LMFDB - Embedded newform 155.7.c.a.154.1

Show commands: Magma / Pari/GP / SageMath

Newspace parameters

comment:Compute space of new eigenforms

gp:[N,k,chi] = [155,7,Mod(154,155)] mf = mfinit([N,k,chi],0) lf = mfeigenbasis(mf)

magma://Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code chi := DirichletCharacter("155.154"); S:= CuspForms(chi, 7); N := Newforms(S);

sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(155, base_ring=CyclotomicField(2)) chi = DirichletCharacter(H, H._module([1, 1])) N = Newforms(chi, 7, names="a")

Level:	$ N $	$=$	$ 155 = 5 \cdot 31 $
Weight:	$ k $	$=$	$ 7 $
Character orbit:	$[\chi]$	$=$	155.c (of order $2$, degree $1$, minimal)

Newform invariants

comment:select newform

sage:traces = [2,0,0,66,246] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(5)] == traces)

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

Self dual:	no
Analytic conductor:	$35.6583829611$
Analytic rank:	$0$
Dimension:	$2$
Coefficient field:	$\Q(\sqrt{-31}) $
comment:defining polynomial gp:f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{2} - x + 8 $ x^2 - x + 8
Coefficient ring:	$\Z[a_1, a_2]$
Coefficient ring index:	$ 2 $
Twist minimal:	yes
Sato-Tate group:	$\mathrm{U}(1)[D_{2}]$

Embedding invariants

Embedding label			154.1
Root			$0.500000 + 2.78388i$ of defining polynomial
Character	$\chi$	$=$	155.154
Dual form			155.7.c.a.154.2

$q$-expansion

comment:q-expansion

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

gp:mfcoefs(f, 20)

$f(q)$	$=$	$q-5.56776i q^{2} +33.0000 q^{4} +(123.000 - 22.2711i) q^{5} -534.505i q^{7} -540.073i q^{8} -729.000 q^{9} +(-124.000 - 684.835i) q^{10} -2976.00 q^{14} -895.000 q^{16} +4058.90i q^{18} -10618.0 q^{19} +(4059.00 - 734.945i) q^{20} +(14633.0 - 5478.68i) q^{25} -17638.7i q^{28} +29791.0 q^{31} -29581.5i q^{32} +(-11904.0 - 65744.2i) q^{35} -24057.0 q^{36} +59118.5i q^{38} +(-12028.0 - 66429.0i) q^{40} -60558.0 q^{41} +(-89667.0 + 16235.6i) q^{45} +116611. i q^{47} -168047. q^{49} +(-30504.0 - 81473.1i) q^{50} -288672. q^{56} +136842. q^{59} -165869. i q^{62} +389654. i q^{63} -221983. q^{64} -586486. i q^{67} +(-366048. + 66278.7i) q^{70} -284178. q^{71} +393713. i q^{72} -350394. q^{76} +(-110085. + 19932.6i) q^{80} +531441. q^{81} +337173. i q^{82} +(90396.0 + 499245. i) q^{90} +649264. q^{94} +(-1.30601e6 + 236474. i) q^{95} +254692. i q^{97} +935646. i q^{98} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 2 q + 66 q^{4} + 246 q^{5} - 1458 q^{9} - 248 q^{10} - 5952 q^{14} - 1790 q^{16} - 21236 q^{19} + 8118 q^{20} + 29266 q^{25} + 59582 q^{31} - 23808 q^{35} - 48114 q^{36} - 24056 q^{40} - 121116 q^{41} - 179334 q^{45}+ \cdots - 2612028 q^{95}+O(q^{100}) $ 2 * q + 66 * q^4 + 246 * q^5 - 1458 * q^9 - 248 * q^10 - 5952 * q^14 - 1790 * q^16 - 21236 * q^19 + 8118 * q^20 + 29266 * q^25 + 59582 * q^31 - 23808 * q^35 - 48114 * q^36 - 24056 * q^40 - 121116 * q^41 - 179334 * q^45 - 336094 * q^49 - 61008 * q^50 - 577344 * q^56 + 273684 * q^59 - 443966 * q^64 - 732096 * q^70 - 568356 * q^71 - 700788 * q^76 - 220170 * q^80 + 1062882 * q^81 + 180792 * q^90 + 1298528 * q^94 - 2612028 * q^95

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/155\mathbb{Z}\right)^\times$.

$n$	$32$	$96$
$\chi(n)$	$-1$	$-1$

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 $

$2$		−	5.56776i		−	0.695971i	−0.937500	−	0.347985i	$-0.886866\pi$
$2$		−	5.56776i		−	0.695971i	0.937500	−	0.347985i	$-0.113134\pi$
$3$		0			0			−	1.00000i	$-0.5\pi$
$3$		0			0				1.00000i	$0.5\pi$
$4$	33.0000			0.515625
$5$	123.000	−	22.2711i	0.984000	−	0.178168i
$5$	123.000	−	22.2711i	0.984000	−	0.178168i
$6$		0			0
$7$		−	534.505i		−	1.55832i	−0.626822	−	0.779162i	$-0.715645\pi$
$7$		−	534.505i		−	1.55832i	0.626822	−	0.779162i	$-0.284355\pi$
$8$		−	540.073i		−	1.05483i
$9$	−729.000			−1.00000
$10$	−124.000	−	684.835i	−0.124000	−	0.684835i
$11$		0			0		1.00000			$0$
$11$		0			0		−1.00000			$\pi$
$12$		0			0
$13$		0			0			−	1.00000i	$-0.5\pi$
$13$		0			0				1.00000i	$0.5\pi$
$14$	−2976.00			−1.08455
$15$		0			0
$16$	−895.000			−0.218506
$17$		0			0			−	1.00000i	$-0.5\pi$
$17$		0			0				1.00000i	$0.5\pi$
$18$			4058.90i			0.695971i
$19$	−10618.0			−1.54804			−0.774020	−	0.633162i	$-0.781757\pi$
$19$	−10618.0			−1.54804			−0.774020	+	0.633162i	$0.781757\pi$
$20$	4059.00	−	734.945i	0.507375	−	0.0918681i
$21$		0			0
$22$		0			0
$23$		0			0			−	1.00000i	$-0.5\pi$
$23$		0			0				1.00000i	$0.5\pi$
$24$		0			0
$25$	14633.0	−	5478.68i	0.936512	−	0.350636i
$26$		0			0
$27$		0			0
$28$		−	17638.7i		−	0.803511i
$29$		0			0		1.00000			$0$
$29$		0			0		−1.00000			$\pi$
$30$		0			0
$31$	29791.0			1.00000
$31$	29791.0			1.00000
$32$		−	29581.5i		−	0.902757i
$33$		0			0
$34$		0			0
$35$	−11904.0	−	65744.2i	−0.277644	−	1.53339i
$36$	−24057.0			−0.515625
$37$		0			0			−	1.00000i	$-0.5\pi$
$37$		0			0				1.00000i	$0.5\pi$
$38$			59118.5i			1.07739i
$39$		0			0
$40$	−12028.0	−	66429.0i	−0.187938	−	1.03795i
$41$	−60558.0			−0.878658			−0.439329	−	0.898326i	$-0.644784\pi$
$41$	−60558.0			−0.878658			−0.439329	+	0.898326i	$0.644784\pi$
$42$		0			0
$43$		0			0			−	1.00000i	$-0.5\pi$
$43$		0			0				1.00000i	$0.5\pi$
$44$		0			0
$45$	−89667.0	+	16235.6i	−0.984000	+	0.178168i
$46$		0			0
$47$			116611.i			1.12317i	0.827418	+	0.561587i	$0.189809\pi$
$47$			116611.i			1.12317i	−0.827418	+	0.561587i	$0.810191\pi$
$48$		0			0
$49$	−168047.			−1.42838
$50$	−30504.0	−	81473.1i	−0.244032	−	0.651785i
$51$		0			0
$52$		0			0
$53$		0			0			−	1.00000i	$-0.5\pi$
$53$		0			0				1.00000i	$0.5\pi$
$54$		0			0
$55$		0			0
$56$	−288672.			−1.64377
$57$		0			0
$58$		0			0
$59$	136842.			0.666290			0.333145	−	0.942876i	$-0.391890\pi$
$59$	136842.			0.666290			0.333145	+	0.942876i	$0.391890\pi$
$60$		0			0
$61$		0			0		1.00000			$0$
$61$		0			0		−1.00000			$\pi$
$62$		−	165869.i		−	0.695971i
$63$			389654.i			1.55832i
$64$	−221983.			−0.846798
$65$		0			0
$66$		0			0
$67$		−	586486.i		−	1.94999i	−0.222218	−	0.974997i	$-0.571330\pi$
$67$		−	586486.i		−	1.94999i	0.222218	−	0.974997i	$-0.428670\pi$
$68$		0			0
$69$		0			0
$70$	−366048.	+	66278.7i	−1.06720	+	0.193232i
$71$	−284178.			−0.793991			−0.396995	−	0.917821i	$-0.629947\pi$
$71$	−284178.			−0.793991			−0.396995	+	0.917821i	$0.629947\pi$
$72$			393713.i			1.05483i
$73$		0			0			−	1.00000i	$-0.5\pi$
$73$		0			0				1.00000i	$0.5\pi$
$74$		0			0
$75$		0			0
$76$	−350394.			−0.798208
$77$		0			0
$78$		0			0
$79$		0			0		1.00000			$0$
$79$		0			0		−1.00000			$\pi$
$80$	−110085.	+	19932.6i	−0.215010	+	0.0389309i
$81$	531441.			1.00000
$82$			337173.i			0.611520i
$83$		0			0			−	1.00000i	$-0.5\pi$
$83$		0			0				1.00000i	$0.5\pi$
$84$		0			0
$85$		0			0
$86$		0			0
$87$		0			0
$88$		0			0
$89$		0			0		1.00000			$0$
$89$		0			0		−1.00000			$\pi$
$90$	90396.0	+	499245.i	0.124000	+	0.684835i
$91$		0			0
$92$		0			0
$93$		0			0
$94$	649264.			0.781696
$95$	−1.30601e6	+	236474.i	−1.52327	+	0.275812i
$96$		0			0
$97$			254692.i			0.279061i	0.990218	+	0.139531i	$0.0445594\pi$
$97$			254692.i			0.279061i	−0.990218	+	0.139531i	$0.955441\pi$
$98$			935646.i			0.994108i
$99$		0			0
$100$	482889.	−	180796.i	0.482889	−	0.180796i
$101$	−1.14350e6			−1.10987			−0.554934	−	0.831894i	$-0.687256\pi$
$101$	−1.14350e6			−1.10987			−0.554934	+	0.831894i	$0.687256\pi$
$102$		0			0
$103$		−	2.11049e6i		−	1.93140i	−0.259658	−	0.965701i	$-0.583610\pi$
$103$		−	2.11049e6i		−	1.93140i	0.259658	−	0.965701i	$-0.416390\pi$
$104$		0			0
$105$		0			0
$106$		0			0
$107$		−	2.44701e6i		−	1.99749i	−0.0500921	−	0.998745i	$-0.515951\pi$
$107$		−	2.44701e6i		−	1.99749i	0.0500921	−	0.998745i	$-0.484049\pi$
$108$		0			0
$109$	199942.			0.154392			0.0771960	−	0.997016i	$-0.475403\pi$
$109$	199942.			0.154392			0.0771960	+	0.997016i	$0.475403\pi$
$110$		0			0
$111$		0			0
$112$			478382.i			0.340503i
$113$		−	106990.i		−	0.0741495i	−0.999312	−	0.0370748i	$-0.988196\pi$
$113$		−	106990.i		−	0.0741495i	0.999312	−	0.0370748i	$-0.0118040\pi$
$114$		0			0
$115$		0			0
$116$		0			0
$117$		0			0
$118$		−	761904.i		−	0.463718i
$119$		0			0
$120$		0			0
$121$	1.77156e6			1.00000
$122$		0			0
$123$		0			0
$124$	983103.			0.515625
$125$	1.67784e6	−	999770.i	0.859056	−	0.511882i
$126$	2.16950e6			1.08455
$127$		0			0			−	1.00000i	$-0.5\pi$
$127$		0			0				1.00000i	$0.5\pi$
$128$		−	657269.i		−	0.313410i
$129$		0			0
$130$		0			0
$131$	4.47658e6			1.99128			0.995641	−	0.0932711i	$-0.0297323\pi$
$131$	4.47658e6			1.99128			0.995641	+	0.0932711i	$0.0297323\pi$
$132$		0			0
$133$			5.67538e6i			2.41235i
$134$	−3.26542e6			−1.35714
$135$		0			0
$136$		0			0
$137$		0			0			−	1.00000i	$-0.5\pi$
$137$		0			0				1.00000i	$0.5\pi$
$138$		0			0
$139$		0			0		1.00000			$0$
$139$		0			0		−1.00000			$\pi$
$140$	−392832.	−	2.16956e6i	−0.143160	−	0.790655i
$141$		0			0
$142$			1.58224e6i			0.552594i
$143$		0			0
$144$	652455.			0.218506
$145$		0			0
$146$		0			0
$147$		0			0
$148$		0			0
$149$	5.42500e6			1.63999			0.819995	−	0.572371i	$-0.193976\pi$
$149$	5.42500e6			1.63999			0.819995	+	0.572371i	$0.193976\pi$
$150$		0			0
$151$		0			0		1.00000			$0$
$151$		0			0		−1.00000			$\pi$
$152$			5.73450e6i			1.63292i
$153$		0			0
$154$		0			0
$155$	3.66429e6	−	663477.i	0.984000	−	0.178168i
$156$		0			0
$157$		−	4.31332e6i		−	1.11459i	−0.830316	−	0.557293i	$-0.811840\pi$
$157$		−	4.31332e6i		−	1.11459i	0.830316	−	0.557293i	$-0.188160\pi$
$158$		0			0
$159$		0			0
$160$	−658812.	−	3.63853e6i	−0.160843	−	0.888313i
$161$		0			0
$162$		−	2.95894e6i		−	0.695971i
$163$			8.46403e6i			1.95440i	0.212313	+	0.977202i	$0.431900\pi$
$163$			8.46403e6i			1.95440i	−0.212313	+	0.977202i	$0.568100\pi$
$164$	−1.99841e6			−0.453058
$165$		0			0
$166$		0			0
$167$		0			0			−	1.00000i	$-0.5\pi$
$167$		0			0				1.00000i	$0.5\pi$
$168$		0			0
$169$	−4.82681e6			−1.00000
$170$		0			0
$171$	7.74052e6			1.54804
$172$		0			0
$173$			2.31632e6i			0.447364i	0.974662	+	0.223682i	$0.0718077\pi$
$173$			2.31632e6i			0.447364i	−0.974662	+	0.223682i	$0.928192\pi$
$174$		0			0
$175$	−2.92838e6	−	7.82142e6i	−0.546404	−	1.45939i
$176$		0			0
$177$		0			0
$178$		0			0
$179$		0			0		1.00000			$0$
$179$		0			0		−1.00000			$\pi$
$180$	−2.95901e6	+	535775.i	−0.507375	+	0.0918681i
$181$		0			0		1.00000			$0$
$181$		0			0		−1.00000			$\pi$
$182$		0			0
$183$		0			0
$184$		0			0
$185$		0			0
$186$		0			0
$187$		0			0
$188$			3.84817e6i			0.579136i
$189$		0			0
$190$	1.31663e6	+	7.27158e6i	0.191957	+	1.06015i
$191$	1.38582e7			1.98888			0.994439	−	0.105316i	$-0.0335855\pi$
$191$	1.38582e7			1.98888			0.994439	+	0.105316i	$0.0335855\pi$
$192$		0			0
$193$			1.29479e7i			1.80105i	0.434803	+	0.900525i	$0.356818\pi$
$193$			1.29479e7i			1.80105i	−0.434803	+	0.900525i	$0.643182\pi$
$194$	1.41806e6			0.194219
$195$		0			0
$196$	−5.54555e6			−0.736506
$197$		0			0			−	1.00000i	$-0.5\pi$
$197$		0			0				1.00000i	$0.5\pi$
$198$		0			0
$199$		0			0		1.00000			$0$
$199$		0			0		−1.00000			$\pi$
$200$	−2.95889e6	−	7.90289e6i	−0.369861	−	0.987861i
$201$		0			0
$202$			6.36673e6i			0.772435i
$203$		0			0
$204$		0			0
$205$	−7.44863e6	+	1.34869e6i	−0.864600	+	0.156549i
$206$	−1.17507e7			−1.34420
$207$		0			0
$208$		0			0
$209$		0			0
$210$		0			0
$211$	1.23363e7			1.31322			0.656608	−	0.754232i	$-0.271991\pi$
$211$	1.23363e7			1.31322			0.656608	+	0.754232i	$0.271991\pi$
$212$		0			0
$213$		0			0
$214$	−1.36244e7			−1.39019
$215$		0			0
$216$		0			0
$217$		−	1.59234e7i		−	1.55832i
$218$		−	1.11323e6i		−	0.107452i
$219$		0			0
$220$		0			0
$221$		0			0
$222$		0			0
$223$		0			0			−	1.00000i	$-0.5\pi$
$223$		0			0				1.00000i	$0.5\pi$
$224$	−1.58115e7			−1.40679
$225$	−1.06675e7	+	3.99396e6i	−0.936512	+	0.350636i
$226$	−595696.			−0.0516059
$227$		−	1.78880e7i		−	1.52927i	−0.644465	−	0.764634i	$-0.722920\pi$
$227$		−	1.78880e7i		−	1.52927i	0.644465	−	0.764634i	$-0.277080\pi$
$228$		0			0
$229$		0			0		1.00000			$0$
$229$		0			0		−1.00000			$\pi$
$230$		0			0
$231$		0			0
$232$		0			0
$233$		−	2.50056e7i		−	1.97683i	−0.151776	−	0.988415i	$-0.548499\pi$
$233$		−	2.50056e7i		−	1.97683i	0.151776	−	0.988415i	$-0.451501\pi$
$234$		0			0
$235$	2.59706e6	+	1.43432e7i	0.200114	+	1.10520i
$236$	4.51579e6			0.343556
$237$		0			0
$238$		0			0
$239$		0			0		1.00000			$0$
$239$		0			0		−1.00000			$\pi$
$240$		0			0
$241$		0			0		1.00000			$0$
$241$		0			0		−1.00000			$\pi$
$242$		−	9.86363e6i		−	0.695971i
$243$		0			0
$244$		0			0
$245$	−2.06698e7	+	3.74258e6i	−1.40552	+	0.254492i
$246$		0			0
$247$		0			0
$248$		−	1.60893e7i		−	1.05483i
$249$		0			0
$250$	−5.56648e6	−	9.34183e6i	−0.356255	−	0.597877i
$251$		0			0		1.00000			$0$
$251$		0			0		−1.00000			$\pi$
$252$			1.28586e7i			0.803511i
$253$		0			0
$254$		0			0
$255$		0			0
$256$	−1.78664e7			−1.06492
$257$			3.01634e7i			1.77697i	0.458903	+	0.888486i	$0.348242\pi$
$257$			3.01634e7i			1.77697i	−0.458903	+	0.888486i	$0.651758\pi$
$258$		0			0
$259$		0			0
$260$		0			0
$261$		0			0
$262$		−	2.49246e7i		−	1.38587i
$263$		0			0			−	1.00000i	$-0.5\pi$
$263$		0			0				1.00000i	$0.5\pi$
$264$		0			0
$265$		0			0
$266$	3.15992e7			1.67892
$267$		0			0
$268$		−	1.93540e7i		−	1.00547i
$269$		0			0		1.00000			$0$
$269$		0			0		−1.00000			$\pi$
$270$		0			0
$271$		0			0		1.00000			$0$
$271$		0			0		−1.00000			$\pi$
$272$		0			0
$273$		0			0
$274$		0			0
$275$		0			0
$276$		0			0
$277$		0			0			−	1.00000i	$-0.5\pi$
$277$		0			0				1.00000i	$0.5\pi$
$278$		0			0
$279$	−2.17176e7			−1.00000
$280$	−3.55067e7	+	6.42903e6i	−1.61747	+	0.292868i
$281$	4.39536e7			1.98096			0.990479	−	0.137663i	$-0.0439591\pi$
$281$	4.39536e7			1.98096			0.990479	+	0.137663i	$0.0439591\pi$
$282$		0			0
$283$			2.97200e7i			1.31126i	0.755082	+	0.655630i	$0.227597\pi$
$283$			2.97200e7i			1.31126i	−0.755082	+	0.655630i	$0.772403\pi$
$284$	−9.37787e6			−0.409401
$285$		0			0
$286$		0			0
$287$			3.23686e7i			1.36923i
$288$			2.15649e7i			0.902757i
$289$	−2.41376e7			−1.00000
$290$		0			0
$291$		0			0
$292$		0			0
$293$		−	4.50204e7i		−	1.78981i	−0.446260	−	0.894903i	$-0.647244\pi$
$293$		−	4.50204e7i		−	1.78981i	0.446260	−	0.894903i	$-0.352756\pi$
$294$		0			0
$295$	1.68316e7	−	3.04762e6i	0.655629	−	0.118712i
$296$		0			0
$297$		0			0
$298$		−	3.02051e7i		−	1.14138i
$299$		0			0
$300$		0			0
$301$		0			0
$302$		0			0
$303$		0			0
$304$	9.50311e6			0.338256
$305$		0			0
$306$		0			0
$307$			4.12199e7i			1.42459i	0.701878	+	0.712297i	$0.252345\pi$
$307$			4.12199e7i			1.42459i	−0.701878	+	0.712297i	$0.747655\pi$
$308$		0			0
$309$		0			0
$310$	−3.69408e6	−	2.04019e7i	−0.124000	−	0.684835i
$311$	4.45334e7			1.48049			0.740243	−	0.672339i	$-0.234710\pi$
$311$	4.45334e7			1.48049			0.740243	+	0.672339i	$0.234710\pi$
$312$		0			0
$313$		0			0			−	1.00000i	$-0.5\pi$
$313$		0			0				1.00000i	$0.5\pi$
$314$	−2.40156e7			−0.775718
$315$	8.67802e6	+	4.79275e7i	0.277644	+	1.53339i
$316$		0			0
$317$			1.52237e7i			0.477907i	0.971031	+	0.238954i	$0.0768044\pi$
$317$			1.52237e7i			0.477907i	−0.971031	+	0.238954i	$0.923196\pi$
$318$		0			0
$319$		0			0
$320$	−2.73039e7	+	4.94380e6i	−0.833249	+	0.150873i
$321$		0			0
$322$		0			0
$323$		0			0
$324$	1.75376e7			0.515625
$325$		0			0
$326$	4.71257e7			1.36021
$327$		0			0
$328$			3.27057e7i			0.926835i
$329$	6.23293e7			1.75027
$330$		0			0
$331$		0			0		1.00000			$0$
$331$		0			0		−1.00000			$\pi$
$332$		0			0
$333$		0			0
$334$		0			0
$335$	−1.30617e7	−	7.21378e7i	−0.347427	−	1.91879i
$336$		0			0
$337$		0			0			−	1.00000i	$-0.5\pi$
$337$		0			0				1.00000i	$0.5\pi$
$338$			2.68745e7i			0.695971i
$339$		0			0
$340$		0			0
$341$		0			0
$342$		−	4.30974e7i		−	1.07739i
$343$			2.69380e7i			0.667549i
$344$		0			0
$345$		0			0
$346$	1.28967e7			0.311352
$347$		0			0			−	1.00000i	$-0.5\pi$
$347$		0			0				1.00000i	$0.5\pi$
$348$		0			0
$349$	−6.55690e7			−1.54249			−0.771245	−	0.636539i	$-0.780366\pi$
$349$	−6.55690e7			−1.54249			−0.771245	+	0.636539i	$0.780366\pi$
$350$	−4.35478e7	+	1.63046e7i	−1.01569	+	0.380281i
$351$		0			0
$352$		0			0
$353$		0			0			−	1.00000i	$-0.5\pi$
$353$		0			0				1.00000i	$0.5\pi$
$354$		0			0
$355$	−3.49539e7	+	6.32894e6i	−0.781287	+	0.141464i
$356$		0			0
$357$		0			0
$358$		0			0
$359$	7.57593e7			1.63739			0.818696	−	0.574227i	$-0.194697\pi$
$359$	7.57593e7			1.63739			0.818696	+	0.574227i	$0.194697\pi$
$360$	8.76841e6	+	4.84267e7i	0.187938	+	1.03795i
$361$	6.56960e7			1.39642
$362$		0			0
$363$		0			0
$364$		0			0
$365$		0			0
$366$		0			0
$367$		0			0			−	1.00000i	$-0.5\pi$
$367$		0			0				1.00000i	$0.5\pi$
$368$		0			0
$369$	4.41468e7			0.878658
$370$		0			0
$371$		0			0
$372$		0			0
$373$		−	2.95313e7i		−	0.569057i	−0.958668	−	0.284529i	$-0.908163\pi$
$373$		−	2.95313e7i		−	0.569057i	0.958668	−	0.284529i	$-0.0918371\pi$
$374$		0			0
$375$		0			0
$376$	6.29786e7			1.18476
$377$		0			0
$378$		0			0
$379$	1.08388e8			1.99096			0.995480	−	0.0949732i	$-0.0302766\pi$
$379$	1.08388e8			1.99096			0.995480	+	0.0949732i	$0.0302766\pi$
$380$	−4.30985e7	+	7.80364e6i	−0.785436	+	0.142215i
$381$		0			0
$382$		−	7.71594e7i		−	1.38420i
$383$		0			0			−	1.00000i	$-0.5\pi$
$383$		0			0				1.00000i	$0.5\pi$
$384$		0			0
$385$		0			0
$386$	7.20906e7			1.25348
$387$		0			0
$388$			8.40483e6i			0.143891i
$389$		0			0		1.00000			$0$
$389$		0			0		−1.00000			$\pi$
$390$		0			0
$391$		0			0
$392$			9.07577e7i			1.50669i
$393$		0			0
$394$		0			0
$395$		0			0
$396$		0			0
$397$			3.86288e7i			0.617362i	0.951166	+	0.308681i	$0.0998876\pi$
$397$			3.86288e7i			0.617362i	−0.951166	+	0.308681i	$0.900112\pi$
$398$		0			0
$399$		0			0
$400$	−1.30965e7	+	4.90342e6i	−0.204633	+	0.0766159i
$401$		0			0		1.00000			$0$
$401$		0			0		−1.00000			$\pi$
$402$		0			0
$403$		0			0
$404$	−3.77354e7			−0.572276
$405$	6.53672e7	−	1.18358e7i	0.984000	−	0.178168i
$406$		0			0
$407$		0			0
$408$		0			0
$409$		0			0		1.00000			$0$
$409$		0			0		−1.00000			$\pi$
$410$	7.50919e6	+	4.14722e7i	0.108954	+	0.601736i
$411$		0			0
$412$		−	6.96463e7i		−	0.995879i
$413$		−	7.31428e7i		−	1.03830i
$414$		0			0
$415$		0			0
$416$		0			0
$417$		0			0
$418$		0			0
$419$	−1.38240e8			−1.87928			−0.939638	−	0.342169i	$-0.888838\pi$
$419$	−1.38240e8			−1.87928			−0.939638	+	0.342169i	$0.888838\pi$
$420$		0			0
$421$	−9.09868e6			−0.121936			−0.0609680	−	0.998140i	$-0.519419\pi$
$421$	−9.09868e6			−0.121936			−0.0609680	+	0.998140i	$0.519419\pi$
$422$		−	6.86854e7i		−	0.913960i
$423$		−	8.50096e7i		−	1.12317i
$424$		0			0
$425$		0			0
$426$		0			0
$427$		0			0
$428$		−	8.07513e7i		−	1.02996i
$429$		0			0
$430$		0			0
$431$	9.32758e6			0.116503			0.0582515	−	0.998302i	$-0.481447\pi$
$431$	9.32758e6			0.116503			0.0582515	+	0.998302i	$0.481447\pi$
$432$		0			0
$433$		0			0			−	1.00000i	$-0.5\pi$
$433$		0			0				1.00000i	$0.5\pi$
$434$	−8.86580e7			−1.08455
$435$		0			0
$436$	6.59809e6			0.0796083
$437$		0			0
$438$		0			0
$439$	−1.65523e8			−1.95643			−0.978214	−	0.207599i	$-0.933435\pi$
$439$	−1.65523e8			−1.95643			−0.978214	+	0.207599i	$0.933435\pi$
$440$		0			0
$441$	1.22506e8			1.42838
$442$		0			0
$443$		−	1.22971e8i		−	1.41447i	−0.706980	−	0.707233i	$-0.749943\pi$
$443$		−	1.22971e8i		−	1.41447i	0.706980	−	0.707233i	$-0.250057\pi$
$444$		0			0
$445$		0			0
$446$		0			0
$447$		0			0
$448$			1.18651e8i			1.31959i
$449$		0			0		1.00000			$0$
$449$		0			0		−1.00000			$\pi$
$450$	2.22374e7	+	5.93939e7i	0.244032	+	0.651785i
$451$		0			0
$452$		−	3.53068e6i		−	0.0382334i
$453$		0			0
$454$	−9.95961e7			−1.06433
$455$		0			0
$456$		0			0
$457$		0			0			−	1.00000i	$-0.5\pi$
$457$		0			0				1.00000i	$0.5\pi$
$458$		0			0
$459$		0			0
$460$		0			0
$461$		0			0		1.00000			$0$
$461$		0			0		−1.00000			$\pi$
$462$		0			0
$463$		0			0			−	1.00000i	$-0.5\pi$
$463$		0			0				1.00000i	$0.5\pi$
$464$		0			0
$465$		0			0
$466$	−1.39225e8			−1.37582
$467$			1.55952e8i			1.53123i	0.643297	+	0.765616i	$0.277566\pi$
$467$			1.55952e8i			1.53123i	−0.643297	+	0.765616i	$0.722434\pi$
$468$		0			0
$469$	−3.13480e8			−3.03872
$470$	7.98595e7	−	1.44598e7i	0.769189	−	0.139274i
$471$		0			0
$472$		−	7.39047e7i		−	0.702823i
$473$		0			0
$474$		0			0
$475$	−1.55373e8	+	5.81726e7i	−1.44976	+	0.542797i
$476$		0			0
$477$		0			0
$478$		0			0
$479$	7.32899e7			0.666865			0.333432	−	0.942774i	$-0.391793\pi$
$479$	7.32899e7			0.666865			0.333432	+	0.942774i	$0.391793\pi$
$480$		0			0
$481$		0			0
$482$		0			0
$483$		0			0
$484$	5.84615e7			0.515625
$485$	5.67226e6	+	3.13271e7i	0.0497199	+	0.274596i
$486$		0			0
$487$		0			0			−	1.00000i	$-0.5\pi$
$487$		0			0				1.00000i	$0.5\pi$
$488$		0			0
$489$		0			0
$490$	2.08378e7	+	1.15084e8i	0.177119	+	0.978202i
$491$		0			0		1.00000			$0$
$491$		0			0		−1.00000			$\pi$
$492$		0			0
$493$		0			0
$494$		0			0
$495$		0			0
$496$	−2.66629e7			−0.218506
$497$			1.51895e8i			1.23730i
$498$		0			0
$499$		0			0		1.00000			$0$
$499$		0			0		−1.00000			$\pi$
$500$	5.53688e7	−	3.29924e7i	0.442951	−	0.263939i
$501$		0			0
$502$		0			0
$503$			1.53188e7i			0.120371i	0.998187	+	0.0601855i	$0.0191692\pi$
$503$			1.53188e7i			0.120371i	−0.998187	+	0.0601855i	$0.980831\pi$
$504$	2.10442e8			1.64377
$505$	−1.40650e8	+	2.54669e7i	−1.09211	+	0.197743i
$506$		0			0
$507$		0			0
$508$		0			0
$509$		0			0		1.00000			$0$
$509$		0			0		−1.00000			$\pi$
$510$		0			0
$511$		0			0
$512$			5.74109e7i			0.427744i
$513$		0			0
$514$	1.67943e8			1.23672
$515$	−4.70029e7	−	2.59591e8i	−0.344115	−	1.90050i
$516$		0			0
$517$		0			0
$518$		0			0
$519$		0			0
$520$		0			0
$521$	6.88046e7			0.486524			0.243262	−	0.969961i	$-0.421782\pi$
$521$	6.88046e7			0.486524			0.243262	+	0.969961i	$0.421782\pi$
$522$		0			0
$523$		0			0			−	1.00000i	$-0.5\pi$
$523$		0			0				1.00000i	$0.5\pi$
$524$	1.47727e8			1.02675
$525$		0			0
$526$		0			0
$527$		0			0
$528$		0			0
$529$	−1.48036e8			−1.00000
$530$		0			0
$531$	−9.97578e7			−0.666290
$532$			1.87287e8i			1.24387i
$533$		0			0
$534$		0			0
$535$	−5.44975e7	−	3.00982e8i	−0.355890	−	1.96553i
$536$	−3.16745e8			−2.05691
$537$		0			0
$538$		0			0
$539$		0			0
$540$		0			0
$541$	−3.01836e8			−1.90625			−0.953124	−	0.302580i	$-0.902152\pi$
$541$	−3.01836e8			−1.90625			−0.953124	+	0.302580i	$0.902152\pi$
$542$		0			0
$543$		0			0
$544$		0			0
$545$	2.45929e7	−	4.45292e6i	0.151922	−	0.0275078i
$546$		0			0
$547$		−	1.45575e8i		−	0.889457i	−0.895665	−	0.444729i	$-0.853300\pi$
$547$		−	1.45575e8i		−	0.889457i	0.895665	−	0.444729i	$-0.146700\pi$
$548$		0			0
$549$		0			0
$550$		0			0
$551$		0			0
$552$		0			0
$553$		0			0
$554$		0			0
$555$		0			0
$556$		0			0
$557$		0			0			−	1.00000i	$-0.5\pi$
$557$		0			0				1.00000i	$0.5\pi$
$558$			1.20919e8i			0.695971i
$559$		0			0
$560$	1.06541e7	+	5.88410e7i	0.0606669	+	0.335055i
$561$		0			0
$562$		−	2.44723e8i		−	1.37869i
$563$			3.26319e8i			1.82859i	0.405046	+	0.914296i	$0.367255\pi$
$563$			3.26319e8i			1.82859i	−0.405046	+	0.914296i	$0.632745\pi$
$564$		0			0
$565$	−2.38278e6	−	1.31598e7i	−0.0132111	−	0.0729632i
$566$	1.65474e8			0.912599
$567$		−	2.84058e8i		−	1.55832i
$568$			1.53477e8i			0.837525i
$569$		0			0		1.00000			$0$
$569$		0			0		−1.00000			$\pi$
$570$		0			0
$571$		0			0		1.00000			$0$
$571$		0			0		−1.00000			$\pi$
$572$		0			0
$573$		0			0
$574$	1.80221e8			0.952947
$575$		0			0
$576$	1.61826e8			0.846798
$577$			2.85403e8i			1.48570i	0.669459	+	0.742849i	$0.266526\pi$
$577$			2.85403e8i			1.48570i	−0.669459	+	0.742849i	$0.733474\pi$
$578$			1.34392e8i			0.695971i
$579$		0			0
$580$		0			0
$581$		0			0
$582$		0			0
$583$		0			0
$584$		0			0
$585$		0			0
$586$	−2.50663e8			−1.24565
$587$		0			0			−	1.00000i	$-0.5\pi$
$587$		0			0				1.00000i	$0.5\pi$
$588$		0			0
$589$	−3.16321e8			−1.54804
$590$	−1.69684e7	−	9.37142e7i	−0.0826200	−	0.456299i
$591$		0			0
$592$		0			0
$593$		−	3.51053e8i		−	1.68348i	−0.539883	−	0.841740i	$-0.681531\pi$
$593$		−	3.51053e8i		−	1.68348i	0.539883	−	0.841740i	$-0.318469\pi$
$594$		0			0
$595$		0			0
$596$	1.79025e8			0.845620
$597$		0			0
$598$		0			0
$599$	2.87376e8			1.33712			0.668558	−	0.743660i	$-0.266912\pi$
$599$	2.87376e8			1.33712			0.668558	+	0.743660i	$0.266912\pi$
$600$		0			0
$601$		0			0		1.00000			$0$
$601$		0			0		−1.00000			$\pi$
$602$		0			0
$603$			4.27548e8i			1.94999i
$604$		0			0
$605$	2.17902e8	−	3.94545e7i	0.984000	−	0.178168i
$606$		0			0
$607$		−	4.38108e8i		−	1.95891i	−0.201657	−	0.979456i	$-0.564633\pi$
$607$		−	4.38108e8i		−	1.95891i	0.201657	−	0.979456i	$-0.435367\pi$
$608$			3.14097e8i			1.39750i
$609$		0			0
$610$		0			0
$611$		0			0
$612$		0			0
$613$		0			0			−	1.00000i	$-0.5\pi$
$613$		0			0				1.00000i	$0.5\pi$
$614$	2.29502e8			0.991476
$615$		0			0
$616$		0			0
$617$		−	1.78161e8i		−	0.758505i	−0.925293	−	0.379252i	$-0.876181\pi$
$617$		−	1.78161e8i		−	0.758505i	0.925293	−	0.379252i	$-0.123819\pi$
$618$		0			0
$619$		0			0		1.00000			$0$
$619$		0			0		−1.00000			$\pi$
$620$	1.20922e8	−	2.18947e7i	0.507375	−	0.0918681i
$621$		0			0
$622$		−	2.47951e8i		−	1.03037i
$623$		0			0
$624$		0			0
$625$	1.84109e8	−	1.60339e8i	0.754109	−	0.656749i
$626$		0			0
$627$		0			0
$628$		−	1.42340e8i		−	0.574708i
$629$		0			0
$630$	2.66849e8	−	4.83171e7i	1.06720	−	0.193232i
$631$		0			0		1.00000			$0$
$631$		0			0		−1.00000			$\pi$
$632$		0			0
$633$		0			0
$634$	8.47622e7			0.332609
$635$		0			0
$636$		0			0
$637$		0			0
$638$		0			0
$639$	2.07166e8			0.793991
$640$	−1.46381e7	−	8.08441e7i	−0.0558398	−	0.308396i
$641$		0			0		1.00000			$0$
$641$		0			0		−1.00000			$\pi$
$642$		0			0
$643$		0			0			−	1.00000i	$-0.5\pi$
$643$		0			0				1.00000i	$0.5\pi$
$644$		0			0
$645$		0			0
$646$		0			0
$647$		0			0			−	1.00000i	$-0.5\pi$
$647$		0			0				1.00000i	$0.5\pi$
$648$		−	2.87017e8i		−	1.05483i
$649$		0			0
$650$		0			0
$651$		0			0
$652$			2.79313e8i			1.00774i
$653$			2.35311e8i			0.845090i	0.906342	+	0.422545i	$0.138863\pi$
$653$			2.35311e8i			0.845090i	−0.906342	+	0.422545i	$0.861137\pi$
$654$		0			0
$655$	5.50620e8	−	9.96982e7i	1.95942	−	0.354784i
$656$	5.41994e7			0.191992
$657$		0			0
$658$		−	3.47035e8i		−	1.21814i
$659$	5.69810e8			1.99101			0.995505	−	0.0947082i	$-0.0301918\pi$
$659$	5.69810e8			1.99101			0.995505	+	0.0947082i	$0.0301918\pi$
$660$		0			0
$661$	−3.43513e8			−1.18943			−0.594715	−	0.803937i	$-0.702735\pi$
$661$	−3.43513e8			−1.18943			−0.594715	+	0.803937i	$0.702735\pi$
$662$		0			0
$663$		0			0
$664$		0			0
$665$	1.26397e8	+	6.98072e8i	0.429804	+	2.37375i
$666$		0			0
$667$		0			0
$668$		0			0
$669$		0			0
$670$	−4.01646e8	+	7.27243e7i	−1.33542	+	0.241799i
$671$		0			0
$672$		0			0
$673$		0			0			−	1.00000i	$-0.5\pi$
$673$		0			0				1.00000i	$0.5\pi$
$674$		0			0
$675$		0			0
$676$	−1.59285e8			−0.515625
$677$		0			0			−	1.00000i	$-0.5\pi$
$677$		0			0				1.00000i	$0.5\pi$
$678$		0			0
$679$	1.36134e8			0.434868
$680$		0			0
$681$		0			0
$682$		0			0
$683$			6.37188e8i			1.99989i	0.0106906	+	0.999943i	$0.496597\pi$
$683$			6.37188e8i			1.99989i	−0.0106906	+	0.999943i	$0.503403\pi$
$684$	2.55437e8			0.798208
$685$		0			0
$686$	1.49984e8			0.464594
$687$		0			0
$688$		0			0
$689$		0			0
$690$		0			0
$691$	3.40452e8			1.03186			0.515931	−	0.856630i	$-0.327446\pi$
$691$	3.40452e8			1.03186			0.515931	+	0.856630i	$0.327446\pi$
$692$			7.64387e7i			0.230672i
$693$		0			0
$694$		0			0
$695$		0			0
$696$		0			0
$697$		0			0
$698$			3.65073e8i			1.07353i
$699$		0			0
$700$	−9.66367e7	−	2.58107e8i	−0.281740	−	0.752498i
$701$	−3.75583e8			−1.09032			−0.545158	−	0.838333i	$-0.683530\pi$
$701$	−3.75583e8			−1.09032			−0.545158	+	0.838333i	$0.683530\pi$
$702$		0			0
$703$		0			0
$704$		0			0
$705$		0			0
$706$		0			0
$707$			6.11206e8i			1.72953i
$708$		0			0
$709$		0			0		1.00000			$0$
$709$		0			0		−1.00000			$\pi$
$710$	3.52381e7	+	1.94615e8i	0.0984548	+	0.543753i
$711$		0			0
$712$		0			0
$713$		0			0
$714$		0			0
$715$		0			0
$716$		0			0
$717$		0			0
$718$		−	4.21810e8i		−	1.13958i
$719$		0			0		1.00000			$0$
$719$		0			0		−1.00000			$\pi$
$720$	8.02520e7	−	1.45309e7i	0.215010	−	0.0389309i
$721$	−1.12807e9			−3.00975
$722$		−	3.65780e8i		−	0.971871i
$723$		0			0
$724$		0			0
$725$		0			0
$726$		0			0
$727$			7.30472e8i			1.90108i	0.310603	+	0.950540i	$0.399469\pi$
$727$			7.30472e8i			1.90108i	−0.310603	+	0.950540i	$0.600531\pi$
$728$		0			0
$729$	−3.87420e8			−1.00000
$730$		0			0
$731$		0			0
$732$		0			0
$733$		−	7.85782e8i		−	1.99522i	−0.0691253	−	0.997608i	$-0.522021\pi$
$733$		−	7.85782e8i		−	1.99522i	0.0691253	−	0.997608i	$-0.477979\pi$
$734$		0			0
$735$		0			0
$736$		0			0
$737$		0			0
$738$		−	2.45799e8i		−	0.611520i
$739$		0			0		1.00000			$0$
$739$		0			0		−1.00000			$\pi$
$740$		0			0
$741$		0			0
$742$		0			0
$743$		0			0			−	1.00000i	$-0.5\pi$
$743$		0			0				1.00000i	$0.5\pi$
$744$		0			0
$745$	6.67275e8	−	1.20821e8i	1.61375	−	0.292194i
$746$	−1.64423e8			−0.396047
$747$		0			0
$748$		0			0
$749$	−1.30794e9			−3.11274
$750$		0			0
$751$	−7.88264e8			−1.86102			−0.930512	−	0.366262i	$-0.880638\pi$
$751$	−7.88264e8			−1.86102			−0.930512	+	0.366262i	$0.880638\pi$
$752$		−	1.04367e8i		−	0.245420i
$753$		0			0
$754$		0			0
$755$		0			0
$756$		0			0
$757$		0			0			−	1.00000i	$-0.5\pi$
$757$		0			0				1.00000i	$0.5\pi$
$758$		−	6.03477e8i		−	1.38565i
$759$		0			0
$760$	1.27713e8	+	7.05343e8i	0.290935	+	1.60679i
$761$		0			0		1.00000			$0$
$761$		0			0		−1.00000			$\pi$
$762$		0			0
$763$		−	1.06870e8i		−	0.240593i
$764$	4.57322e8			1.02551
$765$		0			0
$766$		0			0
$767$		0			0
$768$		0			0
$769$	−2.75984e8			−0.606884			−0.303442	−	0.952850i	$-0.598136\pi$
$769$	−2.75984e8			−0.606884			−0.303442	+	0.952850i	$0.598136\pi$
$770$		0			0
$771$		0			0
$772$			4.27279e8i			0.928667i
$773$		0			0			−	1.00000i	$-0.5\pi$
$773$		0			0				1.00000i	$0.5\pi$
$774$		0			0
$775$	4.35932e8	−	1.63215e8i	0.936512	−	0.350636i
$776$	1.37552e8			0.294362
$777$		0			0
$778$		0			0
$779$	6.43005e8			1.36020
$780$		0			0
$781$		0			0
$782$		0			0
$783$		0			0
$784$	1.50402e8			0.312109
$785$	−9.60623e7	−	5.30539e8i	−0.198584	−	1.09675i
$786$		0			0
$787$		0			0			−	1.00000i	$-0.5\pi$
$787$		0			0				1.00000i	$0.5\pi$
$788$		0			0
$789$		0			0
$790$		0			0
$791$	−5.71868e7			−0.115549
$792$		0			0
$793$		0			0
$794$	2.15076e8			0.429666
$795$		0			0
$796$		0			0
$797$		0			0			−	1.00000i	$-0.5\pi$
$797$		0			0				1.00000i	$0.5\pi$
$798$		0			0
$799$		0			0
$800$	−1.62068e8	−	4.32867e8i	−0.316539	−	0.845442i
$801$		0			0
$802$		0			0
$803$		0			0
$804$		0			0
$805$		0			0
$806$		0			0
$807$		0			0
$808$			6.17573e8i			1.17072i
$809$		0			0		1.00000			$0$
$809$		0			0		−1.00000			$\pi$
$810$	−6.58987e7	−	3.63949e8i	−0.124000	−	0.684835i
$811$	−3.12332e8			−0.585537			−0.292769	−	0.956183i	$-0.594577\pi$
$811$	−3.12332e8			−0.585537			−0.292769	+	0.956183i	$0.594577\pi$
$812$		0			0
$813$		0			0
$814$		0			0
$815$	1.88503e8	+	1.04108e9i	0.348213	+	1.92313i
$816$		0			0
$817$		0			0
$818$		0			0
$819$		0			0
$820$	−2.45805e8	+	4.45068e7i	−0.445809	+	0.0807207i
$821$		0			0		1.00000			$0$
$821$		0			0		−1.00000			$\pi$
$822$		0			0
$823$		0			0			−	1.00000i	$-0.5\pi$
$823$		0			0				1.00000i	$0.5\pi$
$824$	−1.13982e9			−2.03730
$825$		0			0
$826$	−4.07242e8			−0.722624
$827$		0			0			−	1.00000i	$-0.5\pi$
$827$		0			0				1.00000i	$0.5\pi$
$828$		0			0
$829$		0			0		1.00000			$0$
$829$		0			0		−1.00000			$\pi$
$830$		0			0
$831$		0			0
$832$		0			0
$833$		0			0
$834$		0			0
$835$		0			0
$836$		0			0
$837$		0			0
$838$			7.69686e8i			1.30792i
$839$	−1.27524e8			−0.215926			−0.107963	−	0.994155i	$-0.534433\pi$
$839$	−1.27524e8			−0.215926			−0.107963	+	0.994155i	$0.534433\pi$
$840$		0			0
$841$	5.94823e8			1.00000
$842$			5.06593e7i			0.0848639i
$843$		0			0
$844$	4.07097e8			0.677127
$845$	−5.93698e8	+	1.07498e8i	−0.984000	+	0.178168i
$846$	−4.73313e8			−0.781696
$847$		−	9.46909e8i		−	1.55832i
$848$		0			0
$849$		0			0
$850$		0			0
$851$		0			0
$852$		0			0
$853$		−	8.85731e8i		−	1.42710i	−0.700603	−	0.713551i	$-0.747086\pi$
$853$		−	8.85731e8i		−	1.42710i	0.700603	−	0.713551i	$-0.252914\pi$
$854$		0			0
$855$	9.52084e8	−	1.72390e8i	1.52327	−	0.275812i
$856$	−1.32156e9			−2.10701
$857$		−	1.11974e9i		−	1.77899i	−0.456943	−	0.889496i	$-0.651056\pi$
$857$		−	1.11974e9i		−	1.77899i	0.456943	−	0.889496i	$-0.348944\pi$
$858$		0			0
$859$		0			0		1.00000			$0$
$859$		0			0		−1.00000			$\pi$
$860$		0			0
$861$		0			0
$862$		−	5.19338e7i		−	0.0810827i
$863$		0			0			−	1.00000i	$-0.5\pi$
$863$		0			0				1.00000i	$0.5\pi$
$864$		0			0
$865$	5.15870e7	+	2.84908e8i	0.0797061	+	0.440206i
$866$		0			0
$867$		0			0
$868$		−	5.25474e8i		−	0.803511i
$869$		0			0
$870$		0			0
$871$		0			0
$872$		−	1.07983e8i		−	0.162857i
$873$		−	1.85670e8i		−	0.279061i
$874$		0			0
$875$	−5.34382e8	−	8.96816e8i	−0.797679	−	1.33869i
$876$		0			0
$877$			1.34139e9i			1.98865i	0.106396	+	0.994324i	$0.466069\pi$
$877$			1.34139e9i			1.98865i	−0.106396	+	0.994324i	$0.533931\pi$
$878$			9.21591e8i			1.36162i
$879$		0			0
$880$		0			0
$881$		0			0		1.00000			$0$
$881$		0			0		−1.00000			$\pi$
$882$		−	6.82086e8i		−	0.994108i
$883$		0			0			−	1.00000i	$-0.5\pi$
$883$		0			0				1.00000i	$0.5\pi$
$884$		0			0
$885$		0			0
$886$	−6.84675e8			−0.984427
$887$		−	5.82243e8i		−	0.834322i	−0.908833	−	0.417161i	$-0.863025\pi$
$887$		−	5.82243e8i		−	0.834322i	0.908833	−	0.417161i	$-0.136975\pi$
$888$		0			0
$889$		0			0
$890$		0			0
$891$		0			0
$892$		0			0
$893$		−	1.23818e9i		−	1.73872i
$894$		0			0
$895$		0			0
$896$	−3.51314e8			−0.488395
$897$		0			0
$898$		0			0
$899$		0			0
$900$	−3.52026e8	+	1.31801e8i	−0.482889	+	0.180796i
$901$		0			0
$902$		0			0
$903$		0			0
$904$	−5.77825e7			−0.0782152
$905$		0			0
$906$		0			0
$907$			1.01488e9i			1.36017i	0.733133	+	0.680085i	$0.238057\pi$
$907$			1.01488e9i			1.36017i	−0.733133	+	0.680085i	$0.761943\pi$
$908$		−	5.90303e8i		−	0.788529i
$909$	8.33610e8			1.10987
$910$		0			0
$911$		0			0		1.00000			$0$
$911$		0			0		−1.00000			$\pi$
$912$		0			0
$913$		0			0
$914$		0			0
$915$		0			0
$916$		0			0
$917$		−	2.39276e9i		−	3.10306i
$918$		0			0
$919$	−1.18759e9			−1.53010			−0.765051	−	0.643970i	$-0.777286\pi$
$919$	−1.18759e9			−1.53010			−0.765051	+	0.643970i	$0.777286\pi$
$920$		0			0
$921$		0			0
$922$		0			0
$923$		0			0
$924$		0			0
$925$		0			0
$926$		0			0
$927$			1.53855e9i			1.93140i
$928$		0			0
$929$		0			0		1.00000			$0$
$929$		0			0		−1.00000			$\pi$
$930$		0			0
$931$	1.78432e9			2.21118
$932$		−	8.25184e8i		−	1.01930i
$933$		0			0
$934$	8.68306e8			1.06569
$935$		0			0
$936$		0			0
$937$			1.61984e9i			1.96904i	0.175285	+	0.984518i	$0.443915\pi$
$937$			1.61984e9i			1.96904i	−0.175285	+	0.984518i	$0.556085\pi$
$938$			1.74538e9i			2.11486i
$939$		0			0
$940$	8.57028e7	+	4.73325e8i	0.103184	+	0.569870i
$941$		0			0		1.00000			$0$
$941$		0			0		−1.00000			$\pi$
$942$		0			0
$943$		0			0
$944$	−1.22474e8			−0.145588
$945$		0			0
$946$		0			0
$947$		0			0			−	1.00000i	$-0.5\pi$
$947$		0			0				1.00000i	$0.5\pi$
$948$		0			0
$949$		0			0
$950$	3.23891e8	+	8.65081e8i	0.377771	+	1.00899i
$951$		0			0
$952$		0			0
$953$		0			0			−	1.00000i	$-0.5\pi$
$953$		0			0				1.00000i	$0.5\pi$
$954$		0			0
$955$	1.70456e9	−	3.08638e8i	1.95706	−	0.354355i
$956$		0			0
$957$		0			0
$958$		−	4.08061e8i		−	0.464118i
$959$		0			0
$960$		0			0
$961$	8.87504e8			1.00000
$962$		0			0
$963$			1.78387e9i			1.99749i
$964$		0			0
$965$	2.88362e8	+	1.59259e9i	0.320890	+	1.77223i
$966$		0			0
$967$		0			0			−	1.00000i	$-0.5\pi$
$967$		0			0				1.00000i	$0.5\pi$
$968$		−	9.56773e8i		−	1.05483i
$969$		0			0
$970$	1.74422e8	−	3.15818e7i	0.191111	−	0.0346036i
$971$	1.72259e9			1.88159			0.940795	−	0.338976i	$-0.110080\pi$
$971$	1.72259e9			1.88159			0.940795	+	0.338976i	$0.110080\pi$
$972$		0			0
$973$		0			0
$974$		0			0
$975$		0			0
$976$		0			0
$977$			1.31292e9i			1.40784i	0.710277	+	0.703922i	$0.248570\pi$
$977$			1.31292e9i			1.40784i	−0.710277	+	0.703922i	$0.751430\pi$
$978$		0			0
$979$		0			0
$980$	−6.82103e8	+	1.23505e8i	−0.724722	+	0.131222i
$981$	−1.45758e8			−0.154392
$982$		0			0
$983$		0			0			−	1.00000i	$-0.5\pi$
$983$		0			0				1.00000i	$0.5\pi$
$984$		0			0
$985$		0			0
$986$		0			0
$987$		0			0
$988$		0			0
$989$		0			0
$990$		0			0
$991$		0			0		1.00000			$0$
$991$		0			0		−1.00000			$\pi$
$992$		−	8.81263e8i		−	0.902757i
$993$		0			0
$994$	8.45714e8			0.861121
$995$		0			0
$996$		0			0
$997$			1.70630e9i			1.72175i	0.508818	+	0.860874i	$0.330083\pi$
$997$			1.70630e9i			1.72175i	−0.508818	+	0.860874i	$0.669917\pi$
$998$		0			0
$999$		0			0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	155.7.c.a.154.1	✓	2
5.4	even	2	inner	155.7.c.a.154.2	yes	2
31.30	odd	2	CM	155.7.c.a.154.1	✓	2
155.154	odd	2	inner	155.7.c.a.154.2	yes	2

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
155.7.c.a.154.1	✓	2	1.1	even	1	trivial
155.7.c.a.154.1	✓	2	31.30	odd	2	CM
155.7.c.a.154.2	yes	2	5.4	even	2	inner
155.7.c.a.154.2	yes	2	155.154	odd	2	inner

Overview	Random
Universe	Knowledge

Classical	Maass
Hilbert	Bianchi

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

Level:	\( N \)	\(=\)	\( 155 = 5 \cdot 31 \)
Weight:	\( k \)	\(=\)	\( 7 \)
Character orbit:	\([\chi]\)	\(=\)	155.c (of order \(2\), degree \(1\), minimal)

\(f(q)\)	\(=\)	\(q-5.56776i q^{2} +33.0000 q^{4} +(123.000 - 22.2711i) q^{5} -534.505i q^{7} -540.073i q^{8} -729.000 q^{9} +(-124.000 - 684.835i) q^{10} -2976.00 q^{14} -895.000 q^{16} +4058.90i q^{18} -10618.0 q^{19} +(4059.00 - 734.945i) q^{20} +(14633.0 - 5478.68i) q^{25} -17638.7i q^{28} +29791.0 q^{31} -29581.5i q^{32} +(-11904.0 - 65744.2i) q^{35} -24057.0 q^{36} +59118.5i q^{38} +(-12028.0 - 66429.0i) q^{40} -60558.0 q^{41} +(-89667.0 + 16235.6i) q^{45} +116611. i q^{47} -168047. q^{49} +(-30504.0 - 81473.1i) q^{50} -288672. q^{56} +136842. q^{59} -165869. i q^{62} +389654. i q^{63} -221983. q^{64} -586486. i q^{67} +(-366048. + 66278.7i) q^{70} -284178. q^{71} +393713. i q^{72} -350394. q^{76} +(-110085. + 19932.6i) q^{80} +531441. q^{81} +337173. i q^{82} +(90396.0 + 499245. i) q^{90} +649264. q^{94} +(-1.30601e6 + 236474. i) q^{95} +254692. i q^{97} +935646. i q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 2 q + 66 q^{4} + 246 q^{5} - 1458 q^{9} - 248 q^{10} - 5952 q^{14} - 1790 q^{16} - 21236 q^{19} + 8118 q^{20} + 29266 q^{25} + 59582 q^{31} - 23808 q^{35} - 48114 q^{36} - 24056 q^{40} - 121116 q^{41} - 179334 q^{45}+ \cdots - 2612028 q^{95}+O(q^{100}) \) 2 * q + 66 * q^4 + 246 * q^5 - 1458 * q^9 - 248 * q^10 - 5952 * q^14 - 1790 * q^16 - 21236 * q^19 + 8118 * q^20 + 29266 * q^25 + 59582 * q^31 - 23808 * q^35 - 48114 * q^36 - 24056 * q^40 - 121116 * q^41 - 179334 * q^45 - 336094 * q^49 - 61008 * q^50 - 577344 * q^56 + 273684 * q^59 - 443966 * q^64 - 732096 * q^70 - 568356 * q^71 - 700788 * q^76 - 220170 * q^80 + 1062882 * q^81 + 180792 * q^90 + 1298528 * q^94 - 2612028 * q^95

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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