LMFDB - Embedded newform 225.5.d.c.224.1

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)

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);

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")

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			224.1
Root			$2.27380 + 5.48943i$ of defining polynomial
Character	$\chi$	$=$	225.224
Dual form			225.5.d.c.224.2

$q$-expansion

comment:q-expansion

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

gp:mfcoefs(f, 20)

$f(q)$	$=$	$q-7.84548 q^{2} +45.5516 q^{4} -71.8372i q^{7} -231.847 q^{8} +66.0480i q^{11} +143.700i q^{13} +563.598i q^{14} +1090.12 q^{16} -88.1175 q^{17} -397.835 q^{19} -518.178i q^{22} -189.182 q^{23} -1127.39i q^{26} -3272.30i q^{28} -649.431i q^{29} +508.987 q^{31} -4843.01 q^{32} +691.325 q^{34} +1273.43i q^{37} +3121.21 q^{38} -1532.30i q^{41} +1275.35i q^{43} +3008.59i q^{44} +1484.23 q^{46} +3096.33 q^{47} -2759.58 q^{49} +6545.75i q^{52} -940.508 q^{53} +16655.2i q^{56} +5095.10i q^{58} +5071.76i q^{59} -625.387 q^{61} -3993.25 q^{62} +20553.7 q^{64} -2767.03i q^{67} -4013.90 q^{68} +2565.51i q^{71} +7892.47i q^{73} -9990.67i q^{74} -18122.0 q^{76} +4744.70 q^{77} -7074.77 q^{79} +12021.6i q^{82} +10466.7 q^{83} -10005.7i q^{86} -15313.0i q^{88} +14076.6i q^{89} +10323.0 q^{91} -8617.55 q^{92} -24292.2 q^{94} +15128.5i q^{97} +21650.3 q^{98} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 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	$-0.937337\pi$
$2$	−7.84548	−1.96137	−0.980686	+	0.195591i	$0.937337\pi$
$3$	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	$0.0879913\pi$
$11$	66.0480i	0.545851i	−0.962035	+	0.272925i	$0.912009\pi$
$12$	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	$-0.548717\pi$
$17$	−88.1175	−0.304905	−0.152452	+	0.988311i	$0.548717\pi$
$18$	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	$-0.557225\pi$
$23$	−189.182	−0.357622	−0.178811	+	0.983883i	$0.557225\pi$
$24$	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	$-0.873820\pi$
$29$	− 649.431i	− 0.772213i	0.922454	−	0.386106i	$-0.126180\pi$
$30$	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	$-0.849364\pi$
$41$	− 1532.30i	− 0.911541i	0.890097	−	0.455770i	$-0.150636\pi$
$42$	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	$-0.252807\pi$
$47$	3096.33	1.40169	0.700845	+	0.713314i	$0.252807\pi$
$48$	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	$-0.553540\pi$
$53$	−940.508	−0.334820	−0.167410	+	0.985887i	$0.553540\pi$
$54$	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	$0.259778\pi$
$59$	5071.76i	1.45698i	−0.685055	+	0.728491i	$0.740222\pi$
$60$	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	$0.0818991\pi$
$71$	2565.51i	0.508928i	−0.967082	+	0.254464i	$0.918101\pi$
$72$	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	$-0.225360\pi$
$83$	10466.7	1.51934	0.759670	+	0.650308i	$0.225360\pi$
$84$	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	$0.348295\pi$
$89$	14076.6i	1.77712i	−0.458757	+	0.888562i	$0.651705\pi$
$90$	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	$-0.791502\pi$
$101$	− 12428.3i	− 1.21834i	0.793039	−	0.609171i	$-0.208498\pi$
$102$	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	$-0.330907\pi$
$107$	11599.8	1.01317	0.506586	+	0.862190i	$0.330907\pi$
$108$	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	$-0.718866\pi$
$113$	−16208.3	−1.26935	−0.634675	+	0.772779i	$0.718866\pi$
$114$	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	$0.316951\pi$
$131$	28801.6i	1.67831i	−0.543889	+	0.839157i	$0.683049\pi$
$132$	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	$-0.618618\pi$
$137$	−13666.9	−0.728166	−0.364083	+	0.931367i	$0.618618\pi$
$138$	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	$0.225939\pi$
$149$	28936.3i	1.30338i	−0.758486	+	0.651689i	$0.774061\pi$
$150$	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	$-0.525355\pi$
$167$	−4438.33	−0.159143	−0.0795713	+	0.996829i	$0.525355\pi$
$168$	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	$-0.524183\pi$
$173$	−4543.18	−0.151799	−0.0758993	+	0.997115i	$0.524183\pi$
$174$	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	$0.0106641\pi$
$179$	2146.50i	0.0669922i	−0.999439	+	0.0334961i	$0.989336\pi$
$180$	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	$0.0437150\pi$
$191$	9988.75i	0.273807i	−0.990584	+	0.136904i	$0.956285\pi$
$192$	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	$-0.267925\pi$
$197$	51708.2	1.33238	0.666188	+	0.745784i	$0.267925\pi$
$198$	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	$-0.458966\pi$
$227$	13248.7	0.257111	0.128555	+	0.991702i	$0.458966\pi$
$228$	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	$-0.674374\pi$
$233$	−56549.7	−1.04164	−0.520821	+	0.853666i	$0.674374\pi$
$234$	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	$0.0801170\pi$
$239$	28451.5i	0.498092i	−0.968492	+	0.249046i	$0.919883\pi$
$240$	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	$0.0522542\pi$
$251$	20591.9i	0.326850i	−0.986556	+	0.163425i	$0.947746\pi$
$252$	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	$-0.541182\pi$
$257$	−17042.7	−0.258031	−0.129015	+	0.991643i	$0.541182\pi$
$258$	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	$-0.731513\pi$
$263$	−91976.6	−1.32974	−0.664869	+	0.746960i	$0.731513\pi$
$264$	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	$-0.872714\pi$
$269$	− 56341.4i	− 0.778616i	0.921108	−	0.389308i	$-0.127286\pi$
$270$	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	$0.385309\pi$
$281$	147781.i	1.87157i	−0.352566	+	0.935787i	$0.614691\pi$
$282$	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	$-0.541798\pi$
$293$	−22481.5	−0.261872	−0.130936	+	0.991391i	$0.541798\pi$
$294$	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	$-0.842299\pi$
$311$	− 91964.6i	− 0.950823i	0.879763	−	0.475412i	$-0.157701\pi$
$312$	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	$-0.330657\pi$
$317$	101949.	1.01453	0.507263	+	0.861791i	$0.330657\pi$
$318$	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	$-0.742520\pi$
$347$	−166236.	−1.38060	−0.690298	+	0.723526i	$0.742520\pi$
$348$	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	$-0.671647\pi$
$353$	−127971.	−1.02698	−0.513489	+	0.858096i	$0.671647\pi$
$354$	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	$0.132071\pi$
$359$	103906.i	0.806220i	−0.915152	+	0.403110i	$0.867929\pi$
$360$	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	$-0.413774\pi$
$383$	78503.5	0.535170	0.267585	+	0.963534i	$0.413774\pi$
$384$	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	$0.281932\pi$
$389$	234356.i	1.54873i	−0.632737	+	0.774367i	$0.718068\pi$
$390$	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	$-0.765955\pi$
$401$	− 215728.i	− 1.34158i	0.741646	−	0.670791i	$-0.234045\pi$
$402$	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	$0.210207\pi$
$419$	215386.i	1.22684i	−0.789756	+	0.613421i	$0.789793\pi$
$420$	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	$0.168745\pi$
$431$	187858.i	1.01129i	−0.862742	+	0.505644i	$0.831255\pi$
$432$	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	$-0.245607\pi$
$443$	281341.	1.43359	0.716797	+	0.697282i	$0.245607\pi$
$444$	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	$-0.855240\pi$
$449$	− 177112.i	− 0.878526i	0.898358	−	0.439263i	$-0.144760\pi$
$450$	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	$-0.711834\pi$
$461$	− 334343.i	− 1.57322i	0.617448	−	0.786611i	$-0.288166\pi$
$462$	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	$-0.461203\pi$
$467$	53031.5	0.243164	0.121582	+	0.992581i	$0.461203\pi$
$468$	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	$-0.961657\pi$
$479$	− 55142.8i	− 0.240335i	0.992754	−	0.120168i	$-0.0383432\pi$
$480$	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	$-0.873995\pi$
$491$	− 185921.i	− 0.771199i	0.922666	−	0.385599i	$-0.126005\pi$
$492$	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	$-0.407797\pi$
$503$	144534.	0.571262	0.285631	+	0.958340i	$0.407797\pi$
$504$	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	$0.162684\pi$
$509$	253446.i	0.978251i	−0.872214	+	0.489125i	$0.837316\pi$
$510$	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	$-0.882874\pi$
$521$	− 195283.i	− 0.719431i	0.933062	−	0.359716i	$-0.117126\pi$
$522$	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	$-0.357306\pi$
$557$	268937.	0.866842	0.433421	+	0.901191i	$0.357306\pi$
$558$	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	$-0.498042\pi$
$563$	3899.25	0.0123017	0.00615084	+	0.999981i	$0.498042\pi$
$564$	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	$-0.661815\pi$
$569$	− 565640.i	− 1.74709i	0.486742	−	0.873546i	$-0.338185\pi$
$570$	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	$-0.702937\pi$
$587$	−410192.	−1.19045	−0.595225	+	0.803559i	$0.702937\pi$
$588$	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	$-0.608351\pi$
$593$	−234803.	−0.667719	−0.333860	+	0.942623i	$0.608351\pi$
$594$	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	$0.0755020\pi$
$599$	168621.i	0.469957i	−0.972001	+	0.234979i	$0.924498\pi$
$600$	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	$-0.658077\pi$
$617$	−362760.	−0.952904	−0.476452	+	0.879201i	$0.658077\pi$
$618$	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	$0.00469655\pi$
$641$	12124.4i	0.0295082i	−0.999891	+	0.0147541i	$0.995303\pi$
$642$	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	$-0.446931\pi$
$647$	138936.	0.331899	0.165950	+	0.986134i	$0.446931\pi$
$648$	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	$-0.796205\pi$
$653$	−683920.	−1.60391	−0.801953	+	0.597388i	$0.796205\pi$
$654$	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	$0.0455464\pi$
$659$	123857.i	0.285201i	−0.989780	+	0.142601i	$0.954454\pi$
$660$	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	$-0.243724\pi$
$677$	660828.	1.44182	0.720910	+	0.693029i	$0.243724\pi$
$678$	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	$-0.329316\pi$
$683$	476650.	1.02178	0.510891	+	0.859645i	$0.329316\pi$
$684$	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	$0.134831\pi$
$701$	403961.i	0.822061i	−0.911622	+	0.411030i	$0.865169\pi$
$702$	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	$0.0549711\pi$
$719$	177669.i	0.343679i	−0.985125	+	0.171840i	$0.945029\pi$
$720$	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	$-0.921522\pi$
$743$	−1.07071e6	−1.93952	−0.969761	+	0.244056i	$0.921522\pi$
$744$	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	$-0.818025\pi$
$761$	− 626672.i	− 1.08211i	0.840987	−	0.541055i	$-0.181975\pi$
$762$	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	$-0.444295\pi$
$773$	208073.	0.348222	0.174111	+	0.984726i	$0.444295\pi$
$774$	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	$-0.491859\pi$
$797$	32487.5	0.0511445	0.0255723	+	0.999673i	$0.491859\pi$
$798$	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	$-0.785694\pi$
$809$	− 816191.i	− 1.24708i	0.781791	−	0.623540i	$-0.214306\pi$
$810$	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	$0.133021\pi$
$821$	547104.i	0.811678i	−0.913945	+	0.405839i	$0.866979\pi$
$822$	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	$-0.723670\pi$
$827$	−883999.	−1.29253	−0.646266	+	0.763113i	$0.723670\pi$
$828$	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	$0.0574408\pi$
$839$	252676.i	0.358956i	−0.983762	+	0.179478i	$0.942559\pi$
$840$	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	$-0.301436\pi$
$857$	858025.	1.16826	0.584128	+	0.811661i	$0.301436\pi$
$858$	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	$-0.577472\pi$
$863$	−358961.	−0.481977	−0.240988	+	0.970528i	$0.577472\pi$
$864$	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	$0.0969167\pi$
$881$	465371.i	0.599580i	−0.954005	+	0.299790i	$0.903083\pi$
$882$	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	$-0.859318\pi$
$887$	−1.42234e6	−1.80783	−0.903913	+	0.427716i	$0.859318\pi$
$888$	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	$0.183536\pi$
$911$	904900.i	1.09035i	−0.838324	+	0.545173i	$0.816464\pi$
$912$	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	$0.157166\pi$
$929$	818048.i	0.947867i	−0.880561	+	0.473934i	$0.842834\pi$
$930$	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	$-0.851873\pi$
$941$	− 794699.i	− 0.897477i	0.893663	−	0.448739i	$-0.148127\pi$
$942$	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	$-0.313453\pi$
$947$	992010.	1.10616	0.553078	+	0.833130i	$0.313453\pi$
$948$	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	$-0.516509\pi$
$953$	−94165.1	−0.103682	−0.0518411	+	0.998655i	$0.516509\pi$
$954$	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	$-0.929536\pi$
$971$	− 414030.i	− 0.439131i	0.975598	−	0.219565i	$-0.0704639\pi$
$972$	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	$-0.308675\pi$
$977$	1.07961e6	1.13104	0.565520	+	0.824734i	$0.308675\pi$
$978$	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	$-0.588321\pi$
$983$	−529376.	−0.547844	−0.273922	+	0.961752i	$0.588321\pi$
$984$	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.224.1		12
3.2	odd	2	inner	225.5.d.c.224.11		12
5.2	odd	4		225.5.c.d.26.1	yes	6
5.3	odd	4		225.5.c.c.26.6	yes	6
5.4	even	2	inner	225.5.d.c.224.12		12
15.2	even	4		225.5.c.d.26.6	yes	6
15.8	even	4		225.5.c.c.26.1	✓	6
15.14	odd	2	inner	225.5.d.c.224.2		12

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

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