LMFDB - Embedded newform 1104.6.a.o.1.2

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:	$4$
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} - 75x^{2} - 42x + 736 $ x^4 - 75x^2 - 42x + 736
Coefficient ring:	$\Z[a_1, \ldots, a_{11}]$
Coefficient ring index:	$ 2^{4} $
Twist minimal:	no (minimal twist has level 69)
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.2
Root			$8.33314$ 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)

$f(q)$	$=$	$q+9.00000 q^{3} -0.408582 q^{5} +4.34307 q^{7} +81.0000 q^{9} +O(q^{10})$	$q+9.00000 q^{3} -0.408582 q^{5} +4.34307 q^{7} +81.0000 q^{9} -428.488 q^{11} +76.4506 q^{13} -3.67724 q^{15} -71.8173 q^{17} +1731.89 q^{19} +39.0877 q^{21} +529.000 q^{23} -3124.83 q^{25} +729.000 q^{27} -1666.22 q^{29} +3620.99 q^{31} -3856.40 q^{33} -1.77450 q^{35} -1469.54 q^{37} +688.055 q^{39} +7590.26 q^{41} -5304.35 q^{43} -33.0952 q^{45} +4933.30 q^{47} -16788.1 q^{49} -646.356 q^{51} +3095.71 q^{53} +175.073 q^{55} +15587.1 q^{57} +10750.0 q^{59} -21156.5 q^{61} +351.789 q^{63} -31.2364 q^{65} +26405.9 q^{67} +4761.00 q^{69} +58175.6 q^{71} +411.889 q^{73} -28123.5 q^{75} -1860.96 q^{77} +62132.2 q^{79} +6561.00 q^{81} +91527.4 q^{83} +29.3433 q^{85} -14996.0 q^{87} -80408.4 q^{89} +332.031 q^{91} +32588.9 q^{93} -707.621 q^{95} +118915. q^{97} -34707.6 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 4 q + 36 q^{3} + 22 q^{5} + 62 q^{7} + 324 q^{9}+O(q^{10}) $ 4 * q + 36 * q^3 + 22 * q^5 + 62 * q^7 + 324 * q^9	$ 4 q + 36 q^{3} + 22 q^{5} + 62 q^{7} + 324 q^{9} + 1076 q^{11} - 396 q^{13} + 198 q^{15} + 70 q^{17} + 6366 q^{19} + 558 q^{21} + 2116 q^{23} + 1264 q^{25} + 2916 q^{27} + 3948 q^{29} - 3092 q^{31} + 9684 q^{33} - 1304 q^{35} - 17464 q^{37} - 3564 q^{39} + 18680 q^{41} + 25846 q^{43} + 1782 q^{45} - 18392 q^{47} + 7952 q^{49} + 630 q^{51} - 26518 q^{53} + 40848 q^{55} + 57294 q^{57} + 14520 q^{59} - 13688 q^{61} + 5022 q^{63} + 38324 q^{65} + 11098 q^{67} + 19044 q^{69} + 57496 q^{71} - 112272 q^{73} + 11376 q^{75} - 4792 q^{77} + 240754 q^{79} + 26244 q^{81} + 93268 q^{83} - 323204 q^{85} + 35532 q^{87} - 107582 q^{89} + 301532 q^{91} - 27828 q^{93} + 18640 q^{95} - 53076 q^{97} + 87156 q^{99}+O(q^{100}) $ 4 * q + 36 * q^3 + 22 * q^5 + 62 * q^7 + 324 * q^9 + 1076 * q^11 - 396 * q^13 + 198 * q^15 + 70 * q^17 + 6366 * q^19 + 558 * q^21 + 2116 * q^23 + 1264 * q^25 + 2916 * q^27 + 3948 * q^29 - 3092 * q^31 + 9684 * q^33 - 1304 * q^35 - 17464 * q^37 - 3564 * q^39 + 18680 * q^41 + 25846 * q^43 + 1782 * q^45 - 18392 * q^47 + 7952 * q^49 + 630 * q^51 - 26518 * q^53 + 40848 * q^55 + 57294 * q^57 + 14520 * q^59 - 13688 * q^61 + 5022 * q^63 + 38324 * q^65 + 11098 * q^67 + 19044 * q^69 + 57496 * q^71 - 112272 * q^73 + 11376 * q^75 - 4792 * q^77 + 240754 * q^79 + 26244 * q^81 + 93268 * q^83 - 323204 * q^85 + 35532 * q^87 - 107582 * q^89 + 301532 * q^91 - 27828 * q^93 + 18640 * q^95 - 53076 * q^97 + 87156 * 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
$5$	−0.408582	−0.00730894	−0.00365447	−	0.999993i	$-0.501163\pi$
$5$	−0.408582	−0.00730894	−0.00365447	+	0.999993i	$0.501163\pi$
$6$	0	0
$7$	4.34307	0.0335006	0.0167503	−	0.999860i	$-0.494668\pi$
$7$	4.34307	0.0335006	0.0167503	+	0.999860i	$0.494668\pi$
$8$	0	0
$9$	81.0000	0.333333
$10$	0	0
$11$	−428.488	−1.06772	−0.533860	−	0.845573i	$-0.679259\pi$
$11$	−428.488	−1.06772	−0.533860	+	0.845573i	$0.679259\pi$
$12$	0	0
$13$	76.4506	0.125465	0.0627325	−	0.998030i	$-0.480019\pi$
$13$	76.4506	0.125465	0.0627325	+	0.998030i	$0.480019\pi$
$14$	0	0
$15$	−3.67724	−0.00421982
$16$	0	0
$17$	−71.8173	−0.0602708	−0.0301354	−	0.999546i	$-0.509594\pi$
$17$	−71.8173	−0.0602708	−0.0301354	+	0.999546i	$0.509594\pi$
$18$	0	0
$19$	1731.89	1.10062	0.550310	−	0.834960i	$-0.314509\pi$
$19$	1731.89	1.10062	0.550310	+	0.834960i	$0.314509\pi$
$20$	0	0
$21$	39.0877	0.0193416
$22$	0	0
$23$	529.000	0.208514
$23$	529.000	0.208514
$24$	0	0
$25$	−3124.83	−0.999947
$26$	0	0
$27$	729.000	0.192450
$28$	0	0
$29$	−1666.22	−0.367907	−0.183953	−	0.982935i	$-0.558890\pi$
$29$	−1666.22	−0.367907	−0.183953	+	0.982935i	$0.558890\pi$
$30$	0	0
$31$	3620.99	0.676742	0.338371	−	0.941013i	$-0.390124\pi$
$31$	3620.99	0.676742	0.338371	+	0.941013i	$0.390124\pi$
$32$	0	0
$33$	−3856.40	−0.616448
$34$	0	0
$35$	−1.77450	−0.000244854 0
$36$	0	0
$37$	−1469.54	−0.176473	−0.0882365	−	0.996100i	$-0.528123\pi$
$37$	−1469.54	−0.176473	−0.0882365	+	0.996100i	$0.528123\pi$
$38$	0	0
$39$	688.055	0.0724372
$40$	0	0
$41$	7590.26	0.705176	0.352588	−	0.935779i	$-0.385302\pi$
$41$	7590.26	0.705176	0.352588	+	0.935779i	$0.385302\pi$
$42$	0	0
$43$	−5304.35	−0.437483	−0.218741	−	0.975783i	$-0.570195\pi$
$43$	−5304.35	−0.437483	−0.218741	+	0.975783i	$0.570195\pi$
$44$	0	0
$45$	−33.0952	−0.00243631
$46$	0	0
$47$	4933.30	0.325756	0.162878	−	0.986646i	$-0.447922\pi$
$47$	4933.30	0.325756	0.162878	+	0.986646i	$0.447922\pi$
$48$	0	0
$49$	−16788.1	−0.998878
$50$	0	0
$51$	−646.356	−0.0347973
$52$	0	0
$53$	3095.71	0.151381	0.0756904	−	0.997131i	$-0.475884\pi$
$53$	3095.71	0.151381	0.0756904	+	0.997131i	$0.475884\pi$
$54$	0	0
$55$	175.073	0.00780390
$56$	0	0
$57$	15587.1	0.635443
$58$	0	0
$59$	10750.0	0.402047	0.201024	−	0.979586i	$-0.435573\pi$
$59$	10750.0	0.402047	0.201024	+	0.979586i	$0.435573\pi$
$60$	0	0
$61$	−21156.5	−0.727980	−0.363990	−	0.931403i	$-0.618586\pi$
$61$	−21156.5	−0.727980	−0.363990	+	0.931403i	$0.618586\pi$
$62$	0	0
$63$	351.789	0.0111669
$64$	0	0
$65$	−31.2364	−0.000917016 0
$66$	0	0
$67$	26405.9	0.718645	0.359322	−	0.933214i	$-0.383008\pi$
$67$	26405.9	0.718645	0.359322	+	0.933214i	$0.383008\pi$
$68$	0	0
$69$	4761.00	0.120386
$70$	0	0
$71$	58175.6	1.36960	0.684802	−	0.728730i	$-0.259889\pi$
$71$	58175.6	1.36960	0.684802	+	0.728730i	$0.259889\pi$
$72$	0	0
$73$	411.889	0.00904633	0.00452317	−	0.999990i	$-0.498560\pi$
$73$	411.889	0.00904633	0.00452317	+	0.999990i	$0.498560\pi$
$74$	0	0
$75$	−28123.5	−0.577319
$76$	0	0
$77$	−1860.96	−0.0357692
$78$	0	0
$79$	62132.2	1.12008	0.560040	−	0.828465i	$-0.310786\pi$
$79$	62132.2	1.12008	0.560040	+	0.828465i	$0.310786\pi$
$80$	0	0
$81$	6561.00	0.111111
$82$	0	0
$83$	91527.4	1.45833	0.729165	−	0.684338i	$-0.239909\pi$
$83$	91527.4	1.45833	0.729165	+	0.684338i	$0.239909\pi$
$84$	0	0
$85$	29.3433	0.000440515 0
$86$	0	0
$87$	−14996.0	−0.212411
$88$	0	0
$89$	−80408.4	−1.07603	−0.538017	−	0.842934i	$-0.680827\pi$
$89$	−80408.4	−1.07603	−0.538017	+	0.842934i	$0.680827\pi$
$90$	0	0
$91$	332.031	0.00420315
$92$	0	0
$93$	32588.9	0.390717
$94$	0	0
$95$	−707.621	−0.00804437
$96$	0	0
$97$	118915.	1.28323	0.641617	−	0.767025i	$-0.278264\pi$
$97$	118915.	1.28323	0.641617	+	0.767025i	$0.278264\pi$
$98$	0	0
$99$	−34707.6	−0.355907
$100$	0	0
$101$	102784.	1.00258	0.501292	−	0.865278i	$-0.332858\pi$
$101$	102784.	1.00258	0.501292	+	0.865278i	$0.332858\pi$
$102$	0	0
$103$	−120582.	−1.11993	−0.559965	−	0.828516i	$-0.689186\pi$
$103$	−120582.	−1.11993	−0.559965	+	0.828516i	$0.689186\pi$
$104$	0	0
$105$	−15.9705	−0.000141366 0
$106$	0	0
$107$	31567.6	0.266552	0.133276	−	0.991079i	$-0.457450\pi$
$107$	31567.6	0.266552	0.133276	+	0.991079i	$0.457450\pi$
$108$	0	0
$109$	−119904.	−0.966642	−0.483321	−	0.875443i	$-0.660570\pi$
$109$	−119904.	−0.966642	−0.483321	+	0.875443i	$0.660570\pi$
$110$	0	0
$111$	−13225.9	−0.101887
$112$	0	0
$113$	−13937.4	−0.102680	−0.0513399	−	0.998681i	$-0.516349\pi$
$113$	−13937.4	−0.102680	−0.0513399	+	0.998681i	$0.516349\pi$
$114$	0	0
$115$	−216.140	−0.00152402
$116$	0	0
$117$	6192.50	0.0418217
$118$	0	0
$119$	−311.908	−0.00201911
$120$	0	0
$121$	22551.3	0.140026
$122$	0	0
$123$	68312.4	0.407133
$124$	0	0
$125$	2553.57	0.0146175
$126$	0	0
$127$	51910.3	0.285591	0.142795	−	0.989752i	$-0.454391\pi$
$127$	51910.3	0.285591	0.142795	+	0.989752i	$0.454391\pi$
$128$	0	0
$129$	−47739.1	−0.252581
$130$	0	0
$131$	37036.5	0.188561	0.0942805	−	0.995546i	$-0.469945\pi$
$131$	37036.5	0.188561	0.0942805	+	0.995546i	$0.469945\pi$
$132$	0	0
$133$	7521.75	0.0368714
$134$	0	0
$135$	−297.856	−0.00140661
$136$	0	0
$137$	−20689.2	−0.0941762	−0.0470881	−	0.998891i	$-0.514994\pi$
$137$	−20689.2	−0.0941762	−0.0470881	+	0.998891i	$0.514994\pi$
$138$	0	0
$139$	−45101.7	−0.197996	−0.0989978	−	0.995088i	$-0.531564\pi$
$139$	−45101.7	−0.197996	−0.0989978	+	0.995088i	$0.531564\pi$
$140$	0	0
$141$	44399.7	0.188075
$142$	0	0
$143$	−32758.2	−0.133961
$144$	0	0
$145$	680.789	0.00268901
$146$	0	0
$147$	−151093.	−0.576702
$148$	0	0
$149$	429347.	1.58432	0.792159	−	0.610314i	$-0.208957\pi$
$149$	429347.	1.58432	0.792159	+	0.610314i	$0.208957\pi$
$150$	0	0
$151$	153762.	0.548791	0.274395	−	0.961617i	$-0.411522\pi$
$151$	153762.	0.548791	0.274395	+	0.961617i	$0.411522\pi$
$152$	0	0
$153$	−5817.20	−0.0200903
$154$	0	0
$155$	−1479.47	−0.00494627
$156$	0	0
$157$	469295.	1.51949	0.759743	−	0.650224i	$-0.225325\pi$
$157$	469295.	1.51949	0.759743	+	0.650224i	$0.225325\pi$
$158$	0	0
$159$	27861.4	0.0873997
$160$	0	0
$161$	2297.49	0.00698535
$162$	0	0
$163$	300211.	0.885030	0.442515	−	0.896761i	$-0.354086\pi$
$163$	300211.	0.885030	0.442515	+	0.896761i	$0.354086\pi$
$164$	0	0
$165$	1575.65	0.00450558
$166$	0	0
$167$	−83003.5	−0.230306	−0.115153	−	0.993348i	$-0.536736\pi$
$167$	−83003.5	−0.230306	−0.115153	+	0.993348i	$0.536736\pi$
$168$	0	0
$169$	−365448.	−0.984259
$170$	0	0
$171$	140283.	0.366873
$172$	0	0
$173$	−246304.	−0.625686	−0.312843	−	0.949805i	$-0.601281\pi$
$173$	−246304.	−0.625686	−0.312843	+	0.949805i	$0.601281\pi$
$174$	0	0
$175$	−13571.4	−0.0334988
$176$	0	0
$177$	96749.7	0.232122
$178$	0	0
$179$	778110.	1.81513	0.907567	−	0.419907i	$-0.137937\pi$
$179$	778110.	1.81513	0.907567	+	0.419907i	$0.137937\pi$
$180$	0	0
$181$	81753.5	0.185485	0.0927427	−	0.995690i	$-0.470437\pi$
$181$	81753.5	0.185485	0.0927427	+	0.995690i	$0.470437\pi$
$182$	0	0
$183$	−190409.	−0.420299
$184$	0	0
$185$	600.430	0.00128983
$186$	0	0
$187$	30772.9	0.0643523
$188$	0	0
$189$	3166.10	0.00644719
$190$	0	0
$191$	194867.	0.386505	0.193252	−	0.981149i	$-0.438096\pi$
$191$	194867.	0.386505	0.193252	+	0.981149i	$0.438096\pi$
$192$	0	0
$193$	622869.	1.20366	0.601830	−	0.798624i	$-0.294439\pi$
$193$	622869.	1.20366	0.601830	+	0.798624i	$0.294439\pi$
$194$	0	0
$195$	−281.127	−0.000529440 0
$196$	0	0
$197$	−279380.	−0.512896	−0.256448	−	0.966558i	$-0.582552\pi$
$197$	−279380.	−0.512896	−0.256448	+	0.966558i	$0.582552\pi$
$198$	0	0
$199$	715118.	1.28010	0.640052	−	0.768332i	$-0.278913\pi$
$199$	715118.	1.28010	0.640052	+	0.768332i	$0.278913\pi$
$200$	0	0
$201$	237653.	0.414910
$202$	0	0
$203$	−7236.53	−0.0123251
$204$	0	0
$205$	−3101.25	−0.00515409
$206$	0	0
$207$	42849.0	0.0695048
$208$	0	0
$209$	−742097.	−1.17515
$210$	0	0
$211$	−236093.	−0.365071	−0.182535	−	0.983199i	$-0.558430\pi$
$211$	−236093.	−0.365071	−0.182535	+	0.983199i	$0.558430\pi$
$212$	0	0
$213$	523580.	0.790741
$214$	0	0
$215$	2167.26	0.00319754
$216$	0	0
$217$	15726.2	0.0226712
$218$	0	0
$219$	3707.00	0.00522290
$220$	0	0
$221$	−5490.48	−0.00756187
$222$	0	0
$223$	−312172.	−0.420370	−0.210185	−	0.977662i	$-0.567407\pi$
$223$	−312172.	−0.420370	−0.210185	+	0.977662i	$0.567407\pi$
$224$	0	0
$225$	−253111.	−0.333316
$226$	0	0
$227$	−706271.	−0.909718	−0.454859	−	0.890564i	$-0.650310\pi$
$227$	−706271.	−0.909718	−0.454859	+	0.890564i	$0.650310\pi$
$228$	0	0
$229$	−524546.	−0.660990	−0.330495	−	0.943808i	$-0.607216\pi$
$229$	−524546.	−0.660990	−0.330495	+	0.943808i	$0.607216\pi$
$230$	0	0
$231$	−16748.6	−0.0206514
$232$	0	0
$233$	1.44203e6	1.74014	0.870072	−	0.492925i	$-0.164072\pi$
$233$	1.44203e6	1.74014	0.870072	+	0.492925i	$0.164072\pi$
$234$	0	0
$235$	−2015.66	−0.00238093
$236$	0	0
$237$	559190.	0.646679
$238$	0	0
$239$	−447167.	−0.506379	−0.253189	−	0.967417i	$-0.581480\pi$
$239$	−447167.	−0.506379	−0.253189	+	0.967417i	$0.581480\pi$
$240$	0	0
$241$	443110.	0.491438	0.245719	−	0.969341i	$-0.420976\pi$
$241$	443110.	0.491438	0.245719	+	0.969341i	$0.420976\pi$
$242$	0	0
$243$	59049.0	0.0641500
$244$	0	0
$245$	6859.33	0.00730074
$246$	0	0
$247$	132404.	0.138089
$248$	0	0
$249$	823746.	0.841967
$250$	0	0
$251$	−516201.	−0.517172	−0.258586	−	0.965988i	$-0.583256\pi$
$251$	−516201.	−0.517172	−0.258586	+	0.965988i	$0.583256\pi$
$252$	0	0
$253$	−226670.	−0.222635
$254$	0	0
$255$	264.089	0.000254332 0
$256$	0	0
$257$	950802.	0.897961	0.448980	−	0.893542i	$-0.351787\pi$
$257$	950802.	0.897961	0.448980	+	0.893542i	$0.351787\pi$
$258$	0	0
$259$	−6382.34	−0.00591195
$260$	0	0
$261$	−134964.	−0.122636
$262$	0	0
$263$	−1.96344e6	−1.75036	−0.875180	−	0.483797i	$-0.839257\pi$
$263$	−1.96344e6	−1.75036	−0.875180	+	0.483797i	$0.839257\pi$
$264$	0	0
$265$	−1264.85	−0.00110643
$266$	0	0
$267$	−723675.	−0.621249
$268$	0	0
$269$	2.10595e6	1.77447	0.887234	−	0.461320i	$-0.152624\pi$
$269$	2.10595e6	1.77447	0.887234	+	0.461320i	$0.152624\pi$
$270$	0	0
$271$	2.30306e6	1.90494	0.952471	−	0.304629i	$-0.0985323\pi$
$271$	2.30306e6	1.90494	0.952471	+	0.304629i	$0.0985323\pi$
$272$	0	0
$273$	2988.28	0.00242669
$274$	0	0
$275$	1.33895e6	1.06766
$276$	0	0
$277$	−1.76069e6	−1.37875	−0.689374	−	0.724406i	$-0.742114\pi$
$277$	−1.76069e6	−1.37875	−0.689374	+	0.724406i	$0.742114\pi$
$278$	0	0
$279$	293300.	0.225581
$280$	0	0
$281$	1.82573e6	1.37934	0.689669	−	0.724125i	$-0.257756\pi$
$281$	1.82573e6	1.37934	0.689669	+	0.724125i	$0.257756\pi$
$282$	0	0
$283$	−1.72696e6	−1.28179	−0.640893	−	0.767631i	$-0.721436\pi$
$283$	−1.72696e6	−1.28179	−0.640893	+	0.767631i	$0.721436\pi$
$284$	0	0
$285$	−6368.59	−0.00464442
$286$	0	0
$287$	32965.1	0.0236238
$288$	0	0
$289$	−1.41470e6	−0.996367
$290$	0	0
$291$	1.07023e6	0.740876
$292$	0	0
$293$	405581.	0.276000	0.138000	−	0.990432i	$-0.455933\pi$
$293$	405581.	0.276000	0.138000	+	0.990432i	$0.455933\pi$
$294$	0	0
$295$	−4392.25	−0.00293854
$296$	0	0
$297$	−312368.	−0.205483
$298$	0	0
$299$	40442.4	0.0261613
$300$	0	0
$301$	−23037.2	−0.0146559
$302$	0	0
$303$	925053.	0.578842
$304$	0	0
$305$	8644.17	0.00532076
$306$	0	0
$307$	2.61976e6	1.58641	0.793205	−	0.608954i	$-0.208411\pi$
$307$	2.61976e6	1.58641	0.793205	+	0.608954i	$0.208411\pi$
$308$	0	0
$309$	−1.08524e6	−0.646592
$310$	0	0
$311$	−1.54756e6	−0.907293	−0.453646	−	0.891182i	$-0.649877\pi$
$311$	−1.54756e6	−0.907293	−0.453646	+	0.891182i	$0.649877\pi$
$312$	0	0
$313$	369596.	0.213239	0.106619	−	0.994300i	$-0.465997\pi$
$313$	369596.	0.213239	0.106619	+	0.994300i	$0.465997\pi$
$314$	0	0
$315$	−143.735	−8.16179e−5 0
$316$	0	0
$317$	−2.00158e6	−1.11873	−0.559364	−	0.828922i	$-0.688955\pi$
$317$	−2.00158e6	−1.11873	−0.559364	+	0.828922i	$0.688955\pi$
$318$	0	0
$319$	713957.	0.392822
$320$	0	0
$321$	284109.	0.153894
$322$	0	0
$323$	−124380.	−0.0663352
$324$	0	0
$325$	−238895.	−0.125458
$326$	0	0
$327$	−1.07913e6	−0.558091
$328$	0	0
$329$	21425.7	0.0109130
$330$	0	0
$331$	201176.	0.100927	0.0504635	−	0.998726i	$-0.483930\pi$
$331$	201176.	0.100927	0.0504635	+	0.998726i	$0.483930\pi$
$332$	0	0
$333$	−119033.	−0.0588244
$334$	0	0
$335$	−10789.0	−0.00525253
$336$	0	0
$337$	1.33440e6	0.640044	0.320022	−	0.947410i	$-0.396310\pi$
$337$	1.33440e6	0.640044	0.320022	+	0.947410i	$0.396310\pi$
$338$	0	0
$339$	−125436.	−0.0592822
$340$	0	0
$341$	−1.55155e6	−0.722571
$342$	0	0
$343$	−145906.	−0.0669635
$344$	0	0
$345$	−1945.26	−0.000879893 0
$346$	0	0
$347$	4.16940e6	1.85887	0.929436	−	0.368983i	$-0.120294\pi$
$347$	4.16940e6	1.85887	0.929436	+	0.368983i	$0.120294\pi$
$348$	0	0
$349$	−1.02682e6	−0.451263	−0.225631	−	0.974213i	$-0.572444\pi$
$349$	−1.02682e6	−0.451263	−0.225631	+	0.974213i	$0.572444\pi$
$350$	0	0
$351$	55732.5	0.0241457
$352$	0	0
$353$	4.39424e6	1.87693	0.938463	−	0.345379i	$-0.112250\pi$
$353$	4.39424e6	1.87693	0.938463	+	0.345379i	$0.112250\pi$
$354$	0	0
$355$	−23769.5	−0.0100103
$356$	0	0
$357$	−2807.17	−0.00116573
$358$	0	0
$359$	2.65847e6	1.08867	0.544334	−	0.838869i	$-0.316783\pi$
$359$	2.65847e6	1.08867	0.544334	+	0.838869i	$0.316783\pi$
$360$	0	0
$361$	523361.	0.211365
$362$	0	0
$363$	202961.	0.0808439
$364$	0	0
$365$	−168.290	−6.61191e−5 0
$366$	0	0
$367$	3.96638e6	1.53720	0.768598	−	0.639733i	$-0.220955\pi$
$367$	3.96638e6	1.53720	0.768598	+	0.639733i	$0.220955\pi$
$368$	0	0
$369$	614811.	0.235059
$370$	0	0
$371$	13444.9	0.00507134
$372$	0	0
$373$	996522.	0.370864	0.185432	−	0.982657i	$-0.440632\pi$
$373$	996522.	0.370864	0.185432	+	0.982657i	$0.440632\pi$
$374$	0	0
$375$	22982.1	0.00843941
$376$	0	0
$377$	−127384.	−0.0461594
$378$	0	0
$379$	2.86660e6	1.02511	0.512554	−	0.858655i	$-0.328699\pi$
$379$	2.86660e6	1.02511	0.512554	+	0.858655i	$0.328699\pi$
$380$	0	0
$381$	467193.	0.164886
$382$	0	0
$383$	1.89544e6	0.660256	0.330128	−	0.943936i	$-0.392908\pi$
$383$	1.89544e6	0.660256	0.330128	+	0.943936i	$0.392908\pi$
$384$	0	0
$385$	760.354	0.000261435 0
$386$	0	0
$387$	−429652.	−0.145828
$388$	0	0
$389$	1.32321e6	0.443357	0.221678	−	0.975120i	$-0.428847\pi$
$389$	1.32321e6	0.443357	0.221678	+	0.975120i	$0.428847\pi$
$390$	0	0
$391$	−37991.3	−0.0125673
$392$	0	0
$393$	333329.	0.108866
$394$	0	0
$395$	−25386.1	−0.00818660
$396$	0	0
$397$	−2.73318e6	−0.870346	−0.435173	−	0.900347i	$-0.643313\pi$
$397$	−2.73318e6	−0.870346	−0.435173	+	0.900347i	$0.643313\pi$
$398$	0	0
$399$	67695.7	0.0212877
$400$	0	0
$401$	−3.51245e6	−1.09081	−0.545405	−	0.838173i	$-0.683624\pi$
$401$	−3.51245e6	−1.09081	−0.545405	+	0.838173i	$0.683624\pi$
$402$	0	0
$403$	276827.	0.0849074
$404$	0	0
$405$	−2680.71	−0.000812104 0
$406$	0	0
$407$	629683.	0.188424
$408$	0	0
$409$	1.01219e6	0.299195	0.149597	−	0.988747i	$-0.452202\pi$
$409$	1.01219e6	0.299195	0.149597	+	0.988747i	$0.452202\pi$
$410$	0	0
$411$	−186202.	−0.0543727
$412$	0	0
$413$	46687.9	0.0134688
$414$	0	0
$415$	−37396.5	−0.0106588
$416$	0	0
$417$	−405915.	−0.114313
$418$	0	0
$419$	4.96595e6	1.38187	0.690935	−	0.722917i	$-0.257199\pi$
$419$	4.96595e6	1.38187	0.690935	+	0.722917i	$0.257199\pi$
$420$	0	0
$421$	6.28423e6	1.72801	0.864007	−	0.503480i	$-0.167947\pi$
$421$	6.28423e6	1.72801	0.864007	+	0.503480i	$0.167947\pi$
$422$	0	0
$423$	399597.	0.108585
$424$	0	0
$425$	224417.	0.0602675
$426$	0	0
$427$	−91884.3	−0.0243877
$428$	0	0
$429$	−294824.	−0.0773427
$430$	0	0
$431$	2.44198e6	0.633212	0.316606	−	0.948557i	$-0.397457\pi$
$431$	2.44198e6	0.633212	0.316606	+	0.948557i	$0.397457\pi$
$432$	0	0
$433$	3.22990e6	0.827885	0.413942	−	0.910303i	$-0.364151\pi$
$433$	3.22990e6	0.827885	0.413942	+	0.910303i	$0.364151\pi$
$434$	0	0
$435$	6127.10	0.00155250
$436$	0	0
$437$	916172.	0.229495
$438$	0	0
$439$	−1.41329e6	−0.350002	−0.175001	−	0.984568i	$-0.555993\pi$
$439$	−1.41329e6	−0.350002	−0.175001	+	0.984568i	$0.555993\pi$
$440$	0	0
$441$	−1.35984e6	−0.332959
$442$	0	0
$443$	−1.95693e6	−0.473769	−0.236885	−	0.971538i	$-0.576126\pi$
$443$	−1.95693e6	−0.473769	−0.236885	+	0.971538i	$0.576126\pi$
$444$	0	0
$445$	32853.4	0.00786467
$446$	0	0
$447$	3.86412e6	0.914707
$448$	0	0
$449$	3.44271e6	0.805907	0.402954	−	0.915220i	$-0.367984\pi$
$449$	3.44271e6	0.805907	0.402954	+	0.915220i	$0.367984\pi$
$450$	0	0
$451$	−3.25234e6	−0.752930
$452$	0	0
$453$	1.38386e6	0.316844
$454$	0	0
$455$	−135.662	−3.07206e−5 0
$456$	0	0
$457$	−5.91669e6	−1.32522	−0.662610	−	0.748964i	$-0.730551\pi$
$457$	−5.91669e6	−1.32522	−0.662610	+	0.748964i	$0.730551\pi$
$458$	0	0
$459$	−52354.8	−0.0115991
$460$	0	0
$461$	1.76919e6	0.387724	0.193862	−	0.981029i	$-0.437899\pi$
$461$	1.76919e6	0.387724	0.193862	+	0.981029i	$0.437899\pi$
$462$	0	0
$463$	−4.37655e6	−0.948811	−0.474405	−	0.880307i	$-0.657337\pi$
$463$	−4.37655e6	−0.948811	−0.474405	+	0.880307i	$0.657337\pi$
$464$	0	0
$465$	−13315.2	−0.00285573
$466$	0	0
$467$	2.11837e6	0.449478	0.224739	−	0.974419i	$-0.427847\pi$
$467$	2.11837e6	0.449478	0.224739	+	0.974419i	$0.427847\pi$
$468$	0	0
$469$	114683.	0.0240750
$470$	0	0
$471$	4.22365e6	0.877276
$472$	0	0
$473$	2.27285e6	0.467109
$474$	0	0
$475$	−5.41188e6	−1.10056
$476$	0	0
$477$	250753.	0.0504603
$478$	0	0
$479$	−3.77749e6	−0.752255	−0.376127	−	0.926568i	$-0.622744\pi$
$479$	−3.77749e6	−0.752255	−0.376127	+	0.926568i	$0.622744\pi$
$480$	0	0
$481$	−112348.	−0.0221412
$482$	0	0
$483$	20677.4	0.00403299
$484$	0	0
$485$	−48586.4	−0.00937908
$486$	0	0
$487$	−1.00545e7	−1.92105	−0.960524	−	0.278197i	$-0.910263\pi$
$487$	−1.00545e7	−1.92105	−0.960524	+	0.278197i	$0.910263\pi$
$488$	0	0
$489$	2.70190e6	0.510972
$490$	0	0
$491$	−4.65796e6	−0.871950	−0.435975	−	0.899959i	$-0.643596\pi$
$491$	−4.65796e6	−0.871950	−0.435975	+	0.899959i	$0.643596\pi$
$492$	0	0
$493$	119664.	0.0221740
$494$	0	0
$495$	14180.9	0.00260130
$496$	0	0
$497$	252661.	0.0458825
$498$	0	0
$499$	6.85708e6	1.23279	0.616393	−	0.787439i	$-0.288593\pi$
$499$	6.85708e6	1.23279	0.616393	+	0.787439i	$0.288593\pi$
$500$	0	0
$501$	−747032.	−0.132967
$502$	0	0
$503$	5.51954e6	0.972709	0.486354	−	0.873762i	$-0.338326\pi$
$503$	5.51954e6	0.972709	0.486354	+	0.873762i	$0.338326\pi$
$504$	0	0
$505$	−41995.6	−0.00732782
$506$	0	0
$507$	−3.28903e6	−0.568262
$508$	0	0
$509$	9.87497e6	1.68943	0.844717	−	0.535213i	$-0.179769\pi$
$509$	9.87497e6	1.68943	0.844717	+	0.535213i	$0.179769\pi$
$510$	0	0
$511$	1788.86	0.000303057 0
$512$	0	0
$513$	1.26255e6	0.211814
$514$	0	0
$515$	49267.8	0.00818551
$516$	0	0
$517$	−2.11386e6	−0.347816
$518$	0	0
$519$	−2.21674e6	−0.361240
$520$	0	0
$521$	−709124.	−0.114453	−0.0572266	−	0.998361i	$-0.518226\pi$
$521$	−709124.	−0.114453	−0.0572266	+	0.998361i	$0.518226\pi$
$522$	0	0
$523$	1.02536e7	1.63916	0.819581	−	0.572964i	$-0.194207\pi$
$523$	1.02536e7	1.63916	0.819581	+	0.572964i	$0.194207\pi$
$524$	0	0
$525$	−122142.	−0.0193405
$526$	0	0
$527$	−260050.	−0.0407877
$528$	0	0
$529$	279841.	0.0434783
$530$	0	0
$531$	870747.	0.134016
$532$	0	0
$533$	580280.	0.0884748
$534$	0	0
$535$	−12898.0	−0.00194821
$536$	0	0
$537$	7.00299e6	1.04797
$538$	0	0
$539$	7.19352e6	1.06652
$540$	0	0
$541$	−2.50498e6	−0.367969	−0.183984	−	0.982929i	$-0.558900\pi$
$541$	−2.50498e6	−0.367969	−0.183984	+	0.982929i	$0.558900\pi$
$542$	0	0
$543$	735781.	0.107090
$544$	0	0
$545$	48990.5	0.00706513
$546$	0	0
$547$	−5.52668e6	−0.789762	−0.394881	−	0.918732i	$-0.629214\pi$
$547$	−5.52668e6	−0.789762	−0.394881	+	0.918732i	$0.629214\pi$
$548$	0	0
$549$	−1.71368e6	−0.242660
$550$	0	0
$551$	−2.88572e6	−0.404926
$552$	0	0
$553$	269845.	0.0375233
$554$	0	0
$555$	5403.87	0.000744684 0
$556$	0	0
$557$	−1.08390e7	−1.48031	−0.740156	−	0.672436i	$-0.765248\pi$
$557$	−1.08390e7	−1.48031	−0.740156	+	0.672436i	$0.765248\pi$
$558$	0	0
$559$	−405521.	−0.0548888
$560$	0	0
$561$	276956.	0.0371538
$562$	0	0
$563$	5.90889e6	0.785661	0.392831	−	0.919611i	$-0.371496\pi$
$563$	5.90889e6	0.785661	0.392831	+	0.919611i	$0.371496\pi$
$564$	0	0
$565$	5694.56	0.000750480 0
$566$	0	0
$567$	28494.9	0.00372229
$568$	0	0
$569$	−6.51286e6	−0.843317	−0.421659	−	0.906755i	$-0.638552\pi$
$569$	−6.51286e6	−0.843317	−0.421659	+	0.906755i	$0.638552\pi$
$570$	0	0
$571$	−1.08281e7	−1.38984	−0.694918	−	0.719089i	$-0.744560\pi$
$571$	−1.08281e7	−1.38984	−0.694918	+	0.719089i	$0.744560\pi$
$572$	0	0
$573$	1.75380e6	0.223148
$574$	0	0
$575$	−1.65304e6	−0.208503
$576$	0	0
$577$	9.82825e6	1.22896	0.614478	−	0.788934i	$-0.289367\pi$
$577$	9.82825e6	1.22896	0.614478	+	0.788934i	$0.289367\pi$
$578$	0	0
$579$	5.60582e6	0.694933
$580$	0	0
$581$	397510.	0.0488549
$582$	0	0
$583$	−1.32648e6	−0.161632
$584$	0	0
$585$	−2530.14	−0.000305672 0
$586$	0	0
$587$	5.37142e6	0.643419	0.321710	−	0.946838i	$-0.395742\pi$
$587$	5.37142e6	0.643419	0.321710	+	0.946838i	$0.395742\pi$
$588$	0	0
$589$	6.27117e6	0.744836
$590$	0	0
$591$	−2.51442e6	−0.296120
$592$	0	0
$593$	5.42476e6	0.633496	0.316748	−	0.948510i	$-0.397409\pi$
$593$	5.42476e6	0.633496	0.316748	+	0.948510i	$0.397409\pi$
$594$	0	0
$595$	127.440	1.47575e−5 0
$596$	0	0
$597$	6.43606e6	0.739068
$598$	0	0
$599$	−7.31392e6	−0.832882	−0.416441	−	0.909163i	$-0.636723\pi$
$599$	−7.31392e6	−0.832882	−0.416441	+	0.909163i	$0.636723\pi$
$600$	0	0
$601$	−2.65146e6	−0.299433	−0.149716	−	0.988729i	$-0.547836\pi$
$601$	−2.65146e6	−0.299433	−0.149716	+	0.988729i	$0.547836\pi$
$602$	0	0
$603$	2.13888e6	0.239548
$604$	0	0
$605$	−9214.05	−0.00102344
$606$	0	0
$607$	894452.	0.0985338	0.0492669	−	0.998786i	$-0.484312\pi$
$607$	894452.	0.0985338	0.0492669	+	0.998786i	$0.484312\pi$
$608$	0	0
$609$	−65128.7	−0.00711589
$610$	0	0
$611$	377154.	0.0408710
$612$	0	0
$613$	1.51504e7	1.62845	0.814223	−	0.580553i	$-0.197164\pi$
$613$	1.51504e7	1.62845	0.814223	+	0.580553i	$0.197164\pi$
$614$	0	0
$615$	−27911.2	−0.00297571
$616$	0	0
$617$	−3.64371e6	−0.385328	−0.192664	−	0.981265i	$-0.561713\pi$
$617$	−3.64371e6	−0.385328	−0.192664	+	0.981265i	$0.561713\pi$
$618$	0	0
$619$	−1.06699e7	−1.11927	−0.559633	−	0.828741i	$-0.689058\pi$
$619$	−1.06699e7	−1.11927	−0.559633	+	0.828741i	$0.689058\pi$
$620$	0	0
$621$	385641.	0.0401286
$622$	0	0
$623$	−349220.	−0.0360478
$624$	0	0
$625$	9.76406e6	0.999840
$626$	0	0
$627$	−6.67887e6	−0.678476
$628$	0	0
$629$	105539.	0.0106362
$630$	0	0
$631$	−1.27764e7	−1.27742	−0.638712	−	0.769445i	$-0.720533\pi$
$631$	−1.27764e7	−1.27742	−0.638712	+	0.769445i	$0.720533\pi$
$632$	0	0
$633$	−2.12484e6	−0.210774
$634$	0	0
$635$	−21209.6	−0.00208737
$636$	0	0
$637$	−1.28346e6	−0.125324
$638$	0	0
$639$	4.71222e6	0.456534
$640$	0	0
$641$	−463451.	−0.0445511	−0.0222756	−	0.999752i	$-0.507091\pi$
$641$	−463451.	−0.0445511	−0.0222756	+	0.999752i	$0.507091\pi$
$642$	0	0
$643$	1.12992e7	1.07775	0.538877	−	0.842385i	$-0.318849\pi$
$643$	1.12992e7	1.07775	0.538877	+	0.842385i	$0.318849\pi$
$644$	0	0
$645$	19505.4	0.00184610
$646$	0	0
$647$	6.27215e6	0.589055	0.294528	−	0.955643i	$-0.404838\pi$
$647$	6.27215e6	0.589055	0.294528	+	0.955643i	$0.404838\pi$
$648$	0	0
$649$	−4.60624e6	−0.429274
$650$	0	0
$651$	141536.	0.0130892
$652$	0	0
$653$	−6.55444e6	−0.601524	−0.300762	−	0.953699i	$-0.597241\pi$
$653$	−6.55444e6	−0.601524	−0.300762	+	0.953699i	$0.597241\pi$
$654$	0	0
$655$	−15132.5	−0.00137818
$656$	0	0
$657$	33363.0	0.00301544
$658$	0	0
$659$	−5.49875e6	−0.493231	−0.246616	−	0.969113i	$-0.579318\pi$
$659$	−5.49875e6	−0.493231	−0.246616	+	0.969113i	$0.579318\pi$
$660$	0	0
$661$	−1.94455e7	−1.73107	−0.865537	−	0.500845i	$-0.833023\pi$
$661$	−1.94455e7	−1.73107	−0.865537	+	0.500845i	$0.833023\pi$
$662$	0	0
$663$	−49414.3	−0.00436585
$664$	0	0
$665$	−3073.25	−0.000269491 0
$666$	0	0
$667$	−881432.	−0.0767139
$668$	0	0
$669$	−2.80955e6	−0.242701
$670$	0	0
$671$	9.06531e6	0.777278
$672$	0	0
$673$	−1.99517e7	−1.69802	−0.849008	−	0.528381i	$-0.822799\pi$
$673$	−1.99517e7	−1.69802	−0.849008	+	0.528381i	$0.822799\pi$
$674$	0	0
$675$	−2.27800e6	−0.192440
$676$	0	0
$677$	−2.03746e7	−1.70851	−0.854254	−	0.519856i	$-0.825986\pi$
$677$	−2.03746e7	−1.70851	−0.854254	+	0.519856i	$0.825986\pi$
$678$	0	0
$679$	516455.	0.0429891
$680$	0	0
$681$	−6.35644e6	−0.525226
$682$	0	0
$683$	−8.14142e6	−0.667803	−0.333901	−	0.942608i	$-0.608365\pi$
$683$	−8.14142e6	−0.667803	−0.333901	+	0.942608i	$0.608365\pi$
$684$	0	0
$685$	8453.22	0.000688328 0
$686$	0	0
$687$	−4.72092e6	−0.381623
$688$	0	0
$689$	236669.	0.0189930
$690$	0	0
$691$	1.69238e7	1.34835	0.674174	−	0.738573i	$-0.264500\pi$
$691$	1.69238e7	1.34835	0.674174	+	0.738573i	$0.264500\pi$
$692$	0	0
$693$	−150738.	−0.0119231
$694$	0	0
$695$	18427.7	0.00144714
$696$	0	0
$697$	−545112.	−0.0425015
$698$	0	0
$699$	1.29783e7	1.00467
$700$	0	0
$701$	−1.16539e7	−0.895728	−0.447864	−	0.894102i	$-0.647815\pi$
$701$	−1.16539e7	−0.895728	−0.447864	+	0.894102i	$0.647815\pi$
$702$	0	0
$703$	−2.54510e6	−0.194230
$704$	0	0
$705$	−18140.9	−0.00137463
$706$	0	0
$707$	446397.	0.0335871
$708$	0	0
$709$	−1.45958e7	−1.09046	−0.545232	−	0.838285i	$-0.683558\pi$
$709$	−1.45958e7	−1.09046	−0.545232	+	0.838285i	$0.683558\pi$
$710$	0	0
$711$	5.03271e6	0.373360
$712$	0	0
$713$	1.91550e6	0.141110
$714$	0	0
$715$	13384.4	0.000979116 0
$716$	0	0
$717$	−4.02451e6	−0.292358
$718$	0	0
$719$	−1.97942e7	−1.42796	−0.713979	−	0.700168i	$-0.753109\pi$
$719$	−1.97942e7	−1.42796	−0.713979	+	0.700168i	$0.753109\pi$
$720$	0	0
$721$	−523699.	−0.0375183
$722$	0	0
$723$	3.98799e6	0.283732
$724$	0	0
$725$	5.20667e6	0.367887
$726$	0	0
$727$	−2.42059e7	−1.69857	−0.849287	−	0.527932i	$-0.822968\pi$
$727$	−2.42059e7	−1.69857	−0.849287	+	0.527932i	$0.822968\pi$
$728$	0	0
$729$	531441.	0.0370370
$730$	0	0
$731$	380944.	0.0263674
$732$	0	0
$733$	−1.38705e7	−0.953522	−0.476761	−	0.879033i	$-0.658189\pi$
$733$	−1.38705e7	−0.953522	−0.476761	+	0.879033i	$0.658189\pi$
$734$	0	0
$735$	61734.0	0.00421508
$736$	0	0
$737$	−1.13146e7	−0.767311
$738$	0	0
$739$	−1.59262e7	−1.07276	−0.536379	−	0.843977i	$-0.680208\pi$
$739$	−1.59262e7	−1.07276	−0.536379	+	0.843977i	$0.680208\pi$
$740$	0	0
$741$	1.19164e6	0.0797259
$742$	0	0
$743$	−3.17168e6	−0.210774	−0.105387	−	0.994431i	$-0.533608\pi$
$743$	−3.17168e6	−0.210774	−0.105387	+	0.994431i	$0.533608\pi$
$744$	0	0
$745$	−175423.	−0.0115797
$746$	0	0
$747$	7.41372e6	0.486110
$748$	0	0
$749$	137101.	0.00892965
$750$	0	0
$751$	−2.49776e7	−1.61604	−0.808018	−	0.589157i	$-0.799460\pi$
$751$	−2.49776e7	−1.61604	−0.808018	+	0.589157i	$0.799460\pi$
$752$	0	0
$753$	−4.64581e6	−0.298589
$754$	0	0
$755$	−62824.4	−0.00401108
$756$	0	0
$757$	−9.09279e6	−0.576710	−0.288355	−	0.957524i	$-0.593108\pi$
$757$	−9.09279e6	−0.576710	−0.288355	+	0.957524i	$0.593108\pi$
$758$	0	0
$759$	−2.04003e6	−0.128538
$760$	0	0
$761$	1.21933e7	0.763238	0.381619	−	0.924320i	$-0.375367\pi$
$761$	1.21933e7	0.763238	0.381619	+	0.924320i	$0.375367\pi$
$762$	0	0
$763$	−520750.	−0.0323831
$764$	0	0
$765$	2376.80	0.000146838 0
$766$	0	0
$767$	821842.	0.0504429
$768$	0	0
$769$	7.54269e6	0.459950	0.229975	−	0.973197i	$-0.426136\pi$
$769$	7.54269e6	0.459950	0.229975	+	0.973197i	$0.426136\pi$
$770$	0	0
$771$	8.55722e6	0.518438
$772$	0	0
$773$	2.69009e7	1.61927	0.809634	−	0.586935i	$-0.199666\pi$
$773$	2.69009e7	1.61927	0.809634	+	0.586935i	$0.199666\pi$
$774$	0	0
$775$	−1.13150e7	−0.676706
$776$	0	0
$777$	−57441.1	−0.00341327
$778$	0	0
$779$	1.31455e7	0.776131
$780$	0	0
$781$	−2.49276e7	−1.46235
$782$	0	0
$783$	−1.21468e6	−0.0708037
$784$	0	0
$785$	−191746.	−0.0111058
$786$	0	0
$787$	−1.24123e7	−0.714356	−0.357178	−	0.934036i	$-0.616261\pi$
$787$	−1.24123e7	−0.714356	−0.357178	+	0.934036i	$0.616261\pi$
$788$	0	0
$789$	−1.76709e7	−1.01057
$790$	0	0
$791$	−60531.1	−0.00343983
$792$	0	0
$793$	−1.61743e6	−0.0913360
$794$	0	0
$795$	−11383.7	−0.000638799 0
$796$	0	0
$797$	−2.76236e6	−0.154040	−0.0770202	−	0.997030i	$-0.524541\pi$
$797$	−2.76236e6	−0.154040	−0.0770202	+	0.997030i	$0.524541\pi$
$798$	0	0
$799$	−354296.	−0.0196336
$800$	0	0
$801$	−6.51308e6	−0.358678
$802$	0	0
$803$	−176489.	−0.00965895
$804$	0	0
$805$	−938.712	−5.10555e−5 0
$806$	0	0
$807$	1.89536e7	1.02449
$808$	0	0
$809$	−2.05022e7	−1.10136	−0.550681	−	0.834716i	$-0.685632\pi$
$809$	−2.05022e7	−1.10136	−0.550681	+	0.834716i	$0.685632\pi$
$810$	0	0
$811$	−1.35473e7	−0.723272	−0.361636	−	0.932319i	$-0.617782\pi$
$811$	−1.35473e7	−0.723272	−0.361636	+	0.932319i	$0.617782\pi$
$812$	0	0
$813$	2.07275e7	1.09982
$814$	0	0
$815$	−122661.	−0.00646863
$816$	0	0
$817$	−9.18657e6	−0.481502
$818$	0	0
$819$	26894.5	0.00140105
$820$	0	0
$821$	3.14628e7	1.62907	0.814535	−	0.580115i	$-0.196992\pi$
$821$	3.14628e7	1.62907	0.814535	+	0.580115i	$0.196992\pi$
$822$	0	0
$823$	−1.73389e7	−0.892322	−0.446161	−	0.894953i	$-0.647209\pi$
$823$	−1.73389e7	−0.892322	−0.446161	+	0.894953i	$0.647209\pi$
$824$	0	0
$825$	1.20506e7	0.616415
$826$	0	0
$827$	6.54365e6	0.332703	0.166351	−	0.986067i	$-0.446801\pi$
$827$	6.54365e6	0.332703	0.166351	+	0.986067i	$0.446801\pi$
$828$	0	0
$829$	−6.28583e6	−0.317670	−0.158835	−	0.987305i	$-0.550774\pi$
$829$	−6.28583e6	−0.317670	−0.158835	+	0.987305i	$0.550774\pi$
$830$	0	0
$831$	−1.58463e7	−0.796020
$832$	0	0
$833$	1.20568e6	0.0602031
$834$	0	0
$835$	33913.8	0.00168329
$836$	0	0
$837$	2.63970e6	0.130239
$838$	0	0
$839$	−7.73114e6	−0.379174	−0.189587	−	0.981864i	$-0.560715\pi$
$839$	−7.73114e6	−0.379174	−0.189587	+	0.981864i	$0.560715\pi$
$840$	0	0
$841$	−1.77349e7	−0.864645
$842$	0	0
$843$	1.64316e7	0.796361
$844$	0	0
$845$	149316.	0.00719389
$846$	0	0
$847$	97941.9	0.00469094
$848$	0	0
$849$	−1.55426e7	−0.740039
$850$	0	0
$851$	−777389.	−0.0367972
$852$	0	0
$853$	−1.25054e7	−0.588471	−0.294236	−	0.955733i	$-0.595065\pi$
$853$	−1.25054e7	−0.588471	−0.294236	+	0.955733i	$0.595065\pi$
$854$	0	0
$855$	−57317.3	−0.00268146
$856$	0	0
$857$	2.09830e7	0.975924	0.487962	−	0.872865i	$-0.337740\pi$
$857$	2.09830e7	0.975924	0.487962	+	0.872865i	$0.337740\pi$
$858$	0	0
$859$	9.91769e6	0.458593	0.229297	−	0.973357i	$-0.426357\pi$
$859$	9.91769e6	0.458593	0.229297	+	0.973357i	$0.426357\pi$
$860$	0	0
$861$	296686.	0.0136392
$862$	0	0
$863$	−6.48533e6	−0.296418	−0.148209	−	0.988956i	$-0.547351\pi$
$863$	−6.48533e6	−0.296418	−0.148209	+	0.988956i	$0.547351\pi$
$864$	0	0
$865$	100636.	0.00457310
$866$	0	0
$867$	−1.27323e7	−0.575253
$868$	0	0
$869$	−2.66229e7	−1.19593
$870$	0	0
$871$	2.01875e6	0.0901647
$872$	0	0
$873$	9.63208e6	0.427745
$874$	0	0
$875$	11090.3	0.000489694 0
$876$	0	0
$877$	1.00228e7	0.440037	0.220018	−	0.975496i	$-0.429388\pi$
$877$	1.00228e7	0.440037	0.220018	+	0.975496i	$0.429388\pi$
$878$	0	0
$879$	3.65023e6	0.159349
$880$	0	0
$881$	−945657.	−0.0410482	−0.0205241	−	0.999789i	$-0.506533\pi$
$881$	−945657.	−0.0410482	−0.0205241	+	0.999789i	$0.506533\pi$
$882$	0	0
$883$	−8.36917e6	−0.361227	−0.180614	−	0.983554i	$-0.557808\pi$
$883$	−8.36917e6	−0.361227	−0.180614	+	0.983554i	$0.557808\pi$
$884$	0	0
$885$	−39530.2	−0.00169657
$886$	0	0
$887$	2.98445e7	1.27367	0.636833	−	0.771002i	$-0.280244\pi$
$887$	2.98445e7	1.27367	0.636833	+	0.771002i	$0.280244\pi$
$888$	0	0
$889$	225450.	0.00956746
$890$	0	0
$891$	−2.81131e6	−0.118636
$892$	0	0
$893$	8.54396e6	0.358534
$894$	0	0
$895$	−317922.	−0.0132667
$896$	0	0
$897$	363981.	0.0151042
$898$	0	0
$899$	−6.03337e6	−0.248978
$900$	0	0
$901$	−222326.	−0.00912384
$902$	0	0
$903$	−207335.	−0.00846160
$904$	0	0
$905$	−33403.0	−0.00135570
$906$	0	0
$907$	−1.98050e7	−0.799387	−0.399694	−	0.916649i	$-0.630883\pi$
$907$	−1.98050e7	−0.799387	−0.399694	+	0.916649i	$0.630883\pi$
$908$	0	0
$909$	8.32548e6	0.334195
$910$	0	0
$911$	−2.34551e7	−0.936356	−0.468178	−	0.883634i	$-0.655089\pi$
$911$	−2.34551e7	−0.936356	−0.468178	+	0.883634i	$0.655089\pi$
$912$	0	0
$913$	−3.92184e7	−1.55709
$914$	0	0
$915$	77797.5	0.00307194
$916$	0	0
$917$	160852.	0.00631690
$918$	0	0
$919$	2.57051e7	1.00399	0.501996	−	0.864870i	$-0.332599\pi$
$919$	2.57051e7	1.00399	0.501996	+	0.864870i	$0.332599\pi$
$920$	0	0
$921$	2.35778e7	0.915915
$922$	0	0
$923$	4.44756e6	0.171837
$924$	0	0
$925$	4.59208e6	0.176464
$926$	0	0
$927$	−9.76718e6	−0.373310
$928$	0	0
$929$	3.96038e7	1.50556	0.752780	−	0.658273i	$-0.228713\pi$
$929$	3.96038e7	1.50556	0.752780	+	0.658273i	$0.228713\pi$
$930$	0	0
$931$	−2.90753e7	−1.09939
$932$	0	0
$933$	−1.39281e7	−0.523826
$934$	0	0
$935$	−12573.2	−0.000470347 0
$936$	0	0
$937$	−1.91246e7	−0.711611	−0.355805	−	0.934560i	$-0.615793\pi$
$937$	−1.91246e7	−0.711611	−0.355805	+	0.934560i	$0.615793\pi$
$938$	0	0
$939$	3.32636e6	0.123113
$940$	0	0
$941$	−8.52219e6	−0.313745	−0.156873	−	0.987619i	$-0.550141\pi$
$941$	−8.52219e6	−0.313745	−0.156873	+	0.987619i	$0.550141\pi$
$942$	0	0
$943$	4.01525e6	0.147039
$944$	0	0
$945$	−1293.61	−4.71221e−5 0
$946$	0	0
$947$	3.79789e7	1.37616	0.688078	−	0.725636i	$-0.258454\pi$
$947$	3.79789e7	1.37616	0.688078	+	0.725636i	$0.258454\pi$
$948$	0	0
$949$	31489.1	0.00113500
$950$	0	0
$951$	−1.80142e7	−0.645898
$952$	0	0
$953$	236188.	0.00842415	0.00421208	−	0.999991i	$-0.498659\pi$
$953$	236188.	0.00842415	0.00421208	+	0.999991i	$0.498659\pi$
$954$	0	0
$955$	−79619.1	−0.00282494
$956$	0	0
$957$	6.42561e6	0.226796
$958$	0	0
$959$	−89854.5	−0.00315496
$960$	0	0
$961$	−1.55176e7	−0.542021
$962$	0	0
$963$	2.55698e6	0.0888508
$964$	0	0
$965$	−254493.	−0.00879747
$966$	0	0
$967$	−2.02260e7	−0.695573	−0.347787	−	0.937574i	$-0.613067\pi$
$967$	−2.02260e7	−0.695573	−0.347787	+	0.937574i	$0.613067\pi$
$968$	0	0
$969$	−1.11942e6	−0.0382987
$970$	0	0
$971$	−4.63134e7	−1.57637	−0.788185	−	0.615438i	$-0.788979\pi$
$971$	−4.63134e7	−1.57637	−0.788185	+	0.615438i	$0.788979\pi$
$972$	0	0
$973$	−195880.	−0.00663297
$974$	0	0
$975$	−2.15006e6	−0.0724334
$976$	0	0
$977$	−2.97234e7	−0.996237	−0.498118	−	0.867109i	$-0.665976\pi$
$977$	−2.97234e7	−0.996237	−0.498118	+	0.867109i	$0.665976\pi$
$978$	0	0
$979$	3.44541e7	1.14890
$980$	0	0
$981$	−9.71219e6	−0.322214
$982$	0	0
$983$	−3.58345e7	−1.18282	−0.591408	−	0.806372i	$-0.701428\pi$
$983$	−3.58345e7	−1.18282	−0.591408	+	0.806372i	$0.701428\pi$
$984$	0	0
$985$	114150.	0.00374872
$986$	0	0
$987$	192831.	0.00630064
$988$	0	0
$989$	−2.80600e6	−0.0912215
$990$	0	0
$991$	2.93191e7	0.948346	0.474173	−	0.880432i	$-0.342747\pi$
$991$	2.93191e7	0.948346	0.474173	+	0.880432i	$0.342747\pi$
$992$	0	0
$993$	1.81059e6	0.0582702
$994$	0	0
$995$	−292185.	−0.00935620
$996$	0	0
$997$	−5.37510e6	−0.171257	−0.0856286	−	0.996327i	$-0.527290\pi$
$997$	−5.37510e6	−0.171257	−0.0856286	+	0.996327i	$0.527290\pi$
$998$	0	0
$999$	−1.07130e6	−0.0339623

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	1104.6.a.o.1.2		4
4.3	odd	2		69.6.a.d.1.1	✓	4
12.11	even	2		207.6.a.e.1.4		4

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
69.6.a.d.1.1	✓	4	4.3	odd	2
207.6.a.e.1.4		4	12.11	even	2
1104.6.a.o.1.2		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)

\(f(q)\)	\(=\)	\(q+9.00000 q^{3} -0.408582 q^{5} +4.34307 q^{7} +81.0000 q^{9} +O(q^{10})\)	\(q+9.00000 q^{3} -0.408582 q^{5} +4.34307 q^{7} +81.0000 q^{9} -428.488 q^{11} +76.4506 q^{13} -3.67724 q^{15} -71.8173 q^{17} +1731.89 q^{19} +39.0877 q^{21} +529.000 q^{23} -3124.83 q^{25} +729.000 q^{27} -1666.22 q^{29} +3620.99 q^{31} -3856.40 q^{33} -1.77450 q^{35} -1469.54 q^{37} +688.055 q^{39} +7590.26 q^{41} -5304.35 q^{43} -33.0952 q^{45} +4933.30 q^{47} -16788.1 q^{49} -646.356 q^{51} +3095.71 q^{53} +175.073 q^{55} +15587.1 q^{57} +10750.0 q^{59} -21156.5 q^{61} +351.789 q^{63} -31.2364 q^{65} +26405.9 q^{67} +4761.00 q^{69} +58175.6 q^{71} +411.889 q^{73} -28123.5 q^{75} -1860.96 q^{77} +62132.2 q^{79} +6561.00 q^{81} +91527.4 q^{83} +29.3433 q^{85} -14996.0 q^{87} -80408.4 q^{89} +332.031 q^{91} +32588.9 q^{93} -707.621 q^{95} +118915. q^{97} -34707.6 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 4 q + 36 q^{3} + 22 q^{5} + 62 q^{7} + 324 q^{9}+O(q^{10}) \) 4 * q + 36 * q^3 + 22 * q^5 + 62 * q^7 + 324 * q^9	\( 4 q + 36 q^{3} + 22 q^{5} + 62 q^{7} + 324 q^{9} + 1076 q^{11} - 396 q^{13} + 198 q^{15} + 70 q^{17} + 6366 q^{19} + 558 q^{21} + 2116 q^{23} + 1264 q^{25} + 2916 q^{27} + 3948 q^{29} - 3092 q^{31} + 9684 q^{33} - 1304 q^{35} - 17464 q^{37} - 3564 q^{39} + 18680 q^{41} + 25846 q^{43} + 1782 q^{45} - 18392 q^{47} + 7952 q^{49} + 630 q^{51} - 26518 q^{53} + 40848 q^{55} + 57294 q^{57} + 14520 q^{59} - 13688 q^{61} + 5022 q^{63} + 38324 q^{65} + 11098 q^{67} + 19044 q^{69} + 57496 q^{71} - 112272 q^{73} + 11376 q^{75} - 4792 q^{77} + 240754 q^{79} + 26244 q^{81} + 93268 q^{83} - 323204 q^{85} + 35532 q^{87} - 107582 q^{89} + 301532 q^{91} - 27828 q^{93} + 18640 q^{95} - 53076 q^{97} + 87156 q^{99}+O(q^{100}) \) 4 * q + 36 * q^3 + 22 * q^5 + 62 * q^7 + 324 * q^9 + 1076 * q^11 - 396 * q^13 + 198 * q^15 + 70 * q^17 + 6366 * q^19 + 558 * q^21 + 2116 * q^23 + 1264 * q^25 + 2916 * q^27 + 3948 * q^29 - 3092 * q^31 + 9684 * q^33 - 1304 * q^35 - 17464 * q^37 - 3564 * q^39 + 18680 * q^41 + 25846 * q^43 + 1782 * q^45 - 18392 * q^47 + 7952 * q^49 + 630 * q^51 - 26518 * q^53 + 40848 * q^55 + 57294 * q^57 + 14520 * q^59 - 13688 * q^61 + 5022 * q^63 + 38324 * q^65 + 11098 * q^67 + 19044 * q^69 + 57496 * q^71 - 112272 * q^73 + 11376 * q^75 - 4792 * q^77 + 240754 * q^79 + 26244 * q^81 + 93268 * q^83 - 323204 * q^85 + 35532 * q^87 - 107582 * q^89 + 301532 * q^91 - 27828 * q^93 + 18640 * q^95 - 53076 * q^97 + 87156 * q^99

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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