LMFDB - Embedded newform 450.6.c.d.199.2

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [450,6,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, 6, names="a")

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

chi := DirichletCharacter("450.199");

S:= CuspForms(chi, 6);

N := Newforms(S);

Level:	$ N $	$=$	$ 450 = 2 \cdot 3^{2} \cdot 5^{2} $
Weight:	$ k $	$=$	$ 6 $
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:	$72.1727189158$
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 90)
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label			199.2
Root			$-1.00000i$ of defining polynomial
Character	$\chi$	$=$	450.199
Dual form			450.6.c.d.199.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+4.00000i q^{2} -16.0000 q^{4} -98.0000i q^{7} -64.0000i q^{8} +O(q^{10})$	$q+4.00000i q^{2} -16.0000 q^{4} -98.0000i q^{7} -64.0000i q^{8} -354.000 q^{11} +404.000i q^{13} +392.000 q^{14} +256.000 q^{16} +654.000i q^{17} -1796.00 q^{19} -1416.00i q^{22} +1080.00i q^{23} -1616.00 q^{26} +1568.00i q^{28} -5754.00 q^{29} +10196.0 q^{31} +1024.00i q^{32} -2616.00 q^{34} -5552.00i q^{37} -7184.00i q^{38} +12960.0 q^{41} -8968.00i q^{43} +5664.00 q^{44} -4320.00 q^{46} -5400.00i q^{47} +7203.00 q^{49} -6464.00i q^{52} -8214.00i q^{53} -6272.00 q^{56} -23016.0i q^{58} +3954.00 q^{59} +962.000 q^{61} +40784.0i q^{62} -4096.00 q^{64} +17956.0i q^{67} -10464.0i q^{68} +56148.0 q^{71} -85690.0i q^{73} +22208.0 q^{74} +28736.0 q^{76} +34692.0i q^{77} +26044.0 q^{79} +51840.0i q^{82} +93468.0i q^{83} +35872.0 q^{86} +22656.0i q^{88} +73428.0 q^{89} +39592.0 q^{91} -17280.0i q^{92} +21600.0 q^{94} -128978. i q^{97} +28812.0i q^{98} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 2 q - 32 q^{4}+O(q^{10}) $ 2 * q - 32 * q^4	$ 2 q - 32 q^{4} - 708 q^{11} + 784 q^{14} + 512 q^{16} - 3592 q^{19} - 3232 q^{26} - 11508 q^{29} + 20392 q^{31} - 5232 q^{34} + 25920 q^{41} + 11328 q^{44} - 8640 q^{46} + 14406 q^{49} - 12544 q^{56} + 7908 q^{59} + 1924 q^{61} - 8192 q^{64} + 112296 q^{71} + 44416 q^{74} + 57472 q^{76} + 52088 q^{79} + 71744 q^{86} + 146856 q^{89} + 79184 q^{91} + 43200 q^{94}+O(q^{100}) $ 2 * q - 32 * q^4 - 708 * q^11 + 784 * q^14 + 512 * q^16 - 3592 * q^19 - 3232 * q^26 - 11508 * q^29 + 20392 * q^31 - 5232 * q^34 + 25920 * q^41 + 11328 * q^44 - 8640 * q^46 + 14406 * q^49 - 12544 * q^56 + 7908 * q^59 + 1924 * q^61 - 8192 * q^64 + 112296 * q^71 + 44416 * q^74 + 57472 * q^76 + 52088 * q^79 + 71744 * q^86 + 146856 * q^89 + 79184 * q^91 + 43200 * 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 $

$2$	4.00000i	0.707107i
$2$	4.00000i	0.707107i
$3$	0	0
$3$	0	0
$4$	−16.0000	−0.500000
$5$	0	0
$5$	0	0
$6$	0	0
$7$	− 98.0000i	− 0.755929i	−0.925820	−	0.377964i	$-0.876624\pi$
$7$	− 98.0000i	− 0.755929i	0.925820	−	0.377964i	$-0.123376\pi$
$8$	− 64.0000i	− 0.353553i
$9$	0	0
$10$	0	0
$11$	−354.000	−0.882108	−0.441054	−	0.897481i	$-0.645395\pi$
$11$	−354.000	−0.882108	−0.441054	+	0.897481i	$0.645395\pi$
$12$	0	0
$13$	404.000i	0.663014i	0.943453	+	0.331507i	$0.107557\pi$
$13$	404.000i	0.663014i	−0.943453	+	0.331507i	$0.892443\pi$
$14$	392.000	0.534522
$15$	0	0
$16$	256.000	0.250000
$17$	654.000i	0.548852i	0.961608	+	0.274426i	$0.0884879\pi$
$17$	654.000i	0.548852i	−0.961608	+	0.274426i	$0.911512\pi$
$18$	0	0
$19$	−1796.00	−1.14136	−0.570680	−	0.821173i	$-0.693320\pi$
$19$	−1796.00	−1.14136	−0.570680	+	0.821173i	$0.693320\pi$
$20$	0	0
$21$	0	0
$22$	− 1416.00i	− 0.623744i
$23$	1080.00i	0.425701i	0.977085	+	0.212850i	$0.0682746\pi$
$23$	1080.00i	0.425701i	−0.977085	+	0.212850i	$0.931725\pi$
$24$	0	0
$25$	0	0
$26$	−1616.00	−0.468822
$27$	0	0
$28$	1568.00i	0.377964i
$29$	−5754.00	−1.27050	−0.635250	−	0.772306i	$-0.719103\pi$
$29$	−5754.00	−1.27050	−0.635250	+	0.772306i	$0.719103\pi$
$30$	0	0
$31$	10196.0	1.90557	0.952787	−	0.303641i	$-0.0982024\pi$
$31$	10196.0	1.90557	0.952787	+	0.303641i	$0.0982024\pi$
$32$	1024.00i	0.176777i
$33$	0	0
$34$	−2616.00	−0.388097
$35$	0	0
$36$	0	0
$37$	− 5552.00i	− 0.666723i	−0.942799	−	0.333361i	$-0.891817\pi$
$37$	− 5552.00i	− 0.666723i	0.942799	−	0.333361i	$-0.108183\pi$
$38$	− 7184.00i	− 0.807063i
$39$	0	0
$40$	0	0
$41$	12960.0	1.20405	0.602026	−	0.798476i	$-0.294360\pi$
$41$	12960.0	1.20405	0.602026	+	0.798476i	$0.294360\pi$
$42$	0	0
$43$	− 8968.00i	− 0.739647i	−0.929102	−	0.369823i	$-0.879418\pi$
$43$	− 8968.00i	− 0.739647i	0.929102	−	0.369823i	$-0.120582\pi$
$44$	5664.00	0.441054
$45$	0	0
$46$	−4320.00	−0.301016
$47$	− 5400.00i	− 0.356574i	−0.983979	−	0.178287i	$-0.942945\pi$
$47$	− 5400.00i	− 0.356574i	0.983979	−	0.178287i	$-0.0570555\pi$
$48$	0	0
$49$	7203.00	0.428571
$50$	0	0
$51$	0	0
$52$	− 6464.00i	− 0.331507i
$53$	− 8214.00i	− 0.401666i	−0.979625	−	0.200833i	$-0.935635\pi$
$53$	− 8214.00i	− 0.401666i	0.979625	−	0.200833i	$-0.0643649\pi$
$54$	0	0
$55$	0	0
$56$	−6272.00	−0.267261
$57$	0	0
$58$	− 23016.0i	− 0.898380i
$59$	3954.00	0.147879	0.0739395	−	0.997263i	$-0.476443\pi$
$59$	3954.00	0.147879	0.0739395	+	0.997263i	$0.476443\pi$
$60$	0	0
$61$	962.000	0.0331017	0.0165509	−	0.999863i	$-0.494731\pi$
$61$	962.000	0.0331017	0.0165509	+	0.999863i	$0.494731\pi$
$62$	40784.0i	1.34744i
$63$	0	0
$64$	−4096.00	−0.125000
$65$	0	0
$66$	0	0
$67$	17956.0i	0.488678i	0.969690	+	0.244339i	$0.0785709\pi$
$67$	17956.0i	0.488678i	−0.969690	+	0.244339i	$0.921429\pi$
$68$	− 10464.0i	− 0.274426i
$69$	0	0
$70$	0	0
$71$	56148.0	1.32187	0.660935	−	0.750444i	$-0.270160\pi$
$71$	56148.0	1.32187	0.660935	+	0.750444i	$0.270160\pi$
$72$	0	0
$73$	− 85690.0i	− 1.88201i	−0.338386	−	0.941007i	$-0.609881\pi$
$73$	− 85690.0i	− 1.88201i	0.338386	−	0.941007i	$-0.390119\pi$
$74$	22208.0	0.471444
$75$	0	0
$76$	28736.0	0.570680
$77$	34692.0i	0.666811i
$78$	0	0
$79$	26044.0	0.469505	0.234752	−	0.972055i	$-0.424572\pi$
$79$	26044.0	0.469505	0.234752	+	0.972055i	$0.424572\pi$
$80$	0	0
$81$	0	0
$82$	51840.0i	0.851394i
$83$	93468.0i	1.48925i	0.667483	+	0.744625i	$0.267372\pi$
$83$	93468.0i	1.48925i	−0.667483	+	0.744625i	$0.732628\pi$
$84$	0	0
$85$	0	0
$86$	35872.0	0.523009
$87$	0	0
$88$	22656.0i	0.311872i
$89$	73428.0	0.982622	0.491311	−	0.870984i	$-0.336518\pi$
$89$	73428.0	0.982622	0.491311	+	0.870984i	$0.336518\pi$
$90$	0	0
$91$	39592.0	0.501192
$92$	− 17280.0i	− 0.212850i
$93$	0	0
$94$	21600.0	0.252136
$95$	0	0
$96$	0	0
$97$	− 128978.i	− 1.39183i	−0.718124	−	0.695915i	$-0.754999\pi$
$97$	− 128978.i	− 1.39183i	0.718124	−	0.695915i	$-0.245001\pi$
$98$	28812.0i	0.303046i
$99$	0	0
$100$	0	0
$101$	154794.	1.50991	0.754954	−	0.655777i	$-0.227659\pi$
$101$	154794.	1.50991	0.754954	+	0.655777i	$0.227659\pi$
$102$	0	0
$103$	27698.0i	0.257250i	0.991693	+	0.128625i	$0.0410563\pi$
$103$	27698.0i	0.257250i	−0.991693	+	0.128625i	$0.958944\pi$
$104$	25856.0	0.234411
$105$	0	0
$106$	32856.0	0.284021
$107$	221172.i	1.86754i	0.357869	+	0.933772i	$0.383503\pi$
$107$	221172.i	1.86754i	−0.357869	+	0.933772i	$0.616497\pi$
$108$	0	0
$109$	−123122.	−0.992589	−0.496294	−	0.868154i	$-0.665306\pi$
$109$	−123122.	−0.992589	−0.496294	+	0.868154i	$0.665306\pi$
$110$	0	0
$111$	0	0
$112$	− 25088.0i	− 0.188982i
$113$	− 220386.i	− 1.62363i	−0.583913	−	0.811817i	$-0.698479\pi$
$113$	− 220386.i	− 1.62363i	0.583913	−	0.811817i	$-0.301521\pi$
$114$	0	0
$115$	0	0
$116$	92064.0	0.635250
$117$	0	0
$118$	15816.0i	0.104566i
$119$	64092.0	0.414893
$120$	0	0
$121$	−35735.0	−0.221886
$122$	3848.00i	0.0234064i
$123$	0	0
$124$	−163136.	−0.952787
$125$	0	0
$126$	0	0
$127$	− 181298.i	− 0.997433i	−0.866765	−	0.498716i	$-0.833805\pi$
$127$	− 181298.i	− 0.997433i	0.866765	−	0.498716i	$-0.166195\pi$
$128$	− 16384.0i	− 0.0883883i
$129$	0	0
$130$	0	0
$131$	254514.	1.29579	0.647893	−	0.761731i	$-0.275650\pi$
$131$	254514.	1.29579	0.647893	+	0.761731i	$0.275650\pi$
$132$	0	0
$133$	176008.i	0.862786i
$134$	−71824.0	−0.345547
$135$	0	0
$136$	41856.0	0.194049
$137$	415506.i	1.89137i	0.325088	+	0.945684i	$0.394606\pi$
$137$	415506.i	1.89137i	−0.325088	+	0.945684i	$0.605394\pi$
$138$	0	0
$139$	312340.	1.37117	0.685584	−	0.727994i	$-0.259547\pi$
$139$	312340.	1.37117	0.685584	+	0.727994i	$0.259547\pi$
$140$	0	0
$141$	0	0
$142$	224592.i	0.934703i
$143$	− 143016.i	− 0.584850i
$144$	0	0
$145$	0	0
$146$	342760.	1.33079
$147$	0	0
$148$	88832.0i	0.333361i
$149$	300954.	1.11054	0.555270	−	0.831670i	$-0.312615\pi$
$149$	300954.	1.11054	0.555270	+	0.831670i	$0.312615\pi$
$150$	0	0
$151$	−37576.0	−0.134112	−0.0670561	−	0.997749i	$-0.521361\pi$
$151$	−37576.0	−0.134112	−0.0670561	+	0.997749i	$0.521361\pi$
$152$	114944.i	0.403531i
$153$	0	0
$154$	−138768.	−0.471506
$155$	0	0
$156$	0	0
$157$	378316.i	1.22491i	0.790504	+	0.612457i	$0.209819\pi$
$157$	378316.i	1.22491i	−0.790504	+	0.612457i	$0.790181\pi$
$158$	104176.i	0.331990i
$159$	0	0
$160$	0	0
$161$	105840.	0.321799
$162$	0	0
$163$	66980.0i	0.197459i	0.995114	+	0.0987293i	$0.0314778\pi$
$163$	66980.0i	0.197459i	−0.995114	+	0.0987293i	$0.968522\pi$
$164$	−207360.	−0.602026
$165$	0	0
$166$	−373872.	−1.05306
$167$	− 83424.0i	− 0.231473i	−0.993280	−	0.115736i	$-0.963077\pi$
$167$	− 83424.0i	− 0.231473i	0.993280	−	0.115736i	$-0.0369228\pi$
$168$	0	0
$169$	208077.	0.560412
$170$	0	0
$171$	0	0
$172$	143488.i	0.369823i
$173$	− 42318.0i	− 0.107500i	−0.998554	−	0.0537502i	$-0.982883\pi$
$173$	− 42318.0i	− 0.107500i	0.998554	−	0.0537502i	$-0.0171175\pi$
$174$	0	0
$175$	0	0
$176$	−90624.0	−0.220527
$177$	0	0
$178$	293712.i	0.694819i
$179$	−167406.	−0.390516	−0.195258	−	0.980752i	$-0.562554\pi$
$179$	−167406.	−0.390516	−0.195258	+	0.980752i	$0.562554\pi$
$180$	0	0
$181$	−483118.	−1.09612	−0.548058	−	0.836440i	$-0.684633\pi$
$181$	−483118.	−1.09612	−0.548058	+	0.836440i	$0.684633\pi$
$182$	158368.i	0.354396i
$183$	0	0
$184$	69120.0	0.150508
$185$	0	0
$186$	0	0
$187$	− 231516.i	− 0.484147i
$188$	86400.0i	0.178287i
$189$	0	0
$190$	0	0
$191$	772572.	1.53234	0.766171	−	0.642637i	$-0.222160\pi$
$191$	772572.	1.53234	0.766171	+	0.642637i	$0.222160\pi$
$192$	0	0
$193$	− 670570.i	− 1.29584i	−0.761709	−	0.647919i	$-0.775639\pi$
$193$	− 670570.i	− 1.29584i	0.761709	−	0.647919i	$-0.224361\pi$
$194$	515912.	0.984173
$195$	0	0
$196$	−115248.	−0.214286
$197$	851922.i	1.56399i	0.623284	+	0.781996i	$0.285798\pi$
$197$	851922.i	1.56399i	−0.623284	+	0.781996i	$0.714202\pi$
$198$	0	0
$199$	561256.	1.00468	0.502341	−	0.864670i	$-0.332472\pi$
$199$	561256.	1.00468	0.502341	+	0.864670i	$0.332472\pi$
$200$	0	0
$201$	0	0
$202$	619176.i	1.06767i
$203$	563892.i	0.960408i
$204$	0	0
$205$	0	0
$206$	−110792.	−0.181903
$207$	0	0
$208$	103424.i	0.165754i
$209$	635784.	1.00680
$210$	0	0
$211$	−578764.	−0.894943	−0.447471	−	0.894298i	$-0.647675\pi$
$211$	−578764.	−0.894943	−0.447471	+	0.894298i	$0.647675\pi$
$212$	131424.i	0.200833i
$213$	0	0
$214$	−884688.	−1.32055
$215$	0	0
$216$	0	0
$217$	− 999208.i	− 1.44048i
$218$	− 492488.i	− 0.701866i
$219$	0	0
$220$	0	0
$221$	−264216.	−0.363897
$222$	0	0
$223$	863114.i	1.16227i	0.813808	+	0.581134i	$0.197391\pi$
$223$	863114.i	1.16227i	−0.813808	+	0.581134i	$0.802609\pi$
$224$	100352.	0.133631
$225$	0	0
$226$	881544.	1.14808
$227$	− 712800.i	− 0.918128i	−0.888403	−	0.459064i	$-0.848185\pi$
$227$	− 712800.i	− 0.918128i	0.888403	−	0.459064i	$-0.151815\pi$
$228$	0	0
$229$	343666.	0.433060	0.216530	−	0.976276i	$-0.430526\pi$
$229$	343666.	0.433060	0.216530	+	0.976276i	$0.430526\pi$
$230$	0	0
$231$	0	0
$232$	368256.i	0.449190i
$233$	− 218814.i	− 0.264049i	−0.991246	−	0.132025i	$-0.957852\pi$
$233$	− 218814.i	− 0.264049i	0.991246	−	0.132025i	$-0.0421478\pi$
$234$	0	0
$235$	0	0
$236$	−63264.0	−0.0739395
$237$	0	0
$238$	256368.i	0.293374i
$239$	−1.02095e6	−1.15614	−0.578068	−	0.815989i	$-0.696193\pi$
$239$	−1.02095e6	−1.15614	−0.578068	+	0.815989i	$0.696193\pi$
$240$	0	0
$241$	1.37183e6	1.52145	0.760725	−	0.649074i	$-0.224844\pi$
$241$	1.37183e6	1.52145	0.760725	+	0.649074i	$0.224844\pi$
$242$	− 142940.i	− 0.156897i
$243$	0	0
$244$	−15392.0	−0.0165509
$245$	0	0
$246$	0	0
$247$	− 725584.i	− 0.756738i
$248$	− 652544.i	− 0.673722i
$249$	0	0
$250$	0	0
$251$	932058.	0.933810	0.466905	−	0.884307i	$-0.345369\pi$
$251$	932058.	0.933810	0.466905	+	0.884307i	$0.345369\pi$
$252$	0	0
$253$	− 382320.i	− 0.375514i
$254$	725192.	0.705292
$255$	0	0
$256$	65536.0	0.0625000
$257$	− 1.26569e6i	− 1.19535i	−0.801738	−	0.597676i	$-0.796091\pi$
$257$	− 1.26569e6i	− 1.19535i	0.801738	−	0.597676i	$-0.203909\pi$
$258$	0	0
$259$	−544096.	−0.503995
$260$	0	0
$261$	0	0
$262$	1.01806e6i	0.916259i
$263$	699936.i	0.623978i	0.950086	+	0.311989i	$0.100995\pi$
$263$	699936.i	0.623978i	−0.950086	+	0.311989i	$0.899005\pi$
$264$	0	0
$265$	0	0
$266$	−704032.	−0.610082
$267$	0	0
$268$	− 287296.i	− 0.244339i
$269$	−488178.	−0.411337	−0.205668	−	0.978622i	$-0.565937\pi$
$269$	−488178.	−0.411337	−0.205668	+	0.978622i	$0.565937\pi$
$270$	0	0
$271$	525968.	0.435047	0.217523	−	0.976055i	$-0.430202\pi$
$271$	525968.	0.435047	0.217523	+	0.976055i	$0.430202\pi$
$272$	167424.i	0.137213i
$273$	0	0
$274$	−1.66202e6	−1.33740
$275$	0	0
$276$	0	0
$277$	− 2.42931e6i	− 1.90232i	−0.308696	−	0.951161i	$-0.599893\pi$
$277$	− 2.42931e6i	− 1.90232i	0.308696	−	0.951161i	$-0.400107\pi$
$278$	1.24936e6i	0.969562i
$279$	0	0
$280$	0	0
$281$	−2.29031e6	−1.73033	−0.865163	−	0.501490i	$-0.832785\pi$
$281$	−2.29031e6	−1.73033	−0.865163	+	0.501490i	$0.832785\pi$
$282$	0	0
$283$	− 2.09156e6i	− 1.55240i	−0.630487	−	0.776200i	$-0.717145\pi$
$283$	− 2.09156e6i	− 1.55240i	0.630487	−	0.776200i	$-0.282855\pi$
$284$	−898368.	−0.660935
$285$	0	0
$286$	572064.	0.413551
$287$	− 1.27008e6i	− 0.910178i
$288$	0	0
$289$	992141.	0.698761
$290$	0	0
$291$	0	0
$292$	1.37104e6i	0.941007i
$293$	107118.i	0.0728943i	0.999336	+	0.0364471i	$0.0116041\pi$
$293$	107118.i	0.0728943i	−0.999336	+	0.0364471i	$0.988396\pi$
$294$	0	0
$295$	0	0
$296$	−355328.	−0.235722
$297$	0	0
$298$	1.20382e6i	0.785271i
$299$	−436320.	−0.282246
$300$	0	0
$301$	−878864.	−0.559121
$302$	− 150304.i	− 0.0948316i
$303$	0	0
$304$	−459776.	−0.285340
$305$	0	0
$306$	0	0
$307$	− 1.12938e6i	− 0.683900i	−0.939718	−	0.341950i	$-0.888913\pi$
$307$	− 1.12938e6i	− 0.683900i	0.939718	−	0.341950i	$-0.111087\pi$
$308$	− 555072.i	− 0.333405i
$309$	0	0
$310$	0	0
$311$	−3.06145e6	−1.79484	−0.897422	−	0.441174i	$-0.854562\pi$
$311$	−3.06145e6	−1.79484	−0.897422	+	0.441174i	$0.854562\pi$
$312$	0	0
$313$	− 1.23025e6i	− 0.709794i	−0.934905	−	0.354897i	$-0.884516\pi$
$313$	− 1.23025e6i	− 0.709794i	0.934905	−	0.354897i	$-0.115484\pi$
$314$	−1.51326e6	−0.866145
$315$	0	0
$316$	−416704.	−0.234752
$317$	679122.i	0.379577i	0.981825	+	0.189788i	$0.0607802\pi$
$317$	679122.i	0.379577i	−0.981825	+	0.189788i	$0.939220\pi$
$318$	0	0
$319$	2.03692e6	1.12072
$320$	0	0
$321$	0	0
$322$	423360.i	0.227546i
$323$	− 1.17458e6i	− 0.626438i
$324$	0	0
$325$	0	0
$326$	−267920.	−0.139624
$327$	0	0
$328$	− 829440.i	− 0.425697i
$329$	−529200.	−0.269544
$330$	0	0
$331$	98516.0	0.0494239	0.0247119	−	0.999695i	$-0.492133\pi$
$331$	98516.0	0.0494239	0.0247119	+	0.999695i	$0.492133\pi$
$332$	− 1.49549e6i	− 0.744625i
$333$	0	0
$334$	333696.	0.163676
$335$	0	0
$336$	0	0
$337$	1.99779e6i	0.958244i	0.877748	+	0.479122i	$0.159045\pi$
$337$	1.99779e6i	0.958244i	−0.877748	+	0.479122i	$0.840955\pi$
$338$	832308.i	0.396271i
$339$	0	0
$340$	0	0
$341$	−3.60938e6	−1.68092
$342$	0	0
$343$	− 2.35298e6i	− 1.07990i
$344$	−573952.	−0.261505
$345$	0	0
$346$	169272.	0.0760142
$347$	− 2.65032e6i	− 1.18161i	−0.806814	−	0.590806i	$-0.798810\pi$
$347$	− 2.65032e6i	− 1.18161i	0.806814	−	0.590806i	$-0.201190\pi$
$348$	0	0
$349$	2.03378e6	0.893801	0.446900	−	0.894584i	$-0.352528\pi$
$349$	2.03378e6	0.893801	0.446900	+	0.894584i	$0.352528\pi$
$350$	0	0
$351$	0	0
$352$	− 362496.i	− 0.155936i
$353$	− 948630.i	− 0.405191i	−0.979262	−	0.202596i	$-0.935062\pi$
$353$	− 948630.i	− 0.405191i	0.979262	−	0.202596i	$-0.0649377\pi$
$354$	0	0
$355$	0	0
$356$	−1.17485e6	−0.491311
$357$	0	0
$358$	− 669624.i	− 0.276136i
$359$	494616.	0.202550	0.101275	−	0.994858i	$-0.467708\pi$
$359$	494616.	0.202550	0.101275	+	0.994858i	$0.467708\pi$
$360$	0	0
$361$	749517.	0.302701
$362$	− 1.93247e6i	− 0.775072i
$363$	0	0
$364$	−633472.	−0.250596
$365$	0	0
$366$	0	0
$367$	75046.0i	0.0290846i	0.999894	+	0.0145423i	$0.00462911\pi$
$367$	75046.0i	0.0290846i	−0.999894	+	0.0145423i	$0.995371\pi$
$368$	276480.i	0.106425i
$369$	0	0
$370$	0	0
$371$	−804972.	−0.303631
$372$	0	0
$373$	533468.i	0.198535i	0.995061	+	0.0992673i	$0.0316499\pi$
$373$	533468.i	0.198535i	−0.995061	+	0.0992673i	$0.968350\pi$
$374$	926064.	0.342343
$375$	0	0
$376$	−345600.	−0.126068
$377$	− 2.32462e6i	− 0.842360i
$378$	0	0
$379$	199780.	0.0714421	0.0357210	−	0.999362i	$-0.488627\pi$
$379$	199780.	0.0714421	0.0357210	+	0.999362i	$0.488627\pi$
$380$	0	0
$381$	0	0
$382$	3.09029e6i	1.08353i
$383$	869400.i	0.302847i	0.988469	+	0.151423i	$0.0483857\pi$
$383$	869400.i	0.302847i	−0.988469	+	0.151423i	$0.951614\pi$
$384$	0	0
$385$	0	0
$386$	2.68228e6	0.916296
$387$	0	0
$388$	2.06365e6i	0.695915i
$389$	−1.05190e6	−0.352453	−0.176227	−	0.984350i	$-0.556389\pi$
$389$	−1.05190e6	−0.352453	−0.176227	+	0.984350i	$0.556389\pi$
$390$	0	0
$391$	−706320.	−0.233647
$392$	− 460992.i	− 0.151523i
$393$	0	0
$394$	−3.40769e6	−1.10591
$395$	0	0
$396$	0	0
$397$	− 4.19262e6i	− 1.33508i	−0.744572	−	0.667542i	$-0.767346\pi$
$397$	− 4.19262e6i	− 1.33508i	0.744572	−	0.667542i	$-0.232654\pi$
$398$	2.24502e6i	0.710417i
$399$	0	0
$400$	0	0
$401$	1.90292e6	0.590963	0.295482	−	0.955348i	$-0.404520\pi$
$401$	1.90292e6	0.590963	0.295482	+	0.955348i	$0.404520\pi$
$402$	0	0
$403$	4.11918e6i	1.26342i
$404$	−2.47670e6	−0.754954
$405$	0	0
$406$	−2.25557e6	−0.679111
$407$	1.96541e6i	0.588121i
$408$	0	0
$409$	−1.62910e6	−0.481547	−0.240774	−	0.970581i	$-0.577401\pi$
$409$	−1.62910e6	−0.481547	−0.240774	+	0.970581i	$0.577401\pi$
$410$	0	0
$411$	0	0
$412$	− 443168.i	− 0.128625i
$413$	− 387492.i	− 0.111786i
$414$	0	0
$415$	0	0
$416$	−413696.	−0.117206
$417$	0	0
$418$	2.54314e6i	0.711916i
$419$	−2.33027e6	−0.648443	−0.324222	−	0.945981i	$-0.605102\pi$
$419$	−2.33027e6	−0.648443	−0.324222	+	0.945981i	$0.605102\pi$
$420$	0	0
$421$	612974.	0.168553	0.0842766	−	0.996442i	$-0.473142\pi$
$421$	612974.	0.168553	0.0842766	+	0.996442i	$0.473142\pi$
$422$	− 2.31506e6i	− 0.632820i
$423$	0	0
$424$	−525696.	−0.142010
$425$	0	0
$426$	0	0
$427$	− 94276.0i	− 0.0250225i
$428$	− 3.53875e6i	− 0.933772i
$429$	0	0
$430$	0	0
$431$	−6.69384e6	−1.73573	−0.867865	−	0.496800i	$-0.834508\pi$
$431$	−6.69384e6	−1.73573	−0.867865	+	0.496800i	$0.834508\pi$
$432$	0	0
$433$	2.20194e6i	0.564399i	0.959356	+	0.282199i	$0.0910640\pi$
$433$	2.20194e6i	0.564399i	−0.959356	+	0.282199i	$0.908936\pi$
$434$	3.99683e6	1.01857
$435$	0	0
$436$	1.96995e6	0.496294
$437$	− 1.93968e6i	− 0.485877i
$438$	0	0
$439$	−1.10437e6	−0.273497	−0.136748	−	0.990606i	$-0.543665\pi$
$439$	−1.10437e6	−0.273497	−0.136748	+	0.990606i	$0.543665\pi$
$440$	0	0
$441$	0	0
$442$	− 1.05686e6i	− 0.257314i
$443$	3.82202e6i	0.925303i	0.886540	+	0.462652i	$0.153102\pi$
$443$	3.82202e6i	0.925303i	−0.886540	+	0.462652i	$0.846898\pi$
$444$	0	0
$445$	0	0
$446$	−3.45246e6	−0.821847
$447$	0	0
$448$	401408.i	0.0944911i
$449$	3.93985e6	0.922283	0.461141	−	0.887327i	$-0.347440\pi$
$449$	3.93985e6	0.922283	0.461141	+	0.887327i	$0.347440\pi$
$450$	0	0
$451$	−4.58784e6	−1.06210
$452$	3.52618e6i	0.811817i
$453$	0	0
$454$	2.85120e6	0.649214
$455$	0	0
$456$	0	0
$457$	1.01753e6i	0.227906i	0.993486	+	0.113953i	$0.0363513\pi$
$457$	1.01753e6i	0.227906i	−0.993486	+	0.113953i	$0.963649\pi$
$458$	1.37466e6i	0.306220i
$459$	0	0
$460$	0	0
$461$	6.80830e6	1.49206	0.746030	−	0.665912i	$-0.231957\pi$
$461$	6.80830e6	1.49206	0.746030	+	0.665912i	$0.231957\pi$
$462$	0	0
$463$	6.74281e6i	1.46180i	0.682483	+	0.730901i	$0.260900\pi$
$463$	6.74281e6i	1.46180i	−0.682483	+	0.730901i	$0.739100\pi$
$464$	−1.47302e6	−0.317625
$465$	0	0
$466$	875256.	0.186711
$467$	3.22619e6i	0.684538i	0.939602	+	0.342269i	$0.111195\pi$
$467$	3.22619e6i	0.684538i	−0.939602	+	0.342269i	$0.888805\pi$
$468$	0	0
$469$	1.75969e6	0.369406
$470$	0	0
$471$	0	0
$472$	− 253056.i	− 0.0522831i
$473$	3.17467e6i	0.652448i
$474$	0	0
$475$	0	0
$476$	−1.02547e6	−0.207447
$477$	0	0
$478$	− 4.08379e6i	− 0.817512i
$479$	6.21212e6	1.23709	0.618545	−	0.785749i	$-0.287722\pi$
$479$	6.21212e6	1.23709	0.618545	+	0.785749i	$0.287722\pi$
$480$	0	0
$481$	2.24301e6	0.442047
$482$	5.48732e6i	1.07583i
$483$	0	0
$484$	571760.	0.110943
$485$	0	0
$486$	0	0
$487$	1.51326e6i	0.289128i	0.989495	+	0.144564i	$0.0461780\pi$
$487$	1.51326e6i	0.289128i	−0.989495	+	0.144564i	$0.953822\pi$
$488$	− 61568.0i	− 0.0117032i
$489$	0	0
$490$	0	0
$491$	3.43835e6	0.643646	0.321823	−	0.946800i	$-0.395704\pi$
$491$	3.43835e6	0.643646	0.321823	+	0.946800i	$0.395704\pi$
$492$	0	0
$493$	− 3.76312e6i	− 0.697317i
$494$	2.90234e6	0.535094
$495$	0	0
$496$	2.61018e6	0.476393
$497$	− 5.50250e6i	− 0.999239i
$498$	0	0
$499$	−5.27340e6	−0.948067	−0.474033	−	0.880507i	$-0.657202\pi$
$499$	−5.27340e6	−0.948067	−0.474033	+	0.880507i	$0.657202\pi$
$500$	0	0
$501$	0	0
$502$	3.72823e6i	0.660304i
$503$	− 8.81172e6i	− 1.55289i	−0.630185	−	0.776445i	$-0.717021\pi$
$503$	− 8.81172e6i	− 1.55289i	0.630185	−	0.776445i	$-0.282979\pi$
$504$	0	0
$505$	0	0
$506$	1.52928e6	0.265528
$507$	0	0
$508$	2.90077e6i	0.498716i
$509$	−1.03788e7	−1.77563	−0.887817	−	0.460197i	$-0.847779\pi$
$509$	−1.03788e7	−1.77563	−0.887817	+	0.460197i	$0.847779\pi$
$510$	0	0
$511$	−8.39762e6	−1.42267
$512$	262144.i	0.0441942i
$513$	0	0
$514$	5.06278e6	0.845242
$515$	0	0
$516$	0	0
$517$	1.91160e6i	0.314536i
$518$	− 2.17638e6i	− 0.356378i
$519$	0	0
$520$	0	0
$521$	4.72177e6	0.762098	0.381049	−	0.924555i	$-0.375563\pi$
$521$	4.72177e6	0.762098	0.381049	+	0.924555i	$0.375563\pi$
$522$	0	0
$523$	− 4.24340e6i	− 0.678359i	−0.940722	−	0.339179i	$-0.889851\pi$
$523$	− 4.24340e6i	− 0.678359i	0.940722	−	0.339179i	$-0.110149\pi$
$524$	−4.07222e6	−0.647893
$525$	0	0
$526$	−2.79974e6	−0.441219
$527$	6.66818e6i	1.04588i
$528$	0	0
$529$	5.26994e6	0.818779
$530$	0	0
$531$	0	0
$532$	− 2.81613e6i	− 0.431393i
$533$	5.23584e6i	0.798304i
$534$	0	0
$535$	0	0
$536$	1.14918e6	0.172774
$537$	0	0
$538$	− 1.95271e6i	− 0.290859i
$539$	−2.54986e6	−0.378046
$540$	0	0
$541$	−1.30682e6	−0.191966	−0.0959828	−	0.995383i	$-0.530599\pi$
$541$	−1.30682e6	−0.191966	−0.0959828	+	0.995383i	$0.530599\pi$
$542$	2.10387e6i	0.307625i
$543$	0	0
$544$	−669696.	−0.0970243
$545$	0	0
$546$	0	0
$547$	9.69388e6i	1.38525i	0.721296	+	0.692627i	$0.243547\pi$
$547$	9.69388e6i	1.38525i	−0.721296	+	0.692627i	$0.756453\pi$
$548$	− 6.64810e6i	− 0.945684i
$549$	0	0
$550$	0	0
$551$	1.03342e7	1.45010
$552$	0	0
$553$	− 2.55231e6i	− 0.354912i
$554$	9.71725e6	1.34514
$555$	0	0
$556$	−4.99744e6	−0.685584
$557$	8.47666e6i	1.15768i	0.815443	+	0.578838i	$0.196494\pi$
$557$	8.47666e6i	1.15768i	−0.815443	+	0.578838i	$0.803506\pi$
$558$	0	0
$559$	3.62307e6	0.490397
$560$	0	0
$561$	0	0
$562$	− 9.16123e6i	− 1.22353i
$563$	− 1.15102e6i	− 0.153042i	−0.997068	−	0.0765210i	$-0.975619\pi$
$563$	− 1.15102e6i	− 0.153042i	0.997068	−	0.0765210i	$-0.0243812\pi$
$564$	0	0
$565$	0	0
$566$	8.36622e6	1.09771
$567$	0	0
$568$	− 3.59347e6i	− 0.467351i
$569$	−45396.0	−0.00587810	−0.00293905	−	0.999996i	$-0.500936\pi$
$569$	−45396.0	−0.00587810	−0.00293905	+	0.999996i	$0.500936\pi$
$570$	0	0
$571$	6.43696e6	0.826211	0.413105	−	0.910683i	$-0.364444\pi$
$571$	6.43696e6	0.826211	0.413105	+	0.910683i	$0.364444\pi$
$572$	2.28826e6i	0.292425i
$573$	0	0
$574$	5.08032e6	0.643593
$575$	0	0
$576$	0	0
$577$	578758.i	0.0723698i	0.999345	+	0.0361849i	$0.0115205\pi$
$577$	578758.i	0.0723698i	−0.999345	+	0.0361849i	$0.988479\pi$
$578$	3.96856e6i	0.494099i
$579$	0	0
$580$	0	0
$581$	9.15986e6	1.12577
$582$	0	0
$583$	2.90776e6i	0.354313i
$584$	−5.48416e6	−0.665393
$585$	0	0
$586$	−428472.	−0.0515440
$587$	960804.i	0.115091i	0.998343	+	0.0575453i	$0.0183273\pi$
$587$	960804.i	0.115091i	−0.998343	+	0.0575453i	$0.981673\pi$
$588$	0	0
$589$	−1.83120e7	−2.17494
$590$	0	0
$591$	0	0
$592$	− 1.42131e6i	− 0.166681i
$593$	− 5.70220e6i	− 0.665895i	−0.942945	−	0.332948i	$-0.891957\pi$
$593$	− 5.70220e6i	− 0.665895i	0.942945	−	0.332948i	$-0.108043\pi$
$594$	0	0
$595$	0	0
$596$	−4.81526e6	−0.555270
$597$	0	0
$598$	− 1.74528e6i	− 0.199578i
$599$	1.53569e7	1.74879	0.874393	−	0.485219i	$-0.161260\pi$
$599$	1.53569e7	1.74879	0.874393	+	0.485219i	$0.161260\pi$
$600$	0	0
$601$	1.21690e7	1.37425	0.687127	−	0.726537i	$-0.258872\pi$
$601$	1.21690e7	1.37425	0.687127	+	0.726537i	$0.258872\pi$
$602$	− 3.51546e6i	− 0.395358i
$603$	0	0
$604$	601216.	0.0670561
$605$	0	0
$606$	0	0
$607$	2.94630e6i	0.324567i	0.986744	+	0.162284i	$0.0518860\pi$
$607$	2.94630e6i	0.324567i	−0.986744	+	0.162284i	$0.948114\pi$
$608$	− 1.83910e6i	− 0.201766i
$609$	0	0
$610$	0	0
$611$	2.18160e6	0.236413
$612$	0	0
$613$	− 1.06180e7i	− 1.14128i	−0.821199	−	0.570642i	$-0.806695\pi$
$613$	− 1.06180e7i	− 1.14128i	0.821199	−	0.570642i	$-0.193305\pi$
$614$	4.51750e6	0.483590
$615$	0	0
$616$	2.22029e6	0.235753
$617$	1.14174e7i	1.20741i	0.797207	+	0.603706i	$0.206310\pi$
$617$	1.14174e7i	1.20741i	−0.797207	+	0.603706i	$0.793690\pi$
$618$	0	0
$619$	1.28242e6	0.134525	0.0672626	−	0.997735i	$-0.478573\pi$
$619$	1.28242e6	0.134525	0.0672626	+	0.997735i	$0.478573\pi$
$620$	0	0
$621$	0	0
$622$	− 1.22458e7i	− 1.26915i
$623$	− 7.19594e6i	− 0.742793i
$624$	0	0
$625$	0	0
$626$	4.92100e6	0.501900
$627$	0	0
$628$	− 6.05306e6i	− 0.612457i
$629$	3.63101e6	0.365932
$630$	0	0
$631$	7.37894e6	0.737769	0.368885	−	0.929475i	$-0.379740\pi$
$631$	7.37894e6	0.737769	0.368885	+	0.929475i	$0.379740\pi$
$632$	− 1.66682e6i	− 0.165995i
$633$	0	0
$634$	−2.71649e6	−0.268401
$635$	0	0
$636$	0	0
$637$	2.91001e6i	0.284149i
$638$	8.14766e6i	0.792467i
$639$	0	0
$640$	0	0
$641$	1.63802e6	0.157462	0.0787309	−	0.996896i	$-0.474913\pi$
$641$	1.63802e6	0.157462	0.0787309	+	0.996896i	$0.474913\pi$
$642$	0	0
$643$	2.03847e7i	1.94436i	0.234236	+	0.972180i	$0.424741\pi$
$643$	2.03847e7i	1.94436i	−0.234236	+	0.972180i	$0.575259\pi$
$644$	−1.69344e6	−0.160900
$645$	0	0
$646$	4.69834e6	0.442958
$647$	2.10318e7i	1.97522i	0.156939	+	0.987608i	$0.449838\pi$
$647$	2.10318e7i	1.97522i	−0.156939	+	0.987608i	$0.550162\pi$
$648$	0	0
$649$	−1.39972e6	−0.130445
$650$	0	0
$651$	0	0
$652$	− 1.07168e6i	− 0.0987293i
$653$	7.85504e6i	0.720884i	0.932782	+	0.360442i	$0.117374\pi$
$653$	7.85504e6i	0.720884i	−0.932782	+	0.360442i	$0.882626\pi$
$654$	0	0
$655$	0	0
$656$	3.31776e6	0.301013
$657$	0	0
$658$	− 2.11680e6i	− 0.190597i
$659$	9.32727e6	0.836645	0.418322	−	0.908299i	$-0.362618\pi$
$659$	9.32727e6	0.836645	0.418322	+	0.908299i	$0.362618\pi$
$660$	0	0
$661$	1.57325e7	1.40053	0.700267	−	0.713881i	$-0.253064\pi$
$661$	1.57325e7	1.40053	0.700267	+	0.713881i	$0.253064\pi$
$662$	394064.i	0.0349480i
$663$	0	0
$664$	5.98195e6	0.526530
$665$	0	0
$666$	0	0
$667$	− 6.21432e6i	− 0.540853i
$668$	1.33478e6i	0.115736i
$669$	0	0
$670$	0	0
$671$	−340548.	−0.0291993
$672$	0	0
$673$	− 1.26562e7i	− 1.07712i	−0.842586	−	0.538562i	$-0.818968\pi$
$673$	− 1.26562e7i	− 1.07712i	0.842586	−	0.538562i	$-0.181032\pi$
$674$	−7.99118e6	−0.677581
$675$	0	0
$676$	−3.32923e6	−0.280206
$677$	− 2.65061e6i	− 0.222267i	−0.993805	−	0.111133i	$-0.964552\pi$
$677$	− 2.65061e6i	− 0.222267i	0.993805	−	0.111133i	$-0.0354481\pi$
$678$	0	0
$679$	−1.26398e7	−1.05212
$680$	0	0
$681$	0	0
$682$	− 1.44375e7i	− 1.18859i
$683$	5.74475e6i	0.471215i	0.971848	+	0.235608i	$0.0757080\pi$
$683$	5.74475e6i	0.471215i	−0.971848	+	0.235608i	$0.924292\pi$
$684$	0	0
$685$	0	0
$686$	9.41192e6	0.763604
$687$	0	0
$688$	− 2.29581e6i	− 0.184912i
$689$	3.31846e6	0.266310
$690$	0	0
$691$	−1.43749e7	−1.14527	−0.572636	−	0.819809i	$-0.694079\pi$
$691$	−1.43749e7	−1.14527	−0.572636	+	0.819809i	$0.694079\pi$
$692$	677088.i	0.0537502i
$693$	0	0
$694$	1.06013e7	0.835525
$695$	0	0
$696$	0	0
$697$	8.47584e6i	0.660847i
$698$	8.13513e6i	0.632013i
$699$	0	0
$700$	0	0
$701$	−4.26164e6	−0.327553	−0.163776	−	0.986497i	$-0.552368\pi$
$701$	−4.26164e6	−0.327553	−0.163776	+	0.986497i	$0.552368\pi$
$702$	0	0
$703$	9.97139e6i	0.760970i
$704$	1.44998e6	0.110263
$705$	0	0
$706$	3.79452e6	0.286514
$707$	− 1.51698e7i	− 1.14138i
$708$	0	0
$709$	−1.97088e7	−1.47246	−0.736231	−	0.676731i	$-0.763396\pi$
$709$	−1.97088e7	−1.47246	−0.736231	+	0.676731i	$0.763396\pi$
$710$	0	0
$711$	0	0
$712$	− 4.69939e6i	− 0.347409i
$713$	1.10117e7i	0.811203i
$714$	0	0
$715$	0	0
$716$	2.67850e6	0.195258
$717$	0	0
$718$	1.97846e6i	0.143224i
$719$	−1.02211e7	−0.737353	−0.368676	−	0.929558i	$-0.620189\pi$
$719$	−1.02211e7	−0.737353	−0.368676	+	0.929558i	$0.620189\pi$
$720$	0	0
$721$	2.71440e6	0.194463
$722$	2.99807e6i	0.214042i
$723$	0	0
$724$	7.72989e6	0.548058
$725$	0	0
$726$	0	0
$727$	1.32103e7i	0.926996i	0.886098	+	0.463498i	$0.153406\pi$
$727$	1.32103e7i	0.926996i	−0.886098	+	0.463498i	$0.846594\pi$
$728$	− 2.53389e6i	− 0.177198i
$729$	0	0
$730$	0	0
$731$	5.86507e6	0.405957
$732$	0	0
$733$	− 1.30586e7i	− 0.897713i	−0.893604	−	0.448857i	$-0.851831\pi$
$733$	− 1.30586e7i	− 0.897713i	0.893604	−	0.448857i	$-0.148169\pi$
$734$	−300184.	−0.0205659
$735$	0	0
$736$	−1.10592e6	−0.0752539
$737$	− 6.35642e6i	− 0.431066i
$738$	0	0
$739$	1.53767e7	1.03574	0.517870	−	0.855459i	$-0.326725\pi$
$739$	1.53767e7	1.03574	0.517870	+	0.855459i	$0.326725\pi$
$740$	0	0
$741$	0	0
$742$	− 3.21989e6i	− 0.214699i
$743$	− 2.62537e7i	− 1.74469i	−0.488889	−	0.872346i	$-0.662598\pi$
$743$	− 2.62537e7i	− 1.74469i	0.488889	−	0.872346i	$-0.337402\pi$
$744$	0	0
$745$	0	0
$746$	−2.13387e6	−0.140385
$747$	0	0
$748$	3.70426e6i	0.242073i
$749$	2.16749e7	1.41173
$750$	0	0
$751$	−1.36151e7	−0.880886	−0.440443	−	0.897781i	$-0.645179\pi$
$751$	−1.36151e7	−0.880886	−0.440443	+	0.897781i	$0.645179\pi$
$752$	− 1.38240e6i	− 0.0891434i
$753$	0	0
$754$	9.29846e6	0.595639
$755$	0	0
$756$	0	0
$757$	− 8.41208e6i	− 0.533536i	−0.963761	−	0.266768i	$-0.914044\pi$
$757$	− 8.41208e6i	− 0.533536i	0.963761	−	0.266768i	$-0.0859557\pi$
$758$	799120.i	0.0505172i
$759$	0	0
$760$	0	0
$761$	−414684.	−0.0259571	−0.0129785	−	0.999916i	$-0.504131\pi$
$761$	−414684.	−0.0259571	−0.0129785	+	0.999916i	$0.504131\pi$
$762$	0	0
$763$	1.20660e7i	0.750327i
$764$	−1.23612e7	−0.766171
$765$	0	0
$766$	−3.47760e6	−0.214145
$767$	1.59742e6i	0.0980459i
$768$	0	0
$769$	−6.71521e6	−0.409491	−0.204745	−	0.978815i	$-0.565637\pi$
$769$	−6.71521e6	−0.409491	−0.204745	+	0.978815i	$0.565637\pi$
$770$	0	0
$771$	0	0
$772$	1.07291e7i	0.647919i
$773$	− 2.37237e7i	− 1.42802i	−0.700136	−	0.714009i	$-0.746877\pi$
$773$	− 2.37237e7i	− 1.42802i	0.700136	−	0.714009i	$-0.253123\pi$
$774$	0	0
$775$	0	0
$776$	−8.25459e6	−0.492086
$777$	0	0
$778$	− 4.20761e6i	− 0.249222i
$779$	−2.32762e7	−1.37426
$780$	0	0
$781$	−1.98764e7	−1.16603
$782$	− 2.82528e6i	− 0.165213i
$783$	0	0
$784$	1.84397e6	0.107143
$785$	0	0
$786$	0	0
$787$	3.17798e7i	1.82900i	0.404585	+	0.914501i	$0.367416\pi$
$787$	3.17798e7i	1.82900i	−0.404585	+	0.914501i	$0.632584\pi$
$788$	− 1.36308e7i	− 0.781996i
$789$	0	0
$790$	0	0
$791$	−2.15978e7	−1.22735
$792$	0	0
$793$	388648.i	0.0219469i
$794$	1.67705e7	0.944047
$795$	0	0
$796$	−8.98010e6	−0.502341
$797$	− 7.26280e6i	− 0.405003i	−0.979282	−	0.202502i	$-0.935093\pi$
$797$	− 7.26280e6i	− 0.405003i	0.979282	−	0.202502i	$-0.0649071\pi$
$798$	0	0
$799$	3.53160e6	0.195706
$800$	0	0
$801$	0	0
$802$	7.61170e6i	0.417874i
$803$	3.03343e7i	1.66014i
$804$	0	0
$805$	0	0
$806$	−1.64767e7	−0.893375
$807$	0	0
$808$	− 9.90682e6i	− 0.533833i
$809$	−3.22769e7	−1.73388	−0.866942	−	0.498409i	$-0.833918\pi$
$809$	−3.22769e7	−1.73388	−0.866942	+	0.498409i	$0.833918\pi$
$810$	0	0
$811$	−1.00689e7	−0.537563	−0.268782	−	0.963201i	$-0.586621\pi$
$811$	−1.00689e7	−0.537563	−0.268782	+	0.963201i	$0.586621\pi$
$812$	− 9.02227e6i	− 0.480204i
$813$	0	0
$814$	−7.86163e6	−0.415864
$815$	0	0
$816$	0	0
$817$	1.61065e7i	0.844203i
$818$	− 6.51639e6i	− 0.340505i
$819$	0	0
$820$	0	0
$821$	−2.08671e7	−1.08045	−0.540224	−	0.841521i	$-0.681660\pi$
$821$	−2.08671e7	−1.08045	−0.540224	+	0.841521i	$0.681660\pi$
$822$	0	0
$823$	1.40683e7i	0.724005i	0.932177	+	0.362003i	$0.117907\pi$
$823$	1.40683e7i	0.724005i	−0.932177	+	0.362003i	$0.882093\pi$
$824$	1.77267e6	0.0909516
$825$	0	0
$826$	1.54997e6	0.0790447
$827$	− 2.48210e7i	− 1.26199i	−0.775787	−	0.630995i	$-0.782647\pi$
$827$	− 2.48210e7i	− 1.26199i	0.775787	−	0.630995i	$-0.217353\pi$
$828$	0	0
$829$	1.40954e7	0.712347	0.356173	−	0.934420i	$-0.384081\pi$
$829$	1.40954e7	0.712347	0.356173	+	0.934420i	$0.384081\pi$
$830$	0	0
$831$	0	0
$832$	− 1.65478e6i	− 0.0828768i
$833$	4.71076e6i	0.235222i
$834$	0	0
$835$	0	0
$836$	−1.01725e7	−0.503401
$837$	0	0
$838$	− 9.32110e6i	− 0.458519i
$839$	6.47203e6	0.317421	0.158711	−	0.987325i	$-0.449266\pi$
$839$	6.47203e6	0.317421	0.158711	+	0.987325i	$0.449266\pi$
$840$	0	0
$841$	1.25974e7	0.614172
$842$	2.45190e6i	0.119185i
$843$	0	0
$844$	9.26022e6	0.447471
$845$	0	0
$846$	0	0
$847$	3.50203e6i	0.167730i
$848$	− 2.10278e6i	− 0.100416i
$849$	0	0
$850$	0	0
$851$	5.99616e6	0.283824
$852$	0	0
$853$	1.71706e7i	0.808002i	0.914758	+	0.404001i	$0.132381\pi$
$853$	1.71706e7i	0.808002i	−0.914758	+	0.404001i	$0.867619\pi$
$854$	377104.	0.0176936
$855$	0	0
$856$	1.41550e7	0.660276
$857$	2.45494e6i	0.114180i	0.998369	+	0.0570899i	$0.0181822\pi$
$857$	2.45494e6i	0.114180i	−0.998369	+	0.0570899i	$0.981818\pi$
$858$	0	0
$859$	−1.13126e6	−0.0523094	−0.0261547	−	0.999658i	$-0.508326\pi$
$859$	−1.13126e6	−0.0523094	−0.0261547	+	0.999658i	$0.508326\pi$
$860$	0	0
$861$	0	0
$862$	− 2.67754e7i	− 1.22735i
$863$	1.84069e7i	0.841305i	0.907222	+	0.420653i	$0.138199\pi$
$863$	1.84069e7i	0.841305i	−0.907222	+	0.420653i	$0.861801\pi$
$864$	0	0
$865$	0	0
$866$	−8.80777e6	−0.399090
$867$	0	0
$868$	1.59873e7i	0.720239i
$869$	−9.21958e6	−0.414154
$870$	0	0
$871$	−7.25422e6	−0.324000
$872$	7.87981e6i	0.350933i
$873$	0	0
$874$	7.75872e6	0.343567
$875$	0	0
$876$	0	0
$877$	− 2.61539e7i	− 1.14825i	−0.818766	−	0.574127i	$-0.805341\pi$
$877$	− 2.61539e7i	− 1.14825i	0.818766	−	0.574127i	$-0.194659\pi$
$878$	− 4.41747e6i	− 0.193392i
$879$	0	0
$880$	0	0
$881$	−1.97640e7	−0.857898	−0.428949	−	0.903329i	$-0.641116\pi$
$881$	−1.97640e7	−0.857898	−0.428949	+	0.903329i	$0.641116\pi$
$882$	0	0
$883$	3.43551e7i	1.48282i	0.671051	+	0.741411i	$0.265843\pi$
$883$	3.43551e7i	1.48282i	−0.671051	+	0.741411i	$0.734157\pi$
$884$	4.22746e6	0.181948
$885$	0	0
$886$	−1.52881e7	−0.654288
$887$	− 2.11254e7i	− 0.901561i	−0.892635	−	0.450780i	$-0.851146\pi$
$887$	− 2.11254e7i	− 0.901561i	0.892635	−	0.450780i	$-0.148854\pi$
$888$	0	0
$889$	−1.77672e7	−0.753988
$890$	0	0
$891$	0	0
$892$	− 1.38098e7i	− 0.581134i
$893$	9.69840e6i	0.406978i
$894$	0	0
$895$	0	0
$896$	−1.60563e6	−0.0668153
$897$	0	0
$898$	1.57594e7i	0.652152i
$899$	−5.86678e7	−2.42103
$900$	0	0
$901$	5.37196e6	0.220455
$902$	− 1.83514e7i	− 0.751021i
$903$	0	0
$904$	−1.41047e7	−0.574041
$905$	0	0
$906$	0	0
$907$	− 2.61308e6i	− 0.105471i	−0.998609	−	0.0527357i	$-0.983206\pi$
$907$	− 2.61308e6i	− 0.105471i	0.998609	−	0.0527357i	$-0.0167941\pi$
$908$	1.14048e7i	0.459064i
$909$	0	0
$910$	0	0
$911$	−3.27046e7	−1.30561	−0.652803	−	0.757527i	$-0.726407\pi$
$911$	−3.27046e7	−1.30561	−0.652803	+	0.757527i	$0.726407\pi$
$912$	0	0
$913$	− 3.30877e7i	− 1.31368i
$914$	−4.07010e6	−0.161154
$915$	0	0
$916$	−5.49866e6	−0.216530
$917$	− 2.49424e7i	− 0.979522i
$918$	0	0
$919$	−2.06463e7	−0.806406	−0.403203	−	0.915111i	$-0.632103\pi$
$919$	−2.06463e7	−0.806406	−0.403203	+	0.915111i	$0.632103\pi$
$920$	0	0
$921$	0	0
$922$	2.72332e7i	1.05505i
$923$	2.26838e7i	0.876418i
$924$	0	0
$925$	0	0
$926$	−2.69713e7	−1.03365
$927$	0	0
$928$	− 5.89210e6i	− 0.224595i
$929$	2.54318e7	0.966803	0.483401	−	0.875399i	$-0.339401\pi$
$929$	2.54318e7	0.966803	0.483401	+	0.875399i	$0.339401\pi$
$930$	0	0
$931$	−1.29366e7	−0.489154
$932$	3.50102e6i	0.132025i
$933$	0	0
$934$	−1.29048e7	−0.484041
$935$	0	0
$936$	0	0
$937$	− 2.52800e7i	− 0.940652i	−0.882493	−	0.470326i	$-0.844136\pi$
$937$	− 2.52800e7i	− 0.940652i	0.882493	−	0.470326i	$-0.155864\pi$
$938$	7.03875e6i	0.261209i
$939$	0	0
$940$	0	0
$941$	−1.80180e7	−0.663336	−0.331668	−	0.943396i	$-0.607611\pi$
$941$	−1.80180e7	−0.663336	−0.331668	+	0.943396i	$0.607611\pi$
$942$	0	0
$943$	1.39968e7i	0.512566i
$944$	1.01222e6	0.0369698
$945$	0	0
$946$	−1.26987e7	−0.461351
$947$	− 5.10485e7i	− 1.84973i	−0.380299	−	0.924864i	$-0.624179\pi$
$947$	− 5.10485e7i	− 1.84973i	0.380299	−	0.924864i	$-0.375821\pi$
$948$	0	0
$949$	3.46188e7	1.24780
$950$	0	0
$951$	0	0
$952$	− 4.10189e6i	− 0.146687i
$953$	− 4.53610e7i	− 1.61790i	−0.587880	−	0.808948i	$-0.700037\pi$
$953$	− 4.53610e7i	− 1.61790i	0.587880	−	0.808948i	$-0.299963\pi$
$954$	0	0
$955$	0	0
$956$	1.63352e7	0.578068
$957$	0	0
$958$	2.48485e7i	0.874755i
$959$	4.07196e7	1.42974
$960$	0	0
$961$	7.53293e7	2.63121
$962$	8.97203e6i	0.312574i
$963$	0	0
$964$	−2.19493e7	−0.760725
$965$	0	0
$966$	0	0
$967$	1.24701e7i	0.428847i	0.976741	+	0.214424i	$0.0687873\pi$
$967$	1.24701e7i	0.428847i	−0.976741	+	0.214424i	$0.931213\pi$
$968$	2.28704e6i	0.0784486i
$969$	0	0
$970$	0	0
$971$	2.13786e7	0.727664	0.363832	−	0.931465i	$-0.381468\pi$
$971$	2.13786e7	0.727664	0.363832	+	0.931465i	$0.381468\pi$
$972$	0	0
$973$	− 3.06093e7i	− 1.03651i
$974$	−6.05303e6	−0.204445
$975$	0	0
$976$	246272.	0.00827543
$977$	1.31824e7i	0.441833i	0.975293	+	0.220917i	$0.0709049\pi$
$977$	1.31824e7i	0.441833i	−0.975293	+	0.220917i	$0.929095\pi$
$978$	0	0
$979$	−2.59935e7	−0.866779
$980$	0	0
$981$	0	0
$982$	1.37534e7i	0.455126i
$983$	2.98823e7i	0.986349i	0.869930	+	0.493175i	$0.164164\pi$
$983$	2.98823e7i	0.986349i	−0.869930	+	0.493175i	$0.835836\pi$
$984$	0	0
$985$	0	0
$986$	1.50525e7	0.493078
$987$	0	0
$988$	1.16093e7i	0.378369i
$989$	9.68544e6	0.314868
$990$	0	0
$991$	−3.76534e7	−1.21793	−0.608963	−	0.793199i	$-0.708414\pi$
$991$	−3.76534e7	−1.21793	−0.608963	+	0.793199i	$0.708414\pi$
$992$	1.04407e7i	0.336861i
$993$	0	0
$994$	2.20100e7	0.706569
$995$	0	0
$996$	0	0
$997$	− 1.53443e7i	− 0.488887i	−0.969664	−	0.244444i	$-0.921395\pi$
$997$	− 1.53443e7i	− 0.488887i	0.969664	−	0.244444i	$-0.0786053\pi$
$998$	− 2.10936e7i	− 0.670385i
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	450.6.c.d.199.2		2
3.2	odd	2		450.6.c.l.199.1		2
5.2	odd	4		90.6.a.c.1.1	✓	1
5.3	odd	4		450.6.a.o.1.1		1
5.4	even	2	inner	450.6.c.d.199.1		2
15.2	even	4		90.6.a.e.1.1	yes	1
15.8	even	4		450.6.a.d.1.1		1
15.14	odd	2		450.6.c.l.199.2		2
20.7	even	4		720.6.a.o.1.1		1
60.47	odd	4		720.6.a.c.1.1		1

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
90.6.a.c.1.1	✓	1	5.2	odd	4
90.6.a.e.1.1	yes	1	15.2	even	4
450.6.a.d.1.1		1	15.8	even	4
450.6.a.o.1.1		1	5.3	odd	4
450.6.c.d.199.1		2	5.4	even	2	inner
450.6.c.d.199.2		2	1.1	even	1	trivial
450.6.c.l.199.1		2	3.2	odd	2
450.6.c.l.199.2		2	15.14	odd	2
720.6.a.c.1.1		1	60.47	odd	4
720.6.a.o.1.1		1	20.7	even	4

Overview	Random
Universe	Knowledge

Classical	Maass
Hilbert	Bianchi

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

Level:	\( N \)	\(=\)	\( 450 = 2 \cdot 3^{2} \cdot 5^{2} \)
Weight:	\( k \)	\(=\)	\( 6 \)
Character orbit:	\([\chi]\)	\(=\)	450.c (of order \(2\), degree \(1\), not minimal)

\(f(q)\)	\(=\)	\(q+4.00000i q^{2} -16.0000 q^{4} -98.0000i q^{7} -64.0000i q^{8} +O(q^{10})\)	\(q+4.00000i q^{2} -16.0000 q^{4} -98.0000i q^{7} -64.0000i q^{8} -354.000 q^{11} +404.000i q^{13} +392.000 q^{14} +256.000 q^{16} +654.000i q^{17} -1796.00 q^{19} -1416.00i q^{22} +1080.00i q^{23} -1616.00 q^{26} +1568.00i q^{28} -5754.00 q^{29} +10196.0 q^{31} +1024.00i q^{32} -2616.00 q^{34} -5552.00i q^{37} -7184.00i q^{38} +12960.0 q^{41} -8968.00i q^{43} +5664.00 q^{44} -4320.00 q^{46} -5400.00i q^{47} +7203.00 q^{49} -6464.00i q^{52} -8214.00i q^{53} -6272.00 q^{56} -23016.0i q^{58} +3954.00 q^{59} +962.000 q^{61} +40784.0i q^{62} -4096.00 q^{64} +17956.0i q^{67} -10464.0i q^{68} +56148.0 q^{71} -85690.0i q^{73} +22208.0 q^{74} +28736.0 q^{76} +34692.0i q^{77} +26044.0 q^{79} +51840.0i q^{82} +93468.0i q^{83} +35872.0 q^{86} +22656.0i q^{88} +73428.0 q^{89} +39592.0 q^{91} -17280.0i q^{92} +21600.0 q^{94} -128978. i q^{97} +28812.0i q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 2 q - 32 q^{4}+O(q^{10}) \) 2 * q - 32 * q^4	\( 2 q - 32 q^{4} - 708 q^{11} + 784 q^{14} + 512 q^{16} - 3592 q^{19} - 3232 q^{26} - 11508 q^{29} + 20392 q^{31} - 5232 q^{34} + 25920 q^{41} + 11328 q^{44} - 8640 q^{46} + 14406 q^{49} - 12544 q^{56} + 7908 q^{59} + 1924 q^{61} - 8192 q^{64} + 112296 q^{71} + 44416 q^{74} + 57472 q^{76} + 52088 q^{79} + 71744 q^{86} + 146856 q^{89} + 79184 q^{91} + 43200 q^{94}+O(q^{100}) \) 2 * q - 32 * q^4 - 708 * q^11 + 784 * q^14 + 512 * q^16 - 3592 * q^19 - 3232 * q^26 - 11508 * q^29 + 20392 * q^31 - 5232 * q^34 + 25920 * q^41 + 11328 * q^44 - 8640 * q^46 + 14406 * q^49 - 12544 * q^56 + 7908 * q^59 + 1924 * q^61 - 8192 * q^64 + 112296 * q^71 + 44416 * q^74 + 57472 * q^76 + 52088 * q^79 + 71744 * q^86 + 146856 * q^89 + 79184 * q^91 + 43200 * q^94

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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