LMFDB - Embedded newform 225.6.b.e.199.2

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [225,6,Mod(199,225)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(225, base_ring=CyclotomicField(2))

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

N = Newforms(chi, 6, names="a")

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

chi := DirichletCharacter("225.199");

S:= CuspForms(chi, 6);

N := Newforms(S);

Level:	$ N $	$=$	$ 225 = 3^{2} \cdot 5^{2} $
Weight:	$ k $	$=$	$ 6 $
Character orbit:	$[\chi]$	$=$	225.b (of order $2$, degree $1$, not minimal)

Newform invariants

comment: select newform

sage: f = N[0] # Warning: the index may be different

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

Self dual:	no
Analytic conductor:	$36.0863594579$
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 5)
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label			199.2
Root			$1.00000i$ of defining polynomial
Character	$\chi$	$=$	225.199
Dual form			225.6.b.e.199.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+2.00000i q^{2} +28.0000 q^{4} -192.000i q^{7} +120.000i q^{8} +O(q^{10})$	$q+2.00000i q^{2} +28.0000 q^{4} -192.000i q^{7} +120.000i q^{8} +148.000 q^{11} +286.000i q^{13} +384.000 q^{14} +656.000 q^{16} -1678.00i q^{17} -1060.00 q^{19} +296.000i q^{22} -2976.00i q^{23} -572.000 q^{26} -5376.00i q^{28} -3410.00 q^{29} -2448.00 q^{31} +5152.00i q^{32} +3356.00 q^{34} -182.000i q^{37} -2120.00i q^{38} +9398.00 q^{41} -1244.00i q^{43} +4144.00 q^{44} +5952.00 q^{46} -12088.0i q^{47} -20057.0 q^{49} +8008.00i q^{52} -23846.0i q^{53} +23040.0 q^{56} -6820.00i q^{58} -20020.0 q^{59} +32302.0 q^{61} -4896.00i q^{62} +10688.0 q^{64} -60972.0i q^{67} -46984.0i q^{68} +32648.0 q^{71} -38774.0i q^{73} +364.000 q^{74} -29680.0 q^{76} -28416.0i q^{77} +33360.0 q^{79} +18796.0i q^{82} -16716.0i q^{83} +2488.00 q^{86} +17760.0i q^{88} +101370. q^{89} +54912.0 q^{91} -83328.0i q^{92} +24176.0 q^{94} +119038. i q^{97} -40114.0i q^{98} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 2 q + 56 q^{4}+O(q^{10}) $ 2 * q + 56 * q^4	$ 2 q + 56 q^{4} + 296 q^{11} + 768 q^{14} + 1312 q^{16} - 2120 q^{19} - 1144 q^{26} - 6820 q^{29} - 4896 q^{31} + 6712 q^{34} + 18796 q^{41} + 8288 q^{44} + 11904 q^{46} - 40114 q^{49} + 46080 q^{56} - 40040 q^{59} + 64604 q^{61} + 21376 q^{64} + 65296 q^{71} + 728 q^{74} - 59360 q^{76} + 66720 q^{79} + 4976 q^{86} + 202740 q^{89} + 109824 q^{91} + 48352 q^{94}+O(q^{100}) $ 2 * q + 56 * q^4 + 296 * q^11 + 768 * q^14 + 1312 * q^16 - 2120 * q^19 - 1144 * q^26 - 6820 * q^29 - 4896 * q^31 + 6712 * q^34 + 18796 * q^41 + 8288 * q^44 + 11904 * q^46 - 40114 * q^49 + 46080 * q^56 - 40040 * q^59 + 64604 * q^61 + 21376 * q^64 + 65296 * q^71 + 728 * q^74 - 59360 * q^76 + 66720 * q^79 + 4976 * q^86 + 202740 * q^89 + 109824 * q^91 + 48352 * q^94

Display 100 coefficients 10 coefficients

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/225\mathbb{Z}\right)^\times$.

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

Coefficient data

For each $n$ we display the coefficients of the $q$-expansion $a_n$, the Satake parameters $\alpha_p$, and the Satake angles $\theta_p = \textrm{Arg}(\alpha_p)$.

Display $a_p$ with $p$ up to: 50 250 1000 (See $a_n$ instead) (See $a_n$ instead) (See $a_n$ instead) Display $a_n$ with $n$ up to: 50 250 1000 (See only $a_p$) (See only $a_p$) (See only $a_p$)

$n$	$a_n$	$a_n / n^{(k-1)/2}$	$ \alpha_n $			$ \theta_n $
$p$	$a_p$	$a_p / p^{(k-1)/2}$	$ \alpha_p$			$ \theta_p $

$2$	2.00000i	0.353553i	0.984251	+	0.176777i	$0.0565670\pi$
$2$	2.00000i	0.353553i	−0.984251	+	0.176777i	$0.943433\pi$
$3$	0	0
$3$	0	0
$4$	28.0000	0.875000
$5$	0	0
$5$	0	0
$6$	0	0
$7$	− 192.000i	− 1.48100i	−0.672054	−	0.740502i	$-0.734588\pi$
$7$	− 192.000i	− 1.48100i	0.672054	−	0.740502i	$-0.265412\pi$
$8$	120.000i	0.662913i
$9$	0	0
$10$	0	0
$11$	148.000	0.368791	0.184395	−	0.982852i	$-0.440967\pi$
$11$	148.000	0.368791	0.184395	+	0.982852i	$0.440967\pi$
$12$	0	0
$13$	286.000i	0.469362i	0.972072	+	0.234681i	$0.0754045\pi$
$13$	286.000i	0.469362i	−0.972072	+	0.234681i	$0.924595\pi$
$14$	384.000	0.523614
$15$	0	0
$16$	656.000	0.640625
$17$	− 1678.00i	− 1.40822i	−0.710092	−	0.704109i	$-0.751347\pi$
$17$	− 1678.00i	− 1.40822i	0.710092	−	0.704109i	$-0.248653\pi$
$18$	0	0
$19$	−1060.00	−0.673631	−0.336815	−	0.941571i	$-0.609350\pi$
$19$	−1060.00	−0.673631	−0.336815	+	0.941571i	$0.609350\pi$
$20$	0	0
$21$	0	0
$22$	296.000i	0.130387i
$23$	− 2976.00i	− 1.17304i	−0.809934	−	0.586521i	$-0.800497\pi$
$23$	− 2976.00i	− 1.17304i	0.809934	−	0.586521i	$-0.199503\pi$
$24$	0	0
$25$	0	0
$26$	−572.000	−0.165944
$27$	0	0
$28$	− 5376.00i	− 1.29588i
$29$	−3410.00	−0.752938	−0.376469	−	0.926429i	$-0.622862\pi$
$29$	−3410.00	−0.752938	−0.376469	+	0.926429i	$0.622862\pi$
$30$	0	0
$31$	−2448.00	−0.457517	−0.228758	−	0.973483i	$-0.573467\pi$
$31$	−2448.00	−0.457517	−0.228758	+	0.973483i	$0.573467\pi$
$32$	5152.00i	0.889408i
$33$	0	0
$34$	3356.00	0.497880
$35$	0	0
$36$	0	0
$37$	− 182.000i	− 0.0218558i	−0.999940	−	0.0109279i	$-0.996521\pi$
$37$	− 182.000i	− 0.0218558i	0.999940	−	0.0109279i	$-0.00347853\pi$
$38$	− 2120.00i	− 0.238164i
$39$	0	0
$40$	0	0
$41$	9398.00	0.873124	0.436562	−	0.899674i	$-0.356196\pi$
$41$	9398.00	0.873124	0.436562	+	0.899674i	$0.356196\pi$
$42$	0	0
$43$	− 1244.00i	− 0.102600i	−0.998683	−	0.0513002i	$-0.983663\pi$
$43$	− 1244.00i	− 0.102600i	0.998683	−	0.0513002i	$-0.0163365\pi$
$44$	4144.00	0.322692
$45$	0	0
$46$	5952.00	0.414733
$47$	− 12088.0i	− 0.798196i	−0.916908	−	0.399098i	$-0.869323\pi$
$47$	− 12088.0i	− 0.798196i	0.916908	−	0.399098i	$-0.130677\pi$
$48$	0	0
$49$	−20057.0	−1.19337
$50$	0	0
$51$	0	0
$52$	8008.00i	0.410691i
$53$	− 23846.0i	− 1.16607i	−0.812446	−	0.583037i	$-0.801864\pi$
$53$	− 23846.0i	− 1.16607i	0.812446	−	0.583037i	$-0.198136\pi$
$54$	0	0
$55$	0	0
$56$	23040.0	0.981776
$57$	0	0
$58$	− 6820.00i	− 0.266204i
$59$	−20020.0	−0.748745	−0.374373	−	0.927278i	$-0.622142\pi$
$59$	−20020.0	−0.748745	−0.374373	+	0.927278i	$0.622142\pi$
$60$	0	0
$61$	32302.0	1.11149	0.555744	−	0.831353i	$-0.312433\pi$
$61$	32302.0	1.11149	0.555744	+	0.831353i	$0.312433\pi$
$62$	− 4896.00i	− 0.161757i
$63$	0	0
$64$	10688.0	0.326172
$65$	0	0
$66$	0	0
$67$	− 60972.0i	− 1.65937i	−0.558231	−	0.829685i	$-0.688520\pi$
$67$	− 60972.0i	− 1.65937i	0.558231	−	0.829685i	$-0.311480\pi$
$68$	− 46984.0i	− 1.23219i
$69$	0	0
$70$	0	0
$71$	32648.0	0.768618	0.384309	−	0.923204i	$-0.374440\pi$
$71$	32648.0	0.768618	0.384309	+	0.923204i	$0.374440\pi$
$72$	0	0
$73$	− 38774.0i	− 0.851596i	−0.904818	−	0.425798i	$-0.859993\pi$
$73$	− 38774.0i	− 0.851596i	0.904818	−	0.425798i	$-0.140007\pi$
$74$	364.000	0.00772720
$75$	0	0
$76$	−29680.0	−0.589427
$77$	− 28416.0i	− 0.546180i
$78$	0	0
$79$	33360.0	0.601393	0.300696	−	0.953720i	$-0.402781\pi$
$79$	33360.0	0.601393	0.300696	+	0.953720i	$0.402781\pi$
$80$	0	0
$81$	0	0
$82$	18796.0i	0.308696i
$83$	− 16716.0i	− 0.266340i	−0.991093	−	0.133170i	$-0.957484\pi$
$83$	− 16716.0i	− 0.266340i	0.991093	−	0.133170i	$-0.0425157\pi$
$84$	0	0
$85$	0	0
$86$	2488.00	0.0362747
$87$	0	0
$88$	17760.0i	0.244476i
$89$	101370.	1.35655	0.678273	−	0.734810i	$-0.262729\pi$
$89$	101370.	1.35655	0.678273	+	0.734810i	$0.262729\pi$
$90$	0	0
$91$	54912.0	0.695126
$92$	− 83328.0i	− 1.02641i
$93$	0	0
$94$	24176.0	0.282205
$95$	0	0
$96$	0	0
$97$	119038.i	1.28457i	0.766468	+	0.642283i	$0.222013\pi$
$97$	119038.i	1.28457i	−0.766468	+	0.642283i	$0.777987\pi$
$98$	− 40114.0i	− 0.421921i
$99$	0	0
$100$	0	0
$101$	89898.0	0.876893	0.438446	−	0.898757i	$-0.355529\pi$
$101$	89898.0	0.876893	0.438446	+	0.898757i	$0.355529\pi$
$102$	0	0
$103$	− 19504.0i	− 0.181147i	−0.995890	−	0.0905734i	$-0.971130\pi$
$103$	− 19504.0i	− 0.181147i	0.995890	−	0.0905734i	$-0.0288700\pi$
$104$	−34320.0	−0.311146
$105$	0	0
$106$	47692.0	0.412269
$107$	158292.i	1.33659i	0.743895	+	0.668297i	$0.232976\pi$
$107$	158292.i	1.33659i	−0.743895	+	0.668297i	$0.767024\pi$
$108$	0	0
$109$	−36830.0	−0.296917	−0.148459	−	0.988919i	$-0.547431\pi$
$109$	−36830.0	−0.296917	−0.148459	+	0.988919i	$0.547431\pi$
$110$	0	0
$111$	0	0
$112$	− 125952.i	− 0.948768i
$113$	− 11186.0i	− 0.0824098i	−0.999151	−	0.0412049i	$-0.986880\pi$
$113$	− 11186.0i	− 0.0824098i	0.999151	−	0.0412049i	$-0.0131196\pi$
$114$	0	0
$115$	0	0
$116$	−95480.0	−0.658821
$117$	0	0
$118$	− 40040.0i	− 0.264721i
$119$	−322176.	−2.08557
$120$	0	0
$121$	−139147.	−0.863993
$122$	64604.0i	0.392970i
$123$	0	0
$124$	−68544.0	−0.400327
$125$	0	0
$126$	0	0
$127$	− 70552.0i	− 0.388150i	−0.980987	−	0.194075i	$-0.937829\pi$
$127$	− 70552.0i	− 0.388150i	0.980987	−	0.194075i	$-0.0621706\pi$
$128$	186240.i	1.00473i
$129$	0	0
$130$	0	0
$131$	−76452.0	−0.389234	−0.194617	−	0.980879i	$-0.562346\pi$
$131$	−76452.0	−0.389234	−0.194617	+	0.980879i	$0.562346\pi$
$132$	0	0
$133$	203520.i	0.997650i
$134$	121944.	0.586676
$135$	0	0
$136$	201360.	0.933525
$137$	− 144918.i	− 0.659661i	−0.944040	−	0.329831i	$-0.893008\pi$
$137$	− 144918.i	− 0.659661i	0.944040	−	0.329831i	$-0.106992\pi$
$138$	0	0
$139$	−112220.	−0.492644	−0.246322	−	0.969188i	$-0.579222\pi$
$139$	−112220.	−0.492644	−0.246322	+	0.969188i	$0.579222\pi$
$140$	0	0
$141$	0	0
$142$	65296.0i	0.271748i
$143$	42328.0i	0.173096i
$144$	0	0
$145$	0	0
$146$	77548.0	0.301085
$147$	0	0
$148$	− 5096.00i	− 0.0191238i
$149$	403750.	1.48986	0.744932	−	0.667140i	$-0.232482\pi$
$149$	403750.	1.48986	0.744932	+	0.667140i	$0.232482\pi$
$150$	0	0
$151$	−446648.	−1.59413	−0.797064	−	0.603895i	$-0.793615\pi$
$151$	−446648.	−1.59413	−0.797064	+	0.603895i	$0.793615\pi$
$152$	− 127200.i	− 0.446558i
$153$	0	0
$154$	56832.0	0.193104
$155$	0	0
$156$	0	0
$157$	262258.i	0.849141i	0.905395	+	0.424570i	$0.139575\pi$
$157$	262258.i	0.849141i	−0.905395	+	0.424570i	$0.860425\pi$
$158$	66720.0i	0.212625i
$159$	0	0
$160$	0	0
$161$	−571392.	−1.73728
$162$	0	0
$163$	− 154564.i	− 0.455658i	−0.973701	−	0.227829i	$-0.926837\pi$
$163$	− 154564.i	− 0.455658i	0.973701	−	0.227829i	$-0.0731628\pi$
$164$	263144.	0.763983
$165$	0	0
$166$	33432.0	0.0941656
$167$	396672.i	1.10063i	0.834958	+	0.550314i	$0.185492\pi$
$167$	396672.i	1.10063i	−0.834958	+	0.550314i	$0.814508\pi$
$168$	0	0
$169$	289497.	0.779700
$170$	0	0
$171$	0	0
$172$	− 34832.0i	− 0.0897754i
$173$	573474.i	1.45680i	0.685155	+	0.728398i	$0.259735\pi$
$173$	573474.i	1.45680i	−0.685155	+	0.728398i	$0.740265\pi$
$174$	0	0
$175$	0	0
$176$	97088.0	0.236257
$177$	0	0
$178$	202740.i	0.479611i
$179$	−594460.	−1.38672	−0.693362	−	0.720589i	$-0.743871\pi$
$179$	−594460.	−1.38672	−0.693362	+	0.720589i	$0.743871\pi$
$180$	0	0
$181$	−107098.	−0.242988	−0.121494	−	0.992592i	$-0.538769\pi$
$181$	−107098.	−0.242988	−0.121494	+	0.992592i	$0.538769\pi$
$182$	109824.i	0.245764i
$183$	0	0
$184$	357120.	0.777624
$185$	0	0
$186$	0	0
$187$	− 248344.i	− 0.519337i
$188$	− 338464.i	− 0.698422i
$189$	0	0
$190$	0	0
$191$	−469552.	−0.931323	−0.465661	−	0.884963i	$-0.654184\pi$
$191$	−469552.	−0.931323	−0.465661	+	0.884963i	$0.654184\pi$
$192$	0	0
$193$	52706.0i	0.101851i	0.998702	+	0.0509257i	$0.0162172\pi$
$193$	52706.0i	0.101851i	−0.998702	+	0.0509257i	$0.983783\pi$
$194$	−238076.	−0.454163
$195$	0	0
$196$	−561596.	−1.04420
$197$	455862.i	0.836889i	0.908242	+	0.418444i	$0.137425\pi$
$197$	455862.i	0.836889i	−0.908242	+	0.418444i	$0.862575\pi$
$198$	0	0
$199$	−865000.	−1.54840	−0.774200	−	0.632940i	$-0.781848\pi$
$199$	−865000.	−1.54840	−0.774200	+	0.632940i	$0.781848\pi$
$200$	0	0
$201$	0	0
$202$	179796.i	0.310028i
$203$	654720.i	1.11510i
$204$	0	0
$205$	0	0
$206$	39008.0	0.0640451
$207$	0	0
$208$	187616.i	0.300685i
$209$	−156880.	−0.248429
$210$	0	0
$211$	1.10565e6	1.70967	0.854835	−	0.518900i	$-0.173658\pi$
$211$	1.10565e6	1.70967	0.854835	+	0.518900i	$0.173658\pi$
$212$	− 667688.i	− 1.02031i
$213$	0	0
$214$	−316584.	−0.472557
$215$	0	0
$216$	0	0
$217$	470016.i	0.677584i
$218$	− 73660.0i	− 0.104976i
$219$	0	0
$220$	0	0
$221$	479908.	0.660963
$222$	0	0
$223$	1.12158e6i	1.51031i	0.655545	+	0.755156i	$0.272439\pi$
$223$	1.12158e6i	1.51031i	−0.655545	+	0.755156i	$0.727561\pi$
$224$	989184.	1.31722
$225$	0	0
$226$	22372.0	0.0291363
$227$	− 23348.0i	− 0.0300736i	−0.999887	−	0.0150368i	$-0.995213\pi$
$227$	− 23348.0i	− 0.0300736i	0.999887	−	0.0150368i	$-0.00478654\pi$
$228$	0	0
$229$	596010.	0.751043	0.375522	−	0.926814i	$-0.377464\pi$
$229$	596010.	0.751043	0.375522	+	0.926814i	$0.377464\pi$
$230$	0	0
$231$	0	0
$232$	− 409200.i	− 0.499132i
$233$	485334.i	0.585667i	0.956163	+	0.292834i	$0.0945982\pi$
$233$	485334.i	0.585667i	−0.956163	+	0.292834i	$0.905402\pi$
$234$	0	0
$235$	0	0
$236$	−560560.	−0.655152
$237$	0	0
$238$	− 644352.i	− 0.737362i
$239$	−48880.0	−0.0553524	−0.0276762	−	0.999617i	$-0.508811\pi$
$239$	−48880.0	−0.0553524	−0.0276762	+	0.999617i	$0.508811\pi$
$240$	0	0
$241$	−110798.	−0.122882	−0.0614411	−	0.998111i	$-0.519570\pi$
$241$	−110798.	−0.122882	−0.0614411	+	0.998111i	$0.519570\pi$
$242$	− 278294.i	− 0.305468i
$243$	0	0
$244$	904456.	0.972552
$245$	0	0
$246$	0	0
$247$	− 303160.i	− 0.316176i
$248$	− 293760.i	− 0.303294i
$249$	0	0
$250$	0	0
$251$	1.64375e6	1.64684	0.823419	−	0.567434i	$-0.192064\pi$
$251$	1.64375e6	1.64684	0.823419	+	0.567434i	$0.192064\pi$
$252$	0	0
$253$	− 440448.i	− 0.432607i
$254$	141104.	0.137232
$255$	0	0
$256$	−30464.0	−0.0290527
$257$	1.30624e6i	1.23365i	0.787102	+	0.616823i	$0.211581\pi$
$257$	1.30624e6i	1.23365i	−0.787102	+	0.616823i	$0.788419\pi$
$258$	0	0
$259$	−34944.0	−0.0323685
$260$	0	0
$261$	0	0
$262$	− 152904.i	− 0.137615i
$263$	− 2.12834e6i	− 1.89736i	−0.316231	−	0.948682i	$-0.602417\pi$
$263$	− 2.12834e6i	− 1.89736i	0.316231	−	0.948682i	$-0.397583\pi$
$264$	0	0
$265$	0	0
$266$	−407040.	−0.352722
$267$	0	0
$268$	− 1.70722e6i	− 1.45195i
$269$	−1.44109e6	−1.21426	−0.607128	−	0.794604i	$-0.707679\pi$
$269$	−1.44109e6	−1.21426	−0.607128	+	0.794604i	$0.707679\pi$
$270$	0	0
$271$	−93248.0	−0.0771288	−0.0385644	−	0.999256i	$-0.512278\pi$
$271$	−93248.0	−0.0771288	−0.0385644	+	0.999256i	$0.512278\pi$
$272$	− 1.10077e6i	− 0.902139i
$273$	0	0
$274$	289836.	0.233225
$275$	0	0
$276$	0	0
$277$	110298.i	0.0863711i	0.999067	+	0.0431855i	$0.0137507\pi$
$277$	110298.i	0.0863711i	−0.999067	+	0.0431855i	$0.986249\pi$
$278$	− 224440.i	− 0.174176i
$279$	0	0
$280$	0	0
$281$	192198.	0.145205	0.0726027	−	0.997361i	$-0.476869\pi$
$281$	192198.	0.145205	0.0726027	+	0.997361i	$0.476869\pi$
$282$	0	0
$283$	− 331884.i	− 0.246332i	−0.992386	−	0.123166i	$-0.960695\pi$
$283$	− 331884.i	− 0.246332i	0.992386	−	0.123166i	$-0.0393047\pi$
$284$	914144.	0.672541
$285$	0	0
$286$	−84656.0	−0.0611988
$287$	− 1.80442e6i	− 1.29310i
$288$	0	0
$289$	−1.39583e6	−0.983076
$290$	0	0
$291$	0	0
$292$	− 1.08567e6i	− 0.745146i
$293$	− 2.19481e6i	− 1.49358i	−0.665063	−	0.746788i	$-0.731595\pi$
$293$	− 2.19481e6i	− 1.49358i	0.665063	−	0.746788i	$-0.268405\pi$
$294$	0	0
$295$	0	0
$296$	21840.0	0.0144885
$297$	0	0
$298$	807500.i	0.526747i
$299$	851136.	0.550581
$300$	0	0
$301$	−238848.	−0.151952
$302$	− 893296.i	− 0.563609i
$303$	0	0
$304$	−695360.	−0.431545
$305$	0	0
$306$	0	0
$307$	2.37751e6i	1.43971i	0.694123	+	0.719857i	$0.255793\pi$
$307$	2.37751e6i	1.43971i	−0.694123	+	0.719857i	$0.744207\pi$
$308$	− 795648.i	− 0.477908i
$309$	0	0
$310$	0	0
$311$	2.37305e6	1.39125	0.695626	−	0.718405i	$-0.255127\pi$
$311$	2.37305e6	1.39125	0.695626	+	0.718405i	$0.255127\pi$
$312$	0	0
$313$	− 1.42941e6i	− 0.824702i	−0.911025	−	0.412351i	$-0.864708\pi$
$313$	− 1.42941e6i	− 0.824702i	0.911025	−	0.412351i	$-0.135292\pi$
$314$	−524516.	−0.300217
$315$	0	0
$316$	934080.	0.526219
$317$	2.12462e6i	1.18750i	0.804650	+	0.593750i	$0.202353\pi$
$317$	2.12462e6i	1.18750i	−0.804650	+	0.593750i	$0.797647\pi$
$318$	0	0
$319$	−504680.	−0.277677
$320$	0	0
$321$	0	0
$322$	− 1.14278e6i	− 0.614221i
$323$	1.77868e6i	0.948618i
$324$	0	0
$325$	0	0
$326$	309128.	0.161100
$327$	0	0
$328$	1.12776e6i	0.578805i
$329$	−2.32090e6	−1.18213
$330$	0	0
$331$	3.09985e6	1.55515	0.777573	−	0.628793i	$-0.216451\pi$
$331$	3.09985e6	1.55515	0.777573	+	0.628793i	$0.216451\pi$
$332$	− 468048.i	− 0.233048i
$333$	0	0
$334$	−793344.	−0.389131
$335$	0	0
$336$	0	0
$337$	− 2.40008e6i	− 1.15120i	−0.817731	−	0.575601i	$-0.804768\pi$
$337$	− 2.40008e6i	− 1.15120i	0.817731	−	0.575601i	$-0.195232\pi$
$338$	578994.i	0.275665i
$339$	0	0
$340$	0	0
$341$	−362304.	−0.168728
$342$	0	0
$343$	624000.i	0.286384i
$344$	149280.	0.0680151
$345$	0	0
$346$	−1.14695e6	−0.515055
$347$	1.77741e6i	0.792436i	0.918156	+	0.396218i	$0.129678\pi$
$347$	1.77741e6i	0.792436i	−0.918156	+	0.396218i	$0.870322\pi$
$348$	0	0
$349$	2.14805e6	0.944019	0.472010	−	0.881593i	$-0.343529\pi$
$349$	2.14805e6	0.944019	0.472010	+	0.881593i	$0.343529\pi$
$350$	0	0
$351$	0	0
$352$	762496.i	0.328005i
$353$	661854.i	0.282700i	0.989960	+	0.141350i	$0.0451443\pi$
$353$	661854.i	0.282700i	−0.989960	+	0.141350i	$0.954856\pi$
$354$	0	0
$355$	0	0
$356$	2.83836e6	1.18698
$357$	0	0
$358$	− 1.18892e6i	− 0.490281i
$359$	−259320.	−0.106194	−0.0530970	−	0.998589i	$-0.516909\pi$
$359$	−259320.	−0.106194	−0.0530970	+	0.998589i	$0.516909\pi$
$360$	0	0
$361$	−1.35250e6	−0.546222
$362$	− 214196.i	− 0.0859093i
$363$	0	0
$364$	1.53754e6	0.608236
$365$	0	0
$366$	0	0
$367$	1.49993e6i	0.581307i	0.956828	+	0.290653i	$0.0938726\pi$
$367$	1.49993e6i	0.581307i	−0.956828	+	0.290653i	$0.906127\pi$
$368$	− 1.95226e6i	− 0.751480i
$369$	0	0
$370$	0	0
$371$	−4.57843e6	−1.72696
$372$	0	0
$373$	− 2.23807e6i	− 0.832918i	−0.909154	−	0.416459i	$-0.863271\pi$
$373$	− 2.23807e6i	− 0.832918i	0.909154	−	0.416459i	$-0.136729\pi$
$374$	496688.	0.183614
$375$	0	0
$376$	1.45056e6	0.529135
$377$	− 975260.i	− 0.353400i
$378$	0	0
$379$	−3.15934e6	−1.12979	−0.564896	−	0.825162i	$-0.691084\pi$
$379$	−3.15934e6	−1.12979	−0.564896	+	0.825162i	$0.691084\pi$
$380$	0	0
$381$	0	0
$382$	− 939104.i	− 0.329272i
$383$	− 342216.i	− 0.119207i	−0.998222	−	0.0596037i	$-0.981016\pi$
$383$	− 342216.i	− 0.119207i	0.998222	−	0.0596037i	$-0.0189837\pi$
$384$	0	0
$385$	0	0
$386$	−105412.	−0.0360099
$387$	0	0
$388$	3.33306e6i	1.12399i
$389$	88470.0	0.0296430	0.0148215	−	0.999890i	$-0.495282\pi$
$389$	88470.0	0.0296430	0.0148215	+	0.999890i	$0.495282\pi$
$390$	0	0
$391$	−4.99373e6	−1.65190
$392$	− 2.40684e6i	− 0.791101i
$393$	0	0
$394$	−911724.	−0.295885
$395$	0	0
$396$	0	0
$397$	5.45674e6i	1.73763i	0.495138	+	0.868814i	$0.335117\pi$
$397$	5.45674e6i	1.73763i	−0.495138	+	0.868814i	$0.664883\pi$
$398$	− 1.73000e6i	− 0.547442i
$399$	0	0
$400$	0	0
$401$	−4.04680e6	−1.25676	−0.628378	−	0.777908i	$-0.716281\pi$
$401$	−4.04680e6	−1.25676	−0.628378	+	0.777908i	$0.716281\pi$
$402$	0	0
$403$	− 700128.i	− 0.214741i
$404$	2.51714e6	0.767281
$405$	0	0
$406$	−1.30944e6	−0.394249
$407$	− 26936.0i	− 0.00806022i
$408$	0	0
$409$	2.71207e6	0.801664	0.400832	−	0.916151i	$-0.368721\pi$
$409$	2.71207e6	0.801664	0.400832	+	0.916151i	$0.368721\pi$
$410$	0	0
$411$	0	0
$412$	− 546112.i	− 0.158503i
$413$	3.84384e6i	1.10889i
$414$	0	0
$415$	0	0
$416$	−1.47347e6	−0.417454
$417$	0	0
$418$	− 313760.i	− 0.0878328i
$419$	3.71746e6	1.03445	0.517227	−	0.855848i	$-0.326964\pi$
$419$	3.71746e6	1.03445	0.517227	+	0.855848i	$0.326964\pi$
$420$	0	0
$421$	3.55250e6	0.976853	0.488426	−	0.872605i	$-0.337571\pi$
$421$	3.55250e6	0.976853	0.488426	+	0.872605i	$0.337571\pi$
$422$	2.21130e6i	0.604460i
$423$	0	0
$424$	2.86152e6	0.773005
$425$	0	0
$426$	0	0
$427$	− 6.20198e6i	− 1.64612i
$428$	4.43218e6i	1.16952i
$429$	0	0
$430$	0	0
$431$	4.06205e6	1.05330	0.526650	−	0.850082i	$-0.323448\pi$
$431$	4.06205e6	1.05330	0.526650	+	0.850082i	$0.323448\pi$
$432$	0	0
$433$	7.26287e6i	1.86161i	0.365518	+	0.930804i	$0.380892\pi$
$433$	7.26287e6i	1.86161i	−0.365518	+	0.930804i	$0.619108\pi$
$434$	−940032.	−0.239562
$435$	0	0
$436$	−1.03124e6	−0.259803
$437$	3.15456e6i	0.790197i
$438$	0	0
$439$	5.41028e6	1.33986	0.669928	−	0.742426i	$-0.266325\pi$
$439$	5.41028e6	1.33986	0.669928	+	0.742426i	$0.266325\pi$
$440$	0	0
$441$	0	0
$442$	959816.i	0.233686i
$443$	6.51524e6i	1.57733i	0.614826	+	0.788663i	$0.289226\pi$
$443$	6.51524e6i	1.57733i	−0.614826	+	0.788663i	$0.710774\pi$
$444$	0	0
$445$	0	0
$446$	−2.24315e6	−0.533976
$447$	0	0
$448$	− 2.05210e6i	− 0.483062i
$449$	−509950.	−0.119375	−0.0596873	−	0.998217i	$-0.519010\pi$
$449$	−509950.	−0.119375	−0.0596873	+	0.998217i	$0.519010\pi$
$450$	0	0
$451$	1.39090e6	0.322000
$452$	− 313208.i	− 0.0721085i
$453$	0	0
$454$	46696.0	0.0106326
$455$	0	0
$456$	0	0
$457$	− 1.22084e6i	− 0.273444i	−0.990609	−	0.136722i	$-0.956343\pi$
$457$	− 1.22084e6i	− 0.273444i	0.990609	−	0.136722i	$-0.0436568\pi$
$458$	1.19202e6i	0.265534i
$459$	0	0
$460$	0	0
$461$	4.07210e6	0.892413	0.446207	−	0.894930i	$-0.352775\pi$
$461$	4.07210e6	0.892413	0.446207	+	0.894930i	$0.352775\pi$
$462$	0	0
$463$	2.02294e6i	0.438561i	0.975662	+	0.219280i	$0.0703709\pi$
$463$	2.02294e6i	0.438561i	−0.975662	+	0.219280i	$0.929629\pi$
$464$	−2.23696e6	−0.482351
$465$	0	0
$466$	−970668.	−0.207065
$467$	3.25097e6i	0.689797i	0.938640	+	0.344898i	$0.112087\pi$
$467$	3.25097e6i	0.689797i	−0.938640	+	0.344898i	$0.887913\pi$
$468$	0	0
$469$	−1.17066e7	−2.45753
$470$	0	0
$471$	0	0
$472$	− 2.40240e6i	− 0.496353i
$473$	− 184112.i	− 0.0378381i
$474$	0	0
$475$	0	0
$476$	−9.02093e6	−1.82488
$477$	0	0
$478$	− 97760.0i	− 0.0195700i
$479$	−3.27936e6	−0.653056	−0.326528	−	0.945188i	$-0.605879\pi$
$479$	−3.27936e6	−0.653056	−0.326528	+	0.945188i	$0.605879\pi$
$480$	0	0
$481$	52052.0	0.0102583
$482$	− 221596.i	− 0.0434455i
$483$	0	0
$484$	−3.89612e6	−0.755994
$485$	0	0
$486$	0	0
$487$	8.53197e6i	1.63015i	0.579357	+	0.815074i	$0.303304\pi$
$487$	8.53197e6i	1.63015i	−0.579357	+	0.815074i	$0.696696\pi$
$488$	3.87624e6i	0.736819i
$489$	0	0
$490$	0	0
$491$	−1.51265e6	−0.283162	−0.141581	−	0.989927i	$-0.545219\pi$
$491$	−1.51265e6	−0.283162	−0.141581	+	0.989927i	$0.545219\pi$
$492$	0	0
$493$	5.72198e6i	1.06030i
$494$	606320.	0.111785
$495$	0	0
$496$	−1.60589e6	−0.293097
$497$	− 6.26842e6i	− 1.13833i
$498$	0	0
$499$	6.49190e6	1.16713	0.583567	−	0.812065i	$-0.301657\pi$
$499$	6.49190e6	1.16713	0.583567	+	0.812065i	$0.301657\pi$
$500$	0	0
$501$	0	0
$502$	3.28750e6i	0.582245i
$503$	− 8.61770e6i	− 1.51870i	−0.650684	−	0.759349i	$-0.725518\pi$
$503$	− 8.61770e6i	− 1.51870i	0.650684	−	0.759349i	$-0.274482\pi$
$504$	0	0
$505$	0	0
$506$	880896.	0.152950
$507$	0	0
$508$	− 1.97546e6i	− 0.339632i
$509$	2.67323e6	0.457343	0.228671	−	0.973504i	$-0.426562\pi$
$509$	2.67323e6	0.457343	0.228671	+	0.973504i	$0.426562\pi$
$510$	0	0
$511$	−7.44461e6	−1.26122
$512$	5.89875e6i	0.994455i
$513$	0	0
$514$	−2.61248e6	−0.436160
$515$	0	0
$516$	0	0
$517$	− 1.78902e6i	− 0.294367i
$518$	− 69888.0i	− 0.0114440i
$519$	0	0
$520$	0	0
$521$	−6.18500e6	−0.998264	−0.499132	−	0.866526i	$-0.666348\pi$
$521$	−6.18500e6	−0.998264	−0.499132	+	0.866526i	$0.666348\pi$
$522$	0	0
$523$	− 6.89452e6i	− 1.10217i	−0.834448	−	0.551087i	$-0.814213\pi$
$523$	− 6.89452e6i	− 1.10217i	0.834448	−	0.551087i	$-0.185787\pi$
$524$	−2.14066e6	−0.340580
$525$	0	0
$526$	4.25667e6	0.670820
$527$	4.10774e6i	0.644283i
$528$	0	0
$529$	−2.42023e6	−0.376026
$530$	0	0
$531$	0	0
$532$	5.69856e6i	0.872943i
$533$	2.68783e6i	0.409811i
$534$	0	0
$535$	0	0
$536$	7.31664e6	1.10002
$537$	0	0
$538$	− 2.88218e6i	− 0.429304i
$539$	−2.96844e6	−0.440104
$540$	0	0
$541$	155502.	0.0228425	0.0114212	−	0.999935i	$-0.496364\pi$
$541$	155502.	0.0228425	0.0114212	+	0.999935i	$0.496364\pi$
$542$	− 186496.i	− 0.0272691i
$543$	0	0
$544$	8.64506e6	1.25248
$545$	0	0
$546$	0	0
$547$	− 1.26544e7i	− 1.80831i	−0.427201	−	0.904157i	$-0.640500\pi$
$547$	− 1.26544e7i	− 1.80831i	0.427201	−	0.904157i	$-0.359500\pi$
$548$	− 4.05770e6i	− 0.577204i
$549$	0	0
$550$	0	0
$551$	3.61460e6	0.507202
$552$	0	0
$553$	− 6.40512e6i	− 0.890665i
$554$	−220596.	−0.0305368
$555$	0	0
$556$	−3.14216e6	−0.431064
$557$	− 7.07786e6i	− 0.966638i	−0.875444	−	0.483319i	$-0.839431\pi$
$557$	− 7.07786e6i	− 0.966638i	0.875444	−	0.483319i	$-0.160569\pi$
$558$	0	0
$559$	355784.	0.0481567
$560$	0	0
$561$	0	0
$562$	384396.i	0.0513379i
$563$	− 846636.i	− 0.112571i	−0.998415	−	0.0562854i	$-0.982074\pi$
$563$	− 846636.i	− 0.112571i	0.998415	−	0.0562854i	$-0.0179257\pi$
$564$	0	0
$565$	0	0
$566$	663768.	0.0870914
$567$	0	0
$568$	3.91776e6i	0.509527i
$569$	4.96041e6	0.642299	0.321149	−	0.947029i	$-0.395931\pi$
$569$	4.96041e6	0.642299	0.321149	+	0.947029i	$0.395931\pi$
$570$	0	0
$571$	8.96505e6	1.15070	0.575351	−	0.817907i	$-0.304866\pi$
$571$	8.96505e6	1.15070	0.575351	+	0.817907i	$0.304866\pi$
$572$	1.18518e6i	0.151459i
$573$	0	0
$574$	3.60883e6	0.457180
$575$	0	0
$576$	0	0
$577$	2.86080e6i	0.357724i	0.983874	+	0.178862i	$0.0572415\pi$
$577$	2.86080e6i	0.357724i	−0.983874	+	0.178862i	$0.942758\pi$
$578$	− 2.79165e6i	− 0.347570i
$579$	0	0
$580$	0	0
$581$	−3.20947e6	−0.394451
$582$	0	0
$583$	− 3.52921e6i	− 0.430037i
$584$	4.65288e6	0.564534
$585$	0	0
$586$	4.38961e6	0.528059
$587$	− 6.74027e6i	− 0.807387i	−0.914894	−	0.403694i	$-0.867726\pi$
$587$	− 6.74027e6i	− 0.807387i	0.914894	−	0.403694i	$-0.132274\pi$
$588$	0	0
$589$	2.59488e6	0.308197
$590$	0	0
$591$	0	0
$592$	− 119392.i	− 0.0140014i
$593$	1.78609e6i	0.208578i	0.994547	+	0.104289i	$0.0332566\pi$
$593$	1.78609e6i	0.208578i	−0.994547	+	0.104289i	$0.966743\pi$
$594$	0	0
$595$	0	0
$596$	1.13050e7	1.30363
$597$	0	0
$598$	1.70227e6i	0.194660i
$599$	4.94620e6	0.563254	0.281627	−	0.959524i	$-0.409126\pi$
$599$	4.94620e6	0.563254	0.281627	+	0.959524i	$0.409126\pi$
$600$	0	0
$601$	−4.58100e6	−0.517337	−0.258669	−	0.965966i	$-0.583284\pi$
$601$	−4.58100e6	−0.517337	−0.258669	+	0.965966i	$0.583284\pi$
$602$	− 477696.i	− 0.0537230i
$603$	0	0
$604$	−1.25061e7	−1.39486
$605$	0	0
$606$	0	0
$607$	− 7.07999e6i	− 0.779940i	−0.920828	−	0.389970i	$-0.872485\pi$
$607$	− 7.07999e6i	− 0.779940i	0.920828	−	0.389970i	$-0.127515\pi$
$608$	− 5.46112e6i	− 0.599132i
$609$	0	0
$610$	0	0
$611$	3.45717e6	0.374643
$612$	0	0
$613$	5.09609e6i	0.547754i	0.961765	+	0.273877i	$0.0883061\pi$
$613$	5.09609e6i	0.547754i	−0.961765	+	0.273877i	$0.911694\pi$
$614$	−4.75502e6	−0.509016
$615$	0	0
$616$	3.40992e6	0.362070
$617$	− 1.30003e7i	− 1.37480i	−0.726279	−	0.687400i	$-0.758752\pi$
$617$	− 1.30003e7i	− 1.37480i	0.726279	−	0.687400i	$-0.241248\pi$
$618$	0	0
$619$	−4.84406e6	−0.508139	−0.254070	−	0.967186i	$-0.581769\pi$
$619$	−4.84406e6	−0.508139	−0.254070	+	0.967186i	$0.581769\pi$
$620$	0	0
$621$	0	0
$622$	4.74610e6i	0.491882i
$623$	− 1.94630e7i	− 2.00905i
$624$	0	0
$625$	0	0
$626$	2.85883e6	0.291576
$627$	0	0
$628$	7.34322e6i	0.742998i
$629$	−305396.	−0.0307777
$630$	0	0
$631$	6.22775e6	0.622670	0.311335	−	0.950300i	$-0.399224\pi$
$631$	6.22775e6	0.622670	0.311335	+	0.950300i	$0.399224\pi$
$632$	4.00320e6i	0.398671i
$633$	0	0
$634$	−4.24924e6	−0.419845
$635$	0	0
$636$	0	0
$637$	− 5.73630e6i	− 0.560123i
$638$	− 1.00936e6i	− 0.0981735i
$639$	0	0
$640$	0	0
$641$	−1.53280e6	−0.147347	−0.0736734	−	0.997282i	$-0.523472\pi$
$641$	−1.53280e6	−0.147347	−0.0736734	+	0.997282i	$0.523472\pi$
$642$	0	0
$643$	− 1.74382e7i	− 1.66332i	−0.555287	−	0.831659i	$-0.687391\pi$
$643$	− 1.74382e7i	− 1.66332i	0.555287	−	0.831659i	$-0.312609\pi$
$644$	−1.59990e7	−1.52012
$645$	0	0
$646$	−3.55736e6	−0.335387
$647$	− 4.25469e6i	− 0.399583i	−0.979838	−	0.199792i	$-0.935974\pi$
$647$	− 4.25469e6i	− 0.399583i	0.979838	−	0.199792i	$-0.0640265\pi$
$648$	0	0
$649$	−2.96296e6	−0.276130
$650$	0	0
$651$	0	0
$652$	− 4.32779e6i	− 0.398701i
$653$	− 3.01085e6i	− 0.276316i	−0.990410	−	0.138158i	$-0.955882\pi$
$653$	− 3.01085e6i	− 0.276316i	0.990410	−	0.138158i	$-0.0441181\pi$
$654$	0	0
$655$	0	0
$656$	6.16509e6	0.559345
$657$	0	0
$658$	− 4.64179e6i	− 0.417947i
$659$	−8.11462e6	−0.727871	−0.363936	−	0.931424i	$-0.618567\pi$
$659$	−8.11462e6	−0.727871	−0.363936	+	0.931424i	$0.618567\pi$
$660$	0	0
$661$	2.47370e6	0.220213	0.110107	−	0.993920i	$-0.464881\pi$
$661$	2.47370e6	0.220213	0.110107	+	0.993920i	$0.464881\pi$
$662$	6.19970e6i	0.549827i
$663$	0	0
$664$	2.00592e6	0.176560
$665$	0	0
$666$	0	0
$667$	1.01482e7i	0.883228i
$668$	1.11068e7i	0.963049i
$669$	0	0
$670$	0	0
$671$	4.78070e6	0.409907
$672$	0	0
$673$	5.77063e6i	0.491117i	0.969382	+	0.245559i	$0.0789714\pi$
$673$	5.77063e6i	0.491117i	−0.969382	+	0.245559i	$0.921029\pi$
$674$	4.80016e6	0.407011
$675$	0	0
$676$	8.10592e6	0.682237
$677$	1.67197e7i	1.40203i	0.713147	+	0.701014i	$0.247269\pi$
$677$	1.67197e7i	1.40203i	−0.713147	+	0.701014i	$0.752731\pi$
$678$	0	0
$679$	2.28553e7	1.90245
$680$	0	0
$681$	0	0
$682$	− 724608.i	− 0.0596544i
$683$	− 7.14532e6i	− 0.586097i	−0.956098	−	0.293049i	$-0.905330\pi$
$683$	− 7.14532e6i	− 0.586097i	0.956098	−	0.293049i	$-0.0946698\pi$
$684$	0	0
$685$	0	0
$686$	−1.24800e6	−0.101252
$687$	0	0
$688$	− 816064.i	− 0.0657284i
$689$	6.81996e6	0.547310
$690$	0	0
$691$	−8.78395e6	−0.699833	−0.349917	−	0.936781i	$-0.613790\pi$
$691$	−8.78395e6	−0.699833	−0.349917	+	0.936781i	$0.613790\pi$
$692$	1.60573e7i	1.27470i
$693$	0	0
$694$	−3.55482e6	−0.280169
$695$	0	0
$696$	0	0
$697$	− 1.57698e7i	− 1.22955i
$698$	4.29610e6i	0.333761i
$699$	0	0
$700$	0	0
$701$	1.60141e7	1.23086	0.615428	−	0.788193i	$-0.288983\pi$
$701$	1.60141e7	1.23086	0.615428	+	0.788193i	$0.288983\pi$
$702$	0	0
$703$	192920.i	0.0147228i
$704$	1.58182e6	0.120289
$705$	0	0
$706$	−1.32371e6	−0.0999495
$707$	− 1.72604e7i	− 1.29868i
$708$	0	0
$709$	1.91354e7	1.42962	0.714811	−	0.699318i	$-0.246513\pi$
$709$	1.91354e7	1.42962	0.714811	+	0.699318i	$0.246513\pi$
$710$	0	0
$711$	0	0
$712$	1.21644e7i	0.899271i
$713$	7.28525e6i	0.536686i
$714$	0	0
$715$	0	0
$716$	−1.66449e7	−1.21338
$717$	0	0
$718$	− 518640.i	− 0.0375452i
$719$	1.02934e7	0.742566	0.371283	−	0.928520i	$-0.378918\pi$
$719$	1.02934e7	0.742566	0.371283	+	0.928520i	$0.378918\pi$
$720$	0	0
$721$	−3.74477e6	−0.268279
$722$	− 2.70500e6i	− 0.193119i
$723$	0	0
$724$	−2.99874e6	−0.212615
$725$	0	0
$726$	0	0
$727$	1.93264e7i	1.35618i	0.734981	+	0.678088i	$0.237191\pi$
$727$	1.93264e7i	1.35618i	−0.734981	+	0.678088i	$0.762809\pi$
$728$	6.58944e6i	0.460808i
$729$	0	0
$730$	0	0
$731$	−2.08743e6	−0.144484
$732$	0	0
$733$	5.26197e6i	0.361733i	0.983508	+	0.180866i	$0.0578902\pi$
$733$	5.26197e6i	0.361733i	−0.983508	+	0.180866i	$0.942110\pi$
$734$	−2.99986e6	−0.205523
$735$	0	0
$736$	1.53324e7	1.04331
$737$	− 9.02386e6i	− 0.611961i
$738$	0	0
$739$	−2.82944e7	−1.90585	−0.952927	−	0.303199i	$-0.901945\pi$
$739$	−2.82944e7	−1.90585	−0.952927	+	0.303199i	$0.901945\pi$
$740$	0	0
$741$	0	0
$742$	− 9.15686e6i	− 0.610572i
$743$	− 2.09863e7i	− 1.39464i	−0.716759	−	0.697321i	$-0.754375\pi$
$743$	− 2.09863e7i	− 1.39464i	0.716759	−	0.697321i	$-0.245625\pi$
$744$	0	0
$745$	0	0
$746$	4.47615e6	0.294481
$747$	0	0
$748$	− 6.95363e6i	− 0.454420i
$749$	3.03921e7	1.97950
$750$	0	0
$751$	−1.89668e7	−1.22714	−0.613572	−	0.789639i	$-0.710268\pi$
$751$	−1.89668e7	−1.22714	−0.613572	+	0.789639i	$0.710268\pi$
$752$	− 7.92973e6i	− 0.511345i
$753$	0	0
$754$	1.95052e6	0.124946
$755$	0	0
$756$	0	0
$757$	1.08257e7i	0.686617i	0.939223	+	0.343309i	$0.111548\pi$
$757$	1.08257e7i	0.686617i	−0.939223	+	0.343309i	$0.888452\pi$
$758$	− 6.31868e6i	− 0.399442i
$759$	0	0
$760$	0	0
$761$	−1.90534e7	−1.19264	−0.596322	−	0.802745i	$-0.703372\pi$
$761$	−1.90534e7	−1.19264	−0.596322	+	0.802745i	$0.703372\pi$
$762$	0	0
$763$	7.07136e6i	0.439736i
$764$	−1.31475e7	−0.814908
$765$	0	0
$766$	684432.	0.0421462
$767$	− 5.72572e6i	− 0.351432i
$768$	0	0
$769$	1.57826e7	0.962415	0.481208	−	0.876607i	$-0.340198\pi$
$769$	1.57826e7	0.962415	0.481208	+	0.876607i	$0.340198\pi$
$770$	0	0
$771$	0	0
$772$	1.47577e6i	0.0891199i
$773$	2.44049e7i	1.46902i	0.678598	+	0.734510i	$0.262588\pi$
$773$	2.44049e7i	1.46902i	−0.678598	+	0.734510i	$0.737412\pi$
$774$	0	0
$775$	0	0
$776$	−1.42846e7	−0.851555
$777$	0	0
$778$	176940.i	0.0104804i
$779$	−9.96188e6	−0.588163
$780$	0	0
$781$	4.83190e6	0.283459
$782$	− 9.98746e6i	− 0.584034i
$783$	0	0
$784$	−1.31574e7	−0.764504
$785$	0	0
$786$	0	0
$787$	− 3.37607e7i	− 1.94301i	−0.237019	−	0.971505i	$-0.576170\pi$
$787$	− 3.37607e7i	− 1.94301i	0.237019	−	0.971505i	$-0.423830\pi$
$788$	1.27641e7i	0.732278i
$789$	0	0
$790$	0	0
$791$	−2.14771e6	−0.122049
$792$	0	0
$793$	9.23837e6i	0.521690i
$794$	−1.09135e7	−0.614344
$795$	0	0
$796$	−2.42200e7	−1.35485
$797$	2.19885e7i	1.22617i	0.790019	+	0.613083i	$0.210071\pi$
$797$	2.19885e7i	1.22617i	−0.790019	+	0.613083i	$0.789929\pi$
$798$	0	0
$799$	−2.02837e7	−1.12403
$800$	0	0
$801$	0	0
$802$	− 8.09360e6i	− 0.444330i
$803$	− 5.73855e6i	− 0.314061i
$804$	0	0
$805$	0	0
$806$	1.40026e6	0.0759224
$807$	0	0
$808$	1.07878e7i	0.581303i
$809$	−2.93597e7	−1.57717	−0.788587	−	0.614923i	$-0.789187\pi$
$809$	−2.93597e7	−1.57717	−0.788587	+	0.614923i	$0.789187\pi$
$810$	0	0
$811$	3.17703e7	1.69617	0.848083	−	0.529863i	$-0.177757\pi$
$811$	3.17703e7	1.69617	0.848083	+	0.529863i	$0.177757\pi$
$812$	1.83322e7i	0.975716i
$813$	0	0
$814$	53872.0	0.00284972
$815$	0	0
$816$	0	0
$817$	1.31864e6i	0.0691148i
$818$	5.42414e6i	0.283431i
$819$	0	0
$820$	0	0
$821$	2.71430e6	0.140540	0.0702699	−	0.997528i	$-0.477614\pi$
$821$	2.71430e6	0.140540	0.0702699	+	0.997528i	$0.477614\pi$
$822$	0	0
$823$	− 1.25866e7i	− 0.647753i	−0.946099	−	0.323877i	$-0.895014\pi$
$823$	− 1.25866e7i	− 0.647753i	0.946099	−	0.323877i	$-0.104986\pi$
$824$	2.34048e6	0.120084
$825$	0	0
$826$	−7.68768e6	−0.392053
$827$	− 8.72355e6i	− 0.443537i	−0.975099	−	0.221768i	$-0.928817\pi$
$827$	− 8.72355e6i	− 0.443537i	0.975099	−	0.221768i	$-0.0711828\pi$
$828$	0	0
$829$	1.06178e7	0.536597	0.268299	−	0.963336i	$-0.413539\pi$
$829$	1.06178e7	0.536597	0.268299	+	0.963336i	$0.413539\pi$
$830$	0	0
$831$	0	0
$832$	3.05677e6i	0.153093i
$833$	3.36556e7i	1.68053i
$834$	0	0
$835$	0	0
$836$	−4.39264e6	−0.217375
$837$	0	0
$838$	7.43492e6i	0.365735i
$839$	1.67765e7	0.822805	0.411403	−	0.911454i	$-0.365039\pi$
$839$	1.67765e7	0.822805	0.411403	+	0.911454i	$0.365039\pi$
$840$	0	0
$841$	−8.88305e6	−0.433084
$842$	7.10500e6i	0.345370i
$843$	0	0
$844$	3.09583e7	1.49596
$845$	0	0
$846$	0	0
$847$	2.67162e7i	1.27958i
$848$	− 1.56430e7i	− 0.747016i
$849$	0	0
$850$	0	0
$851$	−541632.	−0.0256378
$852$	0	0
$853$	− 2.20186e7i	− 1.03613i	−0.855340	−	0.518067i	$-0.826652\pi$
$853$	− 2.20186e7i	− 1.03613i	0.855340	−	0.518067i	$-0.173348\pi$
$854$	1.24040e7	0.581991
$855$	0	0
$856$	−1.89950e7	−0.886045
$857$	3.16676e7i	1.47287i	0.676510	+	0.736434i	$0.263492\pi$
$857$	3.16676e7i	1.47287i	−0.676510	+	0.736434i	$0.736508\pi$
$858$	0	0
$859$	−1.58064e7	−0.730886	−0.365443	−	0.930834i	$-0.619082\pi$
$859$	−1.58064e7	−0.730886	−0.365443	+	0.930834i	$0.619082\pi$
$860$	0	0
$861$	0	0
$862$	8.12410e6i	0.372398i
$863$	1.44287e7i	0.659476i	0.944072	+	0.329738i	$0.106960\pi$
$863$	1.44287e7i	0.659476i	−0.944072	+	0.329738i	$0.893040\pi$
$864$	0	0
$865$	0	0
$866$	−1.45257e7	−0.658178
$867$	0	0
$868$	1.31604e7i	0.592886i
$869$	4.93728e6	0.221788
$870$	0	0
$871$	1.74380e7	0.778845
$872$	− 4.41960e6i	− 0.196830i
$873$	0	0
$874$	−6.30912e6	−0.279377
$875$	0	0
$876$	0	0
$877$	− 247902.i	− 0.0108838i	−0.999985	−	0.00544191i	$-0.998268\pi$
$877$	− 247902.i	− 0.0108838i	0.999985	−	0.00544191i	$-0.00173222\pi$
$878$	1.08206e7i	0.473711i
$879$	0	0
$880$	0	0
$881$	−4.10268e7	−1.78085	−0.890426	−	0.455128i	$-0.849594\pi$
$881$	−4.10268e7	−1.78085	−0.890426	+	0.455128i	$0.849594\pi$
$882$	0	0
$883$	4.18015e7i	1.80422i	0.431503	+	0.902112i	$0.357984\pi$
$883$	4.18015e7i	1.80422i	−0.431503	+	0.902112i	$0.642016\pi$
$884$	1.34374e7	0.578343
$885$	0	0
$886$	−1.30305e7	−0.557669
$887$	− 2.10476e7i	− 0.898241i	−0.893471	−	0.449120i	$-0.851737\pi$
$887$	− 2.10476e7i	− 0.898241i	0.893471	−	0.449120i	$-0.148263\pi$
$888$	0	0
$889$	−1.35460e7	−0.574852
$890$	0	0
$891$	0	0
$892$	3.14041e7i	1.32152i
$893$	1.28133e7i	0.537690i
$894$	0	0
$895$	0	0
$896$	3.57581e7	1.48800
$897$	0	0
$898$	− 1.01990e6i	− 0.0422053i
$899$	8.34768e6	0.344482
$900$	0	0
$901$	−4.00136e7	−1.64208
$902$	2.78181e6i	0.113844i
$903$	0	0
$904$	1.34232e6	0.0546305
$905$	0	0
$906$	0	0
$907$	− 7.48309e6i	− 0.302039i	−0.988531	−	0.151019i	$-0.951744\pi$
$907$	− 7.48309e6i	− 0.302039i	0.988531	−	0.151019i	$-0.0482556\pi$
$908$	− 653744.i	− 0.0263144i
$909$	0	0
$910$	0	0
$911$	6.63165e6	0.264744	0.132372	−	0.991200i	$-0.457741\pi$
$911$	6.63165e6	0.264744	0.132372	+	0.991200i	$0.457741\pi$
$912$	0	0
$913$	− 2.47397e6i	− 0.0982239i
$914$	2.44168e6	0.0966772
$915$	0	0
$916$	1.66883e7	0.657163
$917$	1.46788e7i	0.576457i
$918$	0	0
$919$	1.68976e7	0.659990	0.329995	−	0.943983i	$-0.392953\pi$
$919$	1.68976e7	0.659990	0.329995	+	0.943983i	$0.392953\pi$
$920$	0	0
$921$	0	0
$922$	8.14420e6i	0.315516i
$923$	9.33733e6i	0.360760i
$924$	0	0
$925$	0	0
$926$	−4.04587e6	−0.155055
$927$	0	0
$928$	− 1.75683e7i	− 0.669669i
$929$	−1.28653e7	−0.489081	−0.244541	−	0.969639i	$-0.578637\pi$
$929$	−1.28653e7	−0.489081	−0.244541	+	0.969639i	$0.578637\pi$
$930$	0	0
$931$	2.12604e7	0.803892
$932$	1.35894e7i	0.512459i
$933$	0	0
$934$	−6.50194e6	−0.243880
$935$	0	0
$936$	0	0
$937$	− 1.06887e7i	− 0.397718i	−0.980028	−	0.198859i	$-0.936276\pi$
$937$	− 1.06887e7i	− 0.397718i	0.980028	−	0.198859i	$-0.0637236\pi$
$938$	− 2.34132e7i	− 0.868870i
$939$	0	0
$940$	0	0
$941$	−2.82455e7	−1.03986	−0.519930	−	0.854209i	$-0.674042\pi$
$941$	−2.82455e7	−1.03986	−0.519930	+	0.854209i	$0.674042\pi$
$942$	0	0
$943$	− 2.79684e7i	− 1.02421i
$944$	−1.31331e7	−0.479665
$945$	0	0
$946$	368224.	0.0133778
$947$	− 1.70892e7i	− 0.619222i	−0.950863	−	0.309611i	$-0.899801\pi$
$947$	− 1.70892e7i	− 0.619222i	0.950863	−	0.309611i	$-0.100199\pi$
$948$	0	0
$949$	1.10894e7	0.399706
$950$	0	0
$951$	0	0
$952$	− 3.86611e7i	− 1.38255i
$953$	− 2.22259e7i	− 0.792735i	−0.918092	−	0.396367i	$-0.870271\pi$
$953$	− 2.22259e7i	− 0.792735i	0.918092	−	0.396367i	$-0.129729\pi$
$954$	0	0
$955$	0	0
$956$	−1.36864e6	−0.0484333
$957$	0	0
$958$	− 6.55872e6i	− 0.230890i
$959$	−2.78243e7	−0.976961
$960$	0	0
$961$	−2.26364e7	−0.790678
$962$	104104.i	0.00362685i
$963$	0	0
$964$	−3.10234e6	−0.107522
$965$	0	0
$966$	0	0
$967$	− 2.41551e7i	− 0.830696i	−0.909663	−	0.415348i	$-0.863660\pi$
$967$	− 2.41551e7i	− 0.830696i	0.909663	−	0.415348i	$-0.136340\pi$
$968$	− 1.66976e7i	− 0.572752i
$969$	0	0
$970$	0	0
$971$	5.48313e7	1.86630	0.933149	−	0.359491i	$-0.117050\pi$
$971$	5.48313e7	1.86630	0.933149	+	0.359491i	$0.117050\pi$
$972$	0	0
$973$	2.15462e7i	0.729608i
$974$	−1.70639e7	−0.576344
$975$	0	0
$976$	2.11901e7	0.712047
$977$	− 1.56612e7i	− 0.524915i	−0.964944	−	0.262457i	$-0.915467\pi$
$977$	− 1.56612e7i	− 0.524915i	0.964944	−	0.262457i	$-0.0845329\pi$
$978$	0	0
$979$	1.50028e7	0.500281
$980$	0	0
$981$	0	0
$982$	− 3.02530e6i	− 0.100113i
$983$	1.63420e7i	0.539412i	0.962943	+	0.269706i	$0.0869266\pi$
$983$	1.63420e7i	0.539412i	−0.962943	+	0.269706i	$0.913073\pi$
$984$	0	0
$985$	0	0
$986$	−1.14440e7	−0.374873
$987$	0	0
$988$	− 8.48848e6i	− 0.276654i
$989$	−3.70214e6	−0.120355
$990$	0	0
$991$	1.37576e7	0.444997	0.222498	−	0.974933i	$-0.428579\pi$
$991$	1.37576e7	0.444997	0.222498	+	0.974933i	$0.428579\pi$
$992$	− 1.26121e7i	− 0.406919i
$993$	0	0
$994$	1.25368e7	0.402459
$995$	0	0
$996$	0	0
$997$	1.29097e7i	0.411320i	0.978624	+	0.205660i	$0.0659341\pi$
$997$	1.29097e7i	0.411320i	−0.978624	+	0.205660i	$0.934066\pi$
$998$	1.29838e7i	0.412644i
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	225.6.b.e.199.2		2
3.2	odd	2		25.6.b.a.24.1		2
5.2	odd	4		45.6.a.b.1.1		1
5.3	odd	4		225.6.a.f.1.1		1
5.4	even	2	inner	225.6.b.e.199.1		2
12.11	even	2		400.6.c.j.49.2		2
15.2	even	4		5.6.a.a.1.1	✓	1
15.8	even	4		25.6.a.a.1.1		1
15.14	odd	2		25.6.b.a.24.2		2
20.7	even	4		720.6.a.a.1.1		1
60.23	odd	4		400.6.a.g.1.1		1
60.47	odd	4		80.6.a.e.1.1		1
60.59	even	2		400.6.c.j.49.1		2
105.62	odd	4		245.6.a.b.1.1		1
120.77	even	4		320.6.a.j.1.1		1
120.107	odd	4		320.6.a.g.1.1		1
165.32	odd	4		605.6.a.a.1.1		1
195.77	even	4		845.6.a.b.1.1		1

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
5.6.a.a.1.1	✓	1	15.2	even	4
25.6.a.a.1.1		1	15.8	even	4
25.6.b.a.24.1		2	3.2	odd	2
25.6.b.a.24.2		2	15.14	odd	2
45.6.a.b.1.1		1	5.2	odd	4
80.6.a.e.1.1		1	60.47	odd	4
225.6.a.f.1.1		1	5.3	odd	4
225.6.b.e.199.1		2	5.4	even	2	inner
225.6.b.e.199.2		2	1.1	even	1	trivial
245.6.a.b.1.1		1	105.62	odd	4
320.6.a.g.1.1		1	120.107	odd	4
320.6.a.j.1.1		1	120.77	even	4
400.6.a.g.1.1		1	60.23	odd	4
400.6.c.j.49.1		2	60.59	even	2
400.6.c.j.49.2		2	12.11	even	2
605.6.a.a.1.1		1	165.32	odd	4
720.6.a.a.1.1		1	20.7	even	4
845.6.a.b.1.1		1	195.77	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:	\( N \)	\(=\)	\( 225 = 3^{2} \cdot 5^{2} \)
Weight:	\( k \)	\(=\)	\( 6 \)
Character orbit:	\([\chi]\)	\(=\)	225.b (of order \(2\), degree \(1\), not minimal)

\(f(q)\)	\(=\)	\(q+2.00000i q^{2} +28.0000 q^{4} -192.000i q^{7} +120.000i q^{8} +O(q^{10})\)	\(q+2.00000i q^{2} +28.0000 q^{4} -192.000i q^{7} +120.000i q^{8} +148.000 q^{11} +286.000i q^{13} +384.000 q^{14} +656.000 q^{16} -1678.00i q^{17} -1060.00 q^{19} +296.000i q^{22} -2976.00i q^{23} -572.000 q^{26} -5376.00i q^{28} -3410.00 q^{29} -2448.00 q^{31} +5152.00i q^{32} +3356.00 q^{34} -182.000i q^{37} -2120.00i q^{38} +9398.00 q^{41} -1244.00i q^{43} +4144.00 q^{44} +5952.00 q^{46} -12088.0i q^{47} -20057.0 q^{49} +8008.00i q^{52} -23846.0i q^{53} +23040.0 q^{56} -6820.00i q^{58} -20020.0 q^{59} +32302.0 q^{61} -4896.00i q^{62} +10688.0 q^{64} -60972.0i q^{67} -46984.0i q^{68} +32648.0 q^{71} -38774.0i q^{73} +364.000 q^{74} -29680.0 q^{76} -28416.0i q^{77} +33360.0 q^{79} +18796.0i q^{82} -16716.0i q^{83} +2488.00 q^{86} +17760.0i q^{88} +101370. q^{89} +54912.0 q^{91} -83328.0i q^{92} +24176.0 q^{94} +119038. i q^{97} -40114.0i q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 2 q + 56 q^{4}+O(q^{10}) \) 2 * q + 56 * q^4	\( 2 q + 56 q^{4} + 296 q^{11} + 768 q^{14} + 1312 q^{16} - 2120 q^{19} - 1144 q^{26} - 6820 q^{29} - 4896 q^{31} + 6712 q^{34} + 18796 q^{41} + 8288 q^{44} + 11904 q^{46} - 40114 q^{49} + 46080 q^{56} - 40040 q^{59} + 64604 q^{61} + 21376 q^{64} + 65296 q^{71} + 728 q^{74} - 59360 q^{76} + 66720 q^{79} + 4976 q^{86} + 202740 q^{89} + 109824 q^{91} + 48352 q^{94}+O(q^{100}) \) 2 * q + 56 * q^4 + 296 * q^11 + 768 * q^14 + 1312 * q^16 - 2120 * q^19 - 1144 * q^26 - 6820 * q^29 - 4896 * q^31 + 6712 * q^34 + 18796 * q^41 + 8288 * q^44 + 11904 * q^46 - 40114 * q^49 + 46080 * q^56 - 40040 * q^59 + 64604 * q^61 + 21376 * q^64 + 65296 * q^71 + 728 * q^74 - 59360 * q^76 + 66720 * q^79 + 4976 * q^86 + 202740 * q^89 + 109824 * q^91 + 48352 * q^94

Introduction

L-functions

Modular forms

Varieties

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Database

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