LMFDB - Embedded newform 228.6.a.b.1.4

Show commands: Magma / Pari/GP / SageMath

Newspace parameters

comment:Compute space of new eigenforms

gp:[N,k,chi] = [228,6,Mod(1,228)] mf = mfinit([N,k,chi],0) lf = mfeigenbasis(mf)

sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(228, base_ring=CyclotomicField(2)) chi = DirichletCharacter(H, H._module([0, 0, 0])) N = Newforms(chi, 6, names="a")

magma://Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code chi := DirichletCharacter("228.1"); S:= CuspForms(chi, 6); N := Newforms(S);

Level:	$ N $	$=$	$ 228 = 2^{2} \cdot 3 \cdot 19 $
Weight:	$ k $	$=$	$ 6 $
Character orbit:	$[\chi]$	$=$	228.a (trivial)

Newform invariants

comment:select newform

sage:traces = [4,0,-36,0,-84] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(5)] == traces)

gp:f = lf[1] \\ Warning: the index may be different

Self dual:	yes
Analytic conductor:	$36.5675109174$
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} - 3227x^{2} + 17265x + 1197450 $ x^4 - x^3 - 3227x^2 + 17265x + 1197450
Coefficient ring:	$\Z[a_1, \ldots, a_{11}]$
Coefficient ring index:	$ 2^{3} $
Twist minimal:	yes
Fricke sign:	$+1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.4
Root			$24.8468$ of defining polynomial
Character	$\chi$	$=$	228.1

$q$-expansion

comment:q-expansion

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

gp:mfcoefs(f, 20)

$f(q)$	$=$	$q-9.00000 q^{3} +64.4605 q^{5} -22.7670 q^{7} +81.0000 q^{9} -164.072 q^{11} -1128.87 q^{13} -580.144 q^{15} +790.656 q^{17} +361.000 q^{19} +204.903 q^{21} +3250.07 q^{23} +1030.16 q^{25} -729.000 q^{27} -3254.56 q^{29} -351.725 q^{31} +1476.65 q^{33} -1467.57 q^{35} -7771.04 q^{37} +10159.8 q^{39} -7767.84 q^{41} -8911.96 q^{43} +5221.30 q^{45} +11149.8 q^{47} -16288.7 q^{49} -7115.91 q^{51} -26529.8 q^{53} -10576.2 q^{55} -3249.00 q^{57} -246.481 q^{59} -31878.9 q^{61} -1844.13 q^{63} -72767.3 q^{65} -57835.0 q^{67} -29250.6 q^{69} +6787.94 q^{71} +4203.82 q^{73} -9271.40 q^{75} +3735.43 q^{77} +39000.3 q^{79} +6561.00 q^{81} +25963.2 q^{83} +50966.1 q^{85} +29291.0 q^{87} -102090. q^{89} +25700.9 q^{91} +3165.52 q^{93} +23270.2 q^{95} +92058.3 q^{97} -13289.8 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 4 q - 36 q^{3} - 84 q^{5} + 54 q^{7} + 324 q^{9} + 354 q^{11} + 46 q^{13} + 756 q^{15} + 2046 q^{17} + 1444 q^{19} - 486 q^{21} + 846 q^{23} + 4018 q^{25} - 2916 q^{27} - 7152 q^{29} + 238 q^{31} - 3186 q^{33}+ \cdots + 28674 q^{99}+O(q^{100}) $ 4 * q - 36 * q^3 - 84 * q^5 + 54 * q^7 + 324 * q^9 + 354 * q^11 + 46 * q^13 + 756 * q^15 + 2046 * q^17 + 1444 * q^19 - 486 * q^21 + 846 * q^23 + 4018 * q^25 - 2916 * q^27 - 7152 * q^29 + 238 * q^31 - 3186 * q^33 - 19878 * q^35 - 10326 * q^37 - 414 * q^39 - 28200 * q^41 - 18590 * q^43 - 6804 * q^45 - 18684 * q^47 - 17946 * q^49 - 18414 * q^51 - 42720 * q^53 - 13818 * q^55 - 12996 * q^57 - 68256 * q^59 - 67654 * q^61 + 4374 * q^63 - 109776 * q^65 - 81836 * q^67 - 7614 * q^69 - 62112 * q^71 - 81966 * q^73 - 36162 * q^75 - 81378 * q^77 + 8766 * q^79 + 26244 * q^81 - 55080 * q^83 - 108030 * q^85 + 64368 * q^87 - 197520 * q^89 + 25820 * q^91 - 2142 * q^93 - 30324 * q^95 - 184228 * q^97 + 28674 * q^99

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
$5$	64.4605	1.15310	0.576552	−	0.817060i	$-0.304398\pi$
$5$	64.4605	1.15310	0.576552	+	0.817060i	$0.304398\pi$
$6$	0	0
$7$	−22.7670	−0.175615	−0.0878073	−	0.996137i	$-0.527986\pi$
$7$	−22.7670	−0.175615	−0.0878073	+	0.996137i	$0.527986\pi$
$8$	0	0
$9$	81.0000	0.333333
$10$	0	0
$11$	−164.072	−0.408840	−0.204420	−	0.978883i	$-0.565531\pi$
$11$	−164.072	−0.408840	−0.204420	+	0.978883i	$0.565531\pi$
$12$	0	0
$13$	−1128.87	−1.85261	−0.926305	−	0.376775i	$-0.877033\pi$
$13$	−1128.87	−1.85261	−0.926305	+	0.376775i	$0.877033\pi$
$14$	0	0
$15$	−580.144	−0.665745
$16$	0	0
$17$	790.656	0.663537	0.331769	−	0.943361i	$-0.392355\pi$
$17$	790.656	0.663537	0.331769	+	0.943361i	$0.392355\pi$
$18$	0	0
$19$	361.000	0.229416
$19$	361.000	0.229416
$20$	0	0
$21$	204.903	0.101391
$22$	0	0
$23$	3250.07	1.28107	0.640535	−	0.767929i	$-0.278713\pi$
$23$	3250.07	1.28107	0.640535	+	0.767929i	$0.278713\pi$
$24$	0	0
$25$	1030.16	0.329650
$26$	0	0
$27$	−729.000	−0.192450
$28$	0	0
$29$	−3254.56	−0.718616	−0.359308	−	0.933219i	$-0.616987\pi$
$29$	−3254.56	−0.718616	−0.359308	+	0.933219i	$0.616987\pi$
$30$	0	0
$31$	−351.725	−0.0657353	−0.0328677	−	0.999460i	$-0.510464\pi$
$31$	−351.725	−0.0657353	−0.0328677	+	0.999460i	$0.510464\pi$
$32$	0	0
$33$	1476.65	0.236044
$34$	0	0
$35$	−1467.57	−0.202502
$36$	0	0
$37$	−7771.04	−0.933200	−0.466600	−	0.884468i	$-0.654521\pi$
$37$	−7771.04	−0.933200	−0.466600	+	0.884468i	$0.654521\pi$
$38$	0	0
$39$	10159.8	1.06960
$40$	0	0
$41$	−7767.84	−0.721673	−0.360837	−	0.932629i	$-0.617509\pi$
$41$	−7767.84	−0.721673	−0.360837	+	0.932629i	$0.617509\pi$
$42$	0	0
$43$	−8911.96	−0.735025	−0.367513	−	0.930019i	$-0.619791\pi$
$43$	−8911.96	−0.735025	−0.367513	+	0.930019i	$0.619791\pi$
$44$	0	0
$45$	5221.30	0.384368
$46$	0	0
$47$	11149.8	0.736247	0.368124	−	0.929777i	$-0.380000\pi$
$47$	11149.8	0.736247	0.368124	+	0.929777i	$0.380000\pi$
$48$	0	0
$49$	−16288.7	−0.969160
$50$	0	0
$51$	−7115.91	−0.383094
$52$	0	0
$53$	−26529.8	−1.29731	−0.648657	−	0.761081i	$-0.724669\pi$
$53$	−26529.8	−1.29731	−0.648657	+	0.761081i	$0.724669\pi$
$54$	0	0
$55$	−10576.2	−0.471435
$56$	0	0
$57$	−3249.00	−0.132453
$58$	0	0
$59$	−246.481	−0.00921834	−0.00460917	−	0.999989i	$-0.501467\pi$
$59$	−246.481	−0.00921834	−0.00460917	+	0.999989i	$0.501467\pi$
$60$	0	0
$61$	−31878.9	−1.09693	−0.548464	−	0.836174i	$-0.684787\pi$
$61$	−31878.9	−1.09693	−0.548464	+	0.836174i	$0.684787\pi$
$62$	0	0
$63$	−1844.13	−0.0585382
$64$	0	0
$65$	−72767.3	−2.13625
$66$	0	0
$67$	−57835.0	−1.57400	−0.786998	−	0.616956i	$-0.788366\pi$
$67$	−57835.0	−1.57400	−0.786998	+	0.616956i	$0.788366\pi$
$68$	0	0
$69$	−29250.6	−0.739626
$70$	0	0
$71$	6787.94	0.159806	0.0799029	−	0.996803i	$-0.474539\pi$
$71$	6787.94	0.159806	0.0799029	+	0.996803i	$0.474539\pi$
$72$	0	0
$73$	4203.82	0.0923288	0.0461644	−	0.998934i	$-0.485300\pi$
$73$	4203.82	0.0923288	0.0461644	+	0.998934i	$0.485300\pi$
$74$	0	0
$75$	−9271.40	−0.190323
$76$	0	0
$77$	3735.43	0.0717982
$78$	0	0
$79$	39000.3	0.703073	0.351536	−	0.936174i	$-0.385659\pi$
$79$	39000.3	0.703073	0.351536	+	0.936174i	$0.385659\pi$
$80$	0	0
$81$	6561.00	0.111111
$82$	0	0
$83$	25963.2	0.413679	0.206839	−	0.978375i	$-0.433682\pi$
$83$	25963.2	0.413679	0.206839	+	0.978375i	$0.433682\pi$
$84$	0	0
$85$	50966.1	0.765128
$86$	0	0
$87$	29291.0	0.414893
$88$	0	0
$89$	−102090.	−1.36618	−0.683089	−	0.730336i	$-0.739364\pi$
$89$	−102090.	−1.36618	−0.683089	+	0.730336i	$0.739364\pi$
$90$	0	0
$91$	25700.9	0.325345
$92$	0	0
$93$	3165.52	0.0379523
$94$	0	0
$95$	23270.2	0.264540
$96$	0	0
$97$	92058.3	0.993421	0.496711	−	0.867916i	$-0.334541\pi$
$97$	92058.3	0.993421	0.496711	+	0.867916i	$0.334541\pi$
$98$	0	0
$99$	−13289.8	−0.136280
$100$	0	0
$101$	−84783.7	−0.827006	−0.413503	−	0.910503i	$-0.635695\pi$
$101$	−84783.7	−0.827006	−0.413503	+	0.910503i	$0.635695\pi$
$102$	0	0
$103$	−90338.7	−0.839036	−0.419518	−	0.907747i	$-0.637801\pi$
$103$	−90338.7	−0.839036	−0.419518	+	0.907747i	$0.637801\pi$
$104$	0	0
$105$	13208.1	0.116914
$106$	0	0
$107$	−219960.	−1.85731	−0.928653	−	0.370950i	$-0.879032\pi$
$107$	−219960.	−1.85731	−0.928653	+	0.370950i	$0.879032\pi$
$108$	0	0
$109$	150565.	1.21383	0.606914	−	0.794768i	$-0.292407\pi$
$109$	150565.	1.21383	0.606914	+	0.794768i	$0.292407\pi$
$110$	0	0
$111$	69939.4	0.538783
$112$	0	0
$113$	51994.3	0.383053	0.191527	−	0.981487i	$-0.438656\pi$
$113$	51994.3	0.383053	0.191527	+	0.981487i	$0.438656\pi$
$114$	0	0
$115$	209501.	1.47721
$116$	0	0
$117$	−91438.1	−0.617537
$118$	0	0
$119$	−18000.9	−0.116527
$120$	0	0
$121$	−134131.	−0.832850
$122$	0	0
$123$	69910.6	0.416658
$124$	0	0
$125$	−135035.	−0.772984
$126$	0	0
$127$	−111316.	−0.612417	−0.306208	−	0.951965i	$-0.599060\pi$
$127$	−111316.	−0.612417	−0.306208	+	0.951965i	$0.599060\pi$
$128$	0	0
$129$	80207.7	0.424367
$130$	0	0
$131$	−64322.5	−0.327480	−0.163740	−	0.986504i	$-0.552356\pi$
$131$	−64322.5	−0.327480	−0.163740	+	0.986504i	$0.552356\pi$
$132$	0	0
$133$	−8218.88	−0.0402887
$134$	0	0
$135$	−46991.7	−0.221915
$136$	0	0
$137$	−6624.53	−0.0301546	−0.0150773	−	0.999886i	$-0.504799\pi$
$137$	−6624.53	−0.0301546	−0.0150773	+	0.999886i	$0.504799\pi$
$138$	0	0
$139$	129168.	0.567044	0.283522	−	0.958966i	$-0.408497\pi$
$139$	129168.	0.567044	0.283522	+	0.958966i	$0.408497\pi$
$140$	0	0
$141$	−100349.	−0.425073
$142$	0	0
$143$	185216.	0.757421
$144$	0	0
$145$	−209790.	−0.828640
$146$	0	0
$147$	146598.	0.559545
$148$	0	0
$149$	275534.	1.01674	0.508369	−	0.861139i	$-0.330248\pi$
$149$	275534.	1.01674	0.508369	+	0.861139i	$0.330248\pi$
$150$	0	0
$151$	250327.	0.893441	0.446721	−	0.894673i	$-0.352592\pi$
$151$	250327.	0.893441	0.446721	+	0.894673i	$0.352592\pi$
$152$	0	0
$153$	64043.2	0.221179
$154$	0	0
$155$	−22672.4	−0.0757997
$156$	0	0
$157$	−353910.	−1.14589	−0.572946	−	0.819593i	$-0.694200\pi$
$157$	−353910.	−1.14589	−0.572946	+	0.819593i	$0.694200\pi$
$158$	0	0
$159$	238769.	0.749004
$160$	0	0
$161$	−73994.2	−0.224974
$162$	0	0
$163$	368413.	1.08609	0.543046	−	0.839703i	$-0.317271\pi$
$163$	368413.	1.08609	0.543046	+	0.839703i	$0.317271\pi$
$164$	0	0
$165$	95185.6	0.272183
$166$	0	0
$167$	−234449.	−0.650516	−0.325258	−	0.945625i	$-0.605451\pi$
$167$	−234449.	−0.650516	−0.325258	+	0.945625i	$0.605451\pi$
$168$	0	0
$169$	903045.	2.43216
$170$	0	0
$171$	29241.0	0.0764719
$172$	0	0
$173$	88770.0	0.225502	0.112751	−	0.993623i	$-0.464034\pi$
$173$	88770.0	0.225502	0.112751	+	0.993623i	$0.464034\pi$
$174$	0	0
$175$	−23453.5	−0.0578913
$176$	0	0
$177$	2218.33	0.00532221
$178$	0	0
$179$	351768.	0.820584	0.410292	−	0.911954i	$-0.365427\pi$
$179$	351768.	0.820584	0.410292	+	0.911954i	$0.365427\pi$
$180$	0	0
$181$	444531.	1.00857	0.504284	−	0.863538i	$-0.331756\pi$
$181$	444531.	1.00857	0.504284	+	0.863538i	$0.331756\pi$
$182$	0	0
$183$	286910.	0.633312
$184$	0	0
$185$	−500925.	−1.07608
$186$	0	0
$187$	−129725.	−0.271281
$188$	0	0
$189$	16597.1	0.0337970
$190$	0	0
$191$	−659450.	−1.30797	−0.653986	−	0.756506i	$-0.726905\pi$
$191$	−659450.	−1.30797	−0.653986	+	0.756506i	$0.726905\pi$
$192$	0	0
$193$	143026.	0.276390	0.138195	−	0.990405i	$-0.455870\pi$
$193$	143026.	0.276390	0.138195	+	0.990405i	$0.455870\pi$
$194$	0	0
$195$	654905.	1.23337
$196$	0	0
$197$	315836.	0.579824	0.289912	−	0.957053i	$-0.406374\pi$
$197$	315836.	0.579824	0.289912	+	0.957053i	$0.406374\pi$
$198$	0	0
$199$	−257395.	−0.460753	−0.230376	−	0.973102i	$-0.573996\pi$
$199$	−257395.	−0.460753	−0.230376	+	0.973102i	$0.573996\pi$
$200$	0	0
$201$	520515.	0.908746
$202$	0	0
$203$	74096.5	0.126199
$204$	0	0
$205$	−500719.	−0.832165
$206$	0	0
$207$	263255.	0.427023
$208$	0	0
$209$	−59230.1	−0.0937943
$210$	0	0
$211$	865762.	1.33873	0.669364	−	0.742934i	$-0.266567\pi$
$211$	865762.	1.33873	0.669364	+	0.742934i	$0.266567\pi$
$212$	0	0
$213$	−61091.5	−0.0922639
$214$	0	0
$215$	−574469.	−0.847561
$216$	0	0
$217$	8007.71	0.0115441
$218$	0	0
$219$	−37834.4	−0.0533061
$220$	0	0
$221$	−892545.	−1.22928
$222$	0	0
$223$	−736281.	−0.991474	−0.495737	−	0.868473i	$-0.665102\pi$
$223$	−736281.	−0.991474	−0.495737	+	0.868473i	$0.665102\pi$
$224$	0	0
$225$	83442.6	0.109883
$226$	0	0
$227$	944475.	1.21654	0.608269	−	0.793731i	$-0.291864\pi$
$227$	944475.	1.21654	0.608269	+	0.793731i	$0.291864\pi$
$228$	0	0
$229$	294052.	0.370540	0.185270	−	0.982688i	$-0.440684\pi$
$229$	294052.	0.370540	0.185270	+	0.982688i	$0.440684\pi$
$230$	0	0
$231$	−33618.9	−0.0414527
$232$	0	0
$233$	−1.58574e6	−1.91356	−0.956779	−	0.290816i	$-0.906073\pi$
$233$	−1.58574e6	−1.91356	−0.956779	+	0.290816i	$0.906073\pi$
$234$	0	0
$235$	718724.	0.848970
$236$	0	0
$237$	−351003.	−0.405919
$238$	0	0
$239$	1.08962e6	1.23391	0.616953	−	0.787000i	$-0.288367\pi$
$239$	1.08962e6	1.23391	0.616953	+	0.787000i	$0.288367\pi$
$240$	0	0
$241$	1.61089e6	1.78658	0.893290	−	0.449482i	$-0.148391\pi$
$241$	1.61089e6	1.78658	0.893290	+	0.449482i	$0.148391\pi$
$242$	0	0
$243$	−59049.0	−0.0641500
$244$	0	0
$245$	−1.04998e6	−1.11754
$246$	0	0
$247$	−407521.	−0.425018
$248$	0	0
$249$	−233669.	−0.238837
$250$	0	0
$251$	356237.	0.356907	0.178453	−	0.983948i	$-0.442891\pi$
$251$	356237.	0.356907	0.178453	+	0.983948i	$0.442891\pi$
$252$	0	0
$253$	−533245.	−0.523752
$254$	0	0
$255$	−458695.	−0.441747
$256$	0	0
$257$	644475.	0.608658	0.304329	−	0.952567i	$-0.401568\pi$
$257$	644475.	0.608658	0.304329	+	0.952567i	$0.401568\pi$
$258$	0	0
$259$	176923.	0.163883
$260$	0	0
$261$	−263619.	−0.239539
$262$	0	0
$263$	−740703.	−0.660321	−0.330160	−	0.943925i	$-0.607103\pi$
$263$	−740703.	−0.660321	−0.330160	+	0.943925i	$0.607103\pi$
$264$	0	0
$265$	−1.71013e6	−1.49594
$266$	0	0
$267$	918807.	0.788763
$268$	0	0
$269$	−2.05874e6	−1.73468	−0.867342	−	0.497713i	$-0.834173\pi$
$269$	−2.05874e6	−1.73468	−0.867342	+	0.497713i	$0.834173\pi$
$270$	0	0
$271$	1.00713e6	0.833031	0.416515	−	0.909129i	$-0.363251\pi$
$271$	1.00713e6	0.833031	0.416515	+	0.909129i	$0.363251\pi$
$272$	0	0
$273$	−231308.	−0.187838
$274$	0	0
$275$	−169020.	−0.134774
$276$	0	0
$277$	−449430.	−0.351935	−0.175968	−	0.984396i	$-0.556305\pi$
$277$	−449430.	−0.351935	−0.175968	+	0.984396i	$0.556305\pi$
$278$	0	0
$279$	−28489.7	−0.0219118
$280$	0	0
$281$	−2.28328e6	−1.72501	−0.862507	−	0.506045i	$-0.831107\pi$
$281$	−2.28328e6	−1.72501	−0.862507	+	0.506045i	$0.831107\pi$
$282$	0	0
$283$	2.38979e6	1.77376	0.886879	−	0.462002i	$-0.152869\pi$
$283$	2.38979e6	1.77376	0.886879	+	0.462002i	$0.152869\pi$
$284$	0	0
$285$	−209432.	−0.152732
$286$	0	0
$287$	176850.	0.126736
$288$	0	0
$289$	−794720.	−0.559718
$290$	0	0
$291$	−828524.	−0.573552
$292$	0	0
$293$	1.58266e6	1.07700	0.538502	−	0.842624i	$-0.318990\pi$
$293$	1.58266e6	1.07700	0.538502	+	0.842624i	$0.318990\pi$
$294$	0	0
$295$	−15888.3	−0.0106297
$296$	0	0
$297$	119609.	0.0786813
$298$	0	0
$299$	−3.66889e6	−2.37332
$300$	0	0
$301$	202898.	0.129081
$302$	0	0
$303$	763053.	0.477472
$304$	0	0
$305$	−2.05493e6	−1.26487
$306$	0	0
$307$	−1.26682e6	−0.767129	−0.383565	−	0.923514i	$-0.625304\pi$
$307$	−1.26682e6	−0.767129	−0.383565	+	0.923514i	$0.625304\pi$
$308$	0	0
$309$	813048.	0.484418
$310$	0	0
$311$	416733.	0.244319	0.122159	−	0.992511i	$-0.461018\pi$
$311$	416733.	0.244319	0.122159	+	0.992511i	$0.461018\pi$
$312$	0	0
$313$	−1.49763e6	−0.864062	−0.432031	−	0.901859i	$-0.642203\pi$
$313$	−1.49763e6	−0.864062	−0.432031	+	0.901859i	$0.642203\pi$
$314$	0	0
$315$	−118873.	−0.0675006
$316$	0	0
$317$	947406.	0.529527	0.264763	−	0.964313i	$-0.414706\pi$
$317$	947406.	0.529527	0.264763	+	0.964313i	$0.414706\pi$
$318$	0	0
$319$	533983.	0.293799
$320$	0	0
$321$	1.97964e6	1.07232
$322$	0	0
$323$	285427.	0.152226
$324$	0	0
$325$	−1.16291e6	−0.610712
$326$	0	0
$327$	−1.35508e6	−0.700804
$328$	0	0
$329$	−253848.	−0.129296
$330$	0	0
$331$	−2.19902e6	−1.10321	−0.551606	−	0.834105i	$-0.685985\pi$
$331$	−2.19902e6	−1.10321	−0.551606	+	0.834105i	$0.685985\pi$
$332$	0	0
$333$	−629454.	−0.311067
$334$	0	0
$335$	−3.72807e6	−1.81498
$336$	0	0
$337$	2.70835e6	1.29906	0.649531	−	0.760335i	$-0.274965\pi$
$337$	2.70835e6	1.29906	0.649531	+	0.760335i	$0.274965\pi$
$338$	0	0
$339$	−467948.	−0.221156
$340$	0	0
$341$	57708.3	0.0268752
$342$	0	0
$343$	753488.	0.345813
$344$	0	0
$345$	−1.88551e6	−0.852865
$346$	0	0
$347$	−277619.	−0.123773	−0.0618864	−	0.998083i	$-0.519712\pi$
$347$	−277619.	−0.123773	−0.0618864	+	0.998083i	$0.519712\pi$
$348$	0	0
$349$	−1.60318e6	−0.704563	−0.352281	−	0.935894i	$-0.614594\pi$
$349$	−1.60318e6	−0.704563	−0.352281	+	0.935894i	$0.614594\pi$
$350$	0	0
$351$	822943.	0.356535
$352$	0	0
$353$	−1.30752e6	−0.558485	−0.279243	−	0.960221i	$-0.590083\pi$
$353$	−1.30752e6	−0.558485	−0.279243	+	0.960221i	$0.590083\pi$
$354$	0	0
$355$	437554.	0.184273
$356$	0	0
$357$	162008.	0.0672768
$358$	0	0
$359$	3.81657e6	1.56292	0.781461	−	0.623954i	$-0.214475\pi$
$359$	3.81657e6	1.56292	0.781461	+	0.623954i	$0.214475\pi$
$360$	0	0
$361$	130321.	0.0526316
$362$	0	0
$363$	1.20718e6	0.480846
$364$	0	0
$365$	270980.	0.106465
$366$	0	0
$367$	1.19970e6	0.464950	0.232475	−	0.972602i	$-0.425318\pi$
$367$	1.19970e6	0.464950	0.232475	+	0.972602i	$0.425318\pi$
$368$	0	0
$369$	−629195.	−0.240558
$370$	0	0
$371$	604004.	0.227827
$372$	0	0
$373$	−4.70268e6	−1.75014	−0.875070	−	0.483996i	$-0.839185\pi$
$373$	−4.70268e6	−1.75014	−0.875070	+	0.483996i	$0.839185\pi$
$374$	0	0
$375$	1.21531e6	0.446282
$376$	0	0
$377$	3.67396e6	1.33132
$378$	0	0
$379$	−795027.	−0.284305	−0.142152	−	0.989845i	$-0.545402\pi$
$379$	−795027.	−0.284305	−0.142152	+	0.989845i	$0.545402\pi$
$380$	0	0
$381$	1.00184e6	0.353579
$382$	0	0
$383$	2.39395e6	0.833908	0.416954	−	0.908928i	$-0.363098\pi$
$383$	2.39395e6	0.833908	0.416954	+	0.908928i	$0.363098\pi$
$384$	0	0
$385$	240788.	0.0827908
$386$	0	0
$387$	−721869.	−0.245008
$388$	0	0
$389$	3.54056e6	1.18631	0.593155	−	0.805088i	$-0.297882\pi$
$389$	3.54056e6	1.18631	0.593155	+	0.805088i	$0.297882\pi$
$390$	0	0
$391$	2.56969e6	0.850037
$392$	0	0
$393$	578903.	0.189071
$394$	0	0
$395$	2.51398e6	0.810716
$396$	0	0
$397$	−3.55073e6	−1.13068	−0.565342	−	0.824857i	$-0.691256\pi$
$397$	−3.55073e6	−1.13068	−0.565342	+	0.824857i	$0.691256\pi$
$398$	0	0
$399$	73969.9	0.0232607
$400$	0	0
$401$	5.24138e6	1.62774	0.813869	−	0.581048i	$-0.197357\pi$
$401$	5.24138e6	1.62774	0.813869	+	0.581048i	$0.197357\pi$
$402$	0	0
$403$	397050.	0.121782
$404$	0	0
$405$	422925.	0.128123
$406$	0	0
$407$	1.27501e6	0.381529
$408$	0	0
$409$	2.30813e6	0.682262	0.341131	−	0.940016i	$-0.389190\pi$
$409$	2.30813e6	0.682262	0.341131	+	0.940016i	$0.389190\pi$
$410$	0	0
$411$	59620.8	0.0174098
$412$	0	0
$413$	5611.62	0.00161887
$414$	0	0
$415$	1.67360e6	0.477015
$416$	0	0
$417$	−1.16251e6	−0.327383
$418$	0	0
$419$	2.10757e6	0.586470	0.293235	−	0.956040i	$-0.405268\pi$
$419$	2.10757e6	0.586470	0.293235	+	0.956040i	$0.405268\pi$
$420$	0	0
$421$	−2.15488e6	−0.592540	−0.296270	−	0.955104i	$-0.595743\pi$
$421$	−2.15488e6	−0.592540	−0.296270	+	0.955104i	$0.595743\pi$
$422$	0	0
$423$	903137.	0.245416
$424$	0	0
$425$	814499.	0.218735
$426$	0	0
$427$	725786.	0.192637
$428$	0	0
$429$	−1.66694e6	−0.437297
$430$	0	0
$431$	−6.32652e6	−1.64048	−0.820242	−	0.572017i	$-0.806161\pi$
$431$	−6.32652e6	−1.64048	−0.820242	+	0.572017i	$0.806161\pi$
$432$	0	0
$433$	−2.03642e6	−0.521972	−0.260986	−	0.965343i	$-0.584048\pi$
$433$	−2.03642e6	−0.521972	−0.260986	+	0.965343i	$0.584048\pi$
$434$	0	0
$435$	1.88811e6	0.478415
$436$	0	0
$437$	1.17327e6	0.293897
$438$	0	0
$439$	1.41688e6	0.350890	0.175445	−	0.984489i	$-0.443864\pi$
$439$	1.41688e6	0.350890	0.175445	+	0.984489i	$0.443864\pi$
$440$	0	0
$441$	−1.31938e6	−0.323053
$442$	0	0
$443$	1.10768e6	0.268166	0.134083	−	0.990970i	$-0.457191\pi$
$443$	1.10768e6	0.268166	0.134083	+	0.990970i	$0.457191\pi$
$444$	0	0
$445$	−6.58075e6	−1.57534
$446$	0	0
$447$	−2.47980e6	−0.587014
$448$	0	0
$449$	6883.75	0.00161142	0.000805710	−	1.00000i	$-0.499744\pi$
$449$	6883.75	0.00161142	0.000805710		1.00000i	$0.499744\pi$
$450$	0	0
$451$	1.27449e6	0.295049
$452$	0	0
$453$	−2.25295e6	−0.515829
$454$	0	0
$455$	1.65669e6	0.375157
$456$	0	0
$457$	2.22170e6	0.497617	0.248809	−	0.968553i	$-0.419961\pi$
$457$	2.22170e6	0.497617	0.248809	+	0.968553i	$0.419961\pi$
$458$	0	0
$459$	−576388.	−0.127698
$460$	0	0
$461$	−750389.	−0.164450	−0.0822250	−	0.996614i	$-0.526203\pi$
$461$	−750389.	−0.164450	−0.0822250	+	0.996614i	$0.526203\pi$
$462$	0	0
$463$	8.31618e6	1.80290	0.901450	−	0.432883i	$-0.142504\pi$
$463$	8.31618e6	1.80290	0.901450	+	0.432883i	$0.142504\pi$
$464$	0	0
$465$	204051.	0.0437630
$466$	0	0
$467$	−6.63371e6	−1.40755	−0.703775	−	0.710423i	$-0.748504\pi$
$467$	−6.63371e6	−1.40755	−0.703775	+	0.710423i	$0.748504\pi$
$468$	0	0
$469$	1.31673e6	0.276416
$470$	0	0
$471$	3.18519e6	0.661582
$472$	0	0
$473$	1.46221e6	0.300508
$474$	0	0
$475$	371886.	0.0756268
$476$	0	0
$477$	−2.14892e6	−0.432438
$478$	0	0
$479$	6.52856e6	1.30011	0.650053	−	0.759889i	$-0.274747\pi$
$479$	6.52856e6	1.30011	0.650053	+	0.759889i	$0.274747\pi$
$480$	0	0
$481$	8.77246e6	1.72886
$482$	0	0
$483$	665948.	0.129889
$484$	0	0
$485$	5.93412e6	1.14552
$486$	0	0
$487$	644478.	0.123136	0.0615681	−	0.998103i	$-0.480390\pi$
$487$	644478.	0.123136	0.0615681	+	0.998103i	$0.480390\pi$
$488$	0	0
$489$	−3.31572e6	−0.627055
$490$	0	0
$491$	−3.74928e6	−0.701849	−0.350925	−	0.936404i	$-0.614133\pi$
$491$	−3.74928e6	−0.701849	−0.350925	+	0.936404i	$0.614133\pi$
$492$	0	0
$493$	−2.57324e6	−0.476829
$494$	0	0
$495$	−856670.	−0.157145
$496$	0	0
$497$	−154541.	−0.0280642
$498$	0	0
$499$	−2.07642e6	−0.373304	−0.186652	−	0.982426i	$-0.559764\pi$
$499$	−2.07642e6	−0.373304	−0.186652	+	0.982426i	$0.559764\pi$
$500$	0	0
$501$	2.11005e6	0.375576
$502$	0	0
$503$	8.25513e6	1.45480	0.727401	−	0.686212i	$-0.240728\pi$
$503$	8.25513e6	1.45480	0.727401	+	0.686212i	$0.240728\pi$
$504$	0	0
$505$	−5.46520e6	−0.953624
$506$	0	0
$507$	−8.12741e6	−1.40421
$508$	0	0
$509$	3.83581e6	0.656240	0.328120	−	0.944636i	$-0.393585\pi$
$509$	3.83581e6	0.656240	0.328120	+	0.944636i	$0.393585\pi$
$510$	0	0
$511$	−95708.3	−0.0162143
$512$	0	0
$513$	−263169.	−0.0441511
$514$	0	0
$515$	−5.82328e6	−0.967496
$516$	0	0
$517$	−1.82938e6	−0.301007
$518$	0	0
$519$	−798930.	−0.130194
$520$	0	0
$521$	6.85680e6	1.10669	0.553346	−	0.832951i	$-0.313351\pi$
$521$	6.85680e6	1.10669	0.553346	+	0.832951i	$0.313351\pi$
$522$	0	0
$523$	4.27671e6	0.683684	0.341842	−	0.939757i	$-0.388949\pi$
$523$	4.27671e6	0.683684	0.341842	+	0.939757i	$0.388949\pi$
$524$	0	0
$525$	211082.	0.0334235
$526$	0	0
$527$	−278093.	−0.0436178
$528$	0	0
$529$	4.12658e6	0.641138
$530$	0	0
$531$	−19964.9	−0.00307278
$532$	0	0
$533$	8.76885e6	1.33698
$534$	0	0
$535$	−1.41787e7	−2.14167
$536$	0	0
$537$	−3.16591e6	−0.473765
$538$	0	0
$539$	2.67252e6	0.396231
$540$	0	0
$541$	2.56801e6	0.377227	0.188614	−	0.982051i	$-0.439601\pi$
$541$	2.56801e6	0.377227	0.188614	+	0.982051i	$0.439601\pi$
$542$	0	0
$543$	−4.00078e6	−0.582297
$544$	0	0
$545$	9.70548e6	1.39967
$546$	0	0
$547$	−2.21579e6	−0.316636	−0.158318	−	0.987388i	$-0.550607\pi$
$547$	−2.21579e6	−0.316636	−0.158318	+	0.987388i	$0.550607\pi$
$548$	0	0
$549$	−2.58219e6	−0.365643
$550$	0	0
$551$	−1.17490e6	−0.164862
$552$	0	0
$553$	−887919.	−0.123470
$554$	0	0
$555$	4.50833e6	0.621273
$556$	0	0
$557$	−2.58631e6	−0.353218	−0.176609	−	0.984281i	$-0.556513\pi$
$557$	−2.58631e6	−0.353218	−0.176609	+	0.984281i	$0.556513\pi$
$558$	0	0
$559$	1.00604e7	1.36171
$560$	0	0
$561$	1.16752e6	0.156624
$562$	0	0
$563$	6.77457e6	0.900763	0.450382	−	0.892836i	$-0.351288\pi$
$563$	6.77457e6	0.900763	0.450382	+	0.892836i	$0.351288\pi$
$564$	0	0
$565$	3.35158e6	0.441701
$566$	0	0
$567$	−149374.	−0.0195127
$568$	0	0
$569$	2.12014e6	0.274526	0.137263	−	0.990535i	$-0.456170\pi$
$569$	2.12014e6	0.274526	0.137263	+	0.990535i	$0.456170\pi$
$570$	0	0
$571$	−1.64115e6	−0.210648	−0.105324	−	0.994438i	$-0.533588\pi$
$571$	−1.64115e6	−0.210648	−0.105324	+	0.994438i	$0.533588\pi$
$572$	0	0
$573$	5.93505e6	0.755158
$574$	0	0
$575$	3.34807e6	0.422304
$576$	0	0
$577$	−1.31588e7	−1.64542	−0.822710	−	0.568461i	$-0.807539\pi$
$577$	−1.31588e7	−1.64542	−0.822710	+	0.568461i	$0.807539\pi$
$578$	0	0
$579$	−1.28723e6	−0.159574
$580$	0	0
$581$	−591104.	−0.0726480
$582$	0	0
$583$	4.35281e6	0.530393
$584$	0	0
$585$	−5.89415e6	−0.712084
$586$	0	0
$587$	−2.09243e6	−0.250643	−0.125321	−	0.992116i	$-0.539996\pi$
$587$	−2.09243e6	−0.250643	−0.125321	+	0.992116i	$0.539996\pi$
$588$	0	0
$589$	−126973.	−0.0150807
$590$	0	0
$591$	−2.84252e6	−0.334761
$592$	0	0
$593$	4.29249e6	0.501271	0.250635	−	0.968082i	$-0.419361\pi$
$593$	4.29249e6	0.501271	0.250635	+	0.968082i	$0.419361\pi$
$594$	0	0
$595$	−1.16034e6	−0.134368
$596$	0	0
$597$	2.31656e6	0.266016
$598$	0	0
$599$	−7.06864e6	−0.804949	−0.402475	−	0.915431i	$-0.631850\pi$
$599$	−7.06864e6	−0.804949	−0.402475	+	0.915431i	$0.631850\pi$
$600$	0	0
$601$	5.59587e6	0.631948	0.315974	−	0.948768i	$-0.397669\pi$
$601$	5.59587e6	0.631948	0.315974	+	0.948768i	$0.397669\pi$
$602$	0	0
$603$	−4.68463e6	−0.524665
$604$	0	0
$605$	−8.64617e6	−0.960363
$606$	0	0
$607$	−1.58098e7	−1.74162	−0.870812	−	0.491617i	$-0.836406\pi$
$607$	−1.58098e7	−1.74162	−0.870812	+	0.491617i	$0.836406\pi$
$608$	0	0
$609$	−666868.	−0.0728613
$610$	0	0
$611$	−1.25867e7	−1.36398
$612$	0	0
$613$	−9.30798e6	−1.00047	−0.500235	−	0.865889i	$-0.666753\pi$
$613$	−9.30798e6	−1.00047	−0.500235	+	0.865889i	$0.666753\pi$
$614$	0	0
$615$	4.50647e6	0.480451
$616$	0	0
$617$	−1.30711e7	−1.38229	−0.691145	−	0.722716i	$-0.742893\pi$
$617$	−1.30711e7	−1.38229	−0.691145	+	0.722716i	$0.742893\pi$
$618$	0	0
$619$	−9.29042e6	−0.974560	−0.487280	−	0.873246i	$-0.662011\pi$
$619$	−9.29042e6	−0.974560	−0.487280	+	0.873246i	$0.662011\pi$
$620$	0	0
$621$	−2.36930e6	−0.246542
$622$	0	0
$623$	2.32427e6	0.239920
$624$	0	0
$625$	−1.19236e7	−1.22098
$626$	0	0
$627$	533071.	0.0541522
$628$	0	0
$629$	−6.14422e6	−0.619213
$630$	0	0
$631$	952359.	0.0952198	0.0476099	−	0.998866i	$-0.484840\pi$
$631$	952359.	0.0952198	0.0476099	+	0.998866i	$0.484840\pi$
$632$	0	0
$633$	−7.79186e6	−0.772915
$634$	0	0
$635$	−7.17546e6	−0.706180
$636$	0	0
$637$	1.83877e7	1.79547
$638$	0	0
$639$	549823.	0.0532686
$640$	0	0
$641$	5.98948e6	0.575763	0.287882	−	0.957666i	$-0.407049\pi$
$641$	5.98948e6	0.575763	0.287882	+	0.957666i	$0.407049\pi$
$642$	0	0
$643$	−1.65895e7	−1.58236	−0.791180	−	0.611583i	$-0.790533\pi$
$643$	−1.65895e7	−1.58236	−0.791180	+	0.611583i	$0.790533\pi$
$644$	0	0
$645$	5.17022e6	0.489339
$646$	0	0
$647$	−3.02741e6	−0.284322	−0.142161	−	0.989844i	$-0.545405\pi$
$647$	−3.02741e6	−0.284322	−0.142161	+	0.989844i	$0.545405\pi$
$648$	0	0
$649$	40440.6	0.00376883
$650$	0	0
$651$	−72069.4	−0.00666497
$652$	0	0
$653$	546890.	0.0501899	0.0250950	−	0.999685i	$-0.492011\pi$
$653$	546890.	0.0501899	0.0250950	+	0.999685i	$0.492011\pi$
$654$	0	0
$655$	−4.14626e6	−0.377619
$656$	0	0
$657$	340510.	0.0307763
$658$	0	0
$659$	−1.62616e7	−1.45865	−0.729324	−	0.684169i	$-0.760165\pi$
$659$	−1.62616e7	−1.45865	−0.729324	+	0.684169i	$0.760165\pi$
$660$	0	0
$661$	−6.26193e6	−0.557448	−0.278724	−	0.960371i	$-0.589912\pi$
$661$	−6.26193e6	−0.557448	−0.278724	+	0.960371i	$0.589912\pi$
$662$	0	0
$663$	8.03291e6	0.709723
$664$	0	0
$665$	−529793.	−0.0464571
$666$	0	0
$667$	−1.05775e7	−0.920597
$668$	0	0
$669$	6.62653e6	0.572428
$670$	0	0
$671$	5.23044e6	0.448468
$672$	0	0
$673$	1.21608e7	1.03496	0.517479	−	0.855696i	$-0.326870\pi$
$673$	1.21608e7	1.03496	0.517479	+	0.855696i	$0.326870\pi$
$674$	0	0
$675$	−750983.	−0.0634411
$676$	0	0
$677$	1.27621e7	1.07016	0.535081	−	0.844801i	$-0.320281\pi$
$677$	1.27621e7	1.07016	0.535081	+	0.844801i	$0.320281\pi$
$678$	0	0
$679$	−2.09589e6	−0.174459
$680$	0	0
$681$	−8.50027e6	−0.702368
$682$	0	0
$683$	−2.36289e7	−1.93817	−0.969084	−	0.246733i	$-0.920643\pi$
$683$	−2.36289e7	−1.93817	−0.969084	+	0.246733i	$0.920643\pi$
$684$	0	0
$685$	−427020.	−0.0347714
$686$	0	0
$687$	−2.64647e6	−0.213931
$688$	0	0
$689$	2.99486e7	2.40342
$690$	0	0
$691$	−2.19421e7	−1.74817	−0.874085	−	0.485774i	$-0.838538\pi$
$691$	−2.19421e7	−1.74817	−0.874085	+	0.485774i	$0.838538\pi$
$692$	0	0
$693$	302570.	0.0239327
$694$	0	0
$695$	8.32620e6	0.653860
$696$	0	0
$697$	−6.14169e6	−0.478857
$698$	0	0
$699$	1.42716e7	1.10479
$700$	0	0
$701$	2.39662e7	1.84206	0.921029	−	0.389494i	$-0.127350\pi$
$701$	2.39662e7	1.84206	0.921029	+	0.389494i	$0.127350\pi$
$702$	0	0
$703$	−2.80535e6	−0.214091
$704$	0	0
$705$	−6.46851e6	−0.490153
$706$	0	0
$707$	1.93027e6	0.145234
$708$	0	0
$709$	1.08716e7	0.812227	0.406114	−	0.913823i	$-0.366884\pi$
$709$	1.08716e7	0.812227	0.406114	+	0.913823i	$0.366884\pi$
$710$	0	0
$711$	3.15902e6	0.234358
$712$	0	0
$713$	−1.14313e6	−0.0842115
$714$	0	0
$715$	1.19391e7	0.873385
$716$	0	0
$717$	−9.80662e6	−0.712396
$718$	0	0
$719$	−1.20672e7	−0.870528	−0.435264	−	0.900303i	$-0.643345\pi$
$719$	−1.20672e7	−0.870528	−0.435264	+	0.900303i	$0.643345\pi$
$720$	0	0
$721$	2.05674e6	0.147347
$722$	0	0
$723$	−1.44980e7	−1.03148
$724$	0	0
$725$	−3.35270e6	−0.236892
$726$	0	0
$727$	−3.82452e6	−0.268374	−0.134187	−	0.990956i	$-0.542842\pi$
$727$	−3.82452e6	−0.268374	−0.134187	+	0.990956i	$0.542842\pi$
$728$	0	0
$729$	531441.	0.0370370
$730$	0	0
$731$	−7.04630e6	−0.487717
$732$	0	0
$733$	−7.95102e6	−0.546591	−0.273296	−	0.961930i	$-0.588114\pi$
$733$	−7.95102e6	−0.546591	−0.273296	+	0.961930i	$0.588114\pi$
$734$	0	0
$735$	9.44978e6	0.645213
$736$	0	0
$737$	9.48911e6	0.643512
$738$	0	0
$739$	−1.40289e7	−0.944956	−0.472478	−	0.881343i	$-0.656640\pi$
$739$	−1.40289e7	−0.944956	−0.472478	+	0.881343i	$0.656640\pi$
$740$	0	0
$741$	3.66769e6	0.245384
$742$	0	0
$743$	−2.00451e7	−1.33210	−0.666048	−	0.745909i	$-0.732015\pi$
$743$	−2.00451e7	−1.33210	−0.666048	+	0.745909i	$0.732015\pi$
$744$	0	0
$745$	1.77610e7	1.17241
$746$	0	0
$747$	2.10302e6	0.137893
$748$	0	0
$749$	5.00782e6	0.326170
$750$	0	0
$751$	1.97350e7	1.27684	0.638420	−	0.769688i	$-0.279588\pi$
$751$	1.97350e7	1.27684	0.638420	+	0.769688i	$0.279588\pi$
$752$	0	0
$753$	−3.20613e6	−0.206060
$754$	0	0
$755$	1.61362e7	1.03023
$756$	0	0
$757$	4.67156e6	0.296293	0.148147	−	0.988965i	$-0.452669\pi$
$757$	4.67156e6	0.296293	0.148147	+	0.988965i	$0.452669\pi$
$758$	0	0
$759$	4.79921e6	0.302388
$760$	0	0
$761$	−2.01453e6	−0.126099	−0.0630496	−	0.998010i	$-0.520083\pi$
$761$	−2.01453e6	−0.126099	−0.0630496	+	0.998010i	$0.520083\pi$
$762$	0	0
$763$	−3.42791e6	−0.213166
$764$	0	0
$765$	4.12825e6	0.255043
$766$	0	0
$767$	278244.	0.0170780
$768$	0	0
$769$	2.21316e7	1.34957	0.674786	−	0.738013i	$-0.264236\pi$
$769$	2.21316e7	1.34957	0.674786	+	0.738013i	$0.264236\pi$
$770$	0	0
$771$	−5.80027e6	−0.351409
$772$	0	0
$773$	5.60195e6	0.337202	0.168601	−	0.985684i	$-0.446075\pi$
$773$	5.60195e6	0.337202	0.168601	+	0.985684i	$0.446075\pi$
$774$	0	0
$775$	−362331.	−0.0216696
$776$	0	0
$777$	−1.59231e6	−0.0946182
$778$	0	0
$779$	−2.80419e6	−0.165563
$780$	0	0
$781$	−1.11371e6	−0.0653350
$782$	0	0
$783$	2.37257e6	0.138298
$784$	0	0
$785$	−2.28132e7	−1.32133
$786$	0	0
$787$	2.56606e7	1.47683	0.738415	−	0.674347i	$-0.235575\pi$
$787$	2.56606e7	1.47683	0.738415	+	0.674347i	$0.235575\pi$
$788$	0	0
$789$	6.66633e6	0.381236
$790$	0	0
$791$	−1.18375e6	−0.0672697
$792$	0	0
$793$	3.59870e7	2.03218
$794$	0	0
$795$	1.53911e7	0.863680
$796$	0	0
$797$	−2.84932e7	−1.58890	−0.794449	−	0.607331i	$-0.792240\pi$
$797$	−2.84932e7	−1.58890	−0.794449	+	0.607331i	$0.792240\pi$
$798$	0	0
$799$	8.81569e6	0.488528
$800$	0	0
$801$	−8.26927e6	−0.455392
$802$	0	0
$803$	−689730.	−0.0377477
$804$	0	0
$805$	−4.76970e6	−0.259419
$806$	0	0
$807$	1.85286e7	1.00152
$808$	0	0
$809$	−2.40603e7	−1.29250	−0.646250	−	0.763126i	$-0.723664\pi$
$809$	−2.40603e7	−1.29250	−0.646250	+	0.763126i	$0.723664\pi$
$810$	0	0
$811$	2.30755e7	1.23197	0.615984	−	0.787759i	$-0.288759\pi$
$811$	2.30755e7	1.23197	0.615984	+	0.787759i	$0.288759\pi$
$812$	0	0
$813$	−9.06414e6	−0.480950
$814$	0	0
$815$	2.37481e7	1.25238
$816$	0	0
$817$	−3.21722e6	−0.168626
$818$	0	0
$819$	2.08177e6	0.108448
$820$	0	0
$821$	1.65486e7	0.856848	0.428424	−	0.903578i	$-0.359069\pi$
$821$	1.65486e7	0.856848	0.428424	+	0.903578i	$0.359069\pi$
$822$	0	0
$823$	1.87603e7	0.965476	0.482738	−	0.875765i	$-0.339642\pi$
$823$	1.87603e7	0.965476	0.482738	+	0.875765i	$0.339642\pi$
$824$	0	0
$825$	1.52118e6	0.0778118
$826$	0	0
$827$	−2.85918e7	−1.45371	−0.726854	−	0.686792i	$-0.759018\pi$
$827$	−2.85918e7	−1.45371	−0.726854	+	0.686792i	$0.759018\pi$
$828$	0	0
$829$	−2.29133e7	−1.15798	−0.578989	−	0.815335i	$-0.696553\pi$
$829$	−2.29133e7	−1.15798	−0.578989	+	0.815335i	$0.696553\pi$
$830$	0	0
$831$	4.04487e6	0.203190
$832$	0	0
$833$	−1.28787e7	−0.643074
$834$	0	0
$835$	−1.51127e7	−0.750113
$836$	0	0
$837$	256407.	0.0126508
$838$	0	0
$839$	−2.25680e7	−1.10685	−0.553423	−	0.832900i	$-0.686679\pi$
$839$	−2.25680e7	−1.10685	−0.553423	+	0.832900i	$0.686679\pi$
$840$	0	0
$841$	−9.91899e6	−0.483590
$842$	0	0
$843$	2.05495e7	0.995937
$844$	0	0
$845$	5.82107e7	2.80454
$846$	0	0
$847$	3.05377e6	0.146261
$848$	0	0
$849$	−2.15081e7	−1.02408
$850$	0	0
$851$	−2.52564e7	−1.19549
$852$	0	0
$853$	5.53978e6	0.260687	0.130344	−	0.991469i	$-0.458392\pi$
$853$	5.53978e6	0.260687	0.130344	+	0.991469i	$0.458392\pi$
$854$	0	0
$855$	1.88489e6	0.0881801
$856$	0	0
$857$	−1.11939e7	−0.520629	−0.260315	−	0.965524i	$-0.583826\pi$
$857$	−1.11939e7	−0.520629	−0.260315	+	0.965524i	$0.583826\pi$
$858$	0	0
$859$	−1.33501e7	−0.617306	−0.308653	−	0.951175i	$-0.599878\pi$
$859$	−1.33501e7	−0.617306	−0.308653	+	0.951175i	$0.599878\pi$
$860$	0	0
$861$	−1.59165e6	−0.0731712
$862$	0	0
$863$	−1.58707e7	−0.725387	−0.362693	−	0.931909i	$-0.618143\pi$
$863$	−1.58707e7	−0.725387	−0.362693	+	0.931909i	$0.618143\pi$
$864$	0	0
$865$	5.72216e6	0.260028
$866$	0	0
$867$	7.15248e6	0.323153
$868$	0	0
$869$	−6.39886e6	−0.287444
$870$	0	0
$871$	6.52879e7	2.91600
$872$	0	0
$873$	7.45672e6	0.331140
$874$	0	0
$875$	3.07433e6	0.135747
$876$	0	0
$877$	1.79695e7	0.788928	0.394464	−	0.918911i	$-0.370930\pi$
$877$	1.79695e7	0.788928	0.394464	+	0.918911i	$0.370930\pi$
$878$	0	0
$879$	−1.42439e7	−0.621809
$880$	0	0
$881$	−2.17141e6	−0.0942546	−0.0471273	−	0.998889i	$-0.515007\pi$
$881$	−2.17141e6	−0.0942546	−0.0471273	+	0.998889i	$0.515007\pi$
$882$	0	0
$883$	3.12727e7	1.34978	0.674891	−	0.737918i	$-0.264191\pi$
$883$	3.12727e7	1.34978	0.674891	+	0.737918i	$0.264191\pi$
$884$	0	0
$885$	142994.	0.00613707
$886$	0	0
$887$	−3.48499e7	−1.48728	−0.743640	−	0.668580i	$-0.766903\pi$
$887$	−3.48499e7	−1.48728	−0.743640	+	0.668580i	$0.766903\pi$
$888$	0	0
$889$	2.53432e6	0.107549
$890$	0	0
$891$	−1.07648e6	−0.0454267
$892$	0	0
$893$	4.02509e6	0.168907
$894$	0	0
$895$	2.26751e7	0.946220
$896$	0	0
$897$	3.30200e7	1.37024
$898$	0	0
$899$	1.14471e6	0.0472385
$900$	0	0
$901$	−2.09760e7	−0.860816
$902$	0	0
$903$	−1.82609e6	−0.0745250
$904$	0	0
$905$	2.86547e7	1.16298
$906$	0	0
$907$	2.15663e7	0.870478	0.435239	−	0.900315i	$-0.356664\pi$
$907$	2.15663e7	0.870478	0.435239	+	0.900315i	$0.356664\pi$
$908$	0	0
$909$	−6.86748e6	−0.275669
$910$	0	0
$911$	3.67371e7	1.46659	0.733295	−	0.679911i	$-0.237981\pi$
$911$	3.67371e7	1.46659	0.733295	+	0.679911i	$0.237981\pi$
$912$	0	0
$913$	−4.25984e6	−0.169128
$914$	0	0
$915$	1.84943e7	0.730275
$916$	0	0
$917$	1.46443e6	0.0575102
$918$	0	0
$919$	1.77809e7	0.694487	0.347243	−	0.937775i	$-0.387118\pi$
$919$	1.77809e7	0.694487	0.347243	+	0.937775i	$0.387118\pi$
$920$	0	0
$921$	1.14014e7	0.442902
$922$	0	0
$923$	−7.66268e6	−0.296058
$924$	0	0
$925$	−8.00538e6	−0.307629
$926$	0	0
$927$	−7.31743e6	−0.279679
$928$	0	0
$929$	1.87505e7	0.712811	0.356405	−	0.934331i	$-0.384002\pi$
$929$	1.87505e7	0.712811	0.356405	+	0.934331i	$0.384002\pi$
$930$	0	0
$931$	−5.88021e6	−0.222340
$932$	0	0
$933$	−3.75059e6	−0.141057
$934$	0	0
$935$	−8.36212e6	−0.312815
$936$	0	0
$937$	−4.46179e7	−1.66020	−0.830100	−	0.557615i	$-0.811717\pi$
$937$	−4.46179e7	−1.66020	−0.830100	+	0.557615i	$0.811717\pi$
$938$	0	0
$939$	1.34787e7	0.498866
$940$	0	0
$941$	9.58472e6	0.352862	0.176431	−	0.984313i	$-0.443545\pi$
$941$	9.58472e6	0.352862	0.176431	+	0.984313i	$0.443545\pi$
$942$	0	0
$943$	−2.52460e7	−0.924514
$944$	0	0
$945$	1.06986e6	0.0389715
$946$	0	0
$947$	1.51375e7	0.548505	0.274252	−	0.961658i	$-0.411570\pi$
$947$	1.51375e7	0.548505	0.274252	+	0.961658i	$0.411570\pi$
$948$	0	0
$949$	−4.74555e6	−0.171049
$950$	0	0
$951$	−8.52665e6	−0.305722
$952$	0	0
$953$	−7.77412e6	−0.277280	−0.138640	−	0.990343i	$-0.544273\pi$
$953$	−7.77412e6	−0.277280	−0.138640	+	0.990343i	$0.544273\pi$
$954$	0	0
$955$	−4.25085e7	−1.50823
$956$	0	0
$957$	−4.80584e6	−0.169625
$958$	0	0
$959$	150821.	0.00529558
$960$	0	0
$961$	−2.85054e7	−0.995679
$962$	0	0
$963$	−1.78167e7	−0.619102
$964$	0	0
$965$	9.21953e6	0.318706
$966$	0	0
$967$	−5.37793e6	−0.184948	−0.0924738	−	0.995715i	$-0.529477\pi$
$967$	−5.37793e6	−0.184948	−0.0924738	+	0.995715i	$0.529477\pi$
$968$	0	0
$969$	−2.56884e6	−0.0878877
$970$	0	0
$971$	−9.19800e6	−0.313073	−0.156536	−	0.987672i	$-0.550033\pi$
$971$	−9.19800e6	−0.313073	−0.156536	+	0.987672i	$0.550033\pi$
$972$	0	0
$973$	−2.94076e6	−0.0995811
$974$	0	0
$975$	1.04662e7	0.352595
$976$	0	0
$977$	−4.28018e7	−1.43458	−0.717292	−	0.696773i	$-0.754619\pi$
$977$	−4.28018e7	−1.43458	−0.717292	+	0.696773i	$0.754619\pi$
$978$	0	0
$979$	1.67501e7	0.558548
$980$	0	0
$981$	1.21957e7	0.404609
$982$	0	0
$983$	2.14081e7	0.706633	0.353317	−	0.935504i	$-0.385054\pi$
$983$	2.14081e7	0.706633	0.353317	+	0.935504i	$0.385054\pi$
$984$	0	0
$985$	2.03589e7	0.668597
$986$	0	0
$987$	2.28463e6	0.0746489
$988$	0	0
$989$	−2.89645e7	−0.941618
$990$	0	0
$991$	−3.45241e7	−1.11670	−0.558352	−	0.829604i	$-0.688566\pi$
$991$	−3.45241e7	−1.11670	−0.558352	+	0.829604i	$0.688566\pi$
$992$	0	0
$993$	1.97912e7	0.636940
$994$	0	0
$995$	−1.65918e7	−0.531296
$996$	0	0
$997$	1.24237e7	0.395833	0.197917	−	0.980219i	$-0.436582\pi$
$997$	1.24237e7	0.395833	0.197917	+	0.980219i	$0.436582\pi$
$998$	0	0
$999$	5.66509e6	0.179594

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	228.6.a.b.1.4	✓	4
3.2	odd	2		684.6.a.d.1.1		4
4.3	odd	2		912.6.a.p.1.4		4

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
228.6.a.b.1.4	✓	4	1.1	even	1	trivial
684.6.a.d.1.1		4	3.2	odd	2
912.6.a.p.1.4		4	4.3	odd	2

Overview	Random
Universe	Knowledge

Classical	Maass
Hilbert	Bianchi

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

Level:	\( N \)	\(=\)	\( 228 = 2^{2} \cdot 3 \cdot 19 \)
Weight:	\( k \)	\(=\)	\( 6 \)
Character orbit:	\([\chi]\)	\(=\)	228.a (trivial)

\(f(q)\)	\(=\)	\(q-9.00000 q^{3} +64.4605 q^{5} -22.7670 q^{7} +81.0000 q^{9} -164.072 q^{11} -1128.87 q^{13} -580.144 q^{15} +790.656 q^{17} +361.000 q^{19} +204.903 q^{21} +3250.07 q^{23} +1030.16 q^{25} -729.000 q^{27} -3254.56 q^{29} -351.725 q^{31} +1476.65 q^{33} -1467.57 q^{35} -7771.04 q^{37} +10159.8 q^{39} -7767.84 q^{41} -8911.96 q^{43} +5221.30 q^{45} +11149.8 q^{47} -16288.7 q^{49} -7115.91 q^{51} -26529.8 q^{53} -10576.2 q^{55} -3249.00 q^{57} -246.481 q^{59} -31878.9 q^{61} -1844.13 q^{63} -72767.3 q^{65} -57835.0 q^{67} -29250.6 q^{69} +6787.94 q^{71} +4203.82 q^{73} -9271.40 q^{75} +3735.43 q^{77} +39000.3 q^{79} +6561.00 q^{81} +25963.2 q^{83} +50966.1 q^{85} +29291.0 q^{87} -102090. q^{89} +25700.9 q^{91} +3165.52 q^{93} +23270.2 q^{95} +92058.3 q^{97} -13289.8 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 4 q - 36 q^{3} - 84 q^{5} + 54 q^{7} + 324 q^{9} + 354 q^{11} + 46 q^{13} + 756 q^{15} + 2046 q^{17} + 1444 q^{19} - 486 q^{21} + 846 q^{23} + 4018 q^{25} - 2916 q^{27} - 7152 q^{29} + 238 q^{31} - 3186 q^{33}+ \cdots + 28674 q^{99}+O(q^{100}) \) 4 * q - 36 * q^3 - 84 * q^5 + 54 * q^7 + 324 * q^9 + 354 * q^11 + 46 * q^13 + 756 * q^15 + 2046 * q^17 + 1444 * q^19 - 486 * q^21 + 846 * q^23 + 4018 * q^25 - 2916 * q^27 - 7152 * q^29 + 238 * q^31 - 3186 * q^33 - 19878 * q^35 - 10326 * q^37 - 414 * q^39 - 28200 * q^41 - 18590 * q^43 - 6804 * q^45 - 18684 * q^47 - 17946 * q^49 - 18414 * q^51 - 42720 * q^53 - 13818 * q^55 - 12996 * q^57 - 68256 * q^59 - 67654 * q^61 + 4374 * q^63 - 109776 * q^65 - 81836 * q^67 - 7614 * q^69 - 62112 * q^71 - 81966 * q^73 - 36162 * q^75 - 81378 * q^77 + 8766 * q^79 + 26244 * q^81 - 55080 * q^83 - 108030 * q^85 + 64368 * q^87 - 197520 * q^89 + 25820 * q^91 - 2142 * q^93 - 30324 * q^95 - 184228 * q^97 + 28674 * q^99

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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