LMFDB - Embedded newform 225.5.d.b.224.3

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [225,5,Mod(224,225)]

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, 1]))

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

//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: 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:	$8$
Coefficient field:	8.0.40960000.1
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{8} + 7x^{4} + 1 $ x^8 + 7*x^4 + 1
Coefficient ring:	$\Z[a_1, \ldots, a_{13}]$
Coefficient ring index:	$ 2^{10}\cdot 3^{4} $
Twist minimal:	no (minimal twist has level 45)
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label			224.3
Root			$-1.14412 - 1.14412i$ of defining polynomial
Character	$\chi$	$=$	225.224
Dual form			225.5.d.b.224.4

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-2.00657 q^{2} -11.9737 q^{4} -54.4078i q^{7} +56.1312 q^{8} +O(q^{10})$	$q-2.00657 q^{2} -11.9737 q^{4} -54.4078i q^{7} +56.1312 q^{8} -104.807i q^{11} +154.566i q^{13} +109.173i q^{14} +78.9473 q^{16} +63.7634 q^{17} +458.974 q^{19} +210.302i q^{22} -665.889 q^{23} -310.148i q^{26} +651.461i q^{28} -1088.09i q^{29} -1495.50 q^{31} -1056.51 q^{32} -127.946 q^{34} +1213.70i q^{37} -920.964 q^{38} -2510.20i q^{41} +2297.81i q^{43} +1254.92i q^{44} +1336.16 q^{46} -1513.12 q^{47} -559.212 q^{49} -1850.72i q^{52} -5351.51 q^{53} -3053.98i q^{56} +2183.34i q^{58} +4414.06i q^{59} -5212.76 q^{61} +3000.83 q^{62} +856.812 q^{64} +2065.68i q^{67} -763.482 q^{68} +2916.84i q^{71} -3715.81i q^{73} -2435.37i q^{74} -5495.60 q^{76} -5702.31 q^{77} -3162.76 q^{79} +5036.90i q^{82} -2860.06 q^{83} -4610.73i q^{86} -5882.93i q^{88} -7432.39i q^{89} +8409.59 q^{91} +7973.14 q^{92} +3036.19 q^{94} -4254.61i q^{97} +1122.10 q^{98} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 8 q + 56 q^{4}+O(q^{10}) $ 8 * q + 56 * q^4	$ 8 q + 56 q^{4} + 328 q^{16} + 3520 q^{19} - 3312 q^{31} + 10816 q^{34} - 11472 q^{46} - 17224 q^{49} - 40336 q^{61} - 23048 q^{64} + 21760 q^{76} - 35472 q^{79} - 44592 q^{91} + 65728 q^{94}+O(q^{100}) $ 8 * q + 56 * q^4 + 328 * q^16 + 3520 * q^19 - 3312 * q^31 + 10816 * q^34 - 11472 * q^46 - 17224 * q^49 - 40336 * q^61 - 23048 * q^64 + 21760 * q^76 - 35472 * q^79 - 44592 * q^91 + 65728 * q^94

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 $

$2$	−2.00657	−0.501643	−0.250822	−	0.968033i	$-0.580701\pi$
$2$	−2.00657	−0.501643	−0.250822	+	0.968033i	$0.580701\pi$
$3$	0	0
$3$	0	0
$4$	−11.9737	−0.748354
$5$	0	0
$5$	0	0
$6$	0	0
$7$	− 54.4078i	− 1.11036i	−0.831729	−	0.555182i	$-0.812649\pi$
$7$	− 54.4078i	− 1.11036i	0.831729	−	0.555182i	$-0.187351\pi$
$8$	56.1312	0.877050
$9$	0	0
$10$	0	0
$11$	− 104.807i	− 0.866172i	−0.901353	−	0.433086i	$-0.857425\pi$
$11$	− 104.807i	− 0.866172i	0.901353	−	0.433086i	$-0.142575\pi$
$12$	0	0
$13$	154.566i	0.914591i	0.889315	+	0.457295i	$0.151182\pi$
$13$	154.566i	0.914591i	−0.889315	+	0.457295i	$0.848818\pi$
$14$	109.173i	0.557006i
$15$	0	0
$16$	78.9473	0.308388
$17$	63.7634	0.220635	0.110317	−	0.993896i	$-0.464813\pi$
$17$	63.7634	0.220635	0.110317	+	0.993896i	$0.464813\pi$
$18$	0	0
$19$	458.974	1.27140	0.635698	−	0.771938i	$-0.280713\pi$
$19$	458.974	1.27140	0.635698	+	0.771938i	$0.280713\pi$
$20$	0	0
$21$	0	0
$22$	210.302i	0.434509i
$23$	−665.889	−1.25877	−0.629385	−	0.777094i	$-0.716693\pi$
$23$	−665.889	−1.25877	−0.629385	+	0.777094i	$0.716693\pi$
$24$	0	0
$25$	0	0
$26$	− 310.148i	− 0.458798i
$27$	0	0
$28$	651.461i	0.830945i
$29$	− 1088.09i	− 1.29381i	−0.762571	−	0.646905i	$-0.776063\pi$
$29$	− 1088.09i	− 1.29381i	0.762571	−	0.646905i	$-0.223937\pi$
$30$	0	0
$31$	−1495.50	−1.55619	−0.778095	−	0.628146i	$-0.783814\pi$
$31$	−1495.50	−1.55619	−0.778095	+	0.628146i	$0.783814\pi$
$32$	−1056.51	−1.03175
$33$	0	0
$34$	−127.946	−0.110680
$35$	0	0
$36$	0	0
$37$	1213.70i	0.886557i	0.896384	+	0.443278i	$0.146185\pi$
$37$	1213.70i	0.886557i	−0.896384	+	0.443278i	$0.853815\pi$
$38$	−920.964	−0.637787
$39$	0	0
$40$	0	0
$41$	− 2510.20i	− 1.49328i	−0.665229	−	0.746640i	$-0.731666\pi$
$41$	− 2510.20i	− 1.49328i	0.665229	−	0.746640i	$-0.268334\pi$
$42$	0	0
$43$	2297.81i	1.24273i	0.783520	+	0.621367i	$0.213422\pi$
$43$	2297.81i	1.24273i	−0.783520	+	0.621367i	$0.786578\pi$
$44$	1254.92i	0.648203i
$45$	0	0
$46$	1336.16	0.631453
$47$	−1513.12	−0.684981	−0.342490	−	0.939521i	$-0.611270\pi$
$47$	−1513.12	−0.684981	−0.342490	+	0.939521i	$0.611270\pi$
$48$	0	0
$49$	−559.212	−0.232908
$50$	0	0
$51$	0	0
$52$	− 1850.72i	− 0.684438i
$53$	−5351.51	−1.90513	−0.952564	−	0.304337i	$-0.901565\pi$
$53$	−5351.51	−1.90513	−0.952564	+	0.304337i	$0.901565\pi$
$54$	0	0
$55$	0	0
$56$	− 3053.98i	− 0.973845i
$57$	0	0
$58$	2183.34i	0.649031i
$59$	4414.06i	1.26804i	0.773315	+	0.634022i	$0.218597\pi$
$59$	4414.06i	1.26804i	−0.773315	+	0.634022i	$0.781403\pi$
$60$	0	0
$61$	−5212.76	−1.40090	−0.700452	−	0.713700i	$-0.747018\pi$
$61$	−5212.76	−1.40090	−0.700452	+	0.713700i	$0.747018\pi$
$62$	3000.83	0.780652
$63$	0	0
$64$	856.812	0.209183
$65$	0	0
$66$	0	0
$67$	2065.68i	0.460165i	0.973171	+	0.230083i	$0.0738997\pi$
$67$	2065.68i	0.460165i	−0.973171	+	0.230083i	$0.926100\pi$
$68$	−763.482	−0.165113
$69$	0	0
$70$	0	0
$71$	2916.84i	0.578624i	0.957235	+	0.289312i	$0.0934264\pi$
$71$	2916.84i	0.578624i	−0.957235	+	0.289312i	$0.906574\pi$
$72$	0	0
$73$	− 3715.81i	− 0.697281i	−0.937256	−	0.348641i	$-0.886643\pi$
$73$	− 3715.81i	− 0.697281i	0.937256	−	0.348641i	$-0.113357\pi$
$74$	− 2435.37i	− 0.444735i
$75$	0	0
$76$	−5495.60	−0.951454
$77$	−5702.31	−0.961766
$78$	0	0
$79$	−3162.76	−0.506772	−0.253386	−	0.967365i	$-0.581544\pi$
$79$	−3162.76	−0.506772	−0.253386	+	0.967365i	$0.581544\pi$
$80$	0	0
$81$	0	0
$82$	5036.90i	0.749094i
$83$	−2860.06	−0.415163	−0.207582	−	0.978218i	$-0.566559\pi$
$83$	−2860.06	−0.415163	−0.207582	+	0.978218i	$0.566559\pi$
$84$	0	0
$85$	0	0
$86$	− 4610.73i	− 0.623409i
$87$	0	0
$88$	− 5882.93i	− 0.759676i
$89$	− 7432.39i	− 0.938315i	−0.883115	−	0.469157i	$-0.844558\pi$
$89$	− 7432.39i	− 0.938315i	0.883115	−	0.469157i	$-0.155442\pi$
$90$	0	0
$91$	8409.59	1.01553
$92$	7973.14	0.942006
$93$	0	0
$94$	3036.19	0.343616
$95$	0	0
$96$	0	0
$97$	− 4254.61i	− 0.452185i	−0.974106	−	0.226092i	$-0.927405\pi$
$97$	− 4254.61i	− 0.452185i	0.974106	−	0.226092i	$-0.0725951\pi$
$98$	1122.10	0.116837
$99$	0	0
$100$	0	0
$101$	6362.56i	0.623720i	0.950128	+	0.311860i	$0.100952\pi$
$101$	6362.56i	0.623720i	−0.950128	+	0.311860i	$0.899048\pi$
$102$	0	0
$103$	16530.4i	1.55815i	0.626930	+	0.779075i	$0.284311\pi$
$103$	16530.4i	1.55815i	−0.626930	+	0.779075i	$0.715689\pi$
$104$	8675.96i	0.802142i
$105$	0	0
$106$	10738.2	0.955695
$107$	−12759.8	−1.11449	−0.557244	−	0.830349i	$-0.688141\pi$
$107$	−12759.8	−1.11449	−0.557244	+	0.830349i	$0.688141\pi$
$108$	0	0
$109$	−8065.54	−0.678861	−0.339430	−	0.940631i	$-0.610234\pi$
$109$	−8065.54	−0.678861	−0.339430	+	0.940631i	$0.610234\pi$
$110$	0	0
$111$	0	0
$112$	− 4295.35i	− 0.342423i
$113$	24841.4	1.94545	0.972724	−	0.231967i	$-0.0745160\pi$
$113$	24841.4	1.94545	0.972724	+	0.231967i	$0.0745160\pi$
$114$	0	0
$115$	0	0
$116$	13028.5i	0.968228i
$117$	0	0
$118$	− 8857.14i	− 0.636106i
$119$	− 3469.23i	− 0.244985i
$120$	0	0
$121$	3656.53	0.249746
$122$	10459.8	0.702754
$123$	0	0
$124$	17906.6	1.16458
$125$	0	0
$126$	0	0
$127$	− 14534.0i	− 0.901108i	−0.892749	−	0.450554i	$-0.851226\pi$
$127$	− 14534.0i	− 0.901108i	0.892749	−	0.450554i	$-0.148774\pi$
$128$	15184.9	0.926816
$129$	0	0
$130$	0	0
$131$	1405.98i	0.0819287i	0.999161	+	0.0409643i	$0.0130430\pi$
$131$	1405.98i	0.0819287i	−0.999161	+	0.0409643i	$0.986957\pi$
$132$	0	0
$133$	− 24971.8i	− 1.41171i
$134$	− 4144.94i	− 0.230839i
$135$	0	0
$136$	3579.12	0.193508
$137$	17530.6	0.934019	0.467010	−	0.884252i	$-0.345331\pi$
$137$	17530.6	0.934019	0.467010	+	0.884252i	$0.345331\pi$
$138$	0	0
$139$	−6890.79	−0.356648	−0.178324	−	0.983972i	$-0.557067\pi$
$139$	−6890.79	−0.356648	−0.178324	+	0.983972i	$0.557067\pi$
$140$	0	0
$141$	0	0
$142$	− 5852.85i	− 0.290263i
$143$	16199.6	0.792193
$144$	0	0
$145$	0	0
$146$	7456.05i	0.349786i
$147$	0	0
$148$	− 14532.4i	− 0.663458i
$149$	− 28238.3i	− 1.27194i	−0.771714	−	0.635970i	$-0.780600\pi$
$149$	− 28238.3i	− 1.27194i	0.771714	−	0.635970i	$-0.219400\pi$
$150$	0	0
$151$	8963.73	0.393129	0.196564	−	0.980491i	$-0.437022\pi$
$151$	8963.73	0.393129	0.196564	+	0.980491i	$0.437022\pi$
$152$	25762.7	1.11508
$153$	0	0
$154$	11442.1	0.482463
$155$	0	0
$156$	0	0
$157$	37910.5i	1.53801i	0.639241	+	0.769006i	$0.279248\pi$
$157$	37910.5i	1.53801i	−0.639241	+	0.769006i	$0.720752\pi$
$158$	6346.32	0.254219
$159$	0	0
$160$	0	0
$161$	36229.6i	1.39769i
$162$	0	0
$163$	− 2933.87i	− 0.110424i	−0.998475	−	0.0552122i	$-0.982416\pi$
$163$	− 2933.87i	− 0.110424i	0.998475	−	0.0552122i	$-0.0175835\pi$
$164$	30056.3i	1.11750i
$165$	0	0
$166$	5738.92	0.208264
$167$	−19694.6	−0.706179	−0.353090	−	0.935589i	$-0.614869\pi$
$167$	−19694.6	−0.706179	−0.353090	+	0.935589i	$0.614869\pi$
$168$	0	0
$169$	4670.40	0.163524
$170$	0	0
$171$	0	0
$172$	− 27513.3i	− 0.930005i
$173$	−29155.8	−0.974166	−0.487083	−	0.873356i	$-0.661939\pi$
$173$	−29155.8	−0.974166	−0.487083	+	0.873356i	$0.661939\pi$
$174$	0	0
$175$	0	0
$176$	− 8274.22i	− 0.267117i
$177$	0	0
$178$	14913.6i	0.470699i
$179$	16308.8i	0.508999i	0.967073	+	0.254499i	$0.0819107\pi$
$179$	16308.8i	0.508999i	−0.967073	+	0.254499i	$0.918089\pi$
$180$	0	0
$181$	−18259.6	−0.557359	−0.278680	−	0.960384i	$-0.589897\pi$
$181$	−18259.6	−0.557359	−0.278680	+	0.960384i	$0.589897\pi$
$182$	−16874.5	−0.509433
$183$	0	0
$184$	−37377.2	−1.10400
$185$	0	0
$186$	0	0
$187$	− 6682.84i	− 0.191108i
$188$	18117.6	0.512608
$189$	0	0
$190$	0	0
$191$	− 26825.8i	− 0.735337i	−0.929957	−	0.367669i	$-0.880156\pi$
$191$	− 26825.8i	− 0.735337i	0.929957	−	0.367669i	$-0.119844\pi$
$192$	0	0
$193$	3540.51i	0.0950499i	0.998870	+	0.0475250i	$0.0151334\pi$
$193$	3540.51i	0.0950499i	−0.998870	+	0.0475250i	$0.984867\pi$
$194$	8537.18i	0.226835i
$195$	0	0
$196$	6695.82	0.174298
$197$	−62733.4	−1.61647	−0.808233	−	0.588863i	$-0.799576\pi$
$197$	−62733.4	−1.61647	−0.808233	+	0.588863i	$0.799576\pi$
$198$	0	0
$199$	−58584.2	−1.47936	−0.739681	−	0.672958i	$-0.765023\pi$
$199$	−58584.2	−1.47936	−0.739681	+	0.672958i	$0.765023\pi$
$200$	0	0
$201$	0	0
$202$	− 12766.9i	− 0.312885i
$203$	−59200.8	−1.43660
$204$	0	0
$205$	0	0
$206$	− 33169.5i	− 0.781636i
$207$	0	0
$208$	12202.6i	0.282049i
$209$	− 48103.6i	− 1.10125i
$210$	0	0
$211$	−499.685	−0.0112236	−0.00561179	−	0.999984i	$-0.501786\pi$
$211$	−499.685	−0.0112236	−0.00561179	+	0.999984i	$0.501786\pi$
$212$	64077.2	1.42571
$213$	0	0
$214$	25603.4	0.559076
$215$	0	0
$216$	0	0
$217$	81366.9i	1.72794i
$218$	16184.1	0.340546
$219$	0	0
$220$	0	0
$221$	9855.65i	0.201790i
$222$	0	0
$223$	13340.6i	0.268266i	0.990963	+	0.134133i	$0.0428249\pi$
$223$	13340.6i	0.268266i	−0.990963	+	0.134133i	$0.957175\pi$
$224$	57482.6i	1.14562i
$225$	0	0
$226$	−49846.1	−0.975920
$227$	−10888.5	−0.211309	−0.105654	−	0.994403i	$-0.533694\pi$
$227$	−10888.5	−0.211309	−0.105654	+	0.994403i	$0.533694\pi$
$228$	0	0
$229$	34935.7	0.666191	0.333096	−	0.942893i	$-0.391907\pi$
$229$	34935.7	0.666191	0.333096	+	0.942893i	$0.391907\pi$
$230$	0	0
$231$	0	0
$232$	− 61076.0i	− 1.13474i
$233$	−46852.0	−0.863012	−0.431506	−	0.902110i	$-0.642018\pi$
$233$	−46852.0	−0.863012	−0.431506	+	0.902110i	$0.642018\pi$
$234$	0	0
$235$	0	0
$236$	− 52852.5i	− 0.948946i
$237$	0	0
$238$	6961.26i	0.122895i
$239$	− 37329.9i	− 0.653524i	−0.945107	−	0.326762i	$-0.894043\pi$
$239$	− 37329.9i	− 0.653524i	0.945107	−	0.326762i	$-0.105957\pi$
$240$	0	0
$241$	−36463.3	−0.627800	−0.313900	−	0.949456i	$-0.601636\pi$
$241$	−36463.3	−0.627800	−0.313900	+	0.949456i	$0.601636\pi$
$242$	−7337.10	−0.125283
$243$	0	0
$244$	62415.9	1.04837
$245$	0	0
$246$	0	0
$247$	70941.6i	1.16281i
$248$	−83944.1	−1.36486
$249$	0	0
$250$	0	0
$251$	− 56356.8i	− 0.894539i	−0.894399	−	0.447269i	$-0.852397\pi$
$251$	− 56356.8i	− 0.894539i	0.894399	−	0.447269i	$-0.147603\pi$
$252$	0	0
$253$	69789.7i	1.09031i
$254$	29163.5i	0.452035i
$255$	0	0
$256$	−44178.7	−0.674113
$257$	15210.2	0.230287	0.115143	−	0.993349i	$-0.463267\pi$
$257$	15210.2	0.230287	0.115143	+	0.993349i	$0.463267\pi$
$258$	0	0
$259$	66034.6	0.984401
$260$	0	0
$261$	0	0
$262$	− 2821.20i	− 0.0410990i
$263$	−35627.8	−0.515084	−0.257542	−	0.966267i	$-0.582913\pi$
$263$	−35627.8	−0.515084	−0.257542	+	0.966267i	$0.582913\pi$
$264$	0	0
$265$	0	0
$266$	50107.7i	0.708175i
$267$	0	0
$268$	− 24733.8i	− 0.344367i
$269$	45736.8i	0.632064i	0.948749	+	0.316032i	$0.102351\pi$
$269$	45736.8i	0.632064i	−0.948749	+	0.316032i	$0.897649\pi$
$270$	0	0
$271$	30592.4	0.416557	0.208279	−	0.978070i	$-0.433214\pi$
$271$	30592.4	0.416557	0.208279	+	0.978070i	$0.433214\pi$
$272$	5033.95	0.0680411
$273$	0	0
$274$	−35176.4	−0.468544
$275$	0	0
$276$	0	0
$277$	− 95851.3i	− 1.24922i	−0.780937	−	0.624610i	$-0.785258\pi$
$277$	− 95851.3i	− 1.24922i	0.780937	−	0.624610i	$-0.214742\pi$
$278$	13826.9	0.178910
$279$	0	0
$280$	0	0
$281$	− 96197.4i	− 1.21829i	−0.793059	−	0.609145i	$-0.791513\pi$
$281$	− 96197.4i	− 1.21829i	0.793059	−	0.609145i	$-0.208487\pi$
$282$	0	0
$283$	113832.i	1.42132i	0.703538	+	0.710658i	$0.251603\pi$
$283$	113832.i	1.42132i	−0.703538	+	0.710658i	$0.748397\pi$
$284$	− 34925.3i	− 0.433015i
$285$	0	0
$286$	−32505.6	−0.397398
$287$	−136575.	−1.65808
$288$	0	0
$289$	−79455.2	−0.951320
$290$	0	0
$291$	0	0
$292$	44491.9i	0.521813i
$293$	35122.3	0.409117	0.204559	−	0.978854i	$-0.434424\pi$
$293$	35122.3	0.409117	0.204559	+	0.978854i	$0.434424\pi$
$294$	0	0
$295$	0	0
$296$	68126.2i	0.777554i
$297$	0	0
$298$	56662.3i	0.638060i
$299$	− 102924.i	− 1.15126i
$300$	0	0
$301$	125019.	1.37989
$302$	−17986.4	−0.197210
$303$	0	0
$304$	36234.7	0.392083
$305$	0	0
$306$	0	0
$307$	100086.i	1.06193i	0.847394	+	0.530964i	$0.178170\pi$
$307$	100086.i	1.06193i	−0.847394	+	0.530964i	$0.821830\pi$
$308$	68277.6	0.719742
$309$	0	0
$310$	0	0
$311$	− 101381.i	− 1.04818i	−0.851664	−	0.524088i	$-0.824406\pi$
$311$	− 101381.i	− 1.04818i	0.851664	−	0.524088i	$-0.175594\pi$
$312$	0	0
$313$	− 25625.8i	− 0.261571i	−0.991411	−	0.130785i	$-0.958250\pi$
$313$	− 25625.8i	− 0.261571i	0.991411	−	0.130785i	$-0.0417499\pi$
$314$	− 76070.1i	− 0.771533i
$315$	0	0
$316$	37869.9	0.379245
$317$	44791.9	0.445740	0.222870	−	0.974848i	$-0.428458\pi$
$317$	44791.9	0.445740	0.222870	+	0.974848i	$0.428458\pi$
$318$	0	0
$319$	−114040.	−1.12066
$320$	0	0
$321$	0	0
$322$	− 72697.3i	− 0.701143i
$323$	29265.7	0.280514
$324$	0	0
$325$	0	0
$326$	5887.02i	0.0553937i
$327$	0	0
$328$	− 140901.i	− 1.30968i
$329$	82325.7i	0.760578i
$330$	0	0
$331$	−112477.	−1.02661	−0.513307	−	0.858205i	$-0.671580\pi$
$331$	−112477.	−1.02661	−0.513307	+	0.858205i	$0.671580\pi$
$332$	34245.4	0.310689
$333$	0	0
$334$	39518.7	0.354250
$335$	0	0
$336$	0	0
$337$	− 58794.0i	− 0.517694i	−0.965918	−	0.258847i	$-0.916657\pi$
$337$	− 58794.0i	− 0.517694i	0.965918	−	0.258847i	$-0.0833426\pi$
$338$	−9371.50	−0.0820306
$339$	0	0
$340$	0	0
$341$	156738.i	1.34793i
$342$	0	0
$343$	− 100208.i	− 0.851751i
$344$	128979.i	1.08994i
$345$	0	0
$346$	58503.3	0.488684
$347$	70987.6	0.589554	0.294777	−	0.955566i	$-0.404755\pi$
$347$	70987.6	0.589554	0.294777	+	0.955566i	$0.404755\pi$
$348$	0	0
$349$	128277.	1.05317	0.526586	−	0.850122i	$-0.323472\pi$
$349$	128277.	1.05317	0.526586	+	0.850122i	$0.323472\pi$
$350$	0	0
$351$	0	0
$352$	110730.i	0.893674i
$353$	46716.0	0.374901	0.187450	−	0.982274i	$-0.439978\pi$
$353$	46716.0	0.374901	0.187450	+	0.982274i	$0.439978\pi$
$354$	0	0
$355$	0	0
$356$	88993.0i	0.702192i
$357$	0	0
$358$	− 32724.8i	− 0.255336i
$359$	− 14079.9i	− 0.109247i	−0.998507	−	0.0546237i	$-0.982604\pi$
$359$	− 14079.9i	− 0.109247i	0.998507	−	0.0546237i	$-0.0173959\pi$
$360$	0	0
$361$	80335.8	0.616446
$362$	36639.3	0.279595
$363$	0	0
$364$	−100694.	−0.759975
$365$	0	0
$366$	0	0
$367$	− 170291.i	− 1.26432i	−0.774836	−	0.632162i	$-0.782168\pi$
$367$	− 170291.i	− 1.26432i	0.774836	−	0.632162i	$-0.217832\pi$
$368$	−52570.2	−0.388190
$369$	0	0
$370$	0	0
$371$	291164.i	2.11539i
$372$	0	0
$373$	− 184961.i	− 1.32942i	−0.747101	−	0.664710i	$-0.768555\pi$
$373$	− 184961.i	− 1.32942i	0.747101	−	0.664710i	$-0.231445\pi$
$374$	13409.6i	0.0958678i
$375$	0	0
$376$	−84933.3	−0.600762
$377$	168182.	1.18331
$378$	0	0
$379$	13059.8	0.0909196	0.0454598	−	0.998966i	$-0.485525\pi$
$379$	13059.8	0.0909196	0.0454598	+	0.998966i	$0.485525\pi$
$380$	0	0
$381$	0	0
$382$	53828.0i	0.368877i
$383$	159528.	1.08752	0.543761	−	0.839240i	$-0.317000\pi$
$383$	159528.	1.08752	0.543761	+	0.839240i	$0.317000\pi$
$384$	0	0
$385$	0	0
$386$	− 7104.30i	− 0.0476811i
$387$	0	0
$388$	50943.3i	0.338394i
$389$	14659.6i	0.0968772i	0.998826	+	0.0484386i	$0.0154245\pi$
$389$	14659.6i	0.0968772i	−0.998826	+	0.0484386i	$0.984575\pi$
$390$	0	0
$391$	−42459.4	−0.277728
$392$	−31389.2	−0.204272
$393$	0	0
$394$	125879.	0.810889
$395$	0	0
$396$	0	0
$397$	− 43901.8i	− 0.278549i	−0.990254	−	0.139274i	$-0.955523\pi$
$397$	− 43901.8i	− 0.278549i	0.990254	−	0.139274i	$-0.0444770\pi$
$398$	117553.	0.742111
$399$	0	0
$400$	0	0
$401$	− 22174.1i	− 0.137898i	−0.997620	−	0.0689488i	$-0.978035\pi$
$401$	− 22174.1i	− 0.137898i	0.997620	−	0.0689488i	$-0.0219645\pi$
$402$	0	0
$403$	− 231153.i	− 1.42328i
$404$	− 76183.2i	− 0.466763i
$405$	0	0
$406$	118791.	0.720660
$407$	127204.	0.767911
$408$	0	0
$409$	−262893.	−1.57157	−0.785783	−	0.618503i	$-0.787740\pi$
$409$	−262893.	−1.57157	−0.785783	+	0.618503i	$0.787740\pi$
$410$	0	0
$411$	0	0
$412$	− 197930.i	− 1.16605i
$413$	240160.	1.40799
$414$	0	0
$415$	0	0
$416$	− 163301.i	− 0.943630i
$417$	0	0
$418$	96523.3i	0.552433i
$419$	69090.3i	0.393540i	0.980450	+	0.196770i	$0.0630452\pi$
$419$	69090.3i	0.393540i	−0.980450	+	0.196770i	$0.936955\pi$
$420$	0	0
$421$	306213.	1.72767	0.863833	−	0.503778i	$-0.168057\pi$
$421$	306213.	1.72767	0.863833	+	0.503778i	$0.168057\pi$
$422$	1002.66	0.00563024
$423$	0	0
$424$	−300386.	−1.67089
$425$	0	0
$426$	0	0
$427$	283615.i	1.55551i
$428$	152781.	0.834032
$429$	0	0
$430$	0	0
$431$	− 37377.2i	− 0.201211i	−0.994926	−	0.100606i	$-0.967922\pi$
$431$	− 37377.2i	− 0.201211i	0.994926	−	0.100606i	$-0.0320780\pi$
$432$	0	0
$433$	14300.7i	0.0762748i	0.999273	+	0.0381374i	$0.0121425\pi$
$433$	14300.7i	0.0762748i	−0.999273	+	0.0381374i	$0.987858\pi$
$434$	− 163269.i	− 0.866808i
$435$	0	0
$436$	96574.1	0.508028
$437$	−305626.	−1.60039
$438$	0	0
$439$	302771.	1.57103	0.785516	−	0.618842i	$-0.212398\pi$
$439$	302771.	1.57103	0.785516	+	0.618842i	$0.212398\pi$
$440$	0	0
$441$	0	0
$442$	− 19776.1i	− 0.101227i
$443$	−340460.	−1.73484	−0.867419	−	0.497578i	$-0.834223\pi$
$443$	−340460.	−1.73484	−0.867419	+	0.497578i	$0.834223\pi$
$444$	0	0
$445$	0	0
$446$	− 26768.9i	− 0.134574i
$447$	0	0
$448$	− 46617.3i	− 0.232269i
$449$	52498.8i	0.260410i	0.991487	+	0.130205i	$0.0415635\pi$
$449$	52498.8i	0.260410i	−0.991487	+	0.130205i	$0.958437\pi$
$450$	0	0
$451$	−263086.	−1.29344
$452$	−297443.	−1.45588
$453$	0	0
$454$	21848.6	0.106002
$455$	0	0
$456$	0	0
$457$	− 278587.i	− 1.33391i	−0.745096	−	0.666957i	$-0.767596\pi$
$457$	− 278587.i	− 1.33391i	0.745096	−	0.666957i	$-0.232404\pi$
$458$	−70101.1	−0.334190
$459$	0	0
$460$	0	0
$461$	351979.i	1.65621i	0.560574	+	0.828104i	$0.310580\pi$
$461$	351979.i	1.65621i	−0.560574	+	0.828104i	$0.689420\pi$
$462$	0	0
$463$	402741.i	1.87873i	0.342922	+	0.939364i	$0.388583\pi$
$463$	402741.i	1.87873i	−0.342922	+	0.939364i	$0.611417\pi$
$464$	− 85902.1i	− 0.398995i
$465$	0	0
$466$	94012.0	0.432924
$467$	269418.	1.23536	0.617678	−	0.786431i	$-0.288073\pi$
$467$	269418.	1.23536	0.617678	+	0.786431i	$0.288073\pi$
$468$	0	0
$469$	112389.	0.510951
$470$	0	0
$471$	0	0
$472$	247767.i	1.11214i
$473$	240827.	1.07642
$474$	0	0
$475$	0	0
$476$	41539.4i	0.183335i
$477$	0	0
$478$	74905.2i	0.327836i
$479$	131839.i	0.574608i	0.957839	+	0.287304i	$0.0927590\pi$
$479$	131839.i	0.574608i	−0.957839	+	0.287304i	$0.907241\pi$
$480$	0	0
$481$	−187596.	−0.810837
$482$	73166.2	0.314932
$483$	0	0
$484$	−43782.1	−0.186898
$485$	0	0
$486$	0	0
$487$	− 52995.4i	− 0.223450i	−0.993739	−	0.111725i	$-0.964362\pi$
$487$	− 52995.4i	− 0.223450i	0.993739	−	0.111725i	$-0.0356376\pi$
$488$	−292599.	−1.22866
$489$	0	0
$490$	0	0
$491$	− 341865.i	− 1.41805i	−0.705182	−	0.709026i	$-0.749135\pi$
$491$	− 341865.i	− 1.41805i	0.705182	−	0.709026i	$-0.250865\pi$
$492$	0	0
$493$	− 69380.6i	− 0.285459i
$494$	− 142350.i	− 0.583314i
$495$	0	0
$496$	−118066.	−0.479910
$497$	158699.	0.642483
$498$	0	0
$499$	358332.	1.43908	0.719540	−	0.694451i	$-0.244353\pi$
$499$	358332.	1.43908	0.719540	+	0.694451i	$0.244353\pi$
$500$	0	0
$501$	0	0
$502$	113084.i	0.448739i
$503$	−362170.	−1.43145	−0.715726	−	0.698382i	$-0.753904\pi$
$503$	−362170.	−1.43145	−0.715726	+	0.698382i	$0.753904\pi$
$504$	0	0
$505$	0	0
$506$	− 140038.i	− 0.546947i
$507$	0	0
$508$	174025.i	0.674348i
$509$	− 70176.0i	− 0.270865i	−0.990787	−	0.135433i	$-0.956758\pi$
$509$	− 70176.0i	− 0.270865i	0.990787	−	0.135433i	$-0.0432424\pi$
$510$	0	0
$511$	−202169.	−0.774236
$512$	−154311.	−0.588651
$513$	0	0
$514$	−30520.4	−0.115522
$515$	0	0
$516$	0	0
$517$	158585.i	0.593311i
$518$	−132503.	−0.493818
$519$	0	0
$520$	0	0
$521$	− 268892.i	− 0.990611i	−0.868719	−	0.495306i	$-0.835056\pi$
$521$	− 268892.i	− 0.990611i	0.868719	−	0.495306i	$-0.164944\pi$
$522$	0	0
$523$	− 138005.i	− 0.504534i	−0.967658	−	0.252267i	$-0.918824\pi$
$523$	− 138005.i	− 0.504534i	0.967658	−	0.252267i	$-0.0811761\pi$
$524$	− 16834.7i	− 0.0613116i
$525$	0	0
$526$	71489.9	0.258388
$527$	−95358.1	−0.343350
$528$	0	0
$529$	163568.	0.584502
$530$	0	0
$531$	0	0
$532$	299004.i	1.05646i
$533$	387992.	1.36574
$534$	0	0
$535$	0	0
$536$	115949.i	0.403588i
$537$	0	0
$538$	− 91774.2i	− 0.317071i
$539$	58609.2i	0.201738i
$540$	0	0
$541$	−49079.9	−0.167691	−0.0838455	−	0.996479i	$-0.526720\pi$
$541$	−49079.9	−0.167691	−0.0838455	+	0.996479i	$0.526720\pi$
$542$	−61385.8	−0.208963
$543$	0	0
$544$	−67366.9	−0.227640
$545$	0	0
$546$	0	0
$547$	− 533201.i	− 1.78204i	−0.453968	−	0.891018i	$-0.649992\pi$
$547$	− 533201.i	− 1.78204i	0.453968	−	0.891018i	$-0.350008\pi$
$548$	−209906.	−0.698977
$549$	0	0
$550$	0	0
$551$	− 499406.i	− 1.64494i
$552$	0	0
$553$	172079.i	0.562701i
$554$	192333.i	0.626662i
$555$	0	0
$556$	82508.1	0.266899
$557$	215094.	0.693294	0.346647	−	0.937996i	$-0.387320\pi$
$557$	215094.	0.693294	0.346647	+	0.937996i	$0.387320\pi$
$558$	0	0
$559$	−355164.	−1.13659
$560$	0	0
$561$	0	0
$562$	193027.i	0.611147i
$563$	−495549.	−1.56340	−0.781700	−	0.623654i	$-0.785647\pi$
$563$	−495549.	−1.56340	−0.781700	+	0.623654i	$0.785647\pi$
$564$	0	0
$565$	0	0
$566$	− 228412.i	− 0.712993i
$567$	0	0
$568$	163726.i	0.507482i
$569$	193944.i	0.599033i	0.954091	+	0.299517i	$0.0968254\pi$
$569$	193944.i	0.599033i	−0.954091	+	0.299517i	$0.903175\pi$
$570$	0	0
$571$	−187645.	−0.575526	−0.287763	−	0.957702i	$-0.592912\pi$
$571$	−187645.	−0.575526	−0.287763	+	0.957702i	$0.592912\pi$
$572$	−193968.	−0.592841
$573$	0	0
$574$	274047.	0.831766
$575$	0	0
$576$	0	0
$577$	− 366387.i	− 1.10050i	−0.835001	−	0.550249i	$-0.814533\pi$
$577$	− 366387.i	− 1.10050i	0.835001	−	0.550249i	$-0.185467\pi$
$578$	159433.	0.477223
$579$	0	0
$580$	0	0
$581$	155610.i	0.460982i
$582$	0	0
$583$	560874.i	1.65017i
$584$	− 208573.i	− 0.611551i
$585$	0	0
$586$	−70475.5	−0.205231
$587$	−298641.	−0.866709	−0.433355	−	0.901224i	$-0.642670\pi$
$587$	−298641.	−0.866709	−0.433355	+	0.901224i	$0.642670\pi$
$588$	0	0
$589$	−686395.	−1.97853
$590$	0	0
$591$	0	0
$592$	95818.1i	0.273403i
$593$	−51059.5	−0.145200	−0.0726001	−	0.997361i	$-0.523130\pi$
$593$	−51059.5	−0.145200	−0.0726001	+	0.997361i	$0.523130\pi$
$594$	0	0
$595$	0	0
$596$	338116.i	0.951861i
$597$	0	0
$598$	206524.i	0.577521i
$599$	624156.i	1.73956i	0.493440	+	0.869780i	$0.335739\pi$
$599$	624156.i	1.73956i	−0.493440	+	0.869780i	$0.664261\pi$
$600$	0	0
$601$	−203319.	−0.562897	−0.281449	−	0.959576i	$-0.590815\pi$
$601$	−203319.	−0.562897	−0.281449	+	0.959576i	$0.590815\pi$
$602$	−250860.	−0.692211
$603$	0	0
$604$	−107329.	−0.294199
$605$	0	0
$606$	0	0
$607$	229991.i	0.624213i	0.950047	+	0.312107i	$0.101035\pi$
$607$	229991.i	0.624213i	−0.950047	+	0.312107i	$0.898965\pi$
$608$	−484911.	−1.31176
$609$	0	0
$610$	0	0
$611$	− 233877.i	− 0.626477i
$612$	0	0
$613$	− 183449.i	− 0.488196i	−0.969751	−	0.244098i	$-0.921508\pi$
$613$	− 183449.i	− 0.488196i	0.969751	−	0.244098i	$-0.0784919\pi$
$614$	− 200829.i	− 0.532709i
$615$	0	0
$616$	−320078.	−0.843517
$617$	351348.	0.922926	0.461463	−	0.887160i	$-0.347325\pi$
$617$	351348.	0.922926	0.461463	+	0.887160i	$0.347325\pi$
$618$	0	0
$619$	263299.	0.687175	0.343588	−	0.939121i	$-0.388358\pi$
$619$	263299.	0.687175	0.343588	+	0.939121i	$0.388358\pi$
$620$	0	0
$621$	0	0
$622$	203427.i	0.525810i
$623$	−404380.	−1.04187
$624$	0	0
$625$	0	0
$626$	51420.1i	0.131215i
$627$	0	0
$628$	− 453927.i	− 1.15098i
$629$	77389.4i	0.195605i
$630$	0	0
$631$	602867.	1.51413	0.757065	−	0.653340i	$-0.226633\pi$
$631$	602867.	1.51413	0.757065	+	0.653340i	$0.226633\pi$
$632$	−177530.	−0.444464
$633$	0	0
$634$	−89878.3	−0.223602
$635$	0	0
$636$	0	0
$637$	− 86435.1i	− 0.213015i
$638$	228829.	0.562172
$639$	0	0
$640$	0	0
$641$	79580.8i	0.193683i	0.995300	+	0.0968417i	$0.0308741\pi$
$641$	79580.8i	0.193683i	−0.995300	+	0.0968417i	$0.969126\pi$
$642$	0	0
$643$	− 160408.i	− 0.387974i	−0.981004	−	0.193987i	$-0.937858\pi$
$643$	− 160408.i	− 0.387974i	0.981004	−	0.193987i	$-0.0621420\pi$
$644$	− 433801.i	− 1.04597i
$645$	0	0
$646$	−58723.8	−0.140718
$647$	−286213.	−0.683723	−0.341861	−	0.939750i	$-0.611057\pi$
$647$	−286213.	−0.683723	−0.341861	+	0.939750i	$0.611057\pi$
$648$	0	0
$649$	462624.	1.09834
$650$	0	0
$651$	0	0
$652$	35129.1i	0.0826366i
$653$	298678.	0.700449	0.350224	−	0.936666i	$-0.386105\pi$
$653$	298678.	0.700449	0.350224	+	0.936666i	$0.386105\pi$
$654$	0	0
$655$	0	0
$656$	− 198174.i	− 0.460510i
$657$	0	0
$658$	− 165192.i	− 0.381539i
$659$	− 74450.5i	− 0.171434i	−0.996320	−	0.0857169i	$-0.972682\pi$
$659$	− 74450.5i	− 0.171434i	0.996320	−	0.0857169i	$-0.0273181\pi$
$660$	0	0
$661$	220326.	0.504271	0.252135	−	0.967692i	$-0.418867\pi$
$661$	220326.	0.504271	0.252135	+	0.967692i	$0.418867\pi$
$662$	225693.	0.514994
$663$	0	0
$664$	−160539.	−0.364119
$665$	0	0
$666$	0	0
$667$	724550.i	1.62861i
$668$	235817.	0.528472
$669$	0	0
$670$	0	0
$671$	546333.i	1.21342i
$672$	0	0
$673$	− 865522.i	− 1.91094i	−0.295081	−	0.955472i	$-0.595347\pi$
$673$	− 865522.i	− 1.91094i	0.295081	−	0.955472i	$-0.404653\pi$
$674$	117974.i	0.259698i
$675$	0	0
$676$	−55921.8	−0.122374
$677$	462235.	1.00852	0.504261	−	0.863551i	$-0.331765\pi$
$677$	462235.	1.00852	0.504261	+	0.863551i	$0.331765\pi$
$678$	0	0
$679$	−231484.	−0.502090
$680$	0	0
$681$	0	0
$682$	− 314507.i	− 0.676179i
$683$	434075.	0.930516	0.465258	−	0.885175i	$-0.345962\pi$
$683$	434075.	0.930516	0.465258	+	0.885175i	$0.345962\pi$
$684$	0	0
$685$	0	0
$686$	201074.i	0.427275i
$687$	0	0
$688$	181406.i	0.383244i
$689$	− 827160.i	− 1.74241i
$690$	0	0
$691$	−534956.	−1.12037	−0.560186	−	0.828367i	$-0.689270\pi$
$691$	−534956.	−1.12037	−0.560186	+	0.828367i	$0.689270\pi$
$692$	349102.	0.729021
$693$	0	0
$694$	−142442.	−0.295746
$695$	0	0
$696$	0	0
$697$	− 160059.i	− 0.329469i
$698$	−257398.	−0.528316
$699$	0	0
$700$	0	0
$701$	268256.i	0.545901i	0.962028	+	0.272950i	$0.0879995\pi$
$701$	268256.i	0.545901i	−0.962028	+	0.272950i	$0.912001\pi$
$702$	0	0
$703$	557055.i	1.12716i
$704$	− 89799.8i	− 0.181188i
$705$	0	0
$706$	−93739.1	−0.188066
$707$	346173.	0.692556
$708$	0	0
$709$	381958.	0.759842	0.379921	−	0.925019i	$-0.375951\pi$
$709$	381958.	0.759842	0.379921	+	0.925019i	$0.375951\pi$
$710$	0	0
$711$	0	0
$712$	− 417189.i	− 0.822949i
$713$	995837.	1.95889
$714$	0	0
$715$	0	0
$716$	− 195276.i	− 0.380911i
$717$	0	0
$718$	28252.4i	0.0548032i
$719$	533463.i	1.03192i	0.856613	+	0.515960i	$0.172565\pi$
$719$	533463.i	1.03192i	−0.856613	+	0.515960i	$0.827435\pi$
$720$	0	0
$721$	899384.	1.73011
$722$	−161200.	−0.309236
$723$	0	0
$724$	218635.	0.417102
$725$	0	0
$726$	0	0
$727$	− 715129.i	− 1.35306i	−0.736417	−	0.676528i	$-0.763484\pi$
$727$	− 715129.i	− 1.35306i	0.736417	−	0.676528i	$-0.236516\pi$
$728$	472040.	0.890669
$729$	0	0
$730$	0	0
$731$	146517.i	0.274190i
$732$	0	0
$733$	− 632035.i	− 1.17634i	−0.808737	−	0.588171i	$-0.799848\pi$
$733$	− 632035.i	− 1.17634i	0.808737	−	0.588171i	$-0.200152\pi$
$734$	341701.i	0.634240i
$735$	0	0
$736$	703520.	1.29874
$737$	216498.	0.398582
$738$	0	0
$739$	169415.	0.310215	0.155108	−	0.987898i	$-0.450428\pi$
$739$	169415.	0.310215	0.155108	+	0.987898i	$0.450428\pi$
$740$	0	0
$741$	0	0
$742$	− 584242.i	− 1.06117i
$743$	481172.	0.871610	0.435805	−	0.900041i	$-0.356464\pi$
$743$	481172.	0.871610	0.435805	+	0.900041i	$0.356464\pi$
$744$	0	0
$745$	0	0
$746$	371138.i	0.666895i
$747$	0	0
$748$	80018.1i	0.143016i
$749$	694232.i	1.23749i
$750$	0	0
$751$	392918.	0.696662	0.348331	−	0.937372i	$-0.386748\pi$
$751$	392918.	0.696662	0.348331	+	0.937372i	$0.386748\pi$
$752$	−119457.	−0.211240
$753$	0	0
$754$	−337470.	−0.593598
$755$	0	0
$756$	0	0
$757$	271769.i	0.474251i	0.971479	+	0.237126i	$0.0762053\pi$
$757$	271769.i	0.474251i	−0.971479	+	0.237126i	$0.923795\pi$
$758$	−26205.4	−0.0456092
$759$	0	0
$760$	0	0
$761$	935198.i	1.61486i	0.589965	+	0.807429i	$0.299141\pi$
$761$	935198.i	1.61486i	−0.589965	+	0.807429i	$0.700859\pi$
$762$	0	0
$763$	438829.i	0.753782i
$764$	321204.i	0.550293i
$765$	0	0
$766$	−320104.	−0.545548
$767$	−682263.	−1.15974
$768$	0	0
$769$	444131.	0.751032	0.375516	−	0.926816i	$-0.377466\pi$
$769$	444131.	0.751032	0.375516	+	0.926816i	$0.377466\pi$
$770$	0	0
$771$	0	0
$772$	− 42392.9i	− 0.0711310i
$773$	−169835.	−0.284229	−0.142114	−	0.989850i	$-0.545390\pi$
$773$	−169835.	−0.284229	−0.142114	+	0.989850i	$0.545390\pi$
$774$	0	0
$775$	0	0
$776$	− 238816.i	− 0.396589i
$777$	0	0
$778$	− 29415.5i	− 0.0485978i
$779$	− 1.15212e6i	− 1.89855i
$780$	0	0
$781$	305705.	0.501187
$782$	85197.8	0.139321
$783$	0	0
$784$	−44148.3	−0.0718260
$785$	0	0
$786$	0	0
$787$	104659.i	0.168977i	0.996424	+	0.0844886i	$0.0269257\pi$
$787$	104659.i	0.168977i	−0.996424	+	0.0844886i	$0.973074\pi$
$788$	751149.	1.20969
$789$	0	0
$790$	0	0
$791$	− 1.35157e6i	− 2.16015i
$792$	0	0
$793$	− 805715.i	− 1.28125i
$794$	88092.1i	0.139732i
$795$	0	0
$796$	701467.	1.10709
$797$	282179.	0.444230	0.222115	−	0.975020i	$-0.428704\pi$
$797$	282179.	0.444230	0.222115	+	0.975020i	$0.428704\pi$
$798$	0	0
$799$	−96481.8	−0.151130
$800$	0	0
$801$	0	0
$802$	44493.9i	0.0691754i
$803$	−389442.	−0.603966
$804$	0	0
$805$	0	0
$806$	463825.i	0.713977i
$807$	0	0
$808$	357138.i	0.547033i
$809$	− 1.27755e6i	− 1.95200i	−0.217762	−	0.976002i	$-0.569875\pi$
$809$	− 1.27755e6i	− 1.95200i	0.217762	−	0.976002i	$-0.430125\pi$
$810$	0	0
$811$	−240897.	−0.366260	−0.183130	−	0.983089i	$-0.558623\pi$
$811$	−240897.	−0.366260	−0.183130	+	0.983089i	$0.558623\pi$
$812$	708851.	1.07509
$813$	0	0
$814$	−255243.	−0.385217
$815$	0	0
$816$	0	0
$817$	1.05464e6i	1.58001i
$818$	527514.	0.788365
$819$	0	0
$820$	0	0
$821$	651543.i	0.966623i	0.875449	+	0.483311i	$0.160566\pi$
$821$	651543.i	0.966623i	−0.875449	+	0.483311i	$0.839434\pi$
$822$	0	0
$823$	− 122158.i	− 0.180353i	−0.995926	−	0.0901764i	$-0.971257\pi$
$823$	− 122158.i	− 0.180353i	0.995926	−	0.0901764i	$-0.0287431\pi$
$824$	927872.i	1.36658i
$825$	0	0
$826$	−481898.	−0.706309
$827$	336627.	0.492196	0.246098	−	0.969245i	$-0.420852\pi$
$827$	336627.	0.492196	0.246098	+	0.969245i	$0.420852\pi$
$828$	0	0
$829$	−173487.	−0.252440	−0.126220	−	0.992002i	$-0.540285\pi$
$829$	−173487.	−0.252440	−0.126220	+	0.992002i	$0.540285\pi$
$830$	0	0
$831$	0	0
$832$	132434.i	0.191317i
$833$	−35657.3	−0.0513876
$834$	0	0
$835$	0	0
$836$	575976.i	0.824123i
$837$	0	0
$838$	− 138635.i	− 0.197417i
$839$	− 739321.i	− 1.05029i	−0.851013	−	0.525145i	$-0.824011\pi$
$839$	− 739321.i	− 1.05029i	0.851013	−	0.525145i	$-0.175989\pi$
$840$	0	0
$841$	−476667.	−0.673943
$842$	−614439.	−0.866672
$843$	0	0
$844$	5983.07	0.00839922
$845$	0	0
$846$	0	0
$847$	− 198944.i	− 0.277309i
$848$	−422487.	−0.587519
$849$	0	0
$850$	0	0
$851$	− 808187.i	− 1.11597i
$852$	0	0
$853$	182365.i	0.250636i	0.992117	+	0.125318i	$0.0399951\pi$
$853$	182365.i	0.250636i	−0.992117	+	0.125318i	$0.960005\pi$
$854$	− 569094.i	− 0.780312i
$855$	0	0
$856$	−716222.	−0.977462
$857$	−471632.	−0.642157	−0.321078	−	0.947053i	$-0.604045\pi$
$857$	−471632.	−0.642157	−0.321078	+	0.947053i	$0.604045\pi$
$858$	0	0
$859$	−498411.	−0.675463	−0.337731	−	0.941243i	$-0.609660\pi$
$859$	−498411.	−0.675463	−0.337731	+	0.941243i	$0.609660\pi$
$860$	0	0
$861$	0	0
$862$	75000.1i	0.100936i
$863$	−797514.	−1.07082	−0.535410	−	0.844592i	$-0.679843\pi$
$863$	−797514.	−1.07082	−0.535410	+	0.844592i	$0.679843\pi$
$864$	0	0
$865$	0	0
$866$	− 28695.4i	− 0.0382628i
$867$	0	0
$868$	− 974260.i	− 1.29311i
$869$	331479.i	0.438952i
$870$	0	0
$871$	−319284.	−0.420863
$872$	−452729.	−0.595395
$873$	0	0
$874$	613260.	0.802827
$875$	0	0
$876$	0	0
$877$	− 143119.i	− 0.186079i	−0.995662	−	0.0930395i	$-0.970342\pi$
$877$	− 143119.i	− 0.186079i	0.995662	−	0.0930395i	$-0.0296583\pi$
$878$	−607532.	−0.788097
$879$	0	0
$880$	0	0
$881$	1.43426e6i	1.84789i	0.382524	+	0.923945i	$0.375055\pi$
$881$	1.43426e6i	1.84789i	−0.382524	+	0.923945i	$0.624945\pi$
$882$	0	0
$883$	248579.i	0.318818i	0.987213	+	0.159409i	$0.0509589\pi$
$883$	248579.i	0.318818i	−0.987213	+	0.159409i	$0.949041\pi$
$884$	− 118008.i	− 0.151011i
$885$	0	0
$886$	683158.	0.870270
$887$	−392111.	−0.498381	−0.249190	−	0.968455i	$-0.580164\pi$
$887$	−392111.	−0.498381	−0.249190	+	0.968455i	$0.580164\pi$
$888$	0	0
$889$	−790762.	−1.00056
$890$	0	0
$891$	0	0
$892$	− 159736.i	− 0.200758i
$893$	−694483.	−0.870881
$894$	0	0
$895$	0	0
$896$	− 826180.i	− 1.02910i
$897$	0	0
$898$	− 105343.i	− 0.130633i
$899$	1.62724e6i	2.01341i
$900$	0	0
$901$	−341230.	−0.420337
$902$	527902.	0.648844
$903$	0	0
$904$	1.39438e6	1.70625
$905$	0	0
$906$	0	0
$907$	1.10626e6i	1.34475i	0.740211	+	0.672375i	$0.234726\pi$
$907$	1.10626e6i	1.34475i	−0.740211	+	0.672375i	$0.765274\pi$
$908$	130376.	0.158134
$909$	0	0
$910$	0	0
$911$	− 1.03385e6i	− 1.24573i	−0.782331	−	0.622863i	$-0.785970\pi$
$911$	− 1.03385e6i	− 1.24573i	0.782331	−	0.622863i	$-0.214030\pi$
$912$	0	0
$913$	299754.i	0.359603i
$914$	559004.i	0.669149i
$915$	0	0
$916$	−418309.	−0.498547
$917$	76496.2	0.0909706
$918$	0	0
$919$	−942074.	−1.11546	−0.557730	−	0.830022i	$-0.688328\pi$
$919$	−942074.	−1.11546	−0.557730	+	0.830022i	$0.688328\pi$
$920$	0	0
$921$	0	0
$922$	− 706271.i	− 0.830826i
$923$	−450844.	−0.529204
$924$	0	0
$925$	0	0
$926$	− 808129.i	− 0.942451i
$927$	0	0
$928$	1.14959e6i	1.33489i
$929$	1.33174e6i	1.54307i	0.636184	+	0.771537i	$0.280512\pi$
$929$	1.33174e6i	1.54307i	−0.636184	+	0.771537i	$0.719488\pi$
$930$	0	0
$931$	−256664.	−0.296118
$932$	560991.	0.645838
$933$	0	0
$934$	−540606.	−0.619708
$935$	0	0
$936$	0	0
$937$	− 1.15212e6i	− 1.31226i	−0.754648	−	0.656130i	$-0.772192\pi$
$937$	− 1.15212e6i	− 1.31226i	0.754648	−	0.656130i	$-0.227808\pi$
$938$	−225517.	−0.256315
$939$	0	0
$940$	0	0
$941$	− 1.61179e6i	− 1.82025i	−0.414338	−	0.910123i	$-0.635987\pi$
$941$	− 1.61179e6i	− 1.82025i	0.414338	−	0.910123i	$-0.364013\pi$
$942$	0	0
$943$	1.67152e6i	1.87970i
$944$	348478.i	0.391050i
$945$	0	0
$946$	−483236.	−0.539979
$947$	−451620.	−0.503585	−0.251793	−	0.967781i	$-0.581020\pi$
$947$	−451620.	−0.503585	−0.251793	+	0.967781i	$0.581020\pi$
$948$	0	0
$949$	574338.	0.637727
$950$	0	0
$951$	0	0
$952$	− 194732.i	− 0.214864i
$953$	−604928.	−0.666067	−0.333033	−	0.942915i	$-0.608072\pi$
$953$	−604928.	−0.666067	−0.333033	+	0.942915i	$0.608072\pi$
$954$	0	0
$955$	0	0
$956$	446976.i	0.489067i
$957$	0	0
$958$	− 264544.i	− 0.288248i
$959$	− 953802.i	− 1.03710i
$960$	0	0
$961$	1.31300e6	1.42173
$962$	376425.	0.406751
$963$	0	0
$964$	436599.	0.469817
$965$	0	0
$966$	0	0
$967$	316089.i	0.338031i	0.985613	+	0.169016i	$0.0540588\pi$
$967$	316089.i	0.338031i	−0.985613	+	0.169016i	$0.945941\pi$
$968$	205245.	0.219040
$969$	0	0
$970$	0	0
$971$	354262.i	0.375739i	0.982194	+	0.187870i	$0.0601582\pi$
$971$	354262.i	0.375739i	−0.982194	+	0.187870i	$0.939842\pi$
$972$	0	0
$973$	374913.i	0.396009i
$974$	106339.i	0.112092i
$975$	0	0
$976$	−411534.	−0.432022
$977$	−1.28016e6	−1.34114	−0.670571	−	0.741846i	$-0.733951\pi$
$977$	−1.28016e6	−1.34114	−0.670571	+	0.741846i	$0.733951\pi$
$978$	0	0
$979$	−778965.	−0.812742
$980$	0	0
$981$	0	0
$982$	685978.i	0.711356i
$983$	−466867.	−0.483154	−0.241577	−	0.970382i	$-0.577665\pi$
$983$	−466867.	−0.483154	−0.241577	+	0.970382i	$0.577665\pi$
$984$	0	0
$985$	0	0
$986$	139217.i	0.143199i
$987$	0	0
$988$	− 849432.i	− 0.870191i
$989$	− 1.53009e6i	− 1.56432i
$990$	0	0
$991$	−1.05636e6	−1.07563	−0.537815	−	0.843063i	$-0.680750\pi$
$991$	−1.05636e6	−1.07563	−0.537815	+	0.843063i	$0.680750\pi$
$992$	1.58001e6	1.60560
$993$	0	0
$994$	−318441.	−0.322297
$995$	0	0
$996$	0	0
$997$	1.55546e6i	1.56483i	0.622756	+	0.782416i	$0.286013\pi$
$997$	1.55546e6i	1.56483i	−0.622756	+	0.782416i	$0.713987\pi$
$998$	−719020.	−0.721904
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	225.5.d.b.224.3		8
3.2	odd	2	inner	225.5.d.b.224.5		8
5.2	odd	4		45.5.c.a.26.2	✓	4
5.3	odd	4		225.5.c.b.26.3		4
5.4	even	2	inner	225.5.d.b.224.6		8
15.2	even	4		45.5.c.a.26.3	yes	4
15.8	even	4		225.5.c.b.26.2		4
15.14	odd	2	inner	225.5.d.b.224.4		8
20.7	even	4		720.5.l.c.161.3		4
60.47	odd	4		720.5.l.c.161.1		4

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
45.5.c.a.26.2	✓	4	5.2	odd	4
45.5.c.a.26.3	yes	4	15.2	even	4
225.5.c.b.26.2		4	15.8	even	4
225.5.c.b.26.3		4	5.3	odd	4
225.5.d.b.224.3		8	1.1	even	1	trivial
225.5.d.b.224.4		8	15.14	odd	2	inner
225.5.d.b.224.5		8	3.2	odd	2	inner
225.5.d.b.224.6		8	5.4	even	2	inner
720.5.l.c.161.1		4	60.47	odd	4
720.5.l.c.161.3		4	20.7	even	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.d (of order \(2\), degree \(1\), not minimal)

\(f(q)\)	\(=\)	\(q-2.00657 q^{2} -11.9737 q^{4} -54.4078i q^{7} +56.1312 q^{8} +O(q^{10})\)	\(q-2.00657 q^{2} -11.9737 q^{4} -54.4078i q^{7} +56.1312 q^{8} -104.807i q^{11} +154.566i q^{13} +109.173i q^{14} +78.9473 q^{16} +63.7634 q^{17} +458.974 q^{19} +210.302i q^{22} -665.889 q^{23} -310.148i q^{26} +651.461i q^{28} -1088.09i q^{29} -1495.50 q^{31} -1056.51 q^{32} -127.946 q^{34} +1213.70i q^{37} -920.964 q^{38} -2510.20i q^{41} +2297.81i q^{43} +1254.92i q^{44} +1336.16 q^{46} -1513.12 q^{47} -559.212 q^{49} -1850.72i q^{52} -5351.51 q^{53} -3053.98i q^{56} +2183.34i q^{58} +4414.06i q^{59} -5212.76 q^{61} +3000.83 q^{62} +856.812 q^{64} +2065.68i q^{67} -763.482 q^{68} +2916.84i q^{71} -3715.81i q^{73} -2435.37i q^{74} -5495.60 q^{76} -5702.31 q^{77} -3162.76 q^{79} +5036.90i q^{82} -2860.06 q^{83} -4610.73i q^{86} -5882.93i q^{88} -7432.39i q^{89} +8409.59 q^{91} +7973.14 q^{92} +3036.19 q^{94} -4254.61i q^{97} +1122.10 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 8 q + 56 q^{4}+O(q^{10}) \) 8 * q + 56 * q^4	\( 8 q + 56 q^{4} + 328 q^{16} + 3520 q^{19} - 3312 q^{31} + 10816 q^{34} - 11472 q^{46} - 17224 q^{49} - 40336 q^{61} - 23048 q^{64} + 21760 q^{76} - 35472 q^{79} - 44592 q^{91} + 65728 q^{94}+O(q^{100}) \) 8 * q + 56 * q^4 + 328 * q^16 + 3520 * q^19 - 3312 * q^31 + 10816 * q^34 - 11472 * q^46 - 17224 * q^49 - 40336 * q^61 - 23048 * q^64 + 21760 * q^76 - 35472 * q^79 - 44592 * q^91 + 65728 * q^94

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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