LMFDB - Embedded newform 4410.2.a.bz.1.1

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:	$\Z[a_1, \ldots, a_{13}]$
Coefficient ring index:	$ 1 $
Twist minimal:	no (minimal twist has level 1470)
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.1
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)

$f(q)$	$=$	$q+1.00000 q^{2} +1.00000 q^{4} +1.00000 q^{5} +1.00000 q^{8} +O(q^{10})$	$q+1.00000 q^{2} +1.00000 q^{4} +1.00000 q^{5} +1.00000 q^{8} +1.00000 q^{10} -2.24264 q^{11} -5.65685 q^{13} +1.00000 q^{16} +2.58579 q^{17} +6.82843 q^{19} +1.00000 q^{20} -2.24264 q^{22} -3.17157 q^{23} +1.00000 q^{25} -5.65685 q^{26} -2.58579 q^{29} +10.2426 q^{31} +1.00000 q^{32} +2.58579 q^{34} -0.242641 q^{37} +6.82843 q^{38} +1.00000 q^{40} +10.4853 q^{41} +9.07107 q^{43} -2.24264 q^{44} -3.17157 q^{46} -2.24264 q^{47} +1.00000 q^{50} -5.65685 q^{52} -0.343146 q^{53} -2.24264 q^{55} -2.58579 q^{58} -3.17157 q^{59} +11.6569 q^{61} +10.2426 q^{62} +1.00000 q^{64} -5.65685 q^{65} +9.07107 q^{67} +2.58579 q^{68} +4.48528 q^{71} +2.00000 q^{73} -0.242641 q^{74} +6.82843 q^{76} +8.00000 q^{79} +1.00000 q^{80} +10.4853 q^{82} +5.17157 q^{83} +2.58579 q^{85} +9.07107 q^{86} -2.24264 q^{88} -0.828427 q^{89} -3.17157 q^{92} -2.24264 q^{94} +6.82843 q^{95} +6.48528 q^{97} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 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} + 2 q^{16} + 8 q^{17} + 8 q^{19} + 2 q^{20} + 4 q^{22} - 12 q^{23} + 2 q^{25} - 8 q^{29} + 12 q^{31} + 2 q^{32} + 8 q^{34} + 8 q^{37} + 8 q^{38} + 2 q^{40} + 4 q^{41} + 4 q^{43} + 4 q^{44} - 12 q^{46} + 4 q^{47} + 2 q^{50} - 12 q^{53} + 4 q^{55} - 8 q^{58} - 12 q^{59} + 12 q^{61} + 12 q^{62} + 2 q^{64} + 4 q^{67} + 8 q^{68} - 8 q^{71} + 4 q^{73} + 8 q^{74} + 8 q^{76} + 16 q^{79} + 2 q^{80} + 4 q^{82} + 16 q^{83} + 8 q^{85} + 4 q^{86} + 4 q^{88} + 4 q^{89} - 12 q^{92} + 4 q^{94} + 8 q^{95} - 4 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 + 2 * q^16 + 8 * q^17 + 8 * q^19 + 2 * q^20 + 4 * q^22 - 12 * q^23 + 2 * q^25 - 8 * q^29 + 12 * q^31 + 2 * q^32 + 8 * q^34 + 8 * q^37 + 8 * q^38 + 2 * q^40 + 4 * q^41 + 4 * q^43 + 4 * q^44 - 12 * q^46 + 4 * q^47 + 2 * q^50 - 12 * q^53 + 4 * q^55 - 8 * q^58 - 12 * q^59 + 12 * q^61 + 12 * q^62 + 2 * q^64 + 4 * q^67 + 8 * q^68 - 8 * q^71 + 4 * q^73 + 8 * q^74 + 8 * q^76 + 16 * q^79 + 2 * q^80 + 4 * q^82 + 16 * q^83 + 8 * q^85 + 4 * q^86 + 4 * q^88 + 4 * q^89 - 12 * q^92 + 4 * q^94 + 8 * q^95 - 4 * 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
$11$	−2.24264	−0.676182	−0.338091	−	0.941113i	$-0.609781\pi$
$11$	−2.24264	−0.676182	−0.338091	+	0.941113i	$0.609781\pi$
$12$	0	0
$13$	−5.65685	−1.56893	−0.784465	−	0.620174i	$-0.787062\pi$
$13$	−5.65685	−1.56893	−0.784465	+	0.620174i	$0.787062\pi$
$14$	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
$19$	6.82843	1.56655	0.783274	−	0.621676i	$-0.213548\pi$
$19$	6.82843	1.56655	0.783274	+	0.621676i	$0.213548\pi$
$20$	1.00000	0.223607
$21$	0	0
$22$	−2.24264	−0.478133
$23$	−3.17157	−0.661319	−0.330659	−	0.943750i	$-0.607271\pi$
$23$	−3.17157	−0.661319	−0.330659	+	0.943750i	$0.607271\pi$
$24$	0	0
$25$	1.00000	0.200000
$26$	−5.65685	−1.10940
$27$	0	0
$28$	0	0
$29$	−2.58579	−0.480168	−0.240084	−	0.970752i	$-0.577175\pi$
$29$	−2.58579	−0.480168	−0.240084	+	0.970752i	$0.577175\pi$
$30$	0	0
$31$	10.2426	1.83963	0.919816	−	0.392349i	$-0.128338\pi$
$31$	10.2426	1.83963	0.919816	+	0.392349i	$0.128338\pi$
$32$	1.00000	0.176777
$33$	0	0
$34$	2.58579	0.443459
$35$	0	0
$36$	0	0
$37$	−0.242641	−0.0398899	−0.0199449	−	0.999801i	$-0.506349\pi$
$37$	−0.242641	−0.0398899	−0.0199449	+	0.999801i	$0.506349\pi$
$38$	6.82843	1.10772
$39$	0	0
$40$	1.00000	0.158114
$41$	10.4853	1.63753	0.818763	−	0.574132i	$-0.194660\pi$
$41$	10.4853	1.63753	0.818763	+	0.574132i	$0.194660\pi$
$42$	0	0
$43$	9.07107	1.38332	0.691662	−	0.722221i	$-0.256879\pi$
$43$	9.07107	1.38332	0.691662	+	0.722221i	$0.256879\pi$
$44$	−2.24264	−0.338091
$45$	0	0
$46$	−3.17157	−0.467623
$47$	−2.24264	−0.327123	−0.163561	−	0.986533i	$-0.552298\pi$
$47$	−2.24264	−0.327123	−0.163561	+	0.986533i	$0.552298\pi$
$48$	0	0
$49$	0	0
$50$	1.00000	0.141421
$51$	0	0
$52$	−5.65685	−0.784465
$53$	−0.343146	−0.0471347	−0.0235673	−	0.999722i	$-0.507502\pi$
$53$	−0.343146	−0.0471347	−0.0235673	+	0.999722i	$0.507502\pi$
$54$	0	0
$55$	−2.24264	−0.302398
$56$	0	0
$57$	0	0
$58$	−2.58579	−0.339530
$59$	−3.17157	−0.412904	−0.206452	−	0.978457i	$-0.566192\pi$
$59$	−3.17157	−0.412904	−0.206452	+	0.978457i	$0.566192\pi$
$60$	0	0
$61$	11.6569	1.49251	0.746254	−	0.665662i	$-0.231851\pi$
$61$	11.6569	1.49251	0.746254	+	0.665662i	$0.231851\pi$
$62$	10.2426	1.30082
$63$	0	0
$64$	1.00000	0.125000
$65$	−5.65685	−0.701646
$66$	0	0
$67$	9.07107	1.10821	0.554104	−	0.832448i	$-0.313061\pi$
$67$	9.07107	1.10821	0.554104	+	0.832448i	$0.313061\pi$
$68$	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
$73$	2.00000	0.234082	0.117041	−	0.993127i	$-0.462659\pi$
$73$	2.00000	0.234082	0.117041	+	0.993127i	$0.462659\pi$
$74$	−0.242641	−0.0282064
$75$	0	0
$76$	6.82843	0.783274
$77$	0	0
$78$	0	0
$79$	8.00000	0.900070	0.450035	−	0.893011i	$-0.351411\pi$
$79$	8.00000	0.900070	0.450035	+	0.893011i	$0.351411\pi$
$80$	1.00000	0.111803
$81$	0	0
$82$	10.4853	1.15791
$83$	5.17157	0.567654	0.283827	−	0.958876i	$-0.408396\pi$
$83$	5.17157	0.567654	0.283827	+	0.958876i	$0.408396\pi$
$84$	0	0
$85$	2.58579	0.280468
$86$	9.07107	0.978158
$87$	0	0
$88$	−2.24264	−0.239066
$89$	−0.828427	−0.0878131	−0.0439065	−	0.999036i	$-0.513980\pi$
$89$	−0.828427	−0.0878131	−0.0439065	+	0.999036i	$0.513980\pi$
$90$	0	0
$91$	0	0
$92$	−3.17157	−0.330659
$93$	0	0
$94$	−2.24264	−0.231311
$95$	6.82843	0.700582
$96$	0	0
$97$	6.48528	0.658481	0.329240	−	0.944246i	$-0.393207\pi$
$97$	6.48528	0.658481	0.329240	+	0.944246i	$0.393207\pi$
$98$	0	0
$99$	0	0
$100$	1.00000	0.100000
$101$	17.6569	1.75692	0.878461	−	0.477813i	$-0.158571\pi$
$101$	17.6569	1.75692	0.878461	+	0.477813i	$0.158571\pi$
$102$	0	0
$103$	−14.8284	−1.46109	−0.730544	−	0.682865i	$-0.760734\pi$
$103$	−14.8284	−1.46109	−0.730544	+	0.682865i	$0.760734\pi$
$104$	−5.65685	−0.554700
$105$	0	0
$106$	−0.343146	−0.0333293
$107$	−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
$109$	−3.65685	−0.350263	−0.175132	−	0.984545i	$-0.556035\pi$
$109$	−3.65685	−0.350263	−0.175132	+	0.984545i	$0.556035\pi$
$110$	−2.24264	−0.213827
$111$	0	0
$112$	0	0
$113$	−10.0000	−0.940721	−0.470360	−	0.882474i	$-0.655876\pi$
$113$	−10.0000	−0.940721	−0.470360	+	0.882474i	$0.655876\pi$
$114$	0	0
$115$	−3.17157	−0.295751
$116$	−2.58579	−0.240084
$117$	0	0
$118$	−3.17157	−0.291967
$119$	0	0
$120$	0	0
$121$	−5.97056	−0.542778
$122$	11.6569	1.05536
$123$	0	0
$124$	10.2426	0.919816
$125$	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
$130$	−5.65685	−0.496139
$131$	−3.17157	−0.277102	−0.138551	−	0.990355i	$-0.544244\pi$
$131$	−3.17157	−0.277102	−0.138551	+	0.990355i	$0.544244\pi$
$132$	0	0
$133$	0	0
$134$	9.07107	0.783621
$135$	0	0
$136$	2.58579	0.221729
$137$	−13.6569	−1.16678	−0.583392	−	0.812191i	$-0.698275\pi$
$137$	−13.6569	−1.16678	−0.583392	+	0.812191i	$0.698275\pi$
$138$	0	0
$139$	2.34315	0.198743	0.0993715	−	0.995050i	$-0.468317\pi$
$139$	2.34315	0.198743	0.0993715	+	0.995050i	$0.468317\pi$
$140$	0	0
$141$	0	0
$142$	4.48528	0.376396
$143$	12.6863	1.06088
$144$	0	0
$145$	−2.58579	−0.214738
$146$	2.00000	0.165521
$147$	0	0
$148$	−0.242641	−0.0199449
$149$	−22.3848	−1.83383	−0.916916	−	0.399080i	$-0.869330\pi$
$149$	−22.3848	−1.83383	−0.916916	+	0.399080i	$0.869330\pi$
$150$	0	0
$151$	20.1421	1.63914	0.819572	−	0.572976i	$-0.194211\pi$
$151$	20.1421	1.63914	0.819572	+	0.572976i	$0.194211\pi$
$152$	6.82843	0.553859
$153$	0	0
$154$	0	0
$155$	10.2426	0.822709
$156$	0	0
$157$	8.34315	0.665856	0.332928	−	0.942952i	$-0.391963\pi$
$157$	8.34315	0.665856	0.332928	+	0.942952i	$0.391963\pi$
$158$	8.00000	0.636446
$159$	0	0
$160$	1.00000	0.0790569
$161$	0	0
$162$	0	0
$163$	−17.0711	−1.33711	−0.668555	−	0.743663i	$-0.733087\pi$
$163$	−17.0711	−1.33711	−0.668555	+	0.743663i	$0.733087\pi$
$164$	10.4853	0.818763
$165$	0	0
$166$	5.17157	0.401392
$167$	2.24264	0.173541	0.0867704	−	0.996228i	$-0.472345\pi$
$167$	2.24264	0.173541	0.0867704	+	0.996228i	$0.472345\pi$
$168$	0	0
$169$	19.0000	1.46154
$170$	2.58579	0.198321
$171$	0	0
$172$	9.07107	0.691662
$173$	20.8284	1.58356	0.791778	−	0.610809i	$-0.209156\pi$
$173$	20.8284	1.58356	0.791778	+	0.610809i	$0.209156\pi$
$174$	0	0
$175$	0	0
$176$	−2.24264	−0.169045
$177$	0	0
$178$	−0.828427	−0.0620932
$179$	−26.2426	−1.96147	−0.980734	−	0.195350i	$-0.937416\pi$
$179$	−26.2426	−1.96147	−0.980734	+	0.195350i	$0.937416\pi$
$180$	0	0
$181$	−18.4853	−1.37400	−0.687000	−	0.726657i	$-0.741073\pi$
$181$	−18.4853	−1.37400	−0.687000	+	0.726657i	$0.741073\pi$
$182$	0	0
$183$	0	0
$184$	−3.17157	−0.233811
$185$	−0.242641	−0.0178393
$186$	0	0
$187$	−5.79899	−0.424064
$188$	−2.24264	−0.163561
$189$	0	0
$190$	6.82843	0.495386
$191$	−4.00000	−0.289430	−0.144715	−	0.989473i	$-0.546227\pi$
$191$	−4.00000	−0.289430	−0.144715	+	0.989473i	$0.546227\pi$
$192$	0	0
$193$	−4.82843	−0.347558	−0.173779	−	0.984785i	$-0.555598\pi$
$193$	−4.82843	−0.347558	−0.173779	+	0.984785i	$0.555598\pi$
$194$	6.48528	0.465616
$195$	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
$199$	5.75736	0.408128	0.204064	−	0.978958i	$-0.434585\pi$
$199$	5.75736	0.408128	0.204064	+	0.978958i	$0.434585\pi$
$200$	1.00000	0.0707107
$201$	0	0
$202$	17.6569	1.24233
$203$	0	0
$204$	0	0
$205$	10.4853	0.732324
$206$	−14.8284	−1.03315
$207$	0	0
$208$	−5.65685	−0.392232
$209$	−15.3137	−1.05927
$210$	0	0
$211$	−23.3137	−1.60498	−0.802491	−	0.596664i	$-0.796492\pi$
$211$	−23.3137	−1.60498	−0.802491	+	0.596664i	$0.796492\pi$
$212$	−0.343146	−0.0235673
$213$	0	0
$214$	−9.65685	−0.660129
$215$	9.07107	0.618642
$216$	0	0
$217$	0	0
$218$	−3.65685	−0.247673
$219$	0	0
$220$	−2.24264	−0.151199
$221$	−14.6274	−0.983947
$222$	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
$226$	−10.0000	−0.665190
$227$	−24.9706	−1.65735	−0.828677	−	0.559727i	$-0.810906\pi$
$227$	−24.9706	−1.65735	−0.828677	+	0.559727i	$0.810906\pi$
$228$	0	0
$229$	16.1421	1.06670	0.533351	−	0.845894i	$-0.320932\pi$
$229$	16.1421	1.06670	0.533351	+	0.845894i	$0.320932\pi$
$230$	−3.17157	−0.209127
$231$	0	0
$232$	−2.58579	−0.169765
$233$	−18.9706	−1.24280	−0.621401	−	0.783492i	$-0.713436\pi$
$233$	−18.9706	−1.24280	−0.621401	+	0.783492i	$0.713436\pi$
$234$	0	0
$235$	−2.24264	−0.146294
$236$	−3.17157	−0.206452
$237$	0	0
$238$	0	0
$239$	19.3137	1.24930	0.624650	−	0.780905i	$-0.285242\pi$
$239$	19.3137	1.24930	0.624650	+	0.780905i	$0.285242\pi$
$240$	0	0
$241$	19.5563	1.25974	0.629868	−	0.776703i	$-0.283109\pi$
$241$	19.5563	1.25974	0.629868	+	0.776703i	$0.283109\pi$
$242$	−5.97056	−0.383802
$243$	0	0
$244$	11.6569	0.746254
$245$	0	0
$246$	0	0
$247$	−38.6274	−2.45780
$248$	10.2426	0.650408
$249$	0	0
$250$	1.00000	0.0632456
$251$	−12.9706	−0.818695	−0.409347	−	0.912379i	$-0.634244\pi$
$251$	−12.9706	−0.818695	−0.409347	+	0.912379i	$0.634244\pi$
$252$	0	0
$253$	7.11270	0.447172
$254$	2.82843	0.177471
$255$	0	0
$256$	1.00000	0.0625000
$257$	−18.5858	−1.15935	−0.579675	−	0.814848i	$-0.696820\pi$
$257$	−18.5858	−1.15935	−0.579675	+	0.814848i	$0.696820\pi$
$258$	0	0
$259$	0	0
$260$	−5.65685	−0.350823
$261$	0	0
$262$	−3.17157	−0.195940
$263$	12.1421	0.748716	0.374358	−	0.927284i	$-0.377863\pi$
$263$	12.1421	0.748716	0.374358	+	0.927284i	$0.377863\pi$
$264$	0	0
$265$	−0.343146	−0.0210793
$266$	0	0
$267$	0	0
$268$	9.07107	0.554104
$269$	12.3431	0.752575	0.376287	−	0.926503i	$-0.377201\pi$
$269$	12.3431	0.752575	0.376287	+	0.926503i	$0.377201\pi$
$270$	0	0
$271$	−31.6985	−1.92555	−0.962773	−	0.270312i	$-0.912873\pi$
$271$	−31.6985	−1.92555	−0.962773	+	0.270312i	$0.912873\pi$
$272$	2.58579	0.156786
$273$	0	0
$274$	−13.6569	−0.825041
$275$	−2.24264	−0.135236
$276$	0	0
$277$	−15.5563	−0.934690	−0.467345	−	0.884075i	$-0.654789\pi$
$277$	−15.5563	−0.934690	−0.467345	+	0.884075i	$0.654789\pi$
$278$	2.34315	0.140533
$279$	0	0
$280$	0	0
$281$	23.4558	1.39926	0.699629	−	0.714506i	$-0.253349\pi$
$281$	23.4558	1.39926	0.699629	+	0.714506i	$0.253349\pi$
$282$	0	0
$283$	−9.51472	−0.565591	−0.282796	−	0.959180i	$-0.591262\pi$
$283$	−9.51472	−0.565591	−0.282796	+	0.959180i	$0.591262\pi$
$284$	4.48528	0.266152
$285$	0	0
$286$	12.6863	0.750156
$287$	0	0
$288$	0	0
$289$	−10.3137	−0.606689
$290$	−2.58579	−0.151843
$291$	0	0
$292$	2.00000	0.117041
$293$	21.3137	1.24516	0.622580	−	0.782556i	$-0.286084\pi$
$293$	21.3137	1.24516	0.622580	+	0.782556i	$0.286084\pi$
$294$	0	0
$295$	−3.17157	−0.184656
$296$	−0.242641	−0.0141032
$297$	0	0
$298$	−22.3848	−1.29672
$299$	17.9411	1.03756
$300$	0	0
$301$	0	0
$302$	20.1421	1.15905
$303$	0	0
$304$	6.82843	0.391637
$305$	11.6569	0.667470
$306$	0	0
$307$	7.31371	0.417415	0.208708	−	0.977978i	$-0.433074\pi$
$307$	7.31371	0.417415	0.208708	+	0.977978i	$0.433074\pi$
$308$	0	0
$309$	0	0
$310$	10.2426	0.581743
$311$	−10.8284	−0.614024	−0.307012	−	0.951706i	$-0.599329\pi$
$311$	−10.8284	−0.614024	−0.307012	+	0.951706i	$0.599329\pi$
$312$	0	0
$313$	−30.4853	−1.72313	−0.861565	−	0.507647i	$-0.830515\pi$
$313$	−30.4853	−1.72313	−0.861565	+	0.507647i	$0.830515\pi$
$314$	8.34315	0.470831
$315$	0	0
$316$	8.00000	0.450035
$317$	25.3137	1.42176	0.710880	−	0.703314i	$-0.248297\pi$
$317$	25.3137	1.42176	0.710880	+	0.703314i	$0.248297\pi$
$318$	0	0
$319$	5.79899	0.324681
$320$	1.00000	0.0559017
$321$	0	0
$322$	0	0
$323$	17.6569	0.982454
$324$	0	0
$325$	−5.65685	−0.313786
$326$	−17.0711	−0.945479
$327$	0	0
$328$	10.4853	0.578953
$329$	0	0
$330$	0	0
$331$	−6.34315	−0.348651	−0.174325	−	0.984688i	$-0.555774\pi$
$331$	−6.34315	−0.348651	−0.174325	+	0.984688i	$0.555774\pi$
$332$	5.17157	0.283827
$333$	0	0
$334$	2.24264	0.122712
$335$	9.07107	0.495605
$336$	0	0
$337$	33.1127	1.80376	0.901882	−	0.431983i	$-0.142186\pi$
$337$	33.1127	1.80376	0.901882	+	0.431983i	$0.142186\pi$
$338$	19.0000	1.03346
$339$	0	0
$340$	2.58579	0.140234
$341$	−22.9706	−1.24393
$342$	0	0
$343$	0	0
$344$	9.07107	0.489079
$345$	0	0
$346$	20.8284	1.11974
$347$	22.3431	1.19944	0.599721	−	0.800209i	$-0.295278\pi$
$347$	22.3431	1.19944	0.599721	+	0.800209i	$0.295278\pi$
$348$	0	0
$349$	−23.4558	−1.25556	−0.627781	−	0.778390i	$-0.716037\pi$
$349$	−23.4558	−1.25556	−0.627781	+	0.778390i	$0.716037\pi$
$350$	0	0
$351$	0	0
$352$	−2.24264	−0.119533
$353$	16.0416	0.853810	0.426905	−	0.904297i	$-0.359604\pi$
$353$	16.0416	0.853810	0.426905	+	0.904297i	$0.359604\pi$
$354$	0	0
$355$	4.48528	0.238054
$356$	−0.828427	−0.0439065
$357$	0	0
$358$	−26.2426	−1.38697
$359$	28.4853	1.50340	0.751698	−	0.659508i	$-0.229235\pi$
$359$	28.4853	1.50340	0.751698	+	0.659508i	$0.229235\pi$
$360$	0	0
$361$	27.6274	1.45407
$362$	−18.4853	−0.971565
$363$	0	0
$364$	0	0
$365$	2.00000	0.104685
$366$	0	0
$367$	10.8284	0.565239	0.282620	−	0.959232i	$-0.408797\pi$
$367$	10.8284	0.565239	0.282620	+	0.959232i	$0.408797\pi$
$368$	−3.17157	−0.165330
$369$	0	0
$370$	−0.242641	−0.0126143
$371$	0	0
$372$	0	0
$373$	6.58579	0.340999	0.170500	−	0.985358i	$-0.445462\pi$
$373$	6.58579	0.340999	0.170500	+	0.985358i	$0.445462\pi$
$374$	−5.79899	−0.299859
$375$	0	0
$376$	−2.24264	−0.115655
$377$	14.6274	0.753350
$378$	0	0
$379$	0.485281	0.0249272	0.0124636	−	0.999922i	$-0.496033\pi$
$379$	0.485281	0.0249272	0.0124636	+	0.999922i	$0.496033\pi$
$380$	6.82843	0.350291
$381$	0	0
$382$	−4.00000	−0.204658
$383$	19.4142	0.992020	0.496010	−	0.868317i	$-0.334798\pi$
$383$	19.4142	0.992020	0.496010	+	0.868317i	$0.334798\pi$
$384$	0	0
$385$	0	0
$386$	−4.82843	−0.245760
$387$	0	0
$388$	6.48528	0.329240
$389$	29.8995	1.51596	0.757982	−	0.652275i	$-0.226185\pi$
$389$	29.8995	1.51596	0.757982	+	0.652275i	$0.226185\pi$
$390$	0	0
$391$	−8.20101	−0.414743
$392$	0	0
$393$	0	0
$394$	25.7990	1.29973
$395$	8.00000	0.402524
$396$	0	0
$397$	−6.68629	−0.335575	−0.167788	−	0.985823i	$-0.553662\pi$
$397$	−6.68629	−0.335575	−0.167788	+	0.985823i	$0.553662\pi$
$398$	5.75736	0.288590
$399$	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
$403$	−57.9411	−2.88625
$404$	17.6569	0.878461
$405$	0	0
$406$	0	0
$407$	0.544156	0.0269728
$408$	0	0
$409$	−4.24264	−0.209785	−0.104893	−	0.994484i	$-0.533450\pi$
$409$	−4.24264	−0.209785	−0.104893	+	0.994484i	$0.533450\pi$
$410$	10.4853	0.517831
$411$	0	0
$412$	−14.8284	−0.730544
$413$	0	0
$414$	0	0
$415$	5.17157	0.253863
$416$	−5.65685	−0.277350
$417$	0	0
$418$	−15.3137	−0.749018
$419$	−36.9706	−1.80613	−0.903065	−	0.429504i	$-0.858689\pi$
$419$	−36.9706	−1.80613	−0.903065	+	0.429504i	$0.858689\pi$
$420$	0	0
$421$	−3.65685	−0.178224	−0.0891121	−	0.996022i	$-0.528403\pi$
$421$	−3.65685	−0.178224	−0.0891121	+	0.996022i	$0.528403\pi$
$422$	−23.3137	−1.13489
$423$	0	0
$424$	−0.343146	−0.0166646
$425$	2.58579	0.125429
$426$	0	0
$427$	0	0
$428$	−9.65685	−0.466782
$429$	0	0
$430$	9.07107	0.437446
$431$	−13.4558	−0.648145	−0.324073	−	0.946032i	$-0.605052\pi$
$431$	−13.4558	−0.648145	−0.324073	+	0.946032i	$0.605052\pi$
$432$	0	0
$433$	17.7990	0.855365	0.427682	−	0.903929i	$-0.359330\pi$
$433$	17.7990	0.855365	0.427682	+	0.903929i	$0.359330\pi$
$434$	0	0
$435$	0	0
$436$	−3.65685	−0.175132
$437$	−21.6569	−1.03599
$438$	0	0
$439$	23.6985	1.13107	0.565533	−	0.824725i	$-0.308670\pi$
$439$	23.6985	1.13107	0.565533	+	0.824725i	$0.308670\pi$
$440$	−2.24264	−0.106914
$441$	0	0
$442$	−14.6274	−0.695755
$443$	−36.2843	−1.72392	−0.861959	−	0.506978i	$-0.830762\pi$
$443$	−36.2843	−1.72392	−0.861959	+	0.506978i	$0.830762\pi$
$444$	0	0
$445$	−0.828427	−0.0392712
$446$	7.31371	0.346314
$447$	0	0
$448$	0	0
$449$	10.4853	0.494831	0.247416	−	0.968909i	$-0.420419\pi$
$449$	10.4853	0.494831	0.247416	+	0.968909i	$0.420419\pi$
$450$	0	0
$451$	−23.5147	−1.10726
$452$	−10.0000	−0.470360
$453$	0	0
$454$	−24.9706	−1.17193
$455$	0	0
$456$	0	0
$457$	2.48528	0.116257	0.0581283	−	0.998309i	$-0.481487\pi$
$457$	2.48528	0.116257	0.0581283	+	0.998309i	$0.481487\pi$
$458$	16.1421	0.754272
$459$	0	0
$460$	−3.17157	−0.147875
$461$	2.00000	0.0931493	0.0465746	−	0.998915i	$-0.485169\pi$
$461$	2.00000	0.0931493	0.0465746	+	0.998915i	$0.485169\pi$
$462$	0	0
$463$	−25.6569	−1.19238	−0.596188	−	0.802845i	$-0.703319\pi$
$463$	−25.6569	−1.19238	−0.596188	+	0.802845i	$0.703319\pi$
$464$	−2.58579	−0.120042
$465$	0	0
$466$	−18.9706	−0.878794
$467$	12.4853	0.577750	0.288875	−	0.957367i	$-0.406719\pi$
$467$	12.4853	0.577750	0.288875	+	0.957367i	$0.406719\pi$
$468$	0	0
$469$	0	0
$470$	−2.24264	−0.103445
$471$	0	0
$472$	−3.17157	−0.145983
$473$	−20.3431	−0.935379
$474$	0	0
$475$	6.82843	0.313310
$476$	0	0
$477$	0	0
$478$	19.3137	0.883388
$479$	−8.48528	−0.387702	−0.193851	−	0.981031i	$-0.562098\pi$
$479$	−8.48528	−0.387702	−0.193851	+	0.981031i	$0.562098\pi$
$480$	0	0
$481$	1.37258	0.0625844
$482$	19.5563	0.890767
$483$	0	0
$484$	−5.97056	−0.271389
$485$	6.48528	0.294481
$486$	0	0
$487$	10.8284	0.490683	0.245341	−	0.969437i	$-0.421100\pi$
$487$	10.8284	0.490683	0.245341	+	0.969437i	$0.421100\pi$
$488$	11.6569	0.527681
$489$	0	0
$490$	0	0
$491$	17.0711	0.770407	0.385203	−	0.922832i	$-0.374131\pi$
$491$	17.0711	0.770407	0.385203	+	0.922832i	$0.374131\pi$
$492$	0	0
$493$	−6.68629	−0.301135
$494$	−38.6274	−1.73793
$495$	0	0
$496$	10.2426	0.459908
$497$	0	0
$498$	0	0
$499$	6.82843	0.305682	0.152841	−	0.988251i	$-0.451158\pi$
$499$	6.82843	0.305682	0.152841	+	0.988251i	$0.451158\pi$
$500$	1.00000	0.0447214
$501$	0	0
$502$	−12.9706	−0.578905
$503$	−4.58579	−0.204470	−0.102235	−	0.994760i	$-0.532599\pi$
$503$	−4.58579	−0.204470	−0.102235	+	0.994760i	$0.532599\pi$
$504$	0	0
$505$	17.6569	0.785720
$506$	7.11270	0.316198
$507$	0	0
$508$	2.82843	0.125491
$509$	−16.9706	−0.752207	−0.376103	−	0.926578i	$-0.622736\pi$
$509$	−16.9706	−0.752207	−0.376103	+	0.926578i	$0.622736\pi$
$510$	0	0
$511$	0	0
$512$	1.00000	0.0441942
$513$	0	0
$514$	−18.5858	−0.819784
$515$	−14.8284	−0.653419
$516$	0	0
$517$	5.02944	0.221194
$518$	0	0
$519$	0	0
$520$	−5.65685	−0.248069
$521$	38.7696	1.69852	0.849262	−	0.527971i	$-0.177047\pi$
$521$	38.7696	1.69852	0.849262	+	0.527971i	$0.177047\pi$
$522$	0	0
$523$	−41.7990	−1.82774	−0.913871	−	0.406004i	$-0.866922\pi$
$523$	−41.7990	−1.82774	−0.913871	+	0.406004i	$0.866922\pi$
$524$	−3.17157	−0.138551
$525$	0	0
$526$	12.1421	0.529422
$527$	26.4853	1.15372
$528$	0	0
$529$	−12.9411	−0.562658
$530$	−0.343146	−0.0149053
$531$	0	0
$532$	0	0
$533$	−59.3137	−2.56916
$534$	0	0
$535$	−9.65685	−0.417502
$536$	9.07107	0.391810
$537$	0	0
$538$	12.3431	0.532151
$539$	0	0
$540$	0	0
$541$	−29.7990	−1.28116	−0.640579	−	0.767892i	$-0.721306\pi$
$541$	−29.7990	−1.28116	−0.640579	+	0.767892i	$0.721306\pi$
$542$	−31.6985	−1.36157
$543$	0	0
$544$	2.58579	0.110865
$545$	−3.65685	−0.156642
$546$	0	0
$547$	−28.3848	−1.21365	−0.606823	−	0.794837i	$-0.707556\pi$
$547$	−28.3848	−1.21365	−0.606823	+	0.794837i	$0.707556\pi$
$548$	−13.6569	−0.583392
$549$	0	0
$550$	−2.24264	−0.0956265
$551$	−17.6569	−0.752207
$552$	0	0
$553$	0	0
$554$	−15.5563	−0.660926
$555$	0	0
$556$	2.34315	0.0993715
$557$	38.0000	1.61011	0.805056	−	0.593199i	$-0.202135\pi$
$557$	38.0000	1.61011	0.805056	+	0.593199i	$0.202135\pi$
$558$	0	0
$559$	−51.3137	−2.17034
$560$	0	0
$561$	0	0
$562$	23.4558	0.989425
$563$	−20.4853	−0.863352	−0.431676	−	0.902029i	$-0.642078\pi$
$563$	−20.4853	−0.863352	−0.431676	+	0.902029i	$0.642078\pi$
$564$	0	0
$565$	−10.0000	−0.420703
$566$	−9.51472	−0.399933
$567$	0	0
$568$	4.48528	0.188198
$569$	−9.02944	−0.378534	−0.189267	−	0.981926i	$-0.560611\pi$
$569$	−9.02944	−0.378534	−0.189267	+	0.981926i	$0.560611\pi$
$570$	0	0
$571$	4.68629	0.196115	0.0980576	−	0.995181i	$-0.468737\pi$
$571$	4.68629	0.196115	0.0980576	+	0.995181i	$0.468737\pi$
$572$	12.6863	0.530440
$573$	0	0
$574$	0	0
$575$	−3.17157	−0.132264
$576$	0	0
$577$	3.85786	0.160605	0.0803025	−	0.996771i	$-0.474411\pi$
$577$	3.85786	0.160605	0.0803025	+	0.996771i	$0.474411\pi$
$578$	−10.3137	−0.428994
$579$	0	0
$580$	−2.58579	−0.107369
$581$	0	0
$582$	0	0
$583$	0.769553	0.0318716
$584$	2.00000	0.0827606
$585$	0	0
$586$	21.3137	0.880461
$587$	−5.17157	−0.213454	−0.106727	−	0.994288i	$-0.534037\pi$
$587$	−5.17157	−0.213454	−0.106727	+	0.994288i	$0.534037\pi$
$588$	0	0
$589$	69.9411	2.88187
$590$	−3.17157	−0.130572
$591$	0	0
$592$	−0.242641	−0.00997247
$593$	−41.2132	−1.69242	−0.846212	−	0.532847i	$-0.821122\pi$
$593$	−41.2132	−1.69242	−0.846212	+	0.532847i	$0.821122\pi$
$594$	0	0
$595$	0	0
$596$	−22.3848	−0.916916
$597$	0	0
$598$	17.9411	0.733667
$599$	−4.00000	−0.163436	−0.0817178	−	0.996656i	$-0.526041\pi$
$599$	−4.00000	−0.163436	−0.0817178	+	0.996656i	$0.526041\pi$
$600$	0	0
$601$	31.7574	1.29541	0.647705	−	0.761891i	$-0.275729\pi$
$601$	31.7574	1.29541	0.647705	+	0.761891i	$0.275729\pi$
$602$	0	0
$603$	0	0
$604$	20.1421	0.819572
$605$	−5.97056	−0.242738
$606$	0	0
$607$	−4.00000	−0.162355	−0.0811775	−	0.996700i	$-0.525868\pi$
$607$	−4.00000	−0.162355	−0.0811775	+	0.996700i	$0.525868\pi$
$608$	6.82843	0.276929
$609$	0	0
$610$	11.6569	0.471972
$611$	12.6863	0.513232
$612$	0	0
$613$	−11.0711	−0.447156	−0.223578	−	0.974686i	$-0.571774\pi$
$613$	−11.0711	−0.447156	−0.223578	+	0.974686i	$0.571774\pi$
$614$	7.31371	0.295157
$615$	0	0
$616$	0	0
$617$	−33.9411	−1.36642	−0.683209	−	0.730223i	$-0.739416\pi$
$617$	−33.9411	−1.36642	−0.683209	+	0.730223i	$0.739416\pi$
$618$	0	0
$619$	−19.1127	−0.768204	−0.384102	−	0.923291i	$-0.625489\pi$
$619$	−19.1127	−0.768204	−0.384102	+	0.923291i	$0.625489\pi$
$620$	10.2426	0.411354
$621$	0	0
$622$	−10.8284	−0.434180
$623$	0	0
$624$	0	0
$625$	1.00000	0.0400000
$626$	−30.4853	−1.21844
$627$	0	0
$628$	8.34315	0.332928
$629$	−0.627417	−0.0250168
$630$	0	0
$631$	−8.14214	−0.324133	−0.162067	−	0.986780i	$-0.551816\pi$
$631$	−8.14214	−0.324133	−0.162067	+	0.986780i	$0.551816\pi$
$632$	8.00000	0.318223
$633$	0	0
$634$	25.3137	1.00534
$635$	2.82843	0.112243
$636$	0	0
$637$	0	0
$638$	5.79899	0.229584
$639$	0	0
$640$	1.00000	0.0395285
$641$	−26.0000	−1.02694	−0.513469	−	0.858108i	$-0.671640\pi$
$641$	−26.0000	−1.02694	−0.513469	+	0.858108i	$0.671640\pi$
$642$	0	0
$643$	−28.8284	−1.13688	−0.568441	−	0.822724i	$-0.692453\pi$
$643$	−28.8284	−1.13688	−0.568441	+	0.822724i	$0.692453\pi$
$644$	0	0
$645$	0	0
$646$	17.6569	0.694700
$647$	8.87006	0.348718	0.174359	−	0.984682i	$-0.444215\pi$
$647$	8.87006	0.348718	0.174359	+	0.984682i	$0.444215\pi$
$648$	0	0
$649$	7.11270	0.279198
$650$	−5.65685	−0.221880
$651$	0	0
$652$	−17.0711	−0.668555
$653$	1.79899	0.0703999	0.0352000	−	0.999380i	$-0.488793\pi$
$653$	1.79899	0.0703999	0.0352000	+	0.999380i	$0.488793\pi$
$654$	0	0
$655$	−3.17157	−0.123924
$656$	10.4853	0.409381
$657$	0	0
$658$	0	0
$659$	−20.3848	−0.794078	−0.397039	−	0.917802i	$-0.629962\pi$
$659$	−20.3848	−0.794078	−0.397039	+	0.917802i	$0.629962\pi$
$660$	0	0
$661$	5.02944	0.195622	0.0978112	−	0.995205i	$-0.468816\pi$
$661$	5.02944	0.195622	0.0978112	+	0.995205i	$0.468816\pi$
$662$	−6.34315	−0.246533
$663$	0	0
$664$	5.17157	0.200696
$665$	0	0
$666$	0	0
$667$	8.20101	0.317544
$668$	2.24264	0.0867704
$669$	0	0
$670$	9.07107	0.350446
$671$	−26.1421	−1.00921
$672$	0	0
$673$	−26.2843	−1.01318	−0.506592	−	0.862186i	$-0.669095\pi$
$673$	−26.2843	−1.01318	−0.506592	+	0.862186i	$0.669095\pi$
$674$	33.1127	1.27545
$675$	0	0
$676$	19.0000	0.730769
$677$	−44.4264	−1.70745	−0.853723	−	0.520728i	$-0.825661\pi$
$677$	−44.4264	−1.70745	−0.853723	+	0.520728i	$0.825661\pi$
$678$	0	0
$679$	0	0
$680$	2.58579	0.0991604
$681$	0	0
$682$	−22.9706	−0.879588
$683$	31.7990	1.21675	0.608377	−	0.793648i	$-0.291821\pi$
$683$	31.7990	1.21675	0.608377	+	0.793648i	$0.291821\pi$
$684$	0	0
$685$	−13.6569	−0.521802
$686$	0	0
$687$	0	0
$688$	9.07107	0.345831
$689$	1.94113	0.0739510
$690$	0	0
$691$	−3.02944	−0.115245	−0.0576226	−	0.998338i	$-0.518352\pi$
$691$	−3.02944	−0.115245	−0.0576226	+	0.998338i	$0.518352\pi$
$692$	20.8284	0.791778
$693$	0	0
$694$	22.3431	0.848134
$695$	2.34315	0.0888806
$696$	0	0
$697$	27.1127	1.02697
$698$	−23.4558	−0.887817
$699$	0	0
$700$	0	0
$701$	17.2132	0.650134	0.325067	−	0.945691i	$-0.394613\pi$
$701$	17.2132	0.650134	0.325067	+	0.945691i	$0.394613\pi$
$702$	0	0
$703$	−1.65685	−0.0624894
$704$	−2.24264	−0.0845227
$705$	0	0
$706$	16.0416	0.603735
$707$	0	0
$708$	0	0
$709$	11.8579	0.445331	0.222666	−	0.974895i	$-0.428524\pi$
$709$	11.8579	0.445331	0.222666	+	0.974895i	$0.428524\pi$
$710$	4.48528	0.168330
$711$	0	0
$712$	−0.828427	−0.0310466
$713$	−32.4853	−1.21658
$714$	0	0
$715$	12.6863	0.474440
$716$	−26.2426	−0.980734
$717$	0	0
$718$	28.4853	1.06306
$719$	17.9411	0.669091	0.334546	−	0.942380i	$-0.391417\pi$
$719$	17.9411	0.669091	0.334546	+	0.942380i	$0.391417\pi$
$720$	0	0
$721$	0	0
$722$	27.6274	1.02819
$723$	0	0
$724$	−18.4853	−0.687000
$725$	−2.58579	−0.0960337
$726$	0	0
$727$	30.6274	1.13591	0.567954	−	0.823060i	$-0.307735\pi$
$727$	30.6274	1.13591	0.567954	+	0.823060i	$0.307735\pi$
$728$	0	0
$729$	0	0
$730$	2.00000	0.0740233
$731$	23.4558	0.867546
$732$	0	0
$733$	34.0000	1.25582	0.627909	−	0.778287i	$-0.283911\pi$
$733$	34.0000	1.25582	0.627909	+	0.778287i	$0.283911\pi$
$734$	10.8284	0.399685
$735$	0	0
$736$	−3.17157	−0.116906
$737$	−20.3431	−0.749349
$738$	0	0
$739$	11.1127	0.408787	0.204394	−	0.978889i	$-0.434478\pi$
$739$	11.1127	0.408787	0.204394	+	0.978889i	$0.434478\pi$
$740$	−0.242641	−0.00891965
$741$	0	0
$742$	0	0
$743$	−28.2843	−1.03765	−0.518825	−	0.854881i	$-0.673630\pi$
$743$	−28.2843	−1.03765	−0.518825	+	0.854881i	$0.673630\pi$
$744$	0	0
$745$	−22.3848	−0.820115
$746$	6.58579	0.241123
$747$	0	0
$748$	−5.79899	−0.212032
$749$	0	0
$750$	0	0
$751$	21.7990	0.795456	0.397728	−	0.917503i	$-0.369799\pi$
$751$	21.7990	0.795456	0.397728	+	0.917503i	$0.369799\pi$
$752$	−2.24264	−0.0817807
$753$	0	0
$754$	14.6274	0.532699
$755$	20.1421	0.733047
$756$	0	0
$757$	25.8995	0.941333	0.470667	−	0.882311i	$-0.344013\pi$
$757$	25.8995	0.941333	0.470667	+	0.882311i	$0.344013\pi$
$758$	0.485281	0.0176262
$759$	0	0
$760$	6.82843	0.247693
$761$	−26.9706	−0.977682	−0.488841	−	0.872373i	$-0.662580\pi$
$761$	−26.9706	−0.977682	−0.488841	+	0.872373i	$0.662580\pi$
$762$	0	0
$763$	0	0
$764$	−4.00000	−0.144715
$765$	0	0
$766$	19.4142	0.701464
$767$	17.9411	0.647816
$768$	0	0
$769$	−4.72792	−0.170493	−0.0852466	−	0.996360i	$-0.527168\pi$
$769$	−4.72792	−0.170493	−0.0852466	+	0.996360i	$0.527168\pi$
$770$	0	0
$771$	0	0
$772$	−4.82843	−0.173779
$773$	−22.9706	−0.826194	−0.413097	−	0.910687i	$-0.635553\pi$
$773$	−22.9706	−0.826194	−0.413097	+	0.910687i	$0.635553\pi$
$774$	0	0
$775$	10.2426	0.367927
$776$	6.48528	0.232808
$777$	0	0
$778$	29.8995	1.07195
$779$	71.5980	2.56526
$780$	0	0
$781$	−10.0589	−0.359935
$782$	−8.20101	−0.293268
$783$	0	0
$784$	0	0
$785$	8.34315	0.297780
$786$	0	0
$787$	−38.3431	−1.36679	−0.683393	−	0.730051i	$-0.739496\pi$
$787$	−38.3431	−1.36679	−0.683393	+	0.730051i	$0.739496\pi$
$788$	25.7990	0.919051
$789$	0	0
$790$	8.00000	0.284627
$791$	0	0
$792$	0	0
$793$	−65.9411	−2.34164
$794$	−6.68629	−0.237288
$795$	0	0
$796$	5.75736	0.204064
$797$	0.142136	0.00503470	0.00251735	−	0.999997i	$-0.499199\pi$
$797$	0.142136	0.00503470	0.00251735	+	0.999997i	$0.499199\pi$
$798$	0	0
$799$	−5.79899	−0.205154
$800$	1.00000	0.0353553
$801$	0	0
$802$	6.00000	0.211867
$803$	−4.48528	−0.158282
$804$	0	0
$805$	0	0
$806$	−57.9411	−2.04089
$807$	0	0
$808$	17.6569	0.621166
$809$	−34.4853	−1.21244	−0.606219	−	0.795298i	$-0.707314\pi$
$809$	−34.4853	−1.21244	−0.606219	+	0.795298i	$0.707314\pi$
$810$	0	0
$811$	−10.3431	−0.363197	−0.181598	−	0.983373i	$-0.558127\pi$
$811$	−10.3431	−0.363197	−0.181598	+	0.983373i	$0.558127\pi$
$812$	0	0
$813$	0	0
$814$	0.544156	0.0190727
$815$	−17.0711	−0.597973
$816$	0	0
$817$	61.9411	2.16705
$818$	−4.24264	−0.148340
$819$	0	0
$820$	10.4853	0.366162
$821$	−1.21320	−0.0423411	−0.0211705	−	0.999776i	$-0.506739\pi$
$821$	−1.21320	−0.0423411	−0.0211705	+	0.999776i	$0.506739\pi$
$822$	0	0
$823$	28.9706	1.00985	0.504925	−	0.863163i	$-0.331520\pi$
$823$	28.9706	1.00985	0.504925	+	0.863163i	$0.331520\pi$
$824$	−14.8284	−0.516573
$825$	0	0
$826$	0	0
$827$	−35.5147	−1.23497	−0.617484	−	0.786584i	$-0.711848\pi$
$827$	−35.5147	−1.23497	−0.617484	+	0.786584i	$0.711848\pi$
$828$	0	0
$829$	−33.3137	−1.15703	−0.578516	−	0.815671i	$-0.696368\pi$
$829$	−33.3137	−1.15703	−0.578516	+	0.815671i	$0.696368\pi$
$830$	5.17157	0.179508
$831$	0	0
$832$	−5.65685	−0.196116
$833$	0	0
$834$	0	0
$835$	2.24264	0.0776098
$836$	−15.3137	−0.529636
$837$	0	0
$838$	−36.9706	−1.27713
$839$	8.48528	0.292944	0.146472	−	0.989215i	$-0.453208\pi$
$839$	8.48528	0.292944	0.146472	+	0.989215i	$0.453208\pi$
$840$	0	0
$841$	−22.3137	−0.769438
$842$	−3.65685	−0.126024
$843$	0	0
$844$	−23.3137	−0.802491
$845$	19.0000	0.653620
$846$	0	0
$847$	0	0
$848$	−0.343146	−0.0117837
$849$	0	0
$850$	2.58579	0.0886917
$851$	0.769553	0.0263799
$852$	0	0
$853$	40.6274	1.39106	0.695528	−	0.718499i	$-0.255170\pi$
$853$	40.6274	1.39106	0.695528	+	0.718499i	$0.255170\pi$
$854$	0	0
$855$	0	0
$856$	−9.65685	−0.330064
$857$	24.7279	0.844690	0.422345	−	0.906435i	$-0.361207\pi$
$857$	24.7279	0.844690	0.422345	+	0.906435i	$0.361207\pi$
$858$	0	0
$859$	35.5980	1.21459	0.607294	−	0.794477i	$-0.292255\pi$
$859$	35.5980	1.21459	0.607294	+	0.794477i	$0.292255\pi$
$860$	9.07107	0.309321
$861$	0	0
$862$	−13.4558	−0.458308
$863$	−45.6569	−1.55418	−0.777089	−	0.629391i	$-0.783304\pi$
$863$	−45.6569	−1.55418	−0.777089	+	0.629391i	$0.783304\pi$
$864$	0	0
$865$	20.8284	0.708188
$866$	17.7990	0.604834
$867$	0	0
$868$	0	0
$869$	−17.9411	−0.608611
$870$	0	0
$871$	−51.3137	−1.73870
$872$	−3.65685	−0.123837
$873$	0	0
$874$	−21.6569	−0.732554
$875$	0	0
$876$	0	0
$877$	4.24264	0.143264	0.0716319	−	0.997431i	$-0.477179\pi$
$877$	4.24264	0.143264	0.0716319	+	0.997431i	$0.477179\pi$
$878$	23.6985	0.799785
$879$	0	0
$880$	−2.24264	−0.0755994
$881$	26.0000	0.875962	0.437981	−	0.898984i	$-0.355694\pi$
$881$	26.0000	0.875962	0.437981	+	0.898984i	$0.355694\pi$
$882$	0	0
$883$	−34.2426	−1.15236	−0.576178	−	0.817324i	$-0.695457\pi$
$883$	−34.2426	−1.15236	−0.576178	+	0.817324i	$0.695457\pi$
$884$	−14.6274	−0.491973
$885$	0	0
$886$	−36.2843	−1.21899
$887$	−32.8701	−1.10367	−0.551834	−	0.833954i	$-0.686072\pi$
$887$	−32.8701	−1.10367	−0.551834	+	0.833954i	$0.686072\pi$
$888$	0	0
$889$	0	0
$890$	−0.828427	−0.0277689
$891$	0	0
$892$	7.31371	0.244881
$893$	−15.3137	−0.512454
$894$	0	0
$895$	−26.2426	−0.877195
$896$	0	0
$897$	0	0
$898$	10.4853	0.349898
$899$	−26.4853	−0.883334
$900$	0	0
$901$	−0.887302	−0.0295603
$902$	−23.5147	−0.782954
$903$	0	0
$904$	−10.0000	−0.332595
$905$	−18.4853	−0.614472
$906$	0	0
$907$	−16.8701	−0.560161	−0.280081	−	0.959977i	$-0.590361\pi$
$907$	−16.8701	−0.560161	−0.280081	+	0.959977i	$0.590361\pi$
$908$	−24.9706	−0.828677
$909$	0	0
$910$	0	0
$911$	−44.9706	−1.48994	−0.744971	−	0.667097i	$-0.767537\pi$
$911$	−44.9706	−1.48994	−0.744971	+	0.667097i	$0.767537\pi$
$912$	0	0
$913$	−11.5980	−0.383837
$914$	2.48528	0.0822058
$915$	0	0
$916$	16.1421	0.533351
$917$	0	0
$918$	0	0
$919$	−30.7696	−1.01499	−0.507497	−	0.861654i	$-0.669429\pi$
$919$	−30.7696	−1.01499	−0.507497	+	0.861654i	$0.669429\pi$
$920$	−3.17157	−0.104564
$921$	0	0
$922$	2.00000	0.0658665
$923$	−25.3726	−0.835149
$924$	0	0
$925$	−0.242641	−0.00797798
$926$	−25.6569	−0.843137
$927$	0	0
$928$	−2.58579	−0.0848826
$929$	55.6569	1.82604	0.913021	−	0.407912i	$-0.133743\pi$
$929$	55.6569	1.82604	0.913021	+	0.407912i	$0.133743\pi$
$930$	0	0
$931$	0	0
$932$	−18.9706	−0.621401
$933$	0	0
$934$	12.4853	0.408531
$935$	−5.79899	−0.189647
$936$	0	0
$937$	−36.3431	−1.18728	−0.593639	−	0.804731i	$-0.702309\pi$
$937$	−36.3431	−1.18728	−0.593639	+	0.804731i	$0.702309\pi$
$938$	0	0
$939$	0	0
$940$	−2.24264	−0.0731469
$941$	−28.2843	−0.922041	−0.461020	−	0.887390i	$-0.652517\pi$
$941$	−28.2843	−0.922041	−0.461020	+	0.887390i	$0.652517\pi$
$942$	0	0
$943$	−33.2548	−1.08293
$944$	−3.17157	−0.103226
$945$	0	0
$946$	−20.3431	−0.661413
$947$	9.17157	0.298036	0.149018	−	0.988834i	$-0.452389\pi$
$947$	9.17157	0.298036	0.149018	+	0.988834i	$0.452389\pi$
$948$	0	0
$949$	−11.3137	−0.367259
$950$	6.82843	0.221543
$951$	0	0
$952$	0	0
$953$	−4.68629	−0.151804	−0.0759019	−	0.997115i	$-0.524184\pi$
$953$	−4.68629	−0.151804	−0.0759019	+	0.997115i	$0.524184\pi$
$954$	0	0
$955$	−4.00000	−0.129437
$956$	19.3137	0.624650
$957$	0	0
$958$	−8.48528	−0.274147
$959$	0	0
$960$	0	0
$961$	73.9117	2.38425
$962$	1.37258	0.0442539
$963$	0	0
$964$	19.5563	0.629868
$965$	−4.82843	−0.155433
$966$	0	0
$967$	−10.3431	−0.332613	−0.166307	−	0.986074i	$-0.553184\pi$
$967$	−10.3431	−0.332613	−0.166307	+	0.986074i	$0.553184\pi$
$968$	−5.97056	−0.191901
$969$	0	0
$970$	6.48528	0.208230
$971$	−4.97056	−0.159513	−0.0797565	−	0.996814i	$-0.525414\pi$
$971$	−4.97056	−0.159513	−0.0797565	+	0.996814i	$0.525414\pi$
$972$	0	0
$973$	0	0
$974$	10.8284	0.346965
$975$	0	0
$976$	11.6569	0.373127
$977$	29.3137	0.937829	0.468914	−	0.883244i	$-0.344645\pi$
$977$	29.3137	0.937829	0.468914	+	0.883244i	$0.344645\pi$
$978$	0	0
$979$	1.85786	0.0593776
$980$	0	0
$981$	0	0
$982$	17.0711	0.544760
$983$	−17.0711	−0.544483	−0.272241	−	0.962229i	$-0.587765\pi$
$983$	−17.0711	−0.544483	−0.272241	+	0.962229i	$0.587765\pi$
$984$	0	0
$985$	25.7990	0.822024
$986$	−6.68629	−0.212935
$987$	0	0
$988$	−38.6274	−1.22890
$989$	−28.7696	−0.914819
$990$	0	0
$991$	−10.2010	−0.324046	−0.162023	−	0.986787i	$-0.551802\pi$
$991$	−10.2010	−0.324046	−0.162023	+	0.986787i	$0.551802\pi$
$992$	10.2426	0.325204
$993$	0	0
$994$	0	0
$995$	5.75736	0.182521
$996$	0	0
$997$	56.6274	1.79341	0.896704	−	0.442630i	$-0.145955\pi$
$997$	56.6274	1.79341	0.896704	+	0.442630i	$0.145955\pi$
$998$	6.82843	0.216150
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	4410.2.a.bz.1.1		2
3.2	odd	2		1470.2.a.t.1.2	yes	2
7.6	odd	2		4410.2.a.bw.1.1		2
15.14	odd	2		7350.2.a.dh.1.2		2
21.2	odd	6		1470.2.i.w.361.1		4
21.5	even	6		1470.2.i.x.361.1		4
21.11	odd	6		1470.2.i.w.961.1		4
21.17	even	6		1470.2.i.x.961.1		4
21.20	even	2		1470.2.a.s.1.2	✓	2
105.104	even	2		7350.2.a.dl.1.2		2

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
1470.2.a.s.1.2	✓	2	21.20	even	2
1470.2.a.t.1.2	yes	2	3.2	odd	2
1470.2.i.w.361.1		4	21.2	odd	6
1470.2.i.w.961.1		4	21.11	odd	6
1470.2.i.x.361.1		4	21.5	even	6
1470.2.i.x.961.1		4	21.17	even	6
4410.2.a.bw.1.1		2	7.6	odd	2
4410.2.a.bz.1.1		2	1.1	even	1	trivial
7350.2.a.dh.1.2		2	15.14	odd	2
7350.2.a.dl.1.2		2	105.104	even	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:	\( N \)	\(=\)	\( 4410 = 2 \cdot 3^{2} \cdot 5 \cdot 7^{2} \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	4410.a (trivial)

\(f(q)\)	\(=\)	\(q+1.00000 q^{2} +1.00000 q^{4} +1.00000 q^{5} +1.00000 q^{8} +O(q^{10})\)	\(q+1.00000 q^{2} +1.00000 q^{4} +1.00000 q^{5} +1.00000 q^{8} +1.00000 q^{10} -2.24264 q^{11} -5.65685 q^{13} +1.00000 q^{16} +2.58579 q^{17} +6.82843 q^{19} +1.00000 q^{20} -2.24264 q^{22} -3.17157 q^{23} +1.00000 q^{25} -5.65685 q^{26} -2.58579 q^{29} +10.2426 q^{31} +1.00000 q^{32} +2.58579 q^{34} -0.242641 q^{37} +6.82843 q^{38} +1.00000 q^{40} +10.4853 q^{41} +9.07107 q^{43} -2.24264 q^{44} -3.17157 q^{46} -2.24264 q^{47} +1.00000 q^{50} -5.65685 q^{52} -0.343146 q^{53} -2.24264 q^{55} -2.58579 q^{58} -3.17157 q^{59} +11.6569 q^{61} +10.2426 q^{62} +1.00000 q^{64} -5.65685 q^{65} +9.07107 q^{67} +2.58579 q^{68} +4.48528 q^{71} +2.00000 q^{73} -0.242641 q^{74} +6.82843 q^{76} +8.00000 q^{79} +1.00000 q^{80} +10.4853 q^{82} +5.17157 q^{83} +2.58579 q^{85} +9.07107 q^{86} -2.24264 q^{88} -0.828427 q^{89} -3.17157 q^{92} -2.24264 q^{94} +6.82843 q^{95} +6.48528 q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 2 q + 2 q^{2} + 2 q^{4} + 2 q^{5} + 2 q^{8}+O(q^{10}) \) 2 * q + 2 * q^2 + 2 * q^4 + 2 * q^5 + 2 * q^8	\( 2 q + 2 q^{2} + 2 q^{4} + 2 q^{5} + 2 q^{8} + 2 q^{10} + 4 q^{11} + 2 q^{16} + 8 q^{17} + 8 q^{19} + 2 q^{20} + 4 q^{22} - 12 q^{23} + 2 q^{25} - 8 q^{29} + 12 q^{31} + 2 q^{32} + 8 q^{34} + 8 q^{37} + 8 q^{38} + 2 q^{40} + 4 q^{41} + 4 q^{43} + 4 q^{44} - 12 q^{46} + 4 q^{47} + 2 q^{50} - 12 q^{53} + 4 q^{55} - 8 q^{58} - 12 q^{59} + 12 q^{61} + 12 q^{62} + 2 q^{64} + 4 q^{67} + 8 q^{68} - 8 q^{71} + 4 q^{73} + 8 q^{74} + 8 q^{76} + 16 q^{79} + 2 q^{80} + 4 q^{82} + 16 q^{83} + 8 q^{85} + 4 q^{86} + 4 q^{88} + 4 q^{89} - 12 q^{92} + 4 q^{94} + 8 q^{95} - 4 q^{97}+O(q^{100}) \) 2 * q + 2 * q^2 + 2 * q^4 + 2 * q^5 + 2 * q^8 + 2 * q^10 + 4 * q^11 + 2 * q^16 + 8 * q^17 + 8 * q^19 + 2 * q^20 + 4 * q^22 - 12 * q^23 + 2 * q^25 - 8 * q^29 + 12 * q^31 + 2 * q^32 + 8 * q^34 + 8 * q^37 + 8 * q^38 + 2 * q^40 + 4 * q^41 + 4 * q^43 + 4 * q^44 - 12 * q^46 + 4 * q^47 + 2 * q^50 - 12 * q^53 + 4 * q^55 - 8 * q^58 - 12 * q^59 + 12 * q^61 + 12 * q^62 + 2 * q^64 + 4 * q^67 + 8 * q^68 - 8 * q^71 + 4 * q^73 + 8 * q^74 + 8 * q^76 + 16 * q^79 + 2 * q^80 + 4 * q^82 + 16 * q^83 + 8 * q^85 + 4 * q^86 + 4 * q^88 + 4 * q^89 - 12 * q^92 + 4 * q^94 + 8 * q^95 - 4 * q^97

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