LMFDB - Embedded newform 450.4.c.e.199.1

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [450,4,Mod(199,450)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(450, base_ring=CyclotomicField(2))

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

N = Newforms(chi, 4, names="a")

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

chi := DirichletCharacter("450.199");

S:= CuspForms(chi, 4);

N := Newforms(S);

Level:	$ N $	$=$	$ 450 = 2 \cdot 3^{2} \cdot 5^{2} $
Weight:	$ k $	$=$	$ 4 $
Character orbit:	$[\chi]$	$=$	450.c (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:	$26.5508595026$
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:	$\Z[a_1, a_2]$
Coefficient ring index:	$ 2 $
Twist minimal:	no (minimal twist has level 6)
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			450.4.c.e.199.2

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-2.00000i q^{2} -4.00000 q^{4} +16.0000i q^{7} +8.00000i q^{8} +O(q^{10})$	$q-2.00000i q^{2} -4.00000 q^{4} +16.0000i q^{7} +8.00000i q^{8} -12.0000 q^{11} +38.0000i q^{13} +32.0000 q^{14} +16.0000 q^{16} -126.000i q^{17} -20.0000 q^{19} +24.0000i q^{22} -168.000i q^{23} +76.0000 q^{26} -64.0000i q^{28} +30.0000 q^{29} -88.0000 q^{31} -32.0000i q^{32} -252.000 q^{34} -254.000i q^{37} +40.0000i q^{38} -42.0000 q^{41} -52.0000i q^{43} +48.0000 q^{44} -336.000 q^{46} -96.0000i q^{47} +87.0000 q^{49} -152.000i q^{52} -198.000i q^{53} -128.000 q^{56} -60.0000i q^{58} -660.000 q^{59} -538.000 q^{61} +176.000i q^{62} -64.0000 q^{64} -884.000i q^{67} +504.000i q^{68} -792.000 q^{71} +218.000i q^{73} -508.000 q^{74} +80.0000 q^{76} -192.000i q^{77} +520.000 q^{79} +84.0000i q^{82} +492.000i q^{83} -104.000 q^{86} -96.0000i q^{88} +810.000 q^{89} -608.000 q^{91} +672.000i q^{92} -192.000 q^{94} -1154.00i q^{97} -174.000i q^{98} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 2 q - 8 q^{4}+O(q^{10}) $ 2 * q - 8 * q^4	$ 2 q - 8 q^{4} - 24 q^{11} + 64 q^{14} + 32 q^{16} - 40 q^{19} + 152 q^{26} + 60 q^{29} - 176 q^{31} - 504 q^{34} - 84 q^{41} + 96 q^{44} - 672 q^{46} + 174 q^{49} - 256 q^{56} - 1320 q^{59} - 1076 q^{61} - 128 q^{64} - 1584 q^{71} - 1016 q^{74} + 160 q^{76} + 1040 q^{79} - 208 q^{86} + 1620 q^{89} - 1216 q^{91} - 384 q^{94}+O(q^{100}) $ 2 * q - 8 * q^4 - 24 * q^11 + 64 * q^14 + 32 * q^16 - 40 * q^19 + 152 * q^26 + 60 * q^29 - 176 * q^31 - 504 * q^34 - 84 * q^41 + 96 * q^44 - 672 * q^46 + 174 * q^49 - 256 * q^56 - 1320 * q^59 - 1076 * q^61 - 128 * q^64 - 1584 * q^71 - 1016 * q^74 + 160 * q^76 + 1040 * q^79 - 208 * q^86 + 1620 * q^89 - 1216 * q^91 - 384 * q^94

Display 100 coefficients 10 coefficients

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 $

$2$	− 2.00000i	− 0.707107i
$2$	− 2.00000i	− 0.707107i
$3$	0	0
$3$	0	0
$4$	−4.00000	−0.500000
$5$	0	0
$5$	0	0
$6$	0	0
$7$	16.0000i	0.863919i	0.901893	+	0.431959i	$0.142178\pi$
$7$	16.0000i	0.863919i	−0.901893	+	0.431959i	$0.857822\pi$
$8$	8.00000i	0.353553i
$9$	0	0
$10$	0	0
$11$	−12.0000	−0.328921	−0.164461	−	0.986384i	$-0.552588\pi$
$11$	−12.0000	−0.328921	−0.164461	+	0.986384i	$0.552588\pi$
$12$	0	0
$13$	38.0000i	0.810716i	0.914158	+	0.405358i	$0.132853\pi$
$13$	38.0000i	0.810716i	−0.914158	+	0.405358i	$0.867147\pi$
$14$	32.0000	0.610883
$15$	0	0
$16$	16.0000	0.250000
$17$	− 126.000i	− 1.79762i	−0.438342	−	0.898808i	$-0.644434\pi$
$17$	− 126.000i	− 1.79762i	0.438342	−	0.898808i	$-0.355566\pi$
$18$	0	0
$19$	−20.0000	−0.241490	−0.120745	−	0.992684i	$-0.538528\pi$
$19$	−20.0000	−0.241490	−0.120745	+	0.992684i	$0.538528\pi$
$20$	0	0
$21$	0	0
$22$	24.0000i	0.232583i
$23$	− 168.000i	− 1.52306i	−0.648129	−	0.761531i	$-0.724448\pi$
$23$	− 168.000i	− 1.52306i	0.648129	−	0.761531i	$-0.275552\pi$
$24$	0	0
$25$	0	0
$26$	76.0000	0.573263
$27$	0	0
$28$	− 64.0000i	− 0.431959i
$29$	30.0000	0.192099	0.0960493	−	0.995377i	$-0.469379\pi$
$29$	30.0000	0.192099	0.0960493	+	0.995377i	$0.469379\pi$
$30$	0	0
$31$	−88.0000	−0.509847	−0.254924	−	0.966961i	$-0.582050\pi$
$31$	−88.0000	−0.509847	−0.254924	+	0.966961i	$0.582050\pi$
$32$	− 32.0000i	− 0.176777i
$33$	0	0
$34$	−252.000	−1.27111
$35$	0	0
$36$	0	0
$37$	− 254.000i	− 1.12858i	−0.825578	−	0.564288i	$-0.809151\pi$
$37$	− 254.000i	− 1.12858i	0.825578	−	0.564288i	$-0.190849\pi$
$38$	40.0000i	0.170759i
$39$	0	0
$40$	0	0
$41$	−42.0000	−0.159983	−0.0799914	−	0.996796i	$-0.525489\pi$
$41$	−42.0000	−0.159983	−0.0799914	+	0.996796i	$0.525489\pi$
$42$	0	0
$43$	− 52.0000i	− 0.184417i	−0.995740	−	0.0922084i	$-0.970607\pi$
$43$	− 52.0000i	− 0.184417i	0.995740	−	0.0922084i	$-0.0293926\pi$
$44$	48.0000	0.164461
$45$	0	0
$46$	−336.000	−1.07697
$47$	− 96.0000i	− 0.297937i	−0.988842	−	0.148969i	$-0.952405\pi$
$47$	− 96.0000i	− 0.297937i	0.988842	−	0.148969i	$-0.0475953\pi$
$48$	0	0
$49$	87.0000	0.253644
$50$	0	0
$51$	0	0
$52$	− 152.000i	− 0.405358i
$53$	− 198.000i	− 0.513158i	−0.966523	−	0.256579i	$-0.917405\pi$
$53$	− 198.000i	− 0.513158i	0.966523	−	0.256579i	$-0.0825954\pi$
$54$	0	0
$55$	0	0
$56$	−128.000	−0.305441
$57$	0	0
$58$	− 60.0000i	− 0.135834i
$59$	−660.000	−1.45635	−0.728175	−	0.685391i	$-0.759631\pi$
$59$	−660.000	−1.45635	−0.728175	+	0.685391i	$0.759631\pi$
$60$	0	0
$61$	−538.000	−1.12924	−0.564622	−	0.825350i	$-0.690978\pi$
$61$	−538.000	−1.12924	−0.564622	+	0.825350i	$0.690978\pi$
$62$	176.000i	0.360516i
$63$	0	0
$64$	−64.0000	−0.125000
$65$	0	0
$66$	0	0
$67$	− 884.000i	− 1.61191i	−0.591979	−	0.805954i	$-0.701653\pi$
$67$	− 884.000i	− 1.61191i	0.591979	−	0.805954i	$-0.298347\pi$
$68$	504.000i	0.898808i
$69$	0	0
$70$	0	0
$71$	−792.000	−1.32385	−0.661923	−	0.749572i	$-0.730260\pi$
$71$	−792.000	−1.32385	−0.661923	+	0.749572i	$0.730260\pi$
$72$	0	0
$73$	218.000i	0.349520i	0.984611	+	0.174760i	$0.0559150\pi$
$73$	218.000i	0.349520i	−0.984611	+	0.174760i	$0.944085\pi$
$74$	−508.000	−0.798024
$75$	0	0
$76$	80.0000	0.120745
$77$	− 192.000i	− 0.284161i
$78$	0	0
$79$	520.000	0.740564	0.370282	−	0.928919i	$-0.379261\pi$
$79$	520.000	0.740564	0.370282	+	0.928919i	$0.379261\pi$
$80$	0	0
$81$	0	0
$82$	84.0000i	0.113125i
$83$	492.000i	0.650651i	0.945602	+	0.325325i	$0.105474\pi$
$83$	492.000i	0.650651i	−0.945602	+	0.325325i	$0.894526\pi$
$84$	0	0
$85$	0	0
$86$	−104.000	−0.130402
$87$	0	0
$88$	− 96.0000i	− 0.116291i
$89$	810.000	0.964717	0.482359	−	0.875974i	$-0.339780\pi$
$89$	810.000	0.964717	0.482359	+	0.875974i	$0.339780\pi$
$90$	0	0
$91$	−608.000	−0.700393
$92$	672.000i	0.761531i
$93$	0	0
$94$	−192.000	−0.210673
$95$	0	0
$96$	0	0
$97$	− 1154.00i	− 1.20795i	−0.797004	−	0.603974i	$-0.793583\pi$
$97$	− 1154.00i	− 1.20795i	0.797004	−	0.603974i	$-0.206417\pi$
$98$	− 174.000i	− 0.179354i
$99$	0	0
$100$	0	0
$101$	618.000	0.608845	0.304422	−	0.952537i	$-0.401537\pi$
$101$	618.000	0.608845	0.304422	+	0.952537i	$0.401537\pi$
$102$	0	0
$103$	128.000i	0.122449i	0.998124	+	0.0612243i	$0.0195005\pi$
$103$	128.000i	0.122449i	−0.998124	+	0.0612243i	$0.980499\pi$
$104$	−304.000	−0.286631
$105$	0	0
$106$	−396.000	−0.362858
$107$	− 1476.00i	− 1.33355i	−0.745257	−	0.666777i	$-0.767673\pi$
$107$	− 1476.00i	− 1.33355i	0.745257	−	0.666777i	$-0.232327\pi$
$108$	0	0
$109$	−1190.00	−1.04570	−0.522850	−	0.852425i	$-0.675131\pi$
$109$	−1190.00	−1.04570	−0.522850	+	0.852425i	$0.675131\pi$
$110$	0	0
$111$	0	0
$112$	256.000i	0.215980i
$113$	462.000i	0.384613i	0.981335	+	0.192307i	$0.0615968\pi$
$113$	462.000i	0.384613i	−0.981335	+	0.192307i	$0.938403\pi$
$114$	0	0
$115$	0	0
$116$	−120.000	−0.0960493
$117$	0	0
$118$	1320.00i	1.02980i
$119$	2016.00	1.55300
$120$	0	0
$121$	−1187.00	−0.891811
$122$	1076.00i	0.798496i
$123$	0	0
$124$	352.000	0.254924
$125$	0	0
$126$	0	0
$127$	2536.00i	1.77192i	0.463763	+	0.885959i	$0.346499\pi$
$127$	2536.00i	1.77192i	−0.463763	+	0.885959i	$0.653501\pi$
$128$	128.000i	0.0883883i
$129$	0	0
$130$	0	0
$131$	−2292.00	−1.52865	−0.764324	−	0.644832i	$-0.776927\pi$
$131$	−2292.00	−1.52865	−0.764324	+	0.644832i	$0.776927\pi$
$132$	0	0
$133$	− 320.000i	− 0.208628i
$134$	−1768.00	−1.13979
$135$	0	0
$136$	1008.00	0.635554
$137$	− 726.000i	− 0.452747i	−0.974041	−	0.226374i	$-0.927313\pi$
$137$	− 726.000i	− 0.452747i	0.974041	−	0.226374i	$-0.0726870\pi$
$138$	0	0
$139$	−380.000	−0.231879	−0.115939	−	0.993256i	$-0.536988\pi$
$139$	−380.000	−0.231879	−0.115939	+	0.993256i	$0.536988\pi$
$140$	0	0
$141$	0	0
$142$	1584.00i	0.936101i
$143$	− 456.000i	− 0.266662i
$144$	0	0
$145$	0	0
$146$	436.000	0.247148
$147$	0	0
$148$	1016.00i	0.564288i
$149$	1590.00	0.874214	0.437107	−	0.899410i	$-0.356003\pi$
$149$	1590.00	0.874214	0.437107	+	0.899410i	$0.356003\pi$
$150$	0	0
$151$	2432.00	1.31068	0.655342	−	0.755332i	$-0.272524\pi$
$151$	2432.00	1.31068	0.655342	+	0.755332i	$0.272524\pi$
$152$	− 160.000i	− 0.0853797i
$153$	0	0
$154$	−384.000	−0.200932
$155$	0	0
$156$	0	0
$157$	− 614.000i	− 0.312118i	−0.987748	−	0.156059i	$-0.950121\pi$
$157$	− 614.000i	− 0.312118i	0.987748	−	0.156059i	$-0.0498790\pi$
$158$	− 1040.00i	− 0.523658i
$159$	0	0
$160$	0	0
$161$	2688.00	1.31580
$162$	0	0
$163$	− 1852.00i	− 0.889938i	−0.895546	−	0.444969i	$-0.853215\pi$
$163$	− 1852.00i	− 0.889938i	0.895546	−	0.444969i	$-0.146785\pi$
$164$	168.000	0.0799914
$165$	0	0
$166$	984.000	0.460080
$167$	− 2136.00i	− 0.989752i	−0.868964	−	0.494876i	$-0.835213\pi$
$167$	− 2136.00i	− 0.989752i	0.868964	−	0.494876i	$-0.164787\pi$
$168$	0	0
$169$	753.000	0.342740
$170$	0	0
$171$	0	0
$172$	208.000i	0.0922084i
$173$	− 1758.00i	− 0.772591i	−0.922375	−	0.386296i	$-0.873754\pi$
$173$	− 1758.00i	− 0.772591i	0.922375	−	0.386296i	$-0.126246\pi$
$174$	0	0
$175$	0	0
$176$	−192.000	−0.0822304
$177$	0	0
$178$	− 1620.00i	− 0.682158i
$179$	−540.000	−0.225483	−0.112742	−	0.993624i	$-0.535963\pi$
$179$	−540.000	−0.225483	−0.112742	+	0.993624i	$0.535963\pi$
$180$	0	0
$181$	1982.00	0.813928	0.406964	−	0.913444i	$-0.366588\pi$
$181$	1982.00	0.813928	0.406964	+	0.913444i	$0.366588\pi$
$182$	1216.00i	0.495252i
$183$	0	0
$184$	1344.00	0.538484
$185$	0	0
$186$	0	0
$187$	1512.00i	0.591275i
$188$	384.000i	0.148969i
$189$	0	0
$190$	0	0
$191$	2688.00	1.01831	0.509154	−	0.860675i	$-0.329958\pi$
$191$	2688.00	1.01831	0.509154	+	0.860675i	$0.329958\pi$
$192$	0	0
$193$	− 2302.00i	− 0.858557i	−0.903172	−	0.429279i	$-0.858768\pi$
$193$	− 2302.00i	− 0.858557i	0.903172	−	0.429279i	$-0.141232\pi$
$194$	−2308.00	−0.854148
$195$	0	0
$196$	−348.000	−0.126822
$197$	4374.00i	1.58190i	0.611880	+	0.790951i	$0.290414\pi$
$197$	4374.00i	1.58190i	−0.611880	+	0.790951i	$0.709586\pi$
$198$	0	0
$199$	1600.00	0.569955	0.284977	−	0.958534i	$-0.408014\pi$
$199$	1600.00	0.569955	0.284977	+	0.958534i	$0.408014\pi$
$200$	0	0
$201$	0	0
$202$	− 1236.00i	− 0.430518i
$203$	480.000i	0.165958i
$204$	0	0
$205$	0	0
$206$	256.000	0.0865843
$207$	0	0
$208$	608.000i	0.202679i
$209$	240.000	0.0794313
$210$	0	0
$211$	3332.00	1.08713	0.543565	−	0.839367i	$-0.317074\pi$
$211$	3332.00	1.08713	0.543565	+	0.839367i	$0.317074\pi$
$212$	792.000i	0.256579i
$213$	0	0
$214$	−2952.00	−0.942965
$215$	0	0
$216$	0	0
$217$	− 1408.00i	− 0.440467i
$218$	2380.00i	0.739422i
$219$	0	0
$220$	0	0
$221$	4788.00	1.45736
$222$	0	0
$223$	2648.00i	0.795171i	0.917565	+	0.397586i	$0.130152\pi$
$223$	2648.00i	0.795171i	−0.917565	+	0.397586i	$0.869848\pi$
$224$	512.000	0.152721
$225$	0	0
$226$	924.000	0.271963
$227$	2244.00i	0.656121i	0.944657	+	0.328061i	$0.106395\pi$
$227$	2244.00i	0.656121i	−0.944657	+	0.328061i	$0.893605\pi$
$228$	0	0
$229$	5650.00	1.63040	0.815202	−	0.579177i	$-0.196626\pi$
$229$	5650.00	1.63040	0.815202	+	0.579177i	$0.196626\pi$
$230$	0	0
$231$	0	0
$232$	240.000i	0.0679171i
$233$	− 4698.00i	− 1.32093i	−0.750858	−	0.660464i	$-0.770360\pi$
$233$	− 4698.00i	− 1.32093i	0.750858	−	0.660464i	$-0.229640\pi$
$234$	0	0
$235$	0	0
$236$	2640.00	0.728175
$237$	0	0
$238$	− 4032.00i	− 1.09813i
$239$	−1200.00	−0.324776	−0.162388	−	0.986727i	$-0.551920\pi$
$239$	−1200.00	−0.324776	−0.162388	+	0.986727i	$0.551920\pi$
$240$	0	0
$241$	−718.000	−0.191911	−0.0959553	−	0.995386i	$-0.530591\pi$
$241$	−718.000	−0.191911	−0.0959553	+	0.995386i	$0.530591\pi$
$242$	2374.00i	0.630605i
$243$	0	0
$244$	2152.00	0.564622
$245$	0	0
$246$	0	0
$247$	− 760.000i	− 0.195780i
$248$	− 704.000i	− 0.180258i
$249$	0	0
$250$	0	0
$251$	−6012.00	−1.51185	−0.755924	−	0.654659i	$-0.772812\pi$
$251$	−6012.00	−1.51185	−0.755924	+	0.654659i	$0.772812\pi$
$252$	0	0
$253$	2016.00i	0.500968i
$254$	5072.00	1.25294
$255$	0	0
$256$	256.000	0.0625000
$257$	− 2046.00i	− 0.496599i	−0.968683	−	0.248300i	$-0.920128\pi$
$257$	− 2046.00i	− 0.496599i	0.968683	−	0.248300i	$-0.0798717\pi$
$258$	0	0
$259$	4064.00	0.974999
$260$	0	0
$261$	0	0
$262$	4584.00i	1.08092i
$263$	6072.00i	1.42363i	0.702365	+	0.711817i	$0.252127\pi$
$263$	6072.00i	1.42363i	−0.702365	+	0.711817i	$0.747873\pi$
$264$	0	0
$265$	0	0
$266$	−640.000	−0.147522
$267$	0	0
$268$	3536.00i	0.805954i
$269$	−6930.00	−1.57074	−0.785371	−	0.619025i	$-0.787528\pi$
$269$	−6930.00	−1.57074	−0.785371	+	0.619025i	$0.787528\pi$
$270$	0	0
$271$	1352.00	0.303056	0.151528	−	0.988453i	$-0.451581\pi$
$271$	1352.00	0.303056	0.151528	+	0.988453i	$0.451581\pi$
$272$	− 2016.00i	− 0.449404i
$273$	0	0
$274$	−1452.00	−0.320141
$275$	0	0
$276$	0	0
$277$	1186.00i	0.257256i	0.991693	+	0.128628i	$0.0410573\pi$
$277$	1186.00i	0.257256i	−0.991693	+	0.128628i	$0.958943\pi$
$278$	760.000i	0.163963i
$279$	0	0
$280$	0	0
$281$	−2442.00	−0.518425	−0.259213	−	0.965820i	$-0.583463\pi$
$281$	−2442.00	−0.518425	−0.259213	+	0.965820i	$0.583463\pi$
$282$	0	0
$283$	2828.00i	0.594018i	0.954875	+	0.297009i	$0.0959892\pi$
$283$	2828.00i	0.594018i	−0.954875	+	0.297009i	$0.904011\pi$
$284$	3168.00	0.661923
$285$	0	0
$286$	−912.000	−0.188558
$287$	− 672.000i	− 0.138212i
$288$	0	0
$289$	−10963.0	−2.23143
$290$	0	0
$291$	0	0
$292$	− 872.000i	− 0.174760i
$293$	− 4758.00i	− 0.948687i	−0.880340	−	0.474344i	$-0.842685\pi$
$293$	− 4758.00i	− 0.948687i	0.880340	−	0.474344i	$-0.157315\pi$
$294$	0	0
$295$	0	0
$296$	2032.00	0.399012
$297$	0	0
$298$	− 3180.00i	− 0.618163i
$299$	6384.00	1.23477
$300$	0	0
$301$	832.000	0.159321
$302$	− 4864.00i	− 0.926794i
$303$	0	0
$304$	−320.000	−0.0603726
$305$	0	0
$306$	0	0
$307$	8476.00i	1.57574i	0.615844	+	0.787868i	$0.288815\pi$
$307$	8476.00i	1.57574i	−0.615844	+	0.787868i	$0.711185\pi$
$308$	768.000i	0.142081i
$309$	0	0
$310$	0	0
$311$	−4632.00	−0.844555	−0.422278	−	0.906467i	$-0.638769\pi$
$311$	−4632.00	−0.844555	−0.422278	+	0.906467i	$0.638769\pi$
$312$	0	0
$313$	− 4822.00i	− 0.870785i	−0.900241	−	0.435392i	$-0.856610\pi$
$313$	− 4822.00i	− 0.870785i	0.900241	−	0.435392i	$-0.143390\pi$
$314$	−1228.00	−0.220701
$315$	0	0
$316$	−2080.00	−0.370282
$317$	− 3426.00i	− 0.607014i	−0.952829	−	0.303507i	$-0.901842\pi$
$317$	− 3426.00i	− 0.607014i	0.952829	−	0.303507i	$-0.0981575\pi$
$318$	0	0
$319$	−360.000	−0.0631854
$320$	0	0
$321$	0	0
$322$	− 5376.00i	− 0.930412i
$323$	2520.00i	0.434107i
$324$	0	0
$325$	0	0
$326$	−3704.00	−0.629281
$327$	0	0
$328$	− 336.000i	− 0.0565625i
$329$	1536.00	0.257393
$330$	0	0
$331$	−2788.00	−0.462968	−0.231484	−	0.972839i	$-0.574358\pi$
$331$	−2788.00	−0.462968	−0.231484	+	0.972839i	$0.574358\pi$
$332$	− 1968.00i	− 0.325325i
$333$	0	0
$334$	−4272.00	−0.699861
$335$	0	0
$336$	0	0
$337$	− 434.000i	− 0.0701528i	−0.999385	−	0.0350764i	$-0.988833\pi$
$337$	− 434.000i	− 0.0701528i	0.999385	−	0.0350764i	$-0.0111675\pi$
$338$	− 1506.00i	− 0.242354i
$339$	0	0
$340$	0	0
$341$	1056.00	0.167700
$342$	0	0
$343$	6880.00i	1.08305i
$344$	416.000	0.0652012
$345$	0	0
$346$	−3516.00	−0.546304
$347$	6684.00i	1.03405i	0.855970	+	0.517026i	$0.172961\pi$
$347$	6684.00i	1.03405i	−0.855970	+	0.517026i	$0.827039\pi$
$348$	0	0
$349$	−2630.00	−0.403383	−0.201692	−	0.979449i	$-0.564644\pi$
$349$	−2630.00	−0.403383	−0.201692	+	0.979449i	$0.564644\pi$
$350$	0	0
$351$	0	0
$352$	384.000i	0.0581456i
$353$	7422.00i	1.11907i	0.828805	+	0.559537i	$0.189021\pi$
$353$	7422.00i	1.11907i	−0.828805	+	0.559537i	$0.810979\pi$
$354$	0	0
$355$	0	0
$356$	−3240.00	−0.482359
$357$	0	0
$358$	1080.00i	0.159441i
$359$	−10440.0	−1.53482	−0.767412	−	0.641154i	$-0.778456\pi$
$359$	−10440.0	−1.53482	−0.767412	+	0.641154i	$0.778456\pi$
$360$	0	0
$361$	−6459.00	−0.941682
$362$	− 3964.00i	− 0.575534i
$363$	0	0
$364$	2432.00	0.350196
$365$	0	0
$366$	0	0
$367$	− 10424.0i	− 1.48264i	−0.671153	−	0.741319i	$-0.734200\pi$
$367$	− 10424.0i	− 1.48264i	0.671153	−	0.741319i	$-0.265800\pi$
$368$	− 2688.00i	− 0.380765i
$369$	0	0
$370$	0	0
$371$	3168.00	0.443327
$372$	0	0
$373$	3278.00i	0.455036i	0.973774	+	0.227518i	$0.0730610\pi$
$373$	3278.00i	0.455036i	−0.973774	+	0.227518i	$0.926939\pi$
$374$	3024.00	0.418094
$375$	0	0
$376$	768.000	0.105337
$377$	1140.00i	0.155737i
$378$	0	0
$379$	−6140.00	−0.832165	−0.416083	−	0.909327i	$-0.636597\pi$
$379$	−6140.00	−0.832165	−0.416083	+	0.909327i	$0.636597\pi$
$380$	0	0
$381$	0	0
$382$	− 5376.00i	− 0.720053i
$383$	3072.00i	0.409848i	0.978778	+	0.204924i	$0.0656948\pi$
$383$	3072.00i	0.409848i	−0.978778	+	0.204924i	$0.934305\pi$
$384$	0	0
$385$	0	0
$386$	−4604.00	−0.607092
$387$	0	0
$388$	4616.00i	0.603974i
$389$	6150.00	0.801587	0.400794	−	0.916168i	$-0.368734\pi$
$389$	6150.00	0.801587	0.400794	+	0.916168i	$0.368734\pi$
$390$	0	0
$391$	−21168.0	−2.73788
$392$	696.000i	0.0896768i
$393$	0	0
$394$	8748.00	1.11857
$395$	0	0
$396$	0	0
$397$	106.000i	0.0134005i	0.999978	+	0.00670024i	$0.00213277\pi$
$397$	106.000i	0.0134005i	−0.999978	+	0.00670024i	$0.997867\pi$
$398$	− 3200.00i	− 0.403019i
$399$	0	0
$400$	0	0
$401$	1758.00	0.218929	0.109464	−	0.993991i	$-0.465086\pi$
$401$	1758.00	0.218929	0.109464	+	0.993991i	$0.465086\pi$
$402$	0	0
$403$	− 3344.00i	− 0.413341i
$404$	−2472.00	−0.304422
$405$	0	0
$406$	960.000	0.117350
$407$	3048.00i	0.371213i
$408$	0	0
$409$	3670.00	0.443691	0.221846	−	0.975082i	$-0.428792\pi$
$409$	3670.00	0.443691	0.221846	+	0.975082i	$0.428792\pi$
$410$	0	0
$411$	0	0
$412$	− 512.000i	− 0.0612243i
$413$	− 10560.0i	− 1.25817i
$414$	0	0
$415$	0	0
$416$	1216.00	0.143316
$417$	0	0
$418$	− 480.000i	− 0.0561664i
$419$	−9660.00	−1.12631	−0.563153	−	0.826353i	$-0.690412\pi$
$419$	−9660.00	−1.12631	−0.563153	+	0.826353i	$0.690412\pi$
$420$	0	0
$421$	8462.00	0.979602	0.489801	−	0.871834i	$-0.337069\pi$
$421$	8462.00	0.979602	0.489801	+	0.871834i	$0.337069\pi$
$422$	− 6664.00i	− 0.768717i
$423$	0	0
$424$	1584.00	0.181429
$425$	0	0
$426$	0	0
$427$	− 8608.00i	− 0.975575i
$428$	5904.00i	0.666777i
$429$	0	0
$430$	0	0
$431$	−9792.00	−1.09435	−0.547174	−	0.837019i	$-0.684296\pi$
$431$	−9792.00	−1.09435	−0.547174	+	0.837019i	$0.684296\pi$
$432$	0	0
$433$	− 7342.00i	− 0.814859i	−0.913237	−	0.407430i	$-0.866425\pi$
$433$	− 7342.00i	− 0.814859i	0.913237	−	0.407430i	$-0.133575\pi$
$434$	−2816.00	−0.311457
$435$	0	0
$436$	4760.00	0.522850
$437$	3360.00i	0.367805i
$438$	0	0
$439$	−10640.0	−1.15676	−0.578382	−	0.815766i	$-0.696316\pi$
$439$	−10640.0	−1.15676	−0.578382	+	0.815766i	$0.696316\pi$
$440$	0	0
$441$	0	0
$442$	− 9576.00i	− 1.03051i
$443$	17412.0i	1.86742i	0.358024	+	0.933712i	$0.383451\pi$
$443$	17412.0i	1.86742i	−0.358024	+	0.933712i	$0.616549\pi$
$444$	0	0
$445$	0	0
$446$	5296.00	0.562271
$447$	0	0
$448$	− 1024.00i	− 0.107990i
$449$	−1710.00	−0.179732	−0.0898662	−	0.995954i	$-0.528644\pi$
$449$	−1710.00	−0.179732	−0.0898662	+	0.995954i	$0.528644\pi$
$450$	0	0
$451$	504.000	0.0526218
$452$	− 1848.00i	− 0.192307i
$453$	0	0
$454$	4488.00	0.463948
$455$	0	0
$456$	0	0
$457$	646.000i	0.0661239i	0.999453	+	0.0330619i	$0.0105259\pi$
$457$	646.000i	0.0661239i	−0.999453	+	0.0330619i	$0.989474\pi$
$458$	− 11300.0i	− 1.15287i
$459$	0	0
$460$	0	0
$461$	6018.00	0.607996	0.303998	−	0.952673i	$-0.401678\pi$
$461$	6018.00	0.607996	0.303998	+	0.952673i	$0.401678\pi$
$462$	0	0
$463$	− 6712.00i	− 0.673722i	−0.941554	−	0.336861i	$-0.890635\pi$
$463$	− 6712.00i	− 0.673722i	0.941554	−	0.336861i	$-0.109365\pi$
$464$	480.000	0.0480247
$465$	0	0
$466$	−9396.00	−0.934037
$467$	5364.00i	0.531512i	0.964040	+	0.265756i	$0.0856216\pi$
$467$	5364.00i	0.531512i	−0.964040	+	0.265756i	$0.914378\pi$
$468$	0	0
$469$	14144.0	1.39256
$470$	0	0
$471$	0	0
$472$	− 5280.00i	− 0.514898i
$473$	624.000i	0.0606587i
$474$	0	0
$475$	0	0
$476$	−8064.00	−0.776498
$477$	0	0
$478$	2400.00i	0.229652i
$479$	9840.00	0.938624	0.469312	−	0.883032i	$-0.344502\pi$
$479$	9840.00	0.938624	0.469312	+	0.883032i	$0.344502\pi$
$480$	0	0
$481$	9652.00	0.914955
$482$	1436.00i	0.135701i
$483$	0	0
$484$	4748.00	0.445905
$485$	0	0
$486$	0	0
$487$	− 1424.00i	− 0.132500i	−0.997803	−	0.0662501i	$-0.978896\pi$
$487$	− 1424.00i	− 0.132500i	0.997803	−	0.0662501i	$-0.0211035\pi$
$488$	− 4304.00i	− 0.399248i
$489$	0	0
$490$	0	0
$491$	4548.00	0.418021	0.209011	−	0.977913i	$-0.432976\pi$
$491$	4548.00	0.418021	0.209011	+	0.977913i	$0.432976\pi$
$492$	0	0
$493$	− 3780.00i	− 0.345320i
$494$	−1520.00	−0.138437
$495$	0	0
$496$	−1408.00	−0.127462
$497$	− 12672.0i	− 1.14370i
$498$	0	0
$499$	−6500.00	−0.583126	−0.291563	−	0.956552i	$-0.594175\pi$
$499$	−6500.00	−0.583126	−0.291563	+	0.956552i	$0.594175\pi$
$500$	0	0
$501$	0	0
$502$	12024.0i	1.06904i
$503$	− 12168.0i	− 1.07862i	−0.842108	−	0.539308i	$-0.818686\pi$
$503$	− 12168.0i	− 1.07862i	0.842108	−	0.539308i	$-0.181314\pi$
$504$	0	0
$505$	0	0
$506$	4032.00	0.354238
$507$	0	0
$508$	− 10144.0i	− 0.885959i
$509$	−21090.0	−1.83654	−0.918269	−	0.395957i	$-0.870413\pi$
$509$	−21090.0	−1.83654	−0.918269	+	0.395957i	$0.870413\pi$
$510$	0	0
$511$	−3488.00	−0.301957
$512$	− 512.000i	− 0.0441942i
$513$	0	0
$514$	−4092.00	−0.351149
$515$	0	0
$516$	0	0
$517$	1152.00i	0.0979979i
$518$	− 8128.00i	− 0.689428i
$519$	0	0
$520$	0	0
$521$	5238.00	0.440462	0.220231	−	0.975448i	$-0.429319\pi$
$521$	5238.00	0.440462	0.220231	+	0.975448i	$0.429319\pi$
$522$	0	0
$523$	8588.00i	0.718025i	0.933333	+	0.359012i	$0.116886\pi$
$523$	8588.00i	0.718025i	−0.933333	+	0.359012i	$0.883114\pi$
$524$	9168.00	0.764324
$525$	0	0
$526$	12144.0	1.00666
$527$	11088.0i	0.916510i
$528$	0	0
$529$	−16057.0	−1.31972
$530$	0	0
$531$	0	0
$532$	1280.00i	0.104314i
$533$	− 1596.00i	− 0.129701i
$534$	0	0
$535$	0	0
$536$	7072.00	0.569895
$537$	0	0
$538$	13860.0i	1.11068i
$539$	−1044.00	−0.0834291
$540$	0	0
$541$	3062.00	0.243338	0.121669	−	0.992571i	$-0.461175\pi$
$541$	3062.00	0.243338	0.121669	+	0.992571i	$0.461175\pi$
$542$	− 2704.00i	− 0.214293i
$543$	0	0
$544$	−4032.00	−0.317777
$545$	0	0
$546$	0	0
$547$	8476.00i	0.662537i	0.943537	+	0.331268i	$0.107477\pi$
$547$	8476.00i	0.662537i	−0.943537	+	0.331268i	$0.892523\pi$
$548$	2904.00i	0.226374i
$549$	0	0
$550$	0	0
$551$	−600.000	−0.0463899
$552$	0	0
$553$	8320.00i	0.639787i
$554$	2372.00	0.181907
$555$	0	0
$556$	1520.00	0.115939
$557$	− 12546.0i	− 0.954383i	−0.878799	−	0.477191i	$-0.841655\pi$
$557$	− 12546.0i	− 0.954383i	0.878799	−	0.477191i	$-0.158345\pi$
$558$	0	0
$559$	1976.00	0.149510
$560$	0	0
$561$	0	0
$562$	4884.00i	0.366582i
$563$	12.0000i	0 0.000898294i	1.00000		0.000449147i	$0.000142968\pi$
$563$	12.0000i	0 0.000898294i	−1.00000		0.000449147i	$0.999857\pi$
$564$	0	0
$565$	0	0
$566$	5656.00	0.420034
$567$	0	0
$568$	− 6336.00i	− 0.468050i
$569$	19290.0	1.42123	0.710614	−	0.703582i	$-0.248417\pi$
$569$	19290.0	1.42123	0.710614	+	0.703582i	$0.248417\pi$
$570$	0	0
$571$	−12148.0	−0.890329	−0.445165	−	0.895449i	$-0.646855\pi$
$571$	−12148.0	−0.890329	−0.445165	+	0.895449i	$0.646855\pi$
$572$	1824.00i	0.133331i
$573$	0	0
$574$	−1344.00	−0.0977308
$575$	0	0
$576$	0	0
$577$	10366.0i	0.747907i	0.927447	+	0.373953i	$0.121998\pi$
$577$	10366.0i	0.747907i	−0.927447	+	0.373953i	$0.878002\pi$
$578$	21926.0i	1.57786i
$579$	0	0
$580$	0	0
$581$	−7872.00	−0.562109
$582$	0	0
$583$	2376.00i	0.168789i
$584$	−1744.00	−0.123574
$585$	0	0
$586$	−9516.00	−0.670823
$587$	7644.00i	0.537482i	0.963213	+	0.268741i	$0.0866075\pi$
$587$	7644.00i	0.537482i	−0.963213	+	0.268741i	$0.913393\pi$
$588$	0	0
$589$	1760.00	0.123123
$590$	0	0
$591$	0	0
$592$	− 4064.00i	− 0.282144i
$593$	− 8658.00i	− 0.599564i	−0.954008	−	0.299782i	$-0.903086\pi$
$593$	− 8658.00i	− 0.599564i	0.954008	−	0.299782i	$-0.0969139\pi$
$594$	0	0
$595$	0	0
$596$	−6360.00	−0.437107
$597$	0	0
$598$	− 12768.0i	− 0.873114i
$599$	25800.0	1.75987	0.879933	−	0.475098i	$-0.157587\pi$
$599$	25800.0	1.75987	0.879933	+	0.475098i	$0.157587\pi$
$600$	0	0
$601$	16202.0	1.09966	0.549828	−	0.835278i	$-0.314693\pi$
$601$	16202.0	1.09966	0.549828	+	0.835278i	$0.314693\pi$
$602$	− 1664.00i	− 0.112657i
$603$	0	0
$604$	−9728.00	−0.655342
$605$	0	0
$606$	0	0
$607$	24136.0i	1.61392i	0.590605	+	0.806960i	$0.298889\pi$
$607$	24136.0i	1.61392i	−0.590605	+	0.806960i	$0.701111\pi$
$608$	640.000i	0.0426898i
$609$	0	0
$610$	0	0
$611$	3648.00	0.241542
$612$	0	0
$613$	− 4642.00i	− 0.305854i	−0.988237	−	0.152927i	$-0.951130\pi$
$613$	− 4642.00i	− 0.305854i	0.988237	−	0.152927i	$-0.0488700\pi$
$614$	16952.0	1.11421
$615$	0	0
$616$	1536.00	0.100466
$617$	− 6726.00i	− 0.438863i	−0.975628	−	0.219432i	$-0.929580\pi$
$617$	− 6726.00i	− 0.438863i	0.975628	−	0.219432i	$-0.0704203\pi$
$618$	0	0
$619$	21220.0	1.37787	0.688937	−	0.724821i	$-0.258078\pi$
$619$	21220.0	1.37787	0.688937	+	0.724821i	$0.258078\pi$
$620$	0	0
$621$	0	0
$622$	9264.00i	0.597191i
$623$	12960.0i	0.833437i
$624$	0	0
$625$	0	0
$626$	−9644.00	−0.615738
$627$	0	0
$628$	2456.00i	0.156059i
$629$	−32004.0	−2.02875
$630$	0	0
$631$	29792.0	1.87956	0.939779	−	0.341783i	$-0.111031\pi$
$631$	29792.0	1.87956	0.939779	+	0.341783i	$0.111031\pi$
$632$	4160.00i	0.261829i
$633$	0	0
$634$	−6852.00	−0.429223
$635$	0	0
$636$	0	0
$637$	3306.00i	0.205633i
$638$	720.000i	0.0446788i
$639$	0	0
$640$	0	0
$641$	10158.0	0.625923	0.312962	−	0.949766i	$-0.398679\pi$
$641$	10158.0	0.625923	0.312962	+	0.949766i	$0.398679\pi$
$642$	0	0
$643$	29828.0i	1.82940i	0.404138	+	0.914698i	$0.367571\pi$
$643$	29828.0i	1.82940i	−0.404138	+	0.914698i	$0.632429\pi$
$644$	−10752.0	−0.657901
$645$	0	0
$646$	5040.00	0.306960
$647$	1944.00i	0.118124i	0.998254	+	0.0590622i	$0.0188110\pi$
$647$	1944.00i	0.118124i	−0.998254	+	0.0590622i	$0.981189\pi$
$648$	0	0
$649$	7920.00	0.479025
$650$	0	0
$651$	0	0
$652$	7408.00i	0.444969i
$653$	− 26718.0i	− 1.60116i	−0.599227	−	0.800579i	$-0.704525\pi$
$653$	− 26718.0i	− 1.60116i	0.599227	−	0.800579i	$-0.295475\pi$
$654$	0	0
$655$	0	0
$656$	−672.000	−0.0399957
$657$	0	0
$658$	− 3072.00i	− 0.182005i
$659$	4260.00	0.251815	0.125907	−	0.992042i	$-0.459816\pi$
$659$	4260.00	0.251815	0.125907	+	0.992042i	$0.459816\pi$
$660$	0	0
$661$	22862.0	1.34528	0.672639	−	0.739971i	$-0.265161\pi$
$661$	22862.0	1.34528	0.672639	+	0.739971i	$0.265161\pi$
$662$	5576.00i	0.327368i
$663$	0	0
$664$	−3936.00	−0.230040
$665$	0	0
$666$	0	0
$667$	− 5040.00i	− 0.292578i
$668$	8544.00i	0.494876i
$669$	0	0
$670$	0	0
$671$	6456.00	0.371432
$672$	0	0
$673$	− 32542.0i	− 1.86390i	−0.362592	−	0.931948i	$-0.618108\pi$
$673$	− 32542.0i	− 1.86390i	0.362592	−	0.931948i	$-0.381892\pi$
$674$	−868.000	−0.0496055
$675$	0	0
$676$	−3012.00	−0.171370
$677$	14214.0i	0.806925i	0.914996	+	0.403463i	$0.132193\pi$
$677$	14214.0i	0.806925i	−0.914996	+	0.403463i	$0.867807\pi$
$678$	0	0
$679$	18464.0	1.04357
$680$	0	0
$681$	0	0
$682$	− 2112.00i	− 0.118582i
$683$	7092.00i	0.397317i	0.980069	+	0.198659i	$0.0636585\pi$
$683$	7092.00i	0.397317i	−0.980069	+	0.198659i	$0.936341\pi$
$684$	0	0
$685$	0	0
$686$	13760.0	0.765830
$687$	0	0
$688$	− 832.000i	− 0.0461042i
$689$	7524.00	0.416026
$690$	0	0
$691$	−13228.0	−0.728244	−0.364122	−	0.931351i	$-0.618631\pi$
$691$	−13228.0	−0.728244	−0.364122	+	0.931351i	$0.618631\pi$
$692$	7032.00i	0.386296i
$693$	0	0
$694$	13368.0	0.731185
$695$	0	0
$696$	0	0
$697$	5292.00i	0.287588i
$698$	5260.00i	0.285235i
$699$	0	0
$700$	0	0
$701$	−28062.0	−1.51196	−0.755982	−	0.654592i	$-0.772840\pi$
$701$	−28062.0	−1.51196	−0.755982	+	0.654592i	$0.772840\pi$
$702$	0	0
$703$	5080.00i	0.272540i
$704$	768.000	0.0411152
$705$	0	0
$706$	14844.0	0.791305
$707$	9888.00i	0.525992i
$708$	0	0
$709$	27250.0	1.44343	0.721717	−	0.692188i	$-0.243353\pi$
$709$	27250.0	1.44343	0.721717	+	0.692188i	$0.243353\pi$
$710$	0	0
$711$	0	0
$712$	6480.00i	0.341079i
$713$	14784.0i	0.776529i
$714$	0	0
$715$	0	0
$716$	2160.00	0.112742
$717$	0	0
$718$	20880.0i	1.08529i
$719$	−14400.0	−0.746912	−0.373456	−	0.927648i	$-0.621827\pi$
$719$	−14400.0	−0.746912	−0.373456	+	0.927648i	$0.621827\pi$
$720$	0	0
$721$	−2048.00	−0.105786
$722$	12918.0i	0.665870i
$723$	0	0
$724$	−7928.00	−0.406964
$725$	0	0
$726$	0	0
$727$	− 17984.0i	− 0.917455i	−0.888577	−	0.458727i	$-0.848305\pi$
$727$	− 17984.0i	− 0.917455i	0.888577	−	0.458727i	$-0.151695\pi$
$728$	− 4864.00i	− 0.247626i
$729$	0	0
$730$	0	0
$731$	−6552.00	−0.331511
$732$	0	0
$733$	16598.0i	0.836373i	0.908361	+	0.418186i	$0.137334\pi$
$733$	16598.0i	0.836373i	−0.908361	+	0.418186i	$0.862666\pi$
$734$	−20848.0	−1.04838
$735$	0	0
$736$	−5376.00	−0.269242
$737$	10608.0i	0.530191i
$738$	0	0
$739$	−1460.00	−0.0726752	−0.0363376	−	0.999340i	$-0.511569\pi$
$739$	−1460.00	−0.0726752	−0.0363376	+	0.999340i	$0.511569\pi$
$740$	0	0
$741$	0	0
$742$	− 6336.00i	− 0.313480i
$743$	30072.0i	1.48484i	0.669936	+	0.742419i	$0.266322\pi$
$743$	30072.0i	1.48484i	−0.669936	+	0.742419i	$0.733678\pi$
$744$	0	0
$745$	0	0
$746$	6556.00	0.321759
$747$	0	0
$748$	− 6048.00i	− 0.295637i
$749$	23616.0	1.15208
$750$	0	0
$751$	−18088.0	−0.878882	−0.439441	−	0.898271i	$-0.644823\pi$
$751$	−18088.0	−0.878882	−0.439441	+	0.898271i	$0.644823\pi$
$752$	− 1536.00i	− 0.0744843i
$753$	0	0
$754$	2280.00	0.110123
$755$	0	0
$756$	0	0
$757$	− 24734.0i	− 1.18755i	−0.804633	−	0.593773i	$-0.797638\pi$
$757$	− 24734.0i	− 1.18755i	0.804633	−	0.593773i	$-0.202362\pi$
$758$	12280.0i	0.588430i
$759$	0	0
$760$	0	0
$761$	22278.0	1.06120	0.530602	−	0.847621i	$-0.321966\pi$
$761$	22278.0	1.06120	0.530602	+	0.847621i	$0.321966\pi$
$762$	0	0
$763$	− 19040.0i	− 0.903400i
$764$	−10752.0	−0.509154
$765$	0	0
$766$	6144.00	0.289806
$767$	− 25080.0i	− 1.18069i
$768$	0	0
$769$	−16130.0	−0.756388	−0.378194	−	0.925726i	$-0.623455\pi$
$769$	−16130.0	−0.756388	−0.378194	+	0.925726i	$0.623455\pi$
$770$	0	0
$771$	0	0
$772$	9208.00i	0.429279i
$773$	− 29718.0i	− 1.38277i	−0.722486	−	0.691386i	$-0.757001\pi$
$773$	− 29718.0i	− 1.38277i	0.722486	−	0.691386i	$-0.242999\pi$
$774$	0	0
$775$	0	0
$776$	9232.00	0.427074
$777$	0	0
$778$	− 12300.0i	− 0.566808i
$779$	840.000	0.0386343
$780$	0	0
$781$	9504.00	0.435442
$782$	42336.0i	1.93597i
$783$	0	0
$784$	1392.00	0.0634111
$785$	0	0
$786$	0	0
$787$	− 9524.00i	− 0.431377i	−0.976462	−	0.215689i	$-0.930800\pi$
$787$	− 9524.00i	− 0.431377i	0.976462	−	0.215689i	$-0.0691996\pi$
$788$	− 17496.0i	− 0.790951i
$789$	0	0
$790$	0	0
$791$	−7392.00	−0.332275
$792$	0	0
$793$	− 20444.0i	− 0.915495i
$794$	212.000	0.00947556
$795$	0	0
$796$	−6400.00	−0.284977
$797$	− 33906.0i	− 1.50692i	−0.657496	−	0.753458i	$-0.728384\pi$
$797$	− 33906.0i	− 1.50692i	0.657496	−	0.753458i	$-0.271616\pi$
$798$	0	0
$799$	−12096.0	−0.535577
$800$	0	0
$801$	0	0
$802$	− 3516.00i	− 0.154806i
$803$	− 2616.00i	− 0.114965i
$804$	0	0
$805$	0	0
$806$	−6688.00	−0.292276
$807$	0	0
$808$	4944.00i	0.215259i
$809$	−630.000	−0.0273790	−0.0136895	−	0.999906i	$-0.504358\pi$
$809$	−630.000	−0.0273790	−0.0136895	+	0.999906i	$0.504358\pi$
$810$	0	0
$811$	−20788.0	−0.900081	−0.450040	−	0.893008i	$-0.648590\pi$
$811$	−20788.0	−0.900081	−0.450040	+	0.893008i	$0.648590\pi$
$812$	− 1920.00i	− 0.0829788i
$813$	0	0
$814$	6096.00	0.262487
$815$	0	0
$816$	0	0
$817$	1040.00i	0.0445349i
$818$	− 7340.00i	− 0.313737i
$819$	0	0
$820$	0	0
$821$	43098.0	1.83207	0.916036	−	0.401097i	$-0.131371\pi$
$821$	43098.0	1.83207	0.916036	+	0.401097i	$0.131371\pi$
$822$	0	0
$823$	− 14272.0i	− 0.604484i	−0.953231	−	0.302242i	$-0.902265\pi$
$823$	− 14272.0i	− 0.604484i	0.953231	−	0.302242i	$-0.0977351\pi$
$824$	−1024.00	−0.0432921
$825$	0	0
$826$	−21120.0	−0.889660
$827$	13644.0i	0.573698i	0.957976	+	0.286849i	$0.0926078\pi$
$827$	13644.0i	0.573698i	−0.957976	+	0.286849i	$0.907392\pi$
$828$	0	0
$829$	2410.00	0.100968	0.0504842	−	0.998725i	$-0.483924\pi$
$829$	2410.00	0.100968	0.0504842	+	0.998725i	$0.483924\pi$
$830$	0	0
$831$	0	0
$832$	− 2432.00i	− 0.101339i
$833$	− 10962.0i	− 0.455955i
$834$	0	0
$835$	0	0
$836$	−960.000	−0.0397157
$837$	0	0
$838$	19320.0i	0.796418i
$839$	23160.0	0.953006	0.476503	−	0.879173i	$-0.341904\pi$
$839$	23160.0	0.953006	0.476503	+	0.879173i	$0.341904\pi$
$840$	0	0
$841$	−23489.0	−0.963098
$842$	− 16924.0i	− 0.692684i
$843$	0	0
$844$	−13328.0	−0.543565
$845$	0	0
$846$	0	0
$847$	− 18992.0i	− 0.770452i
$848$	− 3168.00i	− 0.128290i
$849$	0	0
$850$	0	0
$851$	−42672.0	−1.71889
$852$	0	0
$853$	32078.0i	1.28761i	0.765190	+	0.643804i	$0.222645\pi$
$853$	32078.0i	1.28761i	−0.765190	+	0.643804i	$0.777355\pi$
$854$	−17216.0	−0.689835
$855$	0	0
$856$	11808.0	0.471483
$857$	− 14406.0i	− 0.574212i	−0.957899	−	0.287106i	$-0.907307\pi$
$857$	− 14406.0i	− 0.574212i	0.957899	−	0.287106i	$-0.0926932\pi$
$858$	0	0
$859$	−30620.0	−1.21623	−0.608115	−	0.793849i	$-0.708074\pi$
$859$	−30620.0	−1.21623	−0.608115	+	0.793849i	$0.708074\pi$
$860$	0	0
$861$	0	0
$862$	19584.0i	0.773821i
$863$	− 17568.0i	− 0.692957i	−0.938058	−	0.346478i	$-0.887377\pi$
$863$	− 17568.0i	− 0.692957i	0.938058	−	0.346478i	$-0.112623\pi$
$864$	0	0
$865$	0	0
$866$	−14684.0	−0.576192
$867$	0	0
$868$	5632.00i	0.220233i
$869$	−6240.00	−0.243587
$870$	0	0
$871$	33592.0	1.30680
$872$	− 9520.00i	− 0.369711i
$873$	0	0
$874$	6720.00	0.260077
$875$	0	0
$876$	0	0
$877$	21706.0i	0.835758i	0.908503	+	0.417879i	$0.137226\pi$
$877$	21706.0i	0.835758i	−0.908503	+	0.417879i	$0.862774\pi$
$878$	21280.0i	0.817956i
$879$	0	0
$880$	0	0
$881$	14958.0	0.572018	0.286009	−	0.958227i	$-0.407671\pi$
$881$	14958.0	0.572018	0.286009	+	0.958227i	$0.407671\pi$
$882$	0	0
$883$	− 32812.0i	− 1.25052i	−0.780415	−	0.625261i	$-0.784992\pi$
$883$	− 32812.0i	− 1.25052i	0.780415	−	0.625261i	$-0.215008\pi$
$884$	−19152.0	−0.728678
$885$	0	0
$886$	34824.0	1.32047
$887$	− 38856.0i	− 1.47086i	−0.677598	−	0.735432i	$-0.736979\pi$
$887$	− 38856.0i	− 1.47086i	0.677598	−	0.735432i	$-0.263021\pi$
$888$	0	0
$889$	−40576.0	−1.53079
$890$	0	0
$891$	0	0
$892$	− 10592.0i	− 0.397586i
$893$	1920.00i	0.0719489i
$894$	0	0
$895$	0	0
$896$	−2048.00	−0.0763604
$897$	0	0
$898$	3420.00i	0.127090i
$899$	−2640.00	−0.0979410
$900$	0	0
$901$	−24948.0	−0.922462
$902$	− 1008.00i	− 0.0372092i
$903$	0	0
$904$	−3696.00	−0.135981
$905$	0	0
$906$	0	0
$907$	28276.0i	1.03516i	0.855635	+	0.517579i	$0.173167\pi$
$907$	28276.0i	1.03516i	−0.855635	+	0.517579i	$0.826833\pi$
$908$	− 8976.00i	− 0.328061i
$909$	0	0
$910$	0	0
$911$	−8112.00	−0.295019	−0.147510	−	0.989061i	$-0.547126\pi$
$911$	−8112.00	−0.295019	−0.147510	+	0.989061i	$0.547126\pi$
$912$	0	0
$913$	− 5904.00i	− 0.214013i
$914$	1292.00	0.0467566
$915$	0	0
$916$	−22600.0	−0.815202
$917$	− 36672.0i	− 1.32063i
$918$	0	0
$919$	26080.0	0.936126	0.468063	−	0.883695i	$-0.344952\pi$
$919$	26080.0	0.936126	0.468063	+	0.883695i	$0.344952\pi$
$920$	0	0
$921$	0	0
$922$	− 12036.0i	− 0.429918i
$923$	− 30096.0i	− 1.07326i
$924$	0	0
$925$	0	0
$926$	−13424.0	−0.476393
$927$	0	0
$928$	− 960.000i	− 0.0339586i
$929$	49170.0	1.73651	0.868254	−	0.496120i	$-0.165243\pi$
$929$	49170.0	1.73651	0.868254	+	0.496120i	$0.165243\pi$
$930$	0	0
$931$	−1740.00	−0.0612526
$932$	18792.0i	0.660464i
$933$	0	0
$934$	10728.0	0.375836
$935$	0	0
$936$	0	0
$937$	− 48314.0i	− 1.68447i	−0.539110	−	0.842236i	$-0.681239\pi$
$937$	− 48314.0i	− 1.68447i	0.539110	−	0.842236i	$-0.318761\pi$
$938$	− 28288.0i	− 0.984687i
$939$	0	0
$940$	0	0
$941$	−34782.0	−1.20495	−0.602477	−	0.798137i	$-0.705819\pi$
$941$	−34782.0	−1.20495	−0.602477	+	0.798137i	$0.705819\pi$
$942$	0	0
$943$	7056.00i	0.243664i
$944$	−10560.0	−0.364088
$945$	0	0
$946$	1248.00	0.0428922
$947$	− 25116.0i	− 0.861838i	−0.902391	−	0.430919i	$-0.858190\pi$
$947$	− 25116.0i	− 0.861838i	0.902391	−	0.430919i	$-0.141810\pi$
$948$	0	0
$949$	−8284.00	−0.283361
$950$	0	0
$951$	0	0
$952$	16128.0i	0.549067i
$953$	15462.0i	0.525565i	0.964855	+	0.262782i	$0.0846401\pi$
$953$	15462.0i	0.525565i	−0.964855	+	0.262782i	$0.915360\pi$
$954$	0	0
$955$	0	0
$956$	4800.00	0.162388
$957$	0	0
$958$	− 19680.0i	− 0.663708i
$959$	11616.0	0.391137
$960$	0	0
$961$	−22047.0	−0.740056
$962$	− 19304.0i	− 0.646971i
$963$	0	0
$964$	2872.00	0.0959553
$965$	0	0
$966$	0	0
$967$	736.000i	0.0244759i	0.999925	+	0.0122379i	$0.00389555\pi$
$967$	736.000i	0.0244759i	−0.999925	+	0.0122379i	$0.996104\pi$
$968$	− 9496.00i	− 0.315303i
$969$	0	0
$970$	0	0
$971$	29268.0	0.967307	0.483653	−	0.875260i	$-0.339310\pi$
$971$	29268.0	0.967307	0.483653	+	0.875260i	$0.339310\pi$
$972$	0	0
$973$	− 6080.00i	− 0.200325i
$974$	−2848.00	−0.0936918
$975$	0	0
$976$	−8608.00	−0.282311
$977$	16674.0i	0.546007i	0.962013	+	0.273003i	$0.0880170\pi$
$977$	16674.0i	0.546007i	−0.962013	+	0.273003i	$0.911983\pi$
$978$	0	0
$979$	−9720.00	−0.317316
$980$	0	0
$981$	0	0
$982$	− 9096.00i	− 0.295586i
$983$	31272.0i	1.01467i	0.861749	+	0.507336i	$0.169370\pi$
$983$	31272.0i	1.01467i	−0.861749	+	0.507336i	$0.830630\pi$
$984$	0	0
$985$	0	0
$986$	−7560.00	−0.244178
$987$	0	0
$988$	3040.00i	0.0978900i
$989$	−8736.00	−0.280878
$990$	0	0
$991$	−15928.0	−0.510565	−0.255282	−	0.966867i	$-0.582168\pi$
$991$	−15928.0	−0.510565	−0.255282	+	0.966867i	$0.582168\pi$
$992$	2816.00i	0.0901291i
$993$	0	0
$994$	−25344.0	−0.808715
$995$	0	0
$996$	0	0
$997$	− 42014.0i	− 1.33460i	−0.744789	−	0.667300i	$-0.767450\pi$
$997$	− 42014.0i	− 1.33460i	0.744789	−	0.667300i	$-0.232550\pi$
$998$	13000.0i	0.412332i
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	450.4.c.e.199.1		2
3.2	odd	2		150.4.c.d.49.2		2
5.2	odd	4		18.4.a.a.1.1		1
5.3	odd	4		450.4.a.h.1.1		1
5.4	even	2	inner	450.4.c.e.199.2		2
12.11	even	2		1200.4.f.j.49.2		2
15.2	even	4		6.4.a.a.1.1	✓	1
15.8	even	4		150.4.a.i.1.1		1
15.14	odd	2		150.4.c.d.49.1		2
20.7	even	4		144.4.a.c.1.1		1
35.2	odd	12		882.4.g.i.361.1		2
35.12	even	12		882.4.g.f.361.1		2
35.17	even	12		882.4.g.f.667.1		2
35.27	even	4		882.4.a.n.1.1		1
35.32	odd	12		882.4.g.i.667.1		2
40.27	even	4		576.4.a.r.1.1		1
40.37	odd	4		576.4.a.q.1.1		1
45.2	even	12		162.4.c.f.109.1		2
45.7	odd	12		162.4.c.c.109.1		2
45.22	odd	12		162.4.c.c.55.1		2
45.32	even	12		162.4.c.f.55.1		2
55.32	even	4		2178.4.a.e.1.1		1
60.23	odd	4		1200.4.a.b.1.1		1
60.47	odd	4		48.4.a.c.1.1		1
60.59	even	2		1200.4.f.j.49.1		2
105.2	even	12		294.4.e.h.67.1		2
105.17	odd	12		294.4.e.g.79.1		2
105.32	even	12		294.4.e.h.79.1		2
105.47	odd	12		294.4.e.g.67.1		2
105.62	odd	4		294.4.a.e.1.1		1
120.77	even	4		192.4.a.i.1.1		1
120.107	odd	4		192.4.a.c.1.1		1
165.32	odd	4		726.4.a.f.1.1		1
195.47	odd	4		1014.4.b.d.337.1		2
195.77	even	4		1014.4.a.g.1.1		1
195.122	odd	4		1014.4.b.d.337.2		2
240.77	even	4		768.4.d.n.385.1		2
240.107	odd	4		768.4.d.c.385.1		2
240.197	even	4		768.4.d.n.385.2		2
240.227	odd	4		768.4.d.c.385.2		2
255.152	even	4		1734.4.a.d.1.1		1
285.227	odd	4		2166.4.a.i.1.1		1
420.167	even	4		2352.4.a.e.1.1		1

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
6.4.a.a.1.1	✓	1	15.2	even	4
18.4.a.a.1.1		1	5.2	odd	4
48.4.a.c.1.1		1	60.47	odd	4
144.4.a.c.1.1		1	20.7	even	4
150.4.a.i.1.1		1	15.8	even	4
150.4.c.d.49.1		2	15.14	odd	2
150.4.c.d.49.2		2	3.2	odd	2
162.4.c.c.55.1		2	45.22	odd	12
162.4.c.c.109.1		2	45.7	odd	12
162.4.c.f.55.1		2	45.32	even	12
162.4.c.f.109.1		2	45.2	even	12
192.4.a.c.1.1		1	120.107	odd	4
192.4.a.i.1.1		1	120.77	even	4
294.4.a.e.1.1		1	105.62	odd	4
294.4.e.g.67.1		2	105.47	odd	12
294.4.e.g.79.1		2	105.17	odd	12
294.4.e.h.67.1		2	105.2	even	12
294.4.e.h.79.1		2	105.32	even	12
450.4.a.h.1.1		1	5.3	odd	4
450.4.c.e.199.1		2	1.1	even	1	trivial
450.4.c.e.199.2		2	5.4	even	2	inner
576.4.a.q.1.1		1	40.37	odd	4
576.4.a.r.1.1		1	40.27	even	4
726.4.a.f.1.1		1	165.32	odd	4
768.4.d.c.385.1		2	240.107	odd	4
768.4.d.c.385.2		2	240.227	odd	4
768.4.d.n.385.1		2	240.77	even	4
768.4.d.n.385.2		2	240.197	even	4
882.4.a.n.1.1		1	35.27	even	4
882.4.g.f.361.1		2	35.12	even	12
882.4.g.f.667.1		2	35.17	even	12
882.4.g.i.361.1		2	35.2	odd	12
882.4.g.i.667.1		2	35.32	odd	12
1014.4.a.g.1.1		1	195.77	even	4
1014.4.b.d.337.1		2	195.47	odd	4
1014.4.b.d.337.2		2	195.122	odd	4
1200.4.a.b.1.1		1	60.23	odd	4
1200.4.f.j.49.1		2	60.59	even	2
1200.4.f.j.49.2		2	12.11	even	2
1734.4.a.d.1.1		1	255.152	even	4
2166.4.a.i.1.1		1	285.227	odd	4
2178.4.a.e.1.1		1	55.32	even	4
2352.4.a.e.1.1		1	420.167	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:	\( N \)	\(=\)	\( 450 = 2 \cdot 3^{2} \cdot 5^{2} \)
Weight:	\( k \)	\(=\)	\( 4 \)
Character orbit:	\([\chi]\)	\(=\)	450.c (of order \(2\), degree \(1\), not minimal)

\(f(q)\)	\(=\)	\(q-2.00000i q^{2} -4.00000 q^{4} +16.0000i q^{7} +8.00000i q^{8} +O(q^{10})\)	\(q-2.00000i q^{2} -4.00000 q^{4} +16.0000i q^{7} +8.00000i q^{8} -12.0000 q^{11} +38.0000i q^{13} +32.0000 q^{14} +16.0000 q^{16} -126.000i q^{17} -20.0000 q^{19} +24.0000i q^{22} -168.000i q^{23} +76.0000 q^{26} -64.0000i q^{28} +30.0000 q^{29} -88.0000 q^{31} -32.0000i q^{32} -252.000 q^{34} -254.000i q^{37} +40.0000i q^{38} -42.0000 q^{41} -52.0000i q^{43} +48.0000 q^{44} -336.000 q^{46} -96.0000i q^{47} +87.0000 q^{49} -152.000i q^{52} -198.000i q^{53} -128.000 q^{56} -60.0000i q^{58} -660.000 q^{59} -538.000 q^{61} +176.000i q^{62} -64.0000 q^{64} -884.000i q^{67} +504.000i q^{68} -792.000 q^{71} +218.000i q^{73} -508.000 q^{74} +80.0000 q^{76} -192.000i q^{77} +520.000 q^{79} +84.0000i q^{82} +492.000i q^{83} -104.000 q^{86} -96.0000i q^{88} +810.000 q^{89} -608.000 q^{91} +672.000i q^{92} -192.000 q^{94} -1154.00i q^{97} -174.000i q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 2 q - 8 q^{4}+O(q^{10}) \) 2 * q - 8 * q^4	\( 2 q - 8 q^{4} - 24 q^{11} + 64 q^{14} + 32 q^{16} - 40 q^{19} + 152 q^{26} + 60 q^{29} - 176 q^{31} - 504 q^{34} - 84 q^{41} + 96 q^{44} - 672 q^{46} + 174 q^{49} - 256 q^{56} - 1320 q^{59} - 1076 q^{61} - 128 q^{64} - 1584 q^{71} - 1016 q^{74} + 160 q^{76} + 1040 q^{79} - 208 q^{86} + 1620 q^{89} - 1216 q^{91} - 384 q^{94}+O(q^{100}) \) 2 * q - 8 * q^4 - 24 * q^11 + 64 * q^14 + 32 * q^16 - 40 * q^19 + 152 * q^26 + 60 * q^29 - 176 * q^31 - 504 * q^34 - 84 * q^41 + 96 * q^44 - 672 * q^46 + 174 * q^49 - 256 * q^56 - 1320 * q^59 - 1076 * q^61 - 128 * q^64 - 1584 * q^71 - 1016 * q^74 + 160 * q^76 + 1040 * q^79 - 208 * q^86 + 1620 * q^89 - 1216 * q^91 - 384 * q^94

Introduction

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