LMFDB - Embedded newform 225.10.b.b.199.2

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [225,10,Mod(199,225)]

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([0, 1]))

N = Newforms(chi, 10, names="a")

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

chi := DirichletCharacter("225.199");

S:= CuspForms(chi, 10);

N := Newforms(S);

Level:	$ N $	$=$	$ 225 = 3^{2} \cdot 5^{2} $
Weight:	$ k $	$=$	$ 10 $
Character orbit:	$[\chi]$	$=$	225.b (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:	$115.883063137$
Analytic rank:	$0$
Dimension:	$2$
Coefficient field:	$\Q(i)$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{2} + 1 $ x^2 + 1
Coefficient ring:	$\Z[a_1, \ldots, a_{7}]$
Coefficient ring index:	$ 2 $
Twist minimal:	no (minimal twist has level 15)
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label			199.2
Root			$1.00000i$ of defining polynomial
Character	$\chi$	$=$	225.199
Dual form			225.10.b.b.199.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+22.0000i q^{2} +28.0000 q^{4} +5988.00i q^{7} +11880.0i q^{8} +O(q^{10})$	$q+22.0000i q^{2} +28.0000 q^{4} +5988.00i q^{7} +11880.0i q^{8} +14648.0 q^{11} +37906.0i q^{13} -131736. q^{14} -247024. q^{16} -441098. i q^{17} -441820. q^{19} +322256. i q^{22} -2.26414e6i q^{23} -833932. q^{26} +167664. i q^{28} -1.04935e6 q^{29} -7.91057e6 q^{31} +648032. i q^{32} +9.70416e6 q^{34} +2.09926e7i q^{37} -9.72004e6i q^{38} -1.32856e7 q^{41} -2.31308e7i q^{43} +410144. q^{44} +4.98110e7 q^{46} -1.38737e7i q^{47} +4.49746e6 q^{49} +1.06137e6i q^{52} +5.76352e7i q^{53} -7.11374e7 q^{56} -2.30857e7i q^{58} -3.20421e7 q^{59} +1.10664e8 q^{61} -1.74032e8i q^{62} -1.40733e8 q^{64} +1.18568e8i q^{67} -1.23507e7i q^{68} -2.76680e8 q^{71} -2.64023e8i q^{73} -4.61836e8 q^{74} -1.23710e7 q^{76} +8.77122e7i q^{77} -4.48203e8 q^{79} -2.92282e8i q^{82} -8.51016e8i q^{83} +5.08877e8 q^{86} +1.74018e8i q^{88} +1.89895e8 q^{89} -2.26981e8 q^{91} -6.33958e7i q^{92} +3.05221e8 q^{94} +1.01415e9i q^{97} +9.89442e7i q^{98} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 2 q + 56 q^{4}+O(q^{10}) $ 2 * q + 56 * q^4	$ 2 q + 56 q^{4} + 29296 q^{11} - 263472 q^{14} - 494048 q^{16} - 883640 q^{19} - 1667864 q^{26} - 2098700 q^{29} - 15821136 q^{31} + 19408312 q^{34} - 26571124 q^{41} + 820288 q^{44} + 99621984 q^{46} + 8994926 q^{49} - 142274880 q^{56} - 64084240 q^{59} + 221328044 q^{61} - 281465984 q^{64} - 553359424 q^{71} - 923672552 q^{74} - 24741920 q^{76} - 896405520 q^{79} + 1017753616 q^{86} + 379789860 q^{89} - 453962256 q^{91} + 610442272 q^{94}+O(q^{100}) $ 2 * q + 56 * q^4 + 29296 * q^11 - 263472 * q^14 - 494048 * q^16 - 883640 * q^19 - 1667864 * q^26 - 2098700 * q^29 - 15821136 * q^31 + 19408312 * q^34 - 26571124 * q^41 + 820288 * q^44 + 99621984 * q^46 + 8994926 * q^49 - 142274880 * q^56 - 64084240 * q^59 + 221328044 * q^61 - 281465984 * q^64 - 553359424 * q^71 - 923672552 * q^74 - 24741920 * q^76 - 896405520 * q^79 + 1017753616 * q^86 + 379789860 * q^89 - 453962256 * q^91 + 610442272 * 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 $

$2$	22.0000i	0.972272i	0.873883	+	0.486136i	$0.161594\pi$
$2$	22.0000i	0.972272i	−0.873883	+	0.486136i	$0.838406\pi$
$3$	0	0
$3$	0	0
$4$	28.0000	0.0546875
$5$	0	0
$5$	0	0
$6$	0	0
$7$	5988.00i	0.942629i	0.881965	+	0.471314i	$0.156220\pi$
$7$	5988.00i	0.942629i	−0.881965	+	0.471314i	$0.843780\pi$
$8$	11880.0i	1.02544i
$9$	0	0
$10$	0	0
$11$	14648.0	0.301655	0.150828	−	0.988560i	$-0.451806\pi$
$11$	14648.0	0.301655	0.150828	+	0.988560i	$0.451806\pi$
$12$	0	0
$13$	37906.0i	0.368098i	0.982917	+	0.184049i	$0.0589204\pi$
$13$	37906.0i	0.368098i	−0.982917	+	0.184049i	$0.941080\pi$
$14$	−131736.	−0.916491
$15$	0	0
$16$	−247024.	−0.942322
$17$	− 441098.i	− 1.28090i	−0.768000	−	0.640450i	$-0.778748\pi$
$17$	− 441098.i	− 1.28090i	0.768000	−	0.640450i	$-0.221252\pi$
$18$	0	0
$19$	−441820.	−0.777775	−0.388888	−	0.921285i	$-0.627141\pi$
$19$	−441820.	−0.777775	−0.388888	+	0.921285i	$0.627141\pi$
$20$	0	0
$21$	0	0
$22$	322256.i	0.293291i
$23$	− 2.26414e6i	− 1.68705i	−0.537092	−	0.843524i	$-0.680477\pi$
$23$	− 2.26414e6i	− 1.68705i	0.537092	−	0.843524i	$-0.319523\pi$
$24$	0	0
$25$	0	0
$26$	−833932.	−0.357891
$27$	0	0
$28$	167664.i	0.0515500i
$29$	−1.04935e6	−0.275505	−0.137752	−	0.990467i	$-0.543988\pi$
$29$	−1.04935e6	−0.275505	−0.137752	+	0.990467i	$0.543988\pi$
$30$	0	0
$31$	−7.91057e6	−1.53844	−0.769219	−	0.638985i	$-0.779355\pi$
$31$	−7.91057e6	−1.53844	−0.769219	+	0.638985i	$0.779355\pi$
$32$	648032.i	0.109250i
$33$	0	0
$34$	9.70416e6	1.24538
$35$	0	0
$36$	0	0
$37$	2.09926e7i	1.84144i	0.390224	+	0.920720i	$0.372398\pi$
$37$	2.09926e7i	1.84144i	−0.390224	+	0.920720i	$0.627602\pi$
$38$	− 9.72004e6i	− 0.756209i
$39$	0	0
$40$	0	0
$41$	−1.32856e7	−0.734265	−0.367132	−	0.930169i	$-0.619660\pi$
$41$	−1.32856e7	−0.734265	−0.367132	+	0.930169i	$0.619660\pi$
$42$	0	0
$43$	− 2.31308e7i	− 1.03177i	−0.856659	−	0.515884i	$-0.827464\pi$
$43$	− 2.31308e7i	− 1.03177i	0.856659	−	0.515884i	$-0.172536\pi$
$44$	410144.	0.0164968
$45$	0	0
$46$	4.98110e7	1.64027
$47$	− 1.38737e7i	− 0.414717i	−0.978265	−	0.207358i	$-0.933513\pi$
$47$	− 1.38737e7i	− 0.414717i	0.978265	−	0.207358i	$-0.0664866\pi$
$48$	0	0
$49$	4.49746e6	0.111451
$50$	0	0
$51$	0	0
$52$	1.06137e6i	0.0201303i
$53$	5.76352e7i	1.00334i	0.865060	+	0.501668i	$0.167280\pi$
$53$	5.76352e7i	1.00334i	−0.865060	+	0.501668i	$0.832720\pi$
$54$	0	0
$55$	0	0
$56$	−7.11374e7	−0.966612
$57$	0	0
$58$	− 2.30857e7i	− 0.267866i
$59$	−3.20421e7	−0.344260	−0.172130	−	0.985074i	$-0.555065\pi$
$59$	−3.20421e7	−0.344260	−0.172130	+	0.985074i	$0.555065\pi$
$60$	0	0
$61$	1.10664e8	1.02335	0.511673	−	0.859180i	$-0.329026\pi$
$61$	1.10664e8	1.02335	0.511673	+	0.859180i	$0.329026\pi$
$62$	− 1.74032e8i	− 1.49578i
$63$	0	0
$64$	−1.40733e8	−1.04854
$65$	0	0
$66$	0	0
$67$	1.18568e8i	0.718839i	0.933176	+	0.359420i	$0.117025\pi$
$67$	1.18568e8i	0.718839i	−0.933176	+	0.359420i	$0.882975\pi$
$68$	− 1.23507e7i	− 0.0700492i
$69$	0	0
$70$	0	0
$71$	−2.76680e8	−1.29216	−0.646078	−	0.763272i	$-0.723592\pi$
$71$	−2.76680e8	−1.29216	−0.646078	+	0.763272i	$0.723592\pi$
$72$	0	0
$73$	− 2.64023e8i	− 1.08815i	−0.839036	−	0.544076i	$-0.816880\pi$
$73$	− 2.64023e8i	− 1.08815i	0.839036	−	0.544076i	$-0.183120\pi$
$74$	−4.61836e8	−1.79038
$75$	0	0
$76$	−1.23710e7	−0.0425346
$77$	8.77122e7i	0.284349i
$78$	0	0
$79$	−4.48203e8	−1.29465	−0.647325	−	0.762214i	$-0.724112\pi$
$79$	−4.48203e8	−1.29465	−0.647325	+	0.762214i	$0.724112\pi$
$80$	0	0
$81$	0	0
$82$	− 2.92282e8i	− 0.713905i
$83$	− 8.51016e8i	− 1.96828i	−0.177402	−	0.984138i	$-0.556769\pi$
$83$	− 8.51016e8i	− 1.96828i	0.177402	−	0.984138i	$-0.443231\pi$
$84$	0	0
$85$	0	0
$86$	5.08877e8	1.00316
$87$	0	0
$88$	1.74018e8i	0.309331i
$89$	1.89895e8	0.320818	0.160409	−	0.987051i	$-0.448719\pi$
$89$	1.89895e8	0.320818	0.160409	+	0.987051i	$0.448719\pi$
$90$	0	0
$91$	−2.26981e8	−0.346979
$92$	− 6.33958e7i	− 0.0922604i
$93$	0	0
$94$	3.05221e8	0.403217
$95$	0	0
$96$	0	0
$97$	1.01415e9i	1.16313i	0.813499	+	0.581566i	$0.197560\pi$
$97$	1.01415e9i	1.16313i	−0.813499	+	0.581566i	$0.802440\pi$
$98$	9.89442e7i	0.108361i
$99$	0	0
$100$	0	0
$101$	1.31537e9	1.25777	0.628885	−	0.777498i	$-0.283512\pi$
$101$	1.31537e9	1.25777	0.628885	+	0.777498i	$0.283512\pi$
$102$	0	0
$103$	− 1.82797e9i	− 1.60030i	−0.599798	−	0.800151i	$-0.704752\pi$
$103$	− 1.82797e9i	− 1.60030i	0.599798	−	0.800151i	$-0.295248\pi$
$104$	−4.50323e8	−0.377463
$105$	0	0
$106$	−1.26797e9	−0.975515
$107$	1.85367e8i	0.136712i	0.997661	+	0.0683559i	$0.0217754\pi$
$107$	1.85367e8i	0.136712i	−0.997661	+	0.0683559i	$0.978225\pi$
$108$	0	0
$109$	−1.97869e9	−1.34264	−0.671318	−	0.741169i	$-0.734272\pi$
$109$	−1.97869e9	−1.34264	−0.671318	+	0.741169i	$0.734272\pi$
$110$	0	0
$111$	0	0
$112$	− 1.47918e9i	− 0.888259i
$113$	− 1.57875e9i	− 0.910876i	−0.890267	−	0.455438i	$-0.849483\pi$
$113$	− 1.57875e9i	− 0.910876i	0.890267	−	0.455438i	$-0.150517\pi$
$114$	0	0
$115$	0	0
$116$	−2.93818e7	−0.0150667
$117$	0	0
$118$	− 7.04927e8i	− 0.334715i
$119$	2.64129e9	1.20741
$120$	0	0
$121$	−2.14338e9	−0.909004
$122$	2.43461e9i	0.994970i
$123$	0	0
$124$	−2.21496e8	−0.0841333
$125$	0	0
$126$	0	0
$127$	− 2.40001e9i	− 0.818645i	−0.912390	−	0.409322i	$-0.865765\pi$
$127$	− 2.40001e9i	− 0.818645i	0.912390	−	0.409322i	$-0.134235\pi$
$128$	− 2.76433e9i	− 0.910218i
$129$	0	0
$130$	0	0
$131$	1.96840e9	0.583971	0.291986	−	0.956423i	$-0.405684\pi$
$131$	1.96840e9	0.583971	0.291986	+	0.956423i	$0.405684\pi$
$132$	0	0
$133$	− 2.64562e9i	− 0.733153i
$134$	−2.60850e9	−0.698907
$135$	0	0
$136$	5.24024e9	1.31349
$137$	− 2.02909e9i	− 0.492107i	−0.969256	−	0.246054i	$-0.920866\pi$
$137$	− 2.02909e9i	− 0.492107i	0.969256	−	0.246054i	$-0.0791339\pi$
$138$	0	0
$139$	1.13673e9	0.258280	0.129140	−	0.991626i	$-0.458778\pi$
$139$	1.13673e9	0.258280	0.129140	+	0.991626i	$0.458778\pi$
$140$	0	0
$141$	0	0
$142$	− 6.08695e9i	− 1.25633i
$143$	5.55247e8i	0.111039i
$144$	0	0
$145$	0	0
$146$	5.80851e9	1.05798
$147$	0	0
$148$	5.87792e8i	0.100704i
$149$	−4.73854e9	−0.787601	−0.393800	−	0.919196i	$-0.628840\pi$
$149$	−4.73854e9	−0.787601	−0.393800	+	0.919196i	$0.628840\pi$
$150$	0	0
$151$	2.26216e9	0.354101	0.177051	−	0.984202i	$-0.443344\pi$
$151$	2.26216e9	0.354101	0.177051	+	0.984202i	$0.443344\pi$
$152$	− 5.24882e9i	− 0.797564i
$153$	0	0
$154$	−1.92967e9	−0.276465
$155$	0	0
$156$	0	0
$157$	1.17889e10i	1.54854i	0.632853	+	0.774272i	$0.281884\pi$
$157$	1.17889e10i	1.54854i	−0.632853	+	0.774272i	$0.718116\pi$
$158$	− 9.86046e9i	− 1.25875i
$159$	0	0
$160$	0	0
$161$	1.35576e10	1.59026
$162$	0	0
$163$	1.14608e10i	1.27166i	0.771830	+	0.635829i	$0.219342\pi$
$163$	1.14608e10i	1.27166i	−0.771830	+	0.635829i	$0.780658\pi$
$164$	−3.71996e8	−0.0401551
$165$	0	0
$166$	1.87223e10	1.91370
$167$	− 1.33707e10i	− 1.33024i	−0.746738	−	0.665118i	$-0.768381\pi$
$167$	− 1.33707e10i	− 1.33024i	0.746738	−	0.665118i	$-0.231619\pi$
$168$	0	0
$169$	9.16763e9	0.864504
$170$	0	0
$171$	0	0
$172$	− 6.47661e8i	− 0.0564248i
$173$	− 1.06264e10i	− 0.901939i	−0.892539	−	0.450970i	$-0.851078\pi$
$173$	− 1.06264e10i	− 0.901939i	0.892539	−	0.450970i	$-0.148922\pi$
$174$	0	0
$175$	0	0
$176$	−3.61841e9	−0.284257
$177$	0	0
$178$	4.17769e9i	0.311922i
$179$	2.61254e10	1.90206	0.951031	−	0.309094i	$-0.100026\pi$
$179$	2.61254e10	1.90206	0.951031	+	0.309094i	$0.100026\pi$
$180$	0	0
$181$	2.34689e9	0.162532	0.0812660	−	0.996692i	$-0.474104\pi$
$181$	2.34689e9	0.162532	0.0812660	+	0.996692i	$0.474104\pi$
$182$	− 4.99358e9i	− 0.337358i
$183$	0	0
$184$	2.68979e10	1.72997
$185$	0	0
$186$	0	0
$187$	− 6.46120e9i	− 0.386390i
$188$	− 3.88463e8i	− 0.0226798i
$189$	0	0
$190$	0	0
$191$	−2.24064e10	−1.21821	−0.609105	−	0.793089i	$-0.708471\pi$
$191$	−2.24064e10	−1.21821	−0.609105	+	0.793089i	$0.708471\pi$
$192$	0	0
$193$	− 3.65959e10i	− 1.89856i	−0.314427	−	0.949282i	$-0.601812\pi$
$193$	− 3.65959e10i	− 1.89856i	0.314427	−	0.949282i	$-0.398188\pi$
$194$	−2.23113e10	−1.13088
$195$	0	0
$196$	1.25929e8	0.00609499
$197$	− 5.41546e9i	− 0.256175i	−0.991763	−	0.128088i	$-0.959116\pi$
$197$	− 5.41546e9i	− 0.256175i	0.991763	−	0.128088i	$-0.0408839\pi$
$198$	0	0
$199$	−2.62714e8	−0.0118753	−0.00593764	−	0.999982i	$-0.501890\pi$
$199$	−2.62714e8	−0.0118753	−0.00593764	+	0.999982i	$0.501890\pi$
$200$	0	0
$201$	0	0
$202$	2.89381e10i	1.22289i
$203$	− 6.28351e9i	− 0.259699i
$204$	0	0
$205$	0	0
$206$	4.02154e10	1.55593
$207$	0	0
$208$	− 9.36369e9i	− 0.346866i
$209$	−6.47178e9	−0.234620
$210$	0	0
$211$	1.34493e10	0.467121	0.233560	−	0.972342i	$-0.424962\pi$
$211$	1.34493e10	0.467121	0.233560	+	0.972342i	$0.424962\pi$
$212$	1.61378e9i	0.0548699i
$213$	0	0
$214$	−4.07808e9	−0.132921
$215$	0	0
$216$	0	0
$217$	− 4.73685e10i	− 1.45018i
$218$	− 4.35312e10i	− 1.30541i
$219$	0	0
$220$	0	0
$221$	1.67203e10	0.471496
$222$	0	0
$223$	− 2.66463e10i	− 0.721547i	−0.932654	−	0.360773i	$-0.882513\pi$
$223$	− 2.66463e10i	− 0.721547i	0.932654	−	0.360773i	$-0.117487\pi$
$224$	−3.88042e9	−0.102982
$225$	0	0
$226$	3.47324e10	0.885619
$227$	3.36318e10i	0.840686i	0.907365	+	0.420343i	$0.138090\pi$
$227$	3.36318e10i	0.840686i	−0.907365	+	0.420343i	$0.861910\pi$
$228$	0	0
$229$	−5.00453e10	−1.20255	−0.601276	−	0.799042i	$-0.705341\pi$
$229$	−5.00453e10	−1.20255	−0.601276	+	0.799042i	$0.705341\pi$
$230$	0	0
$231$	0	0
$232$	− 1.24663e10i	− 0.282515i
$233$	3.29626e10i	0.732688i	0.930479	+	0.366344i	$0.119391\pi$
$233$	3.29626e10i	0.732688i	−0.930479	+	0.366344i	$0.880609\pi$
$234$	0	0
$235$	0	0
$236$	−8.97179e8	−0.0188267
$237$	0	0
$238$	5.81085e10i	1.17393i
$239$	−7.95422e9	−0.157691	−0.0788455	−	0.996887i	$-0.525123\pi$
$239$	−7.95422e9	−0.157691	−0.0788455	+	0.996887i	$0.525123\pi$
$240$	0	0
$241$	−7.52477e10	−1.43687	−0.718434	−	0.695595i	$-0.755141\pi$
$241$	−7.52477e10	−1.43687	−0.718434	+	0.695595i	$0.755141\pi$
$242$	− 4.71544e10i	− 0.883799i
$243$	0	0
$244$	3.09859e9	0.0559642
$245$	0	0
$246$	0	0
$247$	− 1.67476e10i	− 0.286297i
$248$	− 9.39775e10i	− 1.57758i
$249$	0	0
$250$	0	0
$251$	9.84631e10	1.56582	0.782910	−	0.622135i	$-0.213734\pi$
$251$	9.84631e10	1.56582	0.782910	+	0.622135i	$0.213734\pi$
$252$	0	0
$253$	− 3.31651e10i	− 0.508907i
$254$	5.28001e10	0.795945
$255$	0	0
$256$	−1.12400e10	−0.163563
$257$	8.52399e9i	0.121883i	0.998141	+	0.0609416i	$0.0194103\pi$
$257$	8.52399e9i	0.121883i	−0.998141	+	0.0609416i	$0.980590\pi$
$258$	0	0
$259$	−1.25703e11	−1.73579
$260$	0	0
$261$	0	0
$262$	4.33047e10i	0.567779i
$263$	− 5.90654e9i	− 0.0761259i	−0.999275	−	0.0380630i	$-0.987881\pi$
$263$	− 5.90654e9i	− 0.0761259i	0.999275	−	0.0380630i	$-0.0121187\pi$
$264$	0	0
$265$	0	0
$266$	5.82036e10	0.712824
$267$	0	0
$268$	3.31991e9i	0.0393115i
$269$	−7.15961e10	−0.833689	−0.416845	−	0.908978i	$-0.636864\pi$
$269$	−7.15961e10	−0.833689	−0.416845	+	0.908978i	$0.636864\pi$
$270$	0	0
$271$	−1.24755e11	−1.40506	−0.702530	−	0.711654i	$-0.747946\pi$
$271$	−1.24755e11	−1.40506	−0.702530	+	0.711654i	$0.747946\pi$
$272$	1.08962e11i	1.20702i
$273$	0	0
$274$	4.46401e10	0.478462
$275$	0	0
$276$	0	0
$277$	− 1.12824e11i	− 1.15145i	−0.817645	−	0.575723i	$-0.804721\pi$
$277$	− 1.12824e11i	− 1.15145i	0.817645	−	0.575723i	$-0.195279\pi$
$278$	2.50080e10i	0.251118i
$279$	0	0
$280$	0	0
$281$	−8.70208e10	−0.832616	−0.416308	−	0.909224i	$-0.636676\pi$
$281$	−8.70208e10	−0.832616	−0.416308	+	0.909224i	$0.636676\pi$
$282$	0	0
$283$	3.27696e9i	0.0303692i	0.999885	+	0.0151846i	$0.00483359\pi$
$283$	3.27696e9i	0.0303692i	−0.999885	+	0.0151846i	$0.995166\pi$
$284$	−7.74703e9	−0.0706647
$285$	0	0
$286$	−1.22154e10	−0.107960
$287$	− 7.95539e10i	− 0.692139i
$288$	0	0
$289$	−7.59796e10	−0.640703
$290$	0	0
$291$	0	0
$292$	− 7.39265e9i	− 0.0595083i
$293$	1.49860e11i	1.18791i	0.804500	+	0.593953i	$0.202433\pi$
$293$	1.49860e11i	1.18791i	−0.804500	+	0.593953i	$0.797567\pi$
$294$	0	0
$295$	0	0
$296$	−2.49392e11	−1.88829
$297$	0	0
$298$	− 1.04248e11i	− 0.765762i
$299$	8.58243e10	0.620998
$300$	0	0
$301$	1.38507e11	0.972574
$302$	4.97676e10i	0.344283i
$303$	0	0
$304$	1.09140e11	0.732915
$305$	0	0
$306$	0	0
$307$	1.84570e11i	1.18587i	0.805249	+	0.592937i	$0.202032\pi$
$307$	1.84570e11i	1.18587i	−0.805249	+	0.592937i	$0.797968\pi$
$308$	2.45594e9i	0.0155503i
$309$	0	0
$310$	0	0
$311$	−9.04650e10	−0.548351	−0.274176	−	0.961680i	$-0.588405\pi$
$311$	−9.04650e10	−0.548351	−0.274176	+	0.961680i	$0.588405\pi$
$312$	0	0
$313$	1.07930e10i	0.0635615i	0.999495	+	0.0317808i	$0.0101178\pi$
$313$	1.07930e10i	0.0635615i	−0.999495	+	0.0317808i	$0.989882\pi$
$314$	−2.59355e11	−1.50560
$315$	0	0
$316$	−1.25497e10	−0.0708012
$317$	4.18319e10i	0.232670i	0.993210	+	0.116335i	$0.0371147\pi$
$317$	4.18319e10i	0.232670i	−0.993210	+	0.116335i	$0.962885\pi$
$318$	0	0
$319$	−1.53709e10	−0.0831076
$320$	0	0
$321$	0	0
$322$	2.98268e11i	1.54616i
$323$	1.94886e11i	0.996252i
$324$	0	0
$325$	0	0
$326$	−2.52137e11	−1.23640
$327$	0	0
$328$	− 1.57832e11i	− 0.752946i
$329$	8.30756e10	0.390924
$330$	0	0
$331$	−1.23310e10	−0.0564640	−0.0282320	−	0.999601i	$-0.508988\pi$
$331$	−1.23310e10	−0.0564640	−0.0282320	+	0.999601i	$0.508988\pi$
$332$	− 2.38284e10i	− 0.107640i
$333$	0	0
$334$	2.94154e11	1.29335
$335$	0	0
$336$	0	0
$337$	2.17976e10i	0.0920606i	0.998940	+	0.0460303i	$0.0146571\pi$
$337$	2.17976e10i	0.0920606i	−0.998940	+	0.0460303i	$0.985343\pi$
$338$	2.01688e11i	0.840533i
$339$	0	0
$340$	0	0
$341$	−1.15874e11	−0.464078
$342$	0	0
$343$	2.68568e11i	1.04769i
$344$	2.74793e11	1.05802
$345$	0	0
$346$	2.33780e11	0.876930
$347$	− 8.43613e10i	− 0.312364i	−0.987728	−	0.156182i	$-0.950081\pi$
$347$	− 8.43613e10i	− 0.312364i	0.987728	−	0.156182i	$-0.0499186\pi$
$348$	0	0
$349$	1.23295e9	0.00444867	0.00222434	−	0.999998i	$-0.499292\pi$
$349$	1.23295e9	0.00444867	0.00222434	+	0.999998i	$0.499292\pi$
$350$	0	0
$351$	0	0
$352$	9.49237e9i	0.0329559i
$353$	− 1.73388e11i	− 0.594336i	−0.954825	−	0.297168i	$-0.903958\pi$
$353$	− 1.73388e11i	− 0.594336i	0.954825	−	0.297168i	$-0.0960422\pi$
$354$	0	0
$355$	0	0
$356$	5.31706e9	0.0175447
$357$	0	0
$358$	5.74760e11i	1.84932i
$359$	6.38159e10	0.202770	0.101385	−	0.994847i	$-0.467673\pi$
$359$	6.38159e10	0.202770	0.101385	+	0.994847i	$0.467673\pi$
$360$	0	0
$361$	−1.27483e11	−0.395066
$362$	5.16316e10i	0.158025i
$363$	0	0
$364$	−6.35547e9	−0.0189754
$365$	0	0
$366$	0	0
$367$	− 1.12242e11i	− 0.322966i	−0.986876	−	0.161483i	$-0.948372\pi$
$367$	− 1.12242e11i	− 0.322966i	0.986876	−	0.161483i	$-0.0516277\pi$
$368$	5.59296e11i	1.58974i
$369$	0	0
$370$	0	0
$371$	−3.45119e11	−0.945773
$372$	0	0
$373$	7.21004e11i	1.92863i	0.264766	+	0.964313i	$0.414705\pi$
$373$	7.21004e11i	1.92863i	−0.264766	+	0.964313i	$0.585295\pi$
$374$	1.42146e11	0.375676
$375$	0	0
$376$	1.64819e11	0.425268
$377$	− 3.97767e10i	− 0.101413i
$378$	0	0
$379$	2.04331e11	0.508695	0.254348	−	0.967113i	$-0.418139\pi$
$379$	2.04331e11	0.508695	0.254348	+	0.967113i	$0.418139\pi$
$380$	0	0
$381$	0	0
$382$	− 4.92941e11i	− 1.18443i
$383$	5.47180e11i	1.29938i	0.760200	+	0.649689i	$0.225101\pi$
$383$	5.47180e11i	1.29938i	−0.760200	+	0.649689i	$0.774899\pi$
$384$	0	0
$385$	0	0
$386$	8.05111e11	1.84592
$387$	0	0
$388$	2.83962e10i	0.0636088i
$389$	−3.46262e11	−0.766711	−0.383356	−	0.923601i	$-0.625232\pi$
$389$	−3.46262e11	−0.766711	−0.383356	+	0.923601i	$0.625232\pi$
$390$	0	0
$391$	−9.98706e11	−2.16094
$392$	5.34299e10i	0.114287i
$393$	0	0
$394$	1.19140e11	0.249072
$395$	0	0
$396$	0	0
$397$	− 2.56758e11i	− 0.518760i	−0.965775	−	0.259380i	$-0.916482\pi$
$397$	− 2.56758e11i	− 0.518760i	0.965775	−	0.259380i	$-0.0835181\pi$
$398$	− 5.77971e9i	− 0.0115460i
$399$	0	0
$400$	0	0
$401$	−2.01679e10	−0.0389504	−0.0194752	−	0.999810i	$-0.506200\pi$
$401$	−2.01679e10	−0.0389504	−0.0194752	+	0.999810i	$0.506200\pi$
$402$	0	0
$403$	− 2.99858e11i	− 0.566295i
$404$	3.68303e10	0.0687843
$405$	0	0
$406$	1.38237e11	0.252498
$407$	3.07499e11i	0.555481i
$408$	0	0
$409$	4.33405e10	0.0765842	0.0382921	−	0.999267i	$-0.487808\pi$
$409$	4.33405e10	0.0765842	0.0382921	+	0.999267i	$0.487808\pi$
$410$	0	0
$411$	0	0
$412$	− 5.11832e10i	− 0.0875166i
$413$	− 1.91868e11i	− 0.324510i
$414$	0	0
$415$	0	0
$416$	−2.45643e10	−0.0402147
$417$	0	0
$418$	− 1.42379e11i	− 0.228115i
$419$	5.10680e11	0.809443	0.404721	−	0.914440i	$-0.367369\pi$
$419$	5.10680e11	0.809443	0.404721	+	0.914440i	$0.367369\pi$
$420$	0	0
$421$	3.21228e11	0.498361	0.249181	−	0.968457i	$-0.419839\pi$
$421$	3.21228e11	0.498361	0.249181	+	0.968457i	$0.419839\pi$
$422$	2.95885e11i	0.454168i
$423$	0	0
$424$	−6.84706e11	−1.02886
$425$	0	0
$426$	0	0
$427$	6.62656e11i	0.964635i
$428$	5.19029e9i	0.00747643i
$429$	0	0
$430$	0	0
$431$	−4.47617e11	−0.624826	−0.312413	−	0.949946i	$-0.601137\pi$
$431$	−4.47617e11	−0.624826	−0.312413	+	0.949946i	$0.601137\pi$
$432$	0	0
$433$	− 1.41186e11i	− 0.193017i	−0.995332	−	0.0965085i	$-0.969233\pi$
$433$	− 1.41186e11i	− 0.193017i	0.995332	−	0.0965085i	$-0.0307675\pi$
$434$	1.04211e12	1.40997
$435$	0	0
$436$	−5.54033e10	−0.0734254
$437$	1.00034e12i	1.31214i
$438$	0	0
$439$	8.50498e11	1.09291	0.546453	−	0.837490i	$-0.315978\pi$
$439$	8.50498e11	1.09291	0.546453	+	0.837490i	$0.315978\pi$
$440$	0	0
$441$	0	0
$442$	3.67846e11i	0.458422i
$443$	− 3.03188e11i	− 0.374020i	−0.982358	−	0.187010i	$-0.940120\pi$
$443$	− 3.03188e11i	− 0.374020i	0.982358	−	0.187010i	$-0.0598797\pi$
$444$	0	0
$445$	0	0
$446$	5.86218e11	0.701540
$447$	0	0
$448$	− 8.42709e11i	− 0.988386i
$449$	−1.40328e12	−1.62943	−0.814714	−	0.579863i	$-0.803106\pi$
$449$	−1.40328e12	−1.62943	−0.814714	+	0.579863i	$0.803106\pi$
$450$	0	0
$451$	−1.94607e11	−0.221495
$452$	− 4.42049e10i	− 0.0498136i
$453$	0	0
$454$	−7.39899e11	−0.817375
$455$	0	0
$456$	0	0
$457$	− 3.35354e11i	− 0.359650i	−0.983699	−	0.179825i	$-0.942447\pi$
$457$	− 3.35354e11i	− 0.359650i	0.983699	−	0.179825i	$-0.0575531\pi$
$458$	− 1.10100e12i	− 1.16921i
$459$	0	0
$460$	0	0
$461$	−1.04275e12	−1.07529	−0.537645	−	0.843171i	$-0.680686\pi$
$461$	−1.04275e12	−1.07529	−0.537645	+	0.843171i	$0.680686\pi$
$462$	0	0
$463$	− 9.74084e11i	− 0.985103i	−0.870283	−	0.492552i	$-0.836064\pi$
$463$	− 9.74084e11i	− 0.985103i	0.870283	−	0.492552i	$-0.163936\pi$
$464$	2.59215e11	0.259614
$465$	0	0
$466$	−7.25176e11	−0.712372
$467$	− 1.97851e11i	− 0.192492i	−0.995358	−	0.0962458i	$-0.969317\pi$
$467$	− 1.97851e11i	− 0.192492i	0.995358	−	0.0962458i	$-0.0306835\pi$
$468$	0	0
$469$	−7.09987e11	−0.677599
$470$	0	0
$471$	0	0
$472$	− 3.80660e11i	− 0.353019i
$473$	− 3.38819e11i	− 0.311238i
$474$	0	0
$475$	0	0
$476$	7.39563e10	0.0660304
$477$	0	0
$478$	− 1.74993e11i	− 0.153319i
$479$	−9.08731e11	−0.788725	−0.394362	−	0.918955i	$-0.629035\pi$
$479$	−9.08731e11	−0.788725	−0.394362	+	0.918955i	$0.629035\pi$
$480$	0	0
$481$	−7.95744e11	−0.677830
$482$	− 1.65545e12i	− 1.39703i
$483$	0	0
$484$	−6.00147e10	−0.0497112
$485$	0	0
$486$	0	0
$487$	− 1.16963e12i	− 0.942254i	−0.882065	−	0.471127i	$-0.843847\pi$
$487$	− 1.16963e12i	− 0.942254i	0.882065	−	0.471127i	$-0.156153\pi$
$488$	1.31469e12i	1.04938i
$489$	0	0
$490$	0	0
$491$	2.49149e12	1.93460	0.967301	−	0.253630i	$-0.0816246\pi$
$491$	2.49149e12	1.93460	0.967301	+	0.253630i	$0.0816246\pi$
$492$	0	0
$493$	4.62866e11i	0.352894i
$494$	3.68448e11	0.278359
$495$	0	0
$496$	1.95410e12	1.44970
$497$	− 1.65676e12i	− 1.21802i
$498$	0	0
$499$	5.57571e11	0.402576	0.201288	−	0.979532i	$-0.435487\pi$
$499$	5.57571e11	0.402576	0.201288	+	0.979532i	$0.435487\pi$
$500$	0	0
$501$	0	0
$502$	2.16619e12i	1.52240i
$503$	− 1.80137e12i	− 1.25472i	−0.778730	−	0.627359i	$-0.784136\pi$
$503$	− 1.80137e12i	− 1.25472i	0.778730	−	0.627359i	$-0.215864\pi$
$504$	0	0
$505$	0	0
$506$	7.29631e11	0.494796
$507$	0	0
$508$	− 6.72001e10i	− 0.0447696i
$509$	−2.40110e12	−1.58555	−0.792774	−	0.609515i	$-0.791364\pi$
$509$	−2.40110e12	−1.58555	−0.792774	+	0.609515i	$0.791364\pi$
$510$	0	0
$511$	1.58097e12	1.02572
$512$	− 1.66262e12i	− 1.06925i
$513$	0	0
$514$	−1.87528e11	−0.118504
$515$	0	0
$516$	0	0
$517$	− 2.03222e11i	− 0.125102i
$518$	− 2.76548e12i	− 1.68766i
$519$	0	0
$520$	0	0
$521$	−1.03008e12	−0.612491	−0.306246	−	0.951953i	$-0.599073\pi$
$521$	−1.03008e12	−0.612491	−0.306246	+	0.951953i	$0.599073\pi$
$522$	0	0
$523$	1.06637e12i	0.623232i	0.950208	+	0.311616i	$0.100870\pi$
$523$	1.06637e12i	0.623232i	−0.950208	+	0.311616i	$0.899130\pi$
$524$	5.51151e10	0.0319359
$525$	0	0
$526$	1.29944e11	0.0740151
$527$	3.48934e12i	1.97058i
$528$	0	0
$529$	−3.32516e12	−1.84613
$530$	0	0
$531$	0	0
$532$	− 7.40773e10i	− 0.0400943i
$533$	− 5.03603e11i	− 0.270281i
$534$	0	0
$535$	0	0
$536$	−1.40859e12	−0.737129
$537$	0	0
$538$	− 1.57511e12i	− 0.810572i
$539$	6.58788e10	0.0336199
$540$	0	0
$541$	2.51749e12	1.26351	0.631756	−	0.775167i	$-0.282334\pi$
$541$	2.51749e12	1.26351	0.631756	+	0.775167i	$0.282334\pi$
$542$	− 2.74460e12i	− 1.36610i
$543$	0	0
$544$	2.85846e11	0.139938
$545$	0	0
$546$	0	0
$547$	1.10353e12i	0.527037i	0.964654	+	0.263518i	$0.0848830\pi$
$547$	1.10353e12i	0.527037i	−0.964654	+	0.263518i	$0.915117\pi$
$548$	− 5.68146e10i	− 0.0269121i
$549$	0	0
$550$	0	0
$551$	4.63624e11	0.214281
$552$	0	0
$553$	− 2.68384e12i	− 1.22037i
$554$	2.48213e12	1.11952
$555$	0	0
$556$	3.18284e10	0.0141247
$557$	− 1.09072e12i	− 0.480137i	−0.970756	−	0.240068i	$-0.922830\pi$
$557$	− 1.09072e12i	− 0.480137i	0.970756	−	0.240068i	$-0.0771699\pi$
$558$	0	0
$559$	8.76795e11	0.379791
$560$	0	0
$561$	0	0
$562$	− 1.91446e12i	− 0.809529i
$563$	− 1.03081e12i	− 0.432403i	−0.976349	−	0.216202i	$-0.930633\pi$
$563$	− 1.03081e12i	− 0.432403i	0.976349	−	0.216202i	$-0.0693668\pi$
$564$	0	0
$565$	0	0
$566$	−7.20932e10	−0.0295271
$567$	0	0
$568$	− 3.28695e12i	− 1.32503i
$569$	−4.06618e12	−1.62623	−0.813113	−	0.582106i	$-0.802229\pi$
$569$	−4.06618e12	−1.62623	−0.813113	+	0.582106i	$0.802229\pi$
$570$	0	0
$571$	9.86144e11	0.388220	0.194110	−	0.980980i	$-0.437818\pi$
$571$	9.86144e11	0.388220	0.194110	+	0.980980i	$0.437818\pi$
$572$	1.55469e10i	0.00607243i
$573$	0	0
$574$	1.75019e12	0.672947
$575$	0	0
$576$	0	0
$577$	2.86947e12i	1.07773i	0.842391	+	0.538866i	$0.181147\pi$
$577$	2.86947e12i	1.07773i	−0.842391	+	0.538866i	$0.818853\pi$
$578$	− 1.67155e12i	− 0.622937i
$579$	0	0
$580$	0	0
$581$	5.09588e12	1.85535
$582$	0	0
$583$	8.44240e11i	0.302662i
$584$	3.13660e12	1.11584
$585$	0	0
$586$	−3.29692e12	−1.15497
$587$	1.14169e12i	0.396898i	0.980111	+	0.198449i	$0.0635904\pi$
$587$	1.14169e12i	0.396898i	−0.980111	+	0.198449i	$0.936410\pi$
$588$	0	0
$589$	3.49505e12	1.19656
$590$	0	0
$591$	0	0
$592$	− 5.18567e12i	− 1.73523i
$593$	2.97176e12i	0.986888i	0.869777	+	0.493444i	$0.164262\pi$
$593$	2.97176e12i	0.986888i	−0.869777	+	0.493444i	$0.835738\pi$
$594$	0	0
$595$	0	0
$596$	−1.32679e11	−0.0430719
$597$	0	0
$598$	1.88814e12i	0.603779i
$599$	1.77262e12	0.562593	0.281297	−	0.959621i	$-0.409236\pi$
$599$	1.77262e12	0.562593	0.281297	+	0.959621i	$0.409236\pi$
$600$	0	0
$601$	1.25838e12	0.393439	0.196719	−	0.980460i	$-0.436971\pi$
$601$	1.25838e12	0.393439	0.196719	+	0.980460i	$0.436971\pi$
$602$	3.04715e12i	0.945606i
$603$	0	0
$604$	6.33405e10	0.0193649
$605$	0	0
$606$	0	0
$607$	− 1.74535e11i	− 0.0521834i	−0.999660	−	0.0260917i	$-0.991694\pi$
$607$	− 1.74535e11i	− 0.0521834i	0.999660	−	0.0260917i	$-0.00830619\pi$
$608$	− 2.86313e11i	− 0.0849720i
$609$	0	0
$610$	0	0
$611$	5.25896e11	0.152656
$612$	0	0
$613$	4.63037e12i	1.32447i	0.749294	+	0.662237i	$0.230393\pi$
$613$	4.63037e12i	1.32447i	−0.749294	+	0.662237i	$0.769607\pi$
$614$	−4.06054e12	−1.15299
$615$	0	0
$616$	−1.04202e12	−0.291584
$617$	2.50518e12i	0.695914i	0.937511	+	0.347957i	$0.113124\pi$
$617$	2.50518e12i	0.695914i	−0.937511	+	0.347957i	$0.886876\pi$
$618$	0	0
$619$	−5.44228e11	−0.148995	−0.0744977	−	0.997221i	$-0.523735\pi$
$619$	−5.44228e11	−0.148995	−0.0744977	+	0.997221i	$0.523735\pi$
$620$	0	0
$621$	0	0
$622$	− 1.99023e12i	− 0.533146i
$623$	1.13709e12i	0.302412i
$624$	0	0
$625$	0	0
$626$	−2.37447e11	−0.0617991
$627$	0	0
$628$	3.30088e11i	0.0846860i
$629$	9.25978e12	2.35870
$630$	0	0
$631$	−6.51975e12	−1.63719	−0.818594	−	0.574372i	$-0.805246\pi$
$631$	−6.51975e12	−1.63719	−0.818594	+	0.574372i	$0.805246\pi$
$632$	− 5.32465e12i	− 1.32759i
$633$	0	0
$634$	−9.20303e11	−0.226219
$635$	0	0
$636$	0	0
$637$	1.70481e11i	0.0410250i
$638$	− 3.38159e11i	− 0.0808032i
$639$	0	0
$640$	0	0
$641$	3.36297e12	0.786794	0.393397	−	0.919369i	$-0.371300\pi$
$641$	3.36297e12	0.786794	0.393397	+	0.919369i	$0.371300\pi$
$642$	0	0
$643$	1.18082e12i	0.272417i	0.990680	+	0.136208i	$0.0434917\pi$
$643$	1.18082e12i	0.272417i	−0.990680	+	0.136208i	$0.956508\pi$
$644$	3.79614e11	0.0869673
$645$	0	0
$646$	−4.28749e12	−0.968628
$647$	6.63176e12i	1.48785i	0.668262	+	0.743926i	$0.267038\pi$
$647$	6.63176e12i	1.48785i	−0.668262	+	0.743926i	$0.732962\pi$
$648$	0	0
$649$	−4.69353e11	−0.103848
$650$	0	0
$651$	0	0
$652$	3.20902e11i	0.0695438i
$653$	− 8.25189e11i	− 0.177601i	−0.996049	−	0.0888003i	$-0.971697\pi$
$653$	− 8.25189e11i	− 0.177601i	0.996049	−	0.0888003i	$-0.0283033\pi$
$654$	0	0
$655$	0	0
$656$	3.28185e12	0.691913
$657$	0	0
$658$	1.82766e12i	0.380084i
$659$	−4.40214e11	−0.0909242	−0.0454621	−	0.998966i	$-0.514476\pi$
$659$	−4.40214e11	−0.0909242	−0.0454621	+	0.998966i	$0.514476\pi$
$660$	0	0
$661$	9.77788e12	1.99222	0.996112	−	0.0880914i	$-0.0280767\pi$
$661$	9.77788e12	1.99222	0.996112	+	0.0880914i	$0.0280767\pi$
$662$	− 2.71281e11i	− 0.0548983i
$663$	0	0
$664$	1.01101e13	2.01836
$665$	0	0
$666$	0	0
$667$	2.37587e12i	0.464790i
$668$	− 3.74378e11i	− 0.0727473i
$669$	0	0
$670$	0	0
$671$	1.62101e12	0.308698
$672$	0	0
$673$	4.22591e12i	0.794058i	0.917806	+	0.397029i	$0.129959\pi$
$673$	4.22591e12i	0.794058i	−0.917806	+	0.397029i	$0.870041\pi$
$674$	−4.79547e11	−0.0895079
$675$	0	0
$676$	2.56694e11	0.0472776
$677$	− 7.56977e12i	− 1.38495i	−0.721442	−	0.692474i	$-0.756521\pi$
$677$	− 7.56977e12i	− 1.38495i	0.721442	−	0.692474i	$-0.243479\pi$
$678$	0	0
$679$	−6.07273e12	−1.09640
$680$	0	0
$681$	0	0
$682$	− 2.54923e12i	− 0.451210i
$683$	1.92029e12i	0.337655i	0.985646	+	0.168827i	$0.0539980\pi$
$683$	1.92029e12i	0.337655i	−0.985646	+	0.168827i	$0.946002\pi$
$684$	0	0
$685$	0	0
$686$	−5.90850e12	−1.01864
$687$	0	0
$688$	5.71385e12i	0.972257i
$689$	−2.18472e12	−0.369325
$690$	0	0
$691$	−3.99839e12	−0.667166	−0.333583	−	0.942721i	$-0.608258\pi$
$691$	−3.99839e12	−0.667166	−0.333583	+	0.942721i	$0.608258\pi$
$692$	− 2.97538e11i	− 0.0493248i
$693$	0	0
$694$	1.85595e12	0.303702
$695$	0	0
$696$	0	0
$697$	5.86023e12i	0.940519i
$698$	2.71249e10i	0.00432532i
$699$	0	0
$700$	0	0
$701$	−1.12736e13	−1.76332	−0.881662	−	0.471881i	$-0.843575\pi$
$701$	−1.12736e13	−1.76332	−0.881662	+	0.471881i	$0.843575\pi$
$702$	0	0
$703$	− 9.27493e12i	− 1.43223i
$704$	−2.06146e12	−0.316299
$705$	0	0
$706$	3.81453e12	0.577856
$707$	7.87643e12i	1.18561i
$708$	0	0
$709$	−1.12679e13	−1.67469	−0.837346	−	0.546673i	$-0.815894\pi$
$709$	−1.12679e13	−1.67469	−0.837346	+	0.546673i	$0.815894\pi$
$710$	0	0
$711$	0	0
$712$	2.25595e12i	0.328980i
$713$	1.79106e13i	2.59542i
$714$	0	0
$715$	0	0
$716$	7.31512e11	0.104019
$717$	0	0
$718$	1.40395e12i	0.197148i
$719$	−9.05247e12	−1.26324	−0.631622	−	0.775277i	$-0.717610\pi$
$719$	−9.05247e12	−1.26324	−0.631622	+	0.775277i	$0.717610\pi$
$720$	0	0
$721$	1.09459e13	1.50849
$722$	− 2.80462e12i	− 0.384111i
$723$	0	0
$724$	6.57129e10	0.00888847
$725$	0	0
$726$	0	0
$727$	− 1.34408e12i	− 0.178452i	−0.996011	−	0.0892260i	$-0.971561\pi$
$727$	− 1.34408e12i	− 0.178452i	0.996011	−	0.0892260i	$-0.0284393\pi$
$728$	− 2.69654e12i	− 0.355807i
$729$	0	0
$730$	0	0
$731$	−1.02029e13	−1.32159
$732$	0	0
$733$	6.57401e11i	0.0841129i	0.999115	+	0.0420564i	$0.0133909\pi$
$733$	6.57401e11i	0.0841129i	−0.999115	+	0.0420564i	$0.986609\pi$
$734$	2.46932e12	0.314011
$735$	0	0
$736$	1.46723e12	0.184310
$737$	1.73679e12i	0.216842i
$738$	0	0
$739$	−2.15587e12	−0.265903	−0.132951	−	0.991123i	$-0.542445\pi$
$739$	−2.15587e12	−0.265903	−0.132951	+	0.991123i	$0.542445\pi$
$740$	0	0
$741$	0	0
$742$	− 7.59263e12i	− 0.919548i
$743$	7.73594e12i	0.931244i	0.884984	+	0.465622i	$0.154169\pi$
$743$	7.73594e12i	0.931244i	−0.884984	+	0.465622i	$0.845831\pi$
$744$	0	0
$745$	0	0
$746$	−1.58621e13	−1.87515
$747$	0	0
$748$	− 1.80914e11i	− 0.0211307i
$749$	−1.10998e12	−0.128869
$750$	0	0
$751$	−1.12448e13	−1.28995	−0.644974	−	0.764205i	$-0.723132\pi$
$751$	−1.12448e13	−1.28995	−0.644974	+	0.764205i	$0.723132\pi$
$752$	3.42713e12i	0.390797i
$753$	0	0
$754$	8.75087e11	0.0986007
$755$	0	0
$756$	0	0
$757$	1.18544e13i	1.31205i	0.754740	+	0.656024i	$0.227763\pi$
$757$	1.18544e13i	1.31205i	−0.754740	+	0.656024i	$0.772237\pi$
$758$	4.49528e12i	0.494590i
$759$	0	0
$760$	0	0
$761$	2.24765e12	0.242939	0.121470	−	0.992595i	$-0.461239\pi$
$761$	2.24765e12	0.242939	0.121470	+	0.992595i	$0.461239\pi$
$762$	0	0
$763$	− 1.18484e13i	− 1.26561i
$764$	−6.27380e11	−0.0666209
$765$	0	0
$766$	−1.20380e13	−1.26335
$767$	− 1.21459e12i	− 0.126721i
$768$	0	0
$769$	1.44505e13	1.49010	0.745049	−	0.667009i	$-0.232426\pi$
$769$	1.44505e13	1.49010	0.745049	+	0.667009i	$0.232426\pi$
$770$	0	0
$771$	0	0
$772$	− 1.02469e12i	− 0.103828i
$773$	− 5.19022e12i	− 0.522852i	−0.965224	−	0.261426i	$-0.915807\pi$
$773$	− 5.19022e12i	− 0.522852i	0.965224	−	0.261426i	$-0.0841927\pi$
$774$	0	0
$775$	0	0
$776$	−1.20481e13	−1.19273
$777$	0	0
$778$	− 7.61777e12i	− 0.745452i
$779$	5.86983e12	0.571093
$780$	0	0
$781$	−4.05280e12	−0.389786
$782$	− 2.19715e13i	− 2.10102i
$783$	0	0
$784$	−1.11098e12	−0.105023
$785$	0	0
$786$	0	0
$787$	1.39938e13i	1.30032i	0.759797	+	0.650161i	$0.225299\pi$
$787$	1.39938e13i	1.30032i	−0.759797	+	0.650161i	$0.774701\pi$
$788$	− 1.51633e11i	− 0.0140096i
$789$	0	0
$790$	0	0
$791$	9.45354e12	0.858618
$792$	0	0
$793$	4.19483e12i	0.376691i
$794$	5.64867e12	0.504375
$795$	0	0
$796$	−7.35599e9	−0.000649430 0
$797$	2.01269e13i	1.76691i	0.468518	+	0.883454i	$0.344788\pi$
$797$	2.01269e13i	1.76691i	−0.468518	+	0.883454i	$0.655212\pi$
$798$	0	0
$799$	−6.11966e12	−0.531210
$800$	0	0
$801$	0	0
$802$	− 4.43695e11i	− 0.0378704i
$803$	− 3.86741e12i	− 0.328247i
$804$	0	0
$805$	0	0
$806$	6.59688e12	0.550593
$807$	0	0
$808$	1.56266e13i	1.28977i
$809$	−8.95775e12	−0.735242	−0.367621	−	0.929976i	$-0.619828\pi$
$809$	−8.95775e12	−0.735242	−0.367621	+	0.929976i	$0.619828\pi$
$810$	0	0
$811$	−1.76446e12	−0.143225	−0.0716123	−	0.997433i	$-0.522814\pi$
$811$	−1.76446e12	−0.143225	−0.0716123	+	0.997433i	$0.522814\pi$
$812$	− 1.75938e11i	− 0.0142023i
$813$	0	0
$814$	−6.76498e12	−0.540078
$815$	0	0
$816$	0	0
$817$	1.02196e13i	0.802483i
$818$	9.53491e11i	0.0744607i
$819$	0	0
$820$	0	0
$821$	−1.48082e13	−1.13752	−0.568758	−	0.822505i	$-0.692576\pi$
$821$	−1.48082e13	−1.13752	−0.568758	+	0.822505i	$0.692576\pi$
$822$	0	0
$823$	− 3.39651e12i	− 0.258067i	−0.991640	−	0.129034i	$-0.958812\pi$
$823$	− 3.39651e12i	− 0.258067i	0.991640	−	0.129034i	$-0.0411876\pi$
$824$	2.17163e13	1.64102
$825$	0	0
$826$	4.22110e12	0.315512
$827$	1.88344e13i	1.40016i	0.714065	+	0.700080i	$0.246852\pi$
$827$	1.88344e13i	1.40016i	−0.714065	+	0.700080i	$0.753148\pi$
$828$	0	0
$829$	−1.18947e13	−0.874695	−0.437348	−	0.899292i	$-0.644082\pi$
$829$	−1.18947e13	−0.874695	−0.437348	+	0.899292i	$0.644082\pi$
$830$	0	0
$831$	0	0
$832$	− 5.33462e12i	− 0.385966i
$833$	− 1.98382e12i	− 0.142758i
$834$	0	0
$835$	0	0
$836$	−1.81210e11	−0.0128308
$837$	0	0
$838$	1.12350e13i	0.786998i
$839$	−1.22881e13	−0.856165	−0.428083	−	0.903740i	$-0.640811\pi$
$839$	−1.22881e13	−0.856165	−0.428083	+	0.903740i	$0.640811\pi$
$840$	0	0
$841$	−1.34060e13	−0.924097
$842$	7.06702e12i	0.484543i
$843$	0	0
$844$	3.76581e11	0.0255457
$845$	0	0
$846$	0	0
$847$	− 1.28346e13i	− 0.856853i
$848$	− 1.42373e13i	− 0.945465i
$849$	0	0
$850$	0	0
$851$	4.75300e13	3.10660
$852$	0	0
$853$	− 4.61355e12i	− 0.298376i	−0.988809	−	0.149188i	$-0.952334\pi$
$853$	− 4.61355e12i	− 0.298376i	0.988809	−	0.149188i	$-0.0476660\pi$
$854$	−1.45784e13	−0.937887
$855$	0	0
$856$	−2.20216e12	−0.140190
$857$	− 4.29363e12i	− 0.271901i	−0.990716	−	0.135950i	$-0.956591\pi$
$857$	− 4.29363e12i	− 0.271901i	0.990716	−	0.135950i	$-0.0434088\pi$
$858$	0	0
$859$	6.40428e12	0.401330	0.200665	−	0.979660i	$-0.435690\pi$
$859$	6.40428e12	0.401330	0.200665	+	0.979660i	$0.435690\pi$
$860$	0	0
$861$	0	0
$862$	− 9.84758e12i	− 0.607501i
$863$	3.24421e12i	0.199095i	0.995033	+	0.0995474i	$0.0317395\pi$
$863$	3.24421e12i	0.199095i	−0.995033	+	0.0995474i	$0.968260\pi$
$864$	0	0
$865$	0	0
$866$	3.10609e12	0.187665
$867$	0	0
$868$	− 1.32632e12i	− 0.0793065i
$869$	−6.56527e12	−0.390539
$870$	0	0
$871$	−4.49445e12	−0.264603
$872$	− 2.35068e13i	− 1.37680i
$873$	0	0
$874$	−2.20075e13	−1.27576
$875$	0	0
$876$	0	0
$877$	− 2.89711e13i	− 1.65374i	−0.562392	−	0.826871i	$-0.690119\pi$
$877$	− 2.89711e13i	− 1.65374i	0.562392	−	0.826871i	$-0.309881\pi$
$878$	1.87109e13i	1.06260i
$879$	0	0
$880$	0	0
$881$	−7.50447e12	−0.419690	−0.209845	−	0.977735i	$-0.567296\pi$
$881$	−7.50447e12	−0.419690	−0.209845	+	0.977735i	$0.567296\pi$
$882$	0	0
$883$	2.87141e13i	1.58954i	0.606908	+	0.794772i	$0.292410\pi$
$883$	2.87141e13i	1.58954i	−0.606908	+	0.794772i	$0.707590\pi$
$884$	4.68167e11	0.0257849
$885$	0	0
$886$	6.67013e12	0.363649
$887$	9.99825e12i	0.542335i	0.962532	+	0.271168i	$0.0874098\pi$
$887$	9.99825e12i	0.542335i	−0.962532	+	0.271168i	$0.912590\pi$
$888$	0	0
$889$	1.43712e13	0.771678
$890$	0	0
$891$	0	0
$892$	− 7.46095e11i	− 0.0394596i
$893$	6.12967e12i	0.322556i
$894$	0	0
$895$	0	0
$896$	1.65528e13	0.857998
$897$	0	0
$898$	− 3.08721e13i	− 1.58425i
$899$	8.30095e12	0.423847
$900$	0	0
$901$	2.54228e13	1.28517
$902$	− 4.28135e12i	− 0.215353i
$903$	0	0
$904$	1.87555e13	0.934052
$905$	0	0
$906$	0	0
$907$	− 4.70846e12i	− 0.231018i	−0.993306	−	0.115509i	$-0.963150\pi$
$907$	− 4.70846e12i	− 0.231018i	0.993306	−	0.115509i	$-0.0368500\pi$
$908$	9.41690e11i	0.0459750i
$909$	0	0
$910$	0	0
$911$	−2.69663e13	−1.29714	−0.648572	−	0.761154i	$-0.724633\pi$
$911$	−2.69663e13	−1.29714	−0.648572	+	0.761154i	$0.724633\pi$
$912$	0	0
$913$	− 1.24657e13i	− 0.593742i
$914$	7.37778e12	0.349678
$915$	0	0
$916$	−1.40127e12	−0.0657645
$917$	1.17868e13i	0.550468i
$918$	0	0
$919$	3.96618e12	0.183422	0.0917112	−	0.995786i	$-0.470766\pi$
$919$	3.96618e12	0.183422	0.0917112	+	0.995786i	$0.470766\pi$
$920$	0	0
$921$	0	0
$922$	− 2.29405e13i	− 1.04547i
$923$	− 1.04878e13i	− 0.475639i
$924$	0	0
$925$	0	0
$926$	2.14298e13	0.957788
$927$	0	0
$928$	− 6.80012e11i	− 0.0300989i
$929$	1.96912e13	0.867365	0.433683	−	0.901066i	$-0.357214\pi$
$929$	1.96912e13	0.867365	0.433683	+	0.901066i	$0.357214\pi$
$930$	0	0
$931$	−1.98707e12	−0.0866841
$932$	9.22952e11i	0.0400689i
$933$	0	0
$934$	4.35271e12	0.187154
$935$	0	0
$936$	0	0
$937$	2.11305e13i	0.895532i	0.894151	+	0.447766i	$0.147780\pi$
$937$	2.11305e13i	0.895532i	−0.894151	+	0.447766i	$0.852220\pi$
$938$	− 1.56197e13i	− 0.658810i
$939$	0	0
$940$	0	0
$941$	−5.41072e12	−0.224958	−0.112479	−	0.993654i	$-0.535879\pi$
$941$	−5.41072e12	−0.224958	−0.112479	+	0.993654i	$0.535879\pi$
$942$	0	0
$943$	3.00803e13i	1.23874i
$944$	7.91517e12	0.324404
$945$	0	0
$946$	7.45403e12	0.302608
$947$	3.44259e12i	0.139095i	0.997579	+	0.0695473i	$0.0221555\pi$
$947$	3.44259e12i	0.139095i	−0.997579	+	0.0695473i	$0.977845\pi$
$948$	0	0
$949$	1.00081e13	0.400546
$950$	0	0
$951$	0	0
$952$	3.13786e13i	1.23813i
$953$	4.18440e12i	0.164329i	0.996619	+	0.0821647i	$0.0261834\pi$
$953$	4.18440e12i	0.164329i	−0.996619	+	0.0821647i	$0.973817\pi$
$954$	0	0
$955$	0	0
$956$	−2.22718e11	−0.00862373
$957$	0	0
$958$	− 1.99921e13i	− 0.766855i
$959$	1.21502e13	0.463874
$960$	0	0
$961$	3.61375e13	1.36679
$962$	− 1.75064e13i	− 0.659035i
$963$	0	0
$964$	−2.10694e12	−0.0785787
$965$	0	0
$966$	0	0
$967$	− 1.49800e13i	− 0.550926i	−0.961312	−	0.275463i	$-0.911169\pi$
$967$	− 1.49800e13i	− 0.550926i	0.961312	−	0.275463i	$-0.0888312\pi$
$968$	− 2.54634e13i	− 0.932132i
$969$	0	0
$970$	0	0
$971$	2.82460e13	1.01970	0.509848	−	0.860264i	$-0.329701\pi$
$971$	2.82460e13	1.01970	0.509848	+	0.860264i	$0.329701\pi$
$972$	0	0
$973$	6.80673e12i	0.243462i
$974$	2.57318e13	0.916127
$975$	0	0
$976$	−2.73367e13	−0.964321
$977$	− 4.62242e13i	− 1.62309i	−0.584287	−	0.811547i	$-0.698626\pi$
$977$	− 4.62242e13i	− 1.62309i	0.584287	−	0.811547i	$-0.301374\pi$
$978$	0	0
$979$	2.78158e12	0.0967764
$980$	0	0
$981$	0	0
$982$	5.48127e13i	1.88096i
$983$	− 4.80363e13i	− 1.64089i	−0.571728	−	0.820443i	$-0.693727\pi$
$983$	− 4.80363e13i	− 1.64089i	0.571728	−	0.820443i	$-0.306273\pi$
$984$	0	0
$985$	0	0
$986$	−1.01831e13	−0.343109
$987$	0	0
$988$	− 4.68934e11i	− 0.0156569i
$989$	−5.23712e13	−1.74064
$990$	0	0
$991$	2.31211e13	0.761512	0.380756	−	0.924675i	$-0.375664\pi$
$991$	2.31211e13	0.761512	0.380756	+	0.924675i	$0.375664\pi$
$992$	− 5.12630e12i	− 0.168074i
$993$	0	0
$994$	3.64487e13	1.18425
$995$	0	0
$996$	0	0
$997$	1.35362e13i	0.433878i	0.976185	+	0.216939i	$0.0696073\pi$
$997$	1.35362e13i	0.433878i	−0.976185	+	0.216939i	$0.930393\pi$
$998$	1.22666e13i	0.391413i
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	225.10.b.b.199.2		2
3.2	odd	2		75.10.b.b.49.1		2
5.2	odd	4		45.10.a.a.1.1		1
5.3	odd	4		225.10.a.f.1.1		1
5.4	even	2	inner	225.10.b.b.199.1		2
15.2	even	4		15.10.a.b.1.1	✓	1
15.8	even	4		75.10.a.a.1.1		1
15.14	odd	2		75.10.b.b.49.2		2
60.47	odd	4		240.10.a.g.1.1		1

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
15.10.a.b.1.1	✓	1	15.2	even	4
45.10.a.a.1.1		1	5.2	odd	4
75.10.a.a.1.1		1	15.8	even	4
75.10.b.b.49.1		2	3.2	odd	2
75.10.b.b.49.2		2	15.14	odd	2
225.10.a.f.1.1		1	5.3	odd	4
225.10.b.b.199.1		2	5.4	even	2	inner
225.10.b.b.199.2		2	1.1	even	1	trivial
240.10.a.g.1.1		1	60.47	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:	\( k \)	\(=\)	\( 10 \)
Character orbit:	\([\chi]\)	\(=\)	225.b (of order \(2\), degree \(1\), not minimal)

\(f(q)\)	\(=\)	\(q+22.0000i q^{2} +28.0000 q^{4} +5988.00i q^{7} +11880.0i q^{8} +O(q^{10})\)	\(q+22.0000i q^{2} +28.0000 q^{4} +5988.00i q^{7} +11880.0i q^{8} +14648.0 q^{11} +37906.0i q^{13} -131736. q^{14} -247024. q^{16} -441098. i q^{17} -441820. q^{19} +322256. i q^{22} -2.26414e6i q^{23} -833932. q^{26} +167664. i q^{28} -1.04935e6 q^{29} -7.91057e6 q^{31} +648032. i q^{32} +9.70416e6 q^{34} +2.09926e7i q^{37} -9.72004e6i q^{38} -1.32856e7 q^{41} -2.31308e7i q^{43} +410144. q^{44} +4.98110e7 q^{46} -1.38737e7i q^{47} +4.49746e6 q^{49} +1.06137e6i q^{52} +5.76352e7i q^{53} -7.11374e7 q^{56} -2.30857e7i q^{58} -3.20421e7 q^{59} +1.10664e8 q^{61} -1.74032e8i q^{62} -1.40733e8 q^{64} +1.18568e8i q^{67} -1.23507e7i q^{68} -2.76680e8 q^{71} -2.64023e8i q^{73} -4.61836e8 q^{74} -1.23710e7 q^{76} +8.77122e7i q^{77} -4.48203e8 q^{79} -2.92282e8i q^{82} -8.51016e8i q^{83} +5.08877e8 q^{86} +1.74018e8i q^{88} +1.89895e8 q^{89} -2.26981e8 q^{91} -6.33958e7i q^{92} +3.05221e8 q^{94} +1.01415e9i q^{97} +9.89442e7i q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 2 q + 56 q^{4}+O(q^{10}) \) 2 * q + 56 * q^4	\( 2 q + 56 q^{4} + 29296 q^{11} - 263472 q^{14} - 494048 q^{16} - 883640 q^{19} - 1667864 q^{26} - 2098700 q^{29} - 15821136 q^{31} + 19408312 q^{34} - 26571124 q^{41} + 820288 q^{44} + 99621984 q^{46} + 8994926 q^{49} - 142274880 q^{56} - 64084240 q^{59} + 221328044 q^{61} - 281465984 q^{64} - 553359424 q^{71} - 923672552 q^{74} - 24741920 q^{76} - 896405520 q^{79} + 1017753616 q^{86} + 379789860 q^{89} - 453962256 q^{91} + 610442272 q^{94}+O(q^{100}) \) 2 * q + 56 * q^4 + 29296 * q^11 - 263472 * q^14 - 494048 * q^16 - 883640 * q^19 - 1667864 * q^26 - 2098700 * q^29 - 15821136 * q^31 + 19408312 * q^34 - 26571124 * q^41 + 820288 * q^44 + 99621984 * q^46 + 8994926 * q^49 - 142274880 * q^56 - 64084240 * q^59 + 221328044 * q^61 - 281465984 * q^64 - 553359424 * q^71 - 923672552 * q^74 - 24741920 * q^76 - 896405520 * q^79 + 1017753616 * q^86 + 379789860 * q^89 - 453962256 * q^91 + 610442272 * q^94

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

Related objects

Downloads

Learn more