LMFDB - Embedded newform 225.5.c.c.26.6

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [225,5,Mod(26,225)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(225, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([1, 0]))

N = Newforms(chi, 5, names="a")

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

chi := DirichletCharacter("225.26");

S:= CuspForms(chi, 5);

N := Newforms(S);

Level:	$ N $	$=$	$ 225 = 3^{2} \cdot 5^{2} $
Weight:	$ k $	$=$	$ 5 $
Character orbit:	$[\chi]$	$=$	225.c (of order $2$, degree $1$, minimal)

Newform invariants

comment: select newform

sage: f = N[0] # Warning: the index may be different

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

Self dual:	no
Analytic conductor:	$23.2582416939$
Analytic rank:	$0$
Dimension:	$6$
Coefficient field:	$\mathbb{Q}[x]/(x^{6} - \cdots)$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{6} - 38x^{4} - 12x^{3} + 496x^{2} + 336x + 1188 $ x^6 - 38x^4 - 12x^3 + 496x^2 + 336x + 1188
Coefficient ring:	$\Z[a_1, \ldots, a_{11}]$
Coefficient ring index:	$ 2^{4}\cdot 3^{5} $
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label			26.6
Root			$-4.54760 + 1.41421i$ of defining polynomial
Character	$\chi$	$=$	225.26
Dual form			225.5.c.c.26.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+7.84548i q^{2} -45.5516 q^{4} -71.8372 q^{7} -231.847i q^{8} +O(q^{10})$	$q+7.84548i q^{2} -45.5516 q^{4} -71.8372 q^{7} -231.847i q^{8} +66.0480i q^{11} -143.700 q^{13} -563.598i q^{14} +1090.12 q^{16} +88.1175i q^{17} +397.835 q^{19} -518.178 q^{22} -189.182i q^{23} -1127.39i q^{26} +3272.30 q^{28} +649.431i q^{29} +508.987 q^{31} +4843.01i q^{32} -691.325 q^{34} +1273.43 q^{37} +3121.21i q^{38} -1532.30i q^{41} -1275.35 q^{43} -3008.59i q^{44} +1484.23 q^{46} -3096.33i q^{47} +2759.58 q^{49} +6545.75 q^{52} -940.508i q^{53} +16655.2i q^{56} -5095.10 q^{58} -5071.76i q^{59} -625.387 q^{61} +3993.25i q^{62} -20553.7 q^{64} -2767.03 q^{67} -4013.90i q^{68} +2565.51i q^{71} -7892.47 q^{73} +9990.67i q^{74} -18122.0 q^{76} -4744.70i q^{77} +7074.77 q^{79} +12021.6 q^{82} +10466.7i q^{83} -10005.7i q^{86} +15313.0 q^{88} -14076.6i q^{89} +10323.0 q^{91} +8617.55i q^{92} +24292.2 q^{94} +15128.5 q^{97} +21650.3i q^{98} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 6 q - 92 q^{4} - 86 q^{7}+O(q^{10}) $ 6 * q - 92 * q^4 - 86 * q^7	$ 6 q - 92 q^{4} - 86 q^{7} - 394 q^{13} + 1816 q^{16} - 250 q^{19} + 1276 q^{22} + 6412 q^{28} + 1846 q^{31} - 3692 q^{34} + 7968 q^{37} + 1382 q^{43} + 9756 q^{46} - 3232 q^{49} + 23268 q^{52} - 2932 q^{58} + 10838 q^{61} - 36144 q^{64} - 6278 q^{67} - 13024 q^{73} - 24780 q^{76} + 3904 q^{79} + 29792 q^{82} + 42744 q^{88} + 17306 q^{91} + 66364 q^{94} + 8262 q^{97}+O(q^{100}) $ 6 * q - 92 * q^4 - 86 * q^7 - 394 * q^13 + 1816 * q^16 - 250 * q^19 + 1276 * q^22 + 6412 * q^28 + 1846 * q^31 - 3692 * q^34 + 7968 * q^37 + 1382 * q^43 + 9756 * q^46 - 3232 * q^49 + 23268 * q^52 - 2932 * q^58 + 10838 * q^61 - 36144 * q^64 - 6278 * q^67 - 13024 * q^73 - 24780 * q^76 + 3904 * q^79 + 29792 * q^82 + 42744 * q^88 + 17306 * q^91 + 66364 * q^94 + 8262 * q^97

Display 100 coefficients 10 coefficients

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 $

$2$	7.84548i	1.96137i	0.195591	+	0.980686i	$0.437337\pi$
$2$	7.84548i	1.96137i	−0.195591	+	0.980686i	$0.562663\pi$
$3$	0	0
$3$	0	0
$4$	−45.5516	−2.84698
$5$	0	0
$5$	0	0
$6$	0	0
$7$	−71.8372	−1.46607	−0.733033	−	0.680194i	$-0.761896\pi$
$7$	−71.8372	−1.46607	−0.733033	+	0.680194i	$0.761896\pi$
$8$	− 231.847i	− 3.62261i
$9$	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
$13$	−143.700	−0.850293	−0.425147	−	0.905124i	$-0.639777\pi$
$13$	−143.700	−0.850293	−0.425147	+	0.905124i	$0.639777\pi$
$14$	− 563.598i	− 2.87550i
$15$	0	0
$16$	1090.12	4.25830
$17$	88.1175i	0.304905i	0.988311	+	0.152452i	$0.0487171\pi$
$17$	88.1175i	0.304905i	−0.988311	+	0.152452i	$0.951283\pi$
$18$	0	0
$19$	397.835	1.10204	0.551018	−	0.834493i	$-0.314239\pi$
$19$	397.835	1.10204	0.551018	+	0.834493i	$0.314239\pi$
$20$	0	0
$21$	0	0
$22$	−518.178	−1.07062
$23$	− 189.182i	− 0.357622i	−0.983883	−	0.178811i	$-0.942775\pi$
$23$	− 189.182i	− 0.357622i	0.983883	−	0.178811i	$-0.0572251\pi$
$24$	0	0
$25$	0	0
$26$	− 1127.39i	− 1.66774i
$27$	0	0
$28$	3272.30	4.17385
$29$	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	$0.414687\pi$
$32$	4843.01i	4.72950i
$33$	0	0
$34$	−691.325	−0.598032
$35$	0	0
$36$	0	0
$37$	1273.43	0.930189	0.465095	−	0.885261i	$-0.346020\pi$
$37$	1273.43	0.930189	0.465095	+	0.885261i	$0.346020\pi$
$38$	3121.21i	2.16150i
$39$	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
$43$	−1275.35	−0.689750	−0.344875	−	0.938649i	$-0.612079\pi$
$43$	−1275.35	−0.689750	−0.344875	+	0.938649i	$0.612079\pi$
$44$	− 3008.59i	− 1.55402i
$45$	0	0
$46$	1484.23	0.701430
$47$	− 3096.33i	− 1.40169i	−0.713314	−	0.700845i	$-0.752807\pi$
$47$	− 3096.33i	− 1.40169i	0.713314	−	0.700845i	$-0.247193\pi$
$48$	0	0
$49$	2759.58	1.14935
$50$	0	0
$51$	0	0
$52$	6545.75	2.42076
$53$	− 940.508i	− 0.334820i	−0.985887	−	0.167410i	$-0.946460\pi$
$53$	− 940.508i	− 0.334820i	0.985887	−	0.167410i	$-0.0535403\pi$
$54$	0	0
$55$	0	0
$56$	16655.2i	5.31098i
$57$	0	0
$58$	−5095.10	−1.51460
$59$	− 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	$0.526781\pi$
$62$	3993.25i	1.03883i
$63$	0	0
$64$	−20553.7	−5.01800
$65$	0	0
$66$	0	0
$67$	−2767.03	−0.616402	−0.308201	−	0.951321i	$-0.599727\pi$
$67$	−2767.03	−0.616402	−0.308201	+	0.951321i	$0.599727\pi$
$68$	− 4013.90i	− 0.868057i
$69$	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
$73$	−7892.47	−1.48104	−0.740521	−	0.672034i	$-0.765421\pi$
$73$	−7892.47	−1.48104	−0.740521	+	0.672034i	$0.765421\pi$
$74$	9990.67i	1.82445i
$75$	0	0
$76$	−18122.0	−3.13747
$77$	− 4744.70i	− 0.800253i
$78$	0	0
$79$	7074.77	1.13359	0.566797	−	0.823857i	$-0.308182\pi$
$79$	7074.77	1.13359	0.566797	+	0.823857i	$0.308182\pi$
$80$	0	0
$81$	0	0
$82$	12021.6	1.78787
$83$	10466.7i	1.51934i	0.650308	+	0.759670i	$0.274640\pi$
$83$	10466.7i	1.51934i	−0.650308	+	0.759670i	$0.725360\pi$
$84$	0	0
$85$	0	0
$86$	− 10005.7i	− 1.35286i
$87$	0	0
$88$	15313.0	1.97740
$89$	− 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
$92$	8617.55i	1.01814i
$93$	0	0
$94$	24292.2	2.74923
$95$	0	0
$96$	0	0
$97$	15128.5	1.60787	0.803936	−	0.594715i	$-0.202735\pi$
$97$	15128.5	1.60787	0.803936	+	0.594715i	$0.202735\pi$
$98$	21650.3i	2.25430i
$99$	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
$103$	−4846.79	−0.456856	−0.228428	−	0.973561i	$-0.573359\pi$
$103$	−4846.79	−0.456856	−0.228428	+	0.973561i	$0.573359\pi$
$104$	33316.3i	3.08028i
$105$	0	0
$106$	7378.74	0.656706
$107$	− 11599.8i	− 1.01317i	−0.862190	−	0.506586i	$-0.830907\pi$
$107$	− 11599.8i	− 1.01317i	0.862190	−	0.506586i	$-0.169093\pi$
$108$	0	0
$109$	3965.61	0.333778	0.166889	−	0.985976i	$-0.446628\pi$
$109$	3965.61	0.333778	0.166889	+	0.985976i	$0.446628\pi$
$110$	0	0
$111$	0	0
$112$	−78311.5	−6.24294
$113$	− 16208.3i	− 1.26935i	−0.772779	−	0.634675i	$-0.781134\pi$
$113$	− 16208.3i	− 1.26935i	0.772779	−	0.634675i	$-0.218866\pi$
$114$	0	0
$115$	0	0
$116$	− 29582.6i	− 2.19847i
$117$	0	0
$118$	39790.4	2.85768
$119$	− 6330.12i	− 0.447010i
$120$	0	0
$121$	10278.7	0.702047
$122$	− 4906.47i	− 0.329647i
$123$	0	0
$124$	−23185.2	−1.50788
$125$	0	0
$126$	0	0
$127$	13904.3	0.862065	0.431033	−	0.902336i	$-0.358149\pi$
$127$	13904.3	0.862065	0.431033	+	0.902336i	$0.358149\pi$
$128$	− 83765.9i	− 5.11266i
$129$	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
$133$	−28579.4	−1.61566
$134$	− 21708.7i	− 1.20899i
$135$	0	0
$136$	20429.8	1.10455
$137$	13666.9i	0.728166i	0.931367	+	0.364083i	$0.118618\pi$
$137$	13666.9i	0.728166i	−0.931367	+	0.364083i	$0.881382\pi$
$138$	0	0
$139$	20672.4	1.06994	0.534971	−	0.844870i	$-0.320323\pi$
$139$	20672.4	1.06994	0.534971	+	0.844870i	$0.320323\pi$
$140$	0	0
$141$	0	0
$142$	−20127.7	−0.998198
$143$	− 9491.06i	− 0.464133i
$144$	0	0
$145$	0	0
$146$	− 61920.2i	− 2.90487i
$147$	0	0
$148$	−58006.8	−2.64823
$149$	− 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	$0.327916\pi$
$152$	− 92236.8i	− 3.99224i
$153$	0	0
$154$	37224.5	1.56959
$155$	0	0
$156$	0	0
$157$	−16398.7	−0.665288	−0.332644	−	0.943053i	$-0.607941\pi$
$157$	−16398.7	−0.665288	−0.332644	+	0.943053i	$0.607941\pi$
$158$	55505.0i	2.22340i
$159$	0	0
$160$	0	0
$161$	13590.3i	0.524297i
$162$	0	0
$163$	−13316.8	−0.501217	−0.250608	−	0.968089i	$-0.580631\pi$
$163$	−13316.8	−0.501217	−0.250608	+	0.968089i	$0.580631\pi$
$164$	69798.8i	2.59514i
$165$	0	0
$166$	−82116.6	−2.97999
$167$	4438.33i	0.159143i	0.996829	+	0.0795713i	$0.0253551\pi$
$167$	4438.33i	0.159143i	−0.996829	+	0.0795713i	$0.974645\pi$
$168$	0	0
$169$	−7911.44	−0.277001
$170$	0	0
$171$	0	0
$172$	58094.1	1.96370
$173$	− 4543.18i	− 0.151799i	−0.997115	−	0.0758993i	$-0.975817\pi$
$173$	− 4543.18i	− 0.151799i	0.997115	−	0.0758993i	$-0.0241827\pi$
$174$	0	0
$175$	0	0
$176$	72000.5i	2.32440i
$177$	0	0
$178$	110438.	3.48560
$179$	− 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	$0.628044\pi$
$182$	80988.7i	2.44502i
$183$	0	0
$184$	−43861.3	−1.29552
$185$	0	0
$186$	0	0
$187$	−5819.98	−0.166433
$188$	141043.i	3.99058i
$189$	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
$193$	16130.0	0.433032	0.216516	−	0.976279i	$-0.430531\pi$
$193$	16130.0	0.433032	0.216516	+	0.976279i	$0.430531\pi$
$194$	118690.i	3.15364i
$195$	0	0
$196$	−125703.	−3.27216
$197$	− 51708.2i	− 1.33238i	−0.745784	−	0.666188i	$-0.767925\pi$
$197$	− 51708.2i	− 1.33238i	0.745784	−	0.666188i	$-0.232075\pi$
$198$	0	0
$199$	−70238.8	−1.77366	−0.886831	−	0.462094i	$-0.847098\pi$
$199$	−70238.8	−1.77366	−0.886831	+	0.462094i	$0.847098\pi$
$200$	0	0
$201$	0	0
$202$	97506.1	2.38962
$203$	− 46653.3i	− 1.13211i
$204$	0	0
$205$	0	0
$206$	− 38025.4i	− 0.896065i
$207$	0	0
$208$	−156650.	−3.62080
$209$	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	$0.404905\pi$
$212$	42841.7i	0.953224i
$213$	0	0
$214$	91006.0	1.98720
$215$	0	0
$216$	0	0
$217$	−36564.2	−0.776491
$218$	31112.1i	0.654662i
$219$	0	0
$220$	0	0
$221$	− 12662.4i	− 0.259259i
$222$	0	0
$223$	−17542.0	−0.352752	−0.176376	−	0.984323i	$-0.556438\pi$
$223$	−17542.0	−0.352752	−0.176376	+	0.984323i	$0.556438\pi$
$224$	− 347908.i	− 6.93375i
$225$	0	0
$226$	127162.	2.48967
$227$	− 13248.7i	− 0.257111i	−0.991702	−	0.128555i	$-0.958966\pi$
$227$	− 13248.7i	− 0.257111i	0.991702	−	0.128555i	$-0.0410340\pi$
$228$	0	0
$229$	−66842.0	−1.27461	−0.637307	−	0.770610i	$-0.719952\pi$
$229$	−66842.0	−1.27461	−0.637307	+	0.770610i	$0.719952\pi$
$230$	0	0
$231$	0	0
$232$	150568.	2.79742
$233$	− 56549.7i	− 1.04164i	−0.853666	−	0.520821i	$-0.825626\pi$
$233$	− 56549.7i	− 1.04164i	0.853666	−	0.520821i	$-0.174374\pi$
$234$	0	0
$235$	0	0
$236$	231027.i	4.14800i
$237$	0	0
$238$	49662.8	0.876753
$239$	− 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	$0.528077\pi$
$242$	80641.1i	1.37697i
$243$	0	0
$244$	28487.4	0.478491
$245$	0	0
$246$	0	0
$247$	−57168.7	−0.937054
$248$	− 118007.i	− 1.91869i
$249$	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
$253$	12495.1	0.195208
$254$	109086.i	1.69083i
$255$	0	0
$256$	328324.	5.00983
$257$	17042.7i	0.258031i	0.991643	+	0.129015i	$0.0411817\pi$
$257$	17042.7i	0.258031i	−0.991643	+	0.129015i	$0.958818\pi$
$258$	0	0
$259$	−91479.6	−1.36372
$260$	0	0
$261$	0	0
$262$	−225962.	−3.29180
$263$	− 91976.6i	− 1.32974i	−0.746960	−	0.664869i	$-0.768487\pi$
$263$	− 91976.6i	− 1.32974i	0.746960	−	0.664869i	$-0.231513\pi$
$264$	0	0
$265$	0	0
$266$	− 224219.i	− 3.16890i
$267$	0	0
$268$	126043.	1.75488
$269$	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	$0.782165\pi$
$272$	96059.1i	1.29838i
$273$	0	0
$274$	−107224.	−1.42820
$275$	0	0
$276$	0	0
$277$	−143482.	−1.86998	−0.934989	−	0.354677i	$-0.884591\pi$
$277$	−143482.	−1.86998	−0.934989	+	0.354677i	$0.884591\pi$
$278$	162185.i	2.09855i
$279$	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
$283$	73418.5	0.916712	0.458356	−	0.888769i	$-0.348439\pi$
$283$	73418.5	0.916712	0.458356	+	0.888769i	$0.348439\pi$
$284$	− 116863.i	− 1.44891i
$285$	0	0
$286$	74462.0	0.910338
$287$	110076.i	1.33638i
$288$	0	0
$289$	75756.3	0.907033
$290$	0	0
$291$	0	0
$292$	359515.	4.21649
$293$	− 22481.5i	− 0.261872i	−0.991391	−	0.130936i	$-0.958202\pi$
$293$	− 22481.5i	− 0.261872i	0.991391	−	0.130936i	$-0.0417983\pi$
$294$	0	0
$295$	0	0
$296$	− 295240.i	− 3.36971i
$297$	0	0
$298$	227019.	2.55641
$299$	27185.4i	0.304084i
$300$	0	0
$301$	91617.4	1.01122
$302$	184132.i	2.01890i
$303$	0	0
$304$	433690.	4.69280
$305$	0	0
$306$	0	0
$307$	38654.6	0.410133	0.205067	−	0.978748i	$-0.434259\pi$
$307$	38654.6	0.410133	0.205067	+	0.978748i	$0.434259\pi$
$308$	216129.i	2.27830i
$309$	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
$313$	−39467.6	−0.402858	−0.201429	−	0.979503i	$-0.564559\pi$
$313$	−39467.6	−0.402858	−0.201429	+	0.979503i	$0.564559\pi$
$314$	− 128656.i	− 1.30488i
$315$	0	0
$316$	−322267.	−3.22732
$317$	− 101949.i	− 1.01453i	−0.861791	−	0.507263i	$-0.830657\pi$
$317$	− 101949.i	− 1.01453i	0.861791	−	0.507263i	$-0.169343\pi$
$318$	0	0
$319$	−42893.6	−0.421513
$320$	0	0
$321$	0	0
$322$	−106623.	−1.02834
$323$	35056.2i	0.336016i
$324$	0	0
$325$	0	0
$326$	− 104477.i	− 0.983072i
$327$	0	0
$328$	−355259.	−3.30215
$329$	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.i	− 4.32553i
$333$	0	0
$334$	−34820.8	−0.312138
$335$	0	0
$336$	0	0
$337$	76923.6	0.677330	0.338665	−	0.940907i	$-0.390025\pi$
$337$	76923.6	0.677330	0.338665	+	0.940907i	$0.390025\pi$
$338$	− 62069.0i	− 0.543302i
$339$	0	0
$340$	0	0
$341$	33617.5i	0.289106i
$342$	0	0
$343$	−25759.6	−0.218953
$344$	295685.i	2.49869i
$345$	0	0
$346$	35643.4	0.297733
$347$	166236.i	1.38060i	0.723526	+	0.690298i	$0.242520\pi$
$347$	166236.i	1.38060i	−0.723526	+	0.690298i	$0.757480\pi$
$348$	0	0
$349$	−172458.	−1.41590	−0.707948	−	0.706264i	$-0.750379\pi$
$349$	−172458.	−1.41590	−0.707948	+	0.706264i	$0.750379\pi$
$350$	0	0
$351$	0	0
$352$	−319871.	−2.58160
$353$	− 127971.i	− 1.02698i	−0.858096	−	0.513489i	$-0.828353\pi$
$353$	− 127971.i	− 1.02698i	0.858096	−	0.513489i	$-0.171647\pi$
$354$	0	0
$355$	0	0
$356$	641212.i	5.05943i
$357$	0	0
$358$	16840.3	0.131397
$359$	− 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.i	− 1.53576i
$363$	0	0
$364$	−470228.	−3.54900
$365$	0	0
$366$	0	0
$367$	146305.	1.08624	0.543121	−	0.839654i	$-0.317242\pi$
$367$	146305.	1.08624	0.543121	+	0.839654i	$0.317242\pi$
$368$	− 206232.i	− 1.52286i
$369$	0	0
$370$	0	0
$371$	67563.5i	0.490867i
$372$	0	0
$373$	114715.	0.824520	0.412260	−	0.911066i	$-0.364740\pi$
$373$	114715.	0.824520	0.412260	+	0.911066i	$0.364740\pi$
$374$	− 45660.6i	− 0.326436i
$375$	0	0
$376$	−717875.	−5.07777
$377$	− 93322.9i	− 0.656607i
$378$	0	0
$379$	−24642.5	−0.171556	−0.0857781	−	0.996314i	$-0.527338\pi$
$379$	−24642.5	−0.171556	−0.0857781	+	0.996314i	$0.527338\pi$
$380$	0	0
$381$	0	0
$382$	−78366.6	−0.537037
$383$	78503.5i	0.535170i	0.963534	+	0.267585i	$0.0862256\pi$
$383$	78503.5i	0.535170i	−0.963534	+	0.267585i	$0.913774\pi$
$384$	0	0
$385$	0	0
$386$	126548.i	0.849336i
$387$	0	0
$388$	−689127.	−4.57758
$389$	− 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.i	− 4.16363i
$393$	0	0
$394$	405676.	2.61328
$395$	0	0
$396$	0	0
$397$	−86679.9	−0.549968	−0.274984	−	0.961449i	$-0.588673\pi$
$397$	−86679.9	−0.549968	−0.274984	+	0.961449i	$0.588673\pi$
$398$	− 551057.i	− 3.47881i
$399$	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
$403$	−73141.1	−0.450352
$404$	566130.i	3.46859i
$405$	0	0
$406$	366018.	2.22050
$407$	84107.4i	0.507745i
$408$	0	0
$409$	319618.	1.91066	0.955332	−	0.295535i	$-0.0954980\pi$
$409$	319618.	1.91066	0.955332	+	0.295535i	$0.0954980\pi$
$410$	0	0
$411$	0	0
$412$	220779.	1.30066
$413$	364341.i	2.13603i
$414$	0	0
$415$	0	0
$416$	− 695938.i	− 4.02146i
$417$	0	0
$418$	−206150.	−1.17986
$419$	− 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.i	1.15456i
$423$	0	0
$424$	−218054.	−1.21292
$425$	0	0
$426$	0	0
$427$	44926.1	0.246401
$428$	528390.i	2.88447i
$429$	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
$433$	−47754.6	−0.254706	−0.127353	−	0.991857i	$-0.540648\pi$
$433$	−47754.6	−0.254706	−0.127353	+	0.991857i	$0.540648\pi$
$434$	− 286864.i	− 1.52299i
$435$	0	0
$436$	−180640.	−0.950257
$437$	− 75263.3i	− 0.394113i
$438$	0	0
$439$	171451.	0.889632	0.444816	−	0.895622i	$-0.353269\pi$
$439$	171451.	0.889632	0.444816	+	0.895622i	$0.353269\pi$
$440$	0	0
$441$	0	0
$442$	99343.0	0.508502
$443$	281341.i	1.43359i	0.697282	+	0.716797i	$0.254393\pi$
$443$	281341.i	1.43359i	−0.697282	+	0.716797i	$0.745607\pi$
$444$	0	0
$445$	0	0
$446$	− 137626.i	− 0.691878i
$447$	0	0
$448$	1.47652e6	7.35672
$449$	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.i	3.61381i
$453$	0	0
$454$	103942.	0.504289
$455$	0	0
$456$	0	0
$457$	29607.9	0.141767	0.0708836	−	0.997485i	$-0.477418\pi$
$457$	29607.9	0.141767	0.0708836	+	0.997485i	$0.477418\pi$
$458$	− 524408.i	− 2.49999i
$459$	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.	0.509022	0.254511	−	0.967070i	$-0.418085\pi$
$463$	109119.	0.509022	0.254511	+	0.967070i	$0.418085\pi$
$464$	707961.i	3.28831i
$465$	0	0
$466$	443660.	2.04305
$467$	− 53031.5i	− 0.243164i	−0.992581	−	0.121582i	$-0.961203\pi$
$467$	− 53031.5i	− 0.243164i	0.992581	−	0.121582i	$-0.0387968\pi$
$468$	0	0
$469$	198775.	0.903685
$470$	0	0
$471$	0	0
$472$	−1.17587e6	−5.27807
$473$	− 84234.1i	− 0.376501i
$474$	0	0
$475$	0	0
$476$	288347.i	1.27263i
$477$	0	0
$478$	223216.	0.976943
$479$	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
$482$	− 80282.0i	− 0.345560i
$483$	0	0
$484$	−468210.	−1.99871
$485$	0	0
$486$	0	0
$487$	285889.	1.20543	0.602713	−	0.797958i	$-0.294087\pi$
$487$	285889.	1.20543	0.602713	+	0.797958i	$0.294087\pi$
$488$	144994.i	0.608850i
$489$	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
$493$	−57226.2	−0.235451
$494$	− 448516.i	− 1.83791i
$495$	0	0
$496$	554859.	2.25538
$497$	− 184299.i	− 0.746122i
$498$	0	0
$499$	204968.	0.823162	0.411581	−	0.911373i	$-0.364977\pi$
$499$	204968.	0.823162	0.411581	+	0.911373i	$0.364977\pi$
$500$	0	0
$501$	0	0
$502$	−161553.	−0.641074
$503$	144534.i	0.571262i	0.958340	+	0.285631i	$0.0922032\pi$
$503$	144534.i	0.571262i	−0.958340	+	0.285631i	$0.907797\pi$
$504$	0	0
$505$	0	0
$506$	98030.1i	0.382876i
$507$	0	0
$508$	−633361.	−2.45428
$509$	− 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
$512$	1.23561e6i	4.71347i
$513$	0	0
$514$	−133708.	−0.506094
$515$	0	0
$516$	0	0
$517$	204506.	0.765114
$518$	− 717701.i	− 2.67476i
$519$	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.	0.492191	0.246095	−	0.969246i	$-0.420852\pi$
$523$	134628.	0.492191	0.246095	+	0.969246i	$0.420852\pi$
$524$	− 1.31196e6i	− 4.77812i
$525$	0	0
$526$	721601.	2.60811
$527$	44850.6i	0.161491i
$528$	0	0
$529$	244051.	0.872106
$530$	0	0
$531$	0	0
$532$	1.30184e6	4.59974
$533$	220191.i	0.775077i
$534$	0	0
$535$	0	0
$536$	641526.i	2.23298i
$537$	0	0
$538$	−442026.	−1.52715
$539$	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.i	− 3.03946i
$543$	0	0
$544$	−426754.	−1.44205
$545$	0	0
$546$	0	0
$547$	306617.	1.02476	0.512380	−	0.858759i	$-0.328764\pi$
$547$	306617.	1.02476	0.512380	+	0.858759i	$0.328764\pi$
$548$	− 622552.i	− 2.07307i
$549$	0	0
$550$	0	0
$551$	258366.i	0.851007i
$552$	0	0
$553$	−508231.	−1.66192
$554$	− 1.12568e6i	− 3.66772i
$555$	0	0
$556$	−941659.	−3.04610
$557$	− 268937.i	− 0.866842i	−0.901191	−	0.433421i	$-0.857306\pi$
$557$	− 268937.i	− 0.866842i	0.901191	−	0.433421i	$-0.142694\pi$
$558$	0	0
$559$	183267.	0.586490
$560$	0	0
$561$	0	0
$562$	−1.15942e6	−3.67085
$563$	3899.25i	0.0123017i	0.999981	+	0.00615084i	$0.00195789\pi$
$563$	3899.25i	0.0123017i	−0.999981	+	0.00615084i	$0.998042\pi$
$564$	0	0
$565$	0	0
$566$	576004.i	1.79801i
$567$	0	0
$568$	594805.	1.84365
$569$	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.i	1.32138i
$573$	0	0
$574$	−863601.	−2.62113
$575$	0	0
$576$	0	0
$577$	−316427.	−0.950434	−0.475217	−	0.879869i	$-0.657630\pi$
$577$	−316427.	−0.950434	−0.475217	+	0.879869i	$0.657630\pi$
$578$	594345.i	1.77903i
$579$	0	0
$580$	0	0
$581$	− 751901.i	− 2.22745i
$582$	0	0
$583$	62118.7	0.182762
$584$	1.82984e6i	5.36523i
$585$	0	0
$586$	176378.	0.513629
$587$	410192.i	1.19045i	0.803559	+	0.595225i	$0.202937\pi$
$587$	410192.i	1.19045i	−0.803559	+	0.595225i	$0.797063\pi$
$588$	0	0
$589$	202493.	0.583686
$590$	0	0
$591$	0	0
$592$	1.38820e6	3.96102
$593$	− 234803.i	− 0.667719i	−0.942623	−	0.333860i	$-0.891649\pi$
$593$	− 234803.i	− 0.667719i	0.942623	−	0.333860i	$-0.108351\pi$
$594$	0	0
$595$	0	0
$596$	1.31810e6i	3.71069i
$597$	0	0
$598$	−213283.	−0.596421
$599$	− 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.i	1.98337i
$603$	0	0
$604$	−1.06909e6	−2.93048
$605$	0	0
$606$	0	0
$607$	−42983.3	−0.116660	−0.0583300	−	0.998297i	$-0.518578\pi$
$607$	−42983.3	−0.116660	−0.0583300	+	0.998297i	$0.518578\pi$
$608$	1.92672e6i	5.21208i
$609$	0	0
$610$	0	0
$611$	444942.i	1.19185i
$612$	0	0
$613$	112093.	0.298302	0.149151	−	0.988814i	$-0.452346\pi$
$613$	112093.	0.298302	0.149151	+	0.988814i	$0.452346\pi$
$614$	303264.i	0.804423i
$615$	0	0
$616$	−1.10004e6	−2.89900
$617$	362760.i	0.952904i	0.879201	+	0.476452i	$0.158077\pi$
$617$	362760.i	0.952904i	−0.879201	+	0.476452i	$0.841923\pi$
$618$	0	0
$619$	42918.9	0.112013	0.0560063	−	0.998430i	$-0.482163\pi$
$619$	42918.9	0.112013	0.0560063	+	0.998430i	$0.482163\pi$
$620$	0	0
$621$	0	0
$622$	721507.	1.86492
$623$	1.01122e6i	2.60538i
$624$	0	0
$625$	0	0
$626$	− 309643.i	− 0.790154i
$627$	0	0
$628$	746986.	1.89406
$629$	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	$0.502333\pi$
$632$	− 1.64026e6i	− 4.10657i
$633$	0	0
$634$	799837.	1.98986
$635$	0	0
$636$	0	0
$637$	−396551.	−0.977282
$638$	− 336521.i	− 0.826744i
$639$	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.	1.40977	0.704885	−	0.709321i	$-0.250998\pi$
$643$	582868.	1.40977	0.704885	+	0.709321i	$0.250998\pi$
$644$	− 619061.i	− 1.49266i
$645$	0	0
$646$	−275033.	−0.659053
$647$	− 138936.i	− 0.331899i	−0.986134	−	0.165950i	$-0.946931\pi$
$647$	− 138936.i	− 0.331899i	0.986134	−	0.165950i	$-0.0530689\pi$
$648$	0	0
$649$	334979.	0.795295
$650$	0	0
$651$	0	0
$652$	606603.	1.42695
$653$	− 683920.i	− 1.60391i	−0.597388	−	0.801953i	$-0.703795\pi$
$653$	− 683920.i	− 1.60391i	0.597388	−	0.801953i	$-0.296205\pi$
$654$	0	0
$655$	0	0
$656$	− 1.67040e6i	− 3.88161i
$657$	0	0
$658$	−1.74509e6	−4.03056
$659$	− 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	$0.731480\pi$
$662$	21077.3i	0.0480948i
$663$	0	0
$664$	2.42668e6	5.50397
$665$	0	0
$666$	0	0
$667$	122861.	0.276160
$668$	− 202173.i	− 0.453075i
$669$	0	0
$670$	0	0
$671$	− 41305.6i	− 0.0917410i
$672$	0	0
$673$	269429.	0.594859	0.297429	−	0.954744i	$-0.403871\pi$
$673$	269429.	0.594859	0.297429	+	0.954744i	$0.403871\pi$
$674$	603503.i	1.32849i
$675$	0	0
$676$	360379.	0.788616
$677$	− 660828.i	− 1.44182i	−0.693029	−	0.720910i	$-0.743724\pi$
$677$	− 660828.i	− 1.44182i	0.693029	−	0.720910i	$-0.256276\pi$
$678$	0	0
$679$	−1.08679e6	−2.35725
$680$	0	0
$681$	0	0
$682$	−263746.	−0.567044
$683$	476650.i	1.02178i	0.859645	+	0.510891i	$0.170684\pi$
$683$	476650.i	1.02178i	−0.859645	+	0.510891i	$0.829316\pi$
$684$	0	0
$685$	0	0
$686$	− 202096.i	− 0.429447i
$687$	0	0
$688$	−1.39029e6	−2.93716
$689$	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.i	0.432167i
$693$	0	0
$694$	−1.30420e6	−2.70786
$695$	0	0
$696$	0	0
$697$	135022.	0.277933
$698$	− 1.35301e6i	− 2.77710i
$699$	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.	1.02510
$704$	− 1.35753e6i	− 2.73908i
$705$	0	0
$706$	1.00399e6	2.01428
$707$	892815.i	1.78617i
$708$	0	0
$709$	−908566.	−1.80744	−0.903720	−	0.428123i	$-0.859175\pi$
$709$	−908566.	−1.80744	−0.903720	+	0.428123i	$0.859175\pi$
$710$	0	0
$711$	0	0
$712$	−3.26361e6	−6.43782
$713$	− 96291.2i	− 0.189412i
$714$	0	0
$715$	0	0
$716$	97776.4i	0.190725i
$717$	0	0
$718$	815196.	1.58130
$719$	− 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.i	0.420684i
$723$	0	0
$724$	1.16849e6	2.22919
$725$	0	0
$726$	0	0
$727$	120930.	0.228805	0.114402	−	0.993434i	$-0.463505\pi$
$727$	120930.	0.228805	0.114402	+	0.993434i	$0.463505\pi$
$728$	− 2.39335e6i	− 4.51589i
$729$	0	0
$730$	0	0
$731$	− 112380.i	− 0.210308i
$732$	0	0
$733$	152351.	0.283555	0.141778	−	0.989899i	$-0.454718\pi$
$733$	152351.	0.283555	0.141778	+	0.989899i	$0.454718\pi$
$734$	1.14783e6i	2.13052i
$735$	0	0
$736$	916210.	1.69137
$737$	− 182757.i	− 0.336463i
$738$	0	0
$739$	−836734.	−1.53214	−0.766070	−	0.642757i	$-0.777790\pi$
$739$	−836734.	−1.53214	−0.766070	+	0.642757i	$0.777790\pi$
$740$	0	0
$741$	0	0
$742$	−530068.	−0.962773
$743$	− 1.07071e6i	− 1.93952i	−0.244056	−	0.969761i	$-0.578478\pi$
$743$	− 1.07071e6i	− 1.93952i	0.244056	−	0.969761i	$-0.421522\pi$
$744$	0	0
$745$	0	0
$746$	899992.i	1.61719i
$747$	0	0
$748$	265110.	0.473830
$749$	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	$0.545630\pi$
$752$	− 3.37539e6i	− 5.96881i
$753$	0	0
$754$	732164.	1.28785
$755$	0	0
$756$	0	0
$757$	−931899.	−1.62621	−0.813106	−	0.582116i	$-0.802225\pi$
$757$	−931899.	−1.62621	−0.813106	+	0.582116i	$0.802225\pi$
$758$	− 193332.i	− 0.336486i
$759$	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.	−0.489340
$764$	− 455004.i	− 0.779522i
$765$	0	0
$766$	−615898.	−1.04967
$767$	728809.i	1.23886i
$768$	0	0
$769$	−274367.	−0.463958	−0.231979	−	0.972721i	$-0.574520\pi$
$769$	−274367.	−0.463958	−0.231979	+	0.972721i	$0.574520\pi$
$770$	0	0
$771$	0	0
$772$	−734748.	−1.23283
$773$	208073.i	0.348222i	0.984726	+	0.174111i	$0.0557051\pi$
$773$	208073.i	0.348222i	−0.984726	+	0.174111i	$0.944295\pi$
$774$	0	0
$775$	0	0
$776$	− 3.50749e6i	− 5.82469i
$777$	0	0
$778$	1.83864e6	3.03764
$779$	− 609603.i	− 1.00455i
$780$	0	0
$781$	−169447.	−0.277799
$782$	130786.i	0.213869i
$783$	0	0
$784$	3.00829e6	4.89426
$785$	0	0
$786$	0	0
$787$	352005.	0.568329	0.284164	−	0.958776i	$-0.408284\pi$
$787$	352005.	0.568329	0.284164	+	0.958776i	$0.408284\pi$
$788$	2.35539e6i	3.79324i
$789$	0	0
$790$	0	0
$791$	1.16436e6i	1.86095i
$792$	0	0
$793$	89867.9	0.142909
$794$	− 680046.i	− 1.07869i
$795$	0	0
$796$	3.19949e6	5.04957
$797$	− 32487.5i	− 0.0511445i	−0.999673	−	0.0255723i	$-0.991859\pi$
$797$	− 32487.5i	− 0.0511445i	0.999673	−	0.0255723i	$-0.00814079\pi$
$798$	0	0
$799$	272841.	0.427382
$800$	0	0
$801$	0	0
$802$	1.69249e6	2.63134
$803$	− 521282.i	− 0.808428i
$804$	0	0
$805$	0	0
$806$	− 573828.i	− 0.883307i
$807$	0	0
$808$	−2.88146e6	−4.41357
$809$	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	$0.716716\pi$
$812$	2.12513e6i	3.22310i
$813$	0	0
$814$	−659863.	−0.995876
$815$	0	0
$816$	0	0
$817$	−507378.	−0.760129
$818$	2.50756e6i	3.74752i
$819$	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.	−0.399662	−0.199831	−	0.979830i	$-0.564039\pi$
$823$	−270702.	−0.399662	−0.199831	+	0.979830i	$0.564039\pi$
$824$	1.12371e6i	1.65501i
$825$	0	0
$826$	−2.85843e6	−4.18955
$827$	883999.i	1.29253i	0.763113	+	0.646266i	$0.223670\pi$
$827$	883999.i	1.29253i	−0.763113	+	0.646266i	$0.776330\pi$
$828$	0	0
$829$	1.01480e6	1.47663	0.738314	−	0.674457i	$-0.235622\pi$
$829$	1.01480e6	1.47663	0.738314	+	0.674457i	$0.235622\pi$
$830$	0	0
$831$	0	0
$832$	2.95356e6	4.26677
$833$	243168.i	0.350442i
$834$	0	0
$835$	0	0
$836$	− 1.19692e6i	− 1.71259i
$837$	0	0
$838$	1.68980e6	2.40629
$839$	− 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
$842$	− 1.15021e6i	− 1.62239i
$843$	0	0
$844$	−1.19379e6	−1.67588
$845$	0	0
$846$	0	0
$847$	−738391.	−1.02925
$848$	− 1.02527e6i	− 1.42576i
$849$	0	0
$850$	0	0
$851$	− 240910.i	− 0.332656i
$852$	0	0
$853$	454214.	0.624256	0.312128	−	0.950040i	$-0.398958\pi$
$853$	454214.	0.624256	0.312128	+	0.950040i	$0.398958\pi$
$854$	352467.i	0.483284i
$855$	0	0
$856$	−2.68938e6	−3.67032
$857$	− 858025.i	− 1.16826i	−0.811661	−	0.584128i	$-0.801436\pi$
$857$	− 858025.i	− 1.16826i	0.811661	−	0.584128i	$-0.198564\pi$
$858$	0	0
$859$	−848325.	−1.14968	−0.574839	−	0.818267i	$-0.694935\pi$
$859$	−848325.	−1.14968	−0.574839	+	0.818267i	$0.694935\pi$
$860$	0	0
$861$	0	0
$862$	−1.47384e6	−1.98351
$863$	− 358961.i	− 0.481977i	−0.970528	−	0.240988i	$-0.922528\pi$
$863$	− 358961.i	− 0.481977i	0.970528	−	0.240988i	$-0.0774715\pi$
$864$	0	0
$865$	0	0
$866$	− 374658.i	− 0.499573i
$867$	0	0
$868$	1.66556e6	2.21065
$869$	467274.i	0.618774i
$870$	0	0
$871$	397621.	0.524122
$872$	− 919414.i	− 1.20914i
$873$	0	0
$874$	590477.	0.773001
$875$	0	0
$876$	0	0
$877$	−382506.	−0.497324	−0.248662	−	0.968590i	$-0.579991\pi$
$877$	−382506.	−0.497324	−0.248662	+	0.968590i	$0.579991\pi$
$878$	1.34511e6i	1.74490i
$879$	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.	−0.411671	−0.205835	−	0.978587i	$-0.565991\pi$
$883$	−320975.	−0.411671	−0.205835	+	0.978587i	$0.565991\pi$
$884$	576795.i	0.738103i
$885$	0	0
$886$	−2.20726e6	−2.81181
$887$	1.42234e6i	1.80783i	0.427716	+	0.903913i	$0.359318\pi$
$887$	1.42234e6i	1.80783i	−0.427716	+	0.903913i	$0.640682\pi$
$888$	0	0
$889$	−998842.	−1.26384
$890$	0	0
$891$	0	0
$892$	799067.	1.00428
$893$	− 1.23183e6i	− 1.54471i
$894$	0	0
$895$	0	0
$896$	6.01751e6i	7.49550i
$897$	0	0
$898$	−1.38953e6	−1.72312
$899$	330552.i	0.408997i
$900$	0	0
$901$	82875.3	0.102088
$902$	794005.i	0.975911i
$903$	0	0
$904$	−3.75785e6	−4.59836
$905$	0	0
$906$	0	0
$907$	1.25408e6	1.52444	0.762218	−	0.647320i	$-0.224110\pi$
$907$	1.25408e6	1.52444	0.762218	+	0.647320i	$0.224110\pi$
$908$	603498.i	0.731988i
$909$	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.	−0.829334
$914$	232288.i	0.278058i
$915$	0	0
$916$	3.04476e6	3.62879
$917$	− 2.06902e6i	− 2.46052i
$918$	0	0
$919$	−442763.	−0.524252	−0.262126	−	0.965034i	$-0.584424\pi$
$919$	−442763.	−0.524252	−0.262126	+	0.965034i	$0.584424\pi$
$920$	0	0
$921$	0	0
$922$	2.62308e6	3.08567
$923$	− 368662.i	− 0.432738i
$924$	0	0
$925$	0	0
$926$	856089.i	0.998382i
$927$	0	0
$928$	−3.14520e6	−3.65218
$929$	− 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
$932$	2.57593e6i	2.96553i
$933$	0	0
$934$	416058.	0.476936
$935$	0	0
$936$	0	0
$937$	−605604.	−0.689778	−0.344889	−	0.938643i	$-0.612083\pi$
$937$	−605604.	−0.689778	−0.344889	+	0.938643i	$0.612083\pi$
$938$	1.55949e6i	1.77246i
$939$	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.	−0.325987
$944$	− 5.52885e6i	− 6.20427i
$945$	0	0
$946$	660857.	0.738457
$947$	− 992010.i	− 1.10616i	−0.833130	−	0.553078i	$-0.813453\pi$
$947$	− 992010.i	− 1.10616i	0.833130	−	0.553078i	$-0.186547\pi$
$948$	0	0
$949$	1.13414e6	1.25932
$950$	0	0
$951$	0	0
$952$	−1.46762e6	−1.61934
$953$	− 94165.1i	− 0.103682i	−0.998655	−	0.0518411i	$-0.983491\pi$
$953$	− 94165.1i	− 0.103682i	0.998655	−	0.0518411i	$-0.0165089\pi$
$954$	0	0
$955$	0	0
$956$	1.29601e6i	1.41806i
$957$	0	0
$958$	−432622.	−0.471387
$959$	− 981795.i	− 1.06754i
$960$	0	0
$961$	−664454.	−0.719479
$962$	− 1.43565e6i	− 1.55131i
$963$	0	0
$964$	466125.	0.501589
$965$	0	0
$966$	0	0
$967$	1.46678e6	1.56860	0.784298	−	0.620385i	$-0.213023\pi$
$967$	1.46678e6	1.56860	0.784298	+	0.620385i	$0.213023\pi$
$968$	− 2.38308e6i	− 2.54324i
$969$	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
$973$	−1.48504e6	−1.56860
$974$	2.24294e6i	2.36429i
$975$	0	0
$976$	−681750.	−0.715691
$977$	− 1.07961e6i	− 1.13104i	−0.824734	−	0.565520i	$-0.808675\pi$
$977$	− 1.07961e6i	− 1.13104i	0.824734	−	0.565520i	$-0.191325\pi$
$978$	0	0
$979$	929731.	0.970045
$980$	0	0
$981$	0	0
$982$	1.45864e6	1.51261
$983$	− 529376.i	− 0.547844i	−0.961752	−	0.273922i	$-0.911679\pi$
$983$	− 529376.i	− 0.547844i	0.961752	−	0.273922i	$-0.0883210\pi$
$984$	0	0
$985$	0	0
$986$	− 448968.i	− 0.461808i
$987$	0	0
$988$	2.60413e6	2.66777
$989$	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	$0.457167\pi$
$992$	2.46502e6i	2.50494i
$993$	0	0
$994$	1.44591e6	1.46342
$995$	0	0
$996$	0	0
$997$	−329724.	−0.331711	−0.165856	−	0.986150i	$-0.553039\pi$
$997$	−329724.	−0.331711	−0.165856	+	0.986150i	$0.553039\pi$
$998$	1.60807e6i	1.61453i
$999$	0	0

Twists

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

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

Overview	Random
Universe	Knowledge

Classical	Maass
Hilbert	Bianchi

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

Level:	\( N \)	\(=\)	\( 225 = 3^{2} \cdot 5^{2} \)
Weight:	\( k \)	\(=\)	\( 5 \)
Character orbit:	\([\chi]\)	\(=\)	225.c (of order \(2\), degree \(1\), minimal)

\(f(q)\)	\(=\)	\(q+7.84548i q^{2} -45.5516 q^{4} -71.8372 q^{7} -231.847i q^{8} +O(q^{10})\)	\(q+7.84548i q^{2} -45.5516 q^{4} -71.8372 q^{7} -231.847i q^{8} +66.0480i q^{11} -143.700 q^{13} -563.598i q^{14} +1090.12 q^{16} +88.1175i q^{17} +397.835 q^{19} -518.178 q^{22} -189.182i q^{23} -1127.39i q^{26} +3272.30 q^{28} +649.431i q^{29} +508.987 q^{31} +4843.01i q^{32} -691.325 q^{34} +1273.43 q^{37} +3121.21i q^{38} -1532.30i q^{41} -1275.35 q^{43} -3008.59i q^{44} +1484.23 q^{46} -3096.33i q^{47} +2759.58 q^{49} +6545.75 q^{52} -940.508i q^{53} +16655.2i q^{56} -5095.10 q^{58} -5071.76i q^{59} -625.387 q^{61} +3993.25i q^{62} -20553.7 q^{64} -2767.03 q^{67} -4013.90i q^{68} +2565.51i q^{71} -7892.47 q^{73} +9990.67i q^{74} -18122.0 q^{76} -4744.70i q^{77} +7074.77 q^{79} +12021.6 q^{82} +10466.7i q^{83} -10005.7i q^{86} +15313.0 q^{88} -14076.6i q^{89} +10323.0 q^{91} +8617.55i q^{92} +24292.2 q^{94} +15128.5 q^{97} +21650.3i q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 6 q - 92 q^{4} - 86 q^{7}+O(q^{10}) \) 6 * q - 92 * q^4 - 86 * q^7	\( 6 q - 92 q^{4} - 86 q^{7} - 394 q^{13} + 1816 q^{16} - 250 q^{19} + 1276 q^{22} + 6412 q^{28} + 1846 q^{31} - 3692 q^{34} + 7968 q^{37} + 1382 q^{43} + 9756 q^{46} - 3232 q^{49} + 23268 q^{52} - 2932 q^{58} + 10838 q^{61} - 36144 q^{64} - 6278 q^{67} - 13024 q^{73} - 24780 q^{76} + 3904 q^{79} + 29792 q^{82} + 42744 q^{88} + 17306 q^{91} + 66364 q^{94} + 8262 q^{97}+O(q^{100}) \) 6 * q - 92 * q^4 - 86 * q^7 - 394 * q^13 + 1816 * q^16 - 250 * q^19 + 1276 * q^22 + 6412 * q^28 + 1846 * q^31 - 3692 * q^34 + 7968 * q^37 + 1382 * q^43 + 9756 * q^46 - 3232 * q^49 + 23268 * q^52 - 2932 * q^58 + 10838 * q^61 - 36144 * q^64 - 6278 * q^67 - 13024 * q^73 - 24780 * q^76 + 3904 * q^79 + 29792 * q^82 + 42744 * q^88 + 17306 * q^91 + 66364 * q^94 + 8262 * q^97

Introduction

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