LMFDB - Embedded newform 450.8.c.l.199.1

Show commands: Magma / Pari/GP / SageMath

Newspace parameters

comment:Compute space of new eigenforms

gp:[N,k,chi] = [450,8,Mod(199,450)] mf = mfinit([N,k,chi],0) lf = mfeigenbasis(mf)

sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(450, base_ring=CyclotomicField(2)) chi = DirichletCharacter(H, H._module([0, 1])) N = Newforms(chi, 8, names="a")

magma://Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code chi := DirichletCharacter("450.199"); S:= CuspForms(chi, 8); N := Newforms(S);

Level:	$ N $	$=$	$ 450 = 2 \cdot 3^{2} \cdot 5^{2} $
Weight:	$ k $	$=$	$ 8 $
Character orbit:	$[\chi]$	$=$	450.c (of order $2$, degree $1$, not minimal)

Newform invariants

comment:select newform

sage:traces = [2,0,0,-128,0,0,0,0,0,0,2166] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(11)] == traces)

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

Self dual:	no
Analytic conductor:	$140.573261468$
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:	$\Z[a_1, \ldots, a_{17}]$
Coefficient ring index:	$ 1 $
Twist minimal:	no (minimal twist has level 50)
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label			199.1
Root			$-1.00000i$ of defining polynomial
Character	$\chi$	$=$	450.199
Dual form			450.8.c.l.199.2

$q$-expansion

comment:q-expansion

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

gp:mfcoefs(f, 20)

$f(q)$	$=$	$q-8.00000i q^{2} -64.0000 q^{4} -1366.00i q^{7} +512.000i q^{8} +1083.00 q^{11} -5468.00i q^{13} -10928.0 q^{14} +4096.00 q^{16} -25269.0i q^{17} -33485.0 q^{19} -8664.00i q^{22} +5838.00i q^{23} -43744.0 q^{26} +87424.0i q^{28} +125280. q^{29} -73798.0 q^{31} -32768.0i q^{32} -202152. q^{34} -395926. i q^{37} +267880. i q^{38} +22683.0 q^{41} -100148. i q^{43} -69312.0 q^{44} +46704.0 q^{46} -1.14524e6i q^{47} -1.04241e6 q^{49} +349952. i q^{52} -354882. i q^{53} +699392. q^{56} -1.00224e6i q^{58} +1.09836e6 q^{59} -422998. q^{61} +590384. i q^{62} -262144. q^{64} +2.55858e6i q^{67} +1.61722e6i q^{68} +2.28743e6 q^{71} -6.37244e6i q^{73} -3.16741e6 q^{74} +2.14304e6 q^{76} -1.47938e6i q^{77} +2.01925e6 q^{79} -181464. i q^{82} +7.97298e6i q^{83} -801184. q^{86} +554496. i q^{88} +2.18594e6 q^{89} -7.46929e6 q^{91} -373632. i q^{92} -9.16195e6 q^{94} -5.82365e6i q^{97} +8.33930e6i q^{98} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 2 q - 128 q^{4} + 2166 q^{11} - 21856 q^{14} + 8192 q^{16} - 66970 q^{19} - 87488 q^{26} + 250560 q^{29} - 147596 q^{31} - 404304 q^{34} + 45366 q^{41} - 138624 q^{44} + 93408 q^{46} - 2084826 q^{49} + 1398784 q^{56}+ \cdots - 18323904 q^{94}+O(q^{100}) $ 2 * q - 128 * q^4 + 2166 * q^11 - 21856 * q^14 + 8192 * q^16 - 66970 * q^19 - 87488 * q^26 + 250560 * q^29 - 147596 * q^31 - 404304 * q^34 + 45366 * q^41 - 138624 * q^44 + 93408 * q^46 - 2084826 * q^49 + 1398784 * q^56 + 2196720 * q^59 - 845996 * q^61 - 524288 * q^64 + 4574856 * q^71 - 6334816 * q^74 + 4286080 * q^76 + 4038500 * q^79 - 1602368 * q^86 + 4371870 * q^89 - 14938576 * q^91 - 18323904 * q^94

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/450\mathbb{Z}\right)^\times$.

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

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 $

$2$	− 8.00000i	− 0.707107i
$2$	− 8.00000i	− 0.707107i
$3$	0	0
$3$	0	0
$4$	−64.0000	−0.500000
$5$	0	0
$5$	0	0
$6$	0	0
$7$	− 1366.00i	− 1.50525i	−0.658452	−	0.752623i	$-0.728788\pi$
$7$	− 1366.00i	− 1.50525i	0.658452	−	0.752623i	$-0.271212\pi$
$8$	512.000i	0.353553i
$9$	0	0
$10$	0	0
$11$	1083.00	0.245332	0.122666	−	0.992448i	$-0.460856\pi$
$11$	1083.00	0.245332	0.122666	+	0.992448i	$0.460856\pi$
$12$	0	0
$13$	− 5468.00i	− 0.690282i	−0.938551	−	0.345141i	$-0.887831\pi$
$13$	− 5468.00i	− 0.690282i	0.938551	−	0.345141i	$-0.112169\pi$
$14$	−10928.0	−1.06437
$15$	0	0
$16$	4096.00	0.250000
$17$	− 25269.0i	− 1.24743i	−0.781651	−	0.623716i	$-0.785622\pi$
$17$	− 25269.0i	− 1.24743i	0.781651	−	0.623716i	$-0.214378\pi$
$18$	0	0
$19$	−33485.0	−1.11999	−0.559993	−	0.828497i	$-0.689196\pi$
$19$	−33485.0	−1.11999	−0.559993	+	0.828497i	$0.689196\pi$
$20$	0	0
$21$	0	0
$22$	− 8664.00i	− 0.173476i
$23$	5838.00i	0.100050i	0.998748	+	0.0500250i	$0.0159301\pi$
$23$	5838.00i	0.100050i	−0.998748	+	0.0500250i	$0.984070\pi$
$24$	0	0
$25$	0	0
$26$	−43744.0	−0.488103
$27$	0	0
$28$	87424.0i	0.752623i
$29$	125280.	0.953869	0.476935	−	0.878939i	$-0.341748\pi$
$29$	125280.	0.953869	0.476935	+	0.878939i	$0.341748\pi$
$30$	0	0
$31$	−73798.0	−0.444917	−0.222458	−	0.974942i	$-0.571408\pi$
$31$	−73798.0	−0.444917	−0.222458	+	0.974942i	$0.571408\pi$
$32$	− 32768.0i	− 0.176777i
$33$	0	0
$34$	−202152.	−0.882068
$35$	0	0
$36$	0	0
$37$	− 395926.i	− 1.28501i	−0.766280	−	0.642507i	$-0.777894\pi$
$37$	− 395926.i	− 1.28501i	0.766280	−	0.642507i	$-0.222106\pi$
$38$	267880.i	0.791950i
$39$	0	0
$40$	0	0
$41$	22683.0	0.0513993	0.0256996	−	0.999670i	$-0.491819\pi$
$41$	22683.0	0.0513993	0.0256996	+	0.999670i	$0.491819\pi$
$42$	0	0
$43$	− 100148.i	− 0.192089i	−0.995377	−	0.0960445i	$-0.969381\pi$
$43$	− 100148.i	− 0.192089i	0.995377	−	0.0960445i	$-0.0306191\pi$
$44$	−69312.0	−0.122666
$45$	0	0
$46$	46704.0	0.0707460
$47$	− 1.14524e6i	− 1.60900i	−0.593954	−	0.804499i	$-0.702434\pi$
$47$	− 1.14524e6i	− 1.60900i	0.593954	−	0.804499i	$-0.297566\pi$
$48$	0	0
$49$	−1.04241e6	−1.26577
$50$	0	0
$51$	0	0
$52$	349952.i	0.345141i
$53$	− 354882.i	− 0.327430i	−0.986508	−	0.163715i	$-0.947652\pi$
$53$	− 354882.i	− 0.327430i	0.986508	−	0.163715i	$-0.0523477\pi$
$54$	0	0
$55$	0	0
$56$	699392.	0.532185
$57$	0	0
$58$	− 1.00224e6i	− 0.674487i
$59$	1.09836e6	0.696246	0.348123	−	0.937449i	$-0.386819\pi$
$59$	1.09836e6	0.696246	0.348123	+	0.937449i	$0.386819\pi$
$60$	0	0
$61$	−422998.	−0.238607	−0.119304	−	0.992858i	$-0.538066\pi$
$61$	−422998.	−0.238607	−0.119304	+	0.992858i	$0.538066\pi$
$62$	590384.i	0.314604i
$63$	0	0
$64$	−262144.	−0.125000
$65$	0	0
$66$	0	0
$67$	2.55858e6i	1.03929i	0.854382	+	0.519645i	$0.173936\pi$
$67$	2.55858e6i	1.03929i	−0.854382	+	0.519645i	$0.826064\pi$
$68$	1.61722e6i	0.623716i
$69$	0	0
$70$	0	0
$71$	2.28743e6	0.758478	0.379239	−	0.925299i	$-0.376186\pi$
$71$	2.28743e6	0.758478	0.379239	+	0.925299i	$0.376186\pi$
$72$	0	0
$73$	− 6.37244e6i	− 1.91724i	−0.284693	−	0.958619i	$-0.591892\pi$
$73$	− 6.37244e6i	− 1.91724i	0.284693	−	0.958619i	$-0.408108\pi$
$74$	−3.16741e6	−0.908642
$75$	0	0
$76$	2.14304e6	0.559993
$77$	− 1.47938e6i	− 0.369285i
$78$	0	0
$79$	2.01925e6	0.460782	0.230391	−	0.973098i	$-0.426000\pi$
$79$	2.01925e6	0.460782	0.230391	+	0.973098i	$0.426000\pi$
$80$	0	0
$81$	0	0
$82$	− 181464.i	− 0.0363448i
$83$	7.97298e6i	1.53055i	0.643703	+	0.765275i	$0.277397\pi$
$83$	7.97298e6i	1.53055i	−0.643703	+	0.765275i	$0.722603\pi$
$84$	0	0
$85$	0	0
$86$	−801184.	−0.135827
$87$	0	0
$88$	554496.i	0.0867379i
$89$	2.18594e6	0.328679	0.164340	−	0.986404i	$-0.447451\pi$
$89$	2.18594e6	0.328679	0.164340	+	0.986404i	$0.447451\pi$
$90$	0	0
$91$	−7.46929e6	−1.03904
$92$	− 373632.i	− 0.0500250i
$93$	0	0
$94$	−9.16195e6	−1.13773
$95$	0	0
$96$	0	0
$97$	− 5.82365e6i	− 0.647879i	−0.946078	−	0.323939i	$-0.894993\pi$
$97$	− 5.82365e6i	− 0.647879i	0.946078	−	0.323939i	$-0.105007\pi$
$98$	8.33930e6i	0.895032i
$99$	0	0
$100$	0	0
$101$	−433062.	−0.0418240	−0.0209120	−	0.999781i	$-0.506657\pi$
$101$	−433062.	−0.0418240	−0.0209120	+	0.999781i	$0.506657\pi$
$102$	0	0
$103$	1.04295e7i	0.940444i	0.882548	+	0.470222i	$0.155826\pi$
$103$	1.04295e7i	0.940444i	−0.882548	+	0.470222i	$0.844174\pi$
$104$	2.79962e6	0.244052
$105$	0	0
$106$	−2.83906e6	−0.231528
$107$	− 2.13164e7i	− 1.68217i	−0.540901	−	0.841086i	$-0.681917\pi$
$107$	− 2.13164e7i	− 1.68217i	0.540901	−	0.841086i	$-0.318083\pi$
$108$	0	0
$109$	−1.60304e7	−1.18564	−0.592819	−	0.805335i	$-0.701985\pi$
$109$	−1.60304e7	−1.18564	−0.592819	+	0.805335i	$0.701985\pi$
$110$	0	0
$111$	0	0
$112$	− 5.59514e6i	− 0.376312i
$113$	905463.i	0.0590332i	0.999564	+	0.0295166i	$0.00939679\pi$
$113$	905463.i	0.0590332i	−0.999564	+	0.0295166i	$0.990603\pi$
$114$	0	0
$115$	0	0
$116$	−8.01792e6	−0.476935
$117$	0	0
$118$	− 8.78688e6i	− 0.492320i
$119$	−3.45175e7	−1.87769
$120$	0	0
$121$	−1.83143e7	−0.939812
$122$	3.38398e6i	0.168721i
$123$	0	0
$124$	4.72307e6	0.222458
$125$	0	0
$126$	0	0
$127$	1.60352e7i	0.694642i	0.937746	+	0.347321i	$0.112909\pi$
$127$	1.60352e7i	0.694642i	−0.937746	+	0.347321i	$0.887091\pi$
$128$	2.09715e6i	0.0883883i
$129$	0	0
$130$	0	0
$131$	−3.83077e6	−0.148880	−0.0744401	−	0.997225i	$-0.523717\pi$
$131$	−3.83077e6	−0.148880	−0.0744401	+	0.997225i	$0.523717\pi$
$132$	0	0
$133$	4.57405e7i	1.68586i
$134$	2.04686e7	0.734889
$135$	0	0
$136$	1.29377e7	0.441034
$137$	− 9.69343e6i	− 0.322074i	−0.986948	−	0.161037i	$-0.948516\pi$
$137$	− 9.69343e6i	− 0.322074i	0.986948	−	0.161037i	$-0.0514838\pi$
$138$	0	0
$139$	−4.49406e7	−1.41934	−0.709671	−	0.704533i	$-0.751156\pi$
$139$	−4.49406e7	−1.41934	−0.709671	+	0.704533i	$0.751156\pi$
$140$	0	0
$141$	0	0
$142$	− 1.82994e7i	− 0.536325i
$143$	− 5.92184e6i	− 0.169348i
$144$	0	0
$145$	0	0
$146$	−5.09795e7	−1.35569
$147$	0	0
$148$	2.53393e7i	0.642507i
$149$	−4.94208e7	−1.22393	−0.611967	−	0.790883i	$-0.709621\pi$
$149$	−4.94208e7	−1.22393	−0.611967	+	0.790883i	$0.709621\pi$
$150$	0	0
$151$	2.01078e7	0.475275	0.237638	−	0.971354i	$-0.423627\pi$
$151$	2.01078e7	0.475275	0.237638	+	0.971354i	$0.423627\pi$
$152$	− 1.71443e7i	− 0.395975i
$153$	0	0
$154$	−1.18350e7	−0.261124
$155$	0	0
$156$	0	0
$157$	7.86694e7i	1.62240i	0.584771	+	0.811199i	$0.301185\pi$
$157$	7.86694e7i	1.62240i	−0.584771	+	0.811199i	$0.698815\pi$
$158$	− 1.61540e7i	− 0.325822i
$159$	0	0
$160$	0	0
$161$	7.97471e6	0.150600
$162$	0	0
$163$	6.99749e7i	1.26557i	0.774328	+	0.632784i	$0.218088\pi$
$163$	6.99749e7i	1.26557i	−0.774328	+	0.632784i	$0.781912\pi$
$164$	−1.45171e6	−0.0256996
$165$	0	0
$166$	6.37839e7	1.08226
$167$	8.38918e7i	1.39384i	0.717151	+	0.696918i	$0.245446\pi$
$167$	8.38918e7i	1.39384i	−0.717151	+	0.696918i	$0.754554\pi$
$168$	0	0
$169$	3.28495e7	0.523510
$170$	0	0
$171$	0	0
$172$	6.40947e6i	0.0960445i
$173$	− 2.73668e7i	− 0.401849i	−0.979607	−	0.200924i	$-0.935605\pi$
$173$	− 2.73668e7i	− 0.401849i	0.979607	−	0.200924i	$-0.0643945\pi$
$174$	0	0
$175$	0	0
$176$	4.43597e6	0.0613330
$177$	0	0
$178$	− 1.74875e7i	− 0.232411i
$179$	−7.97634e7	−1.03948	−0.519742	−	0.854323i	$-0.673972\pi$
$179$	−7.97634e7	−1.03948	−0.519742	+	0.854323i	$0.673972\pi$
$180$	0	0
$181$	5.28714e7	0.662744	0.331372	−	0.943500i	$-0.392488\pi$
$181$	5.28714e7	0.662744	0.331372	+	0.943500i	$0.392488\pi$
$182$	5.97543e7i	0.734716i
$183$	0	0
$184$	−2.98906e6	−0.0353730
$185$	0	0
$186$	0	0
$187$	− 2.73663e7i	− 0.306035i
$188$	7.32956e7i	0.804499i
$189$	0	0
$190$	0	0
$191$	1.03225e8	1.07194	0.535969	−	0.844238i	$-0.319946\pi$
$191$	1.03225e8	1.07194	0.535969	+	0.844238i	$0.319946\pi$
$192$	0	0
$193$	4.99651e7i	0.500283i	0.968209	+	0.250142i	$0.0804772\pi$
$193$	4.99651e7i	0.500283i	−0.968209	+	0.250142i	$0.919523\pi$
$194$	−4.65892e7	−0.458120
$195$	0	0
$196$	6.67144e7	0.632883
$197$	3.33856e7i	0.311120i	0.987826	+	0.155560i	$0.0497182\pi$
$197$	3.33856e7i	0.311120i	−0.987826	+	0.155560i	$0.950282\pi$
$198$	0	0
$199$	−9.25107e7	−0.832158	−0.416079	−	0.909328i	$-0.636596\pi$
$199$	−9.25107e7	−0.832158	−0.416079	+	0.909328i	$0.636596\pi$
$200$	0	0
$201$	0	0
$202$	3.46450e6i	0.0295740i
$203$	− 1.71132e8i	− 1.43581i
$204$	0	0
$205$	0	0
$206$	8.34359e7	0.664994
$207$	0	0
$208$	− 2.23969e7i	− 0.172571i
$209$	−3.62643e7	−0.274768
$210$	0	0
$211$	5.61153e7	0.411238	0.205619	−	0.978632i	$-0.434079\pi$
$211$	5.61153e7	0.411238	0.205619	+	0.978632i	$0.434079\pi$
$212$	2.27124e7i	0.163715i
$213$	0	0
$214$	−1.70531e8	−1.18948
$215$	0	0
$216$	0	0
$217$	1.00808e8i	0.669709i
$218$	1.28243e8i	0.838373i
$219$	0	0
$220$	0	0
$221$	−1.38171e8	−0.861080
$222$	0	0
$223$	7.79936e7i	0.470968i	0.971878	+	0.235484i	$0.0756676\pi$
$223$	7.79936e7i	0.470968i	−0.971878	+	0.235484i	$0.924332\pi$
$224$	−4.47611e7	−0.266092
$225$	0	0
$226$	7.24370e6	0.0417428
$227$	− 1.31342e8i	− 0.745272i	−0.927978	−	0.372636i	$-0.878454\pi$
$227$	− 1.31342e8i	− 0.745272i	0.927978	−	0.372636i	$-0.121546\pi$
$228$	0	0
$229$	−2.28442e8	−1.25705	−0.628525	−	0.777790i	$-0.716341\pi$
$229$	−2.28442e8	−1.25705	−0.628525	+	0.777790i	$0.716341\pi$
$230$	0	0
$231$	0	0
$232$	6.41434e7i	0.337244i
$233$	3.77949e8i	1.95743i	0.205216	+	0.978717i	$0.434210\pi$
$233$	3.77949e8i	1.95743i	−0.205216	+	0.978717i	$0.565790\pi$
$234$	0	0
$235$	0	0
$236$	−7.02950e7	−0.348123
$237$	0	0
$238$	2.76140e8i	1.32773i
$239$	−1.13186e8	−0.536291	−0.268146	−	0.963378i	$-0.586411\pi$
$239$	−1.13186e8	−0.536291	−0.268146	+	0.963378i	$0.586411\pi$
$240$	0	0
$241$	−2.00236e8	−0.921473	−0.460736	−	0.887537i	$-0.652415\pi$
$241$	−2.00236e8	−0.921473	−0.460736	+	0.887537i	$0.652415\pi$
$242$	1.46514e8i	0.664548i
$243$	0	0
$244$	2.70719e7	0.119304
$245$	0	0
$246$	0	0
$247$	1.83096e8i	0.773107i
$248$	− 3.77846e7i	− 0.157302i
$249$	0	0
$250$	0	0
$251$	4.62823e8	1.84738	0.923692	−	0.383136i	$-0.125156\pi$
$251$	4.62823e8	1.84738	0.923692	+	0.383136i	$0.125156\pi$
$252$	0	0
$253$	6.32255e6i	0.0245454i
$254$	1.28282e8	0.491186
$255$	0	0
$256$	1.67772e7	0.0625000
$257$	4.87751e8i	1.79239i	0.443660	+	0.896195i	$0.353680\pi$
$257$	4.87751e8i	1.79239i	−0.443660	+	0.896195i	$0.646320\pi$
$258$	0	0
$259$	−5.40835e8	−1.93426
$260$	0	0
$261$	0	0
$262$	3.06462e7i	0.105274i
$263$	− 4.01821e8i	− 1.36203i	−0.732268	−	0.681017i	$-0.761538\pi$
$263$	− 4.01821e8i	− 1.36203i	0.732268	−	0.681017i	$-0.238462\pi$
$264$	0	0
$265$	0	0
$266$	3.65924e8	1.19208
$267$	0	0
$268$	− 1.63749e8i	− 0.519645i
$269$	4.55386e8	1.42642	0.713209	−	0.700951i	$-0.247241\pi$
$269$	4.55386e8	1.42642	0.713209	+	0.700951i	$0.247241\pi$
$270$	0	0
$271$	5.08459e8	1.55190	0.775949	−	0.630796i	$-0.217271\pi$
$271$	5.08459e8	1.55190	0.775949	+	0.630796i	$0.217271\pi$
$272$	− 1.03502e8i	− 0.311858i
$273$	0	0
$274$	−7.75474e7	−0.227741
$275$	0	0
$276$	0	0
$277$	6.46827e8i	1.82856i	0.405085	+	0.914279i	$0.367242\pi$
$277$	6.46827e8i	1.82856i	−0.405085	+	0.914279i	$0.632758\pi$
$278$	3.59525e8i	1.00363i
$279$	0	0
$280$	0	0
$281$	−1.89546e8	−0.509617	−0.254808	−	0.966992i	$-0.582012\pi$
$281$	−1.89546e8	−0.509617	−0.254808	+	0.966992i	$0.582012\pi$
$282$	0	0
$283$	− 4.63384e8i	− 1.21531i	−0.794200	−	0.607657i	$-0.792110\pi$
$283$	− 4.63384e8i	− 1.21531i	0.794200	−	0.607657i	$-0.207890\pi$
$284$	−1.46395e8	−0.379239
$285$	0	0
$286$	−4.73748e7	−0.119747
$287$	− 3.09850e7i	− 0.0773686i
$288$	0	0
$289$	−2.28184e8	−0.556086
$290$	0	0
$291$	0	0
$292$	4.07836e8i	0.958619i
$293$	1.23173e8i	0.286075i	0.989717	+	0.143038i	$0.0456870\pi$
$293$	1.23173e8i	0.286075i	−0.989717	+	0.143038i	$0.954313\pi$
$294$	0	0
$295$	0	0
$296$	2.02714e8	0.454321
$297$	0	0
$298$	3.95366e8i	0.865452i
$299$	3.19222e7	0.0690627
$300$	0	0
$301$	−1.36802e8	−0.289141
$302$	− 1.60862e8i	− 0.336070i
$303$	0	0
$304$	−1.37155e8	−0.279997
$305$	0	0
$306$	0	0
$307$	5.64202e8i	1.11289i	0.830886	+	0.556443i	$0.187834\pi$
$307$	5.64202e8i	1.11289i	−0.830886	+	0.556443i	$0.812166\pi$
$308$	9.46802e7i	0.184642i
$309$	0	0
$310$	0	0
$311$	−5.51417e8	−1.03949	−0.519743	−	0.854323i	$-0.673972\pi$
$311$	−5.51417e8	−1.03949	−0.519743	+	0.854323i	$0.673972\pi$
$312$	0	0
$313$	1.71416e8i	0.315970i	0.987442	+	0.157985i	$0.0504997\pi$
$313$	1.71416e8i	0.315970i	−0.987442	+	0.157985i	$0.949500\pi$
$314$	6.29355e8	1.14721
$315$	0	0
$316$	−1.29232e8	−0.230391
$317$	1.03349e9i	1.82222i	0.412167	+	0.911108i	$0.364772\pi$
$317$	1.03349e9i	1.82222i	−0.412167	+	0.911108i	$0.635228\pi$
$318$	0	0
$319$	1.35678e8	0.234015
$320$	0	0
$321$	0	0
$322$	− 6.37977e7i	− 0.106490i
$323$	8.46132e8i	1.39711i
$324$	0	0
$325$	0	0
$326$	5.59799e8	0.894892
$327$	0	0
$328$	1.16137e7i	0.0181724i
$329$	−1.56440e9	−2.42194
$330$	0	0
$331$	2.41233e7	0.0365627	0.0182813	−	0.999833i	$-0.494181\pi$
$331$	2.41233e7	0.0365627	0.0182813	+	0.999833i	$0.494181\pi$
$332$	− 5.10271e8i	− 0.765275i
$333$	0	0
$334$	6.71134e8	0.985591
$335$	0	0
$336$	0	0
$337$	− 8.75080e8i	− 1.24550i	−0.782422	−	0.622749i	$-0.786016\pi$
$337$	− 8.75080e8i	− 1.24550i	0.782422	−	0.622749i	$-0.213984\pi$
$338$	− 2.62796e8i	− 0.370178i
$339$	0	0
$340$	0	0
$341$	−7.99232e7	−0.109152
$342$	0	0
$343$	2.98976e8i	0.400044i
$344$	5.12758e7	0.0679137
$345$	0	0
$346$	−2.18934e8	−0.284150
$347$	8.32449e7i	0.106956i	0.998569	+	0.0534779i	$0.0170307\pi$
$347$	8.32449e7i	0.106956i	−0.998569	+	0.0534779i	$0.982969\pi$
$348$	0	0
$349$	−3.97607e8	−0.500685	−0.250343	−	0.968157i	$-0.580543\pi$
$349$	−3.97607e8	−0.500685	−0.250343	+	0.968157i	$0.580543\pi$
$350$	0	0
$351$	0	0
$352$	− 3.54877e7i	− 0.0433690i
$353$	− 2.93710e8i	− 0.355392i	−0.984085	−	0.177696i	$-0.943136\pi$
$353$	− 2.93710e8i	− 0.355392i	0.984085	−	0.177696i	$-0.0568643\pi$
$354$	0	0
$355$	0	0
$356$	−1.39900e8	−0.164340
$357$	0	0
$358$	6.38107e8i	0.735027i
$359$	−1.71667e8	−0.195820	−0.0979098	−	0.995195i	$-0.531216\pi$
$359$	−1.71667e8	−0.195820	−0.0979098	+	0.995195i	$0.531216\pi$
$360$	0	0
$361$	2.27373e8	0.254369
$362$	− 4.22971e8i	− 0.468631i
$363$	0	0
$364$	4.78034e8	0.519522
$365$	0	0
$366$	0	0
$367$	− 9.55463e8i	− 1.00898i	−0.863417	−	0.504490i	$-0.831680\pi$
$367$	− 9.55463e8i	− 1.00898i	0.863417	−	0.504490i	$-0.168320\pi$
$368$	2.39124e7i	0.0250125i
$369$	0	0
$370$	0	0
$371$	−4.84769e8	−0.492863
$372$	0	0
$373$	− 3.44673e8i	− 0.343896i	−0.985106	−	0.171948i	$-0.944994\pi$
$373$	− 3.44673e8i	− 0.343896i	0.985106	−	0.171948i	$-0.0550061\pi$
$374$	−2.18931e8	−0.216399
$375$	0	0
$376$	5.86365e8	0.568867
$377$	− 6.85031e8i	− 0.658439i
$378$	0	0
$379$	−1.45273e9	−1.37071	−0.685356	−	0.728208i	$-0.740353\pi$
$379$	−1.45273e9	−1.37071	−0.685356	+	0.728208i	$0.740353\pi$
$380$	0	0
$381$	0	0
$382$	− 8.25803e8i	− 0.757974i
$383$	− 1.87839e9i	− 1.70840i	−0.519941	−	0.854202i	$-0.674046\pi$
$383$	− 1.87839e9i	− 1.70840i	0.519941	−	0.854202i	$-0.325954\pi$
$384$	0	0
$385$	0	0
$386$	3.99721e8	0.353754
$387$	0	0
$388$	3.72713e8i	0.323939i
$389$	−1.12632e9	−0.970145	−0.485072	−	0.874474i	$-0.661207\pi$
$389$	−1.12632e9	−0.970145	−0.485072	+	0.874474i	$0.661207\pi$
$390$	0	0
$391$	1.47520e8	0.124805
$392$	− 5.33715e8i	− 0.447516i
$393$	0	0
$394$	2.67085e8	0.219995
$395$	0	0
$396$	0	0
$397$	− 1.45733e9i	− 1.16894i	−0.811415	−	0.584470i	$-0.801302\pi$
$397$	− 1.45733e9i	− 1.16894i	0.811415	−	0.584470i	$-0.198698\pi$
$398$	7.40085e8i	0.588425i
$399$	0	0
$400$	0	0
$401$	−1.95849e9	−1.51676	−0.758380	−	0.651813i	$-0.774009\pi$
$401$	−1.95849e9	−1.51676	−0.758380	+	0.651813i	$0.774009\pi$
$402$	0	0
$403$	4.03527e8i	0.307118i
$404$	2.77160e7	0.0209120
$405$	0	0
$406$	−1.36906e9	−1.01527
$407$	− 4.28788e8i	− 0.315255i
$408$	0	0
$409$	1.25594e9	0.907687	0.453844	−	0.891081i	$-0.350052\pi$
$409$	1.25594e9	0.907687	0.453844	+	0.891081i	$0.350052\pi$
$410$	0	0
$411$	0	0
$412$	− 6.67487e8i	− 0.470222i
$413$	− 1.50036e9i	− 1.04802i
$414$	0	0
$415$	0	0
$416$	−1.79175e8	−0.122026
$417$	0	0
$418$	2.90114e8i	0.194291i
$419$	−4.46548e7	−0.0296564	−0.0148282	−	0.999890i	$-0.504720\pi$
$419$	−4.46548e7	−0.0296564	−0.0148282	+	0.999890i	$0.504720\pi$
$420$	0	0
$421$	−5.83121e7	−0.0380865	−0.0190433	−	0.999819i	$-0.506062\pi$
$421$	−5.83121e7	−0.0380865	−0.0190433	+	0.999819i	$0.506062\pi$
$422$	− 4.48923e8i	− 0.290789i
$423$	0	0
$424$	1.81700e8	0.115764
$425$	0	0
$426$	0	0
$427$	5.77815e8i	0.359163i
$428$	1.36425e9i	0.841086i
$429$	0	0
$430$	0	0
$431$	−2.15807e9	−1.29836	−0.649179	−	0.760635i	$-0.724888\pi$
$431$	−2.15807e9	−1.29836	−0.649179	+	0.760635i	$0.724888\pi$
$432$	0	0
$433$	− 2.02867e9i	− 1.20089i	−0.799666	−	0.600446i	$-0.794990\pi$
$433$	− 2.02867e9i	− 1.20089i	0.799666	−	0.600446i	$-0.205010\pi$
$434$	8.06465e8	0.473556
$435$	0	0
$436$	1.02595e9	0.592819
$437$	− 1.95485e8i	− 0.112055i
$438$	0	0
$439$	2.51641e9	1.41956	0.709782	−	0.704422i	$-0.248794\pi$
$439$	2.51641e9	1.41956	0.709782	+	0.704422i	$0.248794\pi$
$440$	0	0
$441$	0	0
$442$	1.10537e9i	0.608876i
$443$	− 2.56720e9i	− 1.40296i	−0.712687	−	0.701482i	$-0.752522\pi$
$443$	− 2.56720e9i	− 1.40296i	0.712687	−	0.701482i	$-0.247478\pi$
$444$	0	0
$445$	0	0
$446$	6.23949e8	0.333025
$447$	0	0
$448$	3.58089e8i	0.188156i
$449$	8.28489e8	0.431941	0.215970	−	0.976400i	$-0.430709\pi$
$449$	8.28489e8	0.431941	0.215970	+	0.976400i	$0.430709\pi$
$450$	0	0
$451$	2.45657e7	0.0126099
$452$	− 5.79496e7i	− 0.0295166i
$453$	0	0
$454$	−1.05074e9	−0.526987
$455$	0	0
$456$	0	0
$457$	3.45538e8i	0.169352i	0.996409	+	0.0846758i	$0.0269855\pi$
$457$	3.45538e8i	0.169352i	−0.996409	+	0.0846758i	$0.973015\pi$
$458$	1.82754e9i	0.888868i
$459$	0	0
$460$	0	0
$461$	−4.81628e8	−0.228959	−0.114480	−	0.993426i	$-0.536520\pi$
$461$	−4.81628e8	−0.228959	−0.114480	+	0.993426i	$0.536520\pi$
$462$	0	0
$463$	− 1.18307e9i	− 0.553958i	−0.960876	−	0.276979i	$-0.910667\pi$
$463$	− 1.18307e9i	− 0.553958i	0.960876	−	0.276979i	$-0.0893333\pi$
$464$	5.13147e8	0.238467
$465$	0	0
$466$	3.02359e9	1.38411
$467$	− 1.01191e9i	− 0.459764i	−0.973219	−	0.229882i	$-0.926166\pi$
$467$	− 1.01191e9i	− 0.459764i	0.973219	−	0.229882i	$-0.0738340\pi$
$468$	0	0
$469$	3.49502e9	1.56439
$470$	0	0
$471$	0	0
$472$	5.62360e8i	0.246160i
$473$	− 1.08460e8i	− 0.0471256i
$474$	0	0
$475$	0	0
$476$	2.20912e9	0.938846
$477$	0	0
$478$	9.05489e8i	0.379215i
$479$	3.78673e9	1.57431	0.787155	−	0.616755i	$-0.211553\pi$
$479$	3.78673e9	1.57431	0.787155	+	0.616755i	$0.211553\pi$
$480$	0	0
$481$	−2.16492e9	−0.887023
$482$	1.60189e9i	0.651579i
$483$	0	0
$484$	1.17211e9	0.469906
$485$	0	0
$486$	0	0
$487$	− 7.20259e8i	− 0.282577i	−0.989968	−	0.141289i	$-0.954875\pi$
$487$	− 7.20259e8i	− 0.282577i	0.989968	−	0.141289i	$-0.0451246\pi$
$488$	− 2.16575e8i	− 0.0843605i
$489$	0	0
$490$	0	0
$491$	1.79667e9	0.684988	0.342494	−	0.939520i	$-0.388728\pi$
$491$	1.79667e9	0.684988	0.342494	+	0.939520i	$0.388728\pi$
$492$	0	0
$493$	− 3.16570e9i	− 1.18989i
$494$	1.46477e9	0.546669
$495$	0	0
$496$	−3.02277e8	−0.111229
$497$	− 3.12463e9i	− 1.14170i
$498$	0	0
$499$	−2.54540e9	−0.917073	−0.458536	−	0.888676i	$-0.651626\pi$
$499$	−2.54540e9	−0.917073	−0.458536	+	0.888676i	$0.651626\pi$
$500$	0	0
$501$	0	0
$502$	− 3.70259e9i	− 1.30630i
$503$	− 1.31401e9i	− 0.460373i	−0.973147	−	0.230187i	$-0.926066\pi$
$503$	− 1.31401e9i	− 0.460373i	0.973147	−	0.230187i	$-0.0739336\pi$
$504$	0	0
$505$	0	0
$506$	5.05804e7	0.0173562
$507$	0	0
$508$	− 1.02625e9i	− 0.347321i
$509$	−2.50386e9	−0.841583	−0.420792	−	0.907157i	$-0.638248\pi$
$509$	−2.50386e9	−0.841583	−0.420792	+	0.907157i	$0.638248\pi$
$510$	0	0
$511$	−8.70476e9	−2.88591
$512$	− 1.34218e8i	− 0.0441942i
$513$	0	0
$514$	3.90201e9	1.26741
$515$	0	0
$516$	0	0
$517$	− 1.24030e9i	− 0.394739i
$518$	4.32668e9i	1.36773i
$519$	0	0
$520$	0	0
$521$	1.22028e9	0.378032	0.189016	−	0.981974i	$-0.439470\pi$
$521$	1.22028e9	0.378032	0.189016	+	0.981974i	$0.439470\pi$
$522$	0	0
$523$	− 1.93318e9i	− 0.590905i	−0.955357	−	0.295452i	$-0.904530\pi$
$523$	− 1.93318e9i	− 0.590905i	0.955357	−	0.295452i	$-0.0954703\pi$
$524$	2.45169e8	0.0744401
$525$	0	0
$526$	−3.21457e9	−0.963103
$527$	1.86480e9i	0.555003i
$528$	0	0
$529$	3.37074e9	0.989990
$530$	0	0
$531$	0	0
$532$	− 2.92739e9i	− 0.842928i
$533$	− 1.24031e8i	− 0.0354800i
$534$	0	0
$535$	0	0
$536$	−1.30999e9	−0.367445
$537$	0	0
$538$	− 3.64309e9i	− 1.00863i
$539$	−1.12893e9	−0.310533
$540$	0	0
$541$	−2.54920e9	−0.692172	−0.346086	−	0.938203i	$-0.612489\pi$
$541$	−2.54920e9	−0.692172	−0.346086	+	0.938203i	$0.612489\pi$
$542$	− 4.06767e9i	− 1.09736i
$543$	0	0
$544$	−8.28015e8	−0.220517
$545$	0	0
$546$	0	0
$547$	4.58291e9i	1.19725i	0.801028	+	0.598626i	$0.204287\pi$
$547$	4.58291e9i	1.19725i	−0.801028	+	0.598626i	$0.795713\pi$
$548$	6.20379e8i	0.161037i
$549$	0	0
$550$	0	0
$551$	−4.19500e9	−1.06832
$552$	0	0
$553$	− 2.75830e9i	− 0.693590i
$554$	5.17461e9	1.29299
$555$	0	0
$556$	2.87620e9	0.709671
$557$	4.94207e9i	1.21176i	0.795557	+	0.605879i	$0.207178\pi$
$557$	4.94207e9i	1.21176i	−0.795557	+	0.605879i	$0.792822\pi$
$558$	0	0
$559$	−5.47609e8	−0.132596
$560$	0	0
$561$	0	0
$562$	1.51637e9i	0.360353i
$563$	7.82258e8i	0.184744i	0.995725	+	0.0923721i	$0.0294449\pi$
$563$	7.82258e8i	0.184744i	−0.995725	+	0.0923721i	$0.970555\pi$
$564$	0	0
$565$	0	0
$566$	−3.70707e9	−0.859356
$567$	0	0
$568$	1.17116e9i	0.268163i
$569$	1.19628e9	0.272232	0.136116	−	0.990693i	$-0.456538\pi$
$569$	1.19628e9	0.272232	0.136116	+	0.990693i	$0.456538\pi$
$570$	0	0
$571$	2.41071e9	0.541899	0.270949	−	0.962594i	$-0.412662\pi$
$571$	2.41071e9	0.541899	0.270949	+	0.962594i	$0.412662\pi$
$572$	3.78998e8i	0.0846741i
$573$	0	0
$574$	−2.47880e8	−0.0547078
$575$	0	0
$576$	0	0
$577$	− 6.45394e9i	− 1.39865i	−0.714803	−	0.699326i	$-0.753484\pi$
$577$	− 6.45394e9i	− 1.39865i	0.714803	−	0.699326i	$-0.246516\pi$
$578$	1.82547e9i	0.393212i
$579$	0	0
$580$	0	0
$581$	1.08911e10	2.30385
$582$	0	0
$583$	− 3.84337e8i	− 0.0803290i
$584$	3.26269e9	0.677846
$585$	0	0
$586$	9.85387e8	0.202286
$587$	− 5.27194e8i	− 0.107581i	−0.998552	−	0.0537907i	$-0.982870\pi$
$587$	− 5.27194e8i	− 0.107581i	0.998552	−	0.0537907i	$-0.0171304\pi$
$588$	0	0
$589$	2.47113e9	0.498301
$590$	0	0
$591$	0	0
$592$	− 1.62171e9i	− 0.321254i
$593$	− 8.99056e9i	− 1.77050i	−0.465117	−	0.885249i	$-0.653988\pi$
$593$	− 8.99056e9i	− 1.77050i	0.465117	−	0.885249i	$-0.346012\pi$
$594$	0	0
$595$	0	0
$596$	3.16293e9	0.611967
$597$	0	0
$598$	− 2.55377e8i	− 0.0488347i
$599$	3.40720e9	0.647745	0.323872	−	0.946101i	$-0.395015\pi$
$599$	3.40720e9	0.647745	0.323872	+	0.946101i	$0.395015\pi$
$600$	0	0
$601$	−4.00808e9	−0.753140	−0.376570	−	0.926388i	$-0.622897\pi$
$601$	−4.00808e9	−0.753140	−0.376570	+	0.926388i	$0.622897\pi$
$602$	1.09442e9i	0.204454i
$603$	0	0
$604$	−1.28690e9	−0.237638
$605$	0	0
$606$	0	0
$607$	3.61151e9i	0.655434i	0.944776	+	0.327717i	$0.106279\pi$
$607$	3.61151e9i	0.655434i	−0.944776	+	0.327717i	$0.893721\pi$
$608$	1.09724e9i	0.197987i
$609$	0	0
$610$	0	0
$611$	−6.26219e9	−1.11066
$612$	0	0
$613$	− 4.16268e8i	− 0.0729897i	−0.999334	−	0.0364948i	$-0.988381\pi$
$613$	− 4.16268e8i	− 0.0729897i	0.999334	−	0.0364948i	$-0.0116192\pi$
$614$	4.51362e9	0.786929
$615$	0	0
$616$	7.57442e8	0.130562
$617$	2.32066e8i	0.0397752i	0.999802	+	0.0198876i	$0.00633084\pi$
$617$	2.32066e8i	0.0397752i	−0.999802	+	0.0198876i	$0.993669\pi$
$618$	0	0
$619$	1.50914e9	0.255748	0.127874	−	0.991790i	$-0.459185\pi$
$619$	1.50914e9	0.255748	0.127874	+	0.991790i	$0.459185\pi$
$620$	0	0
$621$	0	0
$622$	4.41133e9i	0.735027i
$623$	− 2.98599e9i	− 0.494743i
$624$	0	0
$625$	0	0
$626$	1.37133e9	0.223424
$627$	0	0
$628$	− 5.03484e9i	− 0.811199i
$629$	−1.00047e10	−1.60297
$630$	0	0
$631$	1.18512e10	1.87785	0.938923	−	0.344127i	$-0.111825\pi$
$631$	1.18512e10	1.87785	0.938923	+	0.344127i	$0.111825\pi$
$632$	1.03386e9i	0.162911i
$633$	0	0
$634$	8.26794e9	1.28850
$635$	0	0
$636$	0	0
$637$	5.69991e9i	0.873736i
$638$	− 1.08543e9i	− 0.165473i
$639$	0	0
$640$	0	0
$641$	3.89475e9	0.584086	0.292043	−	0.956405i	$-0.405665\pi$
$641$	3.89475e9	0.584086	0.292043	+	0.956405i	$0.405665\pi$
$642$	0	0
$643$	6.87187e8i	0.101938i	0.998700	+	0.0509690i	$0.0162310\pi$
$643$	6.87187e8i	0.101938i	−0.998700	+	0.0509690i	$0.983769\pi$
$644$	−5.10381e8	−0.0752999
$645$	0	0
$646$	6.76906e9	0.987904
$647$	2.24157e9i	0.325378i	0.986677	+	0.162689i	$0.0520167\pi$
$647$	2.24157e9i	0.325378i	−0.986677	+	0.162689i	$0.947983\pi$
$648$	0	0
$649$	1.18952e9	0.170811
$650$	0	0
$651$	0	0
$652$	− 4.47839e9i	− 0.632784i
$653$	3.44569e9i	0.484261i	0.970244	+	0.242131i	$0.0778463\pi$
$653$	3.44569e9i	0.484261i	−0.970244	+	0.242131i	$0.922154\pi$
$654$	0	0
$655$	0	0
$656$	9.29096e7	0.0128498
$657$	0	0
$658$	1.25152e10i	1.71257i
$659$	7.59753e9	1.03413	0.517063	−	0.855947i	$-0.327025\pi$
$659$	7.59753e9	1.03413	0.517063	+	0.855947i	$0.327025\pi$
$660$	0	0
$661$	−5.49901e8	−0.0740592	−0.0370296	−	0.999314i	$-0.511790\pi$
$661$	−5.49901e8	−0.0740592	−0.0370296	+	0.999314i	$0.511790\pi$
$662$	− 1.92986e8i	− 0.0258537i
$663$	0	0
$664$	−4.08217e9	−0.541131
$665$	0	0
$666$	0	0
$667$	7.31385e8i	0.0954345i
$668$	− 5.36907e9i	− 0.696918i
$669$	0	0
$670$	0	0
$671$	−4.58107e8	−0.0585380
$672$	0	0
$673$	− 7.29473e7i	− 0.00922478i	−0.999989	−	0.00461239i	$-0.998532\pi$
$673$	− 7.29473e7i	− 0.00922478i	0.999989	−	0.00461239i	$-0.00146818\pi$
$674$	−7.00064e9	−0.880700
$675$	0	0
$676$	−2.10237e9	−0.261755
$677$	3.97003e9i	0.491738i	0.969303	+	0.245869i	$0.0790733\pi$
$677$	3.97003e9i	0.491738i	−0.969303	+	0.245869i	$0.920927\pi$
$678$	0	0
$679$	−7.95510e9	−0.975217
$680$	0	0
$681$	0	0
$682$	6.39386e8i	0.0771823i
$683$	− 1.18943e10i	− 1.42845i	−0.699914	−	0.714227i	$-0.746779\pi$
$683$	− 1.18943e10i	− 1.42845i	0.699914	−	0.714227i	$-0.253221\pi$
$684$	0	0
$685$	0	0
$686$	2.39181e9	0.282874
$687$	0	0
$688$	− 4.10206e8i	− 0.0480223i
$689$	−1.94049e9	−0.226019
$690$	0	0
$691$	2.08948e9	0.240916	0.120458	−	0.992718i	$-0.461564\pi$
$691$	2.08948e9	0.240916	0.120458	+	0.992718i	$0.461564\pi$
$692$	1.75147e9i	0.200924i
$693$	0	0
$694$	6.65959e8	0.0756292
$695$	0	0
$696$	0	0
$697$	− 5.73177e8i	− 0.0641171i
$698$	3.18085e9i	0.354038i
$699$	0	0
$700$	0	0
$701$	−1.21818e10	−1.33567	−0.667834	−	0.744311i	$-0.732778\pi$
$701$	−1.21818e10	−1.33567	−0.667834	+	0.744311i	$0.732778\pi$
$702$	0	0
$703$	1.32576e10i	1.43920i
$704$	−2.83902e8	−0.0306665
$705$	0	0
$706$	−2.34968e9	−0.251300
$707$	5.91563e8i	0.0629554i
$708$	0	0
$709$	−3.80726e9	−0.401191	−0.200595	−	0.979674i	$-0.564288\pi$
$709$	−3.80726e9	−0.401191	−0.200595	+	0.979674i	$0.564288\pi$
$710$	0	0
$711$	0	0
$712$	1.11920e9i	0.116206i
$713$	− 4.30833e8i	− 0.0445139i
$714$	0	0
$715$	0	0
$716$	5.10486e9	0.519742
$717$	0	0
$718$	1.37334e9i	0.138465i
$719$	2.74426e9	0.275343	0.137671	−	0.990478i	$-0.456038\pi$
$719$	2.74426e9	0.275343	0.137671	+	0.990478i	$0.456038\pi$
$720$	0	0
$721$	1.42467e10	1.41560
$722$	− 1.81899e9i	− 0.179866i
$723$	0	0
$724$	−3.38377e9	−0.331372
$725$	0	0
$726$	0	0
$727$	5.15574e8i	0.0497647i	0.999690	+	0.0248823i	$0.00792111\pi$
$727$	5.15574e8i	0.0497647i	−0.999690	+	0.0248823i	$0.992079\pi$
$728$	− 3.82428e9i	− 0.367358i
$729$	0	0
$730$	0	0
$731$	−2.53064e9	−0.239618
$732$	0	0
$733$	− 9.16791e9i	− 0.859818i	−0.902872	−	0.429909i	$-0.858546\pi$
$733$	− 9.16791e9i	− 0.859818i	0.902872	−	0.429909i	$-0.141454\pi$
$734$	−7.64371e9	−0.713457
$735$	0	0
$736$	1.91300e8	0.0176865
$737$	2.77094e9i	0.254971i
$738$	0	0
$739$	7.95134e9	0.724744	0.362372	−	0.932034i	$-0.381967\pi$
$739$	7.95134e9	0.724744	0.362372	+	0.932034i	$0.381967\pi$
$740$	0	0
$741$	0	0
$742$	3.87815e9i	0.348507i
$743$	1.20333e10i	1.07628i	0.842856	+	0.538139i	$0.180872\pi$
$743$	1.20333e10i	1.07628i	−0.842856	+	0.538139i	$0.819128\pi$
$744$	0	0
$745$	0	0
$746$	−2.75739e9	−0.243171
$747$	0	0
$748$	1.75144e9i	0.153017i
$749$	−2.91182e10	−2.53208
$750$	0	0
$751$	−1.46650e10	−1.26341	−0.631704	−	0.775210i	$-0.717644\pi$
$751$	−1.46650e10	−1.26341	−0.631704	+	0.775210i	$0.717644\pi$
$752$	− 4.69092e9i	− 0.402250i
$753$	0	0
$754$	−5.48025e9	−0.465587
$755$	0	0
$756$	0	0
$757$	− 1.86757e10i	− 1.56474i	−0.622816	−	0.782368i	$-0.714012\pi$
$757$	− 1.86757e10i	− 1.56474i	0.622816	−	0.782368i	$-0.285988\pi$
$758$	1.16218e10i	0.969240i
$759$	0	0
$760$	0	0
$761$	−1.45502e10	−1.19680	−0.598400	−	0.801198i	$-0.704197\pi$
$761$	−1.45502e10	−1.19680	−0.598400	+	0.801198i	$0.704197\pi$
$762$	0	0
$763$	2.18976e10i	1.78468i
$764$	−6.60642e9	−0.535969
$765$	0	0
$766$	−1.50271e10	−1.20802
$767$	− 6.00583e9i	− 0.480606i
$768$	0	0
$769$	−7.45506e9	−0.591166	−0.295583	−	0.955317i	$-0.595514\pi$
$769$	−7.45506e9	−0.591166	−0.295583	+	0.955317i	$0.595514\pi$
$770$	0	0
$771$	0	0
$772$	− 3.19776e9i	− 0.250142i
$773$	1.85671e10i	1.44582i	0.690941	+	0.722911i	$0.257197\pi$
$773$	1.85671e10i	1.44582i	−0.690941	+	0.722911i	$0.742803\pi$
$774$	0	0
$775$	0	0
$776$	2.98171e9	0.229060
$777$	0	0
$778$	9.01053e9i	0.685996i
$779$	−7.59540e8	−0.0575665
$780$	0	0
$781$	2.47728e9	0.186079
$782$	− 1.18016e9i	− 0.0882508i
$783$	0	0
$784$	−4.26972e9	−0.316442
$785$	0	0
$786$	0	0
$787$	− 2.17240e8i	− 0.0158865i	−0.999968	−	0.00794324i	$-0.997472\pi$
$787$	− 2.17240e8i	− 0.0158865i	0.999968	−	0.00794324i	$-0.00252844\pi$
$788$	− 2.13668e9i	− 0.155560i
$789$	0	0
$790$	0	0
$791$	1.23686e9	0.0888595
$792$	0	0
$793$	2.31295e9i	0.164706i
$794$	−1.16587e10	−0.826566
$795$	0	0
$796$	5.92068e9	0.416079
$797$	− 1.04021e10i	− 0.727807i	−0.931437	−	0.363903i	$-0.881444\pi$
$797$	− 1.04021e10i	− 0.727807i	0.931437	−	0.363903i	$-0.118556\pi$
$798$	0	0
$799$	−2.89392e10	−2.00712
$800$	0	0
$801$	0	0
$802$	1.56679e10i	1.07251i
$803$	− 6.90136e9i	− 0.470360i
$804$	0	0
$805$	0	0
$806$	3.22822e9	0.217165
$807$	0	0
$808$	− 2.21728e8i	− 0.0147870i
$809$	1.54020e10	1.02272	0.511360	−	0.859367i	$-0.329142\pi$
$809$	1.54020e10	1.02272	0.511360	+	0.859367i	$0.329142\pi$
$810$	0	0
$811$	9.37914e9	0.617433	0.308717	−	0.951154i	$-0.400101\pi$
$811$	9.37914e9	0.617433	0.308717	+	0.951154i	$0.400101\pi$
$812$	1.09525e10i	0.717904i
$813$	0	0
$814$	−3.43030e9	−0.222919
$815$	0	0
$816$	0	0
$817$	3.35346e9i	0.215137i
$818$	− 1.00475e10i	− 0.641832i
$819$	0	0
$820$	0	0
$821$	1.49190e10	0.940887	0.470443	−	0.882430i	$-0.344094\pi$
$821$	1.49190e10	0.940887	0.470443	+	0.882430i	$0.344094\pi$
$822$	0	0
$823$	− 1.72905e10i	− 1.08121i	−0.841278	−	0.540603i	$-0.818196\pi$
$823$	− 1.72905e10i	− 1.08121i	0.841278	−	0.540603i	$-0.181804\pi$
$824$	−5.33990e9	−0.332497
$825$	0	0
$826$	−1.20029e10	−0.741063
$827$	1.92633e8i	0.0118430i	0.999982	+	0.00592149i	$0.00188488\pi$
$827$	1.92633e8i	0.0118430i	−0.999982	+	0.00592149i	$0.998115\pi$
$828$	0	0
$829$	−1.55199e10	−0.946124	−0.473062	−	0.881029i	$-0.656851\pi$
$829$	−1.55199e10	−0.946124	−0.473062	+	0.881029i	$0.656851\pi$
$830$	0	0
$831$	0	0
$832$	1.43340e9i	0.0862853i
$833$	2.63407e10i	1.57896i
$834$	0	0
$835$	0	0
$836$	2.32091e9	0.137384
$837$	0	0
$838$	3.57238e8i	0.0209703i
$839$	1.04108e10	0.608580	0.304290	−	0.952579i	$-0.401581\pi$
$839$	1.04108e10	0.608580	0.304290	+	0.952579i	$0.401581\pi$
$840$	0	0
$841$	−1.55480e9	−0.0901339
$842$	4.66497e8i	0.0269312i
$843$	0	0
$844$	−3.59138e9	−0.205619
$845$	0	0
$846$	0	0
$847$	2.50173e10i	1.41465i
$848$	− 1.45360e9i	− 0.0818575i
$849$	0	0
$850$	0	0
$851$	2.31142e9	0.128566
$852$	0	0
$853$	1.54506e10i	0.852362i	0.904638	+	0.426181i	$0.140141\pi$
$853$	1.54506e10i	0.852362i	−0.904638	+	0.426181i	$0.859859\pi$
$854$	4.62252e9	0.253967
$855$	0	0
$856$	1.09140e10	0.594738
$857$	− 2.50311e9i	− 0.135846i	−0.997691	−	0.0679231i	$-0.978363\pi$
$857$	− 2.50311e9i	− 0.135846i	0.997691	−	0.0679231i	$-0.0216373\pi$
$858$	0	0
$859$	−2.05842e10	−1.10805	−0.554024	−	0.832500i	$-0.686909\pi$
$859$	−2.05842e10	−1.10805	−0.554024	+	0.832500i	$0.686909\pi$
$860$	0	0
$861$	0	0
$862$	1.72645e10i	0.918078i
$863$	1.91073e10i	1.01196i	0.862546	+	0.505978i	$0.168868\pi$
$863$	1.91073e10i	1.01196i	−0.862546	+	0.505978i	$0.831132\pi$
$864$	0	0
$865$	0	0
$866$	−1.62294e10	−0.849158
$867$	0	0
$868$	− 6.45172e9i	− 0.334855i
$869$	2.18685e9	0.113044
$870$	0	0
$871$	1.39903e10	0.717404
$872$	− 8.20758e9i	− 0.419187i
$873$	0	0
$874$	−1.56388e9	−0.0792345
$875$	0	0
$876$	0	0
$877$	3.33063e10i	1.66736i	0.552251	+	0.833678i	$0.313769\pi$
$877$	3.33063e10i	1.66736i	−0.552251	+	0.833678i	$0.686231\pi$
$878$	− 2.01312e10i	− 1.00378i
$879$	0	0
$880$	0	0
$881$	9.31313e9	0.458860	0.229430	−	0.973325i	$-0.426314\pi$
$881$	9.31313e9	0.458860	0.229430	+	0.973325i	$0.426314\pi$
$882$	0	0
$883$	− 4.38514e9i	− 0.214349i	−0.994240	−	0.107174i	$-0.965820\pi$
$883$	− 4.38514e9i	− 0.214349i	0.994240	−	0.107174i	$-0.0341803\pi$
$884$	8.84294e9	0.430540
$885$	0	0
$886$	−2.05376e10	−0.992045
$887$	1.78764e10i	0.860098i	0.902806	+	0.430049i	$0.141504\pi$
$887$	1.78764e10i	0.860098i	−0.902806	+	0.430049i	$0.858496\pi$
$888$	0	0
$889$	2.19041e10	1.04561
$890$	0	0
$891$	0	0
$892$	− 4.99159e9i	− 0.235484i
$893$	3.83485e10i	1.80206i
$894$	0	0
$895$	0	0
$896$	2.86471e9	0.133046
$897$	0	0
$898$	− 6.62791e9i	− 0.305428i
$899$	−9.24541e9	−0.424392
$900$	0	0
$901$	−8.96751e9	−0.408447
$902$	− 1.96526e8i	− 0.00891653i
$903$	0	0
$904$	−4.63597e8	−0.0208714
$905$	0	0
$906$	0	0
$907$	− 2.84742e9i	− 0.126714i	−0.997991	−	0.0633572i	$-0.979819\pi$
$907$	− 2.84742e9i	− 0.126714i	0.997991	−	0.0633572i	$-0.0201807\pi$
$908$	8.40592e9i	0.372636i
$909$	0	0
$910$	0	0
$911$	3.84643e10	1.68556	0.842779	−	0.538260i	$-0.180918\pi$
$911$	3.84643e10	1.68556	0.842779	+	0.538260i	$0.180918\pi$
$912$	0	0
$913$	8.63474e9i	0.375493i
$914$	2.76431e9	0.119750
$915$	0	0
$916$	1.46203e10	0.628525
$917$	5.23283e9i	0.224101i
$918$	0	0
$919$	−1.07375e10	−0.456352	−0.228176	−	0.973620i	$-0.573276\pi$
$919$	−1.07375e10	−0.456352	−0.228176	+	0.973620i	$0.573276\pi$
$920$	0	0
$921$	0	0
$922$	3.85303e9i	0.161899i
$923$	− 1.25077e10i	− 0.523564i
$924$	0	0
$925$	0	0
$926$	−9.46456e9	−0.391708
$927$	0	0
$928$	− 4.10518e9i	− 0.168622i
$929$	5.34493e9	0.218719	0.109360	−	0.994002i	$-0.465120\pi$
$929$	5.34493e9	0.218719	0.109360	+	0.994002i	$0.465120\pi$
$930$	0	0
$931$	3.49052e10	1.41764
$932$	− 2.41887e10i	− 0.978717i
$933$	0	0
$934$	−8.09531e9	−0.325102
$935$	0	0
$936$	0	0
$937$	− 1.48125e10i	− 0.588218i	−0.955772	−	0.294109i	$-0.904977\pi$
$937$	− 1.48125e10i	− 0.588218i	0.955772	−	0.294109i	$-0.0950229\pi$
$938$	− 2.79602e10i	− 1.10619i
$939$	0	0
$940$	0	0
$941$	−2.18734e10	−0.855763	−0.427882	−	0.903835i	$-0.640740\pi$
$941$	−2.18734e10	−0.855763	−0.427882	+	0.903835i	$0.640740\pi$
$942$	0	0
$943$	1.32423e8i	0.00514249i
$944$	4.49888e9	0.174061
$945$	0	0
$946$	−8.67682e8	−0.0333228
$947$	− 2.45163e10i	− 0.938060i	−0.883182	−	0.469030i	$-0.844604\pi$
$947$	− 2.45163e10i	− 0.938060i	0.883182	−	0.469030i	$-0.155396\pi$
$948$	0	0
$949$	−3.48445e10	−1.32344
$950$	0	0
$951$	0	0
$952$	− 1.76729e10i	− 0.663864i
$953$	1.34056e10i	0.501721i	0.968023	+	0.250860i	$0.0807135\pi$
$953$	1.34056e10i	0.501721i	−0.968023	+	0.250860i	$0.919286\pi$
$954$	0	0
$955$	0	0
$956$	7.24391e9	0.268146
$957$	0	0
$958$	− 3.02939e10i	− 1.11321i
$959$	−1.32412e10	−0.484800
$960$	0	0
$961$	−2.20665e10	−0.802049
$962$	1.73194e10i	0.627220i
$963$	0	0
$964$	1.28151e10	0.460736
$965$	0	0
$966$	0	0
$967$	− 4.64740e9i	− 0.165279i	−0.996580	−	0.0826395i	$-0.973665\pi$
$967$	− 4.64740e9i	− 0.165279i	0.996580	−	0.0826395i	$-0.0263350\pi$
$968$	− 9.37691e9i	− 0.332274i
$969$	0	0
$970$	0	0
$971$	3.24860e10	1.13875	0.569375	−	0.822078i	$-0.307185\pi$
$971$	3.24860e10	1.13875	0.569375	+	0.822078i	$0.307185\pi$
$972$	0	0
$973$	6.13889e10i	2.13646i
$974$	−5.76207e9	−0.199812
$975$	0	0
$976$	−1.73260e9	−0.0596518
$977$	6.06506e9i	0.208067i	0.994574	+	0.104034i	$0.0331750\pi$
$977$	6.06506e9i	0.208067i	−0.994574	+	0.104034i	$0.966825\pi$
$978$	0	0
$979$	2.36737e9	0.0806355
$980$	0	0
$981$	0	0
$982$	− 1.43734e10i	− 0.484360i
$983$	− 2.10842e10i	− 0.707979i	−0.935249	−	0.353989i	$-0.884825\pi$
$983$	− 2.10842e10i	− 0.707979i	0.935249	−	0.353989i	$-0.115175\pi$
$984$	0	0
$985$	0	0
$986$	−2.53256e10	−0.841377
$987$	0	0
$988$	− 1.17181e10i	− 0.386553i
$989$	5.84664e8	0.0192185
$990$	0	0
$991$	4.26476e10	1.39199	0.695997	−	0.718045i	$-0.254963\pi$
$991$	4.26476e10	1.39199	0.695997	+	0.718045i	$0.254963\pi$
$992$	2.41821e9i	0.0786509i
$993$	0	0
$994$	−2.49970e10	−0.807302
$995$	0	0
$996$	0	0
$997$	− 1.59226e10i	− 0.508841i	−0.967094	−	0.254420i	$-0.918115\pi$
$997$	− 1.59226e10i	− 0.508841i	0.967094	−	0.254420i	$-0.0818847\pi$
$998$	2.03632e10i	0.648468i
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	450.8.c.l.199.1		2
3.2	odd	2		50.8.b.a.49.2		2
5.2	odd	4		450.8.a.z.1.1		1
5.3	odd	4		450.8.a.a.1.1		1
5.4	even	2	inner	450.8.c.l.199.2		2
12.11	even	2		400.8.c.a.49.1		2
15.2	even	4		50.8.a.d.1.1	✓	1
15.8	even	4		50.8.a.e.1.1	yes	1
15.14	odd	2		50.8.b.a.49.1		2
60.23	odd	4		400.8.a.s.1.1		1
60.47	odd	4		400.8.a.a.1.1		1
60.59	even	2		400.8.c.a.49.2		2

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
50.8.a.d.1.1	✓	1	15.2	even	4
50.8.a.e.1.1	yes	1	15.8	even	4
50.8.b.a.49.1		2	15.14	odd	2
50.8.b.a.49.2		2	3.2	odd	2
400.8.a.a.1.1		1	60.47	odd	4
400.8.a.s.1.1		1	60.23	odd	4
400.8.c.a.49.1		2	12.11	even	2
400.8.c.a.49.2		2	60.59	even	2
450.8.a.a.1.1		1	5.3	odd	4
450.8.a.z.1.1		1	5.2	odd	4
450.8.c.l.199.1		2	1.1	even	1	trivial
450.8.c.l.199.2		2	5.4	even	2	inner

Overview	Random
Universe	Knowledge

Classical	Maass
Hilbert	Bianchi

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

Level:	\( N \)	\(=\)	\( 450 = 2 \cdot 3^{2} \cdot 5^{2} \)
Weight:	\( k \)	\(=\)	\( 8 \)
Character orbit:	\([\chi]\)	\(=\)	450.c (of order \(2\), degree \(1\), not minimal)

\(f(q)\)	\(=\)	\(q-8.00000i q^{2} -64.0000 q^{4} -1366.00i q^{7} +512.000i q^{8} +1083.00 q^{11} -5468.00i q^{13} -10928.0 q^{14} +4096.00 q^{16} -25269.0i q^{17} -33485.0 q^{19} -8664.00i q^{22} +5838.00i q^{23} -43744.0 q^{26} +87424.0i q^{28} +125280. q^{29} -73798.0 q^{31} -32768.0i q^{32} -202152. q^{34} -395926. i q^{37} +267880. i q^{38} +22683.0 q^{41} -100148. i q^{43} -69312.0 q^{44} +46704.0 q^{46} -1.14524e6i q^{47} -1.04241e6 q^{49} +349952. i q^{52} -354882. i q^{53} +699392. q^{56} -1.00224e6i q^{58} +1.09836e6 q^{59} -422998. q^{61} +590384. i q^{62} -262144. q^{64} +2.55858e6i q^{67} +1.61722e6i q^{68} +2.28743e6 q^{71} -6.37244e6i q^{73} -3.16741e6 q^{74} +2.14304e6 q^{76} -1.47938e6i q^{77} +2.01925e6 q^{79} -181464. i q^{82} +7.97298e6i q^{83} -801184. q^{86} +554496. i q^{88} +2.18594e6 q^{89} -7.46929e6 q^{91} -373632. i q^{92} -9.16195e6 q^{94} -5.82365e6i q^{97} +8.33930e6i q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 2 q - 128 q^{4} + 2166 q^{11} - 21856 q^{14} + 8192 q^{16} - 66970 q^{19} - 87488 q^{26} + 250560 q^{29} - 147596 q^{31} - 404304 q^{34} + 45366 q^{41} - 138624 q^{44} + 93408 q^{46} - 2084826 q^{49} + 1398784 q^{56}+ \cdots - 18323904 q^{94}+O(q^{100}) \) 2 * q - 128 * q^4 + 2166 * q^11 - 21856 * q^14 + 8192 * q^16 - 66970 * q^19 - 87488 * q^26 + 250560 * q^29 - 147596 * q^31 - 404304 * q^34 + 45366 * q^41 - 138624 * q^44 + 93408 * q^46 - 2084826 * q^49 + 1398784 * q^56 + 2196720 * q^59 - 845996 * q^61 - 524288 * q^64 + 4574856 * q^71 - 6334816 * q^74 + 4286080 * q^76 + 4038500 * q^79 - 1602368 * q^86 + 4371870 * q^89 - 14938576 * q^91 - 18323904 * q^94

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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