LMFDB - Embedded newform 45.10.a.e.1.2

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [45,10,Mod(1,45)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(45, base_ring=CyclotomicField(2))

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

N = Newforms(chi, 10, names="a")

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

chi := DirichletCharacter("45.1");

S:= CuspForms(chi, 10);

N := Newforms(S);

Level:	$ N $	$=$	$ 45 = 3^{2} \cdot 5 $
Weight:	$ k $	$=$	$ 10 $
Character orbit:	$[\chi]$	$=$	45.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:	$23.1766126274$
Analytic rank:	$1$
Dimension:	$2$
Coefficient field:	$\Q(\sqrt{4729}) $
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{2} - x - 1182 $ x^2 - x - 1182
Coefficient ring:	$\Z[a_1, a_2]$
Coefficient ring index:	$ 1 $
Twist minimal:	no (minimal twist has level 15)
Fricke sign:	$+1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.2
Root			$-33.8839$ of defining polynomial
Character	$\chi$	$=$	45.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	\(q+24.8839 q^{2} +107.207 q^{4} +625.000 q^{5} -4010.50 q^{7} -10072.8 q^{8} +15552.4 q^{10} -84861.3 q^{11} +119425. q^{13} -99796.8 q^{14} -305541. q^{16} -116934. q^{17} -234932. q^{19} +67004.1 q^{20} -2.11168e6 q^{22} -2.34570e6 q^{23} +390625. q^{25} +2.97176e6 q^{26} -429953. q^{28} +464196. q^{29} -5.11766e6 q^{31} -2.44574e6 q^{32} -2.90976e6 q^{34} -2.50656e6 q^{35} +8.69354e6 q^{37} -5.84601e6 q^{38} -6.29551e6 q^{40} +9.05805e6 q^{41} +8.63491e6 q^{43} -9.09769e6 q^{44} -5.83701e7 q^{46} -3.31511e7 q^{47} -2.42695e7 q^{49} +9.72026e6 q^{50} +1.28032e7 q^{52} +6.41254e7 q^{53} -5.30383e7 q^{55} +4.03971e7 q^{56} +1.15510e7 q^{58} -1.49407e8 q^{59} +1.54634e8 q^{61} -1.27347e8 q^{62} +9.55772e7 q^{64} +7.46408e7 q^{65} +2.72755e8 q^{67} -1.25360e7 q^{68} -6.23730e7 q^{70} +3.56924e8 q^{71} +2.06253e8 q^{73} +2.16329e8 q^{74} -2.51862e7 q^{76} +3.40337e8 q^{77} -4.04380e8 q^{79} -1.90963e8 q^{80} +2.25399e8 q^{82} +5.17034e6 q^{83} -7.30834e7 q^{85} +2.14870e8 q^{86} +8.54793e8 q^{88} -4.32242e8 q^{89} -4.78955e8 q^{91} -2.51475e8 q^{92} -8.24928e8 q^{94} -1.46832e8 q^{95} -1.32066e9 q^{97} -6.03918e8 q^{98} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 2 q - 19 q^{2} + 1521 q^{4} + 1250 q^{5} - 11872 q^{7} - 49647 q^{8} - 11875 q^{10} - 35488 q^{11} + 143676 q^{13} + 245196 q^{14} + 707265 q^{16} - 385156 q^{17} - 403296 q^{19} + 950625 q^{20} - 4278368 q^{22}+ \cdots - 1545205739 q^{98}+O(q^{100}) $ 2 * q - 19 * q^2 + 1521 * q^4 + 1250 * q^5 - 11872 * q^7 - 49647 * q^8 - 11875 * q^10 - 35488 * q^11 + 143676 * q^13 + 245196 * q^14 + 707265 * q^16 - 385156 * q^17 - 403296 * q^19 + 950625 * q^20 - 4278368 * q^22 - 223704 * q^23 + 781250 * q^25 + 1907546 * q^26 - 11544484 * q^28 + 74572 * q^29 - 5027128 * q^31 - 26629583 * q^32 + 8860882 * q^34 - 7420000 * q^35 + 5373628 * q^37 + 1542476 * q^38 - 31029375 * q^40 - 14211332 * q^41 + 27748920 * q^43 + 60705952 * q^44 - 151491648 * q^46 - 95966440 * q^47 - 2819950 * q^49 - 7421875 * q^50 + 47088706 * q^52 + 64305596 * q^53 - 22180000 * q^55 + 351509340 * q^56 + 28649198 * q^58 - 187863136 * q^59 + 154080060 * q^61 - 131320056 * q^62 + 638301089 * q^64 + 89797500 * q^65 + 33592376 * q^67 - 391747238 * q^68 + 153247500 * q^70 + 228270976 * q^71 - 33122316 * q^73 + 362019226 * q^74 - 263218724 * q^76 - 47811456 * q^77 - 932406760 * q^79 + 442040625 * q^80 + 1246549646 * q^82 - 207040152 * q^83 - 240722500 * q^85 - 623926708 * q^86 - 1099114848 * q^88 - 224518164 * q^89 - 669602528 * q^91 + 2748592992 * q^92 + 1931650816 * q^94 - 252060000 * q^95 + 387134596 * q^97 - 1545205739 * q^98

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 $

$2$	24.8839	1.09972	0.549861	−	0.835256i	$-0.314681\pi$
$2$	24.8839	1.09972	0.549861	+	0.835256i	$0.314681\pi$
$3$	0	0
$3$	0	0
$4$	107.207	0.209388
$5$	625.000	0.447214
$5$	625.000	0.447214
$6$	0	0
$7$	−4010.50	−0.631332	−0.315666	−	0.948870i	$-0.602228\pi$
$7$	−4010.50	−0.631332	−0.315666	+	0.948870i	$0.602228\pi$
$8$	−10072.8	−0.869453
$9$	0	0
$10$	15552.4	0.491811
$11$	−84861.3	−1.74760	−0.873801	−	0.486283i	$-0.838352\pi$
$11$	−84861.3	−1.74760	−0.873801	+	0.486283i	$0.838352\pi$
$12$	0	0
$13$	119425.	1.15971	0.579857	−	0.814718i	$-0.303108\pi$
$13$	119425.	1.15971	0.579857	+	0.814718i	$0.303108\pi$
$14$	−99796.8	−0.694289
$15$	0	0
$16$	−305541.	−1.16554
$17$	−116934.	−0.339562	−0.169781	−	0.985482i	$-0.554306\pi$
$17$	−116934.	−0.339562	−0.169781	+	0.985482i	$0.554306\pi$
$18$	0	0
$19$	−234932.	−0.413571	−0.206786	−	0.978386i	$-0.566300\pi$
$19$	−234932.	−0.413571	−0.206786	+	0.978386i	$0.566300\pi$
$20$	67004.1	0.0936411
$21$	0	0
$22$	−2.11168e6	−1.92188
$23$	−2.34570e6	−1.74782	−0.873912	−	0.486084i	$-0.838425\pi$
$23$	−2.34570e6	−1.74782	−0.873912	+	0.486084i	$0.838425\pi$
$24$	0	0
$25$	390625.	0.200000
$26$	2.97176e6	1.27536
$27$	0	0
$28$	−429953.	−0.132193
$29$	464196.	0.121874	0.0609369	−	0.998142i	$-0.480591\pi$
$29$	464196.	0.121874	0.0609369	+	0.998142i	$0.480591\pi$
$30$	0	0
$31$	−5.11766e6	−0.995277	−0.497638	−	0.867385i	$-0.665799\pi$
$31$	−5.11766e6	−0.995277	−0.497638	+	0.867385i	$0.665799\pi$
$32$	−2.44574e6	−0.412321
$33$	0	0
$34$	−2.90976e6	−0.373423
$35$	−2.50656e6	−0.282340
$36$	0	0
$37$	8.69354e6	0.762586	0.381293	−	0.924454i	$-0.375479\pi$
$37$	8.69354e6	0.762586	0.381293	+	0.924454i	$0.375479\pi$
$38$	−5.84601e6	−0.454813
$39$	0	0
$40$	−6.29551e6	−0.388831
$41$	9.05805e6	0.500619	0.250310	−	0.968166i	$-0.419468\pi$
$41$	9.05805e6	0.500619	0.250310	+	0.968166i	$0.419468\pi$
$42$	0	0
$43$	8.63491e6	0.385168	0.192584	−	0.981281i	$-0.438313\pi$
$43$	8.63491e6	0.385168	0.192584	+	0.981281i	$0.438313\pi$
$44$	−9.09769e6	−0.365927
$45$	0	0
$46$	−5.83701e7	−1.92212
$47$	−3.31511e7	−0.990964	−0.495482	−	0.868618i	$-0.665009\pi$
$47$	−3.31511e7	−0.990964	−0.495482	+	0.868618i	$0.665009\pi$
$48$	0	0
$49$	−2.42695e7	−0.601420
$50$	9.72026e6	0.219944
$51$	0	0
$52$	1.28032e7	0.242830
$53$	6.41254e7	1.11632	0.558160	−	0.829733i	$-0.311508\pi$
$53$	6.41254e7	1.11632	0.558160	+	0.829733i	$0.311508\pi$
$54$	0	0
$55$	−5.30383e7	−0.781551
$56$	4.03971e7	0.548914
$57$	0	0
$58$	1.15510e7	0.134027
$59$	−1.49407e8	−1.60523	−0.802613	−	0.596500i	$-0.796558\pi$
$59$	−1.49407e8	−1.60523	−0.802613	+	0.596500i	$0.796558\pi$
$60$	0	0
$61$	1.54634e8	1.42995	0.714973	−	0.699152i	$-0.246439\pi$
$61$	1.54634e8	1.42995	0.714973	+	0.699152i	$0.246439\pi$
$62$	−1.27347e8	−1.09453
$63$	0	0
$64$	9.55772e7	0.712106
$65$	7.46408e7	0.518640
$66$	0	0
$67$	2.72755e8	1.65362	0.826811	−	0.562479i	$-0.190152\pi$
$67$	2.72755e8	1.65362	0.826811	+	0.562479i	$0.190152\pi$
$68$	−1.25360e7	−0.0711001
$69$	0	0
$70$	−6.23730e7	−0.310496
$71$	3.56924e8	1.66691	0.833457	−	0.552584i	$-0.186358\pi$
$71$	3.56924e8	1.66691	0.833457	+	0.552584i	$0.186358\pi$
$72$	0	0
$73$	2.06253e8	0.850057	0.425029	−	0.905180i	$-0.360264\pi$
$73$	2.06253e8	0.850057	0.425029	+	0.905180i	$0.360264\pi$
$74$	2.16329e8	0.838632
$75$	0	0
$76$	−2.51862e7	−0.0865968
$77$	3.40337e8	1.10332
$78$	0	0
$79$	−4.04380e8	−1.16807	−0.584033	−	0.811730i	$-0.698526\pi$
$79$	−4.04380e8	−1.16807	−0.584033	+	0.811730i	$0.698526\pi$
$80$	−1.90963e8	−0.521247
$81$	0	0
$82$	2.25399e8	0.550542
$83$	5.17034e6	0.0119582	0.00597912	−	0.999982i	$-0.498097\pi$
$83$	5.17034e6	0.0119582	0.00597912	+	0.999982i	$0.498097\pi$
$84$	0	0
$85$	−7.30834e7	−0.151857
$86$	2.14870e8	0.423577
$87$	0	0
$88$	8.54793e8	1.51946
$89$	−4.32242e8	−0.730250	−0.365125	−	0.930958i	$-0.618974\pi$
$89$	−4.32242e8	−0.730250	−0.365125	+	0.930958i	$0.618974\pi$
$90$	0	0
$91$	−4.78955e8	−0.732165
$92$	−2.51475e8	−0.365973
$93$	0	0
$94$	−8.24928e8	−1.08978
$95$	−1.46832e8	−0.184955
$96$	0	0
$97$	−1.32066e9	−1.51468	−0.757338	−	0.653023i	$-0.773501\pi$
$97$	−1.32066e9	−1.51468	−0.757338	+	0.653023i	$0.773501\pi$
$98$	−6.03918e8	−0.661395
$99$	0	0
$100$	4.18776e7	0.0418776
$101$	−5.30458e8	−0.507230	−0.253615	−	0.967305i	$-0.581620\pi$
$101$	−5.30458e8	−0.507230	−0.253615	+	0.967305i	$0.581620\pi$
$102$	0	0
$103$	−6.07207e7	−0.0531580	−0.0265790	−	0.999647i	$-0.508461\pi$
$103$	−6.07207e7	−0.0531580	−0.0265790	+	0.999647i	$0.508461\pi$
$104$	−1.20295e9	−1.00832
$105$	0	0
$106$	1.59569e9	1.22764
$107$	1.00828e9	0.743625	0.371812	−	0.928308i	$-0.378736\pi$
$107$	1.00828e9	0.743625	0.371812	+	0.928308i	$0.378736\pi$
$108$	0	0
$109$	−1.77424e9	−1.20391	−0.601954	−	0.798531i	$-0.705611\pi$
$109$	−1.77424e9	−1.20391	−0.601954	+	0.798531i	$0.705611\pi$
$110$	−1.31980e9	−0.859489
$111$	0	0
$112$	1.22537e9	0.735846
$113$	−9.45495e8	−0.545514	−0.272757	−	0.962083i	$-0.587936\pi$
$113$	−9.45495e8	−0.545514	−0.272757	+	0.962083i	$0.587936\pi$
$114$	0	0
$115$	−1.46606e9	−0.781651
$116$	4.97649e7	0.0255189
$117$	0	0
$118$	−3.71782e9	−1.76530
$119$	4.68962e8	0.214376
$120$	0	0
$121$	4.84349e9	2.05411
$122$	3.84788e9	1.57254
$123$	0	0
$124$	−5.48647e8	−0.208399
$125$	2.44141e8	0.0894427
$126$	0	0
$127$	−5.19758e9	−1.77290	−0.886450	−	0.462825i	$-0.846836\pi$
$127$	−5.19758e9	−1.77290	−0.886450	+	0.462825i	$0.846836\pi$
$128$	3.63055e9	1.19544
$129$	0	0
$130$	1.85735e9	0.570360
$131$	−5.28408e8	−0.156765	−0.0783824	−	0.996923i	$-0.524976\pi$
$131$	−5.28408e8	−0.156765	−0.0783824	+	0.996923i	$0.524976\pi$
$132$	0	0
$133$	9.42194e8	0.261101
$134$	6.78720e9	1.81852
$135$	0	0
$136$	1.17785e9	0.295233
$137$	−5.01761e9	−1.21690	−0.608449	−	0.793593i	$-0.708208\pi$
$137$	−5.01761e9	−1.21690	−0.608449	+	0.793593i	$0.708208\pi$
$138$	0	0
$139$	3.51872e9	0.799499	0.399750	−	0.916624i	$-0.369097\pi$
$139$	3.51872e9	0.799499	0.399750	+	0.916624i	$0.369097\pi$
$140$	−2.68720e8	−0.0591186
$141$	0	0
$142$	8.88165e9	1.83314
$143$	−1.01346e10	−2.02672
$144$	0	0
$145$	2.90123e8	0.0545036
$146$	5.13238e9	0.934827
$147$	0	0
$148$	9.32005e8	0.159676
$149$	−4.32815e9	−0.719390	−0.359695	−	0.933070i	$-0.617119\pi$
$149$	−4.32815e9	−0.719390	−0.359695	+	0.933070i	$0.617119\pi$
$150$	0	0
$151$	−5.61832e9	−0.879448	−0.439724	−	0.898133i	$-0.644924\pi$
$151$	−5.61832e9	−0.879448	−0.439724	+	0.898133i	$0.644924\pi$
$152$	2.36642e9	0.359581
$153$	0	0
$154$	8.46889e9	1.21334
$155$	−3.19854e9	−0.445101
$156$	0	0
$157$	1.55603e9	0.204394	0.102197	−	0.994764i	$-0.467413\pi$
$157$	1.55603e9	0.204394	0.102197	+	0.994764i	$0.467413\pi$
$158$	−1.00625e10	−1.28455
$159$	0	0
$160$	−1.52859e9	−0.184396
$161$	9.40745e9	1.10346
$162$	0	0
$163$	−1.15580e10	−1.28245	−0.641224	−	0.767354i	$-0.721573\pi$
$163$	−1.15580e10	−1.28245	−0.641224	+	0.767354i	$0.721573\pi$
$164$	9.71083e8	0.104824
$165$	0	0
$166$	1.28658e8	0.0131507
$167$	1.50486e10	1.49717	0.748585	−	0.663039i	$-0.230734\pi$
$167$	1.50486e10	1.49717	0.748585	+	0.663039i	$0.230734\pi$
$168$	0	0
$169$	3.65789e9	0.344938
$170$	−1.81860e9	−0.167000
$171$	0	0
$172$	9.25719e8	0.0806494
$173$	−2.23157e10	−1.89410	−0.947052	−	0.321081i	$-0.895954\pi$
$173$	−2.23157e10	−1.89410	−0.947052	+	0.321081i	$0.895954\pi$
$174$	0	0
$175$	−1.56660e9	−0.126266
$176$	2.59286e10	2.03691
$177$	0	0
$178$	−1.07558e10	−0.803072
$179$	1.73543e10	1.26348	0.631742	−	0.775179i	$-0.282340\pi$
$179$	1.73543e10	1.26348	0.631742	+	0.775179i	$0.282340\pi$
$180$	0	0
$181$	−1.34040e10	−0.928282	−0.464141	−	0.885761i	$-0.653637\pi$
$181$	−1.34040e10	−0.928282	−0.464141	+	0.885761i	$0.653637\pi$
$182$	−1.19183e10	−0.805178
$183$	0	0
$184$	2.36278e10	1.51965
$185$	5.43346e9	0.341039
$186$	0	0
$187$	9.92313e9	0.593419
$188$	−3.55402e9	−0.207496
$189$	0	0
$190$	−3.65375e9	−0.203399
$191$	6.01312e9	0.326926	0.163463	−	0.986549i	$-0.447734\pi$
$191$	6.01312e9	0.326926	0.163463	+	0.986549i	$0.447734\pi$
$192$	0	0
$193$	−6.91844e9	−0.358922	−0.179461	−	0.983765i	$-0.557435\pi$
$193$	−6.91844e9	−0.358922	−0.179461	+	0.983765i	$0.557435\pi$
$194$	−3.28632e10	−1.66572
$195$	0	0
$196$	−2.60185e9	−0.125930
$197$	−1.66139e10	−0.785909	−0.392955	−	0.919558i	$-0.628547\pi$
$197$	−1.66139e10	−0.785909	−0.392955	+	0.919558i	$0.628547\pi$
$198$	0	0
$199$	−3.06711e10	−1.38641	−0.693204	−	0.720742i	$-0.743801\pi$
$199$	−3.06711e10	−1.38641	−0.693204	+	0.720742i	$0.743801\pi$
$200$	−3.93470e9	−0.173891
$201$	0	0
$202$	−1.31998e10	−0.557812
$203$	−1.86166e9	−0.0769428
$204$	0	0
$205$	5.66128e9	0.223884
$206$	−1.51096e9	−0.0584591
$207$	0	0
$208$	−3.64893e10	−1.35170
$209$	1.99366e10	0.722758
$210$	0	0
$211$	3.85598e10	1.33926	0.669628	−	0.742697i	$-0.266454\pi$
$211$	3.85598e10	1.33926	0.669628	+	0.742697i	$0.266454\pi$
$212$	6.87467e9	0.233744
$213$	0	0
$214$	2.50899e10	0.817780
$215$	5.39682e9	0.172252
$216$	0	0
$217$	2.05244e10	0.628350
$218$	−4.41500e10	−1.32396
$219$	0	0
$220$	−5.68606e9	−0.163647
$221$	−1.39648e10	−0.393795
$222$	0	0
$223$	−3.11357e10	−0.843115	−0.421558	−	0.906802i	$-0.638516\pi$
$223$	−3.11357e10	−0.843115	−0.421558	+	0.906802i	$0.638516\pi$
$224$	9.80866e9	0.260312
$225$	0	0
$226$	−2.35276e10	−0.599914
$227$	1.14589e10	0.286436	0.143218	−	0.989691i	$-0.454255\pi$
$227$	1.14589e10	0.286436	0.143218	+	0.989691i	$0.454255\pi$
$228$	0	0
$229$	−3.04556e10	−0.731825	−0.365913	−	0.930649i	$-0.619243\pi$
$229$	−3.04556e10	−0.731825	−0.365913	+	0.930649i	$0.619243\pi$
$230$	−3.64813e10	−0.859598
$231$	0	0
$232$	−4.67576e9	−0.105964
$233$	−2.83630e9	−0.0630451	−0.0315225	−	0.999503i	$-0.510036\pi$
$233$	−2.83630e9	−0.0630451	−0.0315225	+	0.999503i	$0.510036\pi$
$234$	0	0
$235$	−2.07195e10	−0.443173
$236$	−1.60174e10	−0.336115
$237$	0	0
$238$	1.16696e10	0.235754
$239$	6.25862e10	1.24076	0.620381	−	0.784301i	$-0.286978\pi$
$239$	6.25862e10	1.24076	0.620381	+	0.784301i	$0.286978\pi$
$240$	0	0
$241$	7.24015e10	1.38252	0.691259	−	0.722607i	$-0.257056\pi$
$241$	7.24015e10	1.38252	0.691259	+	0.722607i	$0.257056\pi$
$242$	1.20525e11	2.25895
$243$	0	0
$244$	1.65778e10	0.299414
$245$	−1.51684e10	−0.268963
$246$	0	0
$247$	−2.80568e10	−0.479624
$248$	5.15493e10	0.865347
$249$	0	0
$250$	6.07516e9	0.0983621
$251$	5.65927e10	0.899971	0.449986	−	0.893036i	$-0.351429\pi$
$251$	5.65927e10	0.899971	0.449986	+	0.893036i	$0.351429\pi$
$252$	0	0
$253$	1.99059e11	3.05450
$254$	−1.29336e11	−1.94970
$255$	0	0
$256$	4.14066e10	0.602545
$257$	−4.95688e10	−0.708777	−0.354388	−	0.935098i	$-0.615311\pi$
$257$	−4.95688e10	−0.708777	−0.354388	+	0.935098i	$0.615311\pi$
$258$	0	0
$259$	−3.48655e10	−0.481445
$260$	8.00199e9	0.108597
$261$	0	0
$262$	−1.31488e10	−0.172398
$263$	−3.59498e10	−0.463336	−0.231668	−	0.972795i	$-0.574418\pi$
$263$	−3.59498e10	−0.463336	−0.231668	+	0.972795i	$0.574418\pi$
$264$	0	0
$265$	4.00784e10	0.499233
$266$	2.34454e10	0.287138
$267$	0	0
$268$	2.92412e10	0.346249
$269$	7.09650e10	0.826340	0.413170	−	0.910654i	$-0.364421\pi$
$269$	7.09650e10	0.826340	0.413170	+	0.910654i	$0.364421\pi$
$270$	0	0
$271$	1.47201e11	1.65786	0.828930	−	0.559352i	$-0.188950\pi$
$271$	1.47201e11	1.65786	0.828930	+	0.559352i	$0.188950\pi$
$272$	3.57279e10	0.395774
$273$	0	0
$274$	−1.24857e11	−1.33825
$275$	−3.31489e10	−0.349520
$276$	0	0
$277$	−2.87336e10	−0.293245	−0.146622	−	0.989193i	$-0.546840\pi$
$277$	−2.87336e10	−0.293245	−0.146622	+	0.989193i	$0.546840\pi$
$278$	8.75593e10	0.879227
$279$	0	0
$280$	2.52482e10	0.245482
$281$	−5.17071e10	−0.494734	−0.247367	−	0.968922i	$-0.579565\pi$
$281$	−5.17071e10	−0.494734	−0.247367	+	0.968922i	$0.579565\pi$
$282$	0	0
$283$	−3.22434e9	−0.0298814	−0.0149407	−	0.999888i	$-0.504756\pi$
$283$	−3.22434e9	−0.0298814	−0.0149407	+	0.999888i	$0.504756\pi$
$284$	3.82646e10	0.349032
$285$	0	0
$286$	−2.52188e11	−2.22883
$287$	−3.63273e10	−0.316057
$288$	0	0
$289$	−1.04914e11	−0.884698
$290$	7.21937e9	0.0599388
$291$	0	0
$292$	2.21117e10	0.177992
$293$	1.13591e11	0.900409	0.450204	−	0.892926i	$-0.351351\pi$
$293$	1.13591e11	0.900409	0.450204	+	0.892926i	$0.351351\pi$
$294$	0	0
$295$	−9.33792e10	−0.717879
$296$	−8.75685e10	−0.663033
$297$	0	0
$298$	−1.07701e11	−0.791129
$299$	−2.80136e11	−2.02698
$300$	0	0
$301$	−3.46303e10	−0.243169
$302$	−1.39805e11	−0.967148
$303$	0	0
$304$	7.17811e10	0.482036
$305$	9.66460e10	0.639492
$306$	0	0
$307$	−2.30543e10	−0.148125	−0.0740627	−	0.997254i	$-0.523597\pi$
$307$	−2.30543e10	−0.148125	−0.0740627	+	0.997254i	$0.523597\pi$
$308$	3.64863e10	0.231021
$309$	0	0
$310$	−7.95920e10	−0.489488
$311$	−7.71709e10	−0.467769	−0.233885	−	0.972264i	$-0.575144\pi$
$311$	−7.71709e10	−0.467769	−0.233885	+	0.972264i	$0.575144\pi$
$312$	0	0
$313$	−4.68832e10	−0.276101	−0.138050	−	0.990425i	$-0.544084\pi$
$313$	−4.68832e10	−0.276101	−0.138050	+	0.990425i	$0.544084\pi$
$314$	3.87200e10	0.224777
$315$	0	0
$316$	−4.33522e10	−0.244579
$317$	−2.19420e11	−1.22042	−0.610210	−	0.792240i	$-0.708915\pi$
$317$	−2.19420e11	−1.22042	−0.610210	+	0.792240i	$0.708915\pi$
$318$	0	0
$319$	−3.93923e10	−0.212987
$320$	5.97358e10	0.318463
$321$	0	0
$322$	2.34094e11	1.21350
$323$	2.74714e10	0.140433
$324$	0	0
$325$	4.66505e10	0.231943
$326$	−2.87609e11	−1.41034
$327$	0	0
$328$	−9.12401e10	−0.435265
$329$	1.32953e11	0.625627
$330$	0	0
$331$	3.89075e9	0.0178159	0.00890794	−	0.999960i	$-0.497164\pi$
$331$	3.89075e9	0.0178159	0.00890794	+	0.999960i	$0.497164\pi$
$332$	5.54295e8	0.00250391
$333$	0	0
$334$	3.74466e11	1.64647
$335$	1.70472e11	0.739522
$336$	0	0
$337$	−1.48259e11	−0.626163	−0.313082	−	0.949726i	$-0.601361\pi$
$337$	−1.48259e11	−0.626163	−0.313082	+	0.949726i	$0.601361\pi$
$338$	9.10225e10	0.379336
$339$	0	0
$340$	−7.83503e9	−0.0317969
$341$	4.34291e11	1.73935
$342$	0	0
$343$	2.59171e11	1.01103
$344$	−8.69779e10	−0.334885
$345$	0	0
$346$	−5.55302e11	−2.08299
$347$	1.63695e11	0.606114	0.303057	−	0.952972i	$-0.401993\pi$
$347$	1.63695e11	0.606114	0.303057	+	0.952972i	$0.401993\pi$
$348$	0	0
$349$	7.02057e10	0.253313	0.126657	−	0.991947i	$-0.459575\pi$
$349$	7.02057e10	0.253313	0.126657	+	0.991947i	$0.459575\pi$
$350$	−3.89831e10	−0.138858
$351$	0	0
$352$	2.07549e11	0.720574
$353$	1.55727e10	0.0533798	0.0266899	−	0.999644i	$-0.491503\pi$
$353$	1.55727e10	0.0533798	0.0266899	+	0.999644i	$0.491503\pi$
$354$	0	0
$355$	2.23078e11	0.745467
$356$	−4.63392e10	−0.152906
$357$	0	0
$358$	4.31843e11	1.38948
$359$	−3.28525e10	−0.104386	−0.0521931	−	0.998637i	$-0.516621\pi$
$359$	−3.28525e10	−0.104386	−0.0521931	+	0.998637i	$0.516621\pi$
$360$	0	0
$361$	−2.67495e11	−0.828959
$362$	−3.33543e11	−1.02085
$363$	0	0
$364$	−5.13472e10	−0.153306
$365$	1.28908e11	0.380157
$366$	0	0
$367$	−4.76030e11	−1.36974	−0.684868	−	0.728667i	$-0.740140\pi$
$367$	−4.76030e11	−1.36974	−0.684868	+	0.728667i	$0.740140\pi$
$368$	7.16707e11	2.03717
$369$	0	0
$370$	1.35205e11	0.375048
$371$	−2.57175e11	−0.704768
$372$	0	0
$373$	−9.72745e10	−0.260201	−0.130101	−	0.991501i	$-0.541530\pi$
$373$	−9.72745e10	−0.260201	−0.130101	+	0.991501i	$0.541530\pi$
$374$	2.46926e11	0.652596
$375$	0	0
$376$	3.33925e11	0.861597
$377$	5.54367e10	0.141339
$378$	0	0
$379$	4.45171e11	1.10828	0.554141	−	0.832423i	$-0.313047\pi$
$379$	4.45171e11	1.10828	0.554141	+	0.832423i	$0.313047\pi$
$380$	−1.57414e10	−0.0387273
$381$	0	0
$382$	1.49630e11	0.359528
$383$	−2.19414e11	−0.521039	−0.260519	−	0.965469i	$-0.583894\pi$
$383$	−2.19414e11	−0.521039	−0.260519	+	0.965469i	$0.583894\pi$
$384$	0	0
$385$	2.12710e11	0.493418
$386$	−1.72157e11	−0.394714
$387$	0	0
$388$	−1.41584e11	−0.317155
$389$	3.71956e11	0.823603	0.411801	−	0.911274i	$-0.364900\pi$
$389$	3.71956e11	0.823603	0.411801	+	0.911274i	$0.364900\pi$
$390$	0	0
$391$	2.74291e11	0.593494
$392$	2.44462e11	0.522907
$393$	0	0
$394$	−4.13417e11	−0.864281
$395$	−2.52737e11	−0.522375
$396$	0	0
$397$	−1.38786e11	−0.280406	−0.140203	−	0.990123i	$-0.544776\pi$
$397$	−1.38786e11	−0.280406	−0.140203	+	0.990123i	$0.544776\pi$
$398$	−7.63216e11	−1.52466
$399$	0	0
$400$	−1.19352e11	−0.233109
$401$	−6.95127e11	−1.34250	−0.671251	−	0.741231i	$-0.734243\pi$
$401$	−6.95127e11	−1.34250	−0.671251	+	0.741231i	$0.734243\pi$
$402$	0	0
$403$	−6.11178e11	−1.15424
$404$	−5.68686e10	−0.106208
$405$	0	0
$406$	−4.63253e10	−0.0846157
$407$	−7.37745e11	−1.33270
$408$	0	0
$409$	9.10800e11	1.60942	0.804708	−	0.593671i	$-0.202322\pi$
$409$	9.10800e11	1.60942	0.804708	+	0.593671i	$0.202322\pi$
$410$	1.40875e11	0.246210
$411$	0	0
$412$	−6.50966e9	−0.0111307
$413$	5.99196e11	1.01343
$414$	0	0
$415$	3.23146e9	0.00534789
$416$	−2.92084e11	−0.478175
$417$	0	0
$418$	4.96100e11	0.794832
$419$	−3.11152e11	−0.493184	−0.246592	−	0.969119i	$-0.579311\pi$
$419$	−3.11152e11	−0.493184	−0.246592	+	0.969119i	$0.579311\pi$
$420$	0	0
$421$	7.63941e11	1.18520	0.592598	−	0.805498i	$-0.298102\pi$
$421$	7.63941e11	1.18520	0.592598	+	0.805498i	$0.298102\pi$
$422$	9.59516e11	1.47281
$423$	0	0
$424$	−6.45924e11	−0.970588
$425$	−4.56771e10	−0.0679124
$426$	0	0
$427$	−6.20159e11	−0.902771
$428$	1.08094e11	0.155706
$429$	0	0
$430$	1.34294e11	0.189429
$431$	−6.54629e10	−0.0913792	−0.0456896	−	0.998956i	$-0.514549\pi$
$431$	−6.54629e10	−0.0913792	−0.0456896	+	0.998956i	$0.514549\pi$
$432$	0	0
$433$	−7.87434e11	−1.07651	−0.538256	−	0.842781i	$-0.680917\pi$
$433$	−7.87434e11	−1.07651	−0.538256	+	0.842781i	$0.680917\pi$
$434$	5.10726e11	0.691010
$435$	0	0
$436$	−1.90210e11	−0.252084
$437$	5.51080e11	0.722850
$438$	0	0
$439$	−8.39211e11	−1.07840	−0.539201	−	0.842177i	$-0.681274\pi$
$439$	−8.39211e11	−1.07840	−0.539201	+	0.842177i	$0.681274\pi$
$440$	5.34246e11	0.679523
$441$	0	0
$442$	−3.47499e11	−0.433065
$443$	1.28252e12	1.58215	0.791074	−	0.611720i	$-0.209522\pi$
$443$	1.28252e12	1.58215	0.791074	+	0.611720i	$0.209522\pi$
$444$	0	0
$445$	−2.70151e11	−0.326578
$446$	−7.74777e11	−0.927192
$447$	0	0
$448$	−3.83313e11	−0.449575
$449$	−8.02030e10	−0.0931284	−0.0465642	−	0.998915i	$-0.514827\pi$
$449$	−8.02030e10	−0.0931284	−0.0465642	+	0.998915i	$0.514827\pi$
$450$	0	0
$451$	−7.68678e11	−0.874883
$452$	−1.01363e11	−0.114224
$453$	0	0
$454$	2.85143e11	0.315000
$455$	−2.99347e11	−0.327434
$456$	0	0
$457$	1.27085e11	0.136292	0.0681461	−	0.997675i	$-0.478292\pi$
$457$	1.27085e11	0.136292	0.0681461	+	0.997675i	$0.478292\pi$
$458$	−7.57853e11	−0.804804
$459$	0	0
$460$	−1.57172e11	−0.163668
$461$	7.39880e11	0.762970	0.381485	−	0.924375i	$-0.375413\pi$
$461$	7.39880e11	0.762970	0.381485	+	0.924375i	$0.375413\pi$
$462$	0	0
$463$	5.71025e11	0.577485	0.288742	−	0.957407i	$-0.406763\pi$
$463$	5.71025e11	0.577485	0.288742	+	0.957407i	$0.406763\pi$
$464$	−1.41831e11	−0.142049
$465$	0	0
$466$	−7.05782e10	−0.0693320
$467$	−1.19075e12	−1.15850	−0.579249	−	0.815150i	$-0.696654\pi$
$467$	−1.19075e12	−1.15850	−0.579249	+	0.815150i	$0.696654\pi$
$468$	0	0
$469$	−1.09389e12	−1.04398
$470$	−5.15580e11	−0.487367
$471$	0	0
$472$	1.50495e12	1.39567
$473$	−7.32770e11	−0.673120
$474$	0	0
$475$	−9.17701e10	−0.0827142
$476$	5.02759e10	0.0448878
$477$	0	0
$478$	1.55739e12	1.36449
$479$	−1.46074e12	−1.26784	−0.633919	−	0.773400i	$-0.718555\pi$
$479$	−1.46074e12	−1.26784	−0.633919	+	0.773400i	$0.718555\pi$
$480$	0	0
$481$	1.03823e12	0.884382
$482$	1.80163e12	1.52038
$483$	0	0
$484$	5.19254e11	0.430107
$485$	−8.25416e11	−0.677384
$486$	0	0
$487$	−1.46931e12	−1.18368	−0.591839	−	0.806056i	$-0.701598\pi$
$487$	−1.46931e12	−1.18368	−0.591839	+	0.806056i	$0.701598\pi$
$488$	−1.55760e12	−1.24327
$489$	0	0
$490$	−3.77449e11	−0.295785
$491$	1.76138e12	1.36769	0.683843	−	0.729629i	$-0.260307\pi$
$491$	1.76138e12	1.36769	0.683843	+	0.729629i	$0.260307\pi$
$492$	0	0
$493$	−5.42801e10	−0.0413837
$494$	−6.98161e11	−0.527453
$495$	0	0
$496$	1.56365e12	1.16004
$497$	−1.43145e12	−1.05238
$498$	0	0
$499$	−1.63016e12	−1.17700	−0.588502	−	0.808496i	$-0.700282\pi$
$499$	−1.63016e12	−1.17700	−0.588502	+	0.808496i	$0.700282\pi$
$500$	2.61735e10	0.0187282
$501$	0	0
$502$	1.40825e12	0.989718
$503$	−9.47625e11	−0.660056	−0.330028	−	0.943971i	$-0.607058\pi$
$503$	−9.47625e11	−0.660056	−0.330028	+	0.943971i	$0.607058\pi$
$504$	0	0
$505$	−3.31536e11	−0.226840
$506$	4.95337e12	3.35910
$507$	0	0
$508$	−5.57214e11	−0.371224
$509$	2.90709e11	0.191968	0.0959838	−	0.995383i	$-0.469400\pi$
$509$	2.90709e11	0.191968	0.0959838	+	0.995383i	$0.469400\pi$
$510$	0	0
$511$	−8.27180e11	−0.536668
$512$	−8.28486e11	−0.532808
$513$	0	0
$514$	−1.23346e12	−0.779457
$515$	−3.79504e10	−0.0237730
$516$	0	0
$517$	2.81325e12	1.73181
$518$	−8.67587e11	−0.529455
$519$	0	0
$520$	−7.51843e11	−0.450933
$521$	2.04811e12	1.21782	0.608910	−	0.793240i	$-0.291607\pi$
$521$	2.04811e12	1.21782	0.608910	+	0.793240i	$0.291607\pi$
$522$	0	0
$523$	8.49299e10	0.0496367	0.0248184	−	0.999692i	$-0.492099\pi$
$523$	8.49299e10	0.0496367	0.0248184	+	0.999692i	$0.492099\pi$
$524$	−5.66488e10	−0.0328247
$525$	0	0
$526$	−8.94570e11	−0.509540
$527$	5.98426e11	0.337958
$528$	0	0
$529$	3.70117e12	2.05489
$530$	9.97305e11	0.549018
$531$	0	0
$532$	1.01009e11	0.0546713
$533$	1.08176e12	0.580575
$534$	0	0
$535$	6.30175e11	0.332559
$536$	−2.74741e12	−1.43775
$537$	0	0
$538$	1.76588e12	0.908745
$539$	2.05954e12	1.05104
$540$	0	0
$541$	−1.29101e12	−0.647949	−0.323974	−	0.946066i	$-0.605019\pi$
$541$	−1.29101e12	−0.647949	−0.323974	+	0.946066i	$0.605019\pi$
$542$	3.66292e12	1.82318
$543$	0	0
$544$	2.85989e11	0.140009
$545$	−1.10890e12	−0.538404
$546$	0	0
$547$	−5.30222e11	−0.253230	−0.126615	−	0.991952i	$-0.540411\pi$
$547$	−5.30222e11	−0.253230	−0.126615	+	0.991952i	$0.540411\pi$
$548$	−5.37921e11	−0.254804
$549$	0	0
$550$	−8.24874e11	−0.384375
$551$	−1.09054e11	−0.0504035
$552$	0	0
$553$	1.62177e12	0.737438
$554$	−7.15002e11	−0.322488
$555$	0	0
$556$	3.77230e11	0.167405
$557$	−8.17386e11	−0.359815	−0.179907	−	0.983684i	$-0.557580\pi$
$557$	−8.17386e11	−0.359815	−0.179907	+	0.983684i	$0.557580\pi$
$558$	0	0
$559$	1.03123e12	0.446684
$560$	7.65857e11	0.329080
$561$	0	0
$562$	−1.28667e12	−0.544069
$563$	1.42197e12	0.596488	0.298244	−	0.954490i	$-0.403599\pi$
$563$	1.42197e12	0.596488	0.298244	+	0.954490i	$0.403599\pi$
$564$	0	0
$565$	−5.90934e11	−0.243961
$566$	−8.02339e10	−0.0328613
$567$	0	0
$568$	−3.59523e12	−1.44930
$569$	1.04614e12	0.418394	0.209197	−	0.977874i	$-0.432915\pi$
$569$	1.04614e12	0.418394	0.209197	+	0.977874i	$0.432915\pi$
$570$	0	0
$571$	2.09508e12	0.824780	0.412390	−	0.911007i	$-0.364694\pi$
$571$	2.09508e12	0.824780	0.412390	+	0.911007i	$0.364694\pi$
$572$	−1.08649e12	−0.424371
$573$	0	0
$574$	−9.03965e11	−0.347575
$575$	−9.16290e11	−0.349565
$576$	0	0
$577$	2.68760e12	1.00942	0.504711	−	0.863288i	$-0.331599\pi$
$577$	2.68760e12	1.00942	0.504711	+	0.863288i	$0.331599\pi$
$578$	−2.61068e12	−0.972921
$579$	0	0
$580$	3.11031e10	0.0114124
$581$	−2.07357e10	−0.00754962
$582$	0	0
$583$	−5.44176e12	−1.95088
$584$	−2.07755e12	−0.739085
$585$	0	0
$586$	2.82658e12	0.990199
$587$	−1.61925e12	−0.562913	−0.281457	−	0.959574i	$-0.590818\pi$
$587$	−1.61925e12	−0.562913	−0.281457	+	0.959574i	$0.590818\pi$
$588$	0	0
$589$	1.20230e12	0.411618
$590$	−2.32364e12	−0.789467
$591$	0	0
$592$	−2.65623e12	−0.888828
$593$	−5.82677e12	−1.93500	−0.967502	−	0.252863i	$-0.918628\pi$
$593$	−5.82677e12	−1.93500	−0.967502	+	0.252863i	$0.918628\pi$
$594$	0	0
$595$	2.93101e11	0.0958719
$596$	−4.64007e11	−0.150632
$597$	0	0
$598$	−6.97087e12	−2.22911
$599$	3.89086e12	1.23488	0.617441	−	0.786617i	$-0.288169\pi$
$599$	3.89086e12	1.23488	0.617441	+	0.786617i	$0.288169\pi$
$600$	0	0
$601$	2.13051e12	0.666115	0.333058	−	0.942907i	$-0.391920\pi$
$601$	2.13051e12	0.666115	0.333058	+	0.942907i	$0.391920\pi$
$602$	−8.61737e11	−0.267418
$603$	0	0
$604$	−6.02321e11	−0.184146
$605$	3.02718e12	0.918628
$606$	0	0
$607$	−2.52733e12	−0.755637	−0.377818	−	0.925880i	$-0.623326\pi$
$607$	−2.52733e12	−0.755637	−0.377818	+	0.925880i	$0.623326\pi$
$608$	5.74582e11	0.170524
$609$	0	0
$610$	2.40493e12	0.703263
$611$	−3.95908e12	−1.14924
$612$	0	0
$613$	1.89788e12	0.542870	0.271435	−	0.962457i	$-0.412502\pi$
$613$	1.89788e12	0.542870	0.271435	+	0.962457i	$0.412502\pi$
$614$	−5.73681e11	−0.162897
$615$	0	0
$616$	−3.42815e12	−0.959283
$617$	1.40480e12	0.390240	0.195120	−	0.980779i	$-0.437490\pi$
$617$	1.40480e12	0.390240	0.195120	+	0.980779i	$0.437490\pi$
$618$	0	0
$619$	−2.81250e12	−0.769989	−0.384995	−	0.922919i	$-0.625797\pi$
$619$	−2.81250e12	−0.769989	−0.384995	+	0.922919i	$0.625797\pi$
$620$	−3.42904e11	−0.0931988
$621$	0	0
$622$	−1.92031e12	−0.514416
$623$	1.73351e12	0.461030
$624$	0	0
$625$	1.52588e11	0.0400000
$626$	−1.16664e12	−0.303634
$627$	0	0
$628$	1.66817e11	0.0427977
$629$	−1.01657e12	−0.258945
$630$	0	0
$631$	−1.92653e12	−0.483775	−0.241888	−	0.970304i	$-0.577767\pi$
$631$	−1.92653e12	−0.483775	−0.241888	+	0.970304i	$0.577767\pi$
$632$	4.07325e12	1.01558
$633$	0	0
$634$	−5.46001e12	−1.34212
$635$	−3.24848e12	−0.792865
$636$	0	0
$637$	−2.89839e12	−0.697476
$638$	−9.80232e11	−0.234226
$639$	0	0
$640$	2.26909e12	0.534617
$641$	7.10318e12	1.66185	0.830924	−	0.556385i	$-0.187812\pi$
$641$	7.10318e12	1.66185	0.830924	+	0.556385i	$0.187812\pi$
$642$	0	0
$643$	3.40733e12	0.786077	0.393038	−	0.919522i	$-0.371424\pi$
$643$	3.40733e12	0.786077	0.393038	+	0.919522i	$0.371424\pi$
$644$	1.00854e12	0.231051
$645$	0	0
$646$	6.83594e11	0.154437
$647$	4.31347e12	0.967737	0.483868	−	0.875141i	$-0.339231\pi$
$647$	4.31347e12	0.967737	0.483868	+	0.875141i	$0.339231\pi$
$648$	0	0
$649$	1.26789e13	2.80530
$650$	1.16084e12	0.255073
$651$	0	0
$652$	−1.23910e12	−0.268529
$653$	2.53169e12	0.544880	0.272440	−	0.962173i	$-0.412169\pi$
$653$	2.53169e12	0.544880	0.272440	+	0.962173i	$0.412169\pi$
$654$	0	0
$655$	−3.30255e11	−0.0701074
$656$	−2.76760e12	−0.583494
$657$	0	0
$658$	3.30838e12	0.688016
$659$	4.43263e12	0.915540	0.457770	−	0.889071i	$-0.348648\pi$
$659$	4.43263e12	0.915540	0.457770	+	0.889071i	$0.348648\pi$
$660$	0	0
$661$	9.79827e11	0.199638	0.0998189	−	0.995006i	$-0.468174\pi$
$661$	9.79827e11	0.199638	0.0998189	+	0.995006i	$0.468174\pi$
$662$	9.68168e10	0.0195925
$663$	0	0
$664$	−5.20799e10	−0.0103971
$665$	5.88871e11	0.116768
$666$	0	0
$667$	−1.08887e12	−0.213014
$668$	1.61331e12	0.313489
$669$	0	0
$670$	4.24200e12	0.813269
$671$	−1.31224e13	−2.49898
$672$	0	0
$673$	7.69434e12	1.44578	0.722892	−	0.690961i	$-0.242812\pi$
$673$	7.69434e12	1.44578	0.722892	+	0.690961i	$0.242812\pi$
$674$	−3.68927e12	−0.688605
$675$	0	0
$676$	3.92151e11	0.0722258
$677$	−2.45111e12	−0.448451	−0.224225	−	0.974537i	$-0.571985\pi$
$677$	−2.45111e12	−0.448451	−0.224225	+	0.974537i	$0.571985\pi$
$678$	0	0
$679$	5.29653e12	0.956264
$680$	7.36157e11	0.132032
$681$	0	0
$682$	1.08068e13	1.91280
$683$	−3.13070e12	−0.550488	−0.275244	−	0.961374i	$-0.588759\pi$
$683$	−3.13070e12	−0.550488	−0.275244	+	0.961374i	$0.588759\pi$
$684$	0	0
$685$	−3.13601e12	−0.544213
$686$	6.44918e12	1.11185
$687$	0	0
$688$	−2.63831e12	−0.448930
$689$	7.65819e12	1.29461
$690$	0	0
$691$	−8.92815e12	−1.48974	−0.744870	−	0.667210i	$-0.767488\pi$
$691$	−8.92815e12	−1.48974	−0.744870	+	0.667210i	$0.767488\pi$
$692$	−2.39239e12	−0.396602
$693$	0	0
$694$	4.07338e12	0.666556
$695$	2.19920e12	0.357547
$696$	0	0
$697$	−1.05919e12	−0.169991
$698$	1.74699e12	0.278574
$699$	0	0
$700$	−1.67950e11	−0.0264387
$701$	1.44295e12	0.225695	0.112847	−	0.993612i	$-0.464003\pi$
$701$	1.44295e12	0.225695	0.112847	+	0.993612i	$0.464003\pi$
$702$	0	0
$703$	−2.04239e12	−0.315383
$704$	−8.11081e12	−1.24448
$705$	0	0
$706$	3.87509e11	0.0587030
$707$	2.12740e12	0.320230
$708$	0	0
$709$	−7.40550e12	−1.10064	−0.550321	−	0.834953i	$-0.685495\pi$
$709$	−7.40550e12	−1.10064	−0.550321	+	0.834953i	$0.685495\pi$
$710$	5.55103e12	0.819806
$711$	0	0
$712$	4.35390e12	0.634919
$713$	1.20045e13	1.73957
$714$	0	0
$715$	−6.33411e12	−0.906377
$716$	1.86050e12	0.264558
$717$	0	0
$718$	−8.17496e11	−0.114796
$719$	3.90289e12	0.544636	0.272318	−	0.962207i	$-0.412210\pi$
$719$	3.90289e12	0.544636	0.272318	+	0.962207i	$0.412210\pi$
$720$	0	0
$721$	2.43520e11	0.0335604
$722$	−6.65631e12	−0.911624
$723$	0	0
$724$	−1.43699e12	−0.194371
$725$	1.81327e11	0.0243748
$726$	0	0
$727$	1.33674e12	0.177477	0.0887384	−	0.996055i	$-0.471716\pi$
$727$	1.33674e12	0.177477	0.0887384	+	0.996055i	$0.471716\pi$
$728$	4.82443e12	0.636583
$729$	0	0
$730$	3.20774e12	0.418067
$731$	−1.00971e12	−0.130788
$732$	0	0
$733$	−5.98863e11	−0.0766231	−0.0383116	−	0.999266i	$-0.512198\pi$
$733$	−5.98863e11	−0.0766231	−0.0383116	+	0.999266i	$0.512198\pi$
$734$	−1.18455e13	−1.50633
$735$	0	0
$736$	5.73699e12	0.720665
$737$	−2.31464e13	−2.88987
$738$	0	0
$739$	−1.40697e13	−1.73534	−0.867671	−	0.497138i	$-0.834384\pi$
$739$	−1.40697e13	−1.73534	−0.867671	+	0.497138i	$0.834384\pi$
$740$	5.82503e11	0.0714094
$741$	0	0
$742$	−6.39951e12	−0.775049
$743$	2.52604e12	0.304082	0.152041	−	0.988374i	$-0.451415\pi$
$743$	2.52604e12	0.304082	0.152041	+	0.988374i	$0.451415\pi$
$744$	0	0
$745$	−2.70510e12	−0.321721
$746$	−2.42056e12	−0.286149
$747$	0	0
$748$	1.06383e12	0.124255
$749$	−4.04371e12	−0.469474
$750$	0	0
$751$	3.78704e12	0.434430	0.217215	−	0.976124i	$-0.430303\pi$
$751$	3.78704e12	0.434430	0.217215	+	0.976124i	$0.430303\pi$
$752$	1.01290e13	1.15501
$753$	0	0
$754$	1.37948e12	0.155433
$755$	−3.51145e12	−0.393301
$756$	0	0
$757$	3.55646e12	0.393629	0.196814	−	0.980441i	$-0.436940\pi$
$757$	3.55646e12	0.393629	0.196814	+	0.980441i	$0.436940\pi$
$758$	1.10776e13	1.21880
$759$	0	0
$760$	1.47902e12	0.160809
$761$	−1.67100e13	−1.80612	−0.903058	−	0.429519i	$-0.858683\pi$
$761$	−1.67100e13	−1.80612	−0.903058	+	0.429519i	$0.858683\pi$
$762$	0	0
$763$	7.11560e12	0.760066
$764$	6.44647e11	0.0684544
$765$	0	0
$766$	−5.45987e12	−0.572997
$767$	−1.78429e13	−1.86160
$768$	0	0
$769$	−7.51512e12	−0.774939	−0.387469	−	0.921883i	$-0.626651\pi$
$769$	−7.51512e12	−0.774939	−0.387469	+	0.921883i	$0.626651\pi$
$770$	5.29306e12	0.542623
$771$	0	0
$772$	−7.41702e11	−0.0751540
$773$	8.52580e12	0.858870	0.429435	−	0.903098i	$-0.358713\pi$
$773$	8.52580e12	0.858870	0.429435	+	0.903098i	$0.358713\pi$
$774$	0	0
$775$	−1.99909e12	−0.199055
$776$	1.33028e13	1.31694
$777$	0	0
$778$	9.25569e12	0.905734
$779$	−2.12802e12	−0.207042
$780$	0	0
$781$	−3.02890e13	−2.91310
$782$	6.82543e12	0.652679
$783$	0	0
$784$	7.41531e12	0.700982
$785$	9.72518e11	0.0914080
$786$	0	0
$787$	3.20742e12	0.298037	0.149018	−	0.988834i	$-0.452389\pi$
$787$	3.20742e12	0.298037	0.149018	+	0.988834i	$0.452389\pi$
$788$	−1.78112e12	−0.164560
$789$	0	0
$790$	−6.28908e12	−0.574467
$791$	3.79191e12	0.344401
$792$	0	0
$793$	1.84672e13	1.65833
$794$	−3.45353e12	−0.308369
$795$	0	0
$796$	−3.28815e12	−0.290297
$797$	1.83036e13	1.60684	0.803421	−	0.595412i	$-0.203011\pi$
$797$	1.83036e13	1.60684	0.803421	+	0.595412i	$0.203011\pi$
$798$	0	0
$799$	3.87648e12	0.336494
$800$	−9.55368e11	−0.0824643
$801$	0	0
$802$	−1.72975e13	−1.47638
$803$	−1.75029e13	−1.48556
$804$	0	0
$805$	5.87966e12	0.493481
$806$	−1.52085e13	−1.26934
$807$	0	0
$808$	5.34321e12	0.441013
$809$	−9.38919e12	−0.770655	−0.385327	−	0.922780i	$-0.625911\pi$
$809$	−9.38919e12	−0.770655	−0.385327	+	0.922780i	$0.625911\pi$
$810$	0	0
$811$	2.03949e13	1.65549	0.827746	−	0.561103i	$-0.189623\pi$
$811$	2.03949e13	1.65549	0.827746	+	0.561103i	$0.189623\pi$
$812$	−1.99582e11	−0.0161109
$813$	0	0
$814$	−1.83579e13	−1.46560
$815$	−7.22377e12	−0.573528
$816$	0	0
$817$	−2.02861e12	−0.159294
$818$	2.26642e13	1.76991
$819$	0	0
$820$	6.06927e11	0.0468785
$821$	1.52259e13	1.16961	0.584803	−	0.811176i	$-0.301172\pi$
$821$	1.52259e13	1.16961	0.584803	+	0.811176i	$0.301172\pi$
$822$	0	0
$823$	−2.70705e12	−0.205682	−0.102841	−	0.994698i	$-0.532793\pi$
$823$	−2.70705e12	−0.205682	−0.102841	+	0.994698i	$0.532793\pi$
$824$	6.11629e11	0.0462184
$825$	0	0
$826$	1.49103e13	1.11449
$827$	−5.67203e10	−0.00421661	−0.00210831	−	0.999998i	$-0.500671\pi$
$827$	−5.67203e10	−0.00421661	−0.00210831	+	0.999998i	$0.500671\pi$
$828$	0	0
$829$	−1.43206e13	−1.05309	−0.526547	−	0.850146i	$-0.676513\pi$
$829$	−1.43206e13	−1.05309	−0.526547	+	0.850146i	$0.676513\pi$
$830$	8.04113e10	0.00588119
$831$	0	0
$832$	1.14143e13	0.825839
$833$	2.83791e12	0.204219
$834$	0	0
$835$	9.40535e12	0.669555
$836$	2.13734e12	0.151337
$837$	0	0
$838$	−7.74265e12	−0.542365
$839$	2.13693e13	1.48889	0.744443	−	0.667686i	$-0.232715\pi$
$839$	2.13693e13	1.48889	0.744443	+	0.667686i	$0.232715\pi$
$840$	0	0
$841$	−1.42917e13	−0.985147
$842$	1.90098e13	1.30339
$843$	0	0
$844$	4.13386e12	0.280424
$845$	2.28618e12	0.154261
$846$	0	0
$847$	−1.94248e13	−1.29683
$848$	−1.95929e13	−1.30112
$849$	0	0
$850$	−1.13662e12	−0.0746847
$851$	−2.03925e13	−1.33287
$852$	0	0
$853$	−1.18123e13	−0.763948	−0.381974	−	0.924173i	$-0.624756\pi$
$853$	−1.18123e13	−0.763948	−0.381974	+	0.924173i	$0.624756\pi$
$854$	−1.54319e13	−0.992797
$855$	0	0
$856$	−1.01562e13	−0.646547
$857$	−2.69459e13	−1.70640	−0.853198	−	0.521587i	$-0.825340\pi$
$857$	−2.69459e13	−1.70640	−0.853198	+	0.521587i	$0.825340\pi$
$858$	0	0
$859$	9.66669e12	0.605771	0.302885	−	0.953027i	$-0.402050\pi$
$859$	9.66669e12	0.605771	0.302885	+	0.953027i	$0.402050\pi$
$860$	5.78575e11	0.0360675
$861$	0	0
$862$	−1.62897e12	−0.100492
$863$	−2.80982e12	−0.172437	−0.0862185	−	0.996276i	$-0.527478\pi$
$863$	−2.80982e12	−0.172437	−0.0862185	+	0.996276i	$0.527478\pi$
$864$	0	0
$865$	−1.39473e13	−0.847069
$866$	−1.95944e13	−1.18386
$867$	0	0
$868$	2.20035e12	0.131569
$869$	3.43162e13	2.04132
$870$	0	0
$871$	3.25739e13	1.91773
$872$	1.78716e13	1.04674
$873$	0	0
$874$	1.37130e13	0.794933
$875$	−9.79127e11	−0.0564680
$876$	0	0
$877$	3.37045e13	1.92393	0.961966	−	0.273169i	$-0.0880718\pi$
$877$	3.37045e13	1.92393	0.961966	+	0.273169i	$0.0880718\pi$
$878$	−2.08828e13	−1.18594
$879$	0	0
$880$	1.62054e13	0.910933
$881$	1.20571e13	0.674297	0.337148	−	0.941452i	$-0.390538\pi$
$881$	1.20571e13	0.674297	0.337148	+	0.941452i	$0.390538\pi$
$882$	0	0
$883$	−1.91595e13	−1.06062	−0.530311	−	0.847803i	$-0.677925\pi$
$883$	−1.91595e13	−1.06062	−0.530311	+	0.847803i	$0.677925\pi$
$884$	−1.49712e12	−0.0824559
$885$	0	0
$886$	3.19140e13	1.73992
$887$	2.24991e13	1.22042	0.610209	−	0.792241i	$-0.291086\pi$
$887$	2.24991e13	1.22042	0.610209	+	0.792241i	$0.291086\pi$
$888$	0	0
$889$	2.08449e13	1.11929
$890$	−6.72241e12	−0.359145
$891$	0	0
$892$	−3.33795e12	−0.176538
$893$	7.78825e12	0.409834
$894$	0	0
$895$	1.08465e13	0.565047
$896$	−1.45603e13	−0.754719
$897$	0	0
$898$	−1.99576e12	−0.102415
$899$	−2.37560e12	−0.121298
$900$	0	0
$901$	−7.49841e12	−0.379060
$902$	−1.91277e13	−0.962128
$903$	0	0
$904$	9.52380e12	0.474299
$905$	−8.37748e12	−0.415140
$906$	0	0
$907$	−2.39724e13	−1.17619	−0.588096	−	0.808791i	$-0.700122\pi$
$907$	−2.39724e13	−1.17619	−0.588096	+	0.808791i	$0.700122\pi$
$908$	1.22847e12	0.0599763
$909$	0	0
$910$	−7.44891e12	−0.360086
$911$	6.54470e12	0.314816	0.157408	−	0.987534i	$-0.449686\pi$
$911$	6.54470e12	0.314816	0.157408	+	0.987534i	$0.449686\pi$
$912$	0	0
$913$	−4.38762e11	−0.0208983
$914$	3.16236e12	0.149884
$915$	0	0
$916$	−3.26504e12	−0.153235
$917$	2.11918e12	0.0989706
$918$	0	0
$919$	2.63348e12	0.121789	0.0608947	−	0.998144i	$-0.480605\pi$
$919$	2.63348e12	0.121789	0.0608947	+	0.998144i	$0.480605\pi$
$920$	1.47674e13	0.679609
$921$	0	0
$922$	1.84111e13	0.839055
$923$	4.26258e13	1.93314
$924$	0	0
$925$	3.39591e12	0.152517
$926$	1.42093e13	0.635073
$927$	0	0
$928$	−1.13530e12	−0.0502512
$929$	−2.94271e13	−1.29621	−0.648106	−	0.761550i	$-0.724439\pi$
$929$	−2.94271e13	−1.29621	−0.648106	+	0.761550i	$0.724439\pi$
$930$	0	0
$931$	5.70166e12	0.248730
$932$	−3.04070e11	−0.0132009
$933$	0	0
$934$	−2.96305e13	−1.27403
$935$	6.20196e12	0.265385
$936$	0	0
$937$	−1.71769e13	−0.727974	−0.363987	−	0.931404i	$-0.618585\pi$
$937$	−1.71769e13	−0.727974	−0.363987	+	0.931404i	$0.618585\pi$
$938$	−2.72201e13	−1.14809
$939$	0	0
$940$	−2.22126e12	−0.0927950
$941$	2.46944e13	1.02670	0.513352	−	0.858178i	$-0.328404\pi$
$941$	2.46944e13	1.02670	0.513352	+	0.858178i	$0.328404\pi$
$942$	0	0
$943$	−2.12475e13	−0.874994
$944$	4.56498e13	1.87096
$945$	0	0
$946$	−1.82341e13	−0.740244
$947$	1.19428e12	0.0482537	0.0241268	−	0.999709i	$-0.492319\pi$
$947$	1.19428e12	0.0482537	0.0241268	+	0.999709i	$0.492319\pi$
$948$	0	0
$949$	2.46319e13	0.985824
$950$	−2.28360e12	−0.0909626
$951$	0	0
$952$	−4.72377e12	−0.186390
$953$	−1.47694e13	−0.580023	−0.290011	−	0.957023i	$-0.593659\pi$
$953$	−1.47694e13	−0.580023	−0.290011	+	0.957023i	$0.593659\pi$
$954$	0	0
$955$	3.75820e12	0.146206
$956$	6.70966e12	0.259800
$957$	0	0
$958$	−3.63489e13	−1.39427
$959$	2.01231e13	0.768267
$960$	0	0
$961$	−2.49176e11	−0.00942435
$962$	2.58351e13	0.972574
$963$	0	0
$964$	7.76192e12	0.289483
$965$	−4.32402e12	−0.160515
$966$	0	0
$967$	−8.95584e12	−0.329372	−0.164686	−	0.986346i	$-0.552661\pi$
$967$	−8.95584e12	−0.329372	−0.164686	+	0.986346i	$0.552661\pi$
$968$	−4.87876e13	−1.78596
$969$	0	0
$970$	−2.05395e13	−0.744934
$971$	2.20587e13	0.796330	0.398165	−	0.917314i	$-0.369647\pi$
$971$	2.20587e13	0.796330	0.398165	+	0.917314i	$0.369647\pi$
$972$	0	0
$973$	−1.41118e13	−0.504749
$974$	−3.65621e13	−1.30172
$975$	0	0
$976$	−4.72469e13	−1.66667
$977$	−3.56972e13	−1.25345	−0.626727	−	0.779239i	$-0.715606\pi$
$977$	−3.56972e13	−1.25345	−0.626727	+	0.779239i	$0.715606\pi$
$978$	0	0
$979$	3.66806e13	1.27619
$980$	−1.62615e12	−0.0563176
$981$	0	0
$982$	4.38300e13	1.50408
$983$	−3.19525e13	−1.09147	−0.545737	−	0.837956i	$-0.683750\pi$
$983$	−3.19525e13	−1.09147	−0.545737	+	0.837956i	$0.683750\pi$
$984$	0	0
$985$	−1.03837e13	−0.351469
$986$	−1.35070e12	−0.0455105
$987$	0	0
$988$	−3.00787e12	−0.100428
$989$	−2.02549e13	−0.673205
$990$	0	0
$991$	1.56883e13	0.516706	0.258353	−	0.966051i	$-0.416820\pi$
$991$	1.56883e13	0.516706	0.258353	+	0.966051i	$0.416820\pi$
$992$	1.25165e13	0.410374
$993$	0	0
$994$	−3.56199e13	−1.15732
$995$	−1.91694e13	−0.620020
$996$	0	0
$997$	6.52145e12	0.209033	0.104517	−	0.994523i	$-0.466670\pi$
$997$	6.52145e12	0.209033	0.104517	+	0.994523i	$0.466670\pi$
$998$	−4.05647e13	−1.29438
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	45.10.a.e.1.2		2
3.2	odd	2		15.10.a.c.1.1	✓	2
5.2	odd	4		225.10.b.g.199.3		4
5.3	odd	4		225.10.b.g.199.2		4
5.4	even	2		225.10.a.j.1.1		2
12.11	even	2		240.10.a.m.1.1		2
15.2	even	4		75.10.b.e.49.2		4
15.8	even	4		75.10.b.e.49.3		4
15.14	odd	2		75.10.a.g.1.2		2

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
15.10.a.c.1.1	✓	2	3.2	odd	2
45.10.a.e.1.2		2	1.1	even	1	trivial
75.10.a.g.1.2		2	15.14	odd	2
75.10.b.e.49.2		4	15.2	even	4
75.10.b.e.49.3		4	15.8	even	4
225.10.a.j.1.1		2	5.4	even	2
225.10.b.g.199.2		4	5.3	odd	4
225.10.b.g.199.3		4	5.2	odd	4
240.10.a.m.1.1		2	12.11	even	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 $\F_{q}$

Level:	\( N \)	\(=\)	\( 45 = 3^{2} \cdot 5 \)
Weight:	\( k \)	\(=\)	\( 10 \)
Character orbit:	\([\chi]\)	\(=\)	45.a (trivial)

\(f(q)\)	\(=\)	\(q+24.8839 q^{2} +107.207 q^{4} +625.000 q^{5} -4010.50 q^{7} -10072.8 q^{8} +15552.4 q^{10} -84861.3 q^{11} +119425. q^{13} -99796.8 q^{14} -305541. q^{16} -116934. q^{17} -234932. q^{19} +67004.1 q^{20} -2.11168e6 q^{22} -2.34570e6 q^{23} +390625. q^{25} +2.97176e6 q^{26} -429953. q^{28} +464196. q^{29} -5.11766e6 q^{31} -2.44574e6 q^{32} -2.90976e6 q^{34} -2.50656e6 q^{35} +8.69354e6 q^{37} -5.84601e6 q^{38} -6.29551e6 q^{40} +9.05805e6 q^{41} +8.63491e6 q^{43} -9.09769e6 q^{44} -5.83701e7 q^{46} -3.31511e7 q^{47} -2.42695e7 q^{49} +9.72026e6 q^{50} +1.28032e7 q^{52} +6.41254e7 q^{53} -5.30383e7 q^{55} +4.03971e7 q^{56} +1.15510e7 q^{58} -1.49407e8 q^{59} +1.54634e8 q^{61} -1.27347e8 q^{62} +9.55772e7 q^{64} +7.46408e7 q^{65} +2.72755e8 q^{67} -1.25360e7 q^{68} -6.23730e7 q^{70} +3.56924e8 q^{71} +2.06253e8 q^{73} +2.16329e8 q^{74} -2.51862e7 q^{76} +3.40337e8 q^{77} -4.04380e8 q^{79} -1.90963e8 q^{80} +2.25399e8 q^{82} +5.17034e6 q^{83} -7.30834e7 q^{85} +2.14870e8 q^{86} +8.54793e8 q^{88} -4.32242e8 q^{89} -4.78955e8 q^{91} -2.51475e8 q^{92} -8.24928e8 q^{94} -1.46832e8 q^{95} -1.32066e9 q^{97} -6.03918e8 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 2 q - 19 q^{2} + 1521 q^{4} + 1250 q^{5} - 11872 q^{7} - 49647 q^{8} - 11875 q^{10} - 35488 q^{11} + 143676 q^{13} + 245196 q^{14} + 707265 q^{16} - 385156 q^{17} - 403296 q^{19} + 950625 q^{20} - 4278368 q^{22}+ \cdots - 1545205739 q^{98}+O(q^{100}) \) 2 * q - 19 * q^2 + 1521 * q^4 + 1250 * q^5 - 11872 * q^7 - 49647 * q^8 - 11875 * q^10 - 35488 * q^11 + 143676 * q^13 + 245196 * q^14 + 707265 * q^16 - 385156 * q^17 - 403296 * q^19 + 950625 * q^20 - 4278368 * q^22 - 223704 * q^23 + 781250 * q^25 + 1907546 * q^26 - 11544484 * q^28 + 74572 * q^29 - 5027128 * q^31 - 26629583 * q^32 + 8860882 * q^34 - 7420000 * q^35 + 5373628 * q^37 + 1542476 * q^38 - 31029375 * q^40 - 14211332 * q^41 + 27748920 * q^43 + 60705952 * q^44 - 151491648 * q^46 - 95966440 * q^47 - 2819950 * q^49 - 7421875 * q^50 + 47088706 * q^52 + 64305596 * q^53 - 22180000 * q^55 + 351509340 * q^56 + 28649198 * q^58 - 187863136 * q^59 + 154080060 * q^61 - 131320056 * q^62 + 638301089 * q^64 + 89797500 * q^65 + 33592376 * q^67 - 391747238 * q^68 + 153247500 * q^70 + 228270976 * q^71 - 33122316 * q^73 + 362019226 * q^74 - 263218724 * q^76 - 47811456 * q^77 - 932406760 * q^79 + 442040625 * q^80 + 1246549646 * q^82 - 207040152 * q^83 - 240722500 * q^85 - 623926708 * q^86 - 1099114848 * q^88 - 224518164 * q^89 - 669602528 * q^91 + 2748592992 * q^92 + 1931650816 * q^94 - 252060000 * q^95 + 387134596 * q^97 - 1545205739 * q^98

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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