LMFDB - Embedded newform 225.5.d.c.224.11

Show commands: Magma / Pari/GP / SageMath

Newspace parameters

comment:Compute space of new eigenforms

gp:[N,k,chi] = [225,5,Mod(224,225)] mf = mfinit([N,k,chi],0) lf = mfeigenbasis(mf)

sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(225, base_ring=CyclotomicField(2)) chi = DirichletCharacter(H, H._module([1, 1])) N = Newforms(chi, 5, names="a")

magma://Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code chi := DirichletCharacter("225.224"); S:= CuspForms(chi, 5); N := Newforms(S);

Level:	$ N $	$=$	$ 225 = 3^{2} \cdot 5^{2} $
Weight:	$ k $	$=$	$ 5 $
Character orbit:	$[\chi]$	$=$	225.d (of order $2$, degree $1$, not minimal)

Newform invariants

comment:select newform

sage:traces = [12] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(1)] == traces)

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

Self dual:	no
Analytic conductor:	$23.2582416939$
Analytic rank:	$0$
Dimension:	$12$
Coefficient field:	$\mathbb{Q}[x]/(x^{12} + \cdots)$
comment:defining polynomial gp:f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{12} + 44 x^{10} - 24 x^{9} + 968 x^{8} - 132 x^{7} - 10486 x^{6} + 2904 x^{5} + 56980 x^{4} + \cdots + 162 $ x^12 + 44x^10 - 24x^9 + 968x^8 - 132x^7 - 10486x^6 + 2904x^5 + 56980x^4 - 31836x^3 + 13068x^2 - 2376x + 162
Coefficient ring:	$\Z[a_1, \ldots, a_{13}]$
Coefficient ring index:	$ 2^{12}\cdot 3^{10}\cdot 5^{4} $
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label			224.11
Root			$2.27380 - 0.941838i$ of defining polynomial
Character	$\chi$	$=$	225.224
Dual form			225.5.d.c.224.12

$q$-expansion

comment:q-expansion

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

gp:mfcoefs(f, 20)

$f(q)$	$=$	$q+7.84548 q^{2} +45.5516 q^{4} -71.8372i q^{7} +231.847 q^{8} -66.0480i q^{11} +143.700i q^{13} -563.598i q^{14} +1090.12 q^{16} +88.1175 q^{17} -397.835 q^{19} -518.178i q^{22} +189.182 q^{23} +1127.39i q^{26} -3272.30i q^{28} +649.431i q^{29} +508.987 q^{31} +4843.01 q^{32} +691.325 q^{34} +1273.43i q^{37} -3121.21 q^{38} +1532.30i q^{41} +1275.35i q^{43} -3008.59i q^{44} +1484.23 q^{46} -3096.33 q^{47} -2759.58 q^{49} +6545.75i q^{52} +940.508 q^{53} -16655.2i q^{56} +5095.10i q^{58} -5071.76i q^{59} -625.387 q^{61} +3993.25 q^{62} +20553.7 q^{64} -2767.03i q^{67} +4013.90 q^{68} -2565.51i q^{71} +7892.47i q^{73} +9990.67i q^{74} -18122.0 q^{76} -4744.70 q^{77} -7074.77 q^{79} +12021.6i q^{82} -10466.7 q^{83} +10005.7i q^{86} -15313.0i q^{88} -14076.6i q^{89} +10323.0 q^{91} +8617.55 q^{92} -24292.2 q^{94} +15128.5i q^{97} -21650.3 q^{98} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 12 q + 184 q^{4} + 3632 q^{16} + 500 q^{19} + 3692 q^{31} + 7384 q^{34} + 19512 q^{46} + 6464 q^{49} + 21676 q^{61} + 72288 q^{64} - 49560 q^{76} - 7808 q^{79} + 34612 q^{91} - 132728 q^{94}+O(q^{100}) $ 12 * q + 184 * q^4 + 3632 * q^16 + 500 * q^19 + 3692 * q^31 + 7384 * q^34 + 19512 * q^46 + 6464 * q^49 + 21676 * q^61 + 72288 * q^64 - 49560 * q^76 - 7808 * q^79 + 34612 * q^91 - 132728 * q^94

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/225\mathbb{Z}\right)^\times$.

$n$	$101$	$127$
$\chi(n)$	$-1$	$-1$

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 $

$2$	7.84548	1.96137	0.980686	−	0.195591i	$-0.0626625\pi$
$2$	7.84548	1.96137	0.980686	+	0.195591i	$0.0626625\pi$
$3$	0	0
$3$	0	0
$4$	45.5516	2.84698
$5$	0	0
$5$	0	0
$6$	0	0
$7$	− 71.8372i	− 1.46607i	−0.680194	−	0.733033i	$-0.738104\pi$
$7$	− 71.8372i	− 1.46607i	0.680194	−	0.733033i	$-0.261896\pi$
$8$	231.847	3.62261
$9$	0	0
$10$	0	0
$11$	− 66.0480i	− 0.545851i	−0.962035	−	0.272925i	$-0.912009\pi$
$11$	− 66.0480i	− 0.545851i	0.962035	−	0.272925i	$-0.0879913\pi$
$12$	0	0
$13$	143.700i	0.850293i	0.905124	+	0.425147i	$0.139777\pi$
$13$	143.700i	0.850293i	−0.905124	+	0.425147i	$0.860223\pi$
$14$	− 563.598i	− 2.87550i
$15$	0	0
$16$	1090.12	4.25830
$17$	88.1175	0.304905	0.152452	−	0.988311i	$-0.451283\pi$
$17$	88.1175	0.304905	0.152452	+	0.988311i	$0.451283\pi$
$18$	0	0
$19$	−397.835	−1.10204	−0.551018	−	0.834493i	$-0.685761\pi$
$19$	−397.835	−1.10204	−0.551018	+	0.834493i	$0.685761\pi$
$20$	0	0
$21$	0	0
$22$	− 518.178i	− 1.07062i
$23$	189.182	0.357622	0.178811	−	0.983883i	$-0.442775\pi$
$23$	189.182	0.357622	0.178811	+	0.983883i	$0.442775\pi$
$24$	0	0
$25$	0	0
$26$	1127.39i	1.66774i
$27$	0	0
$28$	− 3272.30i	− 4.17385i
$29$	649.431i	0.772213i	0.922454	+	0.386106i	$0.126180\pi$
$29$	649.431i	0.772213i	−0.922454	+	0.386106i	$0.873820\pi$
$30$	0	0
$31$	508.987	0.529643	0.264821	−	0.964297i	$-0.414687\pi$
$31$	508.987	0.529643	0.264821	+	0.964297i	$0.414687\pi$
$32$	4843.01	4.72950
$33$	0	0
$34$	691.325	0.598032
$35$	0	0
$36$	0	0
$37$	1273.43i	0.930189i	0.885261	+	0.465095i	$0.153980\pi$
$37$	1273.43i	0.930189i	−0.885261	+	0.465095i	$0.846020\pi$
$38$	−3121.21	−2.16150
$39$	0	0
$40$	0	0
$41$	1532.30i	0.911541i	0.890097	+	0.455770i	$0.150636\pi$
$41$	1532.30i	0.911541i	−0.890097	+	0.455770i	$0.849364\pi$
$42$	0	0
$43$	1275.35i	0.689750i	0.938649	+	0.344875i	$0.112079\pi$
$43$	1275.35i	0.689750i	−0.938649	+	0.344875i	$0.887921\pi$
$44$	− 3008.59i	− 1.55402i
$45$	0	0
$46$	1484.23	0.701430
$47$	−3096.33	−1.40169	−0.700845	−	0.713314i	$-0.747193\pi$
$47$	−3096.33	−1.40169	−0.700845	+	0.713314i	$0.747193\pi$
$48$	0	0
$49$	−2759.58	−1.14935
$50$	0	0
$51$	0	0
$52$	6545.75i	2.42076i
$53$	940.508	0.334820	0.167410	−	0.985887i	$-0.446460\pi$
$53$	940.508	0.334820	0.167410	+	0.985887i	$0.446460\pi$
$54$	0	0
$55$	0	0
$56$	− 16655.2i	− 5.31098i
$57$	0	0
$58$	5095.10i	1.51460i
$59$	− 5071.76i	− 1.45698i	−0.685055	−	0.728491i	$-0.740222\pi$
$59$	− 5071.76i	− 1.45698i	0.685055	−	0.728491i	$-0.259778\pi$
$60$	0	0
$61$	−625.387	−0.168070	−0.0840349	−	0.996463i	$-0.526781\pi$
$61$	−625.387	−0.168070	−0.0840349	+	0.996463i	$0.526781\pi$
$62$	3993.25	1.03883
$63$	0	0
$64$	20553.7	5.01800
$65$	0	0
$66$	0	0
$67$	− 2767.03i	− 0.616402i	−0.951321	−	0.308201i	$-0.900273\pi$
$67$	− 2767.03i	− 0.616402i	0.951321	−	0.308201i	$-0.0997269\pi$
$68$	4013.90	0.868057
$69$	0	0
$70$	0	0
$71$	− 2565.51i	− 0.508928i	−0.967082	−	0.254464i	$-0.918101\pi$
$71$	− 2565.51i	− 0.508928i	0.967082	−	0.254464i	$-0.0818991\pi$
$72$	0	0
$73$	7892.47i	1.48104i	0.672034	+	0.740521i	$0.265421\pi$
$73$	7892.47i	1.48104i	−0.672034	+	0.740521i	$0.734579\pi$
$74$	9990.67i	1.82445i
$75$	0	0
$76$	−18122.0	−3.13747
$77$	−4744.70	−0.800253
$78$	0	0
$79$	−7074.77	−1.13359	−0.566797	−	0.823857i	$-0.691818\pi$
$79$	−7074.77	−1.13359	−0.566797	+	0.823857i	$0.691818\pi$
$80$	0	0
$81$	0	0
$82$	12021.6i	1.78787i
$83$	−10466.7	−1.51934	−0.759670	−	0.650308i	$-0.774640\pi$
$83$	−10466.7	−1.51934	−0.759670	+	0.650308i	$0.774640\pi$
$84$	0	0
$85$	0	0
$86$	10005.7i	1.35286i
$87$	0	0
$88$	− 15313.0i	− 1.97740i
$89$	− 14076.6i	− 1.77712i	−0.458757	−	0.888562i	$-0.651705\pi$
$89$	− 14076.6i	− 1.77712i	0.458757	−	0.888562i	$-0.348295\pi$
$90$	0	0
$91$	10323.0	1.24659
$92$	8617.55	1.01814
$93$	0	0
$94$	−24292.2	−2.74923
$95$	0	0
$96$	0	0
$97$	15128.5i	1.60787i	0.594715	+	0.803936i	$0.297265\pi$
$97$	15128.5i	1.60787i	−0.594715	+	0.803936i	$0.702735\pi$
$98$	−21650.3	−2.25430
$99$	0	0
$100$	0	0
$101$	12428.3i	1.21834i	0.793039	+	0.609171i	$0.208498\pi$
$101$	12428.3i	1.21834i	−0.793039	+	0.609171i	$0.791502\pi$
$102$	0	0
$103$	4846.79i	0.456856i	0.973561	+	0.228428i	$0.0733586\pi$
$103$	4846.79i	0.456856i	−0.973561	+	0.228428i	$0.926641\pi$
$104$	33316.3i	3.08028i
$105$	0	0
$106$	7378.74	0.656706
$107$	−11599.8	−1.01317	−0.506586	−	0.862190i	$-0.669093\pi$
$107$	−11599.8	−1.01317	−0.506586	+	0.862190i	$0.669093\pi$
$108$	0	0
$109$	−3965.61	−0.333778	−0.166889	−	0.985976i	$-0.553372\pi$
$109$	−3965.61	−0.333778	−0.166889	+	0.985976i	$0.553372\pi$
$110$	0	0
$111$	0	0
$112$	− 78311.5i	− 6.24294i
$113$	16208.3	1.26935	0.634675	−	0.772779i	$-0.281134\pi$
$113$	16208.3	1.26935	0.634675	+	0.772779i	$0.281134\pi$
$114$	0	0
$115$	0	0
$116$	29582.6i	2.19847i
$117$	0	0
$118$	− 39790.4i	− 2.85768i
$119$	− 6330.12i	− 0.447010i
$120$	0	0
$121$	10278.7	0.702047
$122$	−4906.47	−0.329647
$123$	0	0
$124$	23185.2	1.50788
$125$	0	0
$126$	0	0
$127$	13904.3i	0.862065i	0.902336	+	0.431033i	$0.141851\pi$
$127$	13904.3i	0.862065i	−0.902336	+	0.431033i	$0.858149\pi$
$128$	83765.9	5.11266
$129$	0	0
$130$	0	0
$131$	− 28801.6i	− 1.67831i	−0.543889	−	0.839157i	$-0.683049\pi$
$131$	− 28801.6i	− 1.67831i	0.543889	−	0.839157i	$-0.316951\pi$
$132$	0	0
$133$	28579.4i	1.61566i
$134$	− 21708.7i	− 1.20899i
$135$	0	0
$136$	20429.8	1.10455
$137$	13666.9	0.728166	0.364083	−	0.931367i	$-0.381382\pi$
$137$	13666.9	0.728166	0.364083	+	0.931367i	$0.381382\pi$
$138$	0	0
$139$	−20672.4	−1.06994	−0.534971	−	0.844870i	$-0.679677\pi$
$139$	−20672.4	−1.06994	−0.534971	+	0.844870i	$0.679677\pi$
$140$	0	0
$141$	0	0
$142$	− 20127.7i	− 0.998198i
$143$	9491.06	0.464133
$144$	0	0
$145$	0	0
$146$	61920.2i	2.90487i
$147$	0	0
$148$	58006.8i	2.64823i
$149$	− 28936.3i	− 1.30338i	−0.758486	−	0.651689i	$-0.774061\pi$
$149$	− 28936.3i	− 1.30338i	0.758486	−	0.651689i	$-0.225939\pi$
$150$	0	0
$151$	23469.8	1.02933	0.514666	−	0.857391i	$-0.327916\pi$
$151$	23469.8	1.02933	0.514666	+	0.857391i	$0.327916\pi$
$152$	−92236.8	−3.99224
$153$	0	0
$154$	−37224.5	−1.56959
$155$	0	0
$156$	0	0
$157$	− 16398.7i	− 0.665288i	−0.943053	−	0.332644i	$-0.892059\pi$
$157$	− 16398.7i	− 0.665288i	0.943053	−	0.332644i	$-0.107941\pi$
$158$	−55505.0	−2.22340
$159$	0	0
$160$	0	0
$161$	− 13590.3i	− 0.524297i
$162$	0	0
$163$	13316.8i	0.501217i	0.968089	+	0.250608i	$0.0806306\pi$
$163$	13316.8i	0.501217i	−0.968089	+	0.250608i	$0.919369\pi$
$164$	69798.8i	2.59514i
$165$	0	0
$166$	−82116.6	−2.97999
$167$	4438.33	0.159143	0.0795713	−	0.996829i	$-0.474645\pi$
$167$	4438.33	0.159143	0.0795713	+	0.996829i	$0.474645\pi$
$168$	0	0
$169$	7911.44	0.277001
$170$	0	0
$171$	0	0
$172$	58094.1i	1.96370i
$173$	4543.18	0.151799	0.0758993	−	0.997115i	$-0.475817\pi$
$173$	4543.18	0.151799	0.0758993	+	0.997115i	$0.475817\pi$
$174$	0	0
$175$	0	0
$176$	− 72000.5i	− 2.32440i
$177$	0	0
$178$	− 110438.i	− 3.48560i
$179$	− 2146.50i	− 0.0669922i	−0.999439	−	0.0334961i	$-0.989336\pi$
$179$	− 2146.50i	− 0.0669922i	0.999439	−	0.0334961i	$-0.0106641\pi$
$180$	0	0
$181$	−25652.0	−0.783003	−0.391502	−	0.920177i	$-0.628044\pi$
$181$	−25652.0	−0.783003	−0.391502	+	0.920177i	$0.628044\pi$
$182$	80988.7	2.44502
$183$	0	0
$184$	43861.3	1.29552
$185$	0	0
$186$	0	0
$187$	− 5819.98i	− 0.166433i
$188$	−141043.	−3.99058
$189$	0	0
$190$	0	0
$191$	− 9988.75i	− 0.273807i	−0.990584	−	0.136904i	$-0.956285\pi$
$191$	− 9988.75i	− 0.273807i	0.990584	−	0.136904i	$-0.0437150\pi$
$192$	0	0
$193$	− 16130.0i	− 0.433032i	−0.976279	−	0.216516i	$-0.930531\pi$
$193$	− 16130.0i	− 0.433032i	0.976279	−	0.216516i	$-0.0694693\pi$
$194$	118690.i	3.15364i
$195$	0	0
$196$	−125703.	−3.27216
$197$	−51708.2	−1.33238	−0.666188	−	0.745784i	$-0.732075\pi$
$197$	−51708.2	−1.33238	−0.666188	+	0.745784i	$0.732075\pi$
$198$	0	0
$199$	70238.8	1.77366	0.886831	−	0.462094i	$-0.152902\pi$
$199$	70238.8	1.77366	0.886831	+	0.462094i	$0.152902\pi$
$200$	0	0
$201$	0	0
$202$	97506.1i	2.38962i
$203$	46653.3	1.13211
$204$	0	0
$205$	0	0
$206$	38025.4i	0.896065i
$207$	0	0
$208$	156650.i	3.62080i
$209$	26276.2i	0.601548i
$210$	0	0
$211$	26207.3	0.588651	0.294325	−	0.955705i	$-0.404905\pi$
$211$	26207.3	0.588651	0.294325	+	0.955705i	$0.404905\pi$
$212$	42841.7	0.953224
$213$	0	0
$214$	−91006.0	−1.98720
$215$	0	0
$216$	0	0
$217$	− 36564.2i	− 0.776491i
$218$	−31112.1	−0.654662
$219$	0	0
$220$	0	0
$221$	12662.4i	0.259259i
$222$	0	0
$223$	17542.0i	0.352752i	0.984323	+	0.176376i	$0.0564375\pi$
$223$	17542.0i	0.352752i	−0.984323	+	0.176376i	$0.943562\pi$
$224$	− 347908.i	− 6.93375i
$225$	0	0
$226$	127162.	2.48967
$227$	−13248.7	−0.257111	−0.128555	−	0.991702i	$-0.541034\pi$
$227$	−13248.7	−0.257111	−0.128555	+	0.991702i	$0.541034\pi$
$228$	0	0
$229$	66842.0	1.27461	0.637307	−	0.770610i	$-0.280048\pi$
$229$	66842.0	1.27461	0.637307	+	0.770610i	$0.280048\pi$
$230$	0	0
$231$	0	0
$232$	150568.i	2.79742i
$233$	56549.7	1.04164	0.520821	−	0.853666i	$-0.325626\pi$
$233$	56549.7	1.04164	0.520821	+	0.853666i	$0.325626\pi$
$234$	0	0
$235$	0	0
$236$	− 231027.i	− 4.14800i
$237$	0	0
$238$	− 49662.8i	− 0.876753i
$239$	− 28451.5i	− 0.498092i	−0.968492	−	0.249046i	$-0.919883\pi$
$239$	− 28451.5i	− 0.498092i	0.968492	−	0.249046i	$-0.0801170\pi$
$240$	0	0
$241$	−10232.9	−0.176183	−0.0880915	−	0.996112i	$-0.528077\pi$
$241$	−10232.9	−0.176183	−0.0880915	+	0.996112i	$0.528077\pi$
$242$	80641.1	1.37697
$243$	0	0
$244$	−28487.4	−0.478491
$245$	0	0
$246$	0	0
$247$	− 57168.7i	− 0.937054i
$248$	118007.	1.91869
$249$	0	0
$250$	0	0
$251$	− 20591.9i	− 0.326850i	−0.986556	−	0.163425i	$-0.947746\pi$
$251$	− 20591.9i	− 0.326850i	0.986556	−	0.163425i	$-0.0522542\pi$
$252$	0	0
$253$	− 12495.1i	− 0.195208i
$254$	109086.i	1.69083i
$255$	0	0
$256$	328324.	5.00983
$257$	17042.7	0.258031	0.129015	−	0.991643i	$-0.458818\pi$
$257$	17042.7	0.258031	0.129015	+	0.991643i	$0.458818\pi$
$258$	0	0
$259$	91479.6	1.36372
$260$	0	0
$261$	0	0
$262$	− 225962.i	− 3.29180i
$263$	91976.6	1.32974	0.664869	−	0.746960i	$-0.268487\pi$
$263$	91976.6	1.32974	0.664869	+	0.746960i	$0.268487\pi$
$264$	0	0
$265$	0	0
$266$	224219.i	3.16890i
$267$	0	0
$268$	− 126043.i	− 1.75488i
$269$	56341.4i	0.778616i	0.921108	+	0.389308i	$0.127286\pi$
$269$	56341.4i	0.778616i	−0.921108	+	0.389308i	$0.872714\pi$
$270$	0	0
$271$	−113809.	−1.54966	−0.774830	−	0.632169i	$-0.782165\pi$
$271$	−113809.	−1.54966	−0.774830	+	0.632169i	$0.782165\pi$
$272$	96059.1	1.29838
$273$	0	0
$274$	107224.	1.42820
$275$	0	0
$276$	0	0
$277$	− 143482.i	− 1.86998i	−0.354677	−	0.934989i	$-0.615409\pi$
$277$	− 143482.i	− 1.86998i	0.354677	−	0.934989i	$-0.384591\pi$
$278$	−162185.	−2.09855
$279$	0	0
$280$	0	0
$281$	− 147781.i	− 1.87157i	−0.352566	−	0.935787i	$-0.614691\pi$
$281$	− 147781.i	− 1.87157i	0.352566	−	0.935787i	$-0.385309\pi$
$282$	0	0
$283$	− 73418.5i	− 0.916712i	−0.888769	−	0.458356i	$-0.848439\pi$
$283$	− 73418.5i	− 0.916712i	0.888769	−	0.458356i	$-0.151561\pi$
$284$	− 116863.i	− 1.44891i
$285$	0	0
$286$	74462.0	0.910338
$287$	110076.	1.33638
$288$	0	0
$289$	−75756.3	−0.907033
$290$	0	0
$291$	0	0
$292$	359515.i	4.21649i
$293$	22481.5	0.261872	0.130936	−	0.991391i	$-0.458202\pi$
$293$	22481.5	0.261872	0.130936	+	0.991391i	$0.458202\pi$
$294$	0	0
$295$	0	0
$296$	295240.i	3.36971i
$297$	0	0
$298$	− 227019.i	− 2.55641i
$299$	27185.4i	0.304084i
$300$	0	0
$301$	91617.4	1.01122
$302$	184132.	2.01890
$303$	0	0
$304$	−433690.	−4.69280
$305$	0	0
$306$	0	0
$307$	38654.6i	0.410133i	0.978748	+	0.205067i	$0.0657411\pi$
$307$	38654.6i	0.410133i	−0.978748	+	0.205067i	$0.934259\pi$
$308$	−216129.	−2.27830
$309$	0	0
$310$	0	0
$311$	91964.6i	0.950823i	0.879763	+	0.475412i	$0.157701\pi$
$311$	91964.6i	0.950823i	−0.879763	+	0.475412i	$0.842299\pi$
$312$	0	0
$313$	39467.6i	0.402858i	0.979503	+	0.201429i	$0.0645586\pi$
$313$	39467.6i	0.402858i	−0.979503	+	0.201429i	$0.935441\pi$
$314$	− 128656.i	− 1.30488i
$315$	0	0
$316$	−322267.	−3.22732
$317$	−101949.	−1.01453	−0.507263	−	0.861791i	$-0.669343\pi$
$317$	−101949.	−1.01453	−0.507263	+	0.861791i	$0.669343\pi$
$318$	0	0
$319$	42893.6	0.421513
$320$	0	0
$321$	0	0
$322$	− 106623.i	− 1.02834i
$323$	−35056.2	−0.336016
$324$	0	0
$325$	0	0
$326$	104477.i	0.983072i
$327$	0	0
$328$	355259.i	3.30215i
$329$	222432.i	2.05497i
$330$	0	0
$331$	2686.55	0.0245210	0.0122605	−	0.999925i	$-0.496097\pi$
$331$	2686.55	0.0245210	0.0122605	+	0.999925i	$0.496097\pi$
$332$	−476777.	−4.32553
$333$	0	0
$334$	34820.8	0.312138
$335$	0	0
$336$	0	0
$337$	76923.6i	0.677330i	0.940907	+	0.338665i	$0.109975\pi$
$337$	76923.6i	0.677330i	−0.940907	+	0.338665i	$0.890025\pi$
$338$	62069.0	0.543302
$339$	0	0
$340$	0	0
$341$	− 33617.5i	− 0.289106i
$342$	0	0
$343$	25759.6i	0.218953i
$344$	295685.i	2.49869i
$345$	0	0
$346$	35643.4	0.297733
$347$	166236.	1.38060	0.690298	−	0.723526i	$-0.257480\pi$
$347$	166236.	1.38060	0.690298	+	0.723526i	$0.257480\pi$
$348$	0	0
$349$	172458.	1.41590	0.707948	−	0.706264i	$-0.249621\pi$
$349$	172458.	1.41590	0.707948	+	0.706264i	$0.249621\pi$
$350$	0	0
$351$	0	0
$352$	− 319871.i	− 2.58160i
$353$	127971.	1.02698	0.513489	−	0.858096i	$-0.328353\pi$
$353$	127971.	1.02698	0.513489	+	0.858096i	$0.328353\pi$
$354$	0	0
$355$	0	0
$356$	− 641212.i	− 5.05943i
$357$	0	0
$358$	− 16840.3i	− 0.131397i
$359$	− 103906.i	− 0.806220i	−0.915152	−	0.403110i	$-0.867929\pi$
$359$	− 103906.i	− 0.806220i	0.915152	−	0.403110i	$-0.132071\pi$
$360$	0	0
$361$	27951.8	0.214484
$362$	−201252.	−1.53576
$363$	0	0
$364$	470228.	3.54900
$365$	0	0
$366$	0	0
$367$	146305.i	1.08624i	0.839654	+	0.543121i	$0.182758\pi$
$367$	146305.i	1.08624i	−0.839654	+	0.543121i	$0.817242\pi$
$368$	206232.	1.52286
$369$	0	0
$370$	0	0
$371$	− 67563.5i	− 0.490867i
$372$	0	0
$373$	− 114715.i	− 0.824520i	−0.911066	−	0.412260i	$-0.864740\pi$
$373$	− 114715.i	− 0.824520i	0.911066	−	0.412260i	$-0.135260\pi$
$374$	− 45660.6i	− 0.326436i
$375$	0	0
$376$	−717875.	−5.07777
$377$	−93322.9	−0.656607
$378$	0	0
$379$	24642.5	0.171556	0.0857781	−	0.996314i	$-0.472662\pi$
$379$	24642.5	0.171556	0.0857781	+	0.996314i	$0.472662\pi$
$380$	0	0
$381$	0	0
$382$	− 78366.6i	− 0.537037i
$383$	−78503.5	−0.535170	−0.267585	−	0.963534i	$-0.586226\pi$
$383$	−78503.5	−0.535170	−0.267585	+	0.963534i	$0.586226\pi$
$384$	0	0
$385$	0	0
$386$	− 126548.i	− 0.849336i
$387$	0	0
$388$	689127.i	4.57758i
$389$	− 234356.i	− 1.54873i	−0.632737	−	0.774367i	$-0.718068\pi$
$389$	− 234356.i	− 1.54873i	0.632737	−	0.774367i	$-0.281932\pi$
$390$	0	0
$391$	16670.3	0.109041
$392$	−639800.	−4.16363
$393$	0	0
$394$	−405676.	−2.61328
$395$	0	0
$396$	0	0
$397$	− 86679.9i	− 0.549968i	−0.961449	−	0.274984i	$-0.911327\pi$
$397$	− 86679.9i	− 0.549968i	0.961449	−	0.274984i	$-0.0886725\pi$
$398$	551057.	3.47881
$399$	0	0
$400$	0	0
$401$	215728.i	1.34158i	0.741646	+	0.670791i	$0.234045\pi$
$401$	215728.i	1.34158i	−0.741646	+	0.670791i	$0.765955\pi$
$402$	0	0
$403$	73141.1i	0.450352i
$404$	566130.i	3.46859i
$405$	0	0
$406$	366018.	2.22050
$407$	84107.4	0.507745
$408$	0	0
$409$	−319618.	−1.91066	−0.955332	−	0.295535i	$-0.904502\pi$
$409$	−319618.	−1.91066	−0.955332	+	0.295535i	$0.904502\pi$
$410$	0	0
$411$	0	0
$412$	220779.i	1.30066i
$413$	−364341.	−2.13603
$414$	0	0
$415$	0	0
$416$	695938.i	4.02146i
$417$	0	0
$418$	206150.i	1.17986i
$419$	− 215386.i	− 1.22684i	−0.789756	−	0.613421i	$-0.789793\pi$
$419$	− 215386.i	− 1.22684i	0.789756	−	0.613421i	$-0.210207\pi$
$420$	0	0
$421$	−146608.	−0.827169	−0.413584	−	0.910466i	$-0.635723\pi$
$421$	−146608.	−0.827169	−0.413584	+	0.910466i	$0.635723\pi$
$422$	205609.	1.15456
$423$	0	0
$424$	218054.	1.21292
$425$	0	0
$426$	0	0
$427$	44926.1i	0.246401i
$428$	−528390.	−2.88447
$429$	0	0
$430$	0	0
$431$	− 187858.i	− 1.01129i	−0.862742	−	0.505644i	$-0.831255\pi$
$431$	− 187858.i	− 1.01129i	0.862742	−	0.505644i	$-0.168745\pi$
$432$	0	0
$433$	47754.6i	0.254706i	0.991857	+	0.127353i	$0.0406481\pi$
$433$	47754.6i	0.254706i	−0.991857	+	0.127353i	$0.959352\pi$
$434$	− 286864.i	− 1.52299i
$435$	0	0
$436$	−180640.	−0.950257
$437$	−75263.3	−0.394113
$438$	0	0
$439$	−171451.	−0.889632	−0.444816	−	0.895622i	$-0.646731\pi$
$439$	−171451.	−0.889632	−0.444816	+	0.895622i	$0.646731\pi$
$440$	0	0
$441$	0	0
$442$	99343.0i	0.508502i
$443$	−281341.	−1.43359	−0.716797	−	0.697282i	$-0.754393\pi$
$443$	−281341.	−1.43359	−0.716797	+	0.697282i	$0.754393\pi$
$444$	0	0
$445$	0	0
$446$	137626.i	0.691878i
$447$	0	0
$448$	− 1.47652e6i	− 7.35672i
$449$	177112.i	0.878526i	0.898358	+	0.439263i	$0.144760\pi$
$449$	177112.i	0.878526i	−0.898358	+	0.439263i	$0.855240\pi$
$450$	0	0
$451$	101205.	0.497565
$452$	738316.	3.61381
$453$	0	0
$454$	−103942.	−0.504289
$455$	0	0
$456$	0	0
$457$	29607.9i	0.141767i	0.997485	+	0.0708836i	$0.0225819\pi$
$457$	29607.9i	0.141767i	−0.997485	+	0.0708836i	$0.977418\pi$
$458$	524408.	2.49999
$459$	0	0
$460$	0	0
$461$	334343.i	1.57322i	0.617448	+	0.786611i	$0.288166\pi$
$461$	334343.i	1.57322i	−0.617448	+	0.786611i	$0.711834\pi$
$462$	0	0
$463$	− 109119.i	− 0.509022i	−0.967070	−	0.254511i	$-0.918085\pi$
$463$	− 109119.i	− 0.509022i	0.967070	−	0.254511i	$-0.0819146\pi$
$464$	707961.i	3.28831i
$465$	0	0
$466$	443660.	2.04305
$467$	−53031.5	−0.243164	−0.121582	−	0.992581i	$-0.538797\pi$
$467$	−53031.5	−0.243164	−0.121582	+	0.992581i	$0.538797\pi$
$468$	0	0
$469$	−198775.	−0.903685
$470$	0	0
$471$	0	0
$472$	− 1.17587e6i	− 5.27807i
$473$	84234.1	0.376501
$474$	0	0
$475$	0	0
$476$	− 288347.i	− 1.27263i
$477$	0	0
$478$	− 223216.i	− 0.976943i
$479$	55142.8i	0.240335i	0.992754	+	0.120168i	$0.0383432\pi$
$479$	55142.8i	0.240335i	−0.992754	+	0.120168i	$0.961657\pi$
$480$	0	0
$481$	−182991.	−0.790934
$482$	−80282.0	−0.345560
$483$	0	0
$484$	468210.	1.99871
$485$	0	0
$486$	0	0
$487$	285889.i	1.20543i	0.797958	+	0.602713i	$0.205913\pi$
$487$	285889.i	1.20543i	−0.797958	+	0.602713i	$0.794087\pi$
$488$	−144994.	−0.608850
$489$	0	0
$490$	0	0
$491$	185921.i	0.771199i	0.922666	+	0.385599i	$0.126005\pi$
$491$	185921.i	0.771199i	−0.922666	+	0.385599i	$0.873995\pi$
$492$	0	0
$493$	57226.2i	0.235451i
$494$	− 448516.i	− 1.83791i
$495$	0	0
$496$	554859.	2.25538
$497$	−184299.	−0.746122
$498$	0	0
$499$	−204968.	−0.823162	−0.411581	−	0.911373i	$-0.635023\pi$
$499$	−204968.	−0.823162	−0.411581	+	0.911373i	$0.635023\pi$
$500$	0	0
$501$	0	0
$502$	− 161553.i	− 0.641074i
$503$	−144534.	−0.571262	−0.285631	−	0.958340i	$-0.592203\pi$
$503$	−144534.	−0.571262	−0.285631	+	0.958340i	$0.592203\pi$
$504$	0	0
$505$	0	0
$506$	− 98030.1i	− 0.382876i
$507$	0	0
$508$	633361.i	2.45428i
$509$	− 253446.i	− 0.978251i	−0.872214	−	0.489125i	$-0.837316\pi$
$509$	− 253446.i	− 0.978251i	0.872214	−	0.489125i	$-0.162684\pi$
$510$	0	0
$511$	566973.	2.17130
$512$	1.23561e6	4.71347
$513$	0	0
$514$	133708.	0.506094
$515$	0	0
$516$	0	0
$517$	204506.i	0.765114i
$518$	717701.	2.67476
$519$	0	0
$520$	0	0
$521$	195283.i	0.719431i	0.933062	+	0.359716i	$0.117126\pi$
$521$	195283.i	0.719431i	−0.933062	+	0.359716i	$0.882874\pi$
$522$	0	0
$523$	− 134628.i	− 0.492191i	−0.969246	−	0.246095i	$-0.920852\pi$
$523$	− 134628.i	− 0.492191i	0.969246	−	0.246095i	$-0.0791476\pi$
$524$	− 1.31196e6i	− 4.77812i
$525$	0	0
$526$	721601.	2.60811
$527$	44850.6	0.161491
$528$	0	0
$529$	−244051.	−0.872106
$530$	0	0
$531$	0	0
$532$	1.30184e6i	4.59974i
$533$	−220191.	−0.775077
$534$	0	0
$535$	0	0
$536$	− 641526.i	− 2.23298i
$537$	0	0
$538$	442026.i	1.52715i
$539$	182265.i	0.627372i
$540$	0	0
$541$	−206257.	−0.704715	−0.352358	−	0.935865i	$-0.614620\pi$
$541$	−206257.	−0.704715	−0.352358	+	0.935865i	$0.614620\pi$
$542$	−892884.	−3.03946
$543$	0	0
$544$	426754.	1.44205
$545$	0	0
$546$	0	0
$547$	306617.i	1.02476i	0.858759	+	0.512380i	$0.171236\pi$
$547$	306617.i	1.02476i	−0.858759	+	0.512380i	$0.828764\pi$
$548$	622552.	2.07307
$549$	0	0
$550$	0	0
$551$	− 258366.i	− 0.851007i
$552$	0	0
$553$	508231.i	1.66192i
$554$	− 1.12568e6i	− 3.66772i
$555$	0	0
$556$	−941659.	−3.04610
$557$	−268937.	−0.866842	−0.433421	−	0.901191i	$-0.642694\pi$
$557$	−268937.	−0.866842	−0.433421	+	0.901191i	$0.642694\pi$
$558$	0	0
$559$	−183267.	−0.586490
$560$	0	0
$561$	0	0
$562$	− 1.15942e6i	− 3.67085i
$563$	−3899.25	−0.0123017	−0.00615084	−	0.999981i	$-0.501958\pi$
$563$	−3899.25	−0.0123017	−0.00615084	+	0.999981i	$0.501958\pi$
$564$	0	0
$565$	0	0
$566$	− 576004.i	− 1.79801i
$567$	0	0
$568$	− 594805.i	− 1.84365i
$569$	565640.i	1.74709i	0.486742	+	0.873546i	$0.338185\pi$
$569$	565640.i	1.74709i	−0.486742	+	0.873546i	$0.661815\pi$
$570$	0	0
$571$	−241622.	−0.741078	−0.370539	−	0.928817i	$-0.620827\pi$
$571$	−241622.	−0.741078	−0.370539	+	0.928817i	$0.620827\pi$
$572$	432333.	1.32138
$573$	0	0
$574$	863601.	2.62113
$575$	0	0
$576$	0	0
$577$	− 316427.i	− 0.950434i	−0.879869	−	0.475217i	$-0.842370\pi$
$577$	− 316427.i	− 0.950434i	0.879869	−	0.475217i	$-0.157630\pi$
$578$	−594345.	−1.77903
$579$	0	0
$580$	0	0
$581$	751901.i	2.22745i
$582$	0	0
$583$	− 62118.7i	− 0.182762i
$584$	1.82984e6i	5.36523i
$585$	0	0
$586$	176378.	0.513629
$587$	410192.	1.19045	0.595225	−	0.803559i	$-0.297063\pi$
$587$	410192.	1.19045	0.595225	+	0.803559i	$0.297063\pi$
$588$	0	0
$589$	−202493.	−0.583686
$590$	0	0
$591$	0	0
$592$	1.38820e6i	3.96102i
$593$	234803.	0.667719	0.333860	−	0.942623i	$-0.391649\pi$
$593$	234803.	0.667719	0.333860	+	0.942623i	$0.391649\pi$
$594$	0	0
$595$	0	0
$596$	− 1.31810e6i	− 3.71069i
$597$	0	0
$598$	213283.i	0.596421i
$599$	− 168621.i	− 0.469957i	−0.972001	−	0.234979i	$-0.924498\pi$
$599$	− 168621.i	− 0.469957i	0.972001	−	0.234979i	$-0.0755020\pi$
$600$	0	0
$601$	−215119.	−0.595567	−0.297783	−	0.954633i	$-0.596247\pi$
$601$	−215119.	−0.595567	−0.297783	+	0.954633i	$0.596247\pi$
$602$	718783.	1.98337
$603$	0	0
$604$	1.06909e6	2.93048
$605$	0	0
$606$	0	0
$607$	− 42983.3i	− 0.116660i	−0.998297	−	0.0583300i	$-0.981422\pi$
$607$	− 42983.3i	− 0.116660i	0.998297	−	0.0583300i	$-0.0185776\pi$
$608$	−1.92672e6	−5.21208
$609$	0	0
$610$	0	0
$611$	− 444942.i	− 1.19185i
$612$	0	0
$613$	− 112093.i	− 0.298302i	−0.988814	−	0.149151i	$-0.952346\pi$
$613$	− 112093.i	− 0.298302i	0.988814	−	0.149151i	$-0.0476540\pi$
$614$	303264.i	0.804423i
$615$	0	0
$616$	−1.10004e6	−2.89900
$617$	362760.	0.952904	0.476452	−	0.879201i	$-0.341923\pi$
$617$	362760.	0.952904	0.476452	+	0.879201i	$0.341923\pi$
$618$	0	0
$619$	−42918.9	−0.112013	−0.0560063	−	0.998430i	$-0.517837\pi$
$619$	−42918.9	−0.112013	−0.0560063	+	0.998430i	$0.517837\pi$
$620$	0	0
$621$	0	0
$622$	721507.i	1.86492i
$623$	−1.01122e6	−2.60538
$624$	0	0
$625$	0	0
$626$	309643.i	0.790154i
$627$	0	0
$628$	− 746986.i	− 1.89406i
$629$	112211.i	0.283619i
$630$	0	0
$631$	−5836.63	−0.0146590	−0.00732948	−	0.999973i	$-0.502333\pi$
$631$	−5836.63	−0.0146590	−0.00732948	+	0.999973i	$0.502333\pi$
$632$	−1.64026e6	−4.10657
$633$	0	0
$634$	−799837.	−1.98986
$635$	0	0
$636$	0	0
$637$	− 396551.i	− 0.977282i
$638$	336521.	0.826744
$639$	0	0
$640$	0	0
$641$	− 12124.4i	− 0.0295082i	−0.999891	−	0.0147541i	$-0.995303\pi$
$641$	− 12124.4i	− 0.0295082i	0.999891	−	0.0147541i	$-0.00469655\pi$
$642$	0	0
$643$	− 582868.i	− 1.40977i	−0.709321	−	0.704885i	$-0.750998\pi$
$643$	− 582868.i	− 1.40977i	0.709321	−	0.704885i	$-0.249002\pi$
$644$	− 619061.i	− 1.49266i
$645$	0	0
$646$	−275033.	−0.659053
$647$	−138936.	−0.331899	−0.165950	−	0.986134i	$-0.553069\pi$
$647$	−138936.	−0.331899	−0.165950	+	0.986134i	$0.553069\pi$
$648$	0	0
$649$	−334979.	−0.795295
$650$	0	0
$651$	0	0
$652$	606603.i	1.42695i
$653$	683920.	1.60391	0.801953	−	0.597388i	$-0.203795\pi$
$653$	683920.	1.60391	0.801953	+	0.597388i	$0.203795\pi$
$654$	0	0
$655$	0	0
$656$	1.67040e6i	3.88161i
$657$	0	0
$658$	1.74509e6i	4.03056i
$659$	− 123857.i	− 0.285201i	−0.989780	−	0.142601i	$-0.954454\pi$
$659$	− 123857.i	− 0.285201i	0.989780	−	0.142601i	$-0.0455464\pi$
$660$	0	0
$661$	−580923.	−1.32958	−0.664792	−	0.747029i	$-0.731480\pi$
$661$	−580923.	−1.32958	−0.664792	+	0.747029i	$0.731480\pi$
$662$	21077.3	0.0480948
$663$	0	0
$664$	−2.42668e6	−5.50397
$665$	0	0
$666$	0	0
$667$	122861.i	0.276160i
$668$	202173.	0.453075
$669$	0	0
$670$	0	0
$671$	41305.6i	0.0917410i
$672$	0	0
$673$	− 269429.i	− 0.594859i	−0.954744	−	0.297429i	$-0.903871\pi$
$673$	− 269429.i	− 0.594859i	0.954744	−	0.297429i	$-0.0961293\pi$
$674$	603503.i	1.32849i
$675$	0	0
$676$	360379.	0.788616
$677$	−660828.	−1.44182	−0.720910	−	0.693029i	$-0.756276\pi$
$677$	−660828.	−1.44182	−0.720910	+	0.693029i	$0.756276\pi$
$678$	0	0
$679$	1.08679e6	2.35725
$680$	0	0
$681$	0	0
$682$	− 263746.i	− 0.567044i
$683$	−476650.	−1.02178	−0.510891	−	0.859645i	$-0.670684\pi$
$683$	−476650.	−1.02178	−0.510891	+	0.859645i	$0.670684\pi$
$684$	0	0
$685$	0	0
$686$	202096.i	0.429447i
$687$	0	0
$688$	1.39029e6i	2.93716i
$689$	135151.i	0.284695i
$690$	0	0
$691$	473634.	0.991942	0.495971	−	0.868339i	$-0.334812\pi$
$691$	473634.	0.991942	0.495971	+	0.868339i	$0.334812\pi$
$692$	206949.	0.432167
$693$	0	0
$694$	1.30420e6	2.70786
$695$	0	0
$696$	0	0
$697$	135022.i	0.277933i
$698$	1.35301e6	2.77710
$699$	0	0
$700$	0	0
$701$	− 403961.i	− 0.822061i	−0.911622	−	0.411030i	$-0.865169\pi$
$701$	− 403961.i	− 0.822061i	0.911622	−	0.411030i	$-0.134831\pi$
$702$	0	0
$703$	− 506615.i	− 1.02510i
$704$	− 1.35753e6i	− 2.73908i
$705$	0	0
$706$	1.00399e6	2.01428
$707$	892815.	1.78617
$708$	0	0
$709$	908566.	1.80744	0.903720	−	0.428123i	$-0.140825\pi$
$709$	908566.	1.80744	0.903720	+	0.428123i	$0.140825\pi$
$710$	0	0
$711$	0	0
$712$	− 3.26361e6i	− 6.43782i
$713$	96291.2	0.189412
$714$	0	0
$715$	0	0
$716$	− 97776.4i	− 0.190725i
$717$	0	0
$718$	− 815196.i	− 1.58130i
$719$	− 177669.i	− 0.343679i	−0.985125	−	0.171840i	$-0.945029\pi$
$719$	− 177669.i	− 0.343679i	0.985125	−	0.171840i	$-0.0549711\pi$
$720$	0	0
$721$	348180.	0.669781
$722$	219296.	0.420684
$723$	0	0
$724$	−1.16849e6	−2.22919
$725$	0	0
$726$	0	0
$727$	120930.i	0.228805i	0.993434	+	0.114402i	$0.0364953\pi$
$727$	120930.i	0.228805i	−0.993434	+	0.114402i	$0.963505\pi$
$728$	2.39335e6	4.51589
$729$	0	0
$730$	0	0
$731$	112380.i	0.210308i
$732$	0	0
$733$	− 152351.i	− 0.283555i	−0.989899	−	0.141778i	$-0.954718\pi$
$733$	− 152351.i	− 0.283555i	0.989899	−	0.141778i	$-0.0452818\pi$
$734$	1.14783e6i	2.13052i
$735$	0	0
$736$	916210.	1.69137
$737$	−182757.	−0.336463
$738$	0	0
$739$	836734.	1.53214	0.766070	−	0.642757i	$-0.222210\pi$
$739$	836734.	1.53214	0.766070	+	0.642757i	$0.222210\pi$
$740$	0	0
$741$	0	0
$742$	− 530068.i	− 0.962773i
$743$	1.07071e6	1.93952	0.969761	−	0.244056i	$-0.0784780\pi$
$743$	1.07071e6	1.93952	0.969761	+	0.244056i	$0.0784780\pi$
$744$	0	0
$745$	0	0
$746$	− 899992.i	− 1.61719i
$747$	0	0
$748$	− 265110.i	− 0.473830i
$749$	833297.i	1.48537i
$750$	0	0
$751$	−161149.	−0.285724	−0.142862	−	0.989743i	$-0.545630\pi$
$751$	−161149.	−0.285724	−0.142862	+	0.989743i	$0.545630\pi$
$752$	−3.37539e6	−5.96881
$753$	0	0
$754$	−732164.	−1.28785
$755$	0	0
$756$	0	0
$757$	− 931899.i	− 1.62621i	−0.582116	−	0.813106i	$-0.697775\pi$
$757$	− 931899.i	− 1.62621i	0.582116	−	0.813106i	$-0.302225\pi$
$758$	193332.	0.336486
$759$	0	0
$760$	0	0
$761$	626672.i	1.08211i	0.840987	+	0.541055i	$0.181975\pi$
$761$	626672.i	1.08211i	−0.840987	+	0.541055i	$0.818025\pi$
$762$	0	0
$763$	284878.i	0.489340i
$764$	− 455004.i	− 0.779522i
$765$	0	0
$766$	−615898.	−1.04967
$767$	728809.	1.23886
$768$	0	0
$769$	274367.	0.463958	0.231979	−	0.972721i	$-0.425480\pi$
$769$	274367.	0.463958	0.231979	+	0.972721i	$0.425480\pi$
$770$	0	0
$771$	0	0
$772$	− 734748.i	− 1.23283i
$773$	−208073.	−0.348222	−0.174111	−	0.984726i	$-0.555705\pi$
$773$	−208073.	−0.348222	−0.174111	+	0.984726i	$0.555705\pi$
$774$	0	0
$775$	0	0
$776$	3.50749e6i	5.82469i
$777$	0	0
$778$	− 1.83864e6i	− 3.03764i
$779$	− 609603.i	− 1.00455i
$780$	0	0
$781$	−169447.	−0.277799
$782$	130786.	0.213869
$783$	0	0
$784$	−3.00829e6	−4.89426
$785$	0	0
$786$	0	0
$787$	352005.i	0.568329i	0.958776	+	0.284164i	$0.0917161\pi$
$787$	352005.i	0.568329i	−0.958776	+	0.284164i	$0.908284\pi$
$788$	−2.35539e6	−3.79324
$789$	0	0
$790$	0	0
$791$	− 1.16436e6i	− 1.86095i
$792$	0	0
$793$	− 89867.9i	− 0.142909i
$794$	− 680046.i	− 1.07869i
$795$	0	0
$796$	3.19949e6	5.04957
$797$	−32487.5	−0.0511445	−0.0255723	−	0.999673i	$-0.508141\pi$
$797$	−32487.5	−0.0511445	−0.0255723	+	0.999673i	$0.508141\pi$
$798$	0	0
$799$	−272841.	−0.427382
$800$	0	0
$801$	0	0
$802$	1.69249e6i	2.63134i
$803$	521282.	0.808428
$804$	0	0
$805$	0	0
$806$	573828.i	0.883307i
$807$	0	0
$808$	2.88146e6i	4.41357i
$809$	816191.i	1.24708i	0.781791	+	0.623540i	$0.214306\pi$
$809$	816191.i	1.24708i	−0.781791	+	0.623540i	$0.785694\pi$
$810$	0	0
$811$	−827994.	−1.25888	−0.629442	−	0.777048i	$-0.716716\pi$
$811$	−827994.	−1.25888	−0.629442	+	0.777048i	$0.716716\pi$
$812$	2.12513e6	3.22310
$813$	0	0
$814$	659863.	0.995876
$815$	0	0
$816$	0	0
$817$	− 507378.i	− 0.760129i
$818$	−2.50756e6	−3.74752
$819$	0	0
$820$	0	0
$821$	− 547104.i	− 0.811678i	−0.913945	−	0.405839i	$-0.866979\pi$
$821$	− 547104.i	− 0.811678i	0.913945	−	0.405839i	$-0.133021\pi$
$822$	0	0
$823$	270702.i	0.399662i	0.979830	+	0.199831i	$0.0640393\pi$
$823$	270702.i	0.399662i	−0.979830	+	0.199831i	$0.935961\pi$
$824$	1.12371e6i	1.65501i
$825$	0	0
$826$	−2.85843e6	−4.18955
$827$	883999.	1.29253	0.646266	−	0.763113i	$-0.276330\pi$
$827$	883999.	1.29253	0.646266	+	0.763113i	$0.276330\pi$
$828$	0	0
$829$	−1.01480e6	−1.47663	−0.738314	−	0.674457i	$-0.764378\pi$
$829$	−1.01480e6	−1.47663	−0.738314	+	0.674457i	$0.764378\pi$
$830$	0	0
$831$	0	0
$832$	2.95356e6i	4.26677i
$833$	−243168.	−0.350442
$834$	0	0
$835$	0	0
$836$	1.19692e6i	1.71259i
$837$	0	0
$838$	− 1.68980e6i	− 2.40629i
$839$	− 252676.i	− 0.358956i	−0.983762	−	0.179478i	$-0.942559\pi$
$839$	− 252676.i	− 0.358956i	0.983762	−	0.179478i	$-0.0574408\pi$
$840$	0	0
$841$	285520.	0.403687
$842$	−1.15021e6	−1.62239
$843$	0	0
$844$	1.19379e6	1.67588
$845$	0	0
$846$	0	0
$847$	− 738391.i	− 1.02925i
$848$	1.02527e6	1.42576
$849$	0	0
$850$	0	0
$851$	240910.i	0.332656i
$852$	0	0
$853$	− 454214.i	− 0.624256i	−0.950040	−	0.312128i	$-0.898958\pi$
$853$	− 454214.i	− 0.624256i	0.950040	−	0.312128i	$-0.101042\pi$
$854$	352467.i	0.483284i
$855$	0	0
$856$	−2.68938e6	−3.67032
$857$	−858025.	−1.16826	−0.584128	−	0.811661i	$-0.698564\pi$
$857$	−858025.	−1.16826	−0.584128	+	0.811661i	$0.698564\pi$
$858$	0	0
$859$	848325.	1.14968	0.574839	−	0.818267i	$-0.305065\pi$
$859$	848325.	1.14968	0.574839	+	0.818267i	$0.305065\pi$
$860$	0	0
$861$	0	0
$862$	− 1.47384e6i	− 1.98351i
$863$	358961.	0.481977	0.240988	−	0.970528i	$-0.422528\pi$
$863$	358961.	0.481977	0.240988	+	0.970528i	$0.422528\pi$
$864$	0	0
$865$	0	0
$866$	374658.i	0.499573i
$867$	0	0
$868$	− 1.66556e6i	− 2.21065i
$869$	467274.i	0.618774i
$870$	0	0
$871$	397621.	0.524122
$872$	−919414.	−1.20914
$873$	0	0
$874$	−590477.	−0.773001
$875$	0	0
$876$	0	0
$877$	− 382506.i	− 0.497324i	−0.968590	−	0.248662i	$-0.920009\pi$
$877$	− 382506.i	− 0.497324i	0.968590	−	0.248662i	$-0.0799908\pi$
$878$	−1.34511e6	−1.74490
$879$	0	0
$880$	0	0
$881$	− 465371.i	− 0.599580i	−0.954005	−	0.299790i	$-0.903083\pi$
$881$	− 465371.i	− 0.599580i	0.954005	−	0.299790i	$-0.0969167\pi$
$882$	0	0
$883$	320975.i	0.411671i	0.978587	+	0.205835i	$0.0659911\pi$
$883$	320975.i	0.411671i	−0.978587	+	0.205835i	$0.934009\pi$
$884$	576795.i	0.738103i
$885$	0	0
$886$	−2.20726e6	−2.81181
$887$	1.42234e6	1.80783	0.903913	−	0.427716i	$-0.140682\pi$
$887$	1.42234e6	1.80783	0.903913	+	0.427716i	$0.140682\pi$
$888$	0	0
$889$	998842.	1.26384
$890$	0	0
$891$	0	0
$892$	799067.i	1.00428i
$893$	1.23183e6	1.54471
$894$	0	0
$895$	0	0
$896$	− 6.01751e6i	− 7.49550i
$897$	0	0
$898$	1.38953e6i	1.72312i
$899$	330552.i	0.408997i
$900$	0	0
$901$	82875.3	0.102088
$902$	794005.	0.975911
$903$	0	0
$904$	3.75785e6	4.59836
$905$	0	0
$906$	0	0
$907$	1.25408e6i	1.52444i	0.647320	+	0.762218i	$0.275890\pi$
$907$	1.25408e6i	1.52444i	−0.647320	+	0.762218i	$0.724110\pi$
$908$	−603498.	−0.731988
$909$	0	0
$910$	0	0
$911$	− 904900.i	− 1.09035i	−0.838324	−	0.545173i	$-0.816464\pi$
$911$	− 904900.i	− 1.09035i	0.838324	−	0.545173i	$-0.183536\pi$
$912$	0	0
$913$	691307.i	0.829334i
$914$	232288.i	0.278058i
$915$	0	0
$916$	3.04476e6	3.62879
$917$	−2.06902e6	−2.46052
$918$	0	0
$919$	442763.	0.524252	0.262126	−	0.965034i	$-0.415576\pi$
$919$	442763.	0.524252	0.262126	+	0.965034i	$0.415576\pi$
$920$	0	0
$921$	0	0
$922$	2.62308e6i	3.08567i
$923$	368662.	0.432738
$924$	0	0
$925$	0	0
$926$	− 856089.i	− 0.998382i
$927$	0	0
$928$	3.14520e6i	3.65218i
$929$	− 818048.i	− 0.947867i	−0.880561	−	0.473934i	$-0.842834\pi$
$929$	− 818048.i	− 0.947867i	0.880561	−	0.473934i	$-0.157166\pi$
$930$	0	0
$931$	1.09786e6	1.26662
$932$	2.57593e6	2.96553
$933$	0	0
$934$	−416058.	−0.476936
$935$	0	0
$936$	0	0
$937$	− 605604.i	− 0.689778i	−0.938643	−	0.344889i	$-0.887917\pi$
$937$	− 605604.i	− 0.689778i	0.938643	−	0.344889i	$-0.112083\pi$
$938$	−1.55949e6	−1.77246
$939$	0	0
$940$	0	0
$941$	794699.i	0.897477i	0.893663	+	0.448739i	$0.148127\pi$
$941$	794699.i	0.897477i	−0.893663	+	0.448739i	$0.851873\pi$
$942$	0	0
$943$	289884.i	0.325987i
$944$	− 5.52885e6i	− 6.20427i
$945$	0	0
$946$	660857.	0.738457
$947$	−992010.	−1.10616	−0.553078	−	0.833130i	$-0.686547\pi$
$947$	−992010.	−1.10616	−0.553078	+	0.833130i	$0.686547\pi$
$948$	0	0
$949$	−1.13414e6	−1.25932
$950$	0	0
$951$	0	0
$952$	− 1.46762e6i	− 1.61934i
$953$	94165.1	0.103682	0.0518411	−	0.998655i	$-0.483491\pi$
$953$	94165.1	0.103682	0.0518411	+	0.998655i	$0.483491\pi$
$954$	0	0
$955$	0	0
$956$	− 1.29601e6i	− 1.41806i
$957$	0	0
$958$	432622.i	0.471387i
$959$	− 981795.i	− 1.06754i
$960$	0	0
$961$	−664454.	−0.719479
$962$	−1.43565e6	−1.55131
$963$	0	0
$964$	−466125.	−0.501589
$965$	0	0
$966$	0	0
$967$	1.46678e6i	1.56860i	0.620385	+	0.784298i	$0.286977\pi$
$967$	1.46678e6i	1.56860i	−0.620385	+	0.784298i	$0.713023\pi$
$968$	2.38308e6	2.54324
$969$	0	0
$970$	0	0
$971$	414030.i	0.439131i	0.975598	+	0.219565i	$0.0704639\pi$
$971$	414030.i	0.439131i	−0.975598	+	0.219565i	$0.929536\pi$
$972$	0	0
$973$	1.48504e6i	1.56860i
$974$	2.24294e6i	2.36429i
$975$	0	0
$976$	−681750.	−0.715691
$977$	−1.07961e6	−1.13104	−0.565520	−	0.824734i	$-0.691325\pi$
$977$	−1.07961e6	−1.13104	−0.565520	+	0.824734i	$0.691325\pi$
$978$	0	0
$979$	−929731.	−0.970045
$980$	0	0
$981$	0	0
$982$	1.45864e6i	1.51261i
$983$	529376.	0.547844	0.273922	−	0.961752i	$-0.411679\pi$
$983$	529376.	0.547844	0.273922	+	0.961752i	$0.411679\pi$
$984$	0	0
$985$	0	0
$986$	448968.i	0.461808i
$987$	0	0
$988$	− 2.60413e6i	− 2.66777i
$989$	241273.i	0.246670i
$990$	0	0
$991$	263507.	0.268315	0.134157	−	0.990960i	$-0.457167\pi$
$991$	263507.	0.268315	0.134157	+	0.990960i	$0.457167\pi$
$992$	2.46502e6	2.50494
$993$	0	0
$994$	−1.44591e6	−1.46342
$995$	0	0
$996$	0	0
$997$	− 329724.i	− 0.331711i	−0.986150	−	0.165856i	$-0.946961\pi$
$997$	− 329724.i	− 0.331711i	0.986150	−	0.165856i	$-0.0530386\pi$
$998$	−1.60807e6	−1.61453
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	225.5.d.c.224.11		12
3.2	odd	2	inner	225.5.d.c.224.1		12
5.2	odd	4		225.5.c.d.26.6	yes	6
5.3	odd	4		225.5.c.c.26.1	✓	6
5.4	even	2	inner	225.5.d.c.224.2		12
15.2	even	4		225.5.c.d.26.1	yes	6
15.8	even	4		225.5.c.c.26.6	yes	6
15.14	odd	2	inner	225.5.d.c.224.12		12

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
225.5.c.c.26.1	✓	6	5.3	odd	4
225.5.c.c.26.6	yes	6	15.8	even	4
225.5.c.d.26.1	yes	6	15.2	even	4
225.5.c.d.26.6	yes	6	5.2	odd	4
225.5.d.c.224.1		12	3.2	odd	2	inner
225.5.d.c.224.2		12	5.4	even	2	inner
225.5.d.c.224.11		12	1.1	even	1	trivial
225.5.d.c.224.12		12	15.14	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:	\( N \)	\(=\)	\( 225 = 3^{2} \cdot 5^{2} \)
Weight:	\( k \)	\(=\)	\( 5 \)
Character orbit:	\([\chi]\)	\(=\)	225.d (of order \(2\), degree \(1\), not minimal)

\(f(q)\)	\(=\)	\(q+7.84548 q^{2} +45.5516 q^{4} -71.8372i q^{7} +231.847 q^{8} -66.0480i q^{11} +143.700i q^{13} -563.598i q^{14} +1090.12 q^{16} +88.1175 q^{17} -397.835 q^{19} -518.178i q^{22} +189.182 q^{23} +1127.39i q^{26} -3272.30i q^{28} +649.431i q^{29} +508.987 q^{31} +4843.01 q^{32} +691.325 q^{34} +1273.43i q^{37} -3121.21 q^{38} +1532.30i q^{41} +1275.35i q^{43} -3008.59i q^{44} +1484.23 q^{46} -3096.33 q^{47} -2759.58 q^{49} +6545.75i q^{52} +940.508 q^{53} -16655.2i q^{56} +5095.10i q^{58} -5071.76i q^{59} -625.387 q^{61} +3993.25 q^{62} +20553.7 q^{64} -2767.03i q^{67} +4013.90 q^{68} -2565.51i q^{71} +7892.47i q^{73} +9990.67i q^{74} -18122.0 q^{76} -4744.70 q^{77} -7074.77 q^{79} +12021.6i q^{82} -10466.7 q^{83} +10005.7i q^{86} -15313.0i q^{88} -14076.6i q^{89} +10323.0 q^{91} +8617.55 q^{92} -24292.2 q^{94} +15128.5i q^{97} -21650.3 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 12 q + 184 q^{4} + 3632 q^{16} + 500 q^{19} + 3692 q^{31} + 7384 q^{34} + 19512 q^{46} + 6464 q^{49} + 21676 q^{61} + 72288 q^{64} - 49560 q^{76} - 7808 q^{79} + 34612 q^{91} - 132728 q^{94}+O(q^{100}) \) 12 * q + 184 * q^4 + 3632 * q^16 + 500 * q^19 + 3692 * q^31 + 7384 * q^34 + 19512 * q^46 + 6464 * q^49 + 21676 * q^61 + 72288 * q^64 - 49560 * q^76 - 7808 * q^79 + 34612 * q^91 - 132728 * q^94

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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