LMFDB - Embedded newform 1104.6.a.r.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:	$1$
Dimension:	$5$
Coefficient field:	$\mathbb{Q}[x]/(x^{5} - \cdots)$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{5} - 113x^{3} - 257x^{2} + 1404x + 2197 $ x^5 - 113x^3 - 257x^2 + 1404*x + 2197
Coefficient ring:	$\Z[a_1, \ldots, a_{11}]$
Coefficient ring index:	$ 2^{9} $
Twist minimal:	no (minimal twist has level 69)
Fricke sign:	$1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.1
Root			$11.0973$ 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)

$f(q)$	$=$	$q-9.00000 q^{3} -92.1306 q^{5} +8.98894 q^{7} +81.0000 q^{9} +O(q^{10})$	$q-9.00000 q^{3} -92.1306 q^{5} +8.98894 q^{7} +81.0000 q^{9} -612.746 q^{11} -496.413 q^{13} +829.175 q^{15} +745.481 q^{17} -1299.52 q^{19} -80.9005 q^{21} +529.000 q^{23} +5363.05 q^{25} -729.000 q^{27} +7754.82 q^{29} +1066.62 q^{31} +5514.72 q^{33} -828.157 q^{35} -840.279 q^{37} +4467.72 q^{39} +9821.00 q^{41} +13529.8 q^{43} -7462.58 q^{45} -20485.1 q^{47} -16726.2 q^{49} -6709.33 q^{51} +4816.85 q^{53} +56452.7 q^{55} +11695.7 q^{57} -30528.3 q^{59} +9.53050 q^{61} +728.104 q^{63} +45734.8 q^{65} -9996.72 q^{67} -4761.00 q^{69} +73725.9 q^{71} +55169.6 q^{73} -48267.4 q^{75} -5507.94 q^{77} -62397.4 q^{79} +6561.00 q^{81} +82435.2 q^{83} -68681.6 q^{85} -69793.4 q^{87} +95173.0 q^{89} -4462.23 q^{91} -9599.57 q^{93} +119726. q^{95} +47134.4 q^{97} -49632.4 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 5 q - 45 q^{3} + 94 q^{5} - 272 q^{7} + 405 q^{9}+O(q^{10}) $ 5 * q - 45 * q^3 + 94 * q^5 - 272 * q^7 + 405 * q^9	$ 5 q - 45 q^{3} + 94 q^{5} - 272 q^{7} + 405 q^{9} - 1100 q^{11} - 978 q^{13} - 846 q^{15} + 2522 q^{17} - 2060 q^{19} + 2448 q^{21} + 2645 q^{23} + 12035 q^{25} - 3645 q^{27} + 1526 q^{29} + 7392 q^{31} + 9900 q^{33} - 6056 q^{35} - 8210 q^{37} + 8802 q^{39} + 21250 q^{41} + 4548 q^{43} + 7614 q^{45} - 536 q^{47} - 27979 q^{49} - 22698 q^{51} - 11482 q^{53} + 77064 q^{55} + 18540 q^{57} - 74676 q^{59} - 44618 q^{61} - 22032 q^{63} - 24388 q^{65} + 1412 q^{67} - 23805 q^{69} - 37912 q^{71} + 46546 q^{73} - 108315 q^{75} + 157008 q^{77} - 50544 q^{79} + 32805 q^{81} - 89588 q^{83} + 147892 q^{85} - 13734 q^{87} + 280410 q^{89} + 27416 q^{91} - 66528 q^{93} - 203120 q^{95} + 90074 q^{97} - 89100 q^{99}+O(q^{100}) $ 5 * q - 45 * q^3 + 94 * q^5 - 272 * q^7 + 405 * q^9 - 1100 * q^11 - 978 * q^13 - 846 * q^15 + 2522 * q^17 - 2060 * q^19 + 2448 * q^21 + 2645 * q^23 + 12035 * q^25 - 3645 * q^27 + 1526 * q^29 + 7392 * q^31 + 9900 * q^33 - 6056 * q^35 - 8210 * q^37 + 8802 * q^39 + 21250 * q^41 + 4548 * q^43 + 7614 * q^45 - 536 * q^47 - 27979 * q^49 - 22698 * q^51 - 11482 * q^53 + 77064 * q^55 + 18540 * q^57 - 74676 * q^59 - 44618 * q^61 - 22032 * q^63 - 24388 * q^65 + 1412 * q^67 - 23805 * q^69 - 37912 * q^71 + 46546 * q^73 - 108315 * q^75 + 157008 * q^77 - 50544 * q^79 + 32805 * q^81 - 89588 * q^83 + 147892 * q^85 - 13734 * q^87 + 280410 * q^89 + 27416 * q^91 - 66528 * q^93 - 203120 * q^95 + 90074 * q^97 - 89100 * 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
$5$	−92.1306	−1.64808	−0.824041	−	0.566530i	$-0.808286\pi$
$5$	−92.1306	−1.64808	−0.824041	+	0.566530i	$0.808286\pi$
$6$	0	0
$7$	8.98894	0.0693368	0.0346684	−	0.999399i	$-0.488962\pi$
$7$	8.98894	0.0693368	0.0346684	+	0.999399i	$0.488962\pi$
$8$	0	0
$9$	81.0000	0.333333
$10$	0	0
$11$	−612.746	−1.52686	−0.763430	−	0.645891i	$-0.776486\pi$
$11$	−612.746	−1.52686	−0.763430	+	0.645891i	$0.776486\pi$
$12$	0	0
$13$	−496.413	−0.814676	−0.407338	−	0.913278i	$-0.633543\pi$
$13$	−496.413	−0.814676	−0.407338	+	0.913278i	$0.633543\pi$
$14$	0	0
$15$	829.175	0.951521
$16$	0	0
$17$	745.481	0.625625	0.312813	−	0.949815i	$-0.398729\pi$
$17$	745.481	0.625625	0.312813	+	0.949815i	$0.398729\pi$
$18$	0	0
$19$	−1299.52	−0.825847	−0.412924	−	0.910766i	$-0.635492\pi$
$19$	−1299.52	−0.825847	−0.412924	+	0.910766i	$0.635492\pi$
$20$	0	0
$21$	−80.9005	−0.0400316
$22$	0	0
$23$	529.000	0.208514
$23$	529.000	0.208514
$24$	0	0
$25$	5363.05	1.71618
$26$	0	0
$27$	−729.000	−0.192450
$28$	0	0
$29$	7754.82	1.71229	0.856144	−	0.516738i	$-0.172854\pi$
$29$	7754.82	1.71229	0.856144	+	0.516738i	$0.172854\pi$
$30$	0	0
$31$	1066.62	0.199345	0.0996724	−	0.995020i	$-0.468221\pi$
$31$	1066.62	0.199345	0.0996724	+	0.995020i	$0.468221\pi$
$32$	0	0
$33$	5514.72	0.881533
$34$	0	0
$35$	−828.157	−0.114273
$36$	0	0
$37$	−840.279	−0.100907	−0.0504533	−	0.998726i	$-0.516067\pi$
$37$	−840.279	−0.100907	−0.0504533	+	0.998726i	$0.516067\pi$
$38$	0	0
$39$	4467.72	0.470353
$40$	0	0
$41$	9821.00	0.912422	0.456211	−	0.889872i	$-0.349206\pi$
$41$	9821.00	0.912422	0.456211	+	0.889872i	$0.349206\pi$
$42$	0	0
$43$	13529.8	1.11589	0.557944	−	0.829879i	$-0.311590\pi$
$43$	13529.8	1.11589	0.557944	+	0.829879i	$0.311590\pi$
$44$	0	0
$45$	−7462.58	−0.549361
$46$	0	0
$47$	−20485.1	−1.35268	−0.676338	−	0.736591i	$-0.736434\pi$
$47$	−20485.1	−1.35268	−0.676338	+	0.736591i	$0.736434\pi$
$48$	0	0
$49$	−16726.2	−0.995192
$50$	0	0
$51$	−6709.33	−0.361205
$52$	0	0
$53$	4816.85	0.235545	0.117772	−	0.993041i	$-0.462425\pi$
$53$	4816.85	0.235545	0.117772	+	0.993041i	$0.462425\pi$
$54$	0	0
$55$	56452.7	2.51639
$56$	0	0
$57$	11695.7	0.476803
$58$	0	0
$59$	−30528.3	−1.14175	−0.570877	−	0.821036i	$-0.693397\pi$
$59$	−30528.3	−1.14175	−0.570877	+	0.821036i	$0.693397\pi$
$60$	0	0
$61$	9.53050	0.000327937 0	0.000163969	−	1.00000i	$-0.499948\pi$
$61$	9.53050	0.000327937 0	0.000163969		1.00000i	$0.499948\pi$
$62$	0	0
$63$	728.104	0.0231123
$64$	0	0
$65$	45734.8	1.34265
$66$	0	0
$67$	−9996.72	−0.272064	−0.136032	−	0.990704i	$-0.543435\pi$
$67$	−9996.72	−0.272064	−0.136032	+	0.990704i	$0.543435\pi$
$68$	0	0
$69$	−4761.00	−0.120386
$70$	0	0
$71$	73725.9	1.73570	0.867849	−	0.496828i	$-0.165502\pi$
$71$	73725.9	1.73570	0.867849	+	0.496828i	$0.165502\pi$
$72$	0	0
$73$	55169.6	1.21169	0.605847	−	0.795581i	$-0.292834\pi$
$73$	55169.6	1.21169	0.605847	+	0.795581i	$0.292834\pi$
$74$	0	0
$75$	−48267.4	−0.990834
$76$	0	0
$77$	−5507.94	−0.105867
$78$	0	0
$79$	−62397.4	−1.12486	−0.562431	−	0.826844i	$-0.690134\pi$
$79$	−62397.4	−1.12486	−0.562431	+	0.826844i	$0.690134\pi$
$80$	0	0
$81$	6561.00	0.111111
$82$	0	0
$83$	82435.2	1.31346	0.656731	−	0.754125i	$-0.271939\pi$
$83$	82435.2	1.31346	0.656731	+	0.754125i	$0.271939\pi$
$84$	0	0
$85$	−68681.6	−1.03108
$86$	0	0
$87$	−69793.4	−0.988589
$88$	0	0
$89$	95173.0	1.27362	0.636808	−	0.771022i	$-0.280254\pi$
$89$	95173.0	1.27362	0.636808	+	0.771022i	$0.280254\pi$
$90$	0	0
$91$	−4462.23	−0.0564870
$92$	0	0
$93$	−9599.57	−0.115092
$94$	0	0
$95$	119726.	1.36106
$96$	0	0
$97$	47134.4	0.508638	0.254319	−	0.967120i	$-0.418149\pi$
$97$	47134.4	0.508638	0.254319	+	0.967120i	$0.418149\pi$
$98$	0	0
$99$	−49632.4	−0.508953
$100$	0	0
$101$	−15830.2	−0.154413	−0.0772065	−	0.997015i	$-0.524600\pi$
$101$	−15830.2	−0.154413	−0.0772065	+	0.997015i	$0.524600\pi$
$102$	0	0
$103$	90782.5	0.843158	0.421579	−	0.906792i	$-0.361476\pi$
$103$	90782.5	0.843158	0.421579	+	0.906792i	$0.361476\pi$
$104$	0	0
$105$	7453.41	0.0659754
$106$	0	0
$107$	−209377.	−1.76795	−0.883973	−	0.467538i	$-0.845141\pi$
$107$	−209377.	−1.76795	−0.883973	+	0.467538i	$0.845141\pi$
$108$	0	0
$109$	40116.0	0.323408	0.161704	−	0.986839i	$-0.448301\pi$
$109$	40116.0	0.323408	0.161704	+	0.986839i	$0.448301\pi$
$110$	0	0
$111$	7562.51	0.0582584
$112$	0	0
$113$	248398.	1.83000	0.915001	−	0.403453i	$-0.132190\pi$
$113$	248398.	1.83000	0.915001	+	0.403453i	$0.132190\pi$
$114$	0	0
$115$	−48737.1	−0.343649
$116$	0	0
$117$	−40209.5	−0.271559
$118$	0	0
$119$	6701.09	0.0433788
$120$	0	0
$121$	214407.	1.33130
$122$	0	0
$123$	−88389.0	−0.526787
$124$	0	0
$125$	−206193.	−1.18032
$126$	0	0
$127$	−128385.	−0.706328	−0.353164	−	0.935561i	$-0.614894\pi$
$127$	−128385.	−0.706328	−0.353164	+	0.935561i	$0.614894\pi$
$128$	0	0
$129$	−121768.	−0.644258
$130$	0	0
$131$	361743.	1.84171	0.920857	−	0.389901i	$-0.127491\pi$
$131$	361743.	1.84171	0.920857	+	0.389901i	$0.127491\pi$
$132$	0	0
$133$	−11681.3	−0.0572616
$134$	0	0
$135$	67163.2	0.317174
$136$	0	0
$137$	−116630.	−0.530897	−0.265448	−	0.964125i	$-0.585520\pi$
$137$	−116630.	−0.530897	−0.265448	+	0.964125i	$0.585520\pi$
$138$	0	0
$139$	−94540.9	−0.415033	−0.207517	−	0.978231i	$-0.566538\pi$
$139$	−94540.9	−0.415033	−0.207517	+	0.978231i	$0.566538\pi$
$140$	0	0
$141$	184366.	0.780968
$142$	0	0
$143$	304175.	1.24390
$144$	0	0
$145$	−714456.	−2.82199
$146$	0	0
$147$	150536.	0.574575
$148$	0	0
$149$	275055.	1.01497	0.507486	−	0.861660i	$-0.330575\pi$
$149$	275055.	1.01497	0.507486	+	0.861660i	$0.330575\pi$
$150$	0	0
$151$	47405.4	0.169194	0.0845971	−	0.996415i	$-0.473040\pi$
$151$	47405.4	0.169194	0.0845971	+	0.996415i	$0.473040\pi$
$152$	0	0
$153$	60383.9	0.208542
$154$	0	0
$155$	−98268.2	−0.328537
$156$	0	0
$157$	88519.4	0.286609	0.143304	−	0.989679i	$-0.454227\pi$
$157$	88519.4	0.286609	0.143304	+	0.989679i	$0.454227\pi$
$158$	0	0
$159$	−43351.7	−0.135992
$160$	0	0
$161$	4755.15	0.0144577
$162$	0	0
$163$	−522220.	−1.53952	−0.769759	−	0.638335i	$-0.779623\pi$
$163$	−522220.	−1.53952	−0.769759	+	0.638335i	$0.779623\pi$
$164$	0	0
$165$	−508074.	−1.45284
$166$	0	0
$167$	234704.	0.651223	0.325611	−	0.945504i	$-0.394430\pi$
$167$	234704.	0.651223	0.325611	+	0.945504i	$0.394430\pi$
$168$	0	0
$169$	−124867.	−0.336303
$170$	0	0
$171$	−105261.	−0.275282
$172$	0	0
$173$	−249399.	−0.633547	−0.316773	−	0.948501i	$-0.602599\pi$
$173$	−249399.	−0.633547	−0.316773	+	0.948501i	$0.602599\pi$
$174$	0	0
$175$	48208.1	0.118994
$176$	0	0
$177$	274755.	0.659192
$178$	0	0
$179$	−353019.	−0.823503	−0.411751	−	0.911296i	$-0.635083\pi$
$179$	−353019.	−0.823503	−0.411751	+	0.911296i	$0.635083\pi$
$180$	0	0
$181$	−188804.	−0.428367	−0.214183	−	0.976793i	$-0.568709\pi$
$181$	−188804.	−0.428367	−0.214183	+	0.976793i	$0.568709\pi$
$182$	0	0
$183$	−85.7745	−0.000189335 0
$184$	0	0
$185$	77415.4	0.166302
$186$	0	0
$187$	−456791.	−0.955241
$188$	0	0
$189$	−6552.94	−0.0133439
$190$	0	0
$191$	−374966.	−0.743719	−0.371859	−	0.928289i	$-0.621280\pi$
$191$	−374966.	−0.743719	−0.371859	+	0.928289i	$0.621280\pi$
$192$	0	0
$193$	−761894.	−1.47232	−0.736158	−	0.676810i	$-0.763362\pi$
$193$	−761894.	−1.47232	−0.736158	+	0.676810i	$0.763362\pi$
$194$	0	0
$195$	−411614.	−0.775181
$196$	0	0
$197$	−772963.	−1.41904	−0.709518	−	0.704688i	$-0.751087\pi$
$197$	−772963.	−1.41904	−0.709518	+	0.704688i	$0.751087\pi$
$198$	0	0
$199$	−863431.	−1.54559	−0.772796	−	0.634654i	$-0.781143\pi$
$199$	−863431.	−1.54559	−0.772796	+	0.634654i	$0.781143\pi$
$200$	0	0
$201$	89970.5	0.157076
$202$	0	0
$203$	69707.6	0.118724
$204$	0	0
$205$	−904814.	−1.50375
$206$	0	0
$207$	42849.0	0.0695048
$208$	0	0
$209$	796278.	1.26095
$210$	0	0
$211$	−973725.	−1.50567	−0.752835	−	0.658209i	$-0.771314\pi$
$211$	−973725.	−1.50567	−0.752835	+	0.658209i	$0.771314\pi$
$212$	0	0
$213$	−663533.	−1.00211
$214$	0	0
$215$	−1.24651e6	−1.83908
$216$	0	0
$217$	9587.78	0.0138219
$218$	0	0
$219$	−496527.	−0.699572
$220$	0	0
$221$	−370066.	−0.509682
$222$	0	0
$223$	1.07660e6	1.44974	0.724870	−	0.688885i	$-0.241900\pi$
$223$	1.07660e6	1.44974	0.724870	+	0.688885i	$0.241900\pi$
$224$	0	0
$225$	434407.	0.572059
$226$	0	0
$227$	313794.	0.404185	0.202092	−	0.979366i	$-0.435226\pi$
$227$	313794.	0.404185	0.202092	+	0.979366i	$0.435226\pi$
$228$	0	0
$229$	1.11820e6	1.40907	0.704534	−	0.709671i	$-0.251156\pi$
$229$	1.11820e6	1.40907	0.704534	+	0.709671i	$0.251156\pi$
$230$	0	0
$231$	49571.5	0.0611226
$232$	0	0
$233$	−562380.	−0.678641	−0.339321	−	0.940671i	$-0.610197\pi$
$233$	−562380.	−0.678641	−0.339321	+	0.940671i	$0.610197\pi$
$234$	0	0
$235$	1.88731e6	2.22932
$236$	0	0
$237$	561577.	0.649439
$238$	0	0
$239$	560026.	0.634181	0.317090	−	0.948395i	$-0.397294\pi$
$239$	560026.	0.634181	0.317090	+	0.948395i	$0.397294\pi$
$240$	0	0
$241$	−1.62461e6	−1.80180	−0.900902	−	0.434023i	$-0.857093\pi$
$241$	−1.62461e6	−1.80180	−0.900902	+	0.434023i	$0.857093\pi$
$242$	0	0
$243$	−59049.0	−0.0641500
$244$	0	0
$245$	1.54099e6	1.64016
$246$	0	0
$247$	645100.	0.672798
$248$	0	0
$249$	−741917.	−0.758327
$250$	0	0
$251$	1.09921e6	1.10127	0.550637	−	0.834745i	$-0.314385\pi$
$251$	1.09921e6	1.10127	0.550637	+	0.834745i	$0.314385\pi$
$252$	0	0
$253$	−324143.	−0.318372
$254$	0	0
$255$	618134.	0.595295
$256$	0	0
$257$	−1.36203e6	−1.28633	−0.643167	−	0.765726i	$-0.722380\pi$
$257$	−1.36203e6	−1.28633	−0.643167	+	0.765726i	$0.722380\pi$
$258$	0	0
$259$	−7553.22	−0.00699653
$260$	0	0
$261$	628140.	0.570762
$262$	0	0
$263$	802156.	0.715104	0.357552	−	0.933893i	$-0.383611\pi$
$263$	802156.	0.715104	0.357552	+	0.933893i	$0.383611\pi$
$264$	0	0
$265$	−443780.	−0.388197
$266$	0	0
$267$	−856557.	−0.735323
$268$	0	0
$269$	−352123.	−0.296698	−0.148349	−	0.988935i	$-0.547396\pi$
$269$	−352123.	−0.296698	−0.148349	+	0.988935i	$0.547396\pi$
$270$	0	0
$271$	1.51322e6	1.25164	0.625819	−	0.779968i	$-0.284765\pi$
$271$	1.51322e6	1.25164	0.625819	+	0.779968i	$0.284765\pi$
$272$	0	0
$273$	40160.1	0.0326128
$274$	0	0
$275$	−3.28619e6	−2.62036
$276$	0	0
$277$	−1.26262e6	−0.988720	−0.494360	−	0.869257i	$-0.664598\pi$
$277$	−1.26262e6	−0.988720	−0.494360	+	0.869257i	$0.664598\pi$
$278$	0	0
$279$	86396.1	0.0664483
$280$	0	0
$281$	1.24571e6	0.941133	0.470567	−	0.882365i	$-0.344050\pi$
$281$	1.24571e6	0.941133	0.470567	+	0.882365i	$0.344050\pi$
$282$	0	0
$283$	−372353.	−0.276368	−0.138184	−	0.990407i	$-0.544127\pi$
$283$	−372353.	−0.276368	−0.138184	+	0.990407i	$0.544127\pi$
$284$	0	0
$285$	−1.07753e6	−0.785811
$286$	0	0
$287$	88280.4	0.0632644
$288$	0	0
$289$	−864115.	−0.608593
$290$	0	0
$291$	−424210.	−0.293662
$292$	0	0
$293$	1.74374e6	1.18662	0.593311	−	0.804974i	$-0.297821\pi$
$293$	1.74374e6	1.18662	0.593311	+	0.804974i	$0.297821\pi$
$294$	0	0
$295$	2.81259e6	1.88170
$296$	0	0
$297$	446692.	0.293844
$298$	0	0
$299$	−262603.	−0.169872
$300$	0	0
$301$	121619.	0.0773721
$302$	0	0
$303$	142472.	0.0891504
$304$	0	0
$305$	−878.050	−0.000540468 0
$306$	0	0
$307$	1.60710e6	0.973188	0.486594	−	0.873628i	$-0.338239\pi$
$307$	1.60710e6	0.973188	0.486594	+	0.873628i	$0.338239\pi$
$308$	0	0
$309$	−817042.	−0.486797
$310$	0	0
$311$	−2.21364e6	−1.29780	−0.648898	−	0.760876i	$-0.724770\pi$
$311$	−2.21364e6	−1.29780	−0.648898	+	0.760876i	$0.724770\pi$
$312$	0	0
$313$	480737.	0.277362	0.138681	−	0.990337i	$-0.455714\pi$
$313$	480737.	0.277362	0.138681	+	0.990337i	$0.455714\pi$
$314$	0	0
$315$	−67080.7	−0.0380909
$316$	0	0
$317$	3.23334e6	1.80719	0.903593	−	0.428393i	$-0.140920\pi$
$317$	3.23334e6	1.80719	0.903593	+	0.428393i	$0.140920\pi$
$318$	0	0
$319$	−4.75174e6	−2.61442
$320$	0	0
$321$	1.88439e6	1.02072
$322$	0	0
$323$	−968769.	−0.516671
$324$	0	0
$325$	−2.66229e6	−1.39813
$326$	0	0
$327$	−361044.	−0.186720
$328$	0	0
$329$	−184140.	−0.0937902
$330$	0	0
$331$	−2.61134e6	−1.31007	−0.655034	−	0.755600i	$-0.727346\pi$
$331$	−2.61134e6	−1.31007	−0.655034	+	0.755600i	$0.727346\pi$
$332$	0	0
$333$	−68062.6	−0.0336355
$334$	0	0
$335$	921004.	0.448383
$336$	0	0
$337$	2.76804e6	1.32769	0.663846	−	0.747870i	$-0.268923\pi$
$337$	2.76804e6	1.32769	0.663846	+	0.747870i	$0.268923\pi$
$338$	0	0
$339$	−2.23558e6	−1.05655
$340$	0	0
$341$	−653567.	−0.304371
$342$	0	0
$343$	−301428.	−0.138340
$344$	0	0
$345$	438634.	0.198406
$346$	0	0
$347$	−1.39621e6	−0.622484	−0.311242	−	0.950331i	$-0.600745\pi$
$347$	−1.39621e6	−0.622484	−0.311242	+	0.950331i	$0.600745\pi$
$348$	0	0
$349$	−1.00509e6	−0.441714	−0.220857	−	0.975306i	$-0.570885\pi$
$349$	−1.00509e6	−0.441714	−0.220857	+	0.975306i	$0.570885\pi$
$350$	0	0
$351$	361885.	0.156784
$352$	0	0
$353$	−4.24269e6	−1.81219	−0.906096	−	0.423073i	$-0.860952\pi$
$353$	−4.24269e6	−1.81219	−0.906096	+	0.423073i	$0.860952\pi$
$354$	0	0
$355$	−6.79241e6	−2.86057
$356$	0	0
$357$	−60309.8	−0.0250448
$358$	0	0
$359$	−313836.	−0.128519	−0.0642595	−	0.997933i	$-0.520469\pi$
$359$	−313836.	−0.128519	−0.0642595	+	0.997933i	$0.520469\pi$
$360$	0	0
$361$	−787340.	−0.317976
$362$	0	0
$363$	−1.92966e6	−0.768626
$364$	0	0
$365$	−5.08281e6	−1.99697
$366$	0	0
$367$	−290172.	−0.112458	−0.0562290	−	0.998418i	$-0.517908\pi$
$367$	−290172.	−0.112458	−0.0562290	+	0.998418i	$0.517908\pi$
$368$	0	0
$369$	795501.	0.304141
$370$	0	0
$371$	43298.4	0.0163319
$372$	0	0
$373$	−5.14109e6	−1.91330	−0.956649	−	0.291243i	$-0.905931\pi$
$373$	−5.14109e6	−1.91330	−0.956649	+	0.291243i	$0.905931\pi$
$374$	0	0
$375$	1.85573e6	0.681456
$376$	0	0
$377$	−3.84959e6	−1.39496
$378$	0	0
$379$	631911.	0.225974	0.112987	−	0.993596i	$-0.463958\pi$
$379$	631911.	0.225974	0.112987	+	0.993596i	$0.463958\pi$
$380$	0	0
$381$	1.15547e6	0.407799
$382$	0	0
$383$	118799.	0.0413823	0.0206911	−	0.999786i	$-0.493413\pi$
$383$	118799.	0.0413823	0.0206911	+	0.999786i	$0.493413\pi$
$384$	0	0
$385$	507450.	0.174478
$386$	0	0
$387$	1.09591e6	0.371963
$388$	0	0
$389$	2.58745e6	0.866957	0.433478	−	0.901164i	$-0.357286\pi$
$389$	2.58745e6	0.866957	0.433478	+	0.901164i	$0.357286\pi$
$390$	0	0
$391$	394359.	0.130452
$392$	0	0
$393$	−3.25569e6	−1.06331
$394$	0	0
$395$	5.74871e6	1.85386
$396$	0	0
$397$	3.56801e6	1.13619	0.568093	−	0.822964i	$-0.307681\pi$
$397$	3.56801e6	1.13619	0.568093	+	0.822964i	$0.307681\pi$
$398$	0	0
$399$	105132.	0.0330600
$400$	0	0
$401$	4.28994e6	1.33226	0.666132	−	0.745833i	$-0.267949\pi$
$401$	4.28994e6	1.33226	0.666132	+	0.745833i	$0.267949\pi$
$402$	0	0
$403$	−529483.	−0.162401
$404$	0	0
$405$	−604469.	−0.183120
$406$	0	0
$407$	514878.	0.154070
$408$	0	0
$409$	3.14276e6	0.928973	0.464486	−	0.885580i	$-0.346239\pi$
$409$	3.14276e6	0.928973	0.464486	+	0.885580i	$0.346239\pi$
$410$	0	0
$411$	1.04967e6	0.306513
$412$	0	0
$413$	−274417.	−0.0791655
$414$	0	0
$415$	−7.59480e6	−2.16469
$416$	0	0
$417$	850868.	0.239620
$418$	0	0
$419$	257827.	0.0717454	0.0358727	−	0.999356i	$-0.488579\pi$
$419$	257827.	0.0717454	0.0358727	+	0.999356i	$0.488579\pi$
$420$	0	0
$421$	−1.48632e6	−0.408701	−0.204351	−	0.978898i	$-0.565508\pi$
$421$	−1.48632e6	−0.408701	−0.204351	+	0.978898i	$0.565508\pi$
$422$	0	0
$423$	−1.65929e6	−0.450892
$424$	0	0
$425$	3.99805e6	1.07368
$426$	0	0
$427$	85.6691	2.27381e−5 0
$428$	0	0
$429$	−2.73758e6	−0.718163
$430$	0	0
$431$	−4.98067e6	−1.29150	−0.645750	−	0.763549i	$-0.723455\pi$
$431$	−4.98067e6	−1.29150	−0.645750	+	0.763549i	$0.723455\pi$
$432$	0	0
$433$	1.44742e6	0.371001	0.185500	−	0.982644i	$-0.440609\pi$
$433$	1.44742e6	0.371001	0.185500	+	0.982644i	$0.440609\pi$
$434$	0	0
$435$	6.43010e6	1.62928
$436$	0	0
$437$	−687447.	−0.172201
$438$	0	0
$439$	4.31837e6	1.06944	0.534722	−	0.845028i	$-0.320416\pi$
$439$	4.31837e6	1.06944	0.534722	+	0.845028i	$0.320416\pi$
$440$	0	0
$441$	−1.35482e6	−0.331731
$442$	0	0
$443$	−5.65863e6	−1.36994	−0.684971	−	0.728570i	$-0.740185\pi$
$443$	−5.65863e6	−1.36994	−0.684971	+	0.728570i	$0.740185\pi$
$444$	0	0
$445$	−8.76834e6	−2.09902
$446$	0	0
$447$	−2.47550e6	−0.585994
$448$	0	0
$449$	−1.72793e6	−0.404492	−0.202246	−	0.979335i	$-0.564824\pi$
$449$	−1.72793e6	−0.404492	−0.202246	+	0.979335i	$0.564824\pi$
$450$	0	0
$451$	−6.01778e6	−1.39314
$452$	0	0
$453$	−426649.	−0.0976843
$454$	0	0
$455$	411108.	0.0930952
$456$	0	0
$457$	−3.19765e6	−0.716210	−0.358105	−	0.933681i	$-0.616577\pi$
$457$	−3.19765e6	−0.716210	−0.358105	+	0.933681i	$0.616577\pi$
$458$	0	0
$459$	−543456.	−0.120402
$460$	0	0
$461$	7.02590e6	1.53975	0.769874	−	0.638196i	$-0.220319\pi$
$461$	7.02590e6	1.53975	0.769874	+	0.638196i	$0.220319\pi$
$462$	0	0
$463$	9.07013e6	1.96635	0.983176	−	0.182663i	$-0.0584716\pi$
$463$	9.07013e6	1.96635	0.983176	+	0.182663i	$0.0584716\pi$
$464$	0	0
$465$	884414.	0.189681
$466$	0	0
$467$	446092.	0.0946525	0.0473263	−	0.998879i	$-0.484930\pi$
$467$	446092.	0.0946525	0.0473263	+	0.998879i	$0.484930\pi$
$468$	0	0
$469$	−89860.0	−0.0188640
$470$	0	0
$471$	−796675.	−0.165474
$472$	0	0
$473$	−8.29034e6	−1.70380
$474$	0	0
$475$	−6.96940e6	−1.41730
$476$	0	0
$477$	390165.	0.0785150
$478$	0	0
$479$	2.86711e6	0.570960	0.285480	−	0.958385i	$-0.407847\pi$
$479$	2.86711e6	0.570960	0.285480	+	0.958385i	$0.407847\pi$
$480$	0	0
$481$	417125.	0.0822061
$482$	0	0
$483$	−42796.4	−0.00834717
$484$	0	0
$485$	−4.34252e6	−0.838278
$486$	0	0
$487$	−3.38408e6	−0.646574	−0.323287	−	0.946301i	$-0.604788\pi$
$487$	−3.38408e6	−0.646574	−0.323287	+	0.946301i	$0.604788\pi$
$488$	0	0
$489$	4.69998e6	0.888841
$490$	0	0
$491$	−2.36070e6	−0.441913	−0.220956	−	0.975284i	$-0.570918\pi$
$491$	−2.36070e6	−0.441913	−0.220956	+	0.975284i	$0.570918\pi$
$492$	0	0
$493$	5.78107e6	1.07125
$494$	0	0
$495$	4.57267e6	0.838797
$496$	0	0
$497$	662718.	0.120348
$498$	0	0
$499$	3.42412e6	0.615598	0.307799	−	0.951451i	$-0.400408\pi$
$499$	3.42412e6	0.615598	0.307799	+	0.951451i	$0.400408\pi$
$500$	0	0
$501$	−2.11234e6	−0.375984
$502$	0	0
$503$	−7.49119e6	−1.32017	−0.660087	−	0.751190i	$-0.729480\pi$
$503$	−7.49119e6	−1.32017	−0.660087	+	0.751190i	$0.729480\pi$
$504$	0	0
$505$	1.45845e6	0.254485
$506$	0	0
$507$	1.12380e6	0.194165
$508$	0	0
$509$	−5.34876e6	−0.915079	−0.457539	−	0.889189i	$-0.651269\pi$
$509$	−5.34876e6	−0.915079	−0.457539	+	0.889189i	$0.651269\pi$
$510$	0	0
$511$	495917.	0.0840149
$512$	0	0
$513$	947352.	0.158934
$514$	0	0
$515$	−8.36384e6	−1.38959
$516$	0	0
$517$	1.25522e7	2.06535
$518$	0	0
$519$	2.24459e6	0.365778
$520$	0	0
$521$	−4.73330e6	−0.763958	−0.381979	−	0.924171i	$-0.624757\pi$
$521$	−4.73330e6	−0.763958	−0.381979	+	0.924171i	$0.624757\pi$
$522$	0	0
$523$	2.28134e6	0.364701	0.182350	−	0.983234i	$-0.441630\pi$
$523$	2.28134e6	0.364701	0.182350	+	0.983234i	$0.441630\pi$
$524$	0	0
$525$	−433873.	−0.0687013
$526$	0	0
$527$	795144.	0.124715
$528$	0	0
$529$	279841.	0.0434783
$530$	0	0
$531$	−2.47279e6	−0.380584
$532$	0	0
$533$	−4.87527e6	−0.743329
$534$	0	0
$535$	1.92900e7	2.91372
$536$	0	0
$537$	3.17717e6	0.475449
$538$	0	0
$539$	1.02489e7	1.51952
$540$	0	0
$541$	−2.57849e6	−0.378767	−0.189383	−	0.981903i	$-0.560649\pi$
$541$	−2.57849e6	−0.378767	−0.189383	+	0.981903i	$0.560649\pi$
$542$	0	0
$543$	1.69924e6	0.247318
$544$	0	0
$545$	−3.69591e6	−0.533004
$546$	0	0
$547$	−6.20267e6	−0.886360	−0.443180	−	0.896433i	$-0.646150\pi$
$547$	−6.20267e6	−0.886360	−0.443180	+	0.896433i	$0.646150\pi$
$548$	0	0
$549$	771.970	0.000109312 0
$550$	0	0
$551$	−1.00776e7	−1.41409
$552$	0	0
$553$	−560887.	−0.0779942
$554$	0	0
$555$	−696739.	−0.0960146
$556$	0	0
$557$	−2.67729e6	−0.365643	−0.182821	−	0.983146i	$-0.558523\pi$
$557$	−2.67729e6	−0.365643	−0.182821	+	0.983146i	$0.558523\pi$
$558$	0	0
$559$	−6.71638e6	−0.909087
$560$	0	0
$561$	4.11112e6	0.551509
$562$	0	0
$563$	7.74164e6	1.02935	0.514673	−	0.857386i	$-0.327913\pi$
$563$	7.74164e6	1.02935	0.514673	+	0.857386i	$0.327913\pi$
$564$	0	0
$565$	−2.28850e7	−3.01599
$566$	0	0
$567$	58976.5	0.00770408
$568$	0	0
$569$	1.38677e7	1.79565	0.897827	−	0.440348i	$-0.145145\pi$
$569$	1.38677e7	1.79565	0.897827	+	0.440348i	$0.145145\pi$
$570$	0	0
$571$	3.83164e6	0.491807	0.245904	−	0.969294i	$-0.420915\pi$
$571$	3.83164e6	0.491807	0.245904	+	0.969294i	$0.420915\pi$
$572$	0	0
$573$	3.37470e6	0.429386
$574$	0	0
$575$	2.83705e6	0.357847
$576$	0	0
$577$	−2.35210e6	−0.294114	−0.147057	−	0.989128i	$-0.546980\pi$
$577$	−2.35210e6	−0.294114	−0.147057	+	0.989128i	$0.546980\pi$
$578$	0	0
$579$	6.85704e6	0.850042
$580$	0	0
$581$	741005.	0.0910712
$582$	0	0
$583$	−2.95151e6	−0.359644
$584$	0	0
$585$	3.70452e6	0.447551
$586$	0	0
$587$	5.80653e6	0.695539	0.347770	−	0.937580i	$-0.386939\pi$
$587$	5.80653e6	0.695539	0.347770	+	0.937580i	$0.386939\pi$
$588$	0	0
$589$	−1.38610e6	−0.164628
$590$	0	0
$591$	6.95667e6	0.819281
$592$	0	0
$593$	−5.28179e6	−0.616800	−0.308400	−	0.951257i	$-0.599794\pi$
$593$	−5.28179e6	−0.616800	−0.308400	+	0.951257i	$0.599794\pi$
$594$	0	0
$595$	−617375.	−0.0714919
$596$	0	0
$597$	7.77088e6	0.892348
$598$	0	0
$599$	9.88981e6	1.12621	0.563107	−	0.826384i	$-0.309606\pi$
$599$	9.88981e6	1.12621	0.563107	+	0.826384i	$0.309606\pi$
$600$	0	0
$601$	4.37165e6	0.493695	0.246848	−	0.969054i	$-0.420605\pi$
$601$	4.37165e6	0.493695	0.246848	+	0.969054i	$0.420605\pi$
$602$	0	0
$603$	−809734.	−0.0906879
$604$	0	0
$605$	−1.97534e7	−2.19409
$606$	0	0
$607$	778883.	0.0858026	0.0429013	−	0.999079i	$-0.486340\pi$
$607$	778883.	0.0858026	0.0429013	+	0.999079i	$0.486340\pi$
$608$	0	0
$609$	−627369.	−0.0685456
$610$	0	0
$611$	1.01691e7	1.10199
$612$	0	0
$613$	−5.29772e6	−0.569427	−0.284713	−	0.958613i	$-0.591898\pi$
$613$	−5.29772e6	−0.569427	−0.284713	+	0.958613i	$0.591898\pi$
$614$	0	0
$615$	8.14333e6	0.868189
$616$	0	0
$617$	−1.13961e7	−1.20515	−0.602577	−	0.798060i	$-0.705860\pi$
$617$	−1.13961e7	−1.20515	−0.602577	+	0.798060i	$0.705860\pi$
$618$	0	0
$619$	1.65105e7	1.73194	0.865971	−	0.500094i	$-0.166701\pi$
$619$	1.65105e7	1.73194	0.865971	+	0.500094i	$0.166701\pi$
$620$	0	0
$621$	−385641.	−0.0401286
$622$	0	0
$623$	855504.	0.0883084
$624$	0	0
$625$	2.23714e6	0.229083
$626$	0	0
$627$	−7.16650e6	−0.728011
$628$	0	0
$629$	−626412.	−0.0631296
$630$	0	0
$631$	−1.21667e7	−1.21646	−0.608232	−	0.793760i	$-0.708121\pi$
$631$	−1.21667e7	−1.21646	−0.608232	+	0.793760i	$0.708121\pi$
$632$	0	0
$633$	8.76352e6	0.869299
$634$	0	0
$635$	1.18282e7	1.16409
$636$	0	0
$637$	8.30310e6	0.810759
$638$	0	0
$639$	5.97180e6	0.578566
$640$	0	0
$641$	1.41113e7	1.35651	0.678253	−	0.734829i	$-0.262737\pi$
$641$	1.41113e7	1.35651	0.678253	+	0.734829i	$0.262737\pi$
$642$	0	0
$643$	−5.63004e6	−0.537012	−0.268506	−	0.963278i	$-0.586530\pi$
$643$	−5.63004e6	−0.537012	−0.268506	+	0.963278i	$0.586530\pi$
$644$	0	0
$645$	1.12186e7	1.06179
$646$	0	0
$647$	−7.21927e6	−0.678004	−0.339002	−	0.940786i	$-0.610089\pi$
$647$	−7.21927e6	−0.678004	−0.339002	+	0.940786i	$0.610089\pi$
$648$	0	0
$649$	1.87061e7	1.74330
$650$	0	0
$651$	−86290.0	−0.00798009
$652$	0	0
$653$	1.37204e7	1.25917	0.629586	−	0.776931i	$-0.283225\pi$
$653$	1.37204e7	1.25917	0.629586	+	0.776931i	$0.283225\pi$
$654$	0	0
$655$	−3.33276e7	−3.03530
$656$	0	0
$657$	4.46874e6	0.403898
$658$	0	0
$659$	−1.35053e7	−1.21141	−0.605705	−	0.795689i	$-0.707109\pi$
$659$	−1.35053e7	−1.21141	−0.605705	+	0.795689i	$0.707109\pi$
$660$	0	0
$661$	6.74829e6	0.600745	0.300373	−	0.953822i	$-0.402889\pi$
$661$	6.74829e6	0.600745	0.300373	+	0.953822i	$0.402889\pi$
$662$	0	0
$663$	3.33060e6	0.294265
$664$	0	0
$665$	1.07621e6	0.0943718
$666$	0	0
$667$	4.10230e6	0.357037
$668$	0	0
$669$	−9.68936e6	−0.837008
$670$	0	0
$671$	−5839.78	−0.000500714 0
$672$	0	0
$673$	−1.24941e7	−1.06333	−0.531665	−	0.846955i	$-0.678434\pi$
$673$	−1.24941e7	−1.06333	−0.531665	+	0.846955i	$0.678434\pi$
$674$	0	0
$675$	−3.90966e6	−0.330278
$676$	0	0
$677$	9.64155e6	0.808491	0.404246	−	0.914650i	$-0.367534\pi$
$677$	9.64155e6	0.808491	0.404246	+	0.914650i	$0.367534\pi$
$678$	0	0
$679$	423689.	0.0352673
$680$	0	0
$681$	−2.82415e6	−0.233356
$682$	0	0
$683$	−4.95512e6	−0.406445	−0.203223	−	0.979133i	$-0.565142\pi$
$683$	−4.95512e6	−0.406445	−0.203223	+	0.979133i	$0.565142\pi$
$684$	0	0
$685$	1.07452e7	0.874962
$686$	0	0
$687$	−1.00638e7	−0.813525
$688$	0	0
$689$	−2.39115e6	−0.191893
$690$	0	0
$691$	1.18245e7	0.942079	0.471039	−	0.882112i	$-0.343879\pi$
$691$	1.18245e7	0.942079	0.471039	+	0.882112i	$0.343879\pi$
$692$	0	0
$693$	−446143.	−0.0352892
$694$	0	0
$695$	8.71011e6	0.684009
$696$	0	0
$697$	7.32137e6	0.570834
$698$	0	0
$699$	5.06142e6	0.391814
$700$	0	0
$701$	8.50355e6	0.653590	0.326795	−	0.945095i	$-0.394031\pi$
$701$	8.50355e6	0.653590	0.326795	+	0.945095i	$0.394031\pi$
$702$	0	0
$703$	1.09196e6	0.0833334
$704$	0	0
$705$	−1.69858e7	−1.28710
$706$	0	0
$707$	−142297.	−0.0107065
$708$	0	0
$709$	1.18862e7	0.888029	0.444015	−	0.896020i	$-0.353554\pi$
$709$	1.18862e7	0.888029	0.444015	+	0.896020i	$0.353554\pi$
$710$	0	0
$711$	−5.05419e6	−0.374954
$712$	0	0
$713$	564241.	0.0415663
$714$	0	0
$715$	−2.80239e7	−2.05004
$716$	0	0
$717$	−5.04023e6	−0.366145
$718$	0	0
$719$	3.08137e6	0.222291	0.111145	−	0.993804i	$-0.464548\pi$
$719$	3.08137e6	0.222291	0.111145	+	0.993804i	$0.464548\pi$
$720$	0	0
$721$	816038.	0.0584618
$722$	0	0
$723$	1.46215e7	1.04027
$724$	0	0
$725$	4.15895e7	2.93859
$726$	0	0
$727$	4.11888e6	0.289030	0.144515	−	0.989503i	$-0.453838\pi$
$727$	4.11888e6	0.289030	0.144515	+	0.989503i	$0.453838\pi$
$728$	0	0
$729$	531441.	0.0370370
$730$	0	0
$731$	1.00862e7	0.698128
$732$	0	0
$733$	−2.07922e6	−0.142935	−0.0714677	−	0.997443i	$-0.522768\pi$
$733$	−2.07922e6	−0.142935	−0.0714677	+	0.997443i	$0.522768\pi$
$734$	0	0
$735$	−1.38690e7	−0.946946
$736$	0	0
$737$	6.12545e6	0.415403
$738$	0	0
$739$	9.96819e6	0.671437	0.335719	−	0.941962i	$-0.391021\pi$
$739$	9.96819e6	0.671437	0.335719	+	0.941962i	$0.391021\pi$
$740$	0	0
$741$	−5.80590e6	−0.388440
$742$	0	0
$743$	1.21883e7	0.809977	0.404988	−	0.914322i	$-0.367276\pi$
$743$	1.21883e7	0.809977	0.404988	+	0.914322i	$0.367276\pi$
$744$	0	0
$745$	−2.53410e7	−1.67276
$746$	0	0
$747$	6.67725e6	0.437821
$748$	0	0
$749$	−1.88208e6	−0.122584
$750$	0	0
$751$	−1.30656e7	−0.845334	−0.422667	−	0.906285i	$-0.638906\pi$
$751$	−1.30656e7	−0.845334	−0.422667	+	0.906285i	$0.638906\pi$
$752$	0	0
$753$	−9.89287e6	−0.635821
$754$	0	0
$755$	−4.36749e6	−0.278846
$756$	0	0
$757$	1.30510e7	0.827761	0.413881	−	0.910331i	$-0.364173\pi$
$757$	1.30510e7	0.827761	0.413881	+	0.910331i	$0.364173\pi$
$758$	0	0
$759$	2.91729e6	0.183812
$760$	0	0
$761$	8.03065e6	0.502677	0.251338	−	0.967899i	$-0.419129\pi$
$761$	8.03065e6	0.502677	0.251338	+	0.967899i	$0.419129\pi$
$762$	0	0
$763$	360600.	0.0224241
$764$	0	0
$765$	−5.56321e6	−0.343694
$766$	0	0
$767$	1.51546e7	0.930159
$768$	0	0
$769$	6.12813e6	0.373691	0.186845	−	0.982389i	$-0.440174\pi$
$769$	6.12813e6	0.373691	0.186845	+	0.982389i	$0.440174\pi$
$770$	0	0
$771$	1.22583e7	0.742665
$772$	0	0
$773$	8.04760e6	0.484415	0.242208	−	0.970224i	$-0.422128\pi$
$773$	8.04760e6	0.484415	0.242208	+	0.970224i	$0.422128\pi$
$774$	0	0
$775$	5.72033e6	0.342111
$776$	0	0
$777$	67979.0	0.00403945
$778$	0	0
$779$	−1.27626e7	−0.753522
$780$	0	0
$781$	−4.51753e7	−2.65017
$782$	0	0
$783$	−5.65326e6	−0.329530
$784$	0	0
$785$	−8.15535e6	−0.472355
$786$	0	0
$787$	3.49427e6	0.201103	0.100552	−	0.994932i	$-0.467939\pi$
$787$	3.49427e6	0.201103	0.100552	+	0.994932i	$0.467939\pi$
$788$	0	0
$789$	−7.21940e6	−0.412866
$790$	0	0
$791$	2.23283e6	0.126886
$792$	0	0
$793$	−4731.06	−0.000267163 0
$794$	0	0
$795$	3.99402e6	0.224126
$796$	0	0
$797$	−1.33411e7	−0.743951	−0.371976	−	0.928242i	$-0.621320\pi$
$797$	−1.33411e7	−0.743951	−0.371976	+	0.928242i	$0.621320\pi$
$798$	0	0
$799$	−1.52713e7	−0.846268
$800$	0	0
$801$	7.70901e6	0.424539
$802$	0	0
$803$	−3.38050e7	−1.85009
$804$	0	0
$805$	−438095.	−0.0238275
$806$	0	0
$807$	3.16911e6	0.171298
$808$	0	0
$809$	−5.57231e6	−0.299339	−0.149670	−	0.988736i	$-0.547821\pi$
$809$	−5.57231e6	−0.299339	−0.149670	+	0.988736i	$0.547821\pi$
$810$	0	0
$811$	−8.30913e6	−0.443612	−0.221806	−	0.975091i	$-0.571195\pi$
$811$	−8.30913e6	−0.443612	−0.221806	+	0.975091i	$0.571195\pi$
$812$	0	0
$813$	−1.36190e7	−0.722633
$814$	0	0
$815$	4.81125e7	2.53725
$816$	0	0
$817$	−1.75823e7	−0.921553
$818$	0	0
$819$	−361441.	−0.0188290
$820$	0	0
$821$	1.35323e7	0.700669	0.350334	−	0.936625i	$-0.386068\pi$
$821$	1.35323e7	0.700669	0.350334	+	0.936625i	$0.386068\pi$
$822$	0	0
$823$	−1.20061e7	−0.617879	−0.308939	−	0.951082i	$-0.599974\pi$
$823$	−1.20061e7	−0.617879	−0.308939	+	0.951082i	$0.599974\pi$
$824$	0	0
$825$	2.95757e7	1.51286
$826$	0	0
$827$	2.71374e7	1.37976	0.689881	−	0.723923i	$-0.257663\pi$
$827$	2.71374e7	1.37976	0.689881	+	0.723923i	$0.257663\pi$
$828$	0	0
$829$	−6.90219e6	−0.348819	−0.174410	−	0.984673i	$-0.555802\pi$
$829$	−6.90219e6	−0.348819	−0.174410	+	0.984673i	$0.555802\pi$
$830$	0	0
$831$	1.13636e7	0.570838
$832$	0	0
$833$	−1.24691e7	−0.622617
$834$	0	0
$835$	−2.16234e7	−1.07327
$836$	0	0
$837$	−777565.	−0.0383639
$838$	0	0
$839$	1.57032e6	0.0770164	0.0385082	−	0.999258i	$-0.487739\pi$
$839$	1.57032e6	0.0770164	0.0385082	+	0.999258i	$0.487739\pi$
$840$	0	0
$841$	3.96260e7	1.93193
$842$	0	0
$843$	−1.12114e7	−0.543363
$844$	0	0
$845$	1.15041e7	0.554255
$846$	0	0
$847$	1.92729e6	0.0923079
$848$	0	0
$849$	3.35118e6	0.159561
$850$	0	0
$851$	−444508.	−0.0210405
$852$	0	0
$853$	−2.20053e7	−1.03551	−0.517755	−	0.855529i	$-0.673232\pi$
$853$	−2.20053e7	−1.03551	−0.517755	+	0.855529i	$0.673232\pi$
$854$	0	0
$855$	9.69779e6	0.453688
$856$	0	0
$857$	9.95578e6	0.463045	0.231523	−	0.972830i	$-0.425629\pi$
$857$	9.95578e6	0.463045	0.231523	+	0.972830i	$0.425629\pi$
$858$	0	0
$859$	−1.38922e7	−0.642374	−0.321187	−	0.947016i	$-0.604082\pi$
$859$	−1.38922e7	−0.642374	−0.321187	+	0.947016i	$0.604082\pi$
$860$	0	0
$861$	−794524.	−0.0365257
$862$	0	0
$863$	−7.55200e6	−0.345171	−0.172586	−	0.984995i	$-0.555212\pi$
$863$	−7.55200e6	−0.345171	−0.172586	+	0.984995i	$0.555212\pi$
$864$	0	0
$865$	2.29772e7	1.04414
$866$	0	0
$867$	7.77704e6	0.351371
$868$	0	0
$869$	3.82338e7	1.71750
$870$	0	0
$871$	4.96250e6	0.221644
$872$	0	0
$873$	3.81789e6	0.169546
$874$	0	0
$875$	−1.85346e6	−0.0818393
$876$	0	0
$877$	−2.70277e6	−0.118662	−0.0593309	−	0.998238i	$-0.518897\pi$
$877$	−2.70277e6	−0.118662	−0.0593309	+	0.998238i	$0.518897\pi$
$878$	0	0
$879$	−1.56936e7	−0.685096
$880$	0	0
$881$	−4.41216e7	−1.91519	−0.957595	−	0.288117i	$-0.906971\pi$
$881$	−4.41216e7	−1.91519	−0.957595	+	0.288117i	$0.906971\pi$
$882$	0	0
$883$	−2.72497e7	−1.17614	−0.588071	−	0.808809i	$-0.700112\pi$
$883$	−2.72497e7	−1.17614	−0.588071	+	0.808809i	$0.700112\pi$
$884$	0	0
$885$	−2.53133e7	−1.08640
$886$	0	0
$887$	−1.11988e7	−0.477930	−0.238965	−	0.971028i	$-0.576808\pi$
$887$	−1.11988e7	−0.477930	−0.238965	+	0.971028i	$0.576808\pi$
$888$	0	0
$889$	−1.15405e6	−0.0489745
$890$	0	0
$891$	−4.02023e6	−0.169651
$892$	0	0
$893$	2.66209e7	1.11710
$894$	0	0
$895$	3.25238e7	1.35720
$896$	0	0
$897$	2.36342e6	0.0980754
$898$	0	0
$899$	8.27143e6	0.341336
$900$	0	0
$901$	3.59087e6	0.147363
$902$	0	0
$903$	−1.09457e6	−0.0446708
$904$	0	0
$905$	1.73947e7	0.705984
$906$	0	0
$907$	−2.65141e7	−1.07018	−0.535092	−	0.844794i	$-0.679723\pi$
$907$	−2.65141e7	−1.07018	−0.535092	+	0.844794i	$0.679723\pi$
$908$	0	0
$909$	−1.28225e6	−0.0514710
$910$	0	0
$911$	1.62499e7	0.648715	0.324357	−	0.945935i	$-0.394852\pi$
$911$	1.62499e7	0.648715	0.324357	+	0.945935i	$0.394852\pi$
$912$	0	0
$913$	−5.05119e7	−2.00547
$914$	0	0
$915$	7902.45	0.000312039 0
$916$	0	0
$917$	3.25169e6	0.127698
$918$	0	0
$919$	2.66141e7	1.03950	0.519748	−	0.854319i	$-0.326026\pi$
$919$	2.66141e7	1.03950	0.519748	+	0.854319i	$0.326026\pi$
$920$	0	0
$921$	−1.44639e7	−0.561870
$922$	0	0
$923$	−3.65985e7	−1.41403
$924$	0	0
$925$	−4.50646e6	−0.173173
$926$	0	0
$927$	7.35338e6	0.281053
$928$	0	0
$929$	−1.98073e7	−0.752984	−0.376492	−	0.926420i	$-0.622870\pi$
$929$	−1.98073e7	−0.752984	−0.376492	+	0.926420i	$0.622870\pi$
$930$	0	0
$931$	2.17361e7	0.821877
$932$	0	0
$933$	1.99228e7	0.749282
$934$	0	0
$935$	4.20844e7	1.57432
$936$	0	0
$937$	−9.15888e6	−0.340795	−0.170398	−	0.985375i	$-0.554505\pi$
$937$	−9.15888e6	−0.340795	−0.170398	+	0.985375i	$0.554505\pi$
$938$	0	0
$939$	−4.32664e6	−0.160135
$940$	0	0
$941$	5.38492e7	1.98247	0.991233	−	0.132128i	$-0.0421811\pi$
$941$	5.38492e7	1.98247	0.991233	+	0.132128i	$0.0421811\pi$
$942$	0	0
$943$	5.19531e6	0.190253
$944$	0	0
$945$	603726.	0.0219918
$946$	0	0
$947$	1.89117e7	0.685261	0.342631	−	0.939470i	$-0.388682\pi$
$947$	1.89117e7	0.685261	0.342631	+	0.939470i	$0.388682\pi$
$948$	0	0
$949$	−2.73869e7	−0.987138
$950$	0	0
$951$	−2.91000e7	−1.04338
$952$	0	0
$953$	3.40143e7	1.21319	0.606595	−	0.795011i	$-0.292535\pi$
$953$	3.40143e7	1.21319	0.606595	+	0.795011i	$0.292535\pi$
$954$	0	0
$955$	3.45459e7	1.22571
$956$	0	0
$957$	4.27656e7	1.50944
$958$	0	0
$959$	−1.04838e6	−0.0368107
$960$	0	0
$961$	−2.74915e7	−0.960262
$962$	0	0
$963$	−1.69595e7	−0.589315
$964$	0	0
$965$	7.01937e7	2.42650
$966$	0	0
$967$	6.27887e6	0.215931	0.107966	−	0.994155i	$-0.465566\pi$
$967$	6.27887e6	0.215931	0.107966	+	0.994155i	$0.465566\pi$
$968$	0	0
$969$	8.71892e6	0.298300
$970$	0	0
$971$	2.00595e7	0.682765	0.341383	−	0.939924i	$-0.389105\pi$
$971$	2.00595e7	0.682765	0.341383	+	0.939924i	$0.389105\pi$
$972$	0	0
$973$	−849823.	−0.0287771
$974$	0	0
$975$	2.39606e7	0.807209
$976$	0	0
$977$	−4.81426e7	−1.61359	−0.806795	−	0.590831i	$-0.798800\pi$
$977$	−4.81426e7	−1.61359	−0.806795	+	0.590831i	$0.798800\pi$
$978$	0	0
$979$	−5.83169e7	−1.94463
$980$	0	0
$981$	3.24939e6	0.107803
$982$	0	0
$983$	5.45980e7	1.80216	0.901079	−	0.433656i	$-0.142777\pi$
$983$	5.45980e7	1.80216	0.901079	+	0.433656i	$0.142777\pi$
$984$	0	0
$985$	7.12136e7	2.33869
$986$	0	0
$987$	1.65726e6	0.0541498
$988$	0	0
$989$	7.15727e6	0.232679
$990$	0	0
$991$	−1.21280e7	−0.392288	−0.196144	−	0.980575i	$-0.562842\pi$
$991$	−1.21280e7	−0.392288	−0.196144	+	0.980575i	$0.562842\pi$
$992$	0	0
$993$	2.35021e7	0.756368
$994$	0	0
$995$	7.95484e7	2.54726
$996$	0	0
$997$	−4.49074e6	−0.143080	−0.0715401	−	0.997438i	$-0.522791\pi$
$997$	−4.49074e6	−0.143080	−0.0715401	+	0.997438i	$0.522791\pi$
$998$	0	0
$999$	612563.	0.0194195

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	1104.6.a.r.1.1		5
4.3	odd	2		69.6.a.e.1.2	✓	5
12.11	even	2		207.6.a.f.1.4		5

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
69.6.a.e.1.2	✓	5	4.3	odd	2
207.6.a.f.1.4		5	12.11	even	2
1104.6.a.r.1.1		5	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)

\(f(q)\)	\(=\)	\(q-9.00000 q^{3} -92.1306 q^{5} +8.98894 q^{7} +81.0000 q^{9} +O(q^{10})\)	\(q-9.00000 q^{3} -92.1306 q^{5} +8.98894 q^{7} +81.0000 q^{9} -612.746 q^{11} -496.413 q^{13} +829.175 q^{15} +745.481 q^{17} -1299.52 q^{19} -80.9005 q^{21} +529.000 q^{23} +5363.05 q^{25} -729.000 q^{27} +7754.82 q^{29} +1066.62 q^{31} +5514.72 q^{33} -828.157 q^{35} -840.279 q^{37} +4467.72 q^{39} +9821.00 q^{41} +13529.8 q^{43} -7462.58 q^{45} -20485.1 q^{47} -16726.2 q^{49} -6709.33 q^{51} +4816.85 q^{53} +56452.7 q^{55} +11695.7 q^{57} -30528.3 q^{59} +9.53050 q^{61} +728.104 q^{63} +45734.8 q^{65} -9996.72 q^{67} -4761.00 q^{69} +73725.9 q^{71} +55169.6 q^{73} -48267.4 q^{75} -5507.94 q^{77} -62397.4 q^{79} +6561.00 q^{81} +82435.2 q^{83} -68681.6 q^{85} -69793.4 q^{87} +95173.0 q^{89} -4462.23 q^{91} -9599.57 q^{93} +119726. q^{95} +47134.4 q^{97} -49632.4 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 5 q - 45 q^{3} + 94 q^{5} - 272 q^{7} + 405 q^{9}+O(q^{10}) \) 5 * q - 45 * q^3 + 94 * q^5 - 272 * q^7 + 405 * q^9	\( 5 q - 45 q^{3} + 94 q^{5} - 272 q^{7} + 405 q^{9} - 1100 q^{11} - 978 q^{13} - 846 q^{15} + 2522 q^{17} - 2060 q^{19} + 2448 q^{21} + 2645 q^{23} + 12035 q^{25} - 3645 q^{27} + 1526 q^{29} + 7392 q^{31} + 9900 q^{33} - 6056 q^{35} - 8210 q^{37} + 8802 q^{39} + 21250 q^{41} + 4548 q^{43} + 7614 q^{45} - 536 q^{47} - 27979 q^{49} - 22698 q^{51} - 11482 q^{53} + 77064 q^{55} + 18540 q^{57} - 74676 q^{59} - 44618 q^{61} - 22032 q^{63} - 24388 q^{65} + 1412 q^{67} - 23805 q^{69} - 37912 q^{71} + 46546 q^{73} - 108315 q^{75} + 157008 q^{77} - 50544 q^{79} + 32805 q^{81} - 89588 q^{83} + 147892 q^{85} - 13734 q^{87} + 280410 q^{89} + 27416 q^{91} - 66528 q^{93} - 203120 q^{95} + 90074 q^{97} - 89100 q^{99}+O(q^{100}) \) 5 * q - 45 * q^3 + 94 * q^5 - 272 * q^7 + 405 * q^9 - 1100 * q^11 - 978 * q^13 - 846 * q^15 + 2522 * q^17 - 2060 * q^19 + 2448 * q^21 + 2645 * q^23 + 12035 * q^25 - 3645 * q^27 + 1526 * q^29 + 7392 * q^31 + 9900 * q^33 - 6056 * q^35 - 8210 * q^37 + 8802 * q^39 + 21250 * q^41 + 4548 * q^43 + 7614 * q^45 - 536 * q^47 - 27979 * q^49 - 22698 * q^51 - 11482 * q^53 + 77064 * q^55 + 18540 * q^57 - 74676 * q^59 - 44618 * q^61 - 22032 * q^63 - 24388 * q^65 + 1412 * q^67 - 23805 * q^69 - 37912 * q^71 + 46546 * q^73 - 108315 * q^75 + 157008 * q^77 - 50544 * q^79 + 32805 * q^81 - 89588 * q^83 + 147892 * q^85 - 13734 * q^87 + 280410 * q^89 + 27416 * q^91 - 66528 * q^93 - 203120 * q^95 + 90074 * q^97 - 89100 * q^99

Introduction

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