LMFDB - Embedded newform 1104.6.a.o.1.1

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

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

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

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

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

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

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

chi := DirichletCharacter("1104.1");

S:= CuspForms(chi, 6);

N := Newforms(S);

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

Newform invariants

comment: select newform

sage: f = N[0] # Warning: the index may be different

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

Self dual:	yes
Analytic conductor:	$177.063737074$
Analytic rank:	$0$
Dimension:	$4$
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			1.1
Root			$-7.50608$ 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)

$f(q)$	$=$	$q+9.00000 q^{3} -86.6918 q^{5} +64.0051 q^{7} +81.0000 q^{9} +O(q^{10})$	$q+9.00000 q^{3} -86.6918 q^{5} +64.0051 q^{7} +81.0000 q^{9} +285.170 q^{11} -307.400 q^{13} -780.226 q^{15} +2223.12 q^{17} +1802.44 q^{19} +576.046 q^{21} +529.000 q^{23} +4390.47 q^{25} +729.000 q^{27} -2359.49 q^{29} -8317.38 q^{31} +2566.53 q^{33} -5548.72 q^{35} -9075.00 q^{37} -2766.60 q^{39} +1543.25 q^{41} +15330.5 q^{43} -7022.04 q^{45} -14725.3 q^{47} -12710.3 q^{49} +20008.1 q^{51} -14163.0 q^{53} -24721.9 q^{55} +16222.0 q^{57} -8408.86 q^{59} +26134.4 q^{61} +5184.41 q^{63} +26649.1 q^{65} +13002.9 q^{67} +4761.00 q^{69} -52490.6 q^{71} -16992.9 q^{73} +39514.3 q^{75} +18252.3 q^{77} +100023. q^{79} +6561.00 q^{81} +85251.7 q^{83} -192726. q^{85} -21235.4 q^{87} -83076.2 q^{89} -19675.2 q^{91} -74856.4 q^{93} -156257. q^{95} +31793.1 q^{97} +23098.8 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 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
$5$	−86.6918	−1.55079	−0.775395	−	0.631476i	$-0.782449\pi$
$5$	−86.6918	−1.55079	−0.775395	+	0.631476i	$0.782449\pi$
$6$	0	0
$7$	64.0051	0.493707	0.246854	−	0.969053i	$-0.420603\pi$
$7$	64.0051	0.493707	0.246854	+	0.969053i	$0.420603\pi$
$8$	0	0
$9$	81.0000	0.333333
$10$	0	0
$11$	285.170	0.710595	0.355297	−	0.934753i	$-0.384380\pi$
$11$	285.170	0.710595	0.355297	+	0.934753i	$0.384380\pi$
$12$	0	0
$13$	−307.400	−0.504482	−0.252241	−	0.967664i	$-0.581168\pi$
$13$	−307.400	−0.504482	−0.252241	+	0.967664i	$0.581168\pi$
$14$	0	0
$15$	−780.226	−0.895349
$16$	0	0
$17$	2223.12	1.86569	0.932847	−	0.360273i	$-0.117317\pi$
$17$	2223.12	1.86569	0.932847	+	0.360273i	$0.117317\pi$
$18$	0	0
$19$	1802.44	1.14545	0.572727	−	0.819746i	$-0.305886\pi$
$19$	1802.44	1.14545	0.572727	+	0.819746i	$0.305886\pi$
$20$	0	0
$21$	576.046	0.285042
$22$	0	0
$23$	529.000	0.208514
$23$	529.000	0.208514
$24$	0	0
$25$	4390.47	1.40495
$26$	0	0
$27$	729.000	0.192450
$28$	0	0
$29$	−2359.49	−0.520982	−0.260491	−	0.965476i	$-0.583884\pi$
$29$	−2359.49	−0.520982	−0.260491	+	0.965476i	$0.583884\pi$
$30$	0	0
$31$	−8317.38	−1.55447	−0.777235	−	0.629210i	$-0.783378\pi$
$31$	−8317.38	−1.55447	−0.777235	+	0.629210i	$0.783378\pi$
$32$	0	0
$33$	2566.53	0.410262
$34$	0	0
$35$	−5548.72	−0.765637
$36$	0	0
$37$	−9075.00	−1.08979	−0.544895	−	0.838505i	$-0.683430\pi$
$37$	−9075.00	−1.08979	−0.544895	+	0.838505i	$0.683430\pi$
$38$	0	0
$39$	−2766.60	−0.291263
$40$	0	0
$41$	1543.25	0.143376	0.0716882	−	0.997427i	$-0.477161\pi$
$41$	1543.25	0.143376	0.0716882	+	0.997427i	$0.477161\pi$
$42$	0	0
$43$	15330.5	1.26440	0.632202	−	0.774804i	$-0.282151\pi$
$43$	15330.5	1.26440	0.632202	+	0.774804i	$0.282151\pi$
$44$	0	0
$45$	−7022.04	−0.516930
$46$	0	0
$47$	−14725.3	−0.972343	−0.486172	−	0.873863i	$-0.661607\pi$
$47$	−14725.3	−0.972343	−0.486172	+	0.873863i	$0.661607\pi$
$48$	0	0
$49$	−12710.3	−0.756253
$50$	0	0
$51$	20008.1	1.07716
$52$	0	0
$53$	−14163.0	−0.692572	−0.346286	−	0.938129i	$-0.612557\pi$
$53$	−14163.0	−0.692572	−0.346286	+	0.938129i	$0.612557\pi$
$54$	0	0
$55$	−24721.9	−1.10198
$56$	0	0
$57$	16222.0	0.661328
$58$	0	0
$59$	−8408.86	−0.314490	−0.157245	−	0.987560i	$-0.550261\pi$
$59$	−8408.86	−0.314490	−0.157245	+	0.987560i	$0.550261\pi$
$60$	0	0
$61$	26134.4	0.899267	0.449633	−	0.893213i	$-0.351555\pi$
$61$	26134.4	0.899267	0.449633	+	0.893213i	$0.351555\pi$
$62$	0	0
$63$	5184.41	0.164569
$64$	0	0
$65$	26649.1	0.782346
$66$	0	0
$67$	13002.9	0.353878	0.176939	−	0.984222i	$-0.443380\pi$
$67$	13002.9	0.353878	0.176939	+	0.984222i	$0.443380\pi$
$68$	0	0
$69$	4761.00	0.120386
$70$	0	0
$71$	−52490.6	−1.23576	−0.617882	−	0.786271i	$-0.712009\pi$
$71$	−52490.6	−1.23576	−0.617882	+	0.786271i	$0.712009\pi$
$72$	0	0
$73$	−16992.9	−0.373215	−0.186607	−	0.982435i	$-0.559749\pi$
$73$	−16992.9	−0.373215	−0.186607	+	0.982435i	$0.559749\pi$
$74$	0	0
$75$	39514.3	0.811149
$76$	0	0
$77$	18252.3	0.350826
$78$	0	0
$79$	100023.	1.80316	0.901579	−	0.432614i	$-0.142409\pi$
$79$	100023.	1.80316	0.901579	+	0.432614i	$0.142409\pi$
$80$	0	0
$81$	6561.00	0.111111
$82$	0	0
$83$	85251.7	1.35834	0.679169	−	0.733982i	$-0.262340\pi$
$83$	85251.7	1.35834	0.679169	+	0.733982i	$0.262340\pi$
$84$	0	0
$85$	−192726.	−2.89330
$86$	0	0
$87$	−21235.4	−0.300789
$88$	0	0
$89$	−83076.2	−1.11174	−0.555868	−	0.831271i	$-0.687614\pi$
$89$	−83076.2	−1.11174	−0.555868	+	0.831271i	$0.687614\pi$
$90$	0	0
$91$	−19675.2	−0.249066
$92$	0	0
$93$	−74856.4	−0.897474
$94$	0	0
$95$	−156257.	−1.77636
$96$	0	0
$97$	31793.1	0.343086	0.171543	−	0.985177i	$-0.445125\pi$
$97$	31793.1	0.343086	0.171543	+	0.985177i	$0.445125\pi$
$98$	0	0
$99$	23098.8	0.236865
$100$	0	0
$101$	55155.8	0.538007	0.269003	−	0.963139i	$-0.413306\pi$
$101$	55155.8	0.538007	0.269003	+	0.963139i	$0.413306\pi$
$102$	0	0
$103$	125192.	1.16274	0.581371	−	0.813639i	$-0.302517\pi$
$103$	125192.	1.16274	0.581371	+	0.813639i	$0.302517\pi$
$104$	0	0
$105$	−49938.5	−0.442041
$106$	0	0
$107$	30718.1	0.259379	0.129690	−	0.991555i	$-0.458602\pi$
$107$	30718.1	0.259379	0.129690	+	0.991555i	$0.458602\pi$
$108$	0	0
$109$	−14066.0	−0.113398	−0.0566991	−	0.998391i	$-0.518058\pi$
$109$	−14066.0	−0.113398	−0.0566991	+	0.998391i	$0.518058\pi$
$110$	0	0
$111$	−81675.0	−0.629190
$112$	0	0
$113$	30310.7	0.223305	0.111653	−	0.993747i	$-0.464386\pi$
$113$	30310.7	0.223305	0.111653	+	0.993747i	$0.464386\pi$
$114$	0	0
$115$	−45860.0	−0.323362
$116$	0	0
$117$	−24899.4	−0.168161
$118$	0	0
$119$	142291.	0.921107
$120$	0	0
$121$	−79729.1	−0.495055
$122$	0	0
$123$	13889.3	0.0827784
$124$	0	0
$125$	−109706.	−0.627995
$126$	0	0
$127$	143632.	0.790211	0.395106	−	0.918636i	$-0.370708\pi$
$127$	143632.	0.790211	0.395106	+	0.918636i	$0.370708\pi$
$128$	0	0
$129$	137975.	0.730004
$130$	0	0
$131$	269835.	1.37379	0.686895	−	0.726757i	$-0.258973\pi$
$131$	269835.	1.37379	0.686895	+	0.726757i	$0.258973\pi$
$132$	0	0
$133$	115366.	0.565519
$134$	0	0
$135$	−63198.3	−0.298450
$136$	0	0
$137$	305800.	1.39199	0.695995	−	0.718047i	$-0.254964\pi$
$137$	305800.	1.39199	0.695995	+	0.718047i	$0.254964\pi$
$138$	0	0
$139$	175456.	0.770251	0.385126	−	0.922864i	$-0.374158\pi$
$139$	175456.	0.770251	0.385126	+	0.922864i	$0.374158\pi$
$140$	0	0
$141$	−132528.	−0.561383
$142$	0	0
$143$	−87661.3	−0.358482
$144$	0	0
$145$	204548.	0.807933
$146$	0	0
$147$	−114393.	−0.436623
$148$	0	0
$149$	507949.	1.87436	0.937182	−	0.348840i	$-0.113424\pi$
$149$	507949.	1.87436	0.937182	+	0.348840i	$0.113424\pi$
$150$	0	0
$151$	−214770.	−0.766534	−0.383267	−	0.923638i	$-0.625201\pi$
$151$	−214770.	−0.766534	−0.383267	+	0.923638i	$0.625201\pi$
$152$	0	0
$153$	180073.	0.621898
$154$	0	0
$155$	721049.	2.41066
$156$	0	0
$157$	180401.	0.584103	0.292051	−	0.956403i	$-0.405662\pi$
$157$	180401.	0.584103	0.292051	+	0.956403i	$0.405662\pi$
$158$	0	0
$159$	−127467.	−0.399857
$160$	0	0
$161$	33858.7	0.102945
$162$	0	0
$163$	−461874.	−1.36161	−0.680807	−	0.732463i	$-0.738371\pi$
$163$	−461874.	−1.36161	−0.680807	+	0.732463i	$0.738371\pi$
$164$	0	0
$165$	−222497.	−0.636231
$166$	0	0
$167$	−8395.62	−0.0232949	−0.0116475	−	0.999932i	$-0.503708\pi$
$167$	−8395.62	−0.0232949	−0.0116475	+	0.999932i	$0.503708\pi$
$168$	0	0
$169$	−276798.	−0.745498
$170$	0	0
$171$	145998.	0.381818
$172$	0	0
$173$	675883.	1.71695	0.858473	−	0.512859i	$-0.171414\pi$
$173$	675883.	1.71695	0.858473	+	0.512859i	$0.171414\pi$
$174$	0	0
$175$	281013.	0.693635
$176$	0	0
$177$	−75679.7	−0.181571
$178$	0	0
$179$	−256756.	−0.598947	−0.299474	−	0.954105i	$-0.596811\pi$
$179$	−256756.	−0.598947	−0.299474	+	0.954105i	$0.596811\pi$
$180$	0	0
$181$	650703.	1.47634	0.738170	−	0.674615i	$-0.235690\pi$
$181$	650703.	1.47634	0.738170	+	0.674615i	$0.235690\pi$
$182$	0	0
$183$	235210.	0.519192
$184$	0	0
$185$	786729.	1.69003
$186$	0	0
$187$	633967.	1.32575
$188$	0	0
$189$	46659.7	0.0950140
$190$	0	0
$191$	874077.	1.73367	0.866835	−	0.498595i	$-0.166151\pi$
$191$	874077.	1.73367	0.866835	+	0.498595i	$0.166151\pi$
$192$	0	0
$193$	−700334.	−1.35336	−0.676678	−	0.736279i	$-0.736581\pi$
$193$	−700334.	−1.35336	−0.676678	+	0.736279i	$0.736581\pi$
$194$	0	0
$195$	239842.	0.451688
$196$	0	0
$197$	1.01192e6	1.85773	0.928864	−	0.370420i	$-0.120786\pi$
$197$	1.01192e6	1.85773	0.928864	+	0.370420i	$0.120786\pi$
$198$	0	0
$199$	−751608.	−1.34542	−0.672712	−	0.739905i	$-0.734871\pi$
$199$	−751608.	−1.34542	−0.672712	+	0.739905i	$0.734871\pi$
$200$	0	0
$201$	117026.	0.204312
$202$	0	0
$203$	−151019.	−0.257212
$204$	0	0
$205$	−133787.	−0.222347
$206$	0	0
$207$	42849.0	0.0695048
$208$	0	0
$209$	514003.	0.813954
$210$	0	0
$211$	−200532.	−0.310083	−0.155042	−	0.987908i	$-0.549551\pi$
$211$	−200532.	−0.310083	−0.155042	+	0.987908i	$0.549551\pi$
$212$	0	0
$213$	−472416.	−0.713469
$214$	0	0
$215$	−1.32903e6	−1.96083
$216$	0	0
$217$	−532355.	−0.767453
$218$	0	0
$219$	−152936.	−0.215476
$220$	0	0
$221$	−683387.	−0.941209
$222$	0	0
$223$	−227524.	−0.306383	−0.153192	−	0.988196i	$-0.548955\pi$
$223$	−227524.	−0.306383	−0.153192	+	0.988196i	$0.548955\pi$
$224$	0	0
$225$	355628.	0.468317
$226$	0	0
$227$	136415.	0.175711	0.0878553	−	0.996133i	$-0.471999\pi$
$227$	136415.	0.175711	0.0878553	+	0.996133i	$0.471999\pi$
$228$	0	0
$229$	−1.26300e6	−1.59153	−0.795763	−	0.605608i	$-0.792930\pi$
$229$	−1.26300e6	−1.59153	−0.795763	+	0.605608i	$0.792930\pi$
$230$	0	0
$231$	164271.	0.202549
$232$	0	0
$233$	449189.	0.542050	0.271025	−	0.962572i	$-0.412637\pi$
$233$	449189.	0.542050	0.271025	+	0.962572i	$0.412637\pi$
$234$	0	0
$235$	1.27656e6	1.50790
$236$	0	0
$237$	900211.	1.04105
$238$	0	0
$239$	926816.	1.04954	0.524770	−	0.851244i	$-0.324151\pi$
$239$	926816.	1.04954	0.524770	+	0.851244i	$0.324151\pi$
$240$	0	0
$241$	−1.28019e6	−1.41982	−0.709910	−	0.704292i	$-0.751264\pi$
$241$	−1.28019e6	−1.41982	−0.709910	+	0.704292i	$0.751264\pi$
$242$	0	0
$243$	59049.0	0.0641500
$244$	0	0
$245$	1.10188e6	1.17279
$246$	0	0
$247$	−554071.	−0.577861
$248$	0	0
$249$	767265.	0.784236
$250$	0	0
$251$	−694701.	−0.696007	−0.348003	−	0.937493i	$-0.613140\pi$
$251$	−694701.	−0.696007	−0.348003	+	0.937493i	$0.613140\pi$
$252$	0	0
$253$	150855.	0.148169
$254$	0	0
$255$	−1.73454e6	−1.67045
$256$	0	0
$257$	−994836.	−0.939547	−0.469774	−	0.882787i	$-0.655664\pi$
$257$	−994836.	−0.939547	−0.469774	+	0.882787i	$0.655664\pi$
$258$	0	0
$259$	−580847.	−0.538037
$260$	0	0
$261$	−191118.	−0.173661
$262$	0	0
$263$	−391688.	−0.349182	−0.174591	−	0.984641i	$-0.555860\pi$
$263$	−391688.	−0.349182	−0.174591	+	0.984641i	$0.555860\pi$
$264$	0	0
$265$	1.22781e6	1.07403
$266$	0	0
$267$	−747686.	−0.641861
$268$	0	0
$269$	1.20595e6	1.01613	0.508063	−	0.861320i	$-0.330362\pi$
$269$	1.20595e6	1.01613	0.508063	+	0.861320i	$0.330362\pi$
$270$	0	0
$271$	704840.	0.582999	0.291499	−	0.956571i	$-0.405846\pi$
$271$	704840.	0.582999	0.291499	+	0.956571i	$0.405846\pi$
$272$	0	0
$273$	−177077.	−0.143799
$274$	0	0
$275$	1.25203e6	0.998351
$276$	0	0
$277$	−1.05301e6	−0.824580	−0.412290	−	0.911053i	$-0.635271\pi$
$277$	−1.05301e6	−0.824580	−0.412290	+	0.911053i	$0.635271\pi$
$278$	0	0
$279$	−673708.	−0.518157
$280$	0	0
$281$	−2.32763e6	−1.75852	−0.879260	−	0.476342i	$-0.841962\pi$
$281$	−2.32763e6	−1.75852	−0.879260	+	0.476342i	$0.841962\pi$
$282$	0	0
$283$	2.47456e6	1.83667	0.918335	−	0.395804i	$-0.129534\pi$
$283$	2.47456e6	1.83667	0.918335	+	0.395804i	$0.129534\pi$
$284$	0	0
$285$	−1.40631e6	−1.02558
$286$	0	0
$287$	98776.1	0.0707860
$288$	0	0
$289$	3.52240e6	2.48081
$290$	0	0
$291$	286138.	0.198081
$292$	0	0
$293$	−961746.	−0.654472	−0.327236	−	0.944943i	$-0.606117\pi$
$293$	−961746.	−0.654472	−0.327236	+	0.944943i	$0.606117\pi$
$294$	0	0
$295$	728979.	0.487708
$296$	0	0
$297$	207889.	0.136754
$298$	0	0
$299$	−162615.	−0.105192
$300$	0	0
$301$	981232.	0.624245
$302$	0	0
$303$	496402.	0.310618
$304$	0	0
$305$	−2.26564e6	−1.39457
$306$	0	0
$307$	−837166.	−0.506951	−0.253475	−	0.967342i	$-0.581574\pi$
$307$	−837166.	−0.506951	−0.253475	+	0.967342i	$0.581574\pi$
$308$	0	0
$309$	1.12673e6	0.671309
$310$	0	0
$311$	2.45359e6	1.43847	0.719234	−	0.694768i	$-0.244493\pi$
$311$	2.45359e6	1.43847	0.719234	+	0.694768i	$0.244493\pi$
$312$	0	0
$313$	2.20389e6	1.27154	0.635770	−	0.771879i	$-0.280683\pi$
$313$	2.20389e6	1.27154	0.635770	+	0.771879i	$0.280683\pi$
$314$	0	0
$315$	−449446.	−0.255212
$316$	0	0
$317$	2.16373e6	1.20936	0.604678	−	0.796470i	$-0.293302\pi$
$317$	2.16373e6	1.20936	0.604678	+	0.796470i	$0.293302\pi$
$318$	0	0
$319$	−672855.	−0.370207
$320$	0	0
$321$	276463.	0.149753
$322$	0	0
$323$	4.00705e6	2.13707
$324$	0	0
$325$	−1.34963e6	−0.708773
$326$	0	0
$327$	−126594.	−0.0654704
$328$	0	0
$329$	−942495.	−0.480053
$330$	0	0
$331$	−907798.	−0.455427	−0.227714	−	0.973728i	$-0.573125\pi$
$331$	−907798.	−0.455427	−0.227714	+	0.973728i	$0.573125\pi$
$332$	0	0
$333$	−735075.	−0.363263
$334$	0	0
$335$	−1.12725e6	−0.548791
$336$	0	0
$337$	−2.29895e6	−1.10269	−0.551346	−	0.834277i	$-0.685886\pi$
$337$	−2.29895e6	−1.10269	−0.551346	+	0.834277i	$0.685886\pi$
$338$	0	0
$339$	272796.	0.128925
$340$	0	0
$341$	−2.37187e6	−1.10460
$342$	0	0
$343$	−1.88926e6	−0.867075
$344$	0	0
$345$	−412740.	−0.186693
$346$	0	0
$347$	3.67655e6	1.63914	0.819571	−	0.572977i	$-0.194212\pi$
$347$	3.67655e6	1.63914	0.819571	+	0.572977i	$0.194212\pi$
$348$	0	0
$349$	−4.10991e6	−1.80621	−0.903107	−	0.429416i	$-0.858719\pi$
$349$	−4.10991e6	−1.80621	−0.903107	+	0.429416i	$0.858719\pi$
$350$	0	0
$351$	−224095.	−0.0970876
$352$	0	0
$353$	1.13740e6	0.485819	0.242910	−	0.970049i	$-0.421898\pi$
$353$	1.13740e6	0.485819	0.242910	+	0.970049i	$0.421898\pi$
$354$	0	0
$355$	4.55051e6	1.91641
$356$	0	0
$357$	1.28062e6	0.531801
$358$	0	0
$359$	−1.64861e6	−0.675123	−0.337561	−	0.941304i	$-0.609602\pi$
$359$	−1.64861e6	−0.675123	−0.337561	+	0.941304i	$0.609602\pi$
$360$	0	0
$361$	772702.	0.312064
$362$	0	0
$363$	−717562.	−0.285820
$364$	0	0
$365$	1.47314e6	0.578778
$366$	0	0
$367$	−781323.	−0.302807	−0.151403	−	0.988472i	$-0.548379\pi$
$367$	−781323.	−0.302807	−0.151403	+	0.988472i	$0.548379\pi$
$368$	0	0
$369$	125004.	0.0477921
$370$	0	0
$371$	−906503.	−0.341928
$372$	0	0
$373$	1.51034e6	0.562086	0.281043	−	0.959695i	$-0.409320\pi$
$373$	1.51034e6	0.562086	0.281043	+	0.959695i	$0.409320\pi$
$374$	0	0
$375$	−987356.	−0.362573
$376$	0	0
$377$	725306.	0.262826
$378$	0	0
$379$	2.11472e6	0.756233	0.378117	−	0.925758i	$-0.376572\pi$
$379$	2.11472e6	0.756233	0.378117	+	0.925758i	$0.376572\pi$
$380$	0	0
$381$	1.29269e6	0.456229
$382$	0	0
$383$	1.57857e6	0.549878	0.274939	−	0.961462i	$-0.411342\pi$
$383$	1.57857e6	0.549878	0.274939	+	0.961462i	$0.411342\pi$
$384$	0	0
$385$	−1.58233e6	−0.544058
$386$	0	0
$387$	1.24177e6	0.421468
$388$	0	0
$389$	−579997.	−0.194335	−0.0971677	−	0.995268i	$-0.530978\pi$
$389$	−579997.	−0.194335	−0.0971677	+	0.995268i	$0.530978\pi$
$390$	0	0
$391$	1.17603e6	0.389024
$392$	0	0
$393$	2.42852e6	0.793158
$394$	0	0
$395$	−8.67121e6	−2.79632
$396$	0	0
$397$	−642885.	−0.204719	−0.102359	−	0.994747i	$-0.532639\pi$
$397$	−642885.	−0.204719	−0.102359	+	0.994747i	$0.532639\pi$
$398$	0	0
$399$	1.03829e6	0.326503
$400$	0	0
$401$	4.25371e6	1.32101	0.660506	−	0.750820i	$-0.270342\pi$
$401$	4.25371e6	1.32101	0.660506	+	0.750820i	$0.270342\pi$
$402$	0	0
$403$	2.55676e6	0.784202
$404$	0	0
$405$	−568785.	−0.172310
$406$	0	0
$407$	−2.58792e6	−0.774399
$408$	0	0
$409$	1.78257e6	0.526911	0.263456	−	0.964671i	$-0.415138\pi$
$409$	1.78257e6	0.526911	0.263456	+	0.964671i	$0.415138\pi$
$410$	0	0
$411$	2.75220e6	0.803665
$412$	0	0
$413$	−538210.	−0.155266
$414$	0	0
$415$	−7.39062e6	−2.10650
$416$	0	0
$417$	1.57911e6	0.444705
$418$	0	0
$419$	−1.47722e6	−0.411064	−0.205532	−	0.978650i	$-0.565893\pi$
$419$	−1.47722e6	−0.411064	−0.205532	+	0.978650i	$0.565893\pi$
$420$	0	0
$421$	2.38508e6	0.655840	0.327920	−	0.944705i	$-0.393652\pi$
$421$	2.38508e6	0.655840	0.327920	+	0.944705i	$0.393652\pi$
$422$	0	0
$423$	−1.19275e6	−0.324114
$424$	0	0
$425$	9.76054e6	2.62121
$426$	0	0
$427$	1.67274e6	0.443975
$428$	0	0
$429$	−788951.	−0.206970
$430$	0	0
$431$	7.13658e6	1.85053	0.925267	−	0.379316i	$-0.123841\pi$
$431$	7.13658e6	1.85053	0.925267	+	0.379316i	$0.123841\pi$
$432$	0	0
$433$	5.73488e6	1.46996	0.734979	−	0.678090i	$-0.237192\pi$
$433$	5.73488e6	1.46996	0.734979	+	0.678090i	$0.237192\pi$
$434$	0	0
$435$	1.84093e6	0.466461
$436$	0	0
$437$	953492.	0.238844
$438$	0	0
$439$	3.39829e6	0.841588	0.420794	−	0.907156i	$-0.361751\pi$
$439$	3.39829e6	0.841588	0.420794	+	0.907156i	$0.361751\pi$
$440$	0	0
$441$	−1.02954e6	−0.252084
$442$	0	0
$443$	4.85957e6	1.17649	0.588245	−	0.808683i	$-0.299819\pi$
$443$	4.85957e6	1.17649	0.588245	+	0.808683i	$0.299819\pi$
$444$	0	0
$445$	7.20203e6	1.72407
$446$	0	0
$447$	4.57154e6	1.08217
$448$	0	0
$449$	−4.26858e6	−0.999235	−0.499617	−	0.866246i	$-0.666526\pi$
$449$	−4.26858e6	−0.999235	−0.499617	+	0.866246i	$0.666526\pi$
$450$	0	0
$451$	440089.	0.101882
$452$	0	0
$453$	−1.93293e6	−0.442559
$454$	0	0
$455$	1.70568e6	0.386250
$456$	0	0
$457$	2.34262e6	0.524700	0.262350	−	0.964973i	$-0.415502\pi$
$457$	2.34262e6	0.524700	0.262350	+	0.964973i	$0.415502\pi$
$458$	0	0
$459$	1.62065e6	0.359053
$460$	0	0
$461$	−1.94617e6	−0.426508	−0.213254	−	0.976997i	$-0.568406\pi$
$461$	−1.94617e6	−0.426508	−0.213254	+	0.976997i	$0.568406\pi$
$462$	0	0
$463$	7.57161e6	1.64148	0.820740	−	0.571302i	$-0.193561\pi$
$463$	7.57161e6	1.64148	0.820740	+	0.571302i	$0.193561\pi$
$464$	0	0
$465$	6.48944e6	1.39179
$466$	0	0
$467$	744018.	0.157867	0.0789335	−	0.996880i	$-0.474849\pi$
$467$	744018.	0.157867	0.0789335	+	0.996880i	$0.474849\pi$
$468$	0	0
$469$	832254.	0.174712
$470$	0	0
$471$	1.62361e6	0.337232
$472$	0	0
$473$	4.37180e6	0.898479
$474$	0	0
$475$	7.91358e6	1.60931
$476$	0	0
$477$	−1.14720e6	−0.230857
$478$	0	0
$479$	7.51659e6	1.49686	0.748432	−	0.663212i	$-0.230807\pi$
$479$	7.51659e6	1.49686	0.748432	+	0.663212i	$0.230807\pi$
$480$	0	0
$481$	2.78966e6	0.549779
$482$	0	0
$483$	304728.	0.0594354
$484$	0	0
$485$	−2.75620e6	−0.532055
$486$	0	0
$487$	−1.74559e6	−0.333518	−0.166759	−	0.985998i	$-0.553330\pi$
$487$	−1.74559e6	−0.333518	−0.166759	+	0.985998i	$0.553330\pi$
$488$	0	0
$489$	−4.15686e6	−0.786128
$490$	0	0
$491$	8.33975e6	1.56117	0.780584	−	0.625051i	$-0.214922\pi$
$491$	8.33975e6	1.56117	0.780584	+	0.625051i	$0.214922\pi$
$492$	0	0
$493$	−5.24542e6	−0.971992
$494$	0	0
$495$	−2.00247e6	−0.367328
$496$	0	0
$497$	−3.35967e6	−0.610106
$498$	0	0
$499$	−2.85532e6	−0.513338	−0.256669	−	0.966499i	$-0.582625\pi$
$499$	−2.85532e6	−0.513338	−0.256669	+	0.966499i	$0.582625\pi$
$500$	0	0
$501$	−75560.6	−0.0134493
$502$	0	0
$503$	119267.	0.0210184	0.0105092	−	0.999945i	$-0.496655\pi$
$503$	119267.	0.0210184	0.0105092	+	0.999945i	$0.496655\pi$
$504$	0	0
$505$	−4.78156e6	−0.834336
$506$	0	0
$507$	−2.49118e6	−0.430413
$508$	0	0
$509$	2.88252e6	0.493149	0.246575	−	0.969124i	$-0.420695\pi$
$509$	2.88252e6	0.493149	0.246575	+	0.969124i	$0.420695\pi$
$510$	0	0
$511$	−1.08763e6	−0.184259
$512$	0	0
$513$	1.31398e6	0.220443
$514$	0	0
$515$	−1.08531e7	−1.80317
$516$	0	0
$517$	−4.19921e6	−0.690942
$518$	0	0
$519$	6.08295e6	0.991279
$520$	0	0
$521$	−5.63013e6	−0.908708	−0.454354	−	0.890821i	$-0.650130\pi$
$521$	−5.63013e6	−0.908708	−0.454354	+	0.890821i	$0.650130\pi$
$522$	0	0
$523$	−1.95752e6	−0.312933	−0.156467	−	0.987683i	$-0.550010\pi$
$523$	−1.95752e6	−0.312933	−0.156467	+	0.987683i	$0.550010\pi$
$524$	0	0
$525$	2.52912e6	0.400470
$526$	0	0
$527$	−1.84905e7	−2.90017
$528$	0	0
$529$	279841.	0.0434783
$530$	0	0
$531$	−681117.	−0.104830
$532$	0	0
$533$	−474396.	−0.0723308
$534$	0	0
$535$	−2.66301e6	−0.402243
$536$	0	0
$537$	−2.31081e6	−0.345802
$538$	0	0
$539$	−3.62461e6	−0.537389
$540$	0	0
$541$	4.40593e6	0.647208	0.323604	−	0.946193i	$-0.395105\pi$
$541$	4.40593e6	0.647208	0.323604	+	0.946193i	$0.395105\pi$
$542$	0	0
$543$	5.85633e6	0.852365
$544$	0	0
$545$	1.21941e6	0.175857
$546$	0	0
$547$	−5.08750e6	−0.727003	−0.363502	−	0.931594i	$-0.618419\pi$
$547$	−5.08750e6	−0.727003	−0.363502	+	0.931594i	$0.618419\pi$
$548$	0	0
$549$	2.11689e6	0.299756
$550$	0	0
$551$	−4.25284e6	−0.596760
$552$	0	0
$553$	6.40201e6	0.890233
$554$	0	0
$555$	7.08056e6	0.975742
$556$	0	0
$557$	6.32135e6	0.863320	0.431660	−	0.902036i	$-0.357928\pi$
$557$	6.32135e6	0.863320	0.431660	+	0.902036i	$0.357928\pi$
$558$	0	0
$559$	−4.71260e6	−0.637869
$560$	0	0
$561$	5.70570e6	0.765423
$562$	0	0
$563$	198801.	0.0264331	0.0132165	−	0.999913i	$-0.495793\pi$
$563$	198801.	0.0264331	0.0132165	+	0.999913i	$0.495793\pi$
$564$	0	0
$565$	−2.62769e6	−0.346300
$566$	0	0
$567$	419938.	0.0548564
$568$	0	0
$569$	−1.26952e7	−1.64384	−0.821918	−	0.569605i	$-0.807096\pi$
$569$	−1.26952e7	−1.64384	−0.821918	+	0.569605i	$0.807096\pi$
$570$	0	0
$571$	371684.	0.0477071	0.0238536	−	0.999715i	$-0.492406\pi$
$571$	371684.	0.0477071	0.0238536	+	0.999715i	$0.492406\pi$
$572$	0	0
$573$	7.86670e6	1.00093
$574$	0	0
$575$	2.32256e6	0.292953
$576$	0	0
$577$	−8.62719e6	−1.07877	−0.539386	−	0.842058i	$-0.681344\pi$
$577$	−8.62719e6	−1.07877	−0.539386	+	0.842058i	$0.681344\pi$
$578$	0	0
$579$	−6.30301e6	−0.781361
$580$	0	0
$581$	5.45654e6	0.670621
$582$	0	0
$583$	−4.03886e6	−0.492138
$584$	0	0
$585$	2.15858e6	0.260782
$586$	0	0
$587$	−8.79998e6	−1.05411	−0.527056	−	0.849831i	$-0.676704\pi$
$587$	−8.79998e6	−1.05411	−0.527056	+	0.849831i	$0.676704\pi$
$588$	0	0
$589$	−1.49916e7	−1.78057
$590$	0	0
$591$	9.10731e6	1.07256
$592$	0	0
$593$	6.28515e6	0.733971	0.366985	−	0.930227i	$-0.380390\pi$
$593$	6.28515e6	0.733971	0.366985	+	0.930227i	$0.380390\pi$
$594$	0	0
$595$	−1.23355e7	−1.42844
$596$	0	0
$597$	−6.76448e6	−0.776780
$598$	0	0
$599$	6.18076e6	0.703842	0.351921	−	0.936030i	$-0.385529\pi$
$599$	6.18076e6	0.703842	0.351921	+	0.936030i	$0.385529\pi$
$600$	0	0
$601$	196063.	0.0221417	0.0110708	−	0.999939i	$-0.496476\pi$
$601$	196063.	0.0221417	0.0110708	+	0.999939i	$0.496476\pi$
$602$	0	0
$603$	1.05324e6	0.117959
$604$	0	0
$605$	6.91186e6	0.767727
$606$	0	0
$607$	−2.96914e6	−0.327084	−0.163542	−	0.986536i	$-0.552292\pi$
$607$	−2.96914e6	−0.327084	−0.163542	+	0.986536i	$0.552292\pi$
$608$	0	0
$609$	−1.35917e6	−0.148502
$610$	0	0
$611$	4.52656e6	0.490530
$612$	0	0
$613$	−4.34673e6	−0.467209	−0.233605	−	0.972332i	$-0.575052\pi$
$613$	−4.34673e6	−0.467209	−0.233605	+	0.972332i	$0.575052\pi$
$614$	0	0
$615$	−1.20409e6	−0.128372
$616$	0	0
$617$	1.11692e7	1.18116	0.590580	−	0.806979i	$-0.298899\pi$
$617$	1.11692e7	1.18116	0.590580	+	0.806979i	$0.298899\pi$
$618$	0	0
$619$	−2.31301e6	−0.242633	−0.121317	−	0.992614i	$-0.538712\pi$
$619$	−2.31301e6	−0.242633	−0.121317	+	0.992614i	$0.538712\pi$
$620$	0	0
$621$	385641.	0.0401286
$622$	0	0
$623$	−5.31730e6	−0.548872
$624$	0	0
$625$	−4.20960e6	−0.431063
$626$	0	0
$627$	4.62602e6	0.469936
$628$	0	0
$629$	−2.01748e7	−2.03321
$630$	0	0
$631$	1.11795e7	1.11776	0.558881	−	0.829248i	$-0.311231\pi$
$631$	1.11795e7	1.11776	0.558881	+	0.829248i	$0.311231\pi$
$632$	0	0
$633$	−1.80479e6	−0.179027
$634$	0	0
$635$	−1.24518e7	−1.22545
$636$	0	0
$637$	3.90716e6	0.381516
$638$	0	0
$639$	−4.25174e6	−0.411922
$640$	0	0
$641$	724583.	0.0696536	0.0348268	−	0.999393i	$-0.488912\pi$
$641$	724583.	0.0696536	0.0348268	+	0.999393i	$0.488912\pi$
$642$	0	0
$643$	−9.65287e6	−0.920723	−0.460361	−	0.887732i	$-0.652280\pi$
$643$	−9.65287e6	−0.920723	−0.460361	+	0.887732i	$0.652280\pi$
$644$	0	0
$645$	−1.19613e7	−1.13208
$646$	0	0
$647$	1.20642e7	1.13302	0.566512	−	0.824053i	$-0.308292\pi$
$647$	1.20642e7	1.13302	0.566512	+	0.824053i	$0.308292\pi$
$648$	0	0
$649$	−2.39795e6	−0.223475
$650$	0	0
$651$	−4.79120e6	−0.443089
$652$	0	0
$653$	−1.08359e7	−0.994452	−0.497226	−	0.867621i	$-0.665648\pi$
$653$	−1.08359e7	−0.994452	−0.497226	+	0.867621i	$0.665648\pi$
$654$	0	0
$655$	−2.33925e7	−2.13046
$656$	0	0
$657$	−1.37642e6	−0.124405
$658$	0	0
$659$	8.32711e6	0.746931	0.373466	−	0.927644i	$-0.378169\pi$
$659$	8.32711e6	0.746931	0.373466	+	0.927644i	$0.378169\pi$
$660$	0	0
$661$	1.42602e7	1.26946	0.634732	−	0.772732i	$-0.281110\pi$
$661$	1.42602e7	1.26946	0.634732	+	0.772732i	$0.281110\pi$
$662$	0	0
$663$	−6.15048e6	−0.543407
$664$	0	0
$665$	−1.00013e7	−0.877002
$666$	0	0
$667$	−1.24817e6	−0.108632
$668$	0	0
$669$	−2.04772e6	−0.176891
$670$	0	0
$671$	7.45276e6	0.639014
$672$	0	0
$673$	1.19925e7	1.02064	0.510319	−	0.859985i	$-0.329527\pi$
$673$	1.19925e7	1.02064	0.510319	+	0.859985i	$0.329527\pi$
$674$	0	0
$675$	3.20066e6	0.270383
$676$	0	0
$677$	5.47203e6	0.458857	0.229428	−	0.973326i	$-0.426314\pi$
$677$	5.47203e6	0.458857	0.229428	+	0.973326i	$0.426314\pi$
$678$	0	0
$679$	2.03492e6	0.169384
$680$	0	0
$681$	1.22774e6	0.101447
$682$	0	0
$683$	−1.57601e7	−1.29273	−0.646364	−	0.763029i	$-0.723711\pi$
$683$	−1.57601e7	−1.29273	−0.646364	+	0.763029i	$0.723711\pi$
$684$	0	0
$685$	−2.65104e7	−2.15868
$686$	0	0
$687$	−1.13670e7	−0.918868
$688$	0	0
$689$	4.35370e6	0.349390
$690$	0	0
$691$	−1.53486e7	−1.22285	−0.611424	−	0.791303i	$-0.709403\pi$
$691$	−1.53486e7	−1.22285	−0.611424	+	0.791303i	$0.709403\pi$
$692$	0	0
$693$	1.47844e6	0.116942
$694$	0	0
$695$	−1.52106e7	−1.19450
$696$	0	0
$697$	3.43084e6	0.267496
$698$	0	0
$699$	4.04270e6	0.312953
$700$	0	0
$701$	1.48889e7	1.14437	0.572187	−	0.820123i	$-0.306095\pi$
$701$	1.48889e7	1.14437	0.572187	+	0.820123i	$0.306095\pi$
$702$	0	0
$703$	−1.63572e7	−1.24830
$704$	0	0
$705$	1.14891e7	0.870587
$706$	0	0
$707$	3.53025e6	0.265618
$708$	0	0
$709$	2.04675e6	0.152915	0.0764573	−	0.997073i	$-0.475639\pi$
$709$	2.04675e6	0.152915	0.0764573	+	0.997073i	$0.475639\pi$
$710$	0	0
$711$	8.10190e6	0.601053
$712$	0	0
$713$	−4.39990e6	−0.324129
$714$	0	0
$715$	7.59952e6	0.555931
$716$	0	0
$717$	8.34134e6	0.605952
$718$	0	0
$719$	1.09836e7	0.792357	0.396179	−	0.918173i	$-0.370336\pi$
$719$	1.09836e7	0.792357	0.396179	+	0.918173i	$0.370336\pi$
$720$	0	0
$721$	8.01292e6	0.574054
$722$	0	0
$723$	−1.15218e7	−0.819734
$724$	0	0
$725$	−1.03593e7	−0.731954
$726$	0	0
$727$	6.92842e6	0.486181	0.243091	−	0.970004i	$-0.421839\pi$
$727$	6.92842e6	0.486181	0.243091	+	0.970004i	$0.421839\pi$
$728$	0	0
$729$	531441.	0.0370370
$730$	0	0
$731$	3.40816e7	2.35899
$732$	0	0
$733$	3.44517e6	0.236838	0.118419	−	0.992964i	$-0.462217\pi$
$733$	3.44517e6	0.236838	0.118419	+	0.992964i	$0.462217\pi$
$734$	0	0
$735$	9.91695e6	0.677111
$736$	0	0
$737$	3.70804e6	0.251464
$738$	0	0
$739$	−3.85428e6	−0.259617	−0.129808	−	0.991539i	$-0.541436\pi$
$739$	−3.85428e6	−0.259617	−0.129808	+	0.991539i	$0.541436\pi$
$740$	0	0
$741$	−4.98664e6	−0.333628
$742$	0	0
$743$	−7.82591e6	−0.520071	−0.260035	−	0.965599i	$-0.583734\pi$
$743$	−7.82591e6	−0.520071	−0.260035	+	0.965599i	$0.583734\pi$
$744$	0	0
$745$	−4.40350e7	−2.90675
$746$	0	0
$747$	6.90539e6	0.452779
$748$	0	0
$749$	1.96612e6	0.128057
$750$	0	0
$751$	8.96338e6	0.579925	0.289963	−	0.957038i	$-0.406357\pi$
$751$	8.96338e6	0.579925	0.289963	+	0.957038i	$0.406357\pi$
$752$	0	0
$753$	−6.25231e6	−0.401840
$754$	0	0
$755$	1.86188e7	1.18873
$756$	0	0
$757$	−1.29062e7	−0.818574	−0.409287	−	0.912406i	$-0.634223\pi$
$757$	−1.29062e7	−0.818574	−0.409287	+	0.912406i	$0.634223\pi$
$758$	0	0
$759$	1.35769e6	0.0855456
$760$	0	0
$761$	−1.66351e7	−1.04127	−0.520636	−	0.853779i	$-0.674305\pi$
$761$	−1.66351e7	−1.04127	−0.520636	+	0.853779i	$0.674305\pi$
$762$	0	0
$763$	−900299.	−0.0559855
$764$	0	0
$765$	−1.56108e7	−0.964433
$766$	0	0
$767$	2.58488e6	0.158655
$768$	0	0
$769$	7.10800e6	0.433443	0.216721	−	0.976233i	$-0.430464\pi$
$769$	7.10800e6	0.433443	0.216721	+	0.976233i	$0.430464\pi$
$770$	0	0
$771$	−8.95352e6	−0.542448
$772$	0	0
$773$	1.27704e7	0.768700	0.384350	−	0.923187i	$-0.374426\pi$
$773$	1.27704e7	0.768700	0.384350	+	0.923187i	$0.374426\pi$
$774$	0	0
$775$	−3.65172e7	−2.18396
$776$	0	0
$777$	−5.22762e6	−0.310636
$778$	0	0
$779$	2.78163e6	0.164231
$780$	0	0
$781$	−1.49687e7	−0.878128
$782$	0	0
$783$	−1.72007e6	−0.100263
$784$	0	0
$785$	−1.56393e7	−0.905821
$786$	0	0
$787$	−1.38003e7	−0.794240	−0.397120	−	0.917767i	$-0.629990\pi$
$787$	−1.38003e7	−0.794240	−0.397120	+	0.917767i	$0.629990\pi$
$788$	0	0
$789$	−3.52520e6	−0.201600
$790$	0	0
$791$	1.94004e6	0.110248
$792$	0	0
$793$	−8.03373e6	−0.453664
$794$	0	0
$795$	1.10503e7	0.620094
$796$	0	0
$797$	2.14559e7	1.19647	0.598233	−	0.801322i	$-0.295870\pi$
$797$	2.14559e7	1.19647	0.598233	+	0.801322i	$0.295870\pi$
$798$	0	0
$799$	−3.27361e7	−1.81409
$800$	0	0
$801$	−6.72917e6	−0.370579
$802$	0	0
$803$	−4.84585e6	−0.265205
$804$	0	0
$805$	−2.93527e6	−0.159646
$806$	0	0
$807$	1.08535e7	0.586661
$808$	0	0
$809$	−1.09665e7	−0.589113	−0.294556	−	0.955634i	$-0.595172\pi$
$809$	−1.09665e7	−0.589113	−0.294556	+	0.955634i	$0.595172\pi$
$810$	0	0
$811$	2.58794e7	1.38166	0.690832	−	0.723015i	$-0.257244\pi$
$811$	2.58794e7	1.38166	0.690832	+	0.723015i	$0.257244\pi$
$812$	0	0
$813$	6.34356e6	0.336594
$814$	0	0
$815$	4.00407e7	2.11158
$816$	0	0
$817$	2.76324e7	1.44832
$818$	0	0
$819$	−1.59369e6	−0.0830222
$820$	0	0
$821$	−2.30093e7	−1.19136	−0.595682	−	0.803220i	$-0.703118\pi$
$821$	−2.30093e7	−1.19136	−0.595682	+	0.803220i	$0.703118\pi$
$822$	0	0
$823$	7.51381e6	0.386688	0.193344	−	0.981131i	$-0.438067\pi$
$823$	7.51381e6	0.386688	0.193344	+	0.981131i	$0.438067\pi$
$824$	0	0
$825$	1.12683e7	0.576398
$826$	0	0
$827$	1.08185e7	0.550051	0.275025	−	0.961437i	$-0.411314\pi$
$827$	1.08185e7	0.550051	0.275025	+	0.961437i	$0.411314\pi$
$828$	0	0
$829$	2.11950e6	0.107114	0.0535572	−	0.998565i	$-0.482944\pi$
$829$	2.11950e6	0.107114	0.0535572	+	0.998565i	$0.482944\pi$
$830$	0	0
$831$	−9.47709e6	−0.476072
$832$	0	0
$833$	−2.82566e7	−1.41094
$834$	0	0
$835$	727832.	0.0361256
$836$	0	0
$837$	−6.06337e6	−0.299158
$838$	0	0
$839$	9.53545e6	0.467667	0.233833	−	0.972277i	$-0.424873\pi$
$839$	9.53545e6	0.467667	0.233833	+	0.972277i	$0.424873\pi$
$840$	0	0
$841$	−1.49440e7	−0.728578
$842$	0	0
$843$	−2.09486e7	−1.01528
$844$	0	0
$845$	2.39961e7	1.15611
$846$	0	0
$847$	−5.10307e6	−0.244412
$848$	0	0
$849$	2.22710e7	1.06040
$850$	0	0
$851$	−4.80068e6	−0.227237
$852$	0	0
$853$	2.54943e7	1.19969	0.599846	−	0.800115i	$-0.295228\pi$
$853$	2.54943e7	1.19969	0.599846	+	0.800115i	$0.295228\pi$
$854$	0	0
$855$	−1.26568e7	−0.592120
$856$	0	0
$857$	8.02146e6	0.373079	0.186540	−	0.982447i	$-0.440273\pi$
$857$	8.02146e6	0.373079	0.186540	+	0.982447i	$0.440273\pi$
$858$	0	0
$859$	4.80807e6	0.222325	0.111162	−	0.993802i	$-0.464543\pi$
$859$	4.80807e6	0.222325	0.111162	+	0.993802i	$0.464543\pi$
$860$	0	0
$861$	888985.	0.0408683
$862$	0	0
$863$	−1.98941e7	−0.909280	−0.454640	−	0.890675i	$-0.650232\pi$
$863$	−1.98941e7	−0.909280	−0.454640	+	0.890675i	$0.650232\pi$
$864$	0	0
$865$	−5.85936e7	−2.66262
$866$	0	0
$867$	3.17016e7	1.43230
$868$	0	0
$869$	2.85237e7	1.28132
$870$	0	0
$871$	−3.99710e6	−0.178525
$872$	0	0
$873$	2.57524e6	0.114362
$874$	0	0
$875$	−7.02176e6	−0.310046
$876$	0	0
$877$	−1.69776e7	−0.745381	−0.372691	−	0.927956i	$-0.621565\pi$
$877$	−1.69776e7	−0.745381	−0.372691	+	0.927956i	$0.621565\pi$
$878$	0	0
$879$	−8.65571e6	−0.377860
$880$	0	0
$881$	1.94798e7	0.845562	0.422781	−	0.906232i	$-0.361054\pi$
$881$	1.94798e7	0.845562	0.422781	+	0.906232i	$0.361054\pi$
$882$	0	0
$883$	4.20175e7	1.81354	0.906772	−	0.421621i	$-0.138539\pi$
$883$	4.20175e7	1.81354	0.906772	+	0.421621i	$0.138539\pi$
$884$	0	0
$885$	6.56081e6	0.281579
$886$	0	0
$887$	−3.23258e7	−1.37956	−0.689779	−	0.724020i	$-0.742292\pi$
$887$	−3.23258e7	−1.37956	−0.689779	+	0.724020i	$0.742292\pi$
$888$	0	0
$889$	9.19321e6	0.390133
$890$	0	0
$891$	1.87100e6	0.0789550
$892$	0	0
$893$	−2.65415e7	−1.11377
$894$	0	0
$895$	2.22587e7	0.928842
$896$	0	0
$897$	−1.46353e6	−0.0607325
$898$	0	0
$899$	1.96247e7	0.809850
$900$	0	0
$901$	−3.14860e7	−1.29213
$902$	0	0
$903$	8.83109e6	0.360408
$904$	0	0
$905$	−5.64106e7	−2.28949
$906$	0	0
$907$	−1.37422e7	−0.554676	−0.277338	−	0.960772i	$-0.589452\pi$
$907$	−1.37422e7	−0.554676	−0.277338	+	0.960772i	$0.589452\pi$
$908$	0	0
$909$	4.46762e6	0.179336
$910$	0	0
$911$	1.43255e7	0.571891	0.285945	−	0.958246i	$-0.407692\pi$
$911$	1.43255e7	0.571891	0.285945	+	0.958246i	$0.407692\pi$
$912$	0	0
$913$	2.43112e7	0.965228
$914$	0	0
$915$	−2.03908e7	−0.805158
$916$	0	0
$917$	1.72708e7	0.678250
$918$	0	0
$919$	−4.09852e7	−1.60080	−0.800402	−	0.599463i	$-0.795381\pi$
$919$	−4.09852e7	−1.60080	−0.800402	+	0.599463i	$0.795381\pi$
$920$	0	0
$921$	−7.53450e6	−0.292688
$922$	0	0
$923$	1.61356e7	0.623421
$924$	0	0
$925$	−3.98436e7	−1.53110
$926$	0	0
$927$	1.01405e7	0.387581
$928$	0	0
$929$	2.88621e7	1.09721	0.548603	−	0.836083i	$-0.315160\pi$
$929$	2.88621e7	1.09721	0.548603	+	0.836083i	$0.315160\pi$
$930$	0	0
$931$	−2.29097e7	−0.866253
$932$	0	0
$933$	2.20823e7	0.830500
$934$	0	0
$935$	−5.49597e7	−2.05596
$936$	0	0
$937$	−3.51671e7	−1.30854	−0.654272	−	0.756260i	$-0.727025\pi$
$937$	−3.51671e7	−1.30854	−0.654272	+	0.756260i	$0.727025\pi$
$938$	0	0
$939$	1.98350e7	0.734124
$940$	0	0
$941$	4.23288e7	1.55834	0.779170	−	0.626813i	$-0.215641\pi$
$941$	4.23288e7	1.55834	0.779170	+	0.626813i	$0.215641\pi$
$942$	0	0
$943$	816381.	0.0298960
$944$	0	0
$945$	−4.04502e6	−0.147347
$946$	0	0
$947$	2.17631e7	0.788579	0.394290	−	0.918986i	$-0.370991\pi$
$947$	2.17631e7	0.788579	0.394290	+	0.918986i	$0.370991\pi$
$948$	0	0
$949$	5.22360e6	0.188280
$950$	0	0
$951$	1.94735e7	0.698222
$952$	0	0
$953$	−2.52072e7	−0.899067	−0.449533	−	0.893264i	$-0.648410\pi$
$953$	−2.52072e7	−0.899067	−0.449533	+	0.893264i	$0.648410\pi$
$954$	0	0
$955$	−7.57754e7	−2.68856
$956$	0	0
$957$	−6.05569e6	−0.213739
$958$	0	0
$959$	1.95728e7	0.687235
$960$	0	0
$961$	4.05497e7	1.41638
$962$	0	0
$963$	2.48817e6	0.0864597
$964$	0	0
$965$	6.07133e7	2.09877
$966$	0	0
$967$	−5.15354e6	−0.177231	−0.0886154	−	0.996066i	$-0.528244\pi$
$967$	−5.15354e6	−0.177231	−0.0886154	+	0.996066i	$0.528244\pi$
$968$	0	0
$969$	3.60634e7	1.23384
$970$	0	0
$971$	−4.87285e7	−1.65858	−0.829288	−	0.558822i	$-0.811254\pi$
$971$	−4.87285e7	−1.65858	−0.829288	+	0.558822i	$0.811254\pi$
$972$	0	0
$973$	1.12301e7	0.380279
$974$	0	0
$975$	−1.21467e7	−0.409210
$976$	0	0
$977$	−2.08248e7	−0.697981	−0.348991	−	0.937126i	$-0.613476\pi$
$977$	−2.08248e7	−0.697981	−0.348991	+	0.937126i	$0.613476\pi$
$978$	0	0
$979$	−2.36908e7	−0.789994
$980$	0	0
$981$	−1.13935e6	−0.0377994
$982$	0	0
$983$	−1.68825e7	−0.557255	−0.278627	−	0.960399i	$-0.589879\pi$
$983$	−1.68825e7	−0.557255	−0.278627	+	0.960399i	$0.589879\pi$
$984$	0	0
$985$	−8.77255e7	−2.88095
$986$	0	0
$987$	−8.48245e6	−0.277159
$988$	0	0
$989$	8.10984e6	0.263646
$990$	0	0
$991$	1.28291e6	0.0414965	0.0207482	−	0.999785i	$-0.493395\pi$
$991$	1.28291e6	0.0414965	0.0207482	+	0.999785i	$0.493395\pi$
$992$	0	0
$993$	−8.17018e6	−0.262941
$994$	0	0
$995$	6.51583e7	2.08647
$996$	0	0
$997$	1.88209e7	0.599658	0.299829	−	0.953993i	$-0.403070\pi$
$997$	1.88209e7	0.599658	0.299829	+	0.953993i	$0.403070\pi$
$998$	0	0
$999$	−6.61568e6	−0.209730

Twists

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

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

\(f(q)\)	\(=\)	\(q+9.00000 q^{3} -86.6918 q^{5} +64.0051 q^{7} +81.0000 q^{9} +O(q^{10})\)	\(q+9.00000 q^{3} -86.6918 q^{5} +64.0051 q^{7} +81.0000 q^{9} +285.170 q^{11} -307.400 q^{13} -780.226 q^{15} +2223.12 q^{17} +1802.44 q^{19} +576.046 q^{21} +529.000 q^{23} +4390.47 q^{25} +729.000 q^{27} -2359.49 q^{29} -8317.38 q^{31} +2566.53 q^{33} -5548.72 q^{35} -9075.00 q^{37} -2766.60 q^{39} +1543.25 q^{41} +15330.5 q^{43} -7022.04 q^{45} -14725.3 q^{47} -12710.3 q^{49} +20008.1 q^{51} -14163.0 q^{53} -24721.9 q^{55} +16222.0 q^{57} -8408.86 q^{59} +26134.4 q^{61} +5184.41 q^{63} +26649.1 q^{65} +13002.9 q^{67} +4761.00 q^{69} -52490.6 q^{71} -16992.9 q^{73} +39514.3 q^{75} +18252.3 q^{77} +100023. q^{79} +6561.00 q^{81} +85251.7 q^{83} -192726. q^{85} -21235.4 q^{87} -83076.2 q^{89} -19675.2 q^{91} -74856.4 q^{93} -156257. q^{95} +31793.1 q^{97} +23098.8 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 4 q + 36 q^{3} + 22 q^{5} + 62 q^{7} + 324 q^{9}+O(q^{10}) \) 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

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