LMFDB - Embedded newform 450.6.c.g.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:	$ 1 $
Twist minimal:	no (minimal twist has level 150)
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.g.199.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+4.00000i q^{2} -16.0000 q^{4} +79.0000i q^{7} -64.0000i q^{8} +O(q^{10})$	$q+4.00000i q^{2} -16.0000 q^{4} +79.0000i q^{7} -64.0000i q^{8} -150.000 q^{11} +137.000i q^{13} -316.000 q^{14} +256.000 q^{16} -2034.00i q^{17} +1969.00 q^{19} -600.000i q^{22} +1350.00i q^{23} -548.000 q^{26} -1264.00i q^{28} -2946.00 q^{29} +713.000 q^{31} +1024.00i q^{32} +8136.00 q^{34} +3238.00i q^{37} +7876.00i q^{38} -6564.00 q^{41} -19579.0i q^{43} +2400.00 q^{44} -5400.00 q^{46} -21150.0i q^{47} +10566.0 q^{49} -2192.00i q^{52} +25896.0i q^{53} +5056.00 q^{56} -11784.0i q^{58} +25350.0 q^{59} +50615.0 q^{61} +2852.00i q^{62} -4096.00 q^{64} +22519.0i q^{67} +32544.0i q^{68} -33900.0 q^{71} +82442.0i q^{73} -12952.0 q^{74} -31504.0 q^{76} -11850.0i q^{77} +81472.0 q^{79} -26256.0i q^{82} -25782.0i q^{83} +78316.0 q^{86} +9600.00i q^{88} +103728. q^{89} -10823.0 q^{91} -21600.0i q^{92} +84600.0 q^{94} +57343.0i q^{97} +42264.0i q^{98} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 2 q - 32 q^{4}+O(q^{10}) $ 2 * q - 32 * q^4	$ 2 q - 32 q^{4} - 300 q^{11} - 632 q^{14} + 512 q^{16} + 3938 q^{19} - 1096 q^{26} - 5892 q^{29} + 1426 q^{31} + 16272 q^{34} - 13128 q^{41} + 4800 q^{44} - 10800 q^{46} + 21132 q^{49} + 10112 q^{56} + 50700 q^{59} + 101230 q^{61} - 8192 q^{64} - 67800 q^{71} - 25904 q^{74} - 63008 q^{76} + 162944 q^{79} + 156632 q^{86} + 207456 q^{89} - 21646 q^{91} + 169200 q^{94}+O(q^{100}) $ 2 * q - 32 * q^4 - 300 * q^11 - 632 * q^14 + 512 * q^16 + 3938 * q^19 - 1096 * q^26 - 5892 * q^29 + 1426 * q^31 + 16272 * q^34 - 13128 * q^41 + 4800 * q^44 - 10800 * q^46 + 21132 * q^49 + 10112 * q^56 + 50700 * q^59 + 101230 * q^61 - 8192 * q^64 - 67800 * q^71 - 25904 * q^74 - 63008 * q^76 + 162944 * q^79 + 156632 * q^86 + 207456 * q^89 - 21646 * q^91 + 169200 * 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
$7$	79.0000i	0.609371i	0.952453	+	0.304686i	$0.0985514\pi$
$7$	79.0000i	0.609371i	−0.952453	+	0.304686i	$0.901449\pi$
$8$	− 64.0000i	− 0.353553i
$9$	0	0
$10$	0	0
$11$	−150.000	−0.373774	−0.186887	−	0.982381i	$-0.559840\pi$
$11$	−150.000	−0.373774	−0.186887	+	0.982381i	$0.559840\pi$
$12$	0	0
$13$	137.000i	0.224834i	0.993661	+	0.112417i	$0.0358593\pi$
$13$	137.000i	0.224834i	−0.993661	+	0.112417i	$0.964141\pi$
$14$	−316.000	−0.430891
$15$	0	0
$16$	256.000	0.250000
$17$	− 2034.00i	− 1.70698i	−0.521109	−	0.853490i	$-0.674481\pi$
$17$	− 2034.00i	− 1.70698i	0.521109	−	0.853490i	$-0.325519\pi$
$18$	0	0
$19$	1969.00	1.25130	0.625650	−	0.780104i	$-0.284834\pi$
$19$	1969.00	1.25130	0.625650	+	0.780104i	$0.284834\pi$
$20$	0	0
$21$	0	0
$22$	− 600.000i	− 0.264298i
$23$	1350.00i	0.532126i	0.963956	+	0.266063i	$0.0857229\pi$
$23$	1350.00i	0.532126i	−0.963956	+	0.266063i	$0.914277\pi$
$24$	0	0
$25$	0	0
$26$	−548.000	−0.158982
$27$	0	0
$28$	− 1264.00i	− 0.304686i
$29$	−2946.00	−0.650486	−0.325243	−	0.945631i	$-0.605446\pi$
$29$	−2946.00	−0.650486	−0.325243	+	0.945631i	$0.605446\pi$
$30$	0	0
$31$	713.000	0.133256	0.0666278	−	0.997778i	$-0.478776\pi$
$31$	713.000	0.133256	0.0666278	+	0.997778i	$0.478776\pi$
$32$	1024.00i	0.176777i
$33$	0	0
$34$	8136.00	1.20702
$35$	0	0
$36$	0	0
$37$	3238.00i	0.388841i	0.980918	+	0.194421i	$0.0622827\pi$
$37$	3238.00i	0.388841i	−0.980918	+	0.194421i	$0.937717\pi$
$38$	7876.00i	0.884803i
$39$	0	0
$40$	0	0
$41$	−6564.00	−0.609830	−0.304915	−	0.952380i	$-0.598628\pi$
$41$	−6564.00	−0.609830	−0.304915	+	0.952380i	$0.598628\pi$
$42$	0	0
$43$	− 19579.0i	− 1.61480i	−0.590003	−	0.807401i	$-0.700873\pi$
$43$	− 19579.0i	− 1.61480i	0.590003	−	0.807401i	$-0.299127\pi$
$44$	2400.00	0.186887
$45$	0	0
$46$	−5400.00	−0.376270
$47$	− 21150.0i	− 1.39658i	−0.715815	−	0.698290i	$-0.753945\pi$
$47$	− 21150.0i	− 1.39658i	0.715815	−	0.698290i	$-0.246055\pi$
$48$	0	0
$49$	10566.0	0.628667
$50$	0	0
$51$	0	0
$52$	− 2192.00i	− 0.112417i
$53$	25896.0i	1.26632i	0.774021	+	0.633159i	$0.218242\pi$
$53$	25896.0i	1.26632i	−0.774021	+	0.633159i	$0.781758\pi$
$54$	0	0
$55$	0	0
$56$	5056.00	0.215445
$57$	0	0
$58$	− 11784.0i	− 0.459963i
$59$	25350.0	0.948086	0.474043	−	0.880502i	$-0.342794\pi$
$59$	25350.0	0.948086	0.474043	+	0.880502i	$0.342794\pi$
$60$	0	0
$61$	50615.0	1.74163	0.870813	−	0.491615i	$-0.163593\pi$
$61$	50615.0	1.74163	0.870813	+	0.491615i	$0.163593\pi$
$62$	2852.00i	0.0942259i
$63$	0	0
$64$	−4096.00	−0.125000
$65$	0	0
$66$	0	0
$67$	22519.0i	0.612861i	0.951893	+	0.306431i	$0.0991347\pi$
$67$	22519.0i	0.612861i	−0.951893	+	0.306431i	$0.900865\pi$
$68$	32544.0i	0.853490i
$69$	0	0
$70$	0	0
$71$	−33900.0	−0.798094	−0.399047	−	0.916931i	$-0.630659\pi$
$71$	−33900.0	−0.798094	−0.399047	+	0.916931i	$0.630659\pi$
$72$	0	0
$73$	82442.0i	1.81068i	0.424689	+	0.905339i	$0.360383\pi$
$73$	82442.0i	1.81068i	−0.424689	+	0.905339i	$0.639617\pi$
$74$	−12952.0	−0.274952
$75$	0	0
$76$	−31504.0	−0.625650
$77$	− 11850.0i	− 0.227767i
$78$	0	0
$79$	81472.0	1.46873	0.734363	−	0.678757i	$-0.237481\pi$
$79$	81472.0	1.46873	0.734363	+	0.678757i	$0.237481\pi$
$80$	0	0
$81$	0	0
$82$	− 26256.0i	− 0.431215i
$83$	− 25782.0i	− 0.410791i	−0.978679	−	0.205396i	$-0.934152\pi$
$83$	− 25782.0i	− 0.410791i	0.978679	−	0.205396i	$-0.0658481\pi$
$84$	0	0
$85$	0	0
$86$	78316.0	1.14184
$87$	0	0
$88$	9600.00i	0.132149i
$89$	103728.	1.38810	0.694050	−	0.719926i	$-0.255824\pi$
$89$	103728.	1.38810	0.694050	+	0.719926i	$0.255824\pi$
$90$	0	0
$91$	−10823.0	−0.137007
$92$	− 21600.0i	− 0.266063i
$93$	0	0
$94$	84600.0	0.987531
$95$	0	0
$96$	0	0
$97$	57343.0i	0.618801i	0.950932	+	0.309401i	$0.100128\pi$
$97$	57343.0i	0.618801i	−0.950932	+	0.309401i	$0.899872\pi$
$98$	42264.0i	0.444534i
$99$	0	0
$100$	0	0
$101$	91032.0	0.887954	0.443977	−	0.896038i	$-0.353567\pi$
$101$	91032.0	0.887954	0.443977	+	0.896038i	$0.353567\pi$
$102$	0	0
$103$	191636.i	1.77985i	0.456104	+	0.889926i	$0.349244\pi$
$103$	191636.i	1.77985i	−0.456104	+	0.889926i	$0.650756\pi$
$104$	8768.00	0.0794909
$105$	0	0
$106$	−103584.	−0.895423
$107$	9288.00i	0.0784265i	0.999231	+	0.0392132i	$0.0124852\pi$
$107$	9288.00i	0.0784265i	−0.999231	+	0.0392132i	$0.987515\pi$
$108$	0	0
$109$	20635.0	0.166356	0.0831780	−	0.996535i	$-0.473493\pi$
$109$	20635.0	0.166356	0.0831780	+	0.996535i	$0.473493\pi$
$110$	0	0
$111$	0	0
$112$	20224.0i	0.152343i
$113$	− 176892.i	− 1.30320i	−0.758561	−	0.651602i	$-0.774097\pi$
$113$	− 176892.i	− 1.30320i	0.758561	−	0.651602i	$-0.225903\pi$
$114$	0	0
$115$	0	0
$116$	47136.0	0.325243
$117$	0	0
$118$	101400.i	0.670398i
$119$	160686.	1.04019
$120$	0	0
$121$	−138551.	−0.860293
$122$	202460.i	1.23151i
$123$	0	0
$124$	−11408.0	−0.0666278
$125$	0	0
$126$	0	0
$127$	256480.i	1.41106i	0.708682	+	0.705528i	$0.249290\pi$
$127$	256480.i	1.41106i	−0.708682	+	0.705528i	$0.750710\pi$
$128$	− 16384.0i	− 0.0883883i
$129$	0	0
$130$	0	0
$131$	−177600.	−0.904200	−0.452100	−	0.891967i	$-0.649325\pi$
$131$	−177600.	−0.904200	−0.452100	+	0.891967i	$0.649325\pi$
$132$	0	0
$133$	155551.i	0.762507i
$134$	−90076.0	−0.433358
$135$	0	0
$136$	−130176.	−0.603509
$137$	260886.i	1.18754i	0.804634	+	0.593772i	$0.202362\pi$
$137$	260886.i	1.18754i	−0.804634	+	0.593772i	$0.797638\pi$
$138$	0	0
$139$	217684.	0.955629	0.477815	−	0.878461i	$-0.341429\pi$
$139$	217684.	0.955629	0.477815	+	0.878461i	$0.341429\pi$
$140$	0	0
$141$	0	0
$142$	− 135600.i	− 0.564337i
$143$	− 20550.0i	− 0.0840372i
$144$	0	0
$145$	0	0
$146$	−329768.	−1.28034
$147$	0	0
$148$	− 51808.0i	− 0.194421i
$149$	421794.	1.55645	0.778224	−	0.627987i	$-0.216121\pi$
$149$	421794.	1.55645	0.778224	+	0.627987i	$0.216121\pi$
$150$	0	0
$151$	−101917.	−0.363751	−0.181876	−	0.983322i	$-0.558217\pi$
$151$	−101917.	−0.363751	−0.181876	+	0.983322i	$0.558217\pi$
$152$	− 126016.i	− 0.442402i
$153$	0	0
$154$	47400.0	0.161056
$155$	0	0
$156$	0	0
$157$	101773.i	0.329521i	0.986334	+	0.164761i	$0.0526852\pi$
$157$	101773.i	0.329521i	−0.986334	+	0.164761i	$0.947315\pi$
$158$	325888.i	1.03855i
$159$	0	0
$160$	0	0
$161$	−106650.	−0.324262
$162$	0	0
$163$	− 202249.i	− 0.596235i	−0.954529	−	0.298117i	$-0.903641\pi$
$163$	− 202249.i	− 0.596235i	0.954529	−	0.298117i	$-0.0963587\pi$
$164$	105024.	0.304915
$165$	0	0
$166$	103128.	0.290473
$167$	231600.i	0.642610i	0.946976	+	0.321305i	$0.104121\pi$
$167$	231600.i	0.642610i	−0.946976	+	0.321305i	$0.895879\pi$
$168$	0	0
$169$	352524.	0.949450
$170$	0	0
$171$	0	0
$172$	313264.i	0.807401i
$173$	174222.i	0.442576i	0.975209	+	0.221288i	$0.0710261\pi$
$173$	174222.i	0.442576i	−0.975209	+	0.221288i	$0.928974\pi$
$174$	0	0
$175$	0	0
$176$	−38400.0	−0.0934436
$177$	0	0
$178$	414912.i	0.981535i
$179$	−642066.	−1.49778	−0.748888	−	0.662696i	$-0.769412\pi$
$179$	−642066.	−1.49778	−0.748888	+	0.662696i	$0.769412\pi$
$180$	0	0
$181$	−56071.0	−0.127216	−0.0636080	−	0.997975i	$-0.520261\pi$
$181$	−56071.0	−0.127216	−0.0636080	+	0.997975i	$0.520261\pi$
$182$	− 43292.0i	− 0.0968789i
$183$	0	0
$184$	86400.0	0.188135
$185$	0	0
$186$	0	0
$187$	305100.i	0.638026i
$188$	338400.i	0.698290i
$189$	0	0
$190$	0	0
$191$	209694.	0.415913	0.207957	−	0.978138i	$-0.433319\pi$
$191$	209694.	0.415913	0.207957	+	0.978138i	$0.433319\pi$
$192$	0	0
$193$	− 715597.i	− 1.38285i	−0.722448	−	0.691425i	$-0.756983\pi$
$193$	− 715597.i	− 1.38285i	0.722448	−	0.691425i	$-0.243017\pi$
$194$	−229372.	−0.437558
$195$	0	0
$196$	−169056.	−0.314333
$197$	508854.i	0.934174i	0.884211	+	0.467087i	$0.154696\pi$
$197$	508854.i	0.934174i	−0.884211	+	0.467087i	$0.845304\pi$
$198$	0	0
$199$	986017.	1.76503	0.882514	−	0.470286i	$-0.155849\pi$
$199$	986017.	1.76503	0.882514	+	0.470286i	$0.155849\pi$
$200$	0	0
$201$	0	0
$202$	364128.i	0.627879i
$203$	− 232734.i	− 0.396387i
$204$	0	0
$205$	0	0
$206$	−766544.	−1.25855
$207$	0	0
$208$	35072.0i	0.0562085i
$209$	−295350.	−0.467704
$210$	0	0
$211$	119495.	0.184775	0.0923876	−	0.995723i	$-0.470550\pi$
$211$	119495.	0.184775	0.0923876	+	0.995723i	$0.470550\pi$
$212$	− 414336.i	− 0.633159i
$213$	0	0
$214$	−37152.0	−0.0554559
$215$	0	0
$216$	0	0
$217$	56327.0i	0.0812021i
$218$	82540.0i	0.117631i
$219$	0	0
$220$	0	0
$221$	278658.	0.383788
$222$	0	0
$223$	48545.0i	0.0653706i	0.999466	+	0.0326853i	$0.0104059\pi$
$223$	48545.0i	0.0653706i	−0.999466	+	0.0326853i	$0.989594\pi$
$224$	−80896.0	−0.107723
$225$	0	0
$226$	707568.	0.921504
$227$	− 287652.i	− 0.370512i	−0.982690	−	0.185256i	$-0.940689\pi$
$227$	− 287652.i	− 0.370512i	0.982690	−	0.185256i	$-0.0593115\pi$
$228$	0	0
$229$	−72065.0	−0.0908104	−0.0454052	−	0.998969i	$-0.514458\pi$
$229$	−72065.0	−0.0908104	−0.0454052	+	0.998969i	$0.514458\pi$
$230$	0	0
$231$	0	0
$232$	188544.i	0.229981i
$233$	569148.i	0.686808i	0.939188	+	0.343404i	$0.111580\pi$
$233$	569148.i	0.686808i	−0.939188	+	0.343404i	$0.888420\pi$
$234$	0	0
$235$	0	0
$236$	−405600.	−0.474043
$237$	0	0
$238$	642744.i	0.735522i
$239$	696504.	0.788731	0.394365	−	0.918954i	$-0.370964\pi$
$239$	696504.	0.788731	0.394365	+	0.918954i	$0.370964\pi$
$240$	0	0
$241$	576137.	0.638974	0.319487	−	0.947591i	$-0.396489\pi$
$241$	576137.	0.638974	0.319487	+	0.947591i	$0.396489\pi$
$242$	− 554204.i	− 0.608319i
$243$	0	0
$244$	−809840.	−0.870813
$245$	0	0
$246$	0	0
$247$	269753.i	0.281335i
$248$	− 45632.0i	− 0.0471130i
$249$	0	0
$250$	0	0
$251$	1.22492e6	1.22723	0.613613	−	0.789607i	$-0.289715\pi$
$251$	1.22492e6	1.22723	0.613613	+	0.789607i	$0.289715\pi$
$252$	0	0
$253$	− 202500.i	− 0.198895i
$254$	−1.02592e6	−0.997767
$255$	0	0
$256$	65536.0	0.0625000
$257$	972132.i	0.918105i	0.888409	+	0.459053i	$0.151811\pi$
$257$	972132.i	0.918105i	−0.888409	+	0.459053i	$0.848189\pi$
$258$	0	0
$259$	−255802.	−0.236949
$260$	0	0
$261$	0	0
$262$	− 710400.i	− 0.639366i
$263$	− 1.76190e6i	− 1.57070i	−0.619055	−	0.785348i	$-0.712484\pi$
$263$	− 1.76190e6i	− 1.57070i	0.619055	−	0.785348i	$-0.287516\pi$
$264$	0	0
$265$	0	0
$266$	−622204.	−0.539174
$267$	0	0
$268$	− 360304.i	− 0.306431i
$269$	1.74383e6	1.46935	0.734674	−	0.678421i	$-0.237335\pi$
$269$	1.74383e6	1.46935	0.734674	+	0.678421i	$0.237335\pi$
$270$	0	0
$271$	−1.70426e6	−1.40965	−0.704826	−	0.709381i	$-0.748975\pi$
$271$	−1.70426e6	−1.40965	−0.704826	+	0.709381i	$0.748975\pi$
$272$	− 520704.i	− 0.426745i
$273$	0	0
$274$	−1.04354e6	−0.839720
$275$	0	0
$276$	0	0
$277$	− 1.46972e6i	− 1.15089i	−0.817840	−	0.575446i	$-0.804829\pi$
$277$	− 1.46972e6i	− 1.15089i	0.817840	−	0.575446i	$-0.195171\pi$
$278$	870736.i	0.675732i
$279$	0	0
$280$	0	0
$281$	1.22530e6	0.925715	0.462858	−	0.886433i	$-0.346824\pi$
$281$	1.22530e6	0.925715	0.462858	+	0.886433i	$0.346824\pi$
$282$	0	0
$283$	− 637333.i	− 0.473043i	−0.971626	−	0.236521i	$-0.923993\pi$
$283$	− 637333.i	− 0.473043i	0.971626	−	0.236521i	$-0.0760073\pi$
$284$	542400.	0.399047
$285$	0	0
$286$	82200.0	0.0594233
$287$	− 518556.i	− 0.371613i
$288$	0	0
$289$	−2.71730e6	−1.91378
$290$	0	0
$291$	0	0
$292$	− 1.31907e6i	− 0.905339i
$293$	− 35094.0i	− 0.0238816i	−0.999929	−	0.0119408i	$-0.996199\pi$
$293$	− 35094.0i	− 0.0238816i	0.999929	−	0.0119408i	$-0.00380097\pi$
$294$	0	0
$295$	0	0
$296$	207232.	0.137476
$297$	0	0
$298$	1.68718e6i	1.10058i
$299$	−184950.	−0.119640
$300$	0	0
$301$	1.54674e6	0.984014
$302$	− 407668.i	− 0.257211i
$303$	0	0
$304$	504064.	0.312825
$305$	0	0
$306$	0	0
$307$	− 2.88040e6i	− 1.74424i	−0.489293	−	0.872120i	$-0.662745\pi$
$307$	− 2.88040e6i	− 1.74424i	0.489293	−	0.872120i	$-0.337255\pi$
$308$	189600.i	0.113884i
$309$	0	0
$310$	0	0
$311$	2.38305e6	1.39712	0.698558	−	0.715554i	$-0.253826\pi$
$311$	2.38305e6	1.39712	0.698558	+	0.715554i	$0.253826\pi$
$312$	0	0
$313$	− 1.93081e6i	− 1.11399i	−0.830517	−	0.556993i	$-0.811955\pi$
$313$	− 1.93081e6i	− 1.11399i	0.830517	−	0.556993i	$-0.188045\pi$
$314$	−407092.	−0.233007
$315$	0	0
$316$	−1.30355e6	−0.734363
$317$	− 2.18995e6i	− 1.22401i	−0.790852	−	0.612007i	$-0.790362\pi$
$317$	− 2.18995e6i	− 1.22401i	0.790852	−	0.612007i	$-0.209638\pi$
$318$	0	0
$319$	441900.	0.243135
$320$	0	0
$321$	0	0
$322$	− 426600.i	− 0.229288i
$323$	− 4.00495e6i	− 2.13595i
$324$	0	0
$325$	0	0
$326$	808996.	0.421602
$327$	0	0
$328$	420096.i	0.215608i
$329$	1.67085e6	0.851036
$330$	0	0
$331$	1.41429e6	0.709527	0.354764	−	0.934956i	$-0.384561\pi$
$331$	1.41429e6	0.709527	0.354764	+	0.934956i	$0.384561\pi$
$332$	412512.i	0.205396i
$333$	0	0
$334$	−926400.	−0.454394
$335$	0	0
$336$	0	0
$337$	− 1.38208e6i	− 0.662916i	−0.943470	−	0.331458i	$-0.892459\pi$
$337$	− 1.38208e6i	− 0.662916i	0.943470	−	0.331458i	$-0.107541\pi$
$338$	1.41010e6i	0.671362i
$339$	0	0
$340$	0	0
$341$	−106950.	−0.0498075
$342$	0	0
$343$	2.16247e6i	0.992463i
$344$	−1.25306e6	−0.570919
$345$	0	0
$346$	−696888.	−0.312948
$347$	− 283758.i	− 0.126510i	−0.997997	−	0.0632549i	$-0.979852\pi$
$347$	− 283758.i	− 0.126510i	0.997997	−	0.0632549i	$-0.0201481\pi$
$348$	0	0
$349$	−2.13809e6	−0.939642	−0.469821	−	0.882762i	$-0.655682\pi$
$349$	−2.13809e6	−0.939642	−0.469821	+	0.882762i	$0.655682\pi$
$350$	0	0
$351$	0	0
$352$	− 153600.i	− 0.0660746i
$353$	− 3.16015e6i	− 1.34980i	−0.737908	−	0.674901i	$-0.764186\pi$
$353$	− 3.16015e6i	− 1.34980i	0.737908	−	0.674901i	$-0.235814\pi$
$354$	0	0
$355$	0	0
$356$	−1.65965e6	−0.694050
$357$	0	0
$358$	− 2.56826e6i	− 1.05909i
$359$	−3.59564e6	−1.47245	−0.736225	−	0.676737i	$-0.763393\pi$
$359$	−3.59564e6	−1.47245	−0.736225	+	0.676737i	$0.763393\pi$
$360$	0	0
$361$	1.40086e6	0.565754
$362$	− 224284.i	− 0.0899553i
$363$	0	0
$364$	173168.	0.0685037
$365$	0	0
$366$	0	0
$367$	4.62043e6i	1.79068i	0.445385	+	0.895339i	$0.353067\pi$
$367$	4.62043e6i	1.79068i	−0.445385	+	0.895339i	$0.646933\pi$
$368$	345600.i	0.133031i
$369$	0	0
$370$	0	0
$371$	−2.04578e6	−0.771658
$372$	0	0
$373$	− 3.51983e6i	− 1.30993i	−0.755657	−	0.654967i	$-0.772682\pi$
$373$	− 3.51983e6i	− 1.30993i	0.755657	−	0.654967i	$-0.227318\pi$
$374$	−1.22040e6	−0.451152
$375$	0	0
$376$	−1.35360e6	−0.493765
$377$	− 403602.i	− 0.146251i
$378$	0	0
$379$	−595061.	−0.212796	−0.106398	−	0.994324i	$-0.533932\pi$
$379$	−595061.	−0.212796	−0.106398	+	0.994324i	$0.533932\pi$
$380$	0	0
$381$	0	0
$382$	838776.i	0.294095i
$383$	1.30050e6i	0.453016i	0.974009	+	0.226508i	$0.0727309\pi$
$383$	1.30050e6i	0.453016i	−0.974009	+	0.226508i	$0.927269\pi$
$384$	0	0
$385$	0	0
$386$	2.86239e6	0.977823
$387$	0	0
$388$	− 917488.i	− 0.309401i
$389$	620724.	0.207981	0.103991	−	0.994578i	$-0.466839\pi$
$389$	620724.	0.207981	0.103991	+	0.994578i	$0.466839\pi$
$390$	0	0
$391$	2.74590e6	0.908328
$392$	− 676224.i	− 0.222267i
$393$	0	0
$394$	−2.03542e6	−0.660561
$395$	0	0
$396$	0	0
$397$	− 1.18622e6i	− 0.377737i	−0.982002	−	0.188869i	$-0.939518\pi$
$397$	− 1.18622e6i	− 0.377737i	0.982002	−	0.188869i	$-0.0604821\pi$
$398$	3.94407e6i	1.24806i
$399$	0	0
$400$	0	0
$401$	−3.13334e6	−0.973077	−0.486538	−	0.873659i	$-0.661741\pi$
$401$	−3.13334e6	−0.973077	−0.486538	+	0.873659i	$0.661741\pi$
$402$	0	0
$403$	97681.0i	0.0299604i
$404$	−1.45651e6	−0.443977
$405$	0	0
$406$	930936.	0.280288
$407$	− 485700.i	− 0.145339i
$408$	0	0
$409$	−567581.	−0.167772	−0.0838860	−	0.996475i	$-0.526733\pi$
$409$	−567581.	−0.167772	−0.0838860	+	0.996475i	$0.526733\pi$
$410$	0	0
$411$	0	0
$412$	− 3.06618e6i	− 0.889926i
$413$	2.00265e6i	0.577737i
$414$	0	0
$415$	0	0
$416$	−140288.	−0.0397454
$417$	0	0
$418$	− 1.18140e6i	− 0.330717i
$419$	−6.09000e6	−1.69466	−0.847329	−	0.531068i	$-0.821791\pi$
$419$	−6.09000e6	−1.69466	−0.847329	+	0.531068i	$0.821791\pi$
$420$	0	0
$421$	3.07139e6	0.844558	0.422279	−	0.906466i	$-0.361230\pi$
$421$	3.07139e6	0.844558	0.422279	+	0.906466i	$0.361230\pi$
$422$	477980.i	0.130656i
$423$	0	0
$424$	1.65734e6	0.447711
$425$	0	0
$426$	0	0
$427$	3.99858e6i	1.06130i
$428$	− 148608.i	− 0.0392132i
$429$	0	0
$430$	0	0
$431$	669150.	0.173512	0.0867562	−	0.996230i	$-0.472350\pi$
$431$	669150.	0.173512	0.0867562	+	0.996230i	$0.472350\pi$
$432$	0	0
$433$	− 3.25439e6i	− 0.834161i	−0.908870	−	0.417080i	$-0.863053\pi$
$433$	− 3.25439e6i	− 0.834161i	0.908870	−	0.417080i	$-0.136947\pi$
$434$	−225308.	−0.0574186
$435$	0	0
$436$	−330160.	−0.0831780
$437$	2.65815e6i	0.665849i
$438$	0	0
$439$	−1.58759e6	−0.393168	−0.196584	−	0.980487i	$-0.562985\pi$
$439$	−1.58759e6	−0.393168	−0.196584	+	0.980487i	$0.562985\pi$
$440$	0	0
$441$	0	0
$442$	1.11463e6i	0.271379i
$443$	213888.i	0.0517818i	0.999665	+	0.0258909i	$0.00824225\pi$
$443$	213888.i	0.0517818i	−0.999665	+	0.0258909i	$0.991758\pi$
$444$	0	0
$445$	0	0
$446$	−194180.	−0.0462240
$447$	0	0
$448$	− 323584.i	− 0.0761714i
$449$	3.70724e6	0.867831	0.433916	−	0.900953i	$-0.357132\pi$
$449$	3.70724e6	0.867831	0.433916	+	0.900953i	$0.357132\pi$
$450$	0	0
$451$	984600.	0.227939
$452$	2.83027e6i	0.651602i
$453$	0	0
$454$	1.15061e6	0.261992
$455$	0	0
$456$	0	0
$457$	− 3.83193e6i	− 0.858275i	−0.903239	−	0.429138i	$-0.858818\pi$
$457$	− 3.83193e6i	− 0.858275i	0.903239	−	0.429138i	$-0.141182\pi$
$458$	− 288260.i	− 0.0642127i
$459$	0	0
$460$	0	0
$461$	−5.58672e6	−1.22435	−0.612174	−	0.790723i	$-0.709705\pi$
$461$	−5.58672e6	−1.22435	−0.612174	+	0.790723i	$0.709705\pi$
$462$	0	0
$463$	− 4.40142e6i	− 0.954203i	−0.878848	−	0.477101i	$-0.841687\pi$
$463$	− 4.40142e6i	− 0.954203i	0.878848	−	0.477101i	$-0.158313\pi$
$464$	−754176.	−0.162621
$465$	0	0
$466$	−2.27659e6	−0.485647
$467$	3.35225e6i	0.711287i	0.934622	+	0.355643i	$0.115738\pi$
$467$	3.35225e6i	0.711287i	−0.934622	+	0.355643i	$0.884262\pi$
$468$	0	0
$469$	−1.77900e6	−0.373460
$470$	0	0
$471$	0	0
$472$	− 1.62240e6i	− 0.335199i
$473$	2.93685e6i	0.603572i
$474$	0	0
$475$	0	0
$476$	−2.57098e6	−0.520093
$477$	0	0
$478$	2.78602e6i	0.557717i
$479$	909966.	0.181212	0.0906059	−	0.995887i	$-0.471120\pi$
$479$	909966.	0.181212	0.0906059	+	0.995887i	$0.471120\pi$
$480$	0	0
$481$	−443606.	−0.0874248
$482$	2.30455e6i	0.451823i
$483$	0	0
$484$	2.21682e6	0.430146
$485$	0	0
$486$	0	0
$487$	4.03579e6i	0.771093i	0.922689	+	0.385546i	$0.125987\pi$
$487$	4.03579e6i	0.771093i	−0.922689	+	0.385546i	$0.874013\pi$
$488$	− 3.23936e6i	− 0.615757i
$489$	0	0
$490$	0	0
$491$	−379944.	−0.0711239	−0.0355620	−	0.999367i	$-0.511322\pi$
$491$	−379944.	−0.0711239	−0.0355620	+	0.999367i	$0.511322\pi$
$492$	0	0
$493$	5.99216e6i	1.11037i
$494$	−1.07901e6	−0.198934
$495$	0	0
$496$	182528.	0.0333139
$497$	− 2.67810e6i	− 0.486335i
$498$	0	0
$499$	7.42433e6	1.33477	0.667384	−	0.744714i	$-0.267414\pi$
$499$	7.42433e6	1.33477	0.667384	+	0.744714i	$0.267414\pi$
$500$	0	0
$501$	0	0
$502$	4.89970e6i	0.867780i
$503$	− 2.29988e6i	− 0.405309i	−0.979250	−	0.202654i	$-0.935043\pi$
$503$	− 2.29988e6i	− 0.405309i	0.979250	−	0.202654i	$-0.0649568\pi$
$504$	0	0
$505$	0	0
$506$	810000.	0.140640
$507$	0	0
$508$	− 4.10368e6i	− 0.705528i
$509$	6.73721e6	1.15262	0.576310	−	0.817231i	$-0.304492\pi$
$509$	6.73721e6	1.15262	0.576310	+	0.817231i	$0.304492\pi$
$510$	0	0
$511$	−6.51292e6	−1.10338
$512$	262144.i	0.0441942i
$513$	0	0
$514$	−3.88853e6	−0.649198
$515$	0	0
$516$	0	0
$517$	3.17250e6i	0.522006i
$518$	− 1.02321e6i	− 0.167548i
$519$	0	0
$520$	0	0
$521$	9.26806e6	1.49587	0.747936	−	0.663770i	$-0.231045\pi$
$521$	9.26806e6	1.49587	0.747936	+	0.663770i	$0.231045\pi$
$522$	0	0
$523$	1.10620e7i	1.76839i	0.467113	+	0.884197i	$0.345294\pi$
$523$	1.10620e7i	1.76839i	−0.467113	+	0.884197i	$0.654706\pi$
$524$	2.84160e6	0.452100
$525$	0	0
$526$	7.04760e6	1.11065
$527$	− 1.45024e6i	− 0.227465i
$528$	0	0
$529$	4.61384e6	0.716842
$530$	0	0
$531$	0	0
$532$	− 2.48882e6i	− 0.381253i
$533$	− 899268.i	− 0.137111i
$534$	0	0
$535$	0	0
$536$	1.44122e6	0.216679
$537$	0	0
$538$	6.97534e6i	1.03899i
$539$	−1.58490e6	−0.234979
$540$	0	0
$541$	−1.30754e7	−1.92072	−0.960358	−	0.278771i	$-0.910073\pi$
$541$	−1.30754e7	−1.92072	−0.960358	+	0.278771i	$0.910073\pi$
$542$	− 6.81702e6i	− 0.996774i
$543$	0	0
$544$	2.08282e6	0.301754
$545$	0	0
$546$	0	0
$547$	1.84605e6i	0.263800i	0.991263	+	0.131900i	$0.0421078\pi$
$547$	1.84605e6i	0.263800i	−0.991263	+	0.131900i	$0.957892\pi$
$548$	− 4.17418e6i	− 0.593772i
$549$	0	0
$550$	0	0
$551$	−5.80067e6	−0.813953
$552$	0	0
$553$	6.43629e6i	0.894999i
$554$	5.87887e6	0.813803
$555$	0	0
$556$	−3.48294e6	−0.477815
$557$	6.48782e6i	0.886055i	0.896508	+	0.443027i	$0.146096\pi$
$557$	6.48782e6i	0.886055i	−0.896508	+	0.443027i	$0.853904\pi$
$558$	0	0
$559$	2.68232e6	0.363063
$560$	0	0
$561$	0	0
$562$	4.90121e6i	0.654579i
$563$	− 3.01538e6i	− 0.400932i	−0.979701	−	0.200466i	$-0.935754\pi$
$563$	− 3.01538e6i	− 0.400932i	0.979701	−	0.200466i	$-0.0642456\pi$
$564$	0	0
$565$	0	0
$566$	2.54933e6	0.334492
$567$	0	0
$568$	2.16960e6i	0.282169i
$569$	−3.49810e6	−0.452952	−0.226476	−	0.974017i	$-0.572720\pi$
$569$	−3.49810e6	−0.452952	−0.226476	+	0.974017i	$0.572720\pi$
$570$	0	0
$571$	−1.23749e7	−1.58837	−0.794185	−	0.607676i	$-0.792102\pi$
$571$	−1.23749e7	−1.58837	−0.794185	+	0.607676i	$0.792102\pi$
$572$	328800.i	0.0420186i
$573$	0	0
$574$	2.07422e6	0.262770
$575$	0	0
$576$	0	0
$577$	− 6.80687e6i	− 0.851153i	−0.904922	−	0.425577i	$-0.860071\pi$
$577$	− 6.80687e6i	− 0.851153i	0.904922	−	0.425577i	$-0.139929\pi$
$578$	− 1.08692e7i	− 1.35325i
$579$	0	0
$580$	0	0
$581$	2.03678e6	0.250325
$582$	0	0
$583$	− 3.88440e6i	− 0.473317i
$584$	5.27629e6	0.640172
$585$	0	0
$586$	140376.	0.0168869
$587$	7.77113e6i	0.930870i	0.885082	+	0.465435i	$0.154102\pi$
$587$	7.77113e6i	0.930870i	−0.885082	+	0.465435i	$0.845898\pi$
$588$	0	0
$589$	1.40390e6	0.166743
$590$	0	0
$591$	0	0
$592$	828928.i	0.0972104i
$593$	1.51222e7i	1.76595i	0.469418	+	0.882976i	$0.344464\pi$
$593$	1.51222e7i	1.76595i	−0.469418	+	0.882976i	$0.655536\pi$
$594$	0	0
$595$	0	0
$596$	−6.74870e6	−0.778224
$597$	0	0
$598$	− 739800.i	− 0.0845982i
$599$	−1.67012e6	−0.190187	−0.0950937	−	0.995468i	$-0.530315\pi$
$599$	−1.67012e6	−0.190187	−0.0950937	+	0.995468i	$0.530315\pi$
$600$	0	0
$601$	4.12220e6	0.465525	0.232763	−	0.972534i	$-0.425223\pi$
$601$	4.12220e6	0.465525	0.232763	+	0.972534i	$0.425223\pi$
$602$	6.18696e6i	0.695803i
$603$	0	0
$604$	1.63067e6	0.181876
$605$	0	0
$606$	0	0
$607$	− 6.81870e6i	− 0.751155i	−0.926791	−	0.375578i	$-0.877444\pi$
$607$	− 6.81870e6i	− 0.751155i	0.926791	−	0.375578i	$-0.122556\pi$
$608$	2.01626e6i	0.221201i
$609$	0	0
$610$	0	0
$611$	2.89755e6	0.313999
$612$	0	0
$613$	1.59466e7i	1.71402i	0.515300	+	0.857010i	$0.327681\pi$
$613$	1.59466e7i	1.71402i	−0.515300	+	0.857010i	$0.672319\pi$
$614$	1.15216e7	1.23336
$615$	0	0
$616$	−758400.	−0.0805279
$617$	− 8.21952e6i	− 0.869228i	−0.900617	−	0.434614i	$-0.856885\pi$
$617$	− 8.21952e6i	− 0.869228i	0.900617	−	0.434614i	$-0.143115\pi$
$618$	0	0
$619$	1.23323e7	1.29366	0.646828	−	0.762636i	$-0.276095\pi$
$619$	1.23323e7	1.29366	0.646828	+	0.762636i	$0.276095\pi$
$620$	0	0
$621$	0	0
$622$	9.53220e6i	0.987910i
$623$	8.19451e6i	0.845869i
$624$	0	0
$625$	0	0
$626$	7.72325e6	0.787706
$627$	0	0
$628$	− 1.62837e6i	− 0.164761i
$629$	6.58609e6	0.663745
$630$	0	0
$631$	−3.01912e6	−0.301861	−0.150931	−	0.988544i	$-0.548227\pi$
$631$	−3.01912e6	−0.301861	−0.150931	+	0.988544i	$0.548227\pi$
$632$	− 5.21421e6i	− 0.519273i
$633$	0	0
$634$	8.75981e6	0.865509
$635$	0	0
$636$	0	0
$637$	1.44754e6i	0.141346i
$638$	1.76760e6i	0.171922i
$639$	0	0
$640$	0	0
$641$	1.18052e7	1.13482	0.567411	−	0.823435i	$-0.307945\pi$
$641$	1.18052e7	1.13482	0.567411	+	0.823435i	$0.307945\pi$
$642$	0	0
$643$	− 2.85360e6i	− 0.272186i	−0.990696	−	0.136093i	$-0.956545\pi$
$643$	− 2.85360e6i	− 0.272186i	0.990696	−	0.136093i	$-0.0434546\pi$
$644$	1.70640e6	0.162131
$645$	0	0
$646$	1.60198e7	1.51034
$647$	− 3.40192e6i	− 0.319494i	−0.987158	−	0.159747i	$-0.948932\pi$
$647$	− 3.40192e6i	− 0.319494i	0.987158	−	0.159747i	$-0.0510679\pi$
$648$	0	0
$649$	−3.80250e6	−0.354370
$650$	0	0
$651$	0	0
$652$	3.23598e6i	0.298117i
$653$	9.39166e6i	0.861905i	0.902375	+	0.430953i	$0.141822\pi$
$653$	9.39166e6i	0.861905i	−0.902375	+	0.430953i	$0.858178\pi$
$654$	0	0
$655$	0	0
$656$	−1.68038e6	−0.152458
$657$	0	0
$658$	6.68340e6i	0.601773i
$659$	9.36659e6	0.840171	0.420086	−	0.907484i	$-0.362000\pi$
$659$	9.36659e6	0.840171	0.420086	+	0.907484i	$0.362000\pi$
$660$	0	0
$661$	1.57748e7	1.40430	0.702150	−	0.712029i	$-0.252223\pi$
$661$	1.57748e7	1.40430	0.702150	+	0.712029i	$0.252223\pi$
$662$	5.65717e6i	0.501712i
$663$	0	0
$664$	−1.65005e6	−0.145237
$665$	0	0
$666$	0	0
$667$	− 3.97710e6i	− 0.346140i
$668$	− 3.70560e6i	− 0.321305i
$669$	0	0
$670$	0	0
$671$	−7.59225e6	−0.650975
$672$	0	0
$673$	− 6.64515e6i	− 0.565545i	−0.959187	−	0.282773i	$-0.908746\pi$
$673$	− 6.64515e6i	− 0.565545i	0.959187	−	0.282773i	$-0.0912542\pi$
$674$	5.52832e6	0.468753
$675$	0	0
$676$	−5.64038e6	−0.474725
$677$	− 1.57780e7i	− 1.32306i	−0.749917	−	0.661532i	$-0.769906\pi$
$677$	− 1.57780e7i	− 1.32306i	0.749917	−	0.661532i	$-0.230094\pi$
$678$	0	0
$679$	−4.53010e6	−0.377080
$680$	0	0
$681$	0	0
$682$	− 427800.i	− 0.0352192i
$683$	1.96654e7i	1.61306i	0.591190	+	0.806532i	$0.298658\pi$
$683$	1.96654e7i	1.61306i	−0.591190	+	0.806532i	$0.701342\pi$
$684$	0	0
$685$	0	0
$686$	−8.64987e6	−0.701777
$687$	0	0
$688$	− 5.01222e6i	− 0.403701i
$689$	−3.54775e6	−0.284712
$690$	0	0
$691$	990464.	0.0789121	0.0394560	−	0.999221i	$-0.487437\pi$
$691$	990464.	0.0789121	0.0394560	+	0.999221i	$0.487437\pi$
$692$	− 2.78755e6i	− 0.221288i
$693$	0	0
$694$	1.13503e6	0.0894560
$695$	0	0
$696$	0	0
$697$	1.33512e7i	1.04097i
$698$	− 8.55236e6i	− 0.664427i
$699$	0	0
$700$	0	0
$701$	−6.52919e6	−0.501839	−0.250920	−	0.968008i	$-0.580733\pi$
$701$	−6.52919e6	−0.501839	−0.250920	+	0.968008i	$0.580733\pi$
$702$	0	0
$703$	6.37562e6i	0.486558i
$704$	614400.	0.0467218
$705$	0	0
$706$	1.26406e7	0.954455
$707$	7.19153e6i	0.541094i
$708$	0	0
$709$	2.02106e7	1.50996	0.754979	−	0.655749i	$-0.227647\pi$
$709$	2.02106e7	1.50996	0.754979	+	0.655749i	$0.227647\pi$
$710$	0	0
$711$	0	0
$712$	− 6.63859e6i	− 0.490768i
$713$	962550.i	0.0709087i
$714$	0	0
$715$	0	0
$716$	1.02731e7	0.748888
$717$	0	0
$718$	− 1.43826e7i	− 1.04118i
$719$	−2.27720e7	−1.64277	−0.821387	−	0.570371i	$-0.806800\pi$
$719$	−2.27720e7	−1.64277	−0.821387	+	0.570371i	$0.806800\pi$
$720$	0	0
$721$	−1.51392e7	−1.08459
$722$	5.60345e6i	0.400048i
$723$	0	0
$724$	897136.	0.0636080
$725$	0	0
$726$	0	0
$727$	7.09186e6i	0.497650i	0.968548	+	0.248825i	$0.0800445\pi$
$727$	7.09186e6i	0.497650i	−0.968548	+	0.248825i	$0.919956\pi$
$728$	692672.i	0.0484394i
$729$	0	0
$730$	0	0
$731$	−3.98237e7	−2.75644
$732$	0	0
$733$	− 1.38982e7i	− 0.955432i	−0.878514	−	0.477716i	$-0.841465\pi$
$733$	− 1.38982e7i	− 0.955432i	0.878514	−	0.477716i	$-0.158535\pi$
$734$	−1.84817e7	−1.26620
$735$	0	0
$736$	−1.38240e6	−0.0940674
$737$	− 3.37785e6i	− 0.229072i
$738$	0	0
$739$	−1.54857e7	−1.04309	−0.521543	−	0.853225i	$-0.674643\pi$
$739$	−1.54857e7	−1.04309	−0.521543	+	0.853225i	$0.674643\pi$
$740$	0	0
$741$	0	0
$742$	− 8.18314e6i	− 0.545645i
$743$	− 159276.i	− 0.0105847i	−0.999986	−	0.00529235i	$-0.998315\pi$
$743$	− 159276.i	− 0.0105847i	0.999986	−	0.00529235i	$-0.00168461\pi$
$744$	0	0
$745$	0	0
$746$	1.40793e7	0.926263
$747$	0	0
$748$	− 4.88160e6i	− 0.319013i
$749$	−733752.	−0.0477908
$750$	0	0
$751$	1.15554e7	0.747625	0.373812	−	0.927504i	$-0.378050\pi$
$751$	1.15554e7	0.747625	0.373812	+	0.927504i	$0.378050\pi$
$752$	− 5.41440e6i	− 0.349145i
$753$	0	0
$754$	1.61441e6	0.103415
$755$	0	0
$756$	0	0
$757$	6.34402e6i	0.402369i	0.979553	+	0.201185i	$0.0644792\pi$
$757$	6.34402e6i	0.402369i	−0.979553	+	0.201185i	$0.935521\pi$
$758$	− 2.38024e6i	− 0.150469i
$759$	0	0
$760$	0	0
$761$	1.17748e7	0.737040	0.368520	−	0.929620i	$-0.379865\pi$
$761$	1.17748e7	0.737040	0.368520	+	0.929620i	$0.379865\pi$
$762$	0	0
$763$	1.63016e6i	0.101373i
$764$	−3.35510e6	−0.207957
$765$	0	0
$766$	−5.20200e6	−0.320331
$767$	3.47295e6i	0.213162i
$768$	0	0
$769$	2.47869e7	1.51149	0.755745	−	0.654865i	$-0.227275\pi$
$769$	2.47869e7	1.51149	0.755745	+	0.654865i	$0.227275\pi$
$770$	0	0
$771$	0	0
$772$	1.14496e7i	0.691425i
$773$	1.38017e7i	0.830774i	0.909645	+	0.415387i	$0.136354\pi$
$773$	1.38017e7i	0.830774i	−0.909645	+	0.415387i	$0.863646\pi$
$774$	0	0
$775$	0	0
$776$	3.66995e6	0.218779
$777$	0	0
$778$	2.48290e6i	0.147065i
$779$	−1.29245e7	−0.763081
$780$	0	0
$781$	5.08500e6	0.298307
$782$	1.09836e7i	0.642285i
$783$	0	0
$784$	2.70490e6	0.157167
$785$	0	0
$786$	0	0
$787$	1.53795e7i	0.885129i	0.896737	+	0.442565i	$0.145931\pi$
$787$	1.53795e7i	0.885129i	−0.896737	+	0.442565i	$0.854069\pi$
$788$	− 8.14166e6i	− 0.467087i
$789$	0	0
$790$	0	0
$791$	1.39745e7	0.794135
$792$	0	0
$793$	6.93426e6i	0.391577i
$794$	4.74489e6	0.267101
$795$	0	0
$796$	−1.57763e7	−0.882514
$797$	− 2.33978e7i	− 1.30475i	−0.757894	−	0.652377i	$-0.773772\pi$
$797$	− 2.33978e7i	− 1.30475i	0.757894	−	0.652377i	$-0.226228\pi$
$798$	0	0
$799$	−4.30191e7	−2.38393
$800$	0	0
$801$	0	0
$802$	− 1.25334e7i	− 0.688069i
$803$	− 1.23663e7i	− 0.676785i
$804$	0	0
$805$	0	0
$806$	−390724.	−0.0211852
$807$	0	0
$808$	− 5.82605e6i	− 0.313939i
$809$	−2.80188e7	−1.50515	−0.752573	−	0.658509i	$-0.771187\pi$
$809$	−2.80188e7	−1.50515	−0.752573	+	0.658509i	$0.771187\pi$
$810$	0	0
$811$	−1.65641e6	−0.0884332	−0.0442166	−	0.999022i	$-0.514079\pi$
$811$	−1.65641e6	−0.0884332	−0.0442166	+	0.999022i	$0.514079\pi$
$812$	3.72374e6i	0.198194i
$813$	0	0
$814$	1.94280e6	0.102770
$815$	0	0
$816$	0	0
$817$	− 3.85511e7i	− 2.02060i
$818$	− 2.27032e6i	− 0.118633i
$819$	0	0
$820$	0	0
$821$	−2.97382e7	−1.53978	−0.769888	−	0.638179i	$-0.779688\pi$
$821$	−2.97382e7	−1.53978	−0.769888	+	0.638179i	$0.779688\pi$
$822$	0	0
$823$	− 1.60108e7i	− 0.823972i	−0.911190	−	0.411986i	$-0.864835\pi$
$823$	− 1.60108e7i	− 0.823972i	0.911190	−	0.411986i	$-0.135165\pi$
$824$	1.22647e7	0.629273
$825$	0	0
$826$	−8.01060e6	−0.408522
$827$	2.54748e7i	1.29523i	0.761969	+	0.647614i	$0.224233\pi$
$827$	2.54748e7i	1.29523i	−0.761969	+	0.647614i	$0.775767\pi$
$828$	0	0
$829$	−1.89971e7	−0.960064	−0.480032	−	0.877251i	$-0.659375\pi$
$829$	−1.89971e7	−0.960064	−0.480032	+	0.877251i	$0.659375\pi$
$830$	0	0
$831$	0	0
$832$	− 561152.i	− 0.0281043i
$833$	− 2.14912e7i	− 1.07312i
$834$	0	0
$835$	0	0
$836$	4.72560e6	0.233852
$837$	0	0
$838$	− 2.43600e7i	− 1.19830i
$839$	3.98539e7	1.95463	0.977317	−	0.211782i	$-0.0679267\pi$
$839$	3.98539e7	1.95463	0.977317	+	0.211782i	$0.0679267\pi$
$840$	0	0
$841$	−1.18322e7	−0.576868
$842$	1.22856e7i	0.597193i
$843$	0	0
$844$	−1.91192e6	−0.0923876
$845$	0	0
$846$	0	0
$847$	− 1.09455e7i	− 0.524238i
$848$	6.62938e6i	0.316580i
$849$	0	0
$850$	0	0
$851$	−4.37130e6	−0.206912
$852$	0	0
$853$	2.19179e7i	1.03140i	0.856769	+	0.515700i	$0.172468\pi$
$853$	2.19179e7i	1.03140i	−0.856769	+	0.515700i	$0.827532\pi$
$854$	−1.59943e7	−0.750450
$855$	0	0
$856$	594432.	0.0277280
$857$	− 2.72213e7i	− 1.26607i	−0.774125	−	0.633033i	$-0.781810\pi$
$857$	− 2.72213e7i	− 1.26607i	0.774125	−	0.633033i	$-0.218190\pi$
$858$	0	0
$859$	−3.76214e7	−1.73961	−0.869805	−	0.493396i	$-0.835755\pi$
$859$	−3.76214e7	−1.73961	−0.869805	+	0.493396i	$0.835755\pi$
$860$	0	0
$861$	0	0
$862$	2.67660e6i	0.122692i
$863$	− 2.51995e7i	− 1.15177i	−0.817531	−	0.575885i	$-0.804658\pi$
$863$	− 2.51995e7i	− 1.15177i	0.817531	−	0.575885i	$-0.195342\pi$
$864$	0	0
$865$	0	0
$866$	1.30176e7	0.589841
$867$	0	0
$868$	− 901232.i	− 0.0406011i
$869$	−1.22208e7	−0.548972
$870$	0	0
$871$	−3.08510e6	−0.137792
$872$	− 1.32064e6i	− 0.0588157i
$873$	0	0
$874$	−1.06326e7	−0.470826
$875$	0	0
$876$	0	0
$877$	− 2.77307e6i	− 0.121748i	−0.998145	−	0.0608739i	$-0.980611\pi$
$877$	− 2.77307e6i	− 0.121748i	0.998145	−	0.0608739i	$-0.0193888\pi$
$878$	− 6.35037e6i	− 0.278012i
$879$	0	0
$880$	0	0
$881$	1.37827e7	0.598268	0.299134	−	0.954211i	$-0.403302\pi$
$881$	1.37827e7	0.598268	0.299134	+	0.954211i	$0.403302\pi$
$882$	0	0
$883$	− 4.96322e6i	− 0.214221i	−0.994247	−	0.107110i	$-0.965840\pi$
$883$	− 4.96322e6i	− 0.214221i	0.994247	−	0.107110i	$-0.0341598\pi$
$884$	−4.45853e6	−0.191894
$885$	0	0
$886$	−855552.	−0.0366153
$887$	4.12640e7i	1.76101i	0.474037	+	0.880505i	$0.342796\pi$
$887$	4.12640e7i	1.76101i	−0.474037	+	0.880505i	$0.657204\pi$
$888$	0	0
$889$	−2.02619e7	−0.859857
$890$	0	0
$891$	0	0
$892$	− 776720.i	− 0.0326853i
$893$	− 4.16444e7i	− 1.74754i
$894$	0	0
$895$	0	0
$896$	1.29434e6	0.0538613
$897$	0	0
$898$	1.48290e7i	0.613649i
$899$	−2.10050e6	−0.0866808
$900$	0	0
$901$	5.26725e7	2.16158
$902$	3.93840e6i	0.161177i
$903$	0	0
$904$	−1.13211e7	−0.460752
$905$	0	0
$906$	0	0
$907$	− 1.66308e7i	− 0.671268i	−0.941992	−	0.335634i	$-0.891049\pi$
$907$	− 1.66308e7i	− 0.671268i	0.941992	−	0.335634i	$-0.108951\pi$
$908$	4.60243e6i	0.185256i
$909$	0	0
$910$	0	0
$911$	−3.64995e7	−1.45711	−0.728554	−	0.684989i	$-0.759807\pi$
$911$	−3.64995e7	−1.45711	−0.728554	+	0.684989i	$0.759807\pi$
$912$	0	0
$913$	3.86730e6i	0.153543i
$914$	1.53277e7	0.606892
$915$	0	0
$916$	1.15304e6	0.0454052
$917$	− 1.40304e7i	− 0.550994i
$918$	0	0
$919$	−6.83190e6	−0.266841	−0.133421	−	0.991060i	$-0.542596\pi$
$919$	−6.83190e6	−0.266841	−0.133421	+	0.991060i	$0.542596\pi$
$920$	0	0
$921$	0	0
$922$	− 2.23469e7i	− 0.865744i
$923$	− 4.64430e6i	− 0.179439i
$924$	0	0
$925$	0	0
$926$	1.76057e7	0.674723
$927$	0	0
$928$	− 3.01670e6i	− 0.114991i
$929$	5.05115e6	0.192022	0.0960111	−	0.995380i	$-0.469392\pi$
$929$	5.05115e6	0.192022	0.0960111	+	0.995380i	$0.469392\pi$
$930$	0	0
$931$	2.08045e7	0.786651
$932$	− 9.10637e6i	− 0.343404i
$933$	0	0
$934$	−1.34090e7	−0.502956
$935$	0	0
$936$	0	0
$937$	− 8.30023e6i	− 0.308845i	−0.988005	−	0.154423i	$-0.950648\pi$
$937$	− 8.30023e6i	− 0.308845i	0.988005	−	0.154423i	$-0.0493518\pi$
$938$	− 7.11600e6i	− 0.264076i
$939$	0	0
$940$	0	0
$941$	4.87332e7	1.79412	0.897059	−	0.441910i	$-0.145699\pi$
$941$	4.87332e7	1.79412	0.897059	+	0.441910i	$0.145699\pi$
$942$	0	0
$943$	− 8.86140e6i	− 0.324506i
$944$	6.48960e6	0.237022
$945$	0	0
$946$	−1.17474e7	−0.426790
$947$	− 7.34777e6i	− 0.266245i	−0.991100	−	0.133122i	$-0.957500\pi$
$947$	− 7.34777e6i	− 0.266245i	0.991100	−	0.133122i	$-0.0425003\pi$
$948$	0	0
$949$	−1.12946e7	−0.407102
$950$	0	0
$951$	0	0
$952$	− 1.02839e7i	− 0.367761i
$953$	3.51437e7i	1.25347i	0.779231	+	0.626737i	$0.215610\pi$
$953$	3.51437e7i	1.25347i	−0.779231	+	0.626737i	$0.784390\pi$
$954$	0	0
$955$	0	0
$956$	−1.11441e7	−0.394365
$957$	0	0
$958$	3.63986e6i	0.128136i
$959$	−2.06100e7	−0.723655
$960$	0	0
$961$	−2.81208e7	−0.982243
$962$	− 1.77442e6i	− 0.0618187i
$963$	0	0
$964$	−9.21819e6	−0.319487
$965$	0	0
$966$	0	0
$967$	− 2.31186e6i	− 0.0795050i	−0.999210	−	0.0397525i	$-0.987343\pi$
$967$	− 2.31186e6i	− 0.0795050i	0.999210	−	0.0397525i	$-0.0126570\pi$
$968$	8.86726e6i	0.304159i
$969$	0	0
$970$	0	0
$971$	3.73588e7	1.27158	0.635791	−	0.771861i	$-0.280674\pi$
$971$	3.73588e7	1.27158	0.635791	+	0.771861i	$0.280674\pi$
$972$	0	0
$973$	1.71970e7i	0.582333i
$974$	−1.61432e7	−0.545245
$975$	0	0
$976$	1.29574e7	0.435406
$977$	− 1.02666e7i	− 0.344104i	−0.985088	−	0.172052i	$-0.944960\pi$
$977$	− 1.02666e7i	− 0.344104i	0.985088	−	0.172052i	$-0.0550397\pi$
$978$	0	0
$979$	−1.55592e7	−0.518837
$980$	0	0
$981$	0	0
$982$	− 1.51978e6i	− 0.0502922i
$983$	− 2.98404e7i	− 0.984966i	−0.870322	−	0.492483i	$-0.836089\pi$
$983$	− 2.98404e7i	− 0.984966i	0.870322	−	0.492483i	$-0.163911\pi$
$984$	0	0
$985$	0	0
$986$	−2.39687e7	−0.785148
$987$	0	0
$988$	− 4.31605e6i	− 0.140668i
$989$	2.64316e7	0.859278
$990$	0	0
$991$	−2.34222e7	−0.757605	−0.378803	−	0.925477i	$-0.623664\pi$
$991$	−2.34222e7	−0.757605	−0.378803	+	0.925477i	$0.623664\pi$
$992$	730112.i	0.0235565i
$993$	0	0
$994$	1.07124e7	0.343891
$995$	0	0
$996$	0	0
$997$	2.60709e7i	0.830651i	0.909673	+	0.415326i	$0.136332\pi$
$997$	2.60709e7i	0.830651i	−0.909673	+	0.415326i	$0.863668\pi$
$998$	2.96973e7i	0.943824i
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	450.6.c.g.199.2		2
3.2	odd	2		150.6.c.c.49.1		2
5.2	odd	4		450.6.a.e.1.1		1
5.3	odd	4		450.6.a.t.1.1		1
5.4	even	2	inner	450.6.c.g.199.1		2
15.2	even	4		150.6.a.i.1.1	yes	1
15.8	even	4		150.6.a.g.1.1	✓	1
15.14	odd	2		150.6.c.c.49.2		2

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
150.6.a.g.1.1	✓	1	15.8	even	4
150.6.a.i.1.1	yes	1	15.2	even	4
150.6.c.c.49.1		2	3.2	odd	2
150.6.c.c.49.2		2	15.14	odd	2
450.6.a.e.1.1		1	5.2	odd	4
450.6.a.t.1.1		1	5.3	odd	4
450.6.c.g.199.1		2	5.4	even	2	inner
450.6.c.g.199.2		2	1.1	even	1	trivial

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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)

\(f(q)\)	\(=\)	\(q+4.00000i q^{2} -16.0000 q^{4} +79.0000i q^{7} -64.0000i q^{8} +O(q^{10})\)	\(q+4.00000i q^{2} -16.0000 q^{4} +79.0000i q^{7} -64.0000i q^{8} -150.000 q^{11} +137.000i q^{13} -316.000 q^{14} +256.000 q^{16} -2034.00i q^{17} +1969.00 q^{19} -600.000i q^{22} +1350.00i q^{23} -548.000 q^{26} -1264.00i q^{28} -2946.00 q^{29} +713.000 q^{31} +1024.00i q^{32} +8136.00 q^{34} +3238.00i q^{37} +7876.00i q^{38} -6564.00 q^{41} -19579.0i q^{43} +2400.00 q^{44} -5400.00 q^{46} -21150.0i q^{47} +10566.0 q^{49} -2192.00i q^{52} +25896.0i q^{53} +5056.00 q^{56} -11784.0i q^{58} +25350.0 q^{59} +50615.0 q^{61} +2852.00i q^{62} -4096.00 q^{64} +22519.0i q^{67} +32544.0i q^{68} -33900.0 q^{71} +82442.0i q^{73} -12952.0 q^{74} -31504.0 q^{76} -11850.0i q^{77} +81472.0 q^{79} -26256.0i q^{82} -25782.0i q^{83} +78316.0 q^{86} +9600.00i q^{88} +103728. q^{89} -10823.0 q^{91} -21600.0i q^{92} +84600.0 q^{94} +57343.0i q^{97} +42264.0i q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 2 q - 32 q^{4}+O(q^{10}) \) 2 * q - 32 * q^4	\( 2 q - 32 q^{4} - 300 q^{11} - 632 q^{14} + 512 q^{16} + 3938 q^{19} - 1096 q^{26} - 5892 q^{29} + 1426 q^{31} + 16272 q^{34} - 13128 q^{41} + 4800 q^{44} - 10800 q^{46} + 21132 q^{49} + 10112 q^{56} + 50700 q^{59} + 101230 q^{61} - 8192 q^{64} - 67800 q^{71} - 25904 q^{74} - 63008 q^{76} + 162944 q^{79} + 156632 q^{86} + 207456 q^{89} - 21646 q^{91} + 169200 q^{94}+O(q^{100}) \) 2 * q - 32 * q^4 - 300 * q^11 - 632 * q^14 + 512 * q^16 + 3938 * q^19 - 1096 * q^26 - 5892 * q^29 + 1426 * q^31 + 16272 * q^34 - 13128 * q^41 + 4800 * q^44 - 10800 * q^46 + 21132 * q^49 + 10112 * q^56 + 50700 * q^59 + 101230 * q^61 - 8192 * q^64 - 67800 * q^71 - 25904 * q^74 - 63008 * q^76 + 162944 * q^79 + 156632 * q^86 + 207456 * q^89 - 21646 * q^91 + 169200 * q^94

Introduction

L-functions

Modular forms

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