LMFDB - Embedded newform 441.2.c.b.440.8

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [441,2,Mod(440,441)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(441, base_ring=CyclotomicField(2))

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

N = Newforms(chi, 2, names="a")

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

chi := DirichletCharacter("441.440");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 441 = 3^{2} \cdot 7^{2} $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	441.c (of order $2$, degree $1$, 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:	$3.52140272914$
Analytic rank:	$0$
Dimension:	$8$
Coefficient field:	$\Q(\zeta_{16})$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{8} + 1 $ x^8 + 1
Coefficient ring:	$\Z[a_1, \ldots, a_{5}]$
Coefficient ring index:	$ 2^{3}\cdot 7^{2} $
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label			440.8
Root			$0.382683 - 0.923880i$ of defining polynomial
Character	$\chi$	$=$	441.440
Dual form			441.2.c.b.440.2

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+2.41421i q^{2} -3.82843 q^{4} +3.37849 q^{5} -4.41421i q^{8} +O(q^{10})$	$q+2.41421i q^{2} -3.82843 q^{4} +3.37849 q^{5} -4.41421i q^{8} +8.15640i q^{10} +0.828427i q^{11} +3.37849i q^{13} +3.00000 q^{16} +1.39942 q^{17} +6.75699i q^{19} -12.9343 q^{20} -2.00000 q^{22} +2.00000i q^{23} +6.41421 q^{25} -8.15640 q^{26} -4.82843i q^{29} -6.75699i q^{31} -1.58579i q^{32} +3.37849i q^{34} -2.58579 q^{37} -16.3128 q^{38} -14.9134i q^{40} +8.15640 q^{41} -12.4853 q^{43} -3.17157i q^{44} -4.82843 q^{46} +6.75699 q^{47} +15.4853i q^{50} -12.9343i q^{52} -7.07107i q^{53} +2.79884i q^{55} +11.6569 q^{58} +6.75699 q^{59} -8.15640i q^{61} +16.3128 q^{62} +9.82843 q^{64} +11.4142i q^{65} +8.48528 q^{67} -5.35757 q^{68} +4.82843i q^{71} -1.39942i q^{73} -6.24264i q^{74} -25.8686i q^{76} -9.65685 q^{79} +10.1355 q^{80} +19.6913i q^{82} -13.5140 q^{83} +4.72792 q^{85} -30.1421i q^{86} +3.65685 q^{88} +6.17733 q^{89} -7.65685i q^{92} +16.3128i q^{94} +22.8284i q^{95} +1.39942i q^{97} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 8 q - 8 q^{4}+O(q^{10}) $ 8 * q - 8 * q^4	$ 8 q - 8 q^{4} + 24 q^{16} - 16 q^{22} + 40 q^{25} - 32 q^{37} - 32 q^{43} - 16 q^{46} + 48 q^{58} + 56 q^{64} - 32 q^{79} - 64 q^{85} - 16 q^{88}+O(q^{100}) $ 8 * q - 8 * q^4 + 24 * q^16 - 16 * q^22 + 40 * q^25 - 32 * q^37 - 32 * q^43 - 16 * q^46 + 48 * q^58 + 56 * q^64 - 32 * q^79 - 64 * q^85 - 16 * q^88

Display 100 coefficients 10 coefficients

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/441\mathbb{Z}\right)^\times$.

$n$	$199$	$344$
$\chi(n)$	$-1$	$-1$

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 $

$2$	2.41421i	1.70711i	0.521005	+	0.853553i	$0.325557\pi$
$2$	2.41421i	1.70711i	−0.521005	+	0.853553i	$0.674443\pi$
$3$	0	0
$3$	0	0
$4$	−3.82843	−1.91421
$5$	3.37849	1.51091	0.755454	−	0.655202i	$-0.227416\pi$
$5$	3.37849	1.51091	0.755454	+	0.655202i	$0.227416\pi$
$6$	0	0
$7$	0	0
$7$	0	0
$8$	− 4.41421i	− 1.56066i
$9$	0	0
$10$	8.15640i	2.57928i
$11$	0.828427i	0.249780i	0.992171	+	0.124890i	$0.0398578\pi$
$11$	0.828427i	0.249780i	−0.992171	+	0.124890i	$0.960142\pi$
$12$	0	0
$13$	3.37849i	0.937025i	0.883457	+	0.468513i	$0.155210\pi$
$13$	3.37849i	0.937025i	−0.883457	+	0.468513i	$0.844790\pi$
$14$	0	0
$15$	0	0
$16$	3.00000	0.750000
$17$	1.39942	0.339409	0.169704	−	0.985495i	$-0.445719\pi$
$17$	1.39942	0.339409	0.169704	+	0.985495i	$0.445719\pi$
$18$	0	0
$19$	6.75699i	1.55016i	0.631864	+	0.775079i	$0.282290\pi$
$19$	6.75699i	1.55016i	−0.631864	+	0.775079i	$0.717710\pi$
$20$	−12.9343	−2.89220
$21$	0	0
$22$	−2.00000	−0.426401
$23$	2.00000i	0.417029i	0.978019	+	0.208514i	$0.0668628\pi$
$23$	2.00000i	0.417029i	−0.978019	+	0.208514i	$0.933137\pi$
$24$	0	0
$25$	6.41421	1.28284
$26$	−8.15640	−1.59960
$27$	0	0
$28$	0	0
$29$	− 4.82843i	− 0.896616i	−0.893879	−	0.448308i	$-0.852027\pi$
$29$	− 4.82843i	− 0.896616i	0.893879	−	0.448308i	$-0.147973\pi$
$30$	0	0
$31$	− 6.75699i	− 1.21359i	−0.794858	−	0.606795i	$-0.792455\pi$
$31$	− 6.75699i	− 1.21359i	0.794858	−	0.606795i	$-0.207545\pi$
$32$	− 1.58579i	− 0.280330i
$33$	0	0
$34$	3.37849i	0.579407i
$35$	0	0
$36$	0	0
$37$	−2.58579	−0.425101	−0.212550	−	0.977150i	$-0.568177\pi$
$37$	−2.58579	−0.425101	−0.212550	+	0.977150i	$0.568177\pi$
$38$	−16.3128	−2.64629
$39$	0	0
$40$	− 14.9134i	− 2.35801i
$41$	8.15640	1.27382	0.636908	−	0.770940i	$-0.280213\pi$
$41$	8.15640	1.27382	0.636908	+	0.770940i	$0.280213\pi$
$42$	0	0
$43$	−12.4853	−1.90399	−0.951994	−	0.306117i	$-0.900970\pi$
$43$	−12.4853	−1.90399	−0.951994	+	0.306117i	$0.900970\pi$
$44$	− 3.17157i	− 0.478133i
$45$	0	0
$46$	−4.82843	−0.711913
$47$	6.75699	0.985608	0.492804	−	0.870140i	$-0.335972\pi$
$47$	6.75699	0.985608	0.492804	+	0.870140i	$0.335972\pi$
$48$	0	0
$49$	0	0
$50$	15.4853i	2.18995i
$51$	0	0
$52$	− 12.9343i	− 1.79367i
$53$	− 7.07107i	− 0.971286i	−0.874157	−	0.485643i	$-0.838586\pi$
$53$	− 7.07107i	− 0.971286i	0.874157	−	0.485643i	$-0.161414\pi$
$54$	0	0
$55$	2.79884i	0.377395i
$56$	0	0
$57$	0	0
$58$	11.6569	1.53062
$59$	6.75699	0.879685	0.439842	−	0.898075i	$-0.355034\pi$
$59$	6.75699	0.879685	0.439842	+	0.898075i	$0.355034\pi$
$60$	0	0
$61$	− 8.15640i	− 1.04432i	−0.852847	−	0.522160i	$-0.825126\pi$
$61$	− 8.15640i	− 1.04432i	0.852847	−	0.522160i	$-0.174874\pi$
$62$	16.3128	2.07173
$63$	0	0
$64$	9.82843	1.22855
$65$	11.4142i	1.41576i
$66$	0	0
$67$	8.48528	1.03664	0.518321	−	0.855186i	$-0.326557\pi$
$67$	8.48528	1.03664	0.518321	+	0.855186i	$0.326557\pi$
$68$	−5.35757	−0.649701
$69$	0	0
$70$	0	0
$71$	4.82843i	0.573029i	0.958076	+	0.286514i	$0.0924966\pi$
$71$	4.82843i	0.573029i	−0.958076	+	0.286514i	$0.907503\pi$
$72$	0	0
$73$	− 1.39942i	− 0.163789i	−0.996641	−	0.0818947i	$-0.973903\pi$
$73$	− 1.39942i	− 0.163789i	0.996641	−	0.0818947i	$-0.0260971\pi$
$74$	− 6.24264i	− 0.725692i
$75$	0	0
$76$	− 25.8686i	− 2.96734i
$77$	0	0
$78$	0	0
$79$	−9.65685	−1.08648	−0.543240	−	0.839577i	$-0.682803\pi$
$79$	−9.65685	−1.08648	−0.543240	+	0.839577i	$0.682803\pi$
$80$	10.1355	1.13318
$81$	0	0
$82$	19.6913i	2.17454i
$83$	−13.5140	−1.48335	−0.741676	−	0.670759i	$-0.765969\pi$
$83$	−13.5140	−1.48335	−0.741676	+	0.670759i	$0.765969\pi$
$84$	0	0
$85$	4.72792	0.512815
$86$	− 30.1421i	− 3.25031i
$87$	0	0
$88$	3.65685	0.389822
$89$	6.17733	0.654795	0.327398	−	0.944887i	$-0.393828\pi$
$89$	6.17733	0.654795	0.327398	+	0.944887i	$0.393828\pi$
$90$	0	0
$91$	0	0
$92$	− 7.65685i	− 0.798282i
$93$	0	0
$94$	16.3128i	1.68254i
$95$	22.8284i	2.34215i
$96$	0	0
$97$	1.39942i	0.142089i	0.997473	+	0.0710447i	$0.0226333\pi$
$97$	1.39942i	0.142089i	−0.997473	+	0.0710447i	$0.977367\pi$
$98$	0	0
$99$	0	0
$100$	−24.5563	−2.45563
$101$	12.9343	1.28701	0.643506	−	0.765441i	$-0.277479\pi$
$101$	12.9343	1.28701	0.643506	+	0.765441i	$0.277479\pi$
$102$	0	0
$103$	2.79884i	0.275777i	0.990448	+	0.137889i	$0.0440316\pi$
$103$	2.79884i	0.275777i	−0.990448	+	0.137889i	$0.955968\pi$
$104$	14.9134	1.46238
$105$	0	0
$106$	17.0711	1.65809
$107$	− 9.31371i	− 0.900390i	−0.892930	−	0.450195i	$-0.851354\pi$
$107$	− 9.31371i	− 0.900390i	0.892930	−	0.450195i	$-0.148646\pi$
$108$	0	0
$109$	−2.58579	−0.247673	−0.123837	−	0.992303i	$-0.539520\pi$
$109$	−2.58579	−0.247673	−0.123837	+	0.992303i	$0.539520\pi$
$110$	−6.75699	−0.644253
$111$	0	0
$112$	0	0
$113$	− 5.89949i	− 0.554978i	−0.960729	−	0.277489i	$-0.910498\pi$
$113$	− 5.89949i	− 0.554978i	0.960729	−	0.277489i	$-0.0895022\pi$
$114$	0	0
$115$	6.75699i	0.630092i
$116$	18.4853i	1.71632i
$117$	0	0
$118$	16.3128i	1.50172i
$119$	0	0
$120$	0	0
$121$	10.3137	0.937610
$122$	19.6913	1.78277
$123$	0	0
$124$	25.8686i	2.32307i
$125$	4.77791	0.427349
$126$	0	0
$127$	−0.485281	−0.0430618	−0.0215309	−	0.999768i	$-0.506854\pi$
$127$	−0.485281	−0.0430618	−0.0215309	+	0.999768i	$0.506854\pi$
$128$	20.5563i	1.81694i
$129$	0	0
$130$	−27.5563	−2.41685
$131$	−9.55582	−0.834896	−0.417448	−	0.908701i	$-0.637075\pi$
$131$	−9.55582	−0.834896	−0.417448	+	0.908701i	$0.637075\pi$
$132$	0	0
$133$	0	0
$134$	20.4853i	1.76966i
$135$	0	0
$136$	− 6.17733i	− 0.529701i
$137$	− 0.828427i	− 0.0707773i	−0.999374	−	0.0353887i	$-0.988733\pi$
$137$	− 0.828427i	− 0.0707773i	0.999374	−	0.0353887i	$-0.0112669\pi$
$138$	0	0
$139$	− 9.55582i	− 0.810514i	−0.914203	−	0.405257i	$-0.867182\pi$
$139$	− 9.55582i	− 0.810514i	0.914203	−	0.405257i	$-0.132818\pi$
$140$	0	0
$141$	0	0
$142$	−11.6569	−0.978221
$143$	−2.79884	−0.234050
$144$	0	0
$145$	− 16.3128i	− 1.35470i
$146$	3.37849	0.279606
$147$	0	0
$148$	9.89949	0.813733
$149$	4.24264i	0.347571i	0.984784	+	0.173785i	$0.0555999\pi$
$149$	4.24264i	0.347571i	−0.984784	+	0.173785i	$0.944400\pi$
$150$	0	0
$151$	−12.4853	−1.01604	−0.508019	−	0.861346i	$-0.669622\pi$
$151$	−12.4853	−1.01604	−0.508019	+	0.861346i	$0.669622\pi$
$152$	29.8268	2.41927
$153$	0	0
$154$	0	0
$155$	− 22.8284i	− 1.83362i
$156$	0	0
$157$	− 7.33664i	− 0.585528i	−0.956185	−	0.292764i	$-0.905425\pi$
$157$	− 7.33664i	− 0.585528i	0.956185	−	0.292764i	$-0.0945750\pi$
$158$	− 23.3137i	− 1.85474i
$159$	0	0
$160$	− 5.35757i	− 0.423553i
$161$	0	0
$162$	0	0
$163$	14.8284	1.16145	0.580726	−	0.814099i	$-0.302769\pi$
$163$	14.8284	1.16145	0.580726	+	0.814099i	$0.302769\pi$
$164$	−31.2262	−2.43836
$165$	0	0
$166$	− 32.6256i	− 2.53224i
$167$	2.79884	0.216580	0.108290	−	0.994119i	$-0.465462\pi$
$167$	2.79884	0.216580	0.108290	+	0.994119i	$0.465462\pi$
$168$	0	0
$169$	1.58579	0.121984
$170$	11.4142i	0.875430i
$171$	0	0
$172$	47.7990	3.64464
$173$	8.15640	0.620120	0.310060	−	0.950717i	$-0.399651\pi$
$173$	8.15640	0.620120	0.310060	+	0.950717i	$0.399651\pi$
$174$	0	0
$175$	0	0
$176$	2.48528i	0.187335i
$177$	0	0
$178$	14.9134i	1.11781i
$179$	− 9.51472i	− 0.711163i	−0.934645	−	0.355582i	$-0.884283\pi$
$179$	− 9.51472i	− 0.711163i	0.934645	−	0.355582i	$-0.115717\pi$
$180$	0	0
$181$	− 17.7122i	− 1.31654i	−0.752782	−	0.658270i	$-0.771289\pi$
$181$	− 17.7122i	− 1.31654i	0.752782	−	0.658270i	$-0.228711\pi$
$182$	0	0
$183$	0	0
$184$	8.82843	0.650840
$185$	−8.73606	−0.642288
$186$	0	0
$187$	1.15932i	0.0847775i
$188$	−25.8686	−1.88666
$189$	0	0
$190$	−55.1127	−3.99830
$191$	− 20.8284i	− 1.50709i	−0.657395	−	0.753546i	$-0.728342\pi$
$191$	− 20.8284i	− 1.50709i	0.657395	−	0.753546i	$-0.271658\pi$
$192$	0	0
$193$	−17.6569	−1.27097	−0.635484	−	0.772114i	$-0.719199\pi$
$193$	−17.6569	−1.27097	−0.635484	+	0.772114i	$0.719199\pi$
$194$	−3.37849	−0.242562
$195$	0	0
$196$	0	0
$197$	16.2426i	1.15724i	0.815597	+	0.578620i	$0.196409\pi$
$197$	16.2426i	1.15724i	−0.815597	+	0.578620i	$0.803591\pi$
$198$	0	0
$199$	− 19.1116i	− 1.35479i	−0.735620	−	0.677394i	$-0.763109\pi$
$199$	− 19.1116i	− 1.35479i	0.735620	−	0.677394i	$-0.236891\pi$
$200$	− 28.3137i	− 2.00208i
$201$	0	0
$202$	31.2262i	2.19707i
$203$	0	0
$204$	0	0
$205$	27.5563	1.92462
$206$	−6.75699	−0.470781
$207$	0	0
$208$	10.1355i	0.702769i
$209$	−5.59767	−0.387199
$210$	0	0
$211$	7.31371	0.503496	0.251748	−	0.967793i	$-0.418995\pi$
$211$	7.31371	0.503496	0.251748	+	0.967793i	$0.418995\pi$
$212$	27.0711i	1.85925i
$213$	0	0
$214$	22.4853	1.53706
$215$	−42.1814	−2.87675
$216$	0	0
$217$	0	0
$218$	− 6.24264i	− 0.422805i
$219$	0	0
$220$	− 10.7151i	− 0.722414i
$221$	4.72792i	0.318034i
$222$	0	0
$223$	23.0698i	1.54487i	0.635095	+	0.772434i	$0.280961\pi$
$223$	23.0698i	1.54487i	−0.635095	+	0.772434i	$0.719039\pi$
$224$	0	0
$225$	0	0
$226$	14.2426	0.947406
$227$	2.79884	0.185765	0.0928826	−	0.995677i	$-0.470392\pi$
$227$	2.79884	0.185765	0.0928826	+	0.995677i	$0.470392\pi$
$228$	0	0
$229$	− 12.1146i	− 0.800552i	−0.916395	−	0.400276i	$-0.868914\pi$
$229$	− 12.1146i	− 0.800552i	0.916395	−	0.400276i	$-0.131086\pi$
$230$	−16.3128	−1.07563
$231$	0	0
$232$	−21.3137	−1.39931
$233$	− 10.8284i	− 0.709394i	−0.934981	−	0.354697i	$-0.884584\pi$
$233$	− 10.8284i	− 0.709394i	0.934981	−	0.354697i	$-0.115416\pi$
$234$	0	0
$235$	22.8284	1.48916
$236$	−25.8686	−1.68390
$237$	0	0
$238$	0	0
$239$	− 0.343146i	− 0.0221963i	−0.999938	−	0.0110981i	$-0.996467\pi$
$239$	− 0.343146i	− 0.0221963i	0.999938	−	0.0110981i	$-0.00353272\pi$
$240$	0	0
$241$	8.97616i	0.578205i	0.957298	+	0.289103i	$0.0933569\pi$
$241$	8.97616i	0.578205i	−0.957298	+	0.289103i	$0.906643\pi$
$242$	24.8995i	1.60060i
$243$	0	0
$244$	31.2262i	1.99905i
$245$	0	0
$246$	0	0
$247$	−22.8284	−1.45254
$248$	−29.8268	−1.89400
$249$	0	0
$250$	11.5349i	0.729531i
$251$	−25.8686	−1.63281	−0.816407	−	0.577477i	$-0.804037\pi$
$251$	−25.8686	−1.63281	−0.816407	+	0.577477i	$0.804037\pi$
$252$	0	0
$253$	−1.65685	−0.104166
$254$	− 1.17157i	− 0.0735110i
$255$	0	0
$256$	−29.9706	−1.87316
$257$	21.6704	1.35176	0.675880	−	0.737011i	$-0.263764\pi$
$257$	21.6704	1.35176	0.675880	+	0.737011i	$0.263764\pi$
$258$	0	0
$259$	0	0
$260$	− 43.6985i	− 2.71006i
$261$	0	0
$262$	− 23.0698i	− 1.42526i
$263$	24.1421i	1.48867i	0.667808	+	0.744334i	$0.267233\pi$
$263$	24.1421i	1.48867i	−0.667808	+	0.744334i	$0.732767\pi$
$264$	0	0
$265$	− 23.8896i	− 1.46752i
$266$	0	0
$267$	0	0
$268$	−32.4853	−1.98435
$269$	−31.2262	−1.90389	−0.951947	−	0.306262i	$-0.900922\pi$
$269$	−31.2262	−1.90389	−0.951947	+	0.306262i	$0.900922\pi$
$270$	0	0
$271$	− 6.75699i	− 0.410458i	−0.978714	−	0.205229i	$-0.934206\pi$
$271$	− 6.75699i	− 0.410458i	0.978714	−	0.205229i	$-0.0657939\pi$
$272$	4.19825	0.254556
$273$	0	0
$274$	2.00000	0.120824
$275$	5.31371i	0.320429i
$276$	0	0
$277$	−20.9706	−1.26000	−0.630000	−	0.776596i	$-0.716945\pi$
$277$	−20.9706	−1.26000	−0.630000	+	0.776596i	$0.716945\pi$
$278$	23.0698	1.38363
$279$	0	0
$280$	0	0
$281$	18.8284i	1.12321i	0.827406	+	0.561605i	$0.189816\pi$
$281$	18.8284i	1.12321i	−0.827406	+	0.561605i	$0.810184\pi$
$282$	0	0
$283$	29.8268i	1.77302i	0.462711	+	0.886509i	$0.346877\pi$
$283$	29.8268i	1.77302i	−0.462711	+	0.886509i	$0.653123\pi$
$284$	− 18.4853i	− 1.09690i
$285$	0	0
$286$	− 6.75699i	− 0.399549i
$287$	0	0
$288$	0	0
$289$	−15.0416	−0.884802
$290$	39.3826	2.31263
$291$	0	0
$292$	5.35757i	0.313528i
$293$	7.33664	0.428611	0.214306	−	0.976767i	$-0.431251\pi$
$293$	7.33664	0.428611	0.214306	+	0.976767i	$0.431251\pi$
$294$	0	0
$295$	22.8284	1.32912
$296$	11.4142i	0.663438i
$297$	0	0
$298$	−10.2426	−0.593340
$299$	−6.75699	−0.390767
$300$	0	0
$301$	0	0
$302$	− 30.1421i	− 1.73448i
$303$	0	0
$304$	20.2710i	1.16262i
$305$	− 27.5563i	− 1.57787i
$306$	0	0
$307$	12.3547i	0.705117i	0.935790	+	0.352559i	$0.114688\pi$
$307$	12.3547i	0.705117i	−0.935790	+	0.352559i	$0.885312\pi$
$308$	0	0
$309$	0	0
$310$	55.1127	3.13019
$311$	10.7151	0.607600	0.303800	−	0.952736i	$-0.401745\pi$
$311$	10.7151	0.607600	0.303800	+	0.952736i	$0.401745\pi$
$312$	0	0
$313$	12.9343i	0.731091i	0.930794	+	0.365545i	$0.119117\pi$
$313$	12.9343i	0.731091i	−0.930794	+	0.365545i	$0.880883\pi$
$314$	17.7122	0.999559
$315$	0	0
$316$	36.9706	2.07976
$317$	23.7574i	1.33435i	0.744903	+	0.667173i	$0.232496\pi$
$317$	23.7574i	1.33435i	−0.744903	+	0.667173i	$0.767504\pi$
$318$	0	0
$319$	4.00000	0.223957
$320$	33.2053	1.85623
$321$	0	0
$322$	0	0
$323$	9.45584i	0.526137i
$324$	0	0
$325$	21.6704i	1.20206i
$326$	35.7990i	1.98272i
$327$	0	0
$328$	− 36.0041i	− 1.98799i
$329$	0	0
$330$	0	0
$331$	10.3431	0.568511	0.284255	−	0.958749i	$-0.408254\pi$
$331$	10.3431	0.568511	0.284255	+	0.958749i	$0.408254\pi$
$332$	51.7373	2.83945
$333$	0	0
$334$	6.75699i	0.369726i
$335$	28.6675	1.56627
$336$	0	0
$337$	−12.9289	−0.704284	−0.352142	−	0.935947i	$-0.614547\pi$
$337$	−12.9289	−0.704284	−0.352142	+	0.935947i	$0.614547\pi$
$338$	3.82843i	0.208239i
$339$	0	0
$340$	−18.1005	−0.981638
$341$	5.59767	0.303131
$342$	0	0
$343$	0	0
$344$	55.1127i	2.97148i
$345$	0	0
$346$	19.6913i	1.05861i
$347$	− 18.9706i	− 1.01839i	−0.860650	−	0.509197i	$-0.829943\pi$
$347$	− 18.9706i	− 1.01839i	0.860650	−	0.509197i	$-0.170057\pi$
$348$	0	0
$349$	− 6.17733i	− 0.330665i	−0.986238	−	0.165332i	$-0.947130\pi$
$349$	− 6.17733i	− 0.330665i	0.986238	−	0.165332i	$-0.0528697\pi$
$350$	0	0
$351$	0	0
$352$	1.31371	0.0700209
$353$	23.6494	1.25873	0.629367	−	0.777109i	$-0.283314\pi$
$353$	23.6494	1.25873	0.629367	+	0.777109i	$0.283314\pi$
$354$	0	0
$355$	16.3128i	0.865794i
$356$	−23.6494	−1.25342
$357$	0	0
$358$	22.9706	1.21403
$359$	− 1.02944i	− 0.0543316i	−0.999631	−	0.0271658i	$-0.991352\pi$
$359$	− 1.02944i	− 0.0543316i	0.999631	−	0.0271658i	$-0.00864821\pi$
$360$	0	0
$361$	−26.6569	−1.40299
$362$	42.7611	2.24747
$363$	0	0
$364$	0	0
$365$	− 4.72792i	− 0.247471i
$366$	0	0
$367$	3.95815i	0.206614i	0.994650	+	0.103307i	$0.0329424\pi$
$367$	3.95815i	0.206614i	−0.994650	+	0.103307i	$0.967058\pi$
$368$	6.00000i	0.312772i
$369$	0	0
$370$	− 21.0907i	− 1.09645i
$371$	0	0
$372$	0	0
$373$	−22.6274	−1.17160	−0.585802	−	0.810454i	$-0.699220\pi$
$373$	−22.6274	−1.17160	−0.585802	+	0.810454i	$0.699220\pi$
$374$	−2.79884	−0.144724
$375$	0	0
$376$	− 29.8268i	− 1.53820i
$377$	16.3128	0.840152
$378$	0	0
$379$	0	0		−	1.00000i	$-0.5\pi$
$379$	0	0			1.00000i	$0.5\pi$
$380$	− 87.3970i	− 4.48337i
$381$	0	0
$382$	50.2843	2.57277
$383$	−27.0279	−1.38106	−0.690532	−	0.723302i	$-0.742623\pi$
$383$	−27.0279	−1.38106	−0.690532	+	0.723302i	$0.742623\pi$
$384$	0	0
$385$	0	0
$386$	− 42.6274i	− 2.16968i
$387$	0	0
$388$	− 5.35757i	− 0.271989i
$389$	− 13.7990i	− 0.699637i	−0.936818	−	0.349818i	$-0.886243\pi$
$389$	− 13.7990i	− 0.699637i	0.936818	−	0.349818i	$-0.113757\pi$
$390$	0	0
$391$	2.79884i	0.141543i
$392$	0	0
$393$	0	0
$394$	−39.2132	−1.97553
$395$	−32.6256	−1.64157
$396$	0	0
$397$	27.2680i	1.36854i	0.729227	+	0.684272i	$0.239880\pi$
$397$	27.2680i	1.36854i	−0.729227	+	0.684272i	$0.760120\pi$
$398$	46.1396	2.31277
$399$	0	0
$400$	19.2426	0.962132
$401$	10.8284i	0.540746i	0.962756	+	0.270373i	$0.0871470\pi$
$401$	10.8284i	0.540746i	−0.962756	+	0.270373i	$0.912853\pi$
$402$	0	0
$403$	22.8284	1.13716
$404$	−49.5181	−2.46362
$405$	0	0
$406$	0	0
$407$	− 2.14214i	− 0.106182i
$408$	0	0
$409$	− 17.7122i	− 0.875813i	−0.899020	−	0.437907i	$-0.855720\pi$
$409$	− 17.7122i	− 0.875813i	0.899020	−	0.437907i	$-0.144280\pi$
$410$	66.5269i	3.28553i
$411$	0	0
$412$	− 10.7151i	− 0.527897i
$413$	0	0
$414$	0	0
$415$	−45.6569	−2.24121
$416$	5.35757	0.262676
$417$	0	0
$418$	− 13.5140i	− 0.660990i
$419$	−35.4244	−1.73060	−0.865299	−	0.501256i	$-0.832871\pi$
$419$	−35.4244	−1.73060	−0.865299	+	0.501256i	$0.832871\pi$
$420$	0	0
$421$	−23.3137	−1.13624	−0.568120	−	0.822946i	$-0.692329\pi$
$421$	−23.3137	−1.13624	−0.568120	+	0.822946i	$0.692329\pi$
$422$	17.6569i	0.859522i
$423$	0	0
$424$	−31.2132	−1.51585
$425$	8.97616	0.435408
$426$	0	0
$427$	0	0
$428$	35.6569i	1.72354i
$429$	0	0
$430$	− 101.835i	− 4.91092i
$431$	− 15.4558i	− 0.744482i	−0.928136	−	0.372241i	$-0.878590\pi$
$431$	− 15.4558i	− 0.744482i	0.928136	−	0.372241i	$-0.121410\pi$
$432$	0	0
$433$	8.15640i	0.391972i	0.980607	+	0.195986i	$0.0627907\pi$
$433$	8.15640i	0.391972i	−0.980607	+	0.195986i	$0.937209\pi$
$434$	0	0
$435$	0	0
$436$	9.89949	0.474100
$437$	−13.5140	−0.646461
$438$	0	0
$439$	28.6675i	1.36822i	0.729377	+	0.684112i	$0.239810\pi$
$439$	28.6675i	1.36822i	−0.729377	+	0.684112i	$0.760190\pi$
$440$	12.3547	0.588985
$441$	0	0
$442$	−11.4142	−0.542919
$443$	− 24.8284i	− 1.17963i	−0.807537	−	0.589817i	$-0.799200\pi$
$443$	− 24.8284i	− 1.17963i	0.807537	−	0.589817i	$-0.200800\pi$
$444$	0	0
$445$	20.8701	0.989336
$446$	−55.6954	−2.63725
$447$	0	0
$448$	0	0
$449$	24.2426i	1.14408i	0.820225	+	0.572040i	$0.193848\pi$
$449$	24.2426i	1.14408i	−0.820225	+	0.572040i	$0.806152\pi$
$450$	0	0
$451$	6.75699i	0.318174i
$452$	22.5858i	1.06235i
$453$	0	0
$454$	6.75699i	0.317121i
$455$	0	0
$456$	0	0
$457$	28.6274	1.33913	0.669567	−	0.742752i	$-0.266480\pi$
$457$	28.6274	1.33913	0.669567	+	0.742752i	$0.266480\pi$
$458$	29.2471	1.36663
$459$	0	0
$460$	− 25.8686i	− 1.20613i
$461$	21.6704	1.00929	0.504645	−	0.863327i	$-0.331623\pi$
$461$	21.6704	1.00929	0.504645	+	0.863327i	$0.331623\pi$
$462$	0	0
$463$	3.51472	0.163343	0.0816714	−	0.996659i	$-0.473974\pi$
$463$	3.51472	0.163343	0.0816714	+	0.996659i	$0.473974\pi$
$464$	− 14.4853i	− 0.672462i
$465$	0	0
$466$	26.1421	1.21101
$467$	−2.79884	−0.129515	−0.0647573	−	0.997901i	$-0.520627\pi$
$467$	−2.79884	−0.129515	−0.0647573	+	0.997901i	$0.520627\pi$
$468$	0	0
$469$	0	0
$470$	55.1127i	2.54216i
$471$	0	0
$472$	− 29.8268i	− 1.37289i
$473$	− 10.3431i	− 0.475578i
$474$	0	0
$475$	43.3407i	1.98861i
$476$	0	0
$477$	0	0
$478$	0.828427	0.0378914
$479$	−8.39651	−0.383646	−0.191823	−	0.981430i	$-0.561440\pi$
$479$	−8.39651	−0.383646	−0.191823	+	0.981430i	$0.561440\pi$
$480$	0	0
$481$	− 8.73606i	− 0.398330i
$482$	−21.6704	−0.987059
$483$	0	0
$484$	−39.4853	−1.79479
$485$	4.72792i	0.214684i
$486$	0	0
$487$	9.45584	0.428485	0.214243	−	0.976780i	$-0.431272\pi$
$487$	9.45584	0.428485	0.214243	+	0.976780i	$0.431272\pi$
$488$	−36.0041	−1.62983
$489$	0	0
$490$	0	0
$491$	− 2.68629i	− 0.121231i	−0.998161	−	0.0606153i	$-0.980694\pi$
$491$	− 2.68629i	− 0.121231i	0.998161	−	0.0606153i	$-0.0193063\pi$
$492$	0	0
$493$	− 6.75699i	− 0.304319i
$494$	− 55.1127i	− 2.47964i
$495$	0	0
$496$	− 20.2710i	− 0.910193i
$497$	0	0
$498$	0	0
$499$	−2.14214	−0.0958952	−0.0479476	−	0.998850i	$-0.515268\pi$
$499$	−2.14214	−0.0958952	−0.0479476	+	0.998850i	$0.515268\pi$
$500$	−18.2919	−0.818038
$501$	0	0
$502$	− 62.4524i	− 2.78739i
$503$	32.6256	1.45470	0.727352	−	0.686264i	$-0.240751\pi$
$503$	32.6256	1.45470	0.727352	+	0.686264i	$0.240751\pi$
$504$	0	0
$505$	43.6985	1.94456
$506$	− 4.00000i	− 0.177822i
$507$	0	0
$508$	1.85786	0.0824294
$509$	36.8239	1.63219	0.816095	−	0.577918i	$-0.196135\pi$
$509$	36.8239	1.63219	0.816095	+	0.577918i	$0.196135\pi$
$510$	0	0
$511$	0	0
$512$	− 31.2426i	− 1.38074i
$513$	0	0
$514$	52.3169i	2.30760i
$515$	9.45584i	0.416674i
$516$	0	0
$517$	5.59767i	0.246185i
$518$	0	0
$519$	0	0
$520$	50.3848	2.20952
$521$	−42.7611	−1.87340	−0.936699	−	0.350136i	$-0.886135\pi$
$521$	−42.7611	−1.87340	−0.936699	+	0.350136i	$0.886135\pi$
$522$	0	0
$523$	17.4721i	0.764003i	0.924162	+	0.382001i	$0.124765\pi$
$523$	17.4721i	0.764003i	−0.924162	+	0.382001i	$0.875235\pi$
$524$	36.5838	1.59817
$525$	0	0
$526$	−58.2843	−2.54131
$527$	− 9.45584i	− 0.411903i
$528$	0	0
$529$	19.0000	0.826087
$530$	57.6745	2.50522
$531$	0	0
$532$	0	0
$533$	27.5563i	1.19360i
$534$	0	0
$535$	− 31.4663i	− 1.36041i
$536$	− 37.4558i	− 1.61785i
$537$	0	0
$538$	− 75.3867i	− 3.25015i
$539$	0	0
$540$	0	0
$541$	18.9706	0.815608	0.407804	−	0.913069i	$-0.366295\pi$
$541$	18.9706	0.815608	0.407804	+	0.913069i	$0.366295\pi$
$542$	16.3128	0.700695
$543$	0	0
$544$	− 2.21918i	− 0.0951464i
$545$	−8.73606	−0.374212
$546$	0	0
$547$	−26.6274	−1.13851	−0.569253	−	0.822162i	$-0.692768\pi$
$547$	−26.6274	−1.13851	−0.569253	+	0.822162i	$0.692768\pi$
$548$	3.17157i	0.135483i
$549$	0	0
$550$	−12.8284	−0.547006
$551$	32.6256	1.38990
$552$	0	0
$553$	0	0
$554$	− 50.6274i	− 2.15095i
$555$	0	0
$556$	36.5838i	1.55150i
$557$	− 10.5858i	− 0.448534i	−0.974528	−	0.224267i	$-0.928001\pi$
$557$	− 10.5858i	− 0.448534i	0.974528	−	0.224267i	$-0.0719988\pi$
$558$	0	0
$559$	− 42.1814i	− 1.78408i
$560$	0	0
$561$	0	0
$562$	−45.4558	−1.91744
$563$	2.79884	0.117957	0.0589784	−	0.998259i	$-0.481216\pi$
$563$	2.79884	0.117957	0.0589784	+	0.998259i	$0.481216\pi$
$564$	0	0
$565$	− 19.9314i	− 0.838520i
$566$	−72.0082	−3.02673
$567$	0	0
$568$	21.3137	0.894303
$569$	25.1127i	1.05278i	0.850244	+	0.526390i	$0.176455\pi$
$569$	25.1127i	1.05278i	−0.850244	+	0.526390i	$0.823545\pi$
$570$	0	0
$571$	−11.3137	−0.473464	−0.236732	−	0.971575i	$-0.576076\pi$
$571$	−11.3137	−0.473464	−0.236732	+	0.971575i	$0.576076\pi$
$572$	10.7151	0.448022
$573$	0	0
$574$	0	0
$575$	12.8284i	0.534982i
$576$	0	0
$577$	13.7541i	0.572590i	0.958142	+	0.286295i	$0.0924237\pi$
$577$	13.7541i	0.572590i	−0.958142	+	0.286295i	$0.907576\pi$
$578$	− 36.3137i	− 1.51045i
$579$	0	0
$580$	62.4524i	2.59319i
$581$	0	0
$582$	0	0
$583$	5.85786	0.242608
$584$	−6.17733	−0.255620
$585$	0	0
$586$	17.7122i	0.731685i
$587$	−20.2710	−0.836672	−0.418336	−	0.908292i	$-0.637387\pi$
$587$	−20.2710	−0.836672	−0.418336	+	0.908292i	$0.637387\pi$
$588$	0	0
$589$	45.6569	1.88126
$590$	55.1127i	2.26895i
$591$	0	0
$592$	−7.75736	−0.318826
$593$	−20.5111	−0.842288	−0.421144	−	0.906994i	$-0.638371\pi$
$593$	−20.5111	−0.842288	−0.421144	+	0.906994i	$0.638371\pi$
$594$	0	0
$595$	0	0
$596$	− 16.2426i	− 0.665324i
$597$	0	0
$598$	− 16.3128i	− 0.667080i
$599$	6.48528i	0.264981i	0.991184	+	0.132491i	$0.0422975\pi$
$599$	6.48528i	0.264981i	−0.991184	+	0.132491i	$0.957703\pi$
$600$	0	0
$601$	27.6076i	1.12614i	0.826410	+	0.563069i	$0.190379\pi$
$601$	27.6076i	1.12614i	−0.826410	+	0.563069i	$0.809621\pi$
$602$	0	0
$603$	0	0
$604$	47.7990	1.94491
$605$	34.8448	1.41664
$606$	0	0
$607$	23.0698i	0.936374i	0.883629	+	0.468187i	$0.155093\pi$
$607$	23.0698i	0.936374i	−0.883629	+	0.468187i	$0.844907\pi$
$608$	10.7151	0.434556
$609$	0	0
$610$	66.5269	2.69360
$611$	22.8284i	0.923539i
$612$	0	0
$613$	−10.8701	−0.439037	−0.219519	−	0.975608i	$-0.570449\pi$
$613$	−10.8701	−0.439037	−0.219519	+	0.975608i	$0.570449\pi$
$614$	−29.8268	−1.20371
$615$	0	0
$616$	0	0
$617$	− 30.4853i	− 1.22729i	−0.789582	−	0.613646i	$-0.789702\pi$
$617$	− 30.4853i	− 1.22729i	0.789582	−	0.613646i	$-0.210298\pi$
$618$	0	0
$619$	24.7093i	0.993151i	0.867994	+	0.496576i	$0.165409\pi$
$619$	24.7093i	0.993151i	−0.867994	+	0.496576i	$0.834591\pi$
$620$	87.3970i	3.50995i
$621$	0	0
$622$	25.8686i	1.03724i
$623$	0	0
$624$	0	0
$625$	−15.9289	−0.637157
$626$	−31.2262	−1.24805
$627$	0	0
$628$	28.0878i	1.12083i
$629$	−3.61859	−0.144283
$630$	0	0
$631$	−21.1716	−0.842827	−0.421414	−	0.906869i	$-0.638466\pi$
$631$	−21.1716	−0.842827	−0.421414	+	0.906869i	$0.638466\pi$
$632$	42.6274i	1.69563i
$633$	0	0
$634$	−57.3553	−2.27787
$635$	−1.63952	−0.0650624
$636$	0	0
$637$	0	0
$638$	9.65685i	0.382319i
$639$	0	0
$640$	69.4495i	2.74523i
$641$	7.51472i	0.296814i	0.988926	+	0.148407i	$0.0474145\pi$
$641$	7.51472i	0.296814i	−0.988926	+	0.148407i	$0.952586\pi$
$642$	0	0
$643$	6.75699i	0.266469i	0.991085	+	0.133235i	$0.0425364\pi$
$643$	6.75699i	0.266469i	−0.991085	+	0.133235i	$0.957464\pi$
$644$	0	0
$645$	0	0
$646$	−22.8284	−0.898172
$647$	25.8686	1.01700	0.508500	−	0.861062i	$-0.330200\pi$
$647$	25.8686	1.01700	0.508500	+	0.861062i	$0.330200\pi$
$648$	0	0
$649$	5.59767i	0.219728i
$650$	−52.3169	−2.05204
$651$	0	0
$652$	−56.7696	−2.22327
$653$	− 38.4853i	− 1.50605i	−0.657995	−	0.753023i	$-0.728595\pi$
$653$	− 38.4853i	− 1.50605i	0.657995	−	0.753023i	$-0.271405\pi$
$654$	0	0
$655$	−32.2843	−1.26145
$656$	24.4692	0.955362
$657$	0	0
$658$	0	0
$659$	− 47.4558i	− 1.84862i	−0.381646	−	0.924309i	$-0.624643\pi$
$659$	− 47.4558i	− 1.84862i	0.381646	−	0.924309i	$-0.375357\pi$
$660$	0	0
$661$	7.33664i	0.285362i	0.989769	+	0.142681i	$0.0455724\pi$
$661$	7.33664i	0.285362i	−0.989769	+	0.142681i	$0.954428\pi$
$662$	24.9706i	0.970508i
$663$	0	0
$664$	59.6536i	2.31501i
$665$	0	0
$666$	0	0
$667$	9.65685	0.373915
$668$	−10.7151	−0.414581
$669$	0	0
$670$	69.2094i	2.67379i
$671$	6.75699	0.260851
$672$	0	0
$673$	−37.8995	−1.46092	−0.730459	−	0.682956i	$-0.760694\pi$
$673$	−37.8995	−1.46092	−0.730459	+	0.682956i	$0.760694\pi$
$674$	− 31.2132i	− 1.20229i
$675$	0	0
$676$	−6.07107	−0.233503
$677$	−32.8657	−1.26313	−0.631566	−	0.775322i	$-0.717588\pi$
$677$	−32.8657	−1.26313	−0.631566	+	0.775322i	$0.717588\pi$
$678$	0	0
$679$	0	0
$680$	− 20.8701i	− 0.800330i
$681$	0	0
$682$	13.5140i	0.517477i
$683$	38.7696i	1.48348i	0.670690	+	0.741738i	$0.265998\pi$
$683$	38.7696i	1.48348i	−0.670690	+	0.741738i	$0.734002\pi$
$684$	0	0
$685$	− 2.79884i	− 0.106938i
$686$	0	0
$687$	0	0
$688$	−37.4558	−1.42799
$689$	23.8896	0.910119
$690$	0	0
$691$	− 9.55582i	− 0.363521i	−0.983343	−	0.181760i	$-0.941821\pi$
$691$	− 9.55582i	− 0.363521i	0.983343	−	0.181760i	$-0.0581795\pi$
$692$	−31.2262	−1.18704
$693$	0	0
$694$	45.7990	1.73851
$695$	− 32.2843i	− 1.22461i
$696$	0	0
$697$	11.4142	0.432344
$698$	14.9134	0.564480
$699$	0	0
$700$	0	0
$701$	− 40.7696i	− 1.53984i	−0.638138	−	0.769922i	$-0.720295\pi$
$701$	− 40.7696i	− 1.53984i	0.638138	−	0.769922i	$-0.279705\pi$
$702$	0	0
$703$	− 17.4721i	− 0.658974i
$704$	8.14214i	0.306868i
$705$	0	0
$706$	57.0948i	2.14879i
$707$	0	0
$708$	0	0
$709$	−32.7279	−1.22912	−0.614561	−	0.788869i	$-0.710667\pi$
$709$	−32.7279	−1.22912	−0.614561	+	0.788869i	$0.710667\pi$
$710$	−39.3826	−1.47800
$711$	0	0
$712$	− 27.2680i	− 1.02191i
$713$	13.5140	0.506102
$714$	0	0
$715$	−9.45584	−0.353629
$716$	36.4264i	1.36132i
$717$	0	0
$718$	2.48528	0.0927499
$719$	−13.5140	−0.503986	−0.251993	−	0.967729i	$-0.581086\pi$
$719$	−13.5140	−0.503986	−0.251993	+	0.967729i	$0.581086\pi$
$720$	0	0
$721$	0	0
$722$	− 64.3553i	− 2.39506i
$723$	0	0
$724$	67.8100i	2.52014i
$725$	− 30.9706i	− 1.15022i
$726$	0	0
$727$	− 43.3407i	− 1.60742i	−0.595022	−	0.803710i	$-0.702857\pi$
$727$	− 43.3407i	− 1.60742i	0.595022	−	0.803710i	$-0.297143\pi$
$728$	0	0
$729$	0	0
$730$	11.4142	0.422459
$731$	−17.4721	−0.646230
$732$	0	0
$733$	− 23.3099i	− 0.860971i	−0.902598	−	0.430485i	$-0.858342\pi$
$733$	− 23.3099i	− 0.860971i	0.902598	−	0.430485i	$-0.141658\pi$
$734$	−9.55582	−0.352712
$735$	0	0
$736$	3.17157	0.116906
$737$	7.02944i	0.258933i
$738$	0	0
$739$	−18.8284	−0.692615	−0.346307	−	0.938121i	$-0.612565\pi$
$739$	−18.8284	−0.692615	−0.346307	+	0.938121i	$0.612565\pi$
$740$	33.4454	1.22948
$741$	0	0
$742$	0	0
$743$	52.4264i	1.92334i	0.274212	+	0.961669i	$0.411583\pi$
$743$	52.4264i	1.92334i	−0.274212	+	0.961669i	$0.588417\pi$
$744$	0	0
$745$	14.3337i	0.525147i
$746$	− 54.6274i	− 2.00005i
$747$	0	0
$748$	− 4.43835i	− 0.162282i
$749$	0	0
$750$	0	0
$751$	20.6863	0.754853	0.377427	−	0.926039i	$-0.376809\pi$
$751$	20.6863	0.754853	0.377427	+	0.926039i	$0.376809\pi$
$752$	20.2710	0.739206
$753$	0	0
$754$	39.3826i	1.43423i
$755$	−42.1814	−1.53514
$756$	0	0
$757$	6.87006	0.249696	0.124848	−	0.992176i	$-0.460156\pi$
$757$	6.87006	0.249696	0.124848	+	0.992176i	$0.460156\pi$
$758$	0	0
$759$	0	0
$760$	100.770	3.65530
$761$	−0.579658	−0.0210126	−0.0105063	−	0.999945i	$-0.503344\pi$
$761$	−0.579658	−0.0210126	−0.0105063	+	0.999945i	$0.503344\pi$
$762$	0	0
$763$	0	0
$764$	79.7401i	2.88490i
$765$	0	0
$766$	− 65.2512i	− 2.35762i
$767$	22.8284i	0.824287i
$768$	0	0
$769$	− 45.8995i	− 1.65518i	−0.561335	−	0.827589i	$-0.689712\pi$
$769$	− 45.8995i	− 1.65518i	0.561335	−	0.827589i	$-0.310288\pi$
$770$	0	0
$771$	0	0
$772$	67.5980	2.43290
$773$	18.8715	0.678762	0.339381	−	0.940649i	$-0.389782\pi$
$773$	18.8715	0.678762	0.339381	+	0.940649i	$0.389782\pi$
$774$	0	0
$775$	− 43.3407i	− 1.55685i
$776$	6.17733	0.221753
$777$	0	0
$778$	33.3137	1.19435
$779$	55.1127i	1.97462i
$780$	0	0
$781$	−4.00000	−0.143131
$782$	−6.75699	−0.241629
$783$	0	0
$784$	0	0
$785$	− 24.7868i	− 0.884679i
$786$	0	0
$787$	− 13.5140i	− 0.481721i	−0.970560	−	0.240861i	$-0.922570\pi$
$787$	− 13.5140i	− 0.481721i	0.970560	−	0.240861i	$-0.0774296\pi$
$788$	− 62.1838i	− 2.21520i
$789$	0	0
$790$	− 78.7652i	− 2.80234i
$791$	0	0
$792$	0	0
$793$	27.5563	0.978555
$794$	−65.8309	−2.33625
$795$	0	0
$796$	73.1675i	2.59335i
$797$	−1.73897	−0.0615976	−0.0307988	−	0.999526i	$-0.509805\pi$
$797$	−1.73897	−0.0615976	−0.0307988	+	0.999526i	$0.509805\pi$
$798$	0	0
$799$	9.45584	0.334524
$800$	− 10.1716i	− 0.359619i
$801$	0	0
$802$	−26.1421	−0.923111
$803$	1.15932	0.0409114
$804$	0	0
$805$	0	0
$806$	55.1127i	1.94126i
$807$	0	0
$808$	− 57.0948i	− 2.00859i
$809$	32.0416i	1.12652i	0.826278	+	0.563262i	$0.190454\pi$
$809$	32.0416i	1.12652i	−0.826278	+	0.563262i	$0.809546\pi$
$810$	0	0
$811$	9.55582i	0.335550i	0.985825	+	0.167775i	$0.0536583\pi$
$811$	9.55582i	0.335550i	−0.985825	+	0.167775i	$0.946342\pi$
$812$	0	0
$813$	0	0
$814$	5.17157	0.181264
$815$	50.0977	1.75485
$816$	0	0
$817$	− 84.3629i	− 2.95148i
$818$	42.7611	1.49511
$819$	0	0
$820$	−105.497	−3.68413
$821$	32.2426i	1.12528i	0.826703	+	0.562638i	$0.190213\pi$
$821$	32.2426i	1.12528i	−0.826703	+	0.562638i	$0.809787\pi$
$822$	0	0
$823$	25.9411	0.904251	0.452125	−	0.891954i	$-0.350666\pi$
$823$	25.9411	0.904251	0.452125	+	0.891954i	$0.350666\pi$
$824$	12.3547	0.430395
$825$	0	0
$826$	0	0
$827$	28.3431i	0.985588i	0.870146	+	0.492794i	$0.164024\pi$
$827$	28.3431i	0.985588i	−0.870146	+	0.492794i	$0.835976\pi$
$828$	0	0
$829$	− 34.8448i	− 1.21021i	−0.796146	−	0.605105i	$-0.793131\pi$
$829$	− 34.8448i	− 1.21021i	0.796146	−	0.605105i	$-0.206869\pi$
$830$	− 110.225i	− 3.82598i
$831$	0	0
$832$	33.2053i	1.15119i
$833$	0	0
$834$	0	0
$835$	9.45584	0.327233
$836$	21.4303	0.741181
$837$	0	0
$838$	− 85.5222i	− 2.95431i
$839$	−20.2710	−0.699831	−0.349916	−	0.936781i	$-0.613790\pi$
$839$	−20.2710	−0.699831	−0.349916	+	0.936781i	$0.613790\pi$
$840$	0	0
$841$	5.68629	0.196079
$842$	− 56.2843i	− 1.93968i
$843$	0	0
$844$	−28.0000	−0.963800
$845$	5.35757	0.184306
$846$	0	0
$847$	0	0
$848$	− 21.2132i	− 0.728464i
$849$	0	0
$850$	21.6704i	0.743288i
$851$	− 5.17157i	− 0.177279i
$852$	0	0
$853$	38.8029i	1.32859i	0.747472	+	0.664294i	$0.231268\pi$
$853$	38.8029i	1.32859i	−0.747472	+	0.664294i	$0.768732\pi$
$854$	0	0
$855$	0	0
$856$	−41.1127	−1.40520
$857$	−19.6913	−0.672642	−0.336321	−	0.941747i	$-0.609183\pi$
$857$	−19.6913	−0.672642	−0.336321	+	0.941747i	$0.609183\pi$
$858$	0	0
$859$	− 29.8268i	− 1.01768i	−0.860862	−	0.508838i	$-0.830075\pi$
$859$	− 29.8268i	− 1.01768i	0.860862	−	0.508838i	$-0.169925\pi$
$860$	161.489	5.50671
$861$	0	0
$862$	37.3137	1.27091
$863$	− 33.3137i	− 1.13401i	−0.823714	−	0.567006i	$-0.808102\pi$
$863$	− 33.3137i	− 1.13401i	0.823714	−	0.567006i	$-0.191898\pi$
$864$	0	0
$865$	27.5563	0.936944
$866$	−19.6913	−0.669138
$867$	0	0
$868$	0	0
$869$	− 8.00000i	− 0.271381i
$870$	0	0
$871$	28.6675i	0.971360i
$872$	11.4142i	0.386534i
$873$	0	0
$874$	− 32.6256i	− 1.10358i
$875$	0	0
$876$	0	0
$877$	32.5269	1.09836	0.549178	−	0.835705i	$-0.314941\pi$
$877$	32.5269	1.09836	0.549178	+	0.835705i	$0.314941\pi$
$878$	−69.2094	−2.33570
$879$	0	0
$880$	8.39651i	0.283046i
$881$	20.5111	0.691035	0.345518	−	0.938412i	$-0.387703\pi$
$881$	20.5111	0.691035	0.345518	+	0.938412i	$0.387703\pi$
$882$	0	0
$883$	−37.4558	−1.26049	−0.630245	−	0.776397i	$-0.717045\pi$
$883$	−37.4558	−1.26049	−0.630245	+	0.776397i	$0.717045\pi$
$884$	− 18.1005i	− 0.608786i
$885$	0	0
$886$	59.9411	2.01376
$887$	17.9523	0.602780	0.301390	−	0.953501i	$-0.402549\pi$
$887$	17.9523	0.602780	0.301390	+	0.953501i	$0.402549\pi$
$888$	0	0
$889$	0	0
$890$	50.3848i	1.68890i
$891$	0	0
$892$	− 88.3210i	− 2.95721i
$893$	45.6569i	1.52785i
$894$	0	0
$895$	− 32.1454i	− 1.07450i
$896$	0	0
$897$	0	0
$898$	−58.5269	−1.95307
$899$	−32.6256	−1.08813
$900$	0	0
$901$	− 9.89538i	− 0.329663i
$902$	−16.3128	−0.543157
$903$	0	0
$904$	−26.0416	−0.866132
$905$	− 59.8406i	− 1.98917i
$906$	0	0
$907$	−44.7696	−1.48655	−0.743274	−	0.668987i	$-0.766728\pi$
$907$	−44.7696	−1.48655	−0.743274	+	0.668987i	$0.766728\pi$
$908$	−10.7151	−0.355594
$909$	0	0
$910$	0	0
$911$	25.5147i	0.845340i	0.906284	+	0.422670i	$0.138907\pi$
$911$	25.5147i	0.845340i	−0.906284	+	0.422670i	$0.861093\pi$
$912$	0	0
$913$	− 11.1953i	− 0.370512i
$914$	69.1127i	2.28604i
$915$	0	0
$916$	46.3797i	1.53243i
$917$	0	0
$918$	0	0
$919$	9.45584	0.311920	0.155960	−	0.987763i	$-0.450153\pi$
$919$	9.45584	0.311920	0.155960	+	0.987763i	$0.450153\pi$
$920$	29.8268	0.983360
$921$	0	0
$922$	52.3169i	1.72297i
$923$	−16.3128	−0.536943
$924$	0	0
$925$	−16.5858	−0.545337
$926$	8.48528i	0.278844i
$927$	0	0
$928$	−7.65685	−0.251349
$929$	−19.6913	−0.646051	−0.323025	−	0.946390i	$-0.604700\pi$
$929$	−19.6913	−0.646051	−0.323025	+	0.946390i	$0.604700\pi$
$930$	0	0
$931$	0	0
$932$	41.4558i	1.35793i
$933$	0	0
$934$	− 6.75699i	− 0.221095i
$935$	3.91674i	0.128091i
$936$	0	0
$937$	1.73897i	0.0568098i	0.999596	+	0.0284049i	$0.00904277\pi$
$937$	1.73897i	0.0568098i	−0.999596	+	0.0284049i	$0.990957\pi$
$938$	0	0
$939$	0	0
$940$	−87.3970	−2.85057
$941$	14.9134	0.486163	0.243081	−	0.970006i	$-0.421842\pi$
$941$	14.9134	0.486163	0.243081	+	0.970006i	$0.421842\pi$
$942$	0	0
$943$	16.3128i	0.531218i
$944$	20.2710	0.659763
$945$	0	0
$946$	24.9706	0.811863
$947$	22.9706i	0.746443i	0.927742	+	0.373221i	$0.121747\pi$
$947$	22.9706i	0.746443i	−0.927742	+	0.373221i	$0.878253\pi$
$948$	0	0
$949$	4.72792	0.153475
$950$	−104.634	−3.39477
$951$	0	0
$952$	0	0
$953$	− 26.3848i	− 0.854687i	−0.904089	−	0.427343i	$-0.859450\pi$
$953$	− 26.3848i	− 0.854687i	0.904089	−	0.427343i	$-0.140550\pi$
$954$	0	0
$955$	− 70.3687i	− 2.27708i
$956$	1.31371i	0.0424884i
$957$	0	0
$958$	− 20.2710i	− 0.654925i
$959$	0	0
$960$	0	0
$961$	−14.6569	−0.472802
$962$	21.0907	0.679992
$963$	0	0
$964$	− 34.3646i	− 1.10681i
$965$	−59.6536	−1.92032
$966$	0	0
$967$	43.1127	1.38641	0.693205	−	0.720740i	$-0.256198\pi$
$967$	43.1127	1.38641	0.693205	+	0.720740i	$0.256198\pi$
$968$	− 45.5269i	− 1.46329i
$969$	0	0
$970$	−11.4142	−0.366488
$971$	32.6256	1.04701	0.523503	−	0.852024i	$-0.324625\pi$
$971$	32.6256	1.04701	0.523503	+	0.852024i	$0.324625\pi$
$972$	0	0
$973$	0	0
$974$	22.8284i	0.731470i
$975$	0	0
$976$	− 24.4692i	− 0.783240i
$977$	39.1716i	1.25321i	0.779337	+	0.626605i	$0.215556\pi$
$977$	39.1716i	1.25321i	−0.779337	+	0.626605i	$0.784444\pi$
$978$	0	0
$979$	5.11747i	0.163555i
$980$	0	0
$981$	0	0
$982$	6.48528	0.206954
$983$	17.4721	0.557274	0.278637	−	0.960396i	$-0.410117\pi$
$983$	17.4721	0.557274	0.278637	+	0.960396i	$0.410117\pi$
$984$	0	0
$985$	54.8756i	1.74848i
$986$	16.3128	0.519506
$987$	0	0
$988$	87.3970	2.78047
$989$	− 24.9706i	− 0.794018i
$990$	0	0
$991$	59.3970	1.88681	0.943403	−	0.331647i	$-0.107604\pi$
$991$	59.3970	1.88681	0.943403	+	0.331647i	$0.107604\pi$
$992$	−10.7151	−0.340206
$993$	0	0
$994$	0	0
$995$	− 64.5685i	− 2.04696i
$996$	0	0
$997$	11.7750i	0.372918i	0.982463	+	0.186459i	$0.0597011\pi$
$997$	11.7750i	0.372918i	−0.982463	+	0.186459i	$0.940299\pi$
$998$	− 5.17157i	− 0.163703i
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	441.2.c.b.440.8	yes	8
3.2	odd	2	inner	441.2.c.b.440.1	✓	8
4.3	odd	2		7056.2.k.g.881.7		8
7.2	even	3		441.2.p.c.80.7		16
7.3	odd	6		441.2.p.c.215.2		16
7.4	even	3		441.2.p.c.215.1		16
7.5	odd	6		441.2.p.c.80.8		16
7.6	odd	2	inner	441.2.c.b.440.7	yes	8
12.11	even	2		7056.2.k.g.881.2		8
21.2	odd	6		441.2.p.c.80.2		16
21.5	even	6		441.2.p.c.80.1		16
21.11	odd	6		441.2.p.c.215.8		16
21.17	even	6		441.2.p.c.215.7		16
21.20	even	2	inner	441.2.c.b.440.2	yes	8
28.27	even	2		7056.2.k.g.881.1		8
84.83	odd	2		7056.2.k.g.881.8		8

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
441.2.c.b.440.1	✓	8	3.2	odd	2	inner
441.2.c.b.440.2	yes	8	21.20	even	2	inner
441.2.c.b.440.7	yes	8	7.6	odd	2	inner
441.2.c.b.440.8	yes	8	1.1	even	1	trivial
441.2.p.c.80.1		16	21.5	even	6
441.2.p.c.80.2		16	21.2	odd	6
441.2.p.c.80.7		16	7.2	even	3
441.2.p.c.80.8		16	7.5	odd	6
441.2.p.c.215.1		16	7.4	even	3
441.2.p.c.215.2		16	7.3	odd	6
441.2.p.c.215.7		16	21.17	even	6
441.2.p.c.215.8		16	21.11	odd	6
7056.2.k.g.881.1		8	28.27	even	2
7056.2.k.g.881.2		8	12.11	even	2
7056.2.k.g.881.7		8	4.3	odd	2
7056.2.k.g.881.8		8	84.83	odd	2

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:	\( N \)	\(=\)	\( 441 = 3^{2} \cdot 7^{2} \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	441.c (of order \(2\), degree \(1\), minimal)

\(f(q)\)	\(=\)	\(q+2.41421i q^{2} -3.82843 q^{4} +3.37849 q^{5} -4.41421i q^{8} +O(q^{10})\)	\(q+2.41421i q^{2} -3.82843 q^{4} +3.37849 q^{5} -4.41421i q^{8} +8.15640i q^{10} +0.828427i q^{11} +3.37849i q^{13} +3.00000 q^{16} +1.39942 q^{17} +6.75699i q^{19} -12.9343 q^{20} -2.00000 q^{22} +2.00000i q^{23} +6.41421 q^{25} -8.15640 q^{26} -4.82843i q^{29} -6.75699i q^{31} -1.58579i q^{32} +3.37849i q^{34} -2.58579 q^{37} -16.3128 q^{38} -14.9134i q^{40} +8.15640 q^{41} -12.4853 q^{43} -3.17157i q^{44} -4.82843 q^{46} +6.75699 q^{47} +15.4853i q^{50} -12.9343i q^{52} -7.07107i q^{53} +2.79884i q^{55} +11.6569 q^{58} +6.75699 q^{59} -8.15640i q^{61} +16.3128 q^{62} +9.82843 q^{64} +11.4142i q^{65} +8.48528 q^{67} -5.35757 q^{68} +4.82843i q^{71} -1.39942i q^{73} -6.24264i q^{74} -25.8686i q^{76} -9.65685 q^{79} +10.1355 q^{80} +19.6913i q^{82} -13.5140 q^{83} +4.72792 q^{85} -30.1421i q^{86} +3.65685 q^{88} +6.17733 q^{89} -7.65685i q^{92} +16.3128i q^{94} +22.8284i q^{95} +1.39942i q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 8 q - 8 q^{4}+O(q^{10}) \) 8 * q - 8 * q^4	\( 8 q - 8 q^{4} + 24 q^{16} - 16 q^{22} + 40 q^{25} - 32 q^{37} - 32 q^{43} - 16 q^{46} + 48 q^{58} + 56 q^{64} - 32 q^{79} - 64 q^{85} - 16 q^{88}+O(q^{100}) \) 8 * q - 8 * q^4 + 24 * q^16 - 16 * q^22 + 40 * q^25 - 32 * q^37 - 32 * q^43 - 16 * q^46 + 48 * q^58 + 56 * q^64 - 32 * q^79 - 64 * q^85 - 16 * q^88

Introduction

L-functions

Modular forms

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Database

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