LMFDB - Embedded newform 4410.2.a.by.1.2

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [4410,2,Mod(1,4410)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(4410, base_ring=CyclotomicField(2))

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

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

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

chi := DirichletCharacter("4410.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 4410 = 2 \cdot 3^{2} \cdot 5 \cdot 7^{2} $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	4410.a (trivial)

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:	yes
Analytic conductor:	$35.2140272914$
Analytic rank:	$0$
Dimension:	$2$
Coefficient field:	$\Q(\zeta_{8})^+$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{2} - 2 $ x^2 - 2
Coefficient ring:	$\Z[a_1, \ldots, a_{17}]$
Coefficient ring index:	$ 1 $
Twist minimal:	no (minimal twist has level 490)
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.2
Root			$1.41421$ of defining polynomial
Character	$\chi$	$=$	4410.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+1.00000 q^{2} +1.00000 q^{4} +1.00000 q^{5} +1.00000 q^{8} +O(q^{10})$	$q+1.00000 q^{2} +1.00000 q^{4} +1.00000 q^{5} +1.00000 q^{8} +1.00000 q^{10} +0.828427 q^{11} +4.82843 q^{13} +1.00000 q^{16} +2.58579 q^{17} -0.585786 q^{19} +1.00000 q^{20} +0.828427 q^{22} +1.17157 q^{23} +1.00000 q^{25} +4.82843 q^{26} +4.82843 q^{29} +2.82843 q^{31} +1.00000 q^{32} +2.58579 q^{34} -7.65685 q^{37} -0.585786 q^{38} +1.00000 q^{40} -3.07107 q^{41} -8.82843 q^{43} +0.828427 q^{44} +1.17157 q^{46} +5.17157 q^{47} +1.00000 q^{50} +4.82843 q^{52} -6.48528 q^{53} +0.828427 q^{55} +4.82843 q^{58} +8.58579 q^{59} -9.31371 q^{61} +2.82843 q^{62} +1.00000 q^{64} +4.82843 q^{65} +1.65685 q^{67} +2.58579 q^{68} +4.48528 q^{71} +9.41421 q^{73} -7.65685 q^{74} -0.585786 q^{76} -6.82843 q^{79} +1.00000 q^{80} -3.07107 q^{82} -2.24264 q^{83} +2.58579 q^{85} -8.82843 q^{86} +0.828427 q^{88} +12.7279 q^{89} +1.17157 q^{92} +5.17157 q^{94} -0.585786 q^{95} +7.75736 q^{97} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 2 q + 2 q^{2} + 2 q^{4} + 2 q^{5} + 2 q^{8}+O(q^{10}) $ 2 * q + 2 * q^2 + 2 * q^4 + 2 * q^5 + 2 * q^8	$ 2 q + 2 q^{2} + 2 q^{4} + 2 q^{5} + 2 q^{8} + 2 q^{10} - 4 q^{11} + 4 q^{13} + 2 q^{16} + 8 q^{17} - 4 q^{19} + 2 q^{20} - 4 q^{22} + 8 q^{23} + 2 q^{25} + 4 q^{26} + 4 q^{29} + 2 q^{32} + 8 q^{34} - 4 q^{37} - 4 q^{38} + 2 q^{40} + 8 q^{41} - 12 q^{43} - 4 q^{44} + 8 q^{46} + 16 q^{47} + 2 q^{50} + 4 q^{52} + 4 q^{53} - 4 q^{55} + 4 q^{58} + 20 q^{59} + 4 q^{61} + 2 q^{64} + 4 q^{65} - 8 q^{67} + 8 q^{68} - 8 q^{71} + 16 q^{73} - 4 q^{74} - 4 q^{76} - 8 q^{79} + 2 q^{80} + 8 q^{82} + 4 q^{83} + 8 q^{85} - 12 q^{86} - 4 q^{88} + 8 q^{92} + 16 q^{94} - 4 q^{95} + 24 q^{97}+O(q^{100}) $ 2 * q + 2 * q^2 + 2 * q^4 + 2 * q^5 + 2 * q^8 + 2 * q^10 - 4 * q^11 + 4 * q^13 + 2 * q^16 + 8 * q^17 - 4 * q^19 + 2 * q^20 - 4 * q^22 + 8 * q^23 + 2 * q^25 + 4 * q^26 + 4 * q^29 + 2 * q^32 + 8 * q^34 - 4 * q^37 - 4 * q^38 + 2 * q^40 + 8 * q^41 - 12 * q^43 - 4 * q^44 + 8 * q^46 + 16 * q^47 + 2 * q^50 + 4 * q^52 + 4 * q^53 - 4 * q^55 + 4 * q^58 + 20 * q^59 + 4 * q^61 + 2 * q^64 + 4 * q^65 - 8 * q^67 + 8 * q^68 - 8 * q^71 + 16 * q^73 - 4 * q^74 - 4 * q^76 - 8 * q^79 + 2 * q^80 + 8 * q^82 + 4 * q^83 + 8 * q^85 - 12 * q^86 - 4 * q^88 + 8 * q^92 + 16 * q^94 - 4 * q^95 + 24 * q^97

Display 100 coefficients 10 coefficients

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 $

$2$	1.00000	0.707107
$2$	1.00000	0.707107
$3$	0	0
$3$	0	0
$4$	1.00000	0.500000
$5$	1.00000	0.447214
$5$	1.00000	0.447214
$6$	0	0
$7$	0	0
$7$	0	0
$8$	1.00000	0.353553
$9$	0	0
$10$	1.00000	0.316228
$11$	0.828427	0.249780	0.124890	−	0.992171i	$-0.460142\pi$
$11$	0.828427	0.249780	0.124890	+	0.992171i	$0.460142\pi$
$12$	0	0
$13$	4.82843	1.33916	0.669582	−	0.742738i	$-0.266473\pi$
$13$	4.82843	1.33916	0.669582	+	0.742738i	$0.266473\pi$
$14$	0	0
$15$	0	0
$16$	1.00000	0.250000
$17$	2.58579	0.627145	0.313573	−	0.949564i	$-0.398474\pi$
$17$	2.58579	0.627145	0.313573	+	0.949564i	$0.398474\pi$
$18$	0	0
$19$	−0.585786	−0.134389	−0.0671943	−	0.997740i	$-0.521405\pi$
$19$	−0.585786	−0.134389	−0.0671943	+	0.997740i	$0.521405\pi$
$20$	1.00000	0.223607
$21$	0	0
$22$	0.828427	0.176621
$23$	1.17157	0.244290	0.122145	−	0.992512i	$-0.461023\pi$
$23$	1.17157	0.244290	0.122145	+	0.992512i	$0.461023\pi$
$24$	0	0
$25$	1.00000	0.200000
$26$	4.82843	0.946932
$27$	0	0
$28$	0	0
$29$	4.82843	0.896616	0.448308	−	0.893879i	$-0.352027\pi$
$29$	4.82843	0.896616	0.448308	+	0.893879i	$0.352027\pi$
$30$	0	0
$31$	2.82843	0.508001	0.254000	−	0.967204i	$-0.418254\pi$
$31$	2.82843	0.508001	0.254000	+	0.967204i	$0.418254\pi$
$32$	1.00000	0.176777
$33$	0	0
$34$	2.58579	0.443459
$35$	0	0
$36$	0	0
$37$	−7.65685	−1.25878	−0.629390	−	0.777090i	$-0.716695\pi$
$37$	−7.65685	−1.25878	−0.629390	+	0.777090i	$0.716695\pi$
$38$	−0.585786	−0.0950271
$39$	0	0
$40$	1.00000	0.158114
$41$	−3.07107	−0.479620	−0.239810	−	0.970820i	$-0.577085\pi$
$41$	−3.07107	−0.479620	−0.239810	+	0.970820i	$0.577085\pi$
$42$	0	0
$43$	−8.82843	−1.34632	−0.673161	−	0.739496i	$-0.735064\pi$
$43$	−8.82843	−1.34632	−0.673161	+	0.739496i	$0.735064\pi$
$44$	0.828427	0.124890
$45$	0	0
$46$	1.17157	0.172739
$47$	5.17157	0.754351	0.377176	−	0.926142i	$-0.376895\pi$
$47$	5.17157	0.754351	0.377176	+	0.926142i	$0.376895\pi$
$48$	0	0
$49$	0	0
$50$	1.00000	0.141421
$51$	0	0
$52$	4.82843	0.669582
$53$	−6.48528	−0.890822	−0.445411	−	0.895326i	$-0.646942\pi$
$53$	−6.48528	−0.890822	−0.445411	+	0.895326i	$0.646942\pi$
$54$	0	0
$55$	0.828427	0.111705
$56$	0	0
$57$	0	0
$58$	4.82843	0.634004
$59$	8.58579	1.11777	0.558887	−	0.829244i	$-0.311229\pi$
$59$	8.58579	1.11777	0.558887	+	0.829244i	$0.311229\pi$
$60$	0	0
$61$	−9.31371	−1.19250	−0.596249	−	0.802799i	$-0.703343\pi$
$61$	−9.31371	−1.19250	−0.596249	+	0.802799i	$0.703343\pi$
$62$	2.82843	0.359211
$63$	0	0
$64$	1.00000	0.125000
$65$	4.82843	0.598893
$66$	0	0
$67$	1.65685	0.202417	0.101208	−	0.994865i	$-0.467729\pi$
$67$	1.65685	0.202417	0.101208	+	0.994865i	$0.467729\pi$
$68$	2.58579	0.313573
$69$	0	0
$70$	0	0
$71$	4.48528	0.532305	0.266152	−	0.963931i	$-0.414248\pi$
$71$	4.48528	0.532305	0.266152	+	0.963931i	$0.414248\pi$
$72$	0	0
$73$	9.41421	1.10185	0.550925	−	0.834555i	$-0.314275\pi$
$73$	9.41421	1.10185	0.550925	+	0.834555i	$0.314275\pi$
$74$	−7.65685	−0.890091
$75$	0	0
$76$	−0.585786	−0.0671943
$77$	0	0
$78$	0	0
$79$	−6.82843	−0.768258	−0.384129	−	0.923279i	$-0.625498\pi$
$79$	−6.82843	−0.768258	−0.384129	+	0.923279i	$0.625498\pi$
$80$	1.00000	0.111803
$81$	0	0
$82$	−3.07107	−0.339143
$83$	−2.24264	−0.246162	−0.123081	−	0.992397i	$-0.539277\pi$
$83$	−2.24264	−0.246162	−0.123081	+	0.992397i	$0.539277\pi$
$84$	0	0
$85$	2.58579	0.280468
$86$	−8.82843	−0.951994
$87$	0	0
$88$	0.828427	0.0883106
$89$	12.7279	1.34916	0.674579	−	0.738203i	$-0.264325\pi$
$89$	12.7279	1.34916	0.674579	+	0.738203i	$0.264325\pi$
$90$	0	0
$91$	0	0
$92$	1.17157	0.122145
$93$	0	0
$94$	5.17157	0.533407
$95$	−0.585786	−0.0601004
$96$	0	0
$97$	7.75736	0.787641	0.393820	−	0.919187i	$-0.371153\pi$
$97$	7.75736	0.787641	0.393820	+	0.919187i	$0.371153\pi$
$98$	0	0
$99$	0	0
$100$	1.00000	0.100000
$101$	13.3137	1.32476	0.662382	−	0.749166i	$-0.269546\pi$
$101$	13.3137	1.32476	0.662382	+	0.749166i	$0.269546\pi$
$102$	0	0
$103$	14.8284	1.46109	0.730544	−	0.682865i	$-0.239266\pi$
$103$	14.8284	1.46109	0.730544	+	0.682865i	$0.239266\pi$
$104$	4.82843	0.473466
$105$	0	0
$106$	−6.48528	−0.629906
$107$	−9.65685	−0.933563	−0.466782	−	0.884373i	$-0.654587\pi$
$107$	−9.65685	−0.933563	−0.466782	+	0.884373i	$0.654587\pi$
$108$	0	0
$109$	2.48528	0.238047	0.119023	−	0.992891i	$-0.462024\pi$
$109$	2.48528	0.238047	0.119023	+	0.992891i	$0.462024\pi$
$110$	0.828427	0.0789874
$111$	0	0
$112$	0	0
$113$	15.3137	1.44059	0.720296	−	0.693667i	$-0.244006\pi$
$113$	15.3137	1.44059	0.720296	+	0.693667i	$0.244006\pi$
$114$	0	0
$115$	1.17157	0.109250
$116$	4.82843	0.448308
$117$	0	0
$118$	8.58579	0.790386
$119$	0	0
$120$	0	0
$121$	−10.3137	−0.937610
$122$	−9.31371	−0.843224
$123$	0	0
$124$	2.82843	0.254000
$125$	1.00000	0.0894427
$126$	0	0
$127$	2.82843	0.250982	0.125491	−	0.992095i	$-0.459949\pi$
$127$	2.82843	0.250982	0.125491	+	0.992095i	$0.459949\pi$
$128$	1.00000	0.0883883
$129$	0	0
$130$	4.82843	0.423481
$131$	−6.24264	−0.545422	−0.272711	−	0.962096i	$-0.587920\pi$
$131$	−6.24264	−0.545422	−0.272711	+	0.962096i	$0.587920\pi$
$132$	0	0
$133$	0	0
$134$	1.65685	0.143130
$135$	0	0
$136$	2.58579	0.221729
$137$	16.0000	1.36697	0.683486	−	0.729964i	$-0.260463\pi$
$137$	16.0000	1.36697	0.683486	+	0.729964i	$0.260463\pi$
$138$	0	0
$139$	−19.8995	−1.68785	−0.843927	−	0.536459i	$-0.819762\pi$
$139$	−19.8995	−1.68785	−0.843927	+	0.536459i	$0.819762\pi$
$140$	0	0
$141$	0	0
$142$	4.48528	0.376396
$143$	4.00000	0.334497
$144$	0	0
$145$	4.82843	0.400979
$146$	9.41421	0.779126
$147$	0	0
$148$	−7.65685	−0.629390
$149$	6.00000	0.491539	0.245770	−	0.969328i	$-0.420959\pi$
$149$	6.00000	0.491539	0.245770	+	0.969328i	$0.420959\pi$
$150$	0	0
$151$	−11.3137	−0.920697	−0.460348	−	0.887738i	$-0.652275\pi$
$151$	−11.3137	−0.920697	−0.460348	+	0.887738i	$0.652275\pi$
$152$	−0.585786	−0.0475136
$153$	0	0
$154$	0	0
$155$	2.82843	0.227185
$156$	0	0
$157$	−6.48528	−0.517582	−0.258791	−	0.965933i	$-0.583324\pi$
$157$	−6.48528	−0.517582	−0.258791	+	0.965933i	$0.583324\pi$
$158$	−6.82843	−0.543240
$159$	0	0
$160$	1.00000	0.0790569
$161$	0	0
$162$	0	0
$163$	−20.1421	−1.57765	−0.788827	−	0.614615i	$-0.789311\pi$
$163$	−20.1421	−1.57765	−0.788827	+	0.614615i	$0.789311\pi$
$164$	−3.07107	−0.239810
$165$	0	0
$166$	−2.24264	−0.174063
$167$	15.7990	1.22256	0.611281	−	0.791413i	$-0.290654\pi$
$167$	15.7990	1.22256	0.611281	+	0.791413i	$0.290654\pi$
$168$	0	0
$169$	10.3137	0.793362
$170$	2.58579	0.198321
$171$	0	0
$172$	−8.82843	−0.673161
$173$	−8.82843	−0.671213	−0.335606	−	0.942002i	$-0.608941\pi$
$173$	−8.82843	−0.671213	−0.335606	+	0.942002i	$0.608941\pi$
$174$	0	0
$175$	0	0
$176$	0.828427	0.0624450
$177$	0	0
$178$	12.7279	0.953998
$179$	−4.00000	−0.298974	−0.149487	−	0.988764i	$-0.547762\pi$
$179$	−4.00000	−0.298974	−0.149487	+	0.988764i	$0.547762\pi$
$180$	0	0
$181$	2.48528	0.184730	0.0923648	−	0.995725i	$-0.470557\pi$
$181$	2.48528	0.184730	0.0923648	+	0.995725i	$0.470557\pi$
$182$	0	0
$183$	0	0
$184$	1.17157	0.0863695
$185$	−7.65685	−0.562943
$186$	0	0
$187$	2.14214	0.156648
$188$	5.17157	0.377176
$189$	0	0
$190$	−0.585786	−0.0424974
$191$	−10.1421	−0.733859	−0.366930	−	0.930249i	$-0.619591\pi$
$191$	−10.1421	−0.733859	−0.366930	+	0.930249i	$0.619591\pi$
$192$	0	0
$193$	5.65685	0.407189	0.203595	−	0.979055i	$-0.434738\pi$
$193$	5.65685	0.407189	0.203595	+	0.979055i	$0.434738\pi$
$194$	7.75736	0.556946
$195$	0	0
$196$	0	0
$197$	25.7990	1.83810	0.919051	−	0.394139i	$-0.128957\pi$
$197$	25.7990	1.83810	0.919051	+	0.394139i	$0.128957\pi$
$198$	0	0
$199$	−16.4853	−1.16861	−0.584305	−	0.811534i	$-0.698633\pi$
$199$	−16.4853	−1.16861	−0.584305	+	0.811534i	$0.698633\pi$
$200$	1.00000	0.0707107
$201$	0	0
$202$	13.3137	0.936749
$203$	0	0
$204$	0	0
$205$	−3.07107	−0.214493
$206$	14.8284	1.03315
$207$	0	0
$208$	4.82843	0.334791
$209$	−0.485281	−0.0335676
$210$	0	0
$211$	18.6274	1.28236	0.641182	−	0.767389i	$-0.278444\pi$
$211$	18.6274	1.28236	0.641182	+	0.767389i	$0.278444\pi$
$212$	−6.48528	−0.445411
$213$	0	0
$214$	−9.65685	−0.660129
$215$	−8.82843	−0.602094
$216$	0	0
$217$	0	0
$218$	2.48528	0.168324
$219$	0	0
$220$	0.828427	0.0558525
$221$	12.4853	0.839851
$222$	0	0
$223$	7.31371	0.489762	0.244881	−	0.969553i	$-0.421251\pi$
$223$	7.31371	0.489762	0.244881	+	0.969553i	$0.421251\pi$
$224$	0	0
$225$	0	0
$226$	15.3137	1.01865
$227$	18.2426	1.21081	0.605403	−	0.795919i	$-0.293012\pi$
$227$	18.2426	1.21081	0.605403	+	0.795919i	$0.293012\pi$
$228$	0	0
$229$	16.1421	1.06670	0.533351	−	0.845894i	$-0.320932\pi$
$229$	16.1421	1.06670	0.533351	+	0.845894i	$0.320932\pi$
$230$	1.17157	0.0772512
$231$	0	0
$232$	4.82843	0.317002
$233$	−23.3137	−1.52733	−0.763666	−	0.645612i	$-0.776603\pi$
$233$	−23.3137	−1.52733	−0.763666	+	0.645612i	$0.776603\pi$
$234$	0	0
$235$	5.17157	0.337356
$236$	8.58579	0.558887
$237$	0	0
$238$	0	0
$239$	−1.65685	−0.107173	−0.0535865	−	0.998563i	$-0.517065\pi$
$239$	−1.65685	−0.107173	−0.0535865	+	0.998563i	$0.517065\pi$
$240$	0	0
$241$	13.4142	0.864085	0.432043	−	0.901853i	$-0.357793\pi$
$241$	13.4142	0.864085	0.432043	+	0.901853i	$0.357793\pi$
$242$	−10.3137	−0.662990
$243$	0	0
$244$	−9.31371	−0.596249
$245$	0	0
$246$	0	0
$247$	−2.82843	−0.179969
$248$	2.82843	0.179605
$249$	0	0
$250$	1.00000	0.0632456
$251$	0.585786	0.0369745	0.0184873	−	0.999829i	$-0.494115\pi$
$251$	0.585786	0.0369745	0.0184873	+	0.999829i	$0.494115\pi$
$252$	0	0
$253$	0.970563	0.0610188
$254$	2.82843	0.177471
$255$	0	0
$256$	1.00000	0.0625000
$257$	−9.89949	−0.617514	−0.308757	−	0.951141i	$-0.599913\pi$
$257$	−9.89949	−0.617514	−0.308757	+	0.951141i	$0.599913\pi$
$258$	0	0
$259$	0	0
$260$	4.82843	0.299446
$261$	0	0
$262$	−6.24264	−0.385672
$263$	−28.0000	−1.72655	−0.863277	−	0.504730i	$-0.831592\pi$
$263$	−28.0000	−1.72655	−0.863277	+	0.504730i	$0.831592\pi$
$264$	0	0
$265$	−6.48528	−0.398388
$266$	0	0
$267$	0	0
$268$	1.65685	0.101208
$269$	18.4853	1.12707	0.563534	−	0.826093i	$-0.309441\pi$
$269$	18.4853	1.12707	0.563534	+	0.826093i	$0.309441\pi$
$270$	0	0
$271$	−12.0000	−0.728948	−0.364474	−	0.931214i	$-0.618751\pi$
$271$	−12.0000	−0.728948	−0.364474	+	0.931214i	$0.618751\pi$
$272$	2.58579	0.156786
$273$	0	0
$274$	16.0000	0.966595
$275$	0.828427	0.0499560
$276$	0	0
$277$	−8.14214	−0.489214	−0.244607	−	0.969622i	$-0.578659\pi$
$277$	−8.14214	−0.489214	−0.244607	+	0.969622i	$0.578659\pi$
$278$	−19.8995	−1.19349
$279$	0	0
$280$	0	0
$281$	−8.00000	−0.477240	−0.238620	−	0.971113i	$-0.576695\pi$
$281$	−8.00000	−0.477240	−0.238620	+	0.971113i	$0.576695\pi$
$282$	0	0
$283$	2.24264	0.133311	0.0666556	−	0.997776i	$-0.478767\pi$
$283$	2.24264	0.133311	0.0666556	+	0.997776i	$0.478767\pi$
$284$	4.48528	0.266152
$285$	0	0
$286$	4.00000	0.236525
$287$	0	0
$288$	0	0
$289$	−10.3137	−0.606689
$290$	4.82843	0.283535
$291$	0	0
$292$	9.41421	0.550925
$293$	−8.34315	−0.487412	−0.243706	−	0.969849i	$-0.578363\pi$
$293$	−8.34315	−0.487412	−0.243706	+	0.969849i	$0.578363\pi$
$294$	0	0
$295$	8.58579	0.499884
$296$	−7.65685	−0.445046
$297$	0	0
$298$	6.00000	0.347571
$299$	5.65685	0.327144
$300$	0	0
$301$	0	0
$302$	−11.3137	−0.651031
$303$	0	0
$304$	−0.585786	−0.0335972
$305$	−9.31371	−0.533301
$306$	0	0
$307$	−14.9289	−0.852039	−0.426020	−	0.904714i	$-0.640085\pi$
$307$	−14.9289	−0.852039	−0.426020	+	0.904714i	$0.640085\pi$
$308$	0	0
$309$	0	0
$310$	2.82843	0.160644
$311$	4.00000	0.226819	0.113410	−	0.993548i	$-0.463823\pi$
$311$	4.00000	0.226819	0.113410	+	0.993548i	$0.463823\pi$
$312$	0	0
$313$	−14.3848	−0.813076	−0.406538	−	0.913634i	$-0.633264\pi$
$313$	−14.3848	−0.813076	−0.406538	+	0.913634i	$0.633264\pi$
$314$	−6.48528	−0.365986
$315$	0	0
$316$	−6.82843	−0.384129
$317$	−10.4853	−0.588912	−0.294456	−	0.955665i	$-0.595138\pi$
$317$	−10.4853	−0.588912	−0.294456	+	0.955665i	$0.595138\pi$
$318$	0	0
$319$	4.00000	0.223957
$320$	1.00000	0.0559017
$321$	0	0
$322$	0	0
$323$	−1.51472	−0.0842812
$324$	0	0
$325$	4.82843	0.267833
$326$	−20.1421	−1.11557
$327$	0	0
$328$	−3.07107	−0.169571
$329$	0	0
$330$	0	0
$331$	33.7990	1.85776	0.928880	−	0.370380i	$-0.120773\pi$
$331$	33.7990	1.85776	0.928880	+	0.370380i	$0.120773\pi$
$332$	−2.24264	−0.123081
$333$	0	0
$334$	15.7990	0.864482
$335$	1.65685	0.0905236
$336$	0	0
$337$	6.00000	0.326841	0.163420	−	0.986557i	$-0.447747\pi$
$337$	6.00000	0.326841	0.163420	+	0.986557i	$0.447747\pi$
$338$	10.3137	0.560992
$339$	0	0
$340$	2.58579	0.140234
$341$	2.34315	0.126888
$342$	0	0
$343$	0	0
$344$	−8.82843	−0.475997
$345$	0	0
$346$	−8.82843	−0.474619
$347$	3.17157	0.170259	0.0851295	−	0.996370i	$-0.472870\pi$
$347$	3.17157	0.170259	0.0851295	+	0.996370i	$0.472870\pi$
$348$	0	0
$349$	−2.48528	−0.133034	−0.0665170	−	0.997785i	$-0.521189\pi$
$349$	−2.48528	−0.133034	−0.0665170	+	0.997785i	$0.521189\pi$
$350$	0	0
$351$	0	0
$352$	0.828427	0.0441553
$353$	−2.38478	−0.126929	−0.0634644	−	0.997984i	$-0.520215\pi$
$353$	−2.38478	−0.126929	−0.0634644	+	0.997984i	$0.520215\pi$
$354$	0	0
$355$	4.48528	0.238054
$356$	12.7279	0.674579
$357$	0	0
$358$	−4.00000	−0.211407
$359$	−28.2843	−1.49279	−0.746393	−	0.665505i	$-0.768216\pi$
$359$	−28.2843	−1.49279	−0.746393	+	0.665505i	$0.768216\pi$
$360$	0	0
$361$	−18.6569	−0.981940
$362$	2.48528	0.130623
$363$	0	0
$364$	0	0
$365$	9.41421	0.492762
$366$	0	0
$367$	−24.9706	−1.30345	−0.651726	−	0.758454i	$-0.725955\pi$
$367$	−24.9706	−1.30345	−0.651726	+	0.758454i	$0.725955\pi$
$368$	1.17157	0.0610725
$369$	0	0
$370$	−7.65685	−0.398061
$371$	0	0
$372$	0	0
$373$	−30.4853	−1.57847	−0.789234	−	0.614093i	$-0.789522\pi$
$373$	−30.4853	−1.57847	−0.789234	+	0.614093i	$0.789522\pi$
$374$	2.14214	0.110767
$375$	0	0
$376$	5.17157	0.266704
$377$	23.3137	1.20072
$378$	0	0
$379$	34.4853	1.77139	0.885695	−	0.464268i	$-0.153682\pi$
$379$	34.4853	1.77139	0.885695	+	0.464268i	$0.153682\pi$
$380$	−0.585786	−0.0300502
$381$	0	0
$382$	−10.1421	−0.518917
$383$	−32.4853	−1.65992	−0.829960	−	0.557823i	$-0.811637\pi$
$383$	−32.4853	−1.65992	−0.829960	+	0.557823i	$0.811637\pi$
$384$	0	0
$385$	0	0
$386$	5.65685	0.287926
$387$	0	0
$388$	7.75736	0.393820
$389$	−28.1421	−1.42686	−0.713431	−	0.700725i	$-0.752860\pi$
$389$	−28.1421	−1.42686	−0.713431	+	0.700725i	$0.752860\pi$
$390$	0	0
$391$	3.02944	0.153205
$392$	0	0
$393$	0	0
$394$	25.7990	1.29973
$395$	−6.82843	−0.343575
$396$	0	0
$397$	−33.7990	−1.69632	−0.848161	−	0.529738i	$-0.822290\pi$
$397$	−33.7990	−1.69632	−0.848161	+	0.529738i	$0.822290\pi$
$398$	−16.4853	−0.826332
$399$	0	0
$400$	1.00000	0.0500000
$401$	6.00000	0.299626	0.149813	−	0.988714i	$-0.452133\pi$
$401$	6.00000	0.299626	0.149813	+	0.988714i	$0.452133\pi$
$402$	0	0
$403$	13.6569	0.680296
$404$	13.3137	0.662382
$405$	0	0
$406$	0	0
$407$	−6.34315	−0.314418
$408$	0	0
$409$	10.5858	0.523433	0.261717	−	0.965145i	$-0.415711\pi$
$409$	10.5858	0.523433	0.261717	+	0.965145i	$0.415711\pi$
$410$	−3.07107	−0.151669
$411$	0	0
$412$	14.8284	0.730544
$413$	0	0
$414$	0	0
$415$	−2.24264	−0.110087
$416$	4.82843	0.236733
$417$	0	0
$418$	−0.485281	−0.0237359
$419$	−20.8701	−1.01957	−0.509785	−	0.860302i	$-0.670275\pi$
$419$	−20.8701	−1.01957	−0.509785	+	0.860302i	$0.670275\pi$
$420$	0	0
$421$	17.3137	0.843819	0.421909	−	0.906638i	$-0.361360\pi$
$421$	17.3137	0.843819	0.421909	+	0.906638i	$0.361360\pi$
$422$	18.6274	0.906768
$423$	0	0
$424$	−6.48528	−0.314953
$425$	2.58579	0.125429
$426$	0	0
$427$	0	0
$428$	−9.65685	−0.466782
$429$	0	0
$430$	−8.82843	−0.425745
$431$	22.3431	1.07623	0.538116	−	0.842871i	$-0.319136\pi$
$431$	22.3431	1.07623	0.538116	+	0.842871i	$0.319136\pi$
$432$	0	0
$433$	−10.5858	−0.508720	−0.254360	−	0.967110i	$-0.581865\pi$
$433$	−10.5858	−0.508720	−0.254360	+	0.967110i	$0.581865\pi$
$434$	0	0
$435$	0	0
$436$	2.48528	0.119023
$437$	−0.686292	−0.0328298
$438$	0	0
$439$	24.9706	1.19178	0.595890	−	0.803066i	$-0.296799\pi$
$439$	24.9706	1.19178	0.595890	+	0.803066i	$0.296799\pi$
$440$	0.828427	0.0394937
$441$	0	0
$442$	12.4853	0.593864
$443$	−3.02944	−0.143933	−0.0719665	−	0.997407i	$-0.522927\pi$
$443$	−3.02944	−0.143933	−0.0719665	+	0.997407i	$0.522927\pi$
$444$	0	0
$445$	12.7279	0.603361
$446$	7.31371	0.346314
$447$	0	0
$448$	0	0
$449$	16.6274	0.784696	0.392348	−	0.919817i	$-0.371663\pi$
$449$	16.6274	0.784696	0.392348	+	0.919817i	$0.371663\pi$
$450$	0	0
$451$	−2.54416	−0.119800
$452$	15.3137	0.720296
$453$	0	0
$454$	18.2426	0.856170
$455$	0	0
$456$	0	0
$457$	21.6569	1.01306	0.506532	−	0.862221i	$-0.330927\pi$
$457$	21.6569	1.01306	0.506532	+	0.862221i	$0.330927\pi$
$458$	16.1421	0.754272
$459$	0	0
$460$	1.17157	0.0546249
$461$	−12.8284	−0.597479	−0.298740	−	0.954335i	$-0.596566\pi$
$461$	−12.8284	−0.597479	−0.298740	+	0.954335i	$0.596566\pi$
$462$	0	0
$463$	−16.9706	−0.788689	−0.394344	−	0.918963i	$-0.629028\pi$
$463$	−16.9706	−0.788689	−0.394344	+	0.918963i	$0.629028\pi$
$464$	4.82843	0.224154
$465$	0	0
$466$	−23.3137	−1.07999
$467$	−15.8995	−0.735741	−0.367870	−	0.929877i	$-0.619913\pi$
$467$	−15.8995	−0.735741	−0.367870	+	0.929877i	$0.619913\pi$
$468$	0	0
$469$	0	0
$470$	5.17157	0.238547
$471$	0	0
$472$	8.58579	0.395193
$473$	−7.31371	−0.336285
$474$	0	0
$475$	−0.585786	−0.0268777
$476$	0	0
$477$	0	0
$478$	−1.65685	−0.0757827
$479$	−17.1716	−0.784589	−0.392295	−	0.919840i	$-0.628319\pi$
$479$	−17.1716	−0.784589	−0.392295	+	0.919840i	$0.628319\pi$
$480$	0	0
$481$	−36.9706	−1.68571
$482$	13.4142	0.611001
$483$	0	0
$484$	−10.3137	−0.468805
$485$	7.75736	0.352244
$486$	0	0
$487$	31.7990	1.44095	0.720475	−	0.693481i	$-0.243924\pi$
$487$	31.7990	1.44095	0.720475	+	0.693481i	$0.243924\pi$
$488$	−9.31371	−0.421612
$489$	0	0
$490$	0	0
$491$	−32.2843	−1.45697	−0.728484	−	0.685062i	$-0.759775\pi$
$491$	−32.2843	−1.45697	−0.728484	+	0.685062i	$0.759775\pi$
$492$	0	0
$493$	12.4853	0.562309
$494$	−2.82843	−0.127257
$495$	0	0
$496$	2.82843	0.127000
$497$	0	0
$498$	0	0
$499$	30.3431	1.35835	0.679173	−	0.733978i	$-0.262339\pi$
$499$	30.3431	1.35835	0.679173	+	0.733978i	$0.262339\pi$
$500$	1.00000	0.0447214
$501$	0	0
$502$	0.585786	0.0261449
$503$	17.6569	0.787280	0.393640	−	0.919265i	$-0.371216\pi$
$503$	17.6569	0.787280	0.393640	+	0.919265i	$0.371216\pi$
$504$	0	0
$505$	13.3137	0.592452
$506$	0.970563	0.0431468
$507$	0	0
$508$	2.82843	0.125491
$509$	5.79899	0.257036	0.128518	−	0.991707i	$-0.458978\pi$
$509$	5.79899	0.257036	0.128518	+	0.991707i	$0.458978\pi$
$510$	0	0
$511$	0	0
$512$	1.00000	0.0441942
$513$	0	0
$514$	−9.89949	−0.436648
$515$	14.8284	0.653419
$516$	0	0
$517$	4.28427	0.188422
$518$	0	0
$519$	0	0
$520$	4.82843	0.211741
$521$	19.0711	0.835519	0.417759	−	0.908558i	$-0.362816\pi$
$521$	19.0711	0.835519	0.417759	+	0.908558i	$0.362816\pi$
$522$	0	0
$523$	−23.8995	−1.04505	−0.522526	−	0.852623i	$-0.675010\pi$
$523$	−23.8995	−1.04505	−0.522526	+	0.852623i	$0.675010\pi$
$524$	−6.24264	−0.272711
$525$	0	0
$526$	−28.0000	−1.22086
$527$	7.31371	0.318590
$528$	0	0
$529$	−21.6274	−0.940322
$530$	−6.48528	−0.281703
$531$	0	0
$532$	0	0
$533$	−14.8284	−0.642290
$534$	0	0
$535$	−9.65685	−0.417502
$536$	1.65685	0.0715652
$537$	0	0
$538$	18.4853	0.796957
$539$	0	0
$540$	0	0
$541$	−14.9706	−0.643635	−0.321817	−	0.946802i	$-0.604294\pi$
$541$	−14.9706	−0.643635	−0.321817	+	0.946802i	$0.604294\pi$
$542$	−12.0000	−0.515444
$543$	0	0
$544$	2.58579	0.110865
$545$	2.48528	0.106458
$546$	0	0
$547$	−10.4853	−0.448318	−0.224159	−	0.974553i	$-0.571964\pi$
$547$	−10.4853	−0.448318	−0.224159	+	0.974553i	$0.571964\pi$
$548$	16.0000	0.683486
$549$	0	0
$550$	0.828427	0.0353243
$551$	−2.82843	−0.120495
$552$	0	0
$553$	0	0
$554$	−8.14214	−0.345926
$555$	0	0
$556$	−19.8995	−0.843927
$557$	−15.1716	−0.642840	−0.321420	−	0.946937i	$-0.604160\pi$
$557$	−15.1716	−0.642840	−0.321420	+	0.946937i	$0.604160\pi$
$558$	0	0
$559$	−42.6274	−1.80295
$560$	0	0
$561$	0	0
$562$	−8.00000	−0.337460
$563$	−36.5858	−1.54191	−0.770954	−	0.636891i	$-0.780220\pi$
$563$	−36.5858	−1.54191	−0.770954	+	0.636891i	$0.780220\pi$
$564$	0	0
$565$	15.3137	0.644253
$566$	2.24264	0.0942652
$567$	0	0
$568$	4.48528	0.188198
$569$	29.3137	1.22889	0.614447	−	0.788958i	$-0.289379\pi$
$569$	29.3137	1.22889	0.614447	+	0.788958i	$0.289379\pi$
$570$	0	0
$571$	−2.20101	−0.0921094	−0.0460547	−	0.998939i	$-0.514665\pi$
$571$	−2.20101	−0.0921094	−0.0460547	+	0.998939i	$0.514665\pi$
$572$	4.00000	0.167248
$573$	0	0
$574$	0	0
$575$	1.17157	0.0488580
$576$	0	0
$577$	−6.10051	−0.253967	−0.126984	−	0.991905i	$-0.540530\pi$
$577$	−6.10051	−0.253967	−0.126984	+	0.991905i	$0.540530\pi$
$578$	−10.3137	−0.428994
$579$	0	0
$580$	4.82843	0.200490
$581$	0	0
$582$	0	0
$583$	−5.37258	−0.222510
$584$	9.41421	0.389563
$585$	0	0
$586$	−8.34315	−0.344652
$587$	17.0711	0.704598	0.352299	−	0.935887i	$-0.385400\pi$
$587$	17.0711	0.704598	0.352299	+	0.935887i	$0.385400\pi$
$588$	0	0
$589$	−1.65685	−0.0682695
$590$	8.58579	0.353471
$591$	0	0
$592$	−7.65685	−0.314695
$593$	3.27208	0.134368	0.0671841	−	0.997741i	$-0.478599\pi$
$593$	3.27208	0.134368	0.0671841	+	0.997741i	$0.478599\pi$
$594$	0	0
$595$	0	0
$596$	6.00000	0.245770
$597$	0	0
$598$	5.65685	0.231326
$599$	10.8284	0.442438	0.221219	−	0.975224i	$-0.428997\pi$
$599$	10.8284	0.442438	0.221219	+	0.975224i	$0.428997\pi$
$600$	0	0
$601$	−6.58579	−0.268640	−0.134320	−	0.990938i	$-0.542885\pi$
$601$	−6.58579	−0.268640	−0.134320	+	0.990938i	$0.542885\pi$
$602$	0	0
$603$	0	0
$604$	−11.3137	−0.460348
$605$	−10.3137	−0.419312
$606$	0	0
$607$	−16.2843	−0.660958	−0.330479	−	0.943813i	$-0.607210\pi$
$607$	−16.2843	−0.660958	−0.330479	+	0.943813i	$0.607210\pi$
$608$	−0.585786	−0.0237568
$609$	0	0
$610$	−9.31371	−0.377101
$611$	24.9706	1.01020
$612$	0	0
$613$	−12.3431	−0.498535	−0.249267	−	0.968435i	$-0.580190\pi$
$613$	−12.3431	−0.498535	−0.249267	+	0.968435i	$0.580190\pi$
$614$	−14.9289	−0.602483
$615$	0	0
$616$	0	0
$617$	33.3137	1.34116	0.670580	−	0.741837i	$-0.266045\pi$
$617$	33.3137	1.34116	0.670580	+	0.741837i	$0.266045\pi$
$618$	0	0
$619$	−29.0711	−1.16846	−0.584232	−	0.811586i	$-0.698604\pi$
$619$	−29.0711	−1.16846	−0.584232	+	0.811586i	$0.698604\pi$
$620$	2.82843	0.113592
$621$	0	0
$622$	4.00000	0.160385
$623$	0	0
$624$	0	0
$625$	1.00000	0.0400000
$626$	−14.3848	−0.574931
$627$	0	0
$628$	−6.48528	−0.258791
$629$	−19.7990	−0.789437
$630$	0	0
$631$	−12.4853	−0.497031	−0.248516	−	0.968628i	$-0.579943\pi$
$631$	−12.4853	−0.497031	−0.248516	+	0.968628i	$0.579943\pi$
$632$	−6.82843	−0.271620
$633$	0	0
$634$	−10.4853	−0.416424
$635$	2.82843	0.112243
$636$	0	0
$637$	0	0
$638$	4.00000	0.158362
$639$	0	0
$640$	1.00000	0.0395285
$641$	24.6274	0.972724	0.486362	−	0.873757i	$-0.338324\pi$
$641$	24.6274	0.972724	0.486362	+	0.873757i	$0.338324\pi$
$642$	0	0
$643$	−4.78680	−0.188773	−0.0943864	−	0.995536i	$-0.530089\pi$
$643$	−4.78680	−0.188773	−0.0943864	+	0.995536i	$0.530089\pi$
$644$	0	0
$645$	0	0
$646$	−1.51472	−0.0595958
$647$	−23.1127	−0.908654	−0.454327	−	0.890835i	$-0.650120\pi$
$647$	−23.1127	−0.908654	−0.454327	+	0.890835i	$0.650120\pi$
$648$	0	0
$649$	7.11270	0.279198
$650$	4.82843	0.189386
$651$	0	0
$652$	−20.1421	−0.788827
$653$	−4.34315	−0.169960	−0.0849802	−	0.996383i	$-0.527083\pi$
$653$	−4.34315	−0.169960	−0.0849802	+	0.996383i	$0.527083\pi$
$654$	0	0
$655$	−6.24264	−0.243920
$656$	−3.07107	−0.119905
$657$	0	0
$658$	0	0
$659$	27.1716	1.05845	0.529227	−	0.848480i	$-0.322482\pi$
$659$	27.1716	1.05845	0.529227	+	0.848480i	$0.322482\pi$
$660$	0	0
$661$	38.2843	1.48909	0.744543	−	0.667575i	$-0.232668\pi$
$661$	38.2843	1.48909	0.744543	+	0.667575i	$0.232668\pi$
$662$	33.7990	1.31364
$663$	0	0
$664$	−2.24264	−0.0870313
$665$	0	0
$666$	0	0
$667$	5.65685	0.219034
$668$	15.7990	0.611281
$669$	0	0
$670$	1.65685	0.0640099
$671$	−7.71573	−0.297862
$672$	0	0
$673$	−48.0000	−1.85026	−0.925132	−	0.379646i	$-0.876046\pi$
$673$	−48.0000	−1.85026	−0.925132	+	0.379646i	$0.876046\pi$
$674$	6.00000	0.231111
$675$	0	0
$676$	10.3137	0.396681
$677$	39.4558	1.51641	0.758206	−	0.652015i	$-0.226076\pi$
$677$	39.4558	1.51641	0.758206	+	0.652015i	$0.226076\pi$
$678$	0	0
$679$	0	0
$680$	2.58579	0.0991604
$681$	0	0
$682$	2.34315	0.0897237
$683$	−33.6569	−1.28784	−0.643922	−	0.765091i	$-0.722694\pi$
$683$	−33.6569	−1.28784	−0.643922	+	0.765091i	$0.722694\pi$
$684$	0	0
$685$	16.0000	0.611329
$686$	0	0
$687$	0	0
$688$	−8.82843	−0.336581
$689$	−31.3137	−1.19296
$690$	0	0
$691$	−1.75736	−0.0668531	−0.0334265	−	0.999441i	$-0.510642\pi$
$691$	−1.75736	−0.0668531	−0.0334265	+	0.999441i	$0.510642\pi$
$692$	−8.82843	−0.335606
$693$	0	0
$694$	3.17157	0.120391
$695$	−19.8995	−0.754831
$696$	0	0
$697$	−7.94113	−0.300792
$698$	−2.48528	−0.0940693
$699$	0	0
$700$	0	0
$701$	−2.48528	−0.0938678	−0.0469339	−	0.998898i	$-0.514945\pi$
$701$	−2.48528	−0.0938678	−0.0469339	+	0.998898i	$0.514945\pi$
$702$	0	0
$703$	4.48528	0.169166
$704$	0.828427	0.0312225
$705$	0	0
$706$	−2.38478	−0.0897522
$707$	0	0
$708$	0	0
$709$	45.1127	1.69424	0.847121	−	0.531399i	$-0.178334\pi$
$709$	45.1127	1.69424	0.847121	+	0.531399i	$0.178334\pi$
$710$	4.48528	0.168330
$711$	0	0
$712$	12.7279	0.476999
$713$	3.31371	0.124099
$714$	0	0
$715$	4.00000	0.149592
$716$	−4.00000	−0.149487
$717$	0	0
$718$	−28.2843	−1.05556
$719$	41.4558	1.54604	0.773021	−	0.634380i	$-0.218745\pi$
$719$	41.4558	1.54604	0.773021	+	0.634380i	$0.218745\pi$
$720$	0	0
$721$	0	0
$722$	−18.6569	−0.694336
$723$	0	0
$724$	2.48528	0.0923648
$725$	4.82843	0.179323
$726$	0	0
$727$	3.51472	0.130354	0.0651768	−	0.997874i	$-0.479239\pi$
$727$	3.51472	0.130354	0.0651768	+	0.997874i	$0.479239\pi$
$728$	0	0
$729$	0	0
$730$	9.41421	0.348436
$731$	−22.8284	−0.844340
$732$	0	0
$733$	−34.0000	−1.25582	−0.627909	−	0.778287i	$-0.716089\pi$
$733$	−34.0000	−1.25582	−0.627909	+	0.778287i	$0.716089\pi$
$734$	−24.9706	−0.921680
$735$	0	0
$736$	1.17157	0.0431847
$737$	1.37258	0.0505597
$738$	0	0
$739$	3.17157	0.116668	0.0583341	−	0.998297i	$-0.481421\pi$
$739$	3.17157	0.116668	0.0583341	+	0.998297i	$0.481421\pi$
$740$	−7.65685	−0.281472
$741$	0	0
$742$	0	0
$743$	−51.7990	−1.90032	−0.950160	−	0.311762i	$-0.899081\pi$
$743$	−51.7990	−1.90032	−0.950160	+	0.311762i	$0.899081\pi$
$744$	0	0
$745$	6.00000	0.219823
$746$	−30.4853	−1.11615
$747$	0	0
$748$	2.14214	0.0783242
$749$	0	0
$750$	0	0
$751$	−39.3137	−1.43458	−0.717289	−	0.696776i	$-0.754617\pi$
$751$	−39.3137	−1.43458	−0.717289	+	0.696776i	$0.754617\pi$
$752$	5.17157	0.188588
$753$	0	0
$754$	23.3137	0.849035
$755$	−11.3137	−0.411748
$756$	0	0
$757$	3.65685	0.132911	0.0664553	−	0.997789i	$-0.478831\pi$
$757$	3.65685	0.132911	0.0664553	+	0.997789i	$0.478831\pi$
$758$	34.4853	1.25256
$759$	0	0
$760$	−0.585786	−0.0212487
$761$	22.3848	0.811448	0.405724	−	0.913996i	$-0.367020\pi$
$761$	22.3848	0.811448	0.405724	+	0.913996i	$0.367020\pi$
$762$	0	0
$763$	0	0
$764$	−10.1421	−0.366930
$765$	0	0
$766$	−32.4853	−1.17374
$767$	41.4558	1.49688
$768$	0	0
$769$	−19.5563	−0.705220	−0.352610	−	0.935770i	$-0.614706\pi$
$769$	−19.5563	−0.705220	−0.352610	+	0.935770i	$0.614706\pi$
$770$	0	0
$771$	0	0
$772$	5.65685	0.203595
$773$	−2.00000	−0.0719350	−0.0359675	−	0.999353i	$-0.511451\pi$
$773$	−2.00000	−0.0719350	−0.0359675	+	0.999353i	$0.511451\pi$
$774$	0	0
$775$	2.82843	0.101600
$776$	7.75736	0.278473
$777$	0	0
$778$	−28.1421	−1.00894
$779$	1.79899	0.0644555
$780$	0	0
$781$	3.71573	0.132959
$782$	3.02944	0.108332
$783$	0	0
$784$	0	0
$785$	−6.48528	−0.231470
$786$	0	0
$787$	1.27208	0.0453447	0.0226723	−	0.999743i	$-0.492783\pi$
$787$	1.27208	0.0453447	0.0226723	+	0.999743i	$0.492783\pi$
$788$	25.7990	0.919051
$789$	0	0
$790$	−6.82843	−0.242945
$791$	0	0
$792$	0	0
$793$	−44.9706	−1.59695
$794$	−33.7990	−1.19948
$795$	0	0
$796$	−16.4853	−0.584305
$797$	−41.7990	−1.48060	−0.740298	−	0.672279i	$-0.765316\pi$
$797$	−41.7990	−1.48060	−0.740298	+	0.672279i	$0.765316\pi$
$798$	0	0
$799$	13.3726	0.473088
$800$	1.00000	0.0353553
$801$	0	0
$802$	6.00000	0.211867
$803$	7.79899	0.275220
$804$	0	0
$805$	0	0
$806$	13.6569	0.481042
$807$	0	0
$808$	13.3137	0.468375
$809$	−3.02944	−0.106509	−0.0532547	−	0.998581i	$-0.516960\pi$
$809$	−3.02944	−0.106509	−0.0532547	+	0.998581i	$0.516960\pi$
$810$	0	0
$811$	−32.5858	−1.14424	−0.572121	−	0.820169i	$-0.693879\pi$
$811$	−32.5858	−1.14424	−0.572121	+	0.820169i	$0.693879\pi$
$812$	0	0
$813$	0	0
$814$	−6.34315	−0.222327
$815$	−20.1421	−0.705548
$816$	0	0
$817$	5.17157	0.180930
$818$	10.5858	0.370123
$819$	0	0
$820$	−3.07107	−0.107246
$821$	−17.3137	−0.604253	−0.302126	−	0.953268i	$-0.597696\pi$
$821$	−17.3137	−0.604253	−0.302126	+	0.953268i	$0.597696\pi$
$822$	0	0
$823$	20.2843	0.707065	0.353533	−	0.935422i	$-0.384980\pi$
$823$	20.2843	0.707065	0.353533	+	0.935422i	$0.384980\pi$
$824$	14.8284	0.516573
$825$	0	0
$826$	0	0
$827$	5.37258	0.186823	0.0934115	−	0.995628i	$-0.470223\pi$
$827$	5.37258	0.186823	0.0934115	+	0.995628i	$0.470223\pi$
$828$	0	0
$829$	5.02944	0.174680	0.0873398	−	0.996179i	$-0.472163\pi$
$829$	5.02944	0.174680	0.0873398	+	0.996179i	$0.472163\pi$
$830$	−2.24264	−0.0778432
$831$	0	0
$832$	4.82843	0.167396
$833$	0	0
$834$	0	0
$835$	15.7990	0.546747
$836$	−0.485281	−0.0167838
$837$	0	0
$838$	−20.8701	−0.720944
$839$	−42.1421	−1.45491	−0.727454	−	0.686156i	$-0.759297\pi$
$839$	−42.1421	−1.45491	−0.727454	+	0.686156i	$0.759297\pi$
$840$	0	0
$841$	−5.68629	−0.196079
$842$	17.3137	0.596670
$843$	0	0
$844$	18.6274	0.641182
$845$	10.3137	0.354802
$846$	0	0
$847$	0	0
$848$	−6.48528	−0.222705
$849$	0	0
$850$	2.58579	0.0886917
$851$	−8.97056	−0.307507
$852$	0	0
$853$	43.1716	1.47817	0.739083	−	0.673614i	$-0.235259\pi$
$853$	43.1716	1.47817	0.739083	+	0.673614i	$0.235259\pi$
$854$	0	0
$855$	0	0
$856$	−9.65685	−0.330064
$857$	−4.92893	−0.168369	−0.0841846	−	0.996450i	$-0.526829\pi$
$857$	−4.92893	−0.168369	−0.0841846	+	0.996450i	$0.526829\pi$
$858$	0	0
$859$	7.21320	0.246111	0.123056	−	0.992400i	$-0.460731\pi$
$859$	7.21320	0.246111	0.123056	+	0.992400i	$0.460731\pi$
$860$	−8.82843	−0.301047
$861$	0	0
$862$	22.3431	0.761011
$863$	4.97056	0.169200	0.0846000	−	0.996415i	$-0.473039\pi$
$863$	4.97056	0.169200	0.0846000	+	0.996415i	$0.473039\pi$
$864$	0	0
$865$	−8.82843	−0.300176
$866$	−10.5858	−0.359720
$867$	0	0
$868$	0	0
$869$	−5.65685	−0.191896
$870$	0	0
$871$	8.00000	0.271070
$872$	2.48528	0.0841622
$873$	0	0
$874$	−0.686292	−0.0232142
$875$	0	0
$876$	0	0
$877$	−30.2843	−1.02263	−0.511314	−	0.859394i	$-0.670841\pi$
$877$	−30.2843	−1.02263	−0.511314	+	0.859394i	$0.670841\pi$
$878$	24.9706	0.842716
$879$	0	0
$880$	0.828427	0.0279263
$881$	−2.38478	−0.0803452	−0.0401726	−	0.999193i	$-0.512791\pi$
$881$	−2.38478	−0.0803452	−0.0401726	+	0.999193i	$0.512791\pi$
$882$	0	0
$883$	−41.6569	−1.40186	−0.700932	−	0.713228i	$-0.747233\pi$
$883$	−41.6569	−1.40186	−0.700932	+	0.713228i	$0.747233\pi$
$884$	12.4853	0.419925
$885$	0	0
$886$	−3.02944	−0.101776
$887$	−55.1127	−1.85050	−0.925252	−	0.379354i	$-0.876146\pi$
$887$	−55.1127	−1.85050	−0.925252	+	0.379354i	$0.876146\pi$
$888$	0	0
$889$	0	0
$890$	12.7279	0.426641
$891$	0	0
$892$	7.31371	0.244881
$893$	−3.02944	−0.101376
$894$	0	0
$895$	−4.00000	−0.133705
$896$	0	0
$897$	0	0
$898$	16.6274	0.554864
$899$	13.6569	0.455482
$900$	0	0
$901$	−16.7696	−0.558675
$902$	−2.54416	−0.0847111
$903$	0	0
$904$	15.3137	0.509326
$905$	2.48528	0.0826135
$906$	0	0
$907$	0.284271	0.00943907	0.00471954	−	0.999989i	$-0.498498\pi$
$907$	0.284271	0.00943907	0.00471954	+	0.999989i	$0.498498\pi$
$908$	18.2426	0.605403
$909$	0	0
$910$	0	0
$911$	−36.2843	−1.20215	−0.601076	−	0.799192i	$-0.705261\pi$
$911$	−36.2843	−1.20215	−0.601076	+	0.799192i	$0.705261\pi$
$912$	0	0
$913$	−1.85786	−0.0614863
$914$	21.6569	0.716345
$915$	0	0
$916$	16.1421	0.533351
$917$	0	0
$918$	0	0
$919$	15.5147	0.511783	0.255892	−	0.966705i	$-0.417631\pi$
$919$	15.5147	0.511783	0.255892	+	0.966705i	$0.417631\pi$
$920$	1.17157	0.0386256
$921$	0	0
$922$	−12.8284	−0.422482
$923$	21.6569	0.712844
$924$	0	0
$925$	−7.65685	−0.251756
$926$	−16.9706	−0.557687
$927$	0	0
$928$	4.82843	0.158501
$929$	−17.2132	−0.564747	−0.282373	−	0.959305i	$-0.591122\pi$
$929$	−17.2132	−0.564747	−0.282373	+	0.959305i	$0.591122\pi$
$930$	0	0
$931$	0	0
$932$	−23.3137	−0.763666
$933$	0	0
$934$	−15.8995	−0.520247
$935$	2.14214	0.0700553
$936$	0	0
$937$	−20.2426	−0.661298	−0.330649	−	0.943754i	$-0.607268\pi$
$937$	−20.2426	−0.661298	−0.330649	+	0.943754i	$0.607268\pi$
$938$	0	0
$939$	0	0
$940$	5.17157	0.168678
$941$	−50.0000	−1.62995	−0.814977	−	0.579494i	$-0.803250\pi$
$941$	−50.0000	−1.62995	−0.814977	+	0.579494i	$0.803250\pi$
$942$	0	0
$943$	−3.59798	−0.117166
$944$	8.58579	0.279444
$945$	0	0
$946$	−7.31371	−0.237789
$947$	4.82843	0.156903	0.0784514	−	0.996918i	$-0.475002\pi$
$947$	4.82843	0.156903	0.0784514	+	0.996918i	$0.475002\pi$
$948$	0	0
$949$	45.4558	1.47556
$950$	−0.585786	−0.0190054
$951$	0	0
$952$	0	0
$953$	−0.343146	−0.0111156	−0.00555779	−	0.999985i	$-0.501769\pi$
$953$	−0.343146	−0.0111156	−0.00555779	+	0.999985i	$0.501769\pi$
$954$	0	0
$955$	−10.1421	−0.328192
$956$	−1.65685	−0.0535865
$957$	0	0
$958$	−17.1716	−0.554788
$959$	0	0
$960$	0	0
$961$	−23.0000	−0.741935
$962$	−36.9706	−1.19198
$963$	0	0
$964$	13.4142	0.432043
$965$	5.65685	0.182101
$966$	0	0
$967$	−37.4558	−1.20450	−0.602249	−	0.798308i	$-0.705729\pi$
$967$	−37.4558	−1.20450	−0.602249	+	0.798308i	$0.705729\pi$
$968$	−10.3137	−0.331495
$969$	0	0
$970$	7.75736	0.249074
$971$	−33.3553	−1.07042	−0.535212	−	0.844718i	$-0.679768\pi$
$971$	−33.3553	−1.07042	−0.535212	+	0.844718i	$0.679768\pi$
$972$	0	0
$973$	0	0
$974$	31.7990	1.01891
$975$	0	0
$976$	−9.31371	−0.298125
$977$	12.6863	0.405870	0.202935	−	0.979192i	$-0.434952\pi$
$977$	12.6863	0.405870	0.202935	+	0.979192i	$0.434952\pi$
$978$	0	0
$979$	10.5442	0.336993
$980$	0	0
$981$	0	0
$982$	−32.2843	−1.03023
$983$	−12.2010	−0.389152	−0.194576	−	0.980887i	$-0.562333\pi$
$983$	−12.2010	−0.389152	−0.194576	+	0.980887i	$0.562333\pi$
$984$	0	0
$985$	25.7990	0.822024
$986$	12.4853	0.397612
$987$	0	0
$988$	−2.82843	−0.0899843
$989$	−10.3431	−0.328893
$990$	0	0
$991$	44.7696	1.42215	0.711076	−	0.703115i	$-0.248208\pi$
$991$	44.7696	1.42215	0.711076	+	0.703115i	$0.248208\pi$
$992$	2.82843	0.0898027
$993$	0	0
$994$	0	0
$995$	−16.4853	−0.522619
$996$	0	0
$997$	18.2843	0.579069	0.289534	−	0.957168i	$-0.406500\pi$
$997$	18.2843	0.579069	0.289534	+	0.957168i	$0.406500\pi$
$998$	30.3431	0.960495
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	4410.2.a.by.1.2		2
3.2	odd	2		490.2.a.l.1.1	✓	2
7.6	odd	2		4410.2.a.bt.1.2		2
12.11	even	2		3920.2.a.ca.1.2		2
15.2	even	4		2450.2.c.w.99.2		4
15.8	even	4		2450.2.c.w.99.3		4
15.14	odd	2		2450.2.a.bs.1.2		2
21.2	odd	6		490.2.e.j.361.2		4
21.5	even	6		490.2.e.i.361.1		4
21.11	odd	6		490.2.e.j.471.2		4
21.17	even	6		490.2.e.i.471.1		4
21.20	even	2		490.2.a.m.1.2	yes	2
84.83	odd	2		3920.2.a.bm.1.1		2
105.62	odd	4		2450.2.c.t.99.1		4
105.83	odd	4		2450.2.c.t.99.4		4
105.104	even	2		2450.2.a.bn.1.1		2

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
490.2.a.l.1.1	✓	2	3.2	odd	2
490.2.a.m.1.2	yes	2	21.20	even	2
490.2.e.i.361.1		4	21.5	even	6
490.2.e.i.471.1		4	21.17	even	6
490.2.e.j.361.2		4	21.2	odd	6
490.2.e.j.471.2		4	21.11	odd	6
2450.2.a.bn.1.1		2	105.104	even	2
2450.2.a.bs.1.2		2	15.14	odd	2
2450.2.c.t.99.1		4	105.62	odd	4
2450.2.c.t.99.4		4	105.83	odd	4
2450.2.c.w.99.2		4	15.2	even	4
2450.2.c.w.99.3		4	15.8	even	4
3920.2.a.bm.1.1		2	84.83	odd	2
3920.2.a.ca.1.2		2	12.11	even	2
4410.2.a.bt.1.2		2	7.6	odd	2
4410.2.a.by.1.2		2	1.1	even	1	trivial

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:	\( N \)	\(=\)	\( 4410 = 2 \cdot 3^{2} \cdot 5 \cdot 7^{2} \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	4410.a (trivial)

\(f(q)\)	\(=\)	\(q+1.00000 q^{2} +1.00000 q^{4} +1.00000 q^{5} +1.00000 q^{8} +O(q^{10})\)	\(q+1.00000 q^{2} +1.00000 q^{4} +1.00000 q^{5} +1.00000 q^{8} +1.00000 q^{10} +0.828427 q^{11} +4.82843 q^{13} +1.00000 q^{16} +2.58579 q^{17} -0.585786 q^{19} +1.00000 q^{20} +0.828427 q^{22} +1.17157 q^{23} +1.00000 q^{25} +4.82843 q^{26} +4.82843 q^{29} +2.82843 q^{31} +1.00000 q^{32} +2.58579 q^{34} -7.65685 q^{37} -0.585786 q^{38} +1.00000 q^{40} -3.07107 q^{41} -8.82843 q^{43} +0.828427 q^{44} +1.17157 q^{46} +5.17157 q^{47} +1.00000 q^{50} +4.82843 q^{52} -6.48528 q^{53} +0.828427 q^{55} +4.82843 q^{58} +8.58579 q^{59} -9.31371 q^{61} +2.82843 q^{62} +1.00000 q^{64} +4.82843 q^{65} +1.65685 q^{67} +2.58579 q^{68} +4.48528 q^{71} +9.41421 q^{73} -7.65685 q^{74} -0.585786 q^{76} -6.82843 q^{79} +1.00000 q^{80} -3.07107 q^{82} -2.24264 q^{83} +2.58579 q^{85} -8.82843 q^{86} +0.828427 q^{88} +12.7279 q^{89} +1.17157 q^{92} +5.17157 q^{94} -0.585786 q^{95} +7.75736 q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 2 q + 2 q^{2} + 2 q^{4} + 2 q^{5} + 2 q^{8}+O(q^{10}) \) 2 * q + 2 * q^2 + 2 * q^4 + 2 * q^5 + 2 * q^8	\( 2 q + 2 q^{2} + 2 q^{4} + 2 q^{5} + 2 q^{8} + 2 q^{10} - 4 q^{11} + 4 q^{13} + 2 q^{16} + 8 q^{17} - 4 q^{19} + 2 q^{20} - 4 q^{22} + 8 q^{23} + 2 q^{25} + 4 q^{26} + 4 q^{29} + 2 q^{32} + 8 q^{34} - 4 q^{37} - 4 q^{38} + 2 q^{40} + 8 q^{41} - 12 q^{43} - 4 q^{44} + 8 q^{46} + 16 q^{47} + 2 q^{50} + 4 q^{52} + 4 q^{53} - 4 q^{55} + 4 q^{58} + 20 q^{59} + 4 q^{61} + 2 q^{64} + 4 q^{65} - 8 q^{67} + 8 q^{68} - 8 q^{71} + 16 q^{73} - 4 q^{74} - 4 q^{76} - 8 q^{79} + 2 q^{80} + 8 q^{82} + 4 q^{83} + 8 q^{85} - 12 q^{86} - 4 q^{88} + 8 q^{92} + 16 q^{94} - 4 q^{95} + 24 q^{97}+O(q^{100}) \) 2 * q + 2 * q^2 + 2 * q^4 + 2 * q^5 + 2 * q^8 + 2 * q^10 - 4 * q^11 + 4 * q^13 + 2 * q^16 + 8 * q^17 - 4 * q^19 + 2 * q^20 - 4 * q^22 + 8 * q^23 + 2 * q^25 + 4 * q^26 + 4 * q^29 + 2 * q^32 + 8 * q^34 - 4 * q^37 - 4 * q^38 + 2 * q^40 + 8 * q^41 - 12 * q^43 - 4 * q^44 + 8 * q^46 + 16 * q^47 + 2 * q^50 + 4 * q^52 + 4 * q^53 - 4 * q^55 + 4 * q^58 + 20 * q^59 + 4 * q^61 + 2 * q^64 + 4 * q^65 - 8 * q^67 + 8 * q^68 - 8 * q^71 + 16 * q^73 - 4 * q^74 - 4 * q^76 - 8 * q^79 + 2 * q^80 + 8 * q^82 + 4 * q^83 + 8 * q^85 - 12 * q^86 - 4 * q^88 + 8 * q^92 + 16 * q^94 - 4 * q^95 + 24 * q^97

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