LMFDB - Embedded newform 1104.6.a.p.1.4

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:	$1$
Dimension:	$4$
Coefficient field:	$\mathbb{Q}[x]/(x^{4} - \cdots)$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{4} - x^{3} - 480x^{2} + 3169x + 6509 $ x^4 - x^3 - 480x^2 + 3169x + 6509
Coefficient ring:	$\Z[a_1, \ldots, a_{11}]$
Coefficient ring index:	$ 2^{4} $
Twist minimal:	no (minimal twist has level 276)
Fricke sign:	$1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.4
Root			$-24.0138$ 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)

$f(q)$	$=$	$q+9.00000 q^{3} +104.299 q^{5} +5.92670 q^{7} +81.0000 q^{9} +O(q^{10})$	$q+9.00000 q^{3} +104.299 q^{5} +5.92670 q^{7} +81.0000 q^{9} +174.124 q^{11} -1122.61 q^{13} +938.687 q^{15} -1661.31 q^{17} -559.161 q^{19} +53.3403 q^{21} -529.000 q^{23} +7753.18 q^{25} +729.000 q^{27} -3776.98 q^{29} +1191.66 q^{31} +1567.12 q^{33} +618.146 q^{35} -4654.56 q^{37} -10103.5 q^{39} -8760.43 q^{41} -14658.0 q^{43} +8448.18 q^{45} +7868.26 q^{47} -16771.9 q^{49} -14951.8 q^{51} -16503.2 q^{53} +18160.9 q^{55} -5032.45 q^{57} -34563.6 q^{59} -39373.6 q^{61} +480.063 q^{63} -117086. q^{65} -59059.8 q^{67} -4761.00 q^{69} +68200.1 q^{71} -14311.5 q^{73} +69778.7 q^{75} +1031.98 q^{77} +50134.0 q^{79} +6561.00 q^{81} +101485. q^{83} -173272. q^{85} -33992.8 q^{87} +93653.2 q^{89} -6653.37 q^{91} +10725.0 q^{93} -58319.6 q^{95} +1931.38 q^{97} +14104.1 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 4 q + 36 q^{3} + 22 q^{5} + 154 q^{7} + 324 q^{9}+O(q^{10}) $ 4 * q + 36 * q^3 + 22 * q^5 + 154 * q^7 + 324 * q^9	$ 4 q + 36 q^{3} + 22 q^{5} + 154 q^{7} + 324 q^{9} - 392 q^{11} - 220 q^{13} + 198 q^{15} - 802 q^{17} - 2590 q^{19} + 1386 q^{21} - 2116 q^{23} + 6904 q^{25} + 2916 q^{27} - 5036 q^{29} - 20 q^{31} - 3528 q^{33} + 3752 q^{35} - 10884 q^{37} - 1980 q^{39} - 23312 q^{41} - 4406 q^{43} + 1782 q^{45} - 2472 q^{47} - 24568 q^{49} - 7218 q^{51} - 29222 q^{53} + 45928 q^{55} - 23310 q^{57} - 15136 q^{59} - 113220 q^{61} + 12474 q^{63} - 133892 q^{65} + 23214 q^{67} - 19044 q^{69} + 96096 q^{71} - 95792 q^{73} + 62136 q^{75} - 96584 q^{77} + 190094 q^{79} + 26244 q^{81} + 112280 q^{83} - 276556 q^{85} - 45324 q^{87} - 52206 q^{89} + 141548 q^{91} - 180 q^{93} - 38288 q^{95} + 9636 q^{97} - 31752 q^{99}+O(q^{100}) $ 4 * q + 36 * q^3 + 22 * q^5 + 154 * q^7 + 324 * q^9 - 392 * q^11 - 220 * q^13 + 198 * q^15 - 802 * q^17 - 2590 * q^19 + 1386 * q^21 - 2116 * q^23 + 6904 * q^25 + 2916 * q^27 - 5036 * q^29 - 20 * q^31 - 3528 * q^33 + 3752 * q^35 - 10884 * q^37 - 1980 * q^39 - 23312 * q^41 - 4406 * q^43 + 1782 * q^45 - 2472 * q^47 - 24568 * q^49 - 7218 * q^51 - 29222 * q^53 + 45928 * q^55 - 23310 * q^57 - 15136 * q^59 - 113220 * q^61 + 12474 * q^63 - 133892 * q^65 + 23214 * q^67 - 19044 * q^69 + 96096 * q^71 - 95792 * q^73 + 62136 * q^75 - 96584 * q^77 + 190094 * q^79 + 26244 * q^81 + 112280 * q^83 - 276556 * q^85 - 45324 * q^87 - 52206 * q^89 + 141548 * q^91 - 180 * q^93 - 38288 * q^95 + 9636 * q^97 - 31752 * 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
$5$	104.299	1.86575	0.932874	−	0.360202i	$-0.117292\pi$
$5$	104.299	1.86575	0.932874	+	0.360202i	$0.117292\pi$
$6$	0	0
$7$	5.92670	0.0457160	0.0228580	−	0.999739i	$-0.492723\pi$
$7$	5.92670	0.0457160	0.0228580	+	0.999739i	$0.492723\pi$
$8$	0	0
$9$	81.0000	0.333333
$10$	0	0
$11$	174.124	0.433888	0.216944	−	0.976184i	$-0.430391\pi$
$11$	174.124	0.433888	0.216944	+	0.976184i	$0.430391\pi$
$12$	0	0
$13$	−1122.61	−1.84234	−0.921171	−	0.389159i	$-0.872766\pi$
$13$	−1122.61	−1.84234	−0.921171	+	0.389159i	$0.872766\pi$
$14$	0	0
$15$	938.687	1.07719
$16$	0	0
$17$	−1661.31	−1.39421	−0.697105	−	0.716969i	$-0.745529\pi$
$17$	−1661.31	−1.39421	−0.697105	+	0.716969i	$0.745529\pi$
$18$	0	0
$19$	−559.161	−0.355347	−0.177673	−	0.984090i	$-0.556857\pi$
$19$	−559.161	−0.355347	−0.177673	+	0.984090i	$0.556857\pi$
$20$	0	0
$21$	53.3403	0.0263941
$22$	0	0
$23$	−529.000	−0.208514
$23$	−529.000	−0.208514
$24$	0	0
$25$	7753.18	2.48102
$26$	0	0
$27$	729.000	0.192450
$28$	0	0
$29$	−3776.98	−0.833968	−0.416984	−	0.908914i	$-0.636913\pi$
$29$	−3776.98	−0.833968	−0.416984	+	0.908914i	$0.636913\pi$
$30$	0	0
$31$	1191.66	0.222715	0.111357	−	0.993780i	$-0.464480\pi$
$31$	1191.66	0.222715	0.111357	+	0.993780i	$0.464480\pi$
$32$	0	0
$33$	1567.12	0.250505
$34$	0	0
$35$	618.146	0.0852945
$36$	0	0
$37$	−4654.56	−0.558951	−0.279476	−	0.960153i	$-0.590161\pi$
$37$	−4654.56	−0.558951	−0.279476	+	0.960153i	$0.590161\pi$
$38$	0	0
$39$	−10103.5	−1.06368
$40$	0	0
$41$	−8760.43	−0.813890	−0.406945	−	0.913453i	$-0.633406\pi$
$41$	−8760.43	−0.813890	−0.406945	+	0.913453i	$0.633406\pi$
$42$	0	0
$43$	−14658.0	−1.20893	−0.604467	−	0.796630i	$-0.706614\pi$
$43$	−14658.0	−1.20893	−0.604467	+	0.796630i	$0.706614\pi$
$44$	0	0
$45$	8448.18	0.621916
$46$	0	0
$47$	7868.26	0.519558	0.259779	−	0.965668i	$-0.416350\pi$
$47$	7868.26	0.519558	0.259779	+	0.965668i	$0.416350\pi$
$48$	0	0
$49$	−16771.9	−0.997910
$50$	0	0
$51$	−14951.8	−0.804948
$52$	0	0
$53$	−16503.2	−0.807009	−0.403505	−	0.914978i	$-0.632208\pi$
$53$	−16503.2	−0.807009	−0.403505	+	0.914978i	$0.632208\pi$
$54$	0	0
$55$	18160.9	0.809525
$56$	0	0
$57$	−5032.45	−0.205160
$58$	0	0
$59$	−34563.6	−1.29267	−0.646336	−	0.763053i	$-0.723700\pi$
$59$	−34563.6	−1.29267	−0.646336	+	0.763053i	$0.723700\pi$
$60$	0	0
$61$	−39373.6	−1.35482	−0.677408	−	0.735608i	$-0.736897\pi$
$61$	−39373.6	−1.35482	−0.677408	+	0.735608i	$0.736897\pi$
$62$	0	0
$63$	480.063	0.0152387
$64$	0	0
$65$	−117086.	−3.43735
$66$	0	0
$67$	−59059.8	−1.60733	−0.803664	−	0.595083i	$-0.797119\pi$
$67$	−59059.8	−1.60733	−0.803664	+	0.595083i	$0.797119\pi$
$68$	0	0
$69$	−4761.00	−0.120386
$70$	0	0
$71$	68200.1	1.60561	0.802803	−	0.596244i	$-0.203341\pi$
$71$	68200.1	1.60561	0.802803	+	0.596244i	$0.203341\pi$
$72$	0	0
$73$	−14311.5	−0.314324	−0.157162	−	0.987573i	$-0.550235\pi$
$73$	−14311.5	−0.314324	−0.157162	+	0.987573i	$0.550235\pi$
$74$	0	0
$75$	69778.7	1.43242
$76$	0	0
$77$	1031.98	0.0198356
$78$	0	0
$79$	50134.0	0.903784	0.451892	−	0.892073i	$-0.350749\pi$
$79$	50134.0	0.903784	0.451892	+	0.892073i	$0.350749\pi$
$80$	0	0
$81$	6561.00	0.111111
$82$	0	0
$83$	101485.	1.61699	0.808496	−	0.588501i	$-0.200282\pi$
$83$	101485.	1.61699	0.808496	+	0.588501i	$0.200282\pi$
$84$	0	0
$85$	−173272.	−2.60125
$86$	0	0
$87$	−33992.8	−0.481492
$88$	0	0
$89$	93653.2	1.25328	0.626639	−	0.779310i	$-0.284430\pi$
$89$	93653.2	1.25328	0.626639	+	0.779310i	$0.284430\pi$
$90$	0	0
$91$	−6653.37	−0.0842244
$92$	0	0
$93$	10725.0	0.128584
$94$	0	0
$95$	−58319.6	−0.662988
$96$	0	0
$97$	1931.38	0.0208420	0.0104210	−	0.999946i	$-0.496683\pi$
$97$	1931.38	0.0208420	0.0104210	+	0.999946i	$0.496683\pi$
$98$	0	0
$99$	14104.1	0.144629
$100$	0	0
$101$	−6961.97	−0.0679092	−0.0339546	−	0.999423i	$-0.510810\pi$
$101$	−6961.97	−0.0679092	−0.0339546	+	0.999423i	$0.510810\pi$
$102$	0	0
$103$	−190838.	−1.77244	−0.886219	−	0.463267i	$-0.846677\pi$
$103$	−190838.	−1.77244	−0.886219	+	0.463267i	$0.846677\pi$
$104$	0	0
$105$	5563.32	0.0492448
$106$	0	0
$107$	24642.6	0.208078	0.104039	−	0.994573i	$-0.466823\pi$
$107$	24642.6	0.208078	0.104039	+	0.994573i	$0.466823\pi$
$108$	0	0
$109$	−217503.	−1.75347	−0.876736	−	0.480972i	$-0.840284\pi$
$109$	−217503.	−1.75347	−0.876736	+	0.480972i	$0.840284\pi$
$110$	0	0
$111$	−41891.0	−0.322711
$112$	0	0
$113$	141948.	1.04576	0.522880	−	0.852406i	$-0.324858\pi$
$113$	141948.	1.04576	0.522880	+	0.852406i	$0.324858\pi$
$114$	0	0
$115$	−55173.9	−0.389036
$116$	0	0
$117$	−90931.3	−0.614114
$118$	0	0
$119$	−9846.08	−0.0637377
$120$	0	0
$121$	−130732.	−0.811742
$122$	0	0
$123$	−78843.9	−0.469900
$124$	0	0
$125$	482713.	2.76321
$126$	0	0
$127$	144738.	0.796292	0.398146	−	0.917322i	$-0.369654\pi$
$127$	144738.	0.796292	0.398146	+	0.917322i	$0.369654\pi$
$128$	0	0
$129$	−131922.	−0.697978
$130$	0	0
$131$	−105047.	−0.534815	−0.267408	−	0.963583i	$-0.586167\pi$
$131$	−105047.	−0.534815	−0.267408	+	0.963583i	$0.586167\pi$
$132$	0	0
$133$	−3313.98	−0.0162450
$134$	0	0
$135$	76033.6	0.359064
$136$	0	0
$137$	212.487	0.000967232 0	0.000483616	−	1.00000i	$-0.499846\pi$
$137$	212.487	0.000967232 0	0.000483616		1.00000i	$0.499846\pi$
$138$	0	0
$139$	322980.	1.41788	0.708938	−	0.705270i	$-0.249174\pi$
$139$	322980.	1.41788	0.708938	+	0.705270i	$0.249174\pi$
$140$	0	0
$141$	70814.4	0.299967
$142$	0	0
$143$	−195473.	−0.799369
$144$	0	0
$145$	−393933.	−1.55598
$146$	0	0
$147$	−150947.	−0.576144
$148$	0	0
$149$	−341446.	−1.25996	−0.629980	−	0.776612i	$-0.716937\pi$
$149$	−341446.	−1.25996	−0.629980	+	0.776612i	$0.716937\pi$
$150$	0	0
$151$	138510.	0.494354	0.247177	−	0.968970i	$-0.420497\pi$
$151$	138510.	0.494354	0.247177	+	0.968970i	$0.420497\pi$
$152$	0	0
$153$	−134566.	−0.464737
$154$	0	0
$155$	124289.	0.415530
$156$	0	0
$157$	−31981.7	−0.103551	−0.0517753	−	0.998659i	$-0.516488\pi$
$157$	−31981.7	−0.103551	−0.0517753	+	0.998659i	$0.516488\pi$
$158$	0	0
$159$	−148529.	−0.465927
$160$	0	0
$161$	−3135.22	−0.00953244
$162$	0	0
$163$	314888.	0.928299	0.464149	−	0.885757i	$-0.346360\pi$
$163$	314888.	0.928299	0.464149	+	0.885757i	$0.346360\pi$
$164$	0	0
$165$	163448.	0.467380
$166$	0	0
$167$	−213089.	−0.591247	−0.295624	−	0.955305i	$-0.595527\pi$
$167$	−213089.	−0.591247	−0.295624	+	0.955305i	$0.595527\pi$
$168$	0	0
$169$	888958.	2.39422
$170$	0	0
$171$	−45292.0	−0.118449
$172$	0	0
$173$	512900.	1.30292	0.651459	−	0.758684i	$-0.274157\pi$
$173$	512900.	1.30292	0.651459	+	0.758684i	$0.274157\pi$
$174$	0	0
$175$	45950.8	0.113422
$176$	0	0
$177$	−311072.	−0.746325
$178$	0	0
$179$	495138.	1.15503	0.577516	−	0.816379i	$-0.304022\pi$
$179$	495138.	1.15503	0.577516	+	0.816379i	$0.304022\pi$
$180$	0	0
$181$	−92733.7	−0.210398	−0.105199	−	0.994451i	$-0.533548\pi$
$181$	−92733.7	−0.210398	−0.105199	+	0.994451i	$0.533548\pi$
$182$	0	0
$183$	−354362.	−0.782203
$184$	0	0
$185$	−485463.	−1.04286
$186$	0	0
$187$	−289274.	−0.604930
$188$	0	0
$189$	4320.57	0.00879804
$190$	0	0
$191$	558873.	1.10849	0.554243	−	0.832355i	$-0.313008\pi$
$191$	558873.	1.10849	0.554243	+	0.832355i	$0.313008\pi$
$192$	0	0
$193$	772290.	1.49241	0.746204	−	0.665718i	$-0.231875\pi$
$193$	772290.	1.49241	0.746204	+	0.665718i	$0.231875\pi$
$194$	0	0
$195$	−1.05378e6	−1.98455
$196$	0	0
$197$	731343.	1.34263	0.671314	−	0.741173i	$-0.265730\pi$
$197$	731343.	1.34263	0.671314	+	0.741173i	$0.265730\pi$
$198$	0	0
$199$	−139654.	−0.249988	−0.124994	−	0.992157i	$-0.539891\pi$
$199$	−139654.	−0.249988	−0.124994	+	0.992157i	$0.539891\pi$
$200$	0	0
$201$	−531538.	−0.927992
$202$	0	0
$203$	−22385.0	−0.0381257
$204$	0	0
$205$	−913700.	−1.51851
$206$	0	0
$207$	−42849.0	−0.0695048
$208$	0	0
$209$	−97363.3	−0.154181
$210$	0	0
$211$	−715174.	−1.10587	−0.552937	−	0.833223i	$-0.686493\pi$
$211$	−715174.	−1.10587	−0.552937	+	0.833223i	$0.686493\pi$
$212$	0	0
$213$	613801.	0.926998
$214$	0	0
$215$	−1.52880e6	−2.25557
$216$	0	0
$217$	7062.63	0.0101816
$218$	0	0
$219$	−128804.	−0.181475
$220$	0	0
$221$	1.86500e6	2.56861
$222$	0	0
$223$	−341714.	−0.460151	−0.230075	−	0.973173i	$-0.573897\pi$
$223$	−341714.	−0.460151	−0.230075	+	0.973173i	$0.573897\pi$
$224$	0	0
$225$	628008.	0.827006
$226$	0	0
$227$	−375339.	−0.483458	−0.241729	−	0.970344i	$-0.577714\pi$
$227$	−375339.	−0.483458	−0.241729	+	0.970344i	$0.577714\pi$
$228$	0	0
$229$	−935168.	−1.17842	−0.589211	−	0.807979i	$-0.700561\pi$
$229$	−935168.	−1.17842	−0.589211	+	0.807979i	$0.700561\pi$
$230$	0	0
$231$	9287.83	0.0114521
$232$	0	0
$233$	−256545.	−0.309581	−0.154790	−	0.987947i	$-0.549470\pi$
$233$	−256545.	−0.309581	−0.154790	+	0.987947i	$0.549470\pi$
$234$	0	0
$235$	820648.	0.969365
$236$	0	0
$237$	451206.	0.521800
$238$	0	0
$239$	669809.	0.758501	0.379250	−	0.925294i	$-0.376182\pi$
$239$	669809.	0.758501	0.379250	+	0.925294i	$0.376182\pi$
$240$	0	0
$241$	−471096.	−0.522476	−0.261238	−	0.965274i	$-0.584131\pi$
$241$	−471096.	−0.522476	−0.261238	+	0.965274i	$0.584131\pi$
$242$	0	0
$243$	59049.0	0.0641500
$244$	0	0
$245$	−1.74928e6	−1.86185
$246$	0	0
$247$	627719.	0.654670
$248$	0	0
$249$	913368.	0.933571
$250$	0	0
$251$	−192344.	−0.192706	−0.0963529	−	0.995347i	$-0.530718\pi$
$251$	−192344.	−0.192706	−0.0963529	+	0.995347i	$0.530718\pi$
$252$	0	0
$253$	−92111.7	−0.0904718
$254$	0	0
$255$	−1.55945e6	−1.50183
$256$	0	0
$257$	−804037.	−0.759352	−0.379676	−	0.925119i	$-0.623965\pi$
$257$	−804037.	−0.759352	−0.379676	+	0.925119i	$0.623965\pi$
$258$	0	0
$259$	−27586.2	−0.0255530
$260$	0	0
$261$	−305935.	−0.277989
$262$	0	0
$263$	−1.59730e6	−1.42396	−0.711979	−	0.702201i	$-0.752201\pi$
$263$	−1.59730e6	−1.42396	−0.711979	+	0.702201i	$0.752201\pi$
$264$	0	0
$265$	−1.72126e6	−1.50568
$266$	0	0
$267$	842879.	0.723581
$268$	0	0
$269$	2.01461e6	1.69750	0.848751	−	0.528793i	$-0.177355\pi$
$269$	2.01461e6	1.69750	0.848751	+	0.528793i	$0.177355\pi$
$270$	0	0
$271$	−841314.	−0.695881	−0.347940	−	0.937517i	$-0.613119\pi$
$271$	−841314.	−0.695881	−0.347940	+	0.937517i	$0.613119\pi$
$272$	0	0
$273$	−59880.3	−0.0486270
$274$	0	0
$275$	1.35002e6	1.07648
$276$	0	0
$277$	−1.17541e6	−0.920431	−0.460215	−	0.887807i	$-0.652228\pi$
$277$	−1.17541e6	−0.920431	−0.460215	+	0.887807i	$0.652228\pi$
$278$	0	0
$279$	96524.7	0.0742383
$280$	0	0
$281$	293393.	0.221659	0.110829	−	0.993839i	$-0.464649\pi$
$281$	293393.	0.221659	0.110829	+	0.993839i	$0.464649\pi$
$282$	0	0
$283$	1.36967e6	1.01660	0.508299	−	0.861181i	$-0.330274\pi$
$283$	1.36967e6	1.01660	0.508299	+	0.861181i	$0.330274\pi$
$284$	0	0
$285$	−524877.	−0.382776
$286$	0	0
$287$	−51920.4	−0.0372078
$288$	0	0
$289$	1.34009e6	0.943822
$290$	0	0
$291$	17382.5	0.0120331
$292$	0	0
$293$	694352.	0.472509	0.236255	−	0.971691i	$-0.424080\pi$
$293$	694352.	0.472509	0.236255	+	0.971691i	$0.424080\pi$
$294$	0	0
$295$	−3.60493e6	−2.41180
$296$	0	0
$297$	126936.	0.0835017
$298$	0	0
$299$	593860.	0.384155
$300$	0	0
$301$	−86873.4	−0.0552676
$302$	0	0
$303$	−62657.7	−0.0392074
$304$	0	0
$305$	−4.10660e6	−2.52774
$306$	0	0
$307$	1.42934e6	0.865547	0.432773	−	0.901503i	$-0.357535\pi$
$307$	1.42934e6	0.865547	0.432773	+	0.901503i	$0.357535\pi$
$308$	0	0
$309$	−1.71754e6	−1.02332
$310$	0	0
$311$	−2.20400e6	−1.29214	−0.646072	−	0.763277i	$-0.723589\pi$
$311$	−2.20400e6	−1.29214	−0.646072	+	0.763277i	$0.723589\pi$
$312$	0	0
$313$	103256.	0.0595736	0.0297868	−	0.999556i	$-0.490517\pi$
$313$	103256.	0.0595736	0.0297868	+	0.999556i	$0.490517\pi$
$314$	0	0
$315$	50069.8	0.0284315
$316$	0	0
$317$	2.10433e6	1.17616	0.588080	−	0.808803i	$-0.299884\pi$
$317$	2.10433e6	1.17616	0.588080	+	0.808803i	$0.299884\pi$
$318$	0	0
$319$	−657663.	−0.361849
$320$	0	0
$321$	221783.	0.120134
$322$	0	0
$323$	928939.	0.495428
$324$	0	0
$325$	−8.70379e6	−4.57088
$326$	0	0
$327$	−1.95753e6	−1.01237
$328$	0	0
$329$	46632.8	0.0237521
$330$	0	0
$331$	−863565.	−0.433237	−0.216618	−	0.976256i	$-0.569503\pi$
$331$	−863565.	−0.433237	−0.216618	+	0.976256i	$0.569503\pi$
$332$	0	0
$333$	−377019.	−0.186317
$334$	0	0
$335$	−6.15985e6	−2.99887
$336$	0	0
$337$	−3.57269e6	−1.71364	−0.856821	−	0.515613i	$-0.827564\pi$
$337$	−3.57269e6	−1.71364	−0.856821	+	0.515613i	$0.827564\pi$
$338$	0	0
$339$	1.27753e6	0.603770
$340$	0	0
$341$	207497.	0.0966332
$342$	0	0
$343$	−199012.	−0.0913364
$344$	0	0
$345$	−496565.	−0.224610
$346$	0	0
$347$	−2.74908e6	−1.22564	−0.612822	−	0.790221i	$-0.709966\pi$
$347$	−2.74908e6	−1.22564	−0.612822	+	0.790221i	$0.709966\pi$
$348$	0	0
$349$	2.36732e6	1.04038	0.520191	−	0.854050i	$-0.325861\pi$
$349$	2.36732e6	1.04038	0.520191	+	0.854050i	$0.325861\pi$
$350$	0	0
$351$	−818382.	−0.354559
$352$	0	0
$353$	−730286.	−0.311929	−0.155965	−	0.987763i	$-0.549849\pi$
$353$	−730286.	−0.311929	−0.155965	+	0.987763i	$0.549849\pi$
$354$	0	0
$355$	7.11317e6	2.99566
$356$	0	0
$357$	−88614.8	−0.0367990
$358$	0	0
$359$	798828.	0.327127	0.163564	−	0.986533i	$-0.447701\pi$
$359$	798828.	0.327127	0.163564	+	0.986533i	$0.447701\pi$
$360$	0	0
$361$	−2.16344e6	−0.873729
$362$	0	0
$363$	−1.17659e6	−0.468659
$364$	0	0
$365$	−1.49267e6	−0.586450
$366$	0	0
$367$	1.70555e6	0.660998	0.330499	−	0.943806i	$-0.392783\pi$
$367$	1.70555e6	0.660998	0.330499	+	0.943806i	$0.392783\pi$
$368$	0	0
$369$	−709595.	−0.271297
$370$	0	0
$371$	−97809.5	−0.0368932
$372$	0	0
$373$	950449.	0.353718	0.176859	−	0.984236i	$-0.443406\pi$
$373$	950449.	0.353718	0.176859	+	0.984236i	$0.443406\pi$
$374$	0	0
$375$	4.34442e6	1.59534
$376$	0	0
$377$	4.24007e6	1.53645
$378$	0	0
$379$	−2.69681e6	−0.964390	−0.482195	−	0.876064i	$-0.660160\pi$
$379$	−2.69681e6	−0.964390	−0.482195	+	0.876064i	$0.660160\pi$
$380$	0	0
$381$	1.30264e6	0.459739
$382$	0	0
$383$	4.97134e6	1.73172	0.865858	−	0.500290i	$-0.166773\pi$
$383$	4.97134e6	1.73172	0.865858	+	0.500290i	$0.166773\pi$
$384$	0	0
$385$	107634.	0.0370082
$386$	0	0
$387$	−1.18730e6	−0.402978
$388$	0	0
$389$	−4.53626e6	−1.51993	−0.759965	−	0.649964i	$-0.774784\pi$
$389$	−4.53626e6	−1.51993	−0.759965	+	0.649964i	$0.774784\pi$
$390$	0	0
$391$	878833.	0.290713
$392$	0	0
$393$	−945420.	−0.308776
$394$	0	0
$395$	5.22890e6	1.68623
$396$	0	0
$397$	529593.	0.168642	0.0843210	−	0.996439i	$-0.473128\pi$
$397$	529593.	0.168642	0.0843210	+	0.996439i	$0.473128\pi$
$398$	0	0
$399$	−29825.8	−0.00937907
$400$	0	0
$401$	1.82979e6	0.568250	0.284125	−	0.958787i	$-0.408297\pi$
$401$	1.82979e6	0.568250	0.284125	+	0.958787i	$0.408297\pi$
$402$	0	0
$403$	−1.33777e6	−0.410317
$404$	0	0
$405$	684303.	0.207305
$406$	0	0
$407$	−810471.	−0.242522
$408$	0	0
$409$	−1.95901e6	−0.579067	−0.289534	−	0.957168i	$-0.593500\pi$
$409$	−1.95901e6	−0.579067	−0.289534	+	0.957168i	$0.593500\pi$
$410$	0	0
$411$	1912.38	0.000558432 0
$412$	0	0
$413$	−204848.	−0.0590958
$414$	0	0
$415$	1.05848e7	3.01690
$416$	0	0
$417$	2.90682e6	0.818612
$418$	0	0
$419$	−3.20941e6	−0.893080	−0.446540	−	0.894764i	$-0.647344\pi$
$419$	−3.20941e6	−0.893080	−0.446540	+	0.894764i	$0.647344\pi$
$420$	0	0
$421$	−2.06326e6	−0.567346	−0.283673	−	0.958921i	$-0.591553\pi$
$421$	−2.06326e6	−0.567346	−0.283673	+	0.958921i	$0.591553\pi$
$422$	0	0
$423$	637329.	0.173186
$424$	0	0
$425$	−1.28804e7	−3.45906
$426$	0	0
$427$	−233355.	−0.0619367
$428$	0	0
$429$	−1.75926e6	−0.461516
$430$	0	0
$431$	999698.	0.259224	0.129612	−	0.991565i	$-0.458627\pi$
$431$	999698.	0.259224	0.129612	+	0.991565i	$0.458627\pi$
$432$	0	0
$433$	4.96855e6	1.27353	0.636766	−	0.771057i	$-0.280272\pi$
$433$	4.96855e6	1.27353	0.636766	+	0.771057i	$0.280272\pi$
$434$	0	0
$435$	−3.54540e6	−0.898343
$436$	0	0
$437$	295796.	0.0740950
$438$	0	0
$439$	−7.46695e6	−1.84919	−0.924596	−	0.380949i	$-0.875597\pi$
$439$	−7.46695e6	−1.84919	−0.924596	+	0.380949i	$0.875597\pi$
$440$	0	0
$441$	−1.35852e6	−0.332637
$442$	0	0
$443$	1.74165e6	0.421651	0.210825	−	0.977524i	$-0.432385\pi$
$443$	1.74165e6	0.421651	0.210825	+	0.977524i	$0.432385\pi$
$444$	0	0
$445$	9.76789e6	2.33830
$446$	0	0
$447$	−3.07302e6	−0.727438
$448$	0	0
$449$	2.64114e6	0.618265	0.309133	−	0.951019i	$-0.399961\pi$
$449$	2.64114e6	0.618265	0.309133	+	0.951019i	$0.399961\pi$
$450$	0	0
$451$	−1.52540e6	−0.353137
$452$	0	0
$453$	1.24659e6	0.285416
$454$	0	0
$455$	−693936.	−0.157142
$456$	0	0
$457$	3.64392e6	0.816165	0.408082	−	0.912945i	$-0.366198\pi$
$457$	3.64392e6	0.816165	0.408082	+	0.912945i	$0.366198\pi$
$458$	0	0
$459$	−1.21109e6	−0.268316
$460$	0	0
$461$	525643.	0.115196	0.0575982	−	0.998340i	$-0.481656\pi$
$461$	525643.	0.115196	0.0575982	+	0.998340i	$0.481656\pi$
$462$	0	0
$463$	−1.16294e6	−0.252118	−0.126059	−	0.992023i	$-0.540233\pi$
$463$	−1.16294e6	−0.252118	−0.126059	+	0.992023i	$0.540233\pi$
$464$	0	0
$465$	1.11860e6	0.239906
$466$	0	0
$467$	783210.	0.166183	0.0830914	−	0.996542i	$-0.473521\pi$
$467$	783210.	0.166183	0.0830914	+	0.996542i	$0.473521\pi$
$468$	0	0
$469$	−350030.	−0.0734806
$470$	0	0
$471$	−287835.	−0.0597849
$472$	0	0
$473$	−2.55231e6	−0.524542
$474$	0	0
$475$	−4.33527e6	−0.881622
$476$	0	0
$477$	−1.33676e6	−0.269003
$478$	0	0
$479$	−6.97619e6	−1.38925	−0.694623	−	0.719373i	$-0.744429\pi$
$479$	−6.97619e6	−1.38925	−0.694623	+	0.719373i	$0.744429\pi$
$480$	0	0
$481$	5.22525e6	1.02978
$482$	0	0
$483$	−28217.0	−0.00550356
$484$	0	0
$485$	201441.	0.0388859
$486$	0	0
$487$	3.81402e6	0.728719	0.364359	−	0.931258i	$-0.381288\pi$
$487$	3.81402e6	0.728719	0.364359	+	0.931258i	$0.381288\pi$
$488$	0	0
$489$	2.83400e6	0.535953
$490$	0	0
$491$	−3.20975e6	−0.600853	−0.300426	−	0.953805i	$-0.597129\pi$
$491$	−3.20975e6	−0.600853	−0.300426	+	0.953805i	$0.597129\pi$
$492$	0	0
$493$	6.27473e6	1.16273
$494$	0	0
$495$	1.47103e6	0.269842
$496$	0	0
$497$	404202.	0.0734019
$498$	0	0
$499$	−8.84078e6	−1.58942	−0.794711	−	0.606988i	$-0.792378\pi$
$499$	−8.84078e6	−1.58942	−0.794711	+	0.606988i	$0.792378\pi$
$500$	0	0
$501$	−1.91780e6	−0.341357
$502$	0	0
$503$	1.03646e7	1.82656	0.913280	−	0.407332i	$-0.133541\pi$
$503$	1.03646e7	1.82656	0.913280	+	0.407332i	$0.133541\pi$
$504$	0	0
$505$	−726123.	−0.126702
$506$	0	0
$507$	8.00062e6	1.38230
$508$	0	0
$509$	−4.33570e6	−0.741762	−0.370881	−	0.928680i	$-0.620944\pi$
$509$	−4.33570e6	−0.741762	−0.370881	+	0.928680i	$0.620944\pi$
$510$	0	0
$511$	−84820.0	−0.0143696
$512$	0	0
$513$	−407628.	−0.0683865
$514$	0	0
$515$	−1.99041e7	−3.30692
$516$	0	0
$517$	1.37005e6	0.225430
$518$	0	0
$519$	4.61610e6	0.752240
$520$	0	0
$521$	−1.10759e7	−1.78766	−0.893828	−	0.448409i	$-0.851991\pi$
$521$	−1.10759e7	−1.78766	−0.893828	+	0.448409i	$0.851991\pi$
$522$	0	0
$523$	−7.51750e6	−1.20176	−0.600882	−	0.799338i	$-0.705184\pi$
$523$	−7.51750e6	−1.20176	−0.600882	+	0.799338i	$0.705184\pi$
$524$	0	0
$525$	413557.	0.0654843
$526$	0	0
$527$	−1.97972e6	−0.310511
$528$	0	0
$529$	279841.	0.0434783
$530$	0	0
$531$	−2.79965e6	−0.430891
$532$	0	0
$533$	9.83453e6	1.49946
$534$	0	0
$535$	2.57018e6	0.388222
$536$	0	0
$537$	4.45624e6	0.666858
$538$	0	0
$539$	−2.92039e6	−0.432981
$540$	0	0
$541$	−9.64770e6	−1.41720	−0.708599	−	0.705611i	$-0.750672\pi$
$541$	−9.64770e6	−1.41720	−0.708599	+	0.705611i	$0.750672\pi$
$542$	0	0
$543$	−834604.	−0.121473
$544$	0	0
$545$	−2.26852e7	−3.27154
$546$	0	0
$547$	−664811.	−0.0950014	−0.0475007	−	0.998871i	$-0.515126\pi$
$547$	−664811.	−0.0950014	−0.0475007	+	0.998871i	$0.515126\pi$
$548$	0	0
$549$	−3.18926e6	−0.451605
$550$	0	0
$551$	2.11194e6	0.296348
$552$	0	0
$553$	297129.	0.0413174
$554$	0	0
$555$	−4.36917e6	−0.602097
$556$	0	0
$557$	5.91528e6	0.807862	0.403931	−	0.914789i	$-0.367644\pi$
$557$	5.91528e6	0.807862	0.403931	+	0.914789i	$0.367644\pi$
$558$	0	0
$559$	1.64552e7	2.22727
$560$	0	0
$561$	−2.60347e6	−0.349257
$562$	0	0
$563$	−8.31290e6	−1.10530	−0.552652	−	0.833412i	$-0.686384\pi$
$563$	−8.31290e6	−1.10530	−0.552652	+	0.833412i	$0.686384\pi$
$564$	0	0
$565$	1.48049e7	1.95113
$566$	0	0
$567$	38885.1	0.00507955
$568$	0	0
$569$	−1.12928e7	−1.46224	−0.731122	−	0.682246i	$-0.761003\pi$
$569$	−1.12928e7	−1.46224	−0.731122	+	0.682246i	$0.761003\pi$
$570$	0	0
$571$	1.15115e6	0.147755	0.0738773	−	0.997267i	$-0.476463\pi$
$571$	1.15115e6	0.147755	0.0738773	+	0.997267i	$0.476463\pi$
$572$	0	0
$573$	5.02986e6	0.639984
$574$	0	0
$575$	−4.10143e6	−0.517328
$576$	0	0
$577$	−758214.	−0.0948096	−0.0474048	−	0.998876i	$-0.515095\pi$
$577$	−758214.	−0.0948096	−0.0474048	+	0.998876i	$0.515095\pi$
$578$	0	0
$579$	6.95061e6	0.861642
$580$	0	0
$581$	601473.	0.0739224
$582$	0	0
$583$	−2.87361e6	−0.350151
$584$	0	0
$585$	−9.48400e6	−1.14578
$586$	0	0
$587$	−1.64923e7	−1.97554	−0.987770	−	0.155919i	$-0.950166\pi$
$587$	−1.64923e7	−1.97554	−0.987770	+	0.155919i	$0.950166\pi$
$588$	0	0
$589$	−666331.	−0.0791410
$590$	0	0
$591$	6.58208e6	0.775166
$592$	0	0
$593$	9.06649e6	1.05877	0.529386	−	0.848381i	$-0.322422\pi$
$593$	9.06649e6	1.05877	0.529386	+	0.848381i	$0.322422\pi$
$594$	0	0
$595$	−1.02693e6	−0.118918
$596$	0	0
$597$	−1.25688e6	−0.144331
$598$	0	0
$599$	−2.17387e6	−0.247552	−0.123776	−	0.992310i	$-0.539500\pi$
$599$	−2.17387e6	−0.247552	−0.123776	+	0.992310i	$0.539500\pi$
$600$	0	0
$601$	6.51506e6	0.735754	0.367877	−	0.929875i	$-0.380085\pi$
$601$	6.51506e6	0.735754	0.367877	+	0.929875i	$0.380085\pi$
$602$	0	0
$603$	−4.78384e6	−0.535776
$604$	0	0
$605$	−1.36351e7	−1.51451
$606$	0	0
$607$	4.55975e6	0.502307	0.251154	−	0.967947i	$-0.419190\pi$
$607$	4.55975e6	0.502307	0.251154	+	0.967947i	$0.419190\pi$
$608$	0	0
$609$	−201465.	−0.0220119
$610$	0	0
$611$	−8.83298e6	−0.957204
$612$	0	0
$613$	1.57272e6	0.169044	0.0845220	−	0.996422i	$-0.473064\pi$
$613$	1.57272e6	0.169044	0.0845220	+	0.996422i	$0.473064\pi$
$614$	0	0
$615$	−8.22330e6	−0.876715
$616$	0	0
$617$	−2.08022e6	−0.219987	−0.109993	−	0.993932i	$-0.535083\pi$
$617$	−2.08022e6	−0.219987	−0.109993	+	0.993932i	$0.535083\pi$
$618$	0	0
$619$	8.51878e6	0.893616	0.446808	−	0.894630i	$-0.352561\pi$
$619$	8.51878e6	0.893616	0.446808	+	0.894630i	$0.352561\pi$
$620$	0	0
$621$	−385641.	−0.0401286
$622$	0	0
$623$	555054.	0.0572948
$624$	0	0
$625$	2.61175e7	2.67444
$626$	0	0
$627$	−876270.	−0.0890162
$628$	0	0
$629$	7.73266e6	0.779296
$630$	0	0
$631$	−1.74770e7	−1.74741	−0.873703	−	0.486460i	$-0.838288\pi$
$631$	−1.74770e7	−1.74741	−0.873703	+	0.486460i	$0.838288\pi$
$632$	0	0
$633$	−6.43657e6	−0.638477
$634$	0	0
$635$	1.50959e7	1.48568
$636$	0	0
$637$	1.88283e7	1.83849
$638$	0	0
$639$	5.52421e6	0.535202
$640$	0	0
$641$	1.51030e7	1.45184	0.725921	−	0.687778i	$-0.241414\pi$
$641$	1.51030e7	1.45184	0.725921	+	0.687778i	$0.241414\pi$
$642$	0	0
$643$	1.48265e7	1.41420	0.707098	−	0.707115i	$-0.250004\pi$
$643$	1.48265e7	1.41420	0.707098	+	0.707115i	$0.250004\pi$
$644$	0	0
$645$	−1.37592e7	−1.30225
$646$	0	0
$647$	−8.18344e6	−0.768556	−0.384278	−	0.923217i	$-0.625550\pi$
$647$	−8.18344e6	−0.768556	−0.384278	+	0.923217i	$0.625550\pi$
$648$	0	0
$649$	−6.01835e6	−0.560874
$650$	0	0
$651$	63563.7	0.00587836
$652$	0	0
$653$	−6.48026e6	−0.594716	−0.297358	−	0.954766i	$-0.596106\pi$
$653$	−6.48026e6	−0.594716	−0.297358	+	0.954766i	$0.596106\pi$
$654$	0	0
$655$	−1.09562e7	−0.997831
$656$	0	0
$657$	−1.15923e6	−0.104775
$658$	0	0
$659$	1.41446e7	1.26876	0.634378	−	0.773023i	$-0.281256\pi$
$659$	1.41446e7	1.26876	0.634378	+	0.773023i	$0.281256\pi$
$660$	0	0
$661$	7.57909e6	0.674704	0.337352	−	0.941379i	$-0.390469\pi$
$661$	7.57909e6	0.674704	0.337352	+	0.941379i	$0.390469\pi$
$662$	0	0
$663$	1.67850e7	1.48299
$664$	0	0
$665$	−345643.	−0.0303091
$666$	0	0
$667$	1.99802e6	0.173894
$668$	0	0
$669$	−3.07542e6	−0.265668
$670$	0	0
$671$	−6.85589e6	−0.587837
$672$	0	0
$673$	−657285.	−0.0559392	−0.0279696	−	0.999609i	$-0.508904\pi$
$673$	−657285.	−0.0559392	−0.0279696	+	0.999609i	$0.508904\pi$
$674$	0	0
$675$	5.65207e6	0.477472
$676$	0	0
$677$	1.16844e7	0.979797	0.489898	−	0.871780i	$-0.337034\pi$
$677$	1.16844e7	0.979797	0.489898	+	0.871780i	$0.337034\pi$
$678$	0	0
$679$	11446.7	0.000952812 0
$680$	0	0
$681$	−3.37805e6	−0.279125
$682$	0	0
$683$	−6.65441e6	−0.545830	−0.272915	−	0.962038i	$-0.587988\pi$
$683$	−6.65441e6	−0.545830	−0.272915	+	0.962038i	$0.587988\pi$
$684$	0	0
$685$	22162.1	0.00180461
$686$	0	0
$687$	−8.41651e6	−0.680362
$688$	0	0
$689$	1.85266e7	1.48679
$690$	0	0
$691$	−2.31727e7	−1.84621	−0.923106	−	0.384546i	$-0.874358\pi$
$691$	−2.31727e7	−1.84621	−0.923106	+	0.384546i	$0.874358\pi$
$692$	0	0
$693$	83590.5	0.00661186
$694$	0	0
$695$	3.36863e7	2.64540
$696$	0	0
$697$	1.45538e7	1.13473
$698$	0	0
$699$	−2.30891e6	−0.178736
$700$	0	0
$701$	1.32786e7	1.02060	0.510301	−	0.859996i	$-0.329534\pi$
$701$	1.32786e7	1.02060	0.510301	+	0.859996i	$0.329534\pi$
$702$	0	0
$703$	2.60264e6	0.198622
$704$	0	0
$705$	7.38584e6	0.559663
$706$	0	0
$707$	−41261.5	−0.00310454
$708$	0	0
$709$	−5.78702e6	−0.432354	−0.216177	−	0.976354i	$-0.569359\pi$
$709$	−5.78702e6	−0.432354	−0.216177	+	0.976354i	$0.569359\pi$
$710$	0	0
$711$	4.06085e6	0.301261
$712$	0	0
$713$	−630390.	−0.0464393
$714$	0	0
$715$	−2.03876e7	−1.49142
$716$	0	0
$717$	6.02828e6	0.437921
$718$	0	0
$719$	1.77831e7	1.28288	0.641439	−	0.767174i	$-0.278338\pi$
$719$	1.77831e7	1.28288	0.641439	+	0.767174i	$0.278338\pi$
$720$	0	0
$721$	−1.13104e6	−0.0810287
$722$	0	0
$723$	−4.23986e6	−0.301652
$724$	0	0
$725$	−2.92836e7	−2.06909
$726$	0	0
$727$	2.24997e7	1.57885	0.789425	−	0.613847i	$-0.210379\pi$
$727$	2.24997e7	1.57885	0.789425	+	0.613847i	$0.210379\pi$
$728$	0	0
$729$	531441.	0.0370370
$730$	0	0
$731$	2.43514e7	1.68551
$732$	0	0
$733$	−1.00277e7	−0.689349	−0.344675	−	0.938722i	$-0.612011\pi$
$733$	−1.00277e7	−0.689349	−0.344675	+	0.938722i	$0.612011\pi$
$734$	0	0
$735$	−1.57435e7	−1.07494
$736$	0	0
$737$	−1.02837e7	−0.697400
$738$	0	0
$739$	1.87137e7	1.26052	0.630259	−	0.776385i	$-0.282949\pi$
$739$	1.87137e7	1.26052	0.630259	+	0.776385i	$0.282949\pi$
$740$	0	0
$741$	5.64947e6	0.377974
$742$	0	0
$743$	−2.02030e7	−1.34259	−0.671294	−	0.741191i	$-0.734261\pi$
$743$	−2.02030e7	−1.34259	−0.671294	+	0.741191i	$0.734261\pi$
$744$	0	0
$745$	−3.56123e7	−2.35077
$746$	0	0
$747$	8.22031e6	0.538998
$748$	0	0
$749$	146049.	0.00951249
$750$	0	0
$751$	1.62929e7	1.05414	0.527071	−	0.849821i	$-0.323290\pi$
$751$	1.62929e7	1.05414	0.527071	+	0.849821i	$0.323290\pi$
$752$	0	0
$753$	−1.73110e6	−0.111259
$754$	0	0
$755$	1.44464e7	0.922341
$756$	0	0
$757$	−2.28174e7	−1.44720	−0.723598	−	0.690222i	$-0.757513\pi$
$757$	−2.28174e7	−1.44720	−0.723598	+	0.690222i	$0.757513\pi$
$758$	0	0
$759$	−829005.	−0.0522339
$760$	0	0
$761$	−1.56817e7	−0.981593	−0.490797	−	0.871274i	$-0.663294\pi$
$761$	−1.56817e7	−0.981593	−0.490797	+	0.871274i	$0.663294\pi$
$762$	0	0
$763$	−1.28907e6	−0.0801617
$764$	0	0
$765$	−1.40350e7	−0.867082
$766$	0	0
$767$	3.88014e7	2.38154
$768$	0	0
$769$	−6.84509e6	−0.417411	−0.208705	−	0.977979i	$-0.566925\pi$
$769$	−6.84509e6	−0.417411	−0.208705	+	0.977979i	$0.566925\pi$
$770$	0	0
$771$	−7.23634e6	−0.438412
$772$	0	0
$773$	−354932.	−0.0213647	−0.0106823	−	0.999943i	$-0.503400\pi$
$773$	−354932.	−0.0213647	−0.0106823	+	0.999943i	$0.503400\pi$
$774$	0	0
$775$	9.23918e6	0.552560
$776$	0	0
$777$	−248275.	−0.0147530
$778$	0	0
$779$	4.89849e6	0.289213
$780$	0	0
$781$	1.18753e7	0.696653
$782$	0	0
$783$	−2.75342e6	−0.160497
$784$	0	0
$785$	−3.33564e6	−0.193199
$786$	0	0
$787$	2.66769e7	1.53532	0.767658	−	0.640859i	$-0.221422\pi$
$787$	2.66769e7	1.53532	0.767658	+	0.640859i	$0.221422\pi$
$788$	0	0
$789$	−1.43757e7	−0.822122
$790$	0	0
$791$	841281.	0.0478079
$792$	0	0
$793$	4.42011e7	2.49603
$794$	0	0
$795$	−1.54913e7	−0.869303
$796$	0	0
$797$	1.95008e7	1.08744	0.543721	−	0.839266i	$-0.317015\pi$
$797$	1.95008e7	1.08744	0.543721	+	0.839266i	$0.317015\pi$
$798$	0	0
$799$	−1.30716e7	−0.724373
$800$	0	0
$801$	7.58591e6	0.417759
$802$	0	0
$803$	−2.49198e6	−0.136381
$804$	0	0
$805$	−326999.	−0.0177851
$806$	0	0
$807$	1.81315e7	0.980053
$808$	0	0
$809$	−2.45630e7	−1.31950	−0.659752	−	0.751483i	$-0.729339\pi$
$809$	−2.45630e7	−1.31950	−0.659752	+	0.751483i	$0.729339\pi$
$810$	0	0
$811$	−2.88283e7	−1.53910	−0.769550	−	0.638586i	$-0.779520\pi$
$811$	−2.88283e7	−1.53910	−0.769550	+	0.638586i	$0.779520\pi$
$812$	0	0
$813$	−7.57182e6	−0.401767
$814$	0	0
$815$	3.28424e7	1.73197
$816$	0	0
$817$	8.19616e6	0.429591
$818$	0	0
$819$	−538923.	−0.0280748
$820$	0	0
$821$	−3.19571e7	−1.65466	−0.827331	−	0.561715i	$-0.810142\pi$
$821$	−3.19571e7	−1.65466	−0.827331	+	0.561715i	$0.810142\pi$
$822$	0	0
$823$	2.78834e7	1.43498	0.717491	−	0.696568i	$-0.245291\pi$
$823$	2.78834e7	1.43498	0.717491	+	0.696568i	$0.245291\pi$
$824$	0	0
$825$	1.21501e7	0.621508
$826$	0	0
$827$	6.63417e6	0.337305	0.168653	−	0.985676i	$-0.446058\pi$
$827$	6.63417e6	0.337305	0.168653	+	0.985676i	$0.446058\pi$
$828$	0	0
$829$	1.98731e7	1.00434	0.502169	−	0.864769i	$-0.332535\pi$
$829$	1.98731e7	1.00434	0.502169	+	0.864769i	$0.332535\pi$
$830$	0	0
$831$	−1.05787e7	−0.531411
$832$	0	0
$833$	2.78633e7	1.39130
$834$	0	0
$835$	−2.22248e7	−1.10312
$836$	0	0
$837$	868722.	0.0428615
$838$	0	0
$839$	3.89399e7	1.90981	0.954904	−	0.296914i	$-0.0959574\pi$
$839$	3.89399e7	1.90981	0.954904	+	0.296914i	$0.0959574\pi$
$840$	0	0
$841$	−6.24558e6	−0.304497
$842$	0	0
$843$	2.64054e6	0.127975
$844$	0	0
$845$	9.27170e7	4.46702
$846$	0	0
$847$	−774808.	−0.0371096
$848$	0	0
$849$	1.23270e7	0.586933
$850$	0	0
$851$	2.46226e6	0.116549
$852$	0	0
$853$	6.16423e6	0.290073	0.145036	−	0.989426i	$-0.453670\pi$
$853$	6.16423e6	0.290073	0.145036	+	0.989426i	$0.453670\pi$
$854$	0	0
$855$	−4.72389e6	−0.220996
$856$	0	0
$857$	2.58928e7	1.20428	0.602139	−	0.798391i	$-0.294315\pi$
$857$	2.58928e7	1.20428	0.602139	+	0.798391i	$0.294315\pi$
$858$	0	0
$859$	1.04919e7	0.485146	0.242573	−	0.970133i	$-0.422009\pi$
$859$	1.04919e7	0.485146	0.242573	+	0.970133i	$0.422009\pi$
$860$	0	0
$861$	−467284.	−0.0214819
$862$	0	0
$863$	3.11963e7	1.42586	0.712929	−	0.701237i	$-0.247368\pi$
$863$	3.11963e7	1.42586	0.712929	+	0.701237i	$0.247368\pi$
$864$	0	0
$865$	5.34947e7	2.43092
$866$	0	0
$867$	1.20608e7	0.544916
$868$	0	0
$869$	8.72954e6	0.392141
$870$	0	0
$871$	6.63010e7	2.96125
$872$	0	0
$873$	156442.	0.00694734
$874$	0	0
$875$	2.86089e6	0.126323
$876$	0	0
$877$	2.08261e7	0.914343	0.457171	−	0.889379i	$-0.348863\pi$
$877$	2.08261e7	0.914343	0.457171	+	0.889379i	$0.348863\pi$
$878$	0	0
$879$	6.24917e6	0.272803
$880$	0	0
$881$	−3.02148e6	−0.131154	−0.0655768	−	0.997848i	$-0.520889\pi$
$881$	−3.02148e6	−0.131154	−0.0655768	+	0.997848i	$0.520889\pi$
$882$	0	0
$883$	4.75951e6	0.205428	0.102714	−	0.994711i	$-0.467247\pi$
$883$	4.75951e6	0.205428	0.102714	+	0.994711i	$0.467247\pi$
$884$	0	0
$885$	−3.24444e7	−1.39245
$886$	0	0
$887$	4.69051e6	0.200176	0.100088	−	0.994979i	$-0.468088\pi$
$887$	4.69051e6	0.200176	0.100088	+	0.994979i	$0.468088\pi$
$888$	0	0
$889$	857817.	0.0364032
$890$	0	0
$891$	1.14243e6	0.0482097
$892$	0	0
$893$	−4.39962e6	−0.184623
$894$	0	0
$895$	5.16422e7	2.15500
$896$	0	0
$897$	5.34474e6	0.221792
$898$	0	0
$899$	−4.50089e6	−0.185737
$900$	0	0
$901$	2.74169e7	1.12514
$902$	0	0
$903$	−781861.	−0.0319088
$904$	0	0
$905$	−9.67199e6	−0.392550
$906$	0	0
$907$	−2.43774e7	−0.983941	−0.491971	−	0.870612i	$-0.663723\pi$
$907$	−2.43774e7	−0.983941	−0.491971	+	0.870612i	$0.663723\pi$
$908$	0	0
$909$	−563920.	−0.0226364
$910$	0	0
$911$	−3.57967e7	−1.42905	−0.714525	−	0.699610i	$-0.753357\pi$
$911$	−3.57967e7	−1.42905	−0.714525	+	0.699610i	$0.753357\pi$
$912$	0	0
$913$	1.76710e7	0.701593
$914$	0	0
$915$	−3.69594e7	−1.45939
$916$	0	0
$917$	−622580.	−0.0244496
$918$	0	0
$919$	6.89162e6	0.269174	0.134587	−	0.990902i	$-0.457029\pi$
$919$	6.89162e6	0.269174	0.134587	+	0.990902i	$0.457029\pi$
$920$	0	0
$921$	1.28641e7	0.499724
$922$	0	0
$923$	−7.65621e7	−2.95808
$924$	0	0
$925$	−3.60876e7	−1.38677
$926$	0	0
$927$	−1.54578e7	−0.590812
$928$	0	0
$929$	−3.37079e7	−1.28142	−0.640711	−	0.767782i	$-0.721361\pi$
$929$	−3.37079e7	−1.28142	−0.640711	+	0.767782i	$0.721361\pi$
$930$	0	0
$931$	9.37817e6	0.354604
$932$	0	0
$933$	−1.98360e7	−0.746019
$934$	0	0
$935$	−3.01709e7	−1.12865
$936$	0	0
$937$	−1.47245e7	−0.547886	−0.273943	−	0.961746i	$-0.588328\pi$
$937$	−1.47245e7	−0.547886	−0.273943	+	0.961746i	$0.588328\pi$
$938$	0	0
$939$	929303.	0.0343949
$940$	0	0
$941$	−1.25779e7	−0.463058	−0.231529	−	0.972828i	$-0.574373\pi$
$941$	−1.25779e7	−0.463058	−0.231529	+	0.972828i	$0.574373\pi$
$942$	0	0
$943$	4.63427e6	0.169708
$944$	0	0
$945$	450629.	0.0164149
$946$	0	0
$947$	−3.68473e6	−0.133515	−0.0667576	−	0.997769i	$-0.521265\pi$
$947$	−3.68473e6	−0.133515	−0.0667576	+	0.997769i	$0.521265\pi$
$948$	0	0
$949$	1.60662e7	0.579093
$950$	0	0
$951$	1.89390e7	0.679056
$952$	0	0
$953$	5.22323e6	0.186297	0.0931487	−	0.995652i	$-0.470307\pi$
$953$	5.22323e6	0.186297	0.0931487	+	0.995652i	$0.470307\pi$
$954$	0	0
$955$	5.82897e7	2.06816
$956$	0	0
$957$	−5.91897e6	−0.208913
$958$	0	0
$959$	1259.35	4.42179e−5 0
$960$	0	0
$961$	−2.72091e7	−0.950398
$962$	0	0
$963$	1.99605e6	0.0693594
$964$	0	0
$965$	8.05488e7	2.78446
$966$	0	0
$967$	−4.09410e7	−1.40797	−0.703984	−	0.710216i	$-0.748597\pi$
$967$	−4.09410e7	−1.40797	−0.703984	+	0.710216i	$0.748597\pi$
$968$	0	0
$969$	8.36045e6	0.286036
$970$	0	0
$971$	−3.59135e7	−1.22239	−0.611194	−	0.791480i	$-0.709311\pi$
$971$	−3.59135e7	−1.22239	−0.611194	+	0.791480i	$0.709311\pi$
$972$	0	0
$973$	1.91421e6	0.0648196
$974$	0	0
$975$	−7.83341e7	−2.63900
$976$	0	0
$977$	−1.03696e7	−0.347556	−0.173778	−	0.984785i	$-0.555598\pi$
$977$	−1.03696e7	−0.347556	−0.173778	+	0.984785i	$0.555598\pi$
$978$	0	0
$979$	1.63073e7	0.543782
$980$	0	0
$981$	−1.76177e7	−0.584491
$982$	0	0
$983$	2.95165e6	0.0974275	0.0487137	−	0.998813i	$-0.484488\pi$
$983$	2.95165e6	0.0974275	0.0487137	+	0.998813i	$0.484488\pi$
$984$	0	0
$985$	7.62780e7	2.50500
$986$	0	0
$987$	419696.	0.0137133
$988$	0	0
$989$	7.75407e6	0.252080
$990$	0	0
$991$	−2.49530e7	−0.807121	−0.403561	−	0.914953i	$-0.632228\pi$
$991$	−2.49530e7	−0.807121	−0.403561	+	0.914953i	$0.632228\pi$
$992$	0	0
$993$	−7.77209e6	−0.250129
$994$	0	0
$995$	−1.45657e7	−0.466415
$996$	0	0
$997$	−4.87774e7	−1.55411	−0.777054	−	0.629434i	$-0.783287\pi$
$997$	−4.87774e7	−1.55411	−0.777054	+	0.629434i	$0.783287\pi$
$998$	0	0
$999$	−3.39317e6	−0.107570

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	1104.6.a.p.1.4		4
4.3	odd	2		276.6.a.a.1.4	✓	4
12.11	even	2		828.6.a.a.1.1		4

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
276.6.a.a.1.4	✓	4	4.3	odd	2
828.6.a.a.1.1		4	12.11	even	2
1104.6.a.p.1.4		4	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)

\(f(q)\)	\(=\)	\(q+9.00000 q^{3} +104.299 q^{5} +5.92670 q^{7} +81.0000 q^{9} +O(q^{10})\)	\(q+9.00000 q^{3} +104.299 q^{5} +5.92670 q^{7} +81.0000 q^{9} +174.124 q^{11} -1122.61 q^{13} +938.687 q^{15} -1661.31 q^{17} -559.161 q^{19} +53.3403 q^{21} -529.000 q^{23} +7753.18 q^{25} +729.000 q^{27} -3776.98 q^{29} +1191.66 q^{31} +1567.12 q^{33} +618.146 q^{35} -4654.56 q^{37} -10103.5 q^{39} -8760.43 q^{41} -14658.0 q^{43} +8448.18 q^{45} +7868.26 q^{47} -16771.9 q^{49} -14951.8 q^{51} -16503.2 q^{53} +18160.9 q^{55} -5032.45 q^{57} -34563.6 q^{59} -39373.6 q^{61} +480.063 q^{63} -117086. q^{65} -59059.8 q^{67} -4761.00 q^{69} +68200.1 q^{71} -14311.5 q^{73} +69778.7 q^{75} +1031.98 q^{77} +50134.0 q^{79} +6561.00 q^{81} +101485. q^{83} -173272. q^{85} -33992.8 q^{87} +93653.2 q^{89} -6653.37 q^{91} +10725.0 q^{93} -58319.6 q^{95} +1931.38 q^{97} +14104.1 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 4 q + 36 q^{3} + 22 q^{5} + 154 q^{7} + 324 q^{9}+O(q^{10}) \) 4 * q + 36 * q^3 + 22 * q^5 + 154 * q^7 + 324 * q^9	\( 4 q + 36 q^{3} + 22 q^{5} + 154 q^{7} + 324 q^{9} - 392 q^{11} - 220 q^{13} + 198 q^{15} - 802 q^{17} - 2590 q^{19} + 1386 q^{21} - 2116 q^{23} + 6904 q^{25} + 2916 q^{27} - 5036 q^{29} - 20 q^{31} - 3528 q^{33} + 3752 q^{35} - 10884 q^{37} - 1980 q^{39} - 23312 q^{41} - 4406 q^{43} + 1782 q^{45} - 2472 q^{47} - 24568 q^{49} - 7218 q^{51} - 29222 q^{53} + 45928 q^{55} - 23310 q^{57} - 15136 q^{59} - 113220 q^{61} + 12474 q^{63} - 133892 q^{65} + 23214 q^{67} - 19044 q^{69} + 96096 q^{71} - 95792 q^{73} + 62136 q^{75} - 96584 q^{77} + 190094 q^{79} + 26244 q^{81} + 112280 q^{83} - 276556 q^{85} - 45324 q^{87} - 52206 q^{89} + 141548 q^{91} - 180 q^{93} - 38288 q^{95} + 9636 q^{97} - 31752 q^{99}+O(q^{100}) \) 4 * q + 36 * q^3 + 22 * q^5 + 154 * q^7 + 324 * q^9 - 392 * q^11 - 220 * q^13 + 198 * q^15 - 802 * q^17 - 2590 * q^19 + 1386 * q^21 - 2116 * q^23 + 6904 * q^25 + 2916 * q^27 - 5036 * q^29 - 20 * q^31 - 3528 * q^33 + 3752 * q^35 - 10884 * q^37 - 1980 * q^39 - 23312 * q^41 - 4406 * q^43 + 1782 * q^45 - 2472 * q^47 - 24568 * q^49 - 7218 * q^51 - 29222 * q^53 + 45928 * q^55 - 23310 * q^57 - 15136 * q^59 - 113220 * q^61 + 12474 * q^63 - 133892 * q^65 + 23214 * q^67 - 19044 * q^69 + 96096 * q^71 - 95792 * q^73 + 62136 * q^75 - 96584 * q^77 + 190094 * q^79 + 26244 * q^81 + 112280 * q^83 - 276556 * q^85 - 45324 * q^87 - 52206 * q^89 + 141548 * q^91 - 180 * q^93 - 38288 * q^95 + 9636 * q^97 - 31752 * q^99

Introduction

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