.657417	−	0.753527i	$-0.271649\pi$
$277$	1.67908e6	1.31483	0.657417	+	0.753527i	$0.271649\pi$
$278$	0	0
$279$	232488.	0.178809
$280$	0	0
$281$	−301892.	−0.228079	−0.114040	−	0.993476i	$-0.536379\pi$
$281$	−301892.	−0.228079	−0.114040	+	0.993476i	$0.536379\pi$
$282$	0	0
$283$	675339.	0.501252	0.250626	−	0.968084i	$-0.419364\pi$
$283$	675339.	0.501252	0.250626	+	0.968084i	$0.419364\pi$
$284$	0	0
$285$	837517.	0.610775
$286$	0	0
$287$	304505.	0.218218
$288$	0	0
$289$	−962861.	−0.678140
$290$	0	0
$291$	−1.56292e6	−1.08194
$292$	0	0
$293$	−2.49740e6	−1.69949	−0.849745	−	0.527194i	$-0.823244\pi$
$293$	−2.49740e6	−1.69949	−0.849745	+	0.527194i	$0.823244\pi$
$294$	0	0
$295$	−1.36537e6	−0.913470
$296$	0	0
$297$	−190541.	−0.125342
$298$	0	0
$299$	455002.	0.294331
$300$	0	0
$301$	−1.49929e6	−0.953825
$302$	0	0
$303$	491383.	0.307477
$304$	0	0
$305$	87343.9	0.0537630
$306$	0	0
$307$	770733.	0.466722	0.233361	−	0.972390i	$-0.425028\pi$
$307$	770733.	0.466722	0.233361	+	0.972390i	$0.425028\pi$
$308$	0	0
$309$	−936233.	−0.557812
$310$	0	0
$311$	1.49099e6	0.874126	0.437063	−	0.899431i	$-0.356019\pi$
$311$	1.49099e6	0.874126	0.437063	+	0.899431i	$0.356019\pi$
$312$	0	0
$313$	−1.86294e6	−1.07483	−0.537414	−	0.843318i	$-0.680599\pi$
$313$	−1.86294e6	−1.07483	−0.537414	+	0.843318i	$0.680599\pi$
$314$	0	0
$315$	−635330.	−0.360764
$316$	0	0
$317$	510991.	0.285604	0.142802	−	0.989751i	$-0.454389\pi$
$317$	510991.	0.285604	0.142802	+	0.989751i	$0.454389\pi$
$318$	0	0
$319$	−1.29706e6	−0.713648
$320$	0	0
$321$	220722.	0.119559
$322$	0	0
$323$	−1.31019e6	−0.698758
$324$	0	0
$325$	−704942.	−0.370207
$326$	0	0
$327$	−1.10358e6	−0.570737
$328$	0	0
$329$	−1.07822e6	−0.549183
$330$	0	0
$331$	1.85686e6	0.931555	0.465778	−	0.884902i	$-0.345775\pi$
$331$	1.85686e6	0.931555	0.465778	+	0.884902i	$0.345775\pi$
$332$	0	0
$333$	566121.	0.279768
$334$	0	0
$335$	855374.	0.416432
$336$	0	0
$337$	3.99442e6	1.91593	0.957963	−	0.286892i	$-0.0926220\pi$
$337$	3.99442e6	1.91593	0.957963	+	0.286892i	$0.0926220\pi$
$338$	0	0
$339$	1.05154e6	0.496964
$340$	0	0
$341$	−750198.	−0.349374
$342$	0	0
$343$	−1.13178e6	−0.519429
$344$	0	0
$345$	−228598.	−0.103401
$346$	0	0
$347$	330363.	0.147288	0.0736441	−	0.997285i	$-0.476537\pi$
$347$	330363.	0.147288	0.0736441	+	0.997285i	$0.476537\pi$
$348$	0	0
$349$	−790332.	−0.347333	−0.173666	−	0.984805i	$-0.555561\pi$
$349$	−790332.	−0.347333	−0.173666	+	0.984805i	$0.555561\pi$
$350$	0	0
$351$	627026.	0.271655
$352$	0	0
$353$	−2.07394e6	−0.885847	−0.442923	−	0.896559i	$-0.646059\pi$
$353$	−2.07394e6	−0.885847	−0.442923	+	0.896559i	$0.646059\pi$
$354$	0	0
$355$	−1.28007e6	−0.539093
$356$	0	0
$357$	993891.	0.412732
$358$	0	0
$359$	−873196.	−0.357582	−0.178791	−	0.983887i	$-0.557219\pi$
$359$	−873196.	−0.357582	−0.178791	+	0.983887i	$0.557219\pi$
$360$	0	0
$361$	1.28014e6	0.517000
$362$	0	0
$363$	−834619.	−0.332446
$364$	0	0
$365$	1.97340e6	0.775324
$366$	0	0
$367$	3.24356e6	1.25706	0.628531	−	0.777785i	$-0.283657\pi$
$367$	3.24356e6	1.25706	0.628531	+	0.777785i	$0.283657\pi$
$368$	0	0
$369$	150987.	0.0577264
$370$	0	0
$371$	4.40306e6	1.66081
$372$	0	0
$373$	−530660.	−0.197489	−0.0987447	−	0.995113i	$-0.531483\pi$
$373$	−530660.	−0.197489	−0.0987447	+	0.995113i	$0.531483\pi$
$374$	0	0
$375$	1.70458e6	0.625951
$376$	0	0
$377$	4.26833e6	1.54670
$378$	0	0
$379$	875207.	0.312977	0.156489	−	0.987680i	$-0.449983\pi$
$379$	875207.	0.312977	0.156489	+	0.987680i	$0.449983\pi$
$380$	0	0
$381$	831670.	0.293520
$382$	0	0
$383$	3.71404e6	1.29375	0.646874	−	0.762596i	$-0.276076\pi$
$383$	3.71404e6	1.29375	0.646874	+	0.762596i	$0.276076\pi$
$384$	0	0
$385$	2.05010e6	0.704892
$386$	0	0
$387$	−743413.	−0.252321
$388$	0	0
$389$	632002.	0.211760	0.105880	−	0.994379i	$-0.466234\pi$
$389$	632002.	0.211760	0.105880	+	0.994379i	$0.466234\pi$
$390$	0	0
$391$	357612.	0.118296
$392$	0	0
$393$	2.41033e6	0.787218
$394$	0	0
$395$	1.89870e6	0.612300
$396$	0	0
$397$	2.29539e6	0.730939	0.365469	−	0.930823i	$-0.380909\pi$
$397$	2.29539e6	0.730939	0.365469	+	0.930823i	$0.380909\pi$
$398$	0	0
$399$	−2.84944e6	−0.896039
$400$	0	0
$401$	3.38917e6	1.05252	0.526262	−	0.850322i	$-0.323593\pi$
$401$	3.38917e6	1.05252	0.526262	+	0.850322i	$0.323593\pi$
$402$	0	0
$403$	2.46873e6	0.757201
$404$	0	0
$405$	−315024.	−0.0954348
$406$	0	0
$407$	−1.82677e6	−0.546636
$408$	0	0
$409$	6.56293e6	1.93995	0.969973	−	0.243211i	$-0.0782007\pi$
$409$	6.56293e6	1.93995	0.969973	+	0.243211i	$0.0782007\pi$
$410$	0	0
$411$	235704.	0.0688275
$412$	0	0
$413$	4.64531e6	1.34011
$414$	0	0
$415$	3.03672e6	0.865536
$416$	0	0
$417$	2.82093e6	0.794423
$418$	0	0
$419$	−715025.	−0.198969	−0.0994847	−	0.995039i	$-0.531719\pi$
$419$	−715025.	−0.198969	−0.0994847	+	0.995039i	$0.531719\pi$
$420$	0	0
$421$	−5.87847e6	−1.61644	−0.808219	−	0.588882i	$-0.799568\pi$
$421$	−5.87847e6	−1.61644	−0.808219	+	0.588882i	$0.799568\pi$
$422$	0	0
$423$	−534628.	−0.145278
$424$	0	0
$425$	−554054.	−0.148792
$426$	0	0
$427$	−297166.	−0.0788731
$428$	0	0
$429$	−2.02330e6	−0.530783
$430$	0	0
$431$	562006.	0.145730	0.0728648	−	0.997342i	$-0.476786\pi$
$431$	562006.	0.145730	0.0728648	+	0.997342i	$0.476786\pi$
$432$	0	0
$433$	−780601.	−0.200083	−0.100041	−	0.994983i	$-0.531897\pi$
$433$	−780601.	−0.200083	−0.100041	+	0.994983i	$0.531897\pi$
$434$	0	0
$435$	−2.14446e6	−0.543368
$436$	0	0
$437$	−1.02526e6	−0.256820
$438$	0	0
$439$	3.39573e6	0.840953	0.420477	−	0.907303i	$-0.361863\pi$
$439$	3.39573e6	0.840953	0.420477	+	0.907303i	$0.361863\pi$
$440$	0	0
$441$	800182.	0.195926
$442$	0	0
$443$	3.19295e6	0.773006	0.386503	−	0.922288i	$-0.373683\pi$
$443$	3.19295e6	0.773006	0.386503	+	0.922288i	$0.373683\pi$
$444$	0	0
$445$	−6.65939e6	−1.59417
$446$	0	0
$447$	3.25212e6	0.769835
$448$	0	0
$449$	82392.7	0.0192874	0.00964368	−	0.999953i	$-0.496930\pi$
$449$	82392.7	0.0192874	0.00964368	+	0.999953i	$0.496930\pi$
$450$	0	0
$451$	−487209.	−0.112791
$452$	0	0
$453$	−1.87907e6	−0.430227
$454$	0	0
$455$	−6.74640e6	−1.52772
$456$	0	0
$457$	4.93121e6	1.10449	0.552246	−	0.833681i	$-0.313771\pi$
$457$	4.93121e6	1.10449	0.552246	+	0.833681i	$0.313771\pi$
$458$	0	0
$459$	492815.	0.109182
$460$	0	0
$461$	494913.	0.108462	0.0542309	−	0.998528i	$-0.482729\pi$
$461$	494913.	0.108462	0.0542309	+	0.998528i	$0.482729\pi$
$462$	0	0
$463$	7.57909e6	1.64310	0.821551	−	0.570135i	$-0.193109\pi$
$463$	7.57909e6	1.64310	0.821551	+	0.570135i	$0.193109\pi$
$464$	0	0
$465$	−1.24032e6	−0.266011
$466$	0	0
$467$	−2.13302e6	−0.452588	−0.226294	−	0.974059i	$-0.572661\pi$
$467$	−2.13302e6	−0.452588	−0.226294	+	0.974059i	$0.572661\pi$
$468$	0	0
$469$	−2.91019e6	−0.610928
$470$	0	0
$471$	3.43371e6	0.713200
$472$	0	0
$473$	2.39886e6	0.493006
$474$	0	0
$475$	1.58845e6	0.323027
$476$	0	0
$477$	2.18323e6	0.439343
$478$	0	0
$479$	4.46780e6	0.889724	0.444862	−	0.895599i	$-0.353253\pi$
$479$	4.46780e6	0.889724	0.444862	+	0.895599i	$0.353253\pi$
$480$	0	0
$481$	6.01149e6	1.18473
$482$	0	0
$483$	777747.	0.151695
$484$	0	0
$485$	8.33811e6	1.60958
$486$	0	0
$487$	−2.49660e6	−0.477010	−0.238505	−	0.971141i	$-0.576657\pi$
$487$	−2.49660e6	−0.477010	−0.238505	+	0.971141i	$0.576657\pi$
$488$	0	0
$489$	−1.12078e6	−0.211958
$490$	0	0
$491$	−6.23986e6	−1.16808	−0.584038	−	0.811727i	$-0.698528\pi$
$491$	−6.23986e6	−1.16808	−0.584038	+	0.811727i	$0.698528\pi$
$492$	0	0
$493$	3.35472e6	0.621641
$494$	0	0
$495$	1.01653e6	0.186469
$496$	0	0
$497$	4.35512e6	0.790877
$498$	0	0
$499$	8.10512e6	1.45716	0.728582	−	0.684959i	$-0.240180\pi$
$499$	8.10512e6	1.45716	0.728582	+	0.684959i	$0.240180\pi$
$500$	0	0
$501$	−2.06761e6	−0.368023
$502$	0	0
$503$	569351.	0.100337	0.0501684	−	0.998741i	$-0.484024\pi$
$503$	569351.	0.100337	0.0501684	+	0.998741i	$0.484024\pi$
$504$	0	0
$505$	−2.62151e6	−0.457428
$506$	0	0
$507$	3.31658e6	0.573022
$508$	0	0
$509$	−1.05346e7	−1.80229	−0.901147	−	0.433514i	$-0.857274\pi$
$509$	−1.05346e7	−1.80229	−0.901147	+	0.433514i	$0.857274\pi$
$510$	0	0
$511$	−6.71400e6	−1.13744
$512$	0	0
$513$	−1.41288e6	−0.237034
$514$	0	0
$515$	4.99477e6	0.829846
$516$	0	0
$517$	1.72515e6	0.283857
$518$	0	0
$519$	6.72662e6	1.09617
$520$	0	0
$521$	−4.13736e6	−0.667772	−0.333886	−	0.942613i	$-0.608360\pi$
$521$	−4.13736e6	−0.667772	−0.333886	+	0.942613i	$0.608360\pi$
$522$	0	0
$523$	−2.56172e6	−0.409522	−0.204761	−	0.978812i	$-0.565642\pi$
$523$	−2.56172e6	−0.409522	−0.204761	+	0.978812i	$0.565642\pi$
$524$	0	0
$525$	−1.20498e6	−0.190801
$526$	0	0
$527$	1.94031e6	0.304331
$528$	0	0
$529$	279841.	0.0434783
$530$	0	0
$531$	2.30335e6	0.354506
$532$	0	0
$533$	1.60329e6	0.244453
$534$	0	0
$535$	−1.17754e6	−0.177866
$536$	0	0
$537$	−767911.	−0.114915
$538$	0	0
$539$	−2.58204e6	−0.382817
$540$	0	0
$541$	1.09529e7	1.60892	0.804459	−	0.594008i	$-0.202455\pi$
$541$	1.09529e7	1.60892	0.804459	+	0.594008i	$0.202455\pi$
$542$	0	0
$543$	991753.	0.144346
$544$	0	0
$545$	5.88758e6	0.849074
$546$	0	0
$547$	−8.55360e6	−1.22231	−0.611154	−	0.791512i	$-0.709294\pi$
$547$	−8.55360e6	−1.22231	−0.611154	+	0.791512i	$0.709294\pi$
$548$	0	0
$549$	−147348.	−0.0208647
$550$	0	0
$551$	−9.61784e6	−1.34958
$552$	0	0
$553$	−6.45986e6	−0.898277
$554$	0	0
$555$	−3.02024e6	−0.416206
$556$	0	0
$557$	−5.60095e6	−0.764933	−0.382467	−	0.923969i	$-0.624925\pi$
$557$	−5.60095e6	−0.764933	−0.382467	+	0.923969i	$0.624925\pi$
$558$	0	0
$559$	−7.89410e6	−1.06850
$560$	0	0
$561$	−1.59023e6	−0.213330
$562$	0	0
$563$	−9.43239e6	−1.25415	−0.627077	−	0.778957i	$-0.715749\pi$
$563$	−9.43239e6	−1.25415	−0.627077	+	0.778957i	$0.715749\pi$
$564$	0	0
$565$	−5.60991e6	−0.739323
$566$	0	0
$567$	1.07179e6	0.140008
$568$	0	0
$569$	618306.	0.0800614	0.0400307	−	0.999198i	$-0.487254\pi$
$569$	618306.	0.0800614	0.0400307	+	0.999198i	$0.487254\pi$
$570$	0	0
$571$	5.85911e6	0.752040	0.376020	−	0.926611i	$-0.377292\pi$
$571$	5.85911e6	0.752040	0.376020	+	0.926611i	$0.377292\pi$
$572$	0	0
$573$	−3.19492e6	−0.406512
$574$	0	0
$575$	−433562.	−0.0546867
$576$	0	0
$577$	1.41215e7	1.76580	0.882899	−	0.469563i	$-0.155589\pi$
$577$	1.41215e7	1.76580	0.882899	+	0.469563i	$0.155589\pi$
$578$	0	0
$579$	3.38274e6	0.419345
$580$	0	0
$581$	−1.03317e7	−1.26979
$582$	0	0
$583$	−7.04490e6	−0.858428
$584$	0	0
$585$	−3.34516e6	−0.404136
$586$	0	0
$587$	8.35876e6	1.00126	0.500629	−	0.865662i	$-0.333102\pi$
$587$	8.35876e6	1.00126	0.500629	+	0.865662i	$0.333102\pi$
$588$	0	0
$589$	−5.56279e6	−0.660700
$590$	0	0
$591$	6.23924e6	0.734790
$592$	0	0
$593$	−3.25922e6	−0.380607	−0.190303	−	0.981725i	$-0.560947\pi$
$593$	−3.25922e6	−0.380607	−0.190303	+	0.981725i	$0.560947\pi$
$594$	0	0
$595$	−5.30237e6	−0.614014
$596$	0	0
$597$	8.62696e6	0.990654
$598$	0	0
$599$	−1.37527e7	−1.56611	−0.783053	−	0.621955i	$-0.786339\pi$
$599$	−1.37527e7	−1.56611	−0.783053	+	0.621955i	$0.786339\pi$
$600$	0	0
$601$	−1.41017e7	−1.59252	−0.796261	−	0.604953i	$-0.793192\pi$
$601$	−1.41017e7	−1.59252	−0.796261	+	0.604953i	$0.793192\pi$
$602$	0	0
$603$	−1.44300e6	−0.161612
$604$	0	0
$605$	4.45266e6	0.494574
$606$	0	0
$607$	7.09889e6	0.782022	0.391011	−	0.920386i	$-0.372126\pi$
$607$	7.09889e6	0.782022	0.391011	+	0.920386i	$0.372126\pi$
$608$	0	0
$609$	7.29597e6	0.797149
$610$	0	0
$611$	−5.67707e6	−0.615207
$612$	0	0
$613$	3.85640e6	0.414506	0.207253	−	0.978287i	$-0.433548\pi$
$613$	3.85640e6	0.414506	0.207253	+	0.978287i	$0.433548\pi$
$614$	0	0
$615$	−805512.	−0.0858784
$616$	0	0
$617$	−1.48791e7	−1.57349	−0.786747	−	0.617276i	$-0.788236\pi$
$617$	−1.48791e7	−1.57349	−0.786747	+	0.617276i	$0.788236\pi$
$618$	0	0
$619$	−1.39271e7	−1.46094	−0.730471	−	0.682944i	$-0.760699\pi$
$619$	−1.39271e7	−1.46094	−0.730471	+	0.682944i	$0.760699\pi$
$620$	0	0
$621$	385641.	0.0401286
$622$	0	0
$623$	2.26569e7	2.33873
$624$	0	0
$625$	−6.53269e6	−0.668947
$626$	0	0
$627$	4.55910e6	0.463138
$628$	0	0
$629$	4.72477e6	0.476161
$630$	0	0
$631$	−1.02554e7	−1.02537	−0.512684	−	0.858577i	$-0.671349\pi$
$631$	−1.02554e7	−1.02537	−0.512684	+	0.858577i	$0.671349\pi$
$632$	0	0
$633$	9.18591e6	0.911199
$634$	0	0
$635$	−4.43693e6	−0.436665
$636$	0	0
$637$	8.49692e6	0.829684
$638$	0	0
$639$	2.15946e6	0.209215
$640$	0	0
$641$	−2.42650e6	−0.233258	−0.116629	−	0.993176i	$-0.537209\pi$
$641$	−2.42650e6	−0.233258	−0.116629	+	0.993176i	$0.537209\pi$
$642$	0	0
$643$	502897.	0.0479680	0.0239840	−	0.999712i	$-0.492365\pi$
$643$	502897.	0.0479680	0.0239840	+	0.999712i	$0.492365\pi$
$644$	0	0
$645$	3.96608e6	0.375373
$646$	0	0
$647$	4.43851e6	0.416847	0.208423	−	0.978039i	$-0.433167\pi$
$647$	4.43851e6	0.416847	0.208423	+	0.978039i	$0.433167\pi$
$648$	0	0
$649$	−7.43250e6	−0.692665
$650$	0	0
$651$	4.21986e6	0.390253
$652$	0	0
$653$	3.96100e6	0.363514	0.181757	−	0.983343i	$-0.441822\pi$
$653$	3.96100e6	0.363514	0.181757	+	0.983343i	$0.441822\pi$
$654$	0	0
$655$	−1.28590e7	−1.17113
$656$	0	0
$657$	−3.32910e6	−0.300893
$658$	0	0
$659$	−694464.	−0.0622925	−0.0311463	−	0.999515i	$-0.509916\pi$
$659$	−694464.	−0.0622925	−0.0311463	+	0.999515i	$0.509916\pi$
$660$	0	0
$661$	−1.61914e7	−1.44139	−0.720693	−	0.693254i	$-0.756176\pi$
$661$	−1.61914e7	−1.44139	−0.720693	+	0.693254i	$0.756176\pi$
$662$	0	0
$663$	5.23307e6	0.462352
$664$	0	0
$665$	1.52017e7	1.33302
$666$	0	0
$667$	2.62516e6	0.228477
$668$	0	0
$669$	7.01368e6	0.605871
$670$	0	0
$671$	475465.	0.0407673
$672$	0	0
$673$	−9.39067e6	−0.799207	−0.399603	−	0.916688i	$-0.630852\pi$
$673$	−9.39067e6	−0.799207	−0.399603	+	0.916688i	$0.630852\pi$
$674$	0	0
$675$	−597480.	−0.0504735
$676$	0	0
$677$	4.57480e6	0.383620	0.191810	−	0.981432i	$-0.438564\pi$
$677$	4.57480e6	0.383620	0.191810	+	0.981432i	$0.438564\pi$
$678$	0	0
$679$	−2.83683e7	−2.36134
$680$	0	0
$681$	−5.41698e6	−0.447600
$682$	0	0
$683$	1.41204e7	1.15823	0.579117	−	0.815244i	$-0.303397\pi$
$683$	1.41204e7	1.15823	0.579117	+	0.815244i	$0.303397\pi$
$684$	0	0
$685$	−1.25747e6	−0.102393
$686$	0	0
$687$	−1.00971e7	−0.816213
$688$	0	0
$689$	2.31832e7	1.86048
$690$	0	0
$691$	−2.06559e7	−1.64570	−0.822848	−	0.568262i	$-0.807616\pi$
$691$	−2.06559e7	−1.64570	−0.822848	+	0.568262i	$0.807616\pi$
$692$	0	0
$693$	−3.45848e6	−0.273560
$694$	0	0
$695$	−1.50495e7	−1.18185
$696$	0	0
$697$	1.26012e6	0.0982493
$698$	0	0
$699$	6.55550e6	0.507473
$700$	0	0
$701$	5.53145e6	0.425152	0.212576	−	0.977145i	$-0.431815\pi$
$701$	5.53145e6	0.425152	0.212576	+	0.977145i	$0.431815\pi$
$702$	0	0
$703$	−1.35457e7	−1.03374
$704$	0	0
$705$	2.85222e6	0.216128
$706$	0	0
$707$	8.91902e6	0.671072
$708$	0	0
$709$	−1.84065e6	−0.137517	−0.0687586	−	0.997633i	$-0.521904\pi$
$709$	−1.84065e6	−0.137517	−0.0687586	+	0.997633i	$0.521904\pi$
$710$	0	0
$711$	−3.20308e6	−0.237626
$712$	0	0
$713$	1.51835e6	0.111853
$714$	0	0
$715$	1.07942e7	0.789636
$716$	0	0
$717$	−1.07007e7	−0.777348
$718$	0	0
$719$	−9.26007e6	−0.668024	−0.334012	−	0.942569i	$-0.608403\pi$
$719$	−9.26007e6	−0.668024	−0.334012	+	0.942569i	$0.608403\pi$
$720$	0	0
$721$	−1.69934e7	−1.21743
$722$	0	0
$723$	1.10280e7	0.784605
$724$	0	0
$725$	−4.06721e6	−0.287376
$726$	0	0
$727$	−1.47751e7	−1.03680	−0.518401	−	0.855138i	$-0.673472\pi$
$727$	−1.47751e7	−1.03680	−0.518401	+	0.855138i	$0.673472\pi$
$728$	0	0
$729$	531441.	0.0370370
$730$	0	0
$731$	−6.20442e6	−0.429445
$732$	0	0
$733$	8.65422e6	0.594933	0.297467	−	0.954732i	$-0.403858\pi$
$733$	8.65422e6	0.594933	0.297467	+	0.954732i	$0.403858\pi$
$734$	0	0
$735$	−4.26894e6	−0.291475
$736$	0	0
$737$	4.65631e6	0.315772
$738$	0	0
$739$	1.56586e7	1.05473	0.527364	−	0.849639i	$-0.323180\pi$
$739$	1.56586e7	1.05473	0.527364	+	0.849639i	$0.323180\pi$
$740$	0	0
$741$	−1.50030e7	−1.00376
$742$	0	0
$743$	3.54958e6	0.235887	0.117944	−	0.993020i	$-0.462370\pi$
$743$	3.54958e6	0.235887	0.117944	+	0.993020i	$0.462370\pi$
$744$	0	0
$745$	−1.73500e7	−1.14527
$746$	0	0
$747$	−5.12290e6	−0.335903
$748$	0	0
$749$	4.00630e6	0.260939
$750$	0	0
$751$	−1.11896e7	−0.723961	−0.361981	−	0.932186i	$-0.617899\pi$
$751$	−1.11896e7	−0.723961	−0.361981	+	0.932186i	$0.617899\pi$
$752$	0	0
$753$	1.46197e7	0.939616
$754$	0	0
$755$	1.00248e7	0.640040
$756$	0	0
$757$	−1.16311e7	−0.737703	−0.368851	−	0.929488i	$-0.620249\pi$
$757$	−1.16311e7	−0.737703	−0.368851	+	0.929488i	$0.620249\pi$
$758$	0	0
$759$	−1.24439e6	−0.0784068
$760$	0	0
$761$	−3.13563e7	−1.96274	−0.981372	−	0.192119i	$-0.938464\pi$
$761$	−3.13563e7	−1.96274	−0.981372	+	0.192119i	$0.938464\pi$
$762$	0	0
$763$	−2.00310e7	−1.24564
$764$	0	0
$765$	−2.62915e6	−0.162428
$766$	0	0
$767$	2.44587e7	1.50122
$768$	0	0
$769$	−2.45626e7	−1.49782	−0.748908	−	0.662674i	$-0.769422\pi$
$769$	−2.45626e7	−1.49782	−0.748908	+	0.662674i	$0.769422\pi$
$770$	0	0
$771$	−1.52972e7	−0.926779
$772$	0	0
$773$	−2.54785e7	−1.53365	−0.766824	−	0.641857i	$-0.778164\pi$
$773$	−2.54785e7	−1.53365	−0.766824	+	0.641857i	$0.778164\pi$
$774$	0	0
$775$	−2.35240e6	−0.140688
$776$	0	0
$777$	1.02756e7	0.610596
$778$	0	0
$779$	−3.61270e6	−0.213299
$780$	0	0
$781$	−6.96819e6	−0.408783
$782$	0	0
$783$	3.61766e6	0.210874
$784$	0	0
$785$	−1.83187e7	−1.06101
$786$	0	0
$787$	7.96753e6	0.458550	0.229275	−	0.973362i	$-0.426364\pi$
$787$	7.96753e6	0.458550	0.229275	+	0.973362i	$0.426364\pi$
$788$	0	0
$789$	9.92509e6	0.567599
$790$	0	0
$791$	1.90863e7	1.08463
$792$	0	0
$793$	−1.56465e6	−0.0883554
$794$	0	0
$795$	−1.16475e7	−0.653603
$796$	0	0
$797$	−2.29664e6	−0.128070	−0.0640350	−	0.997948i	$-0.520397\pi$
$797$	−2.29664e6	−0.128070	−0.0640350	+	0.997948i	$0.520397\pi$
$798$	0	0
$799$	−4.46193e6	−0.247261
$800$	0	0
$801$	1.12343e7	0.618677
$802$	0	0
$803$	1.07424e7	0.587912
$804$	0	0
$805$	−4.14925e6	−0.225673
$806$	0	0
$807$	5.16595e6	0.279233
$808$	0	0
$809$	−1.50093e7	−0.806287	−0.403144	−	0.915137i	$-0.632082\pi$
$809$	−1.50093e7	−0.806287	−0.403144	+	0.915137i	$0.632082\pi$
$810$	0	0
$811$	2.24636e7	1.19930	0.599650	−	0.800263i	$-0.295307\pi$
$811$	2.24636e7	1.19930	0.599650	+	0.800263i	$0.295307\pi$
$812$	0	0
$813$	−1.52797e7	−0.810752
$814$	0	0
$815$	5.97935e6	0.315326
$816$	0	0
$817$	1.77878e7	0.932324
$818$	0	0
$819$	1.13811e7	0.592889
$820$	0	0
$821$	1.21792e7	0.630612	0.315306	−	0.948990i	$-0.397893\pi$
$821$	1.21792e7	0.630612	0.315306	+	0.948990i	$0.397893\pi$
$822$	0	0
$823$	3.17358e7	1.63324	0.816620	−	0.577176i	$-0.195845\pi$
$823$	3.17358e7	1.63324	0.816620	+	0.577176i	$0.195845\pi$
$824$	0	0
$825$	1.92796e6	0.0986196
$826$	0	0
$827$	−1.62922e7	−0.828353	−0.414176	−	0.910197i	$-0.635930\pi$
$827$	−1.62922e7	−0.828353	−0.414176	+	0.910197i	$0.635930\pi$
$828$	0	0
$829$	−2.29328e7	−1.15896	−0.579482	−	0.814985i	$-0.696745\pi$
$829$	−2.29328e7	−1.15896	−0.579482	+	0.814985i	$0.696745\pi$
$830$	0	0
$831$	1.51117e7	0.759120
$832$	0	0
$833$	6.67820e6	0.333463
$834$	0	0
$835$	1.10307e7	0.547501
$836$	0	0
$837$	2.09239e6	0.103236
$838$	0	0
$839$	1.29476e7	0.635014	0.317507	−	0.948256i	$-0.397154\pi$
$839$	1.29476e7	0.635014	0.317507	+	0.948256i	$0.397154\pi$
$840$	0	0
$841$	4.11526e6	0.200635
$842$	0	0
$843$	−2.71703e6	−0.131682
$844$	0	0
$845$	−1.76939e7	−0.852474
$846$	0	0
$847$	−1.51491e7	−0.725566
$848$	0	0
$849$	6.07805e6	0.289398
$850$	0	0
$851$	3.69726e6	0.175007
$852$	0	0
$853$	1.07870e7	0.507606	0.253803	−	0.967256i	$-0.418319\pi$
$853$	1.07870e7	0.507606	0.253803	+	0.967256i	$0.418319\pi$
$854$	0	0
$855$	7.53765e6	0.352631
$856$	0	0
$857$	−3.72942e7	−1.73456	−0.867281	−	0.497819i	$-0.834134\pi$
$857$	−3.72942e7	−1.73456	−0.867281	+	0.497819i	$0.834134\pi$
$858$	0	0
$859$	2.67499e7	1.23691	0.618456	−	0.785819i	$-0.287759\pi$
$859$	2.67499e7	1.23691	0.618456	+	0.785819i	$0.287759\pi$
$860$	0	0
$861$	2.74055e6	0.125988
$862$	0	0
$863$	−1.69440e7	−0.774441	−0.387221	−	0.921987i	$-0.626565\pi$
$863$	−1.69440e7	−0.774441	−0.387221	+	0.921987i	$0.626565\pi$
$864$	0	0
$865$	−3.58863e7	−1.63075
$866$	0	0
$867$	−8.66575e6	−0.391524
$868$	0	0
$869$	1.03358e7	0.464294
$870$	0	0
$871$	−1.53229e7	−0.684375
$872$	0	0
$873$	−1.40662e7	−0.624658
$874$	0	0
$875$	3.09397e7	1.36614
$876$	0	0
$877$	1.29556e6	0.0568800	0.0284400	−	0.999596i	$-0.490946\pi$
$877$	1.29556e6	0.0568800	0.0284400	+	0.999596i	$0.490946\pi$
$878$	0	0
$879$	−2.24766e7	−0.981201
$880$	0	0
$881$	1.10103e7	0.477923	0.238961	−	0.971029i	$-0.423193\pi$
$881$	1.10103e7	0.477923	0.238961	+	0.971029i	$0.423193\pi$
$882$	0	0
$883$	−1.17572e7	−0.507462	−0.253731	−	0.967275i	$-0.581658\pi$
$883$	−1.17572e7	−0.507462	−0.253731	+	0.967275i	$0.581658\pi$
$884$	0	0
$885$	−1.22883e7	−0.527392
$886$	0	0
$887$	−2.03967e7	−0.870464	−0.435232	−	0.900318i	$-0.643334\pi$
$887$	−2.03967e7	−0.870464	−0.435232	+	0.900318i	$0.643334\pi$
$888$	0	0
$889$	1.50955e7	0.640610
$890$	0	0
$891$	−1.71487e6	−0.0723662
$892$	0	0
$893$	1.27921e7	0.536803
$894$	0	0
$895$	4.09678e6	0.170956
$896$	0	0
$897$	4.09502e6	0.169932
$898$	0	0
$899$	1.42435e7	0.587783
$900$	0	0
$901$	1.82210e7	0.747755
$902$	0	0
$903$	−1.34936e7	−0.550691
$904$	0	0
$905$	−5.29097e6	−0.214740
$906$	0	0
$907$	−2.22154e7	−0.896676	−0.448338	−	0.893864i	$-0.647984\pi$
$907$	−2.22154e7	−0.896676	−0.448338	+	0.893864i	$0.647984\pi$
$908$	0	0
$909$	4.42244e6	0.177522
$910$	0	0
$911$	3.42737e7	1.36825	0.684124	−	0.729366i	$-0.260185\pi$
$911$	3.42737e7	1.36825	0.684124	+	0.729366i	$0.260185\pi$
$912$	0	0
$913$	1.65307e7	0.656317
$914$	0	0
$915$	786095.	0.0310401
$916$	0	0
$917$	4.37496e7	1.71811
$918$	0	0
$919$	−3.31398e7	−1.29438	−0.647188	−	0.762330i	$-0.724055\pi$
$919$	−3.31398e7	−1.29438	−0.647188	+	0.762330i	$0.724055\pi$
$920$	0	0
$921$	6.93660e6	0.269462
$922$	0	0
$923$	2.29307e7	0.885959
$924$	0	0
$925$	−5.72822e6	−0.220123
$926$	0	0
$927$	−8.42610e6	−0.322053
$928$	0	0
$929$	−2.33011e6	−0.0885802	−0.0442901	−	0.999019i	$-0.514103\pi$
$929$	−2.33011e6	−0.0885802	−0.0442901	+	0.999019i	$0.514103\pi$
$930$	0	0
$931$	−1.91461e7	−0.723946
$932$	0	0
$933$	1.34189e7	0.504677
$934$	0	0
$935$	8.48380e6	0.317367
$936$	0	0
$937$	4.69146e7	1.74566	0.872829	−	0.488027i	$-0.162283\pi$
$937$	4.69146e7	1.74566	0.872829	+	0.488027i	$0.162283\pi$
$938$	0	0
$939$	−1.67665e7	−0.620552
$940$	0	0
$941$	2.83548e7	1.04389	0.521943	−	0.852980i	$-0.325207\pi$
$941$	2.83548e7	1.04389	0.521943	+	0.852980i	$0.325207\pi$
$942$	0	0
$943$	986077.	0.0361103
$944$	0	0
$945$	−5.71797e6	−0.208287
$946$	0	0
$947$	−5.07751e7	−1.83982	−0.919912	−	0.392125i	$-0.871740\pi$
$947$	−5.07751e7	−1.83982	−0.919912	+	0.392125i	$0.871740\pi$
$948$	0	0
$949$	−3.53508e7	−1.27419
$950$	0	0
$951$	4.59892e6	0.164894
$952$	0	0
$953$	2.16062e7	0.770631	0.385315	−	0.922785i	$-0.374093\pi$
$953$	2.16062e7	0.770631	0.385315	+	0.922785i	$0.374093\pi$
$954$	0	0
$955$	1.70448e7	0.604760
$956$	0	0
$957$	−1.16736e7	−0.412025
$958$	0	0
$959$	4.27823e6	0.150217
$960$	0	0
$961$	−2.03910e7	−0.712245
$962$	0	0
$963$	1.98650e6	0.0690275
$964$	0	0
$965$	−1.80468e7	−0.623852
$966$	0	0
$967$	−5.34811e7	−1.83922	−0.919611	−	0.392831i	$-0.871496\pi$
$967$	−5.34811e7	−1.83922	−0.919611	+	0.392831i	$0.871496\pi$
$968$	0	0
$969$	−1.17917e7	−0.403428
$970$	0	0
$971$	−2.76540e7	−0.941262	−0.470631	−	0.882330i	$-0.655974\pi$
$971$	−2.76540e7	−0.941262	−0.470631	+	0.882330i	$0.655974\pi$
$972$	0	0
$973$	5.12023e7	1.73383
$974$	0	0
$975$	−6.34448e6	−0.213739
$976$	0	0
$977$	−1.12752e7	−0.377910	−0.188955	−	0.981986i	$-0.560510\pi$
$977$	−1.12752e7	−0.377910	−0.188955	+	0.981986i	$0.560510\pi$
$978$	0	0
$979$	−3.62510e7	−1.20883
$980$	0	0
$981$	−9.93225e6	−0.329515
$982$	0	0
$983$	−2.00662e7	−0.662341	−0.331170	−	0.943571i	$-0.607443\pi$
$983$	−2.00662e7	−0.662341	−0.331170	+	0.943571i	$0.607443\pi$
$984$	0	0
$985$	−3.32861e7	−1.09313
$986$	0	0
$987$	−9.70396e6	−0.317071
$988$	0	0
$989$	−4.85513e6	−0.157837
$990$	0	0
$991$	−1.35753e7	−0.439100	−0.219550	−	0.975601i	$-0.570459\pi$
$991$	−1.35753e7	−0.439100	−0.219550	+	0.975601i	$0.570459\pi$
$992$	0	0
$993$	1.67117e7	0.537834
$994$	0	0
$995$	−4.60245e7	−1.47378
$996$	0	0
$997$	−2.20333e7	−0.702006	−0.351003	−	0.936374i	$-0.614159\pi$
$997$	−2.20333e7	−0.702006	−0.351003	+	0.936374i	$0.614159\pi$
$998$	0	0
$999$	5.09509e6	0.161524

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	1104.6.a.g.1.1		2
4.3	odd	2		138.6.a.f.1.1	✓	2
12.11	even	2		414.6.a.e.1.2		2

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
138.6.a.f.1.1	✓	2	4.3	odd	2
414.6.a.e.1.2		2	12.11	even	2
1104.6.a.g.1.1		2	1.1	even	1	trivial

Overview	Random
Universe	Knowledge

Classical	Maass
Hilbert	Bianchi

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

Level:	\( N \)	\(=\)	\( 1104 = 2^{4} \cdot 3 \cdot 23 \)
Weight:	\( k \)	\(=\)	\( 6 \)
Character orbit:	\([\chi]\)	\(=\)	1104.a (trivial)

\(f(q)\)	\(=\)	\(q+9.00000 q^{3} -48.0147 q^{5} +163.358 q^{7} +81.0000 q^{9} +O(q^{10})\)	\(q+9.00000 q^{3} -48.0147 q^{5} +163.358 q^{7} +81.0000 q^{9} -261.373 q^{11} +860.118 q^{13} -432.132 q^{15} +676.015 q^{17} -1938.10 q^{19} +1470.22 q^{21} +529.000 q^{23} -819.588 q^{25} +729.000 q^{27} +4962.50 q^{29} +2870.23 q^{31} -2352.35 q^{33} -7843.58 q^{35} +6989.15 q^{37} +7741.06 q^{39} +1864.04 q^{41} -9177.94 q^{43} -3889.19 q^{45} -6600.34 q^{47} +9878.78 q^{49} +6084.13 q^{51} +26953.5 q^{53} +12549.7 q^{55} -17442.9 q^{57} +28436.4 q^{59} -1819.11 q^{61} +13232.0 q^{63} -41298.3 q^{65} -17814.8 q^{67} +4761.00 q^{69} +26660.0 q^{71} -41099.9 q^{73} -7376.29 q^{75} -42697.3 q^{77} -39544.2 q^{79} +6561.00 q^{81} -63245.7 q^{83} -32458.6 q^{85} +44662.5 q^{87} +138695. q^{89} +140507. q^{91} +25832.0 q^{93} +93057.4 q^{95} -173657. q^{97} -21171.2 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 2 q + 18 q^{3} + 40 q^{5} + 100 q^{7} + 162 q^{9}+O(q^{10}) \) 2 * q + 18 * q^3 + 40 * q^5 + 100 * q^7 + 162 * q^9	\( 2 q + 18 q^{3} + 40 q^{5} + 100 q^{7} + 162 q^{9} - 160 q^{11} + 632 q^{13} + 360 q^{15} + 1216 q^{17} - 2924 q^{19} + 900 q^{21} + 1058 q^{23} + 3802 q^{25} + 1458 q^{27} + 5300 q^{29} - 2512 q^{31} - 1440 q^{33} - 13420 q^{35} + 21868 q^{37} + 5688 q^{39} + 9532 q^{41} - 6612 q^{43} + 3240 q^{45} - 3860 q^{47} - 2914 q^{49} + 10944 q^{51} + 27744 q^{53} + 21472 q^{55} - 26316 q^{57} + 59140 q^{59} - 5724 q^{61} + 8100 q^{63} - 61376 q^{65} + 46124 q^{67} + 9522 q^{69} + 16320 q^{71} + 9756 q^{73} + 34218 q^{75} - 49120 q^{77} + 3028 q^{79} + 13122 q^{81} - 61560 q^{83} + 15068 q^{85} + 47700 q^{87} + 65592 q^{89} + 154960 q^{91} - 22608 q^{93} + 6284 q^{95} - 106724 q^{97} - 12960 q^{99}+O(q^{100}) \) 2 * q + 18 * q^3 + 40 * q^5 + 100 * q^7 + 162 * q^9 - 160 * q^11 + 632 * q^13 + 360 * q^15 + 1216 * q^17 - 2924 * q^19 + 900 * q^21 + 1058 * q^23 + 3802 * q^25 + 1458 * q^27 + 5300 * q^29 - 2512 * q^31 - 1440 * q^33 - 13420 * q^35 + 21868 * q^37 + 5688 * q^39 + 9532 * q^41 - 6612 * q^43 + 3240 * q^45 - 3860 * q^47 - 2914 * q^49 + 10944 * q^51 + 27744 * q^53 + 21472 * q^55 - 26316 * q^57 + 59140 * q^59 - 5724 * q^61 + 8100 * q^63 - 61376 * q^65 + 46124 * q^67 + 9522 * q^69 + 16320 * q^71 + 9756 * q^73 + 34218 * q^75 - 49120 * q^77 + 3028 * q^79 + 13122 * q^81 - 61560 * q^83 + 15068 * q^85 + 47700 * q^87 + 65592 * q^89 + 154960 * q^91 - 22608 * q^93 + 6284 * q^95 - 106724 * q^97 - 12960 * q^99

Introduction

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