LMFDB - Embedded newform 4410.2.b.c.881.1

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [4410,2,Mod(881,4410)]

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([1, 0, 1]))

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

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

chi := DirichletCharacter("4410.881");

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.b (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:	$35.2140272914$
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:	$\Z[a_1, \ldots, a_{13}]$
Coefficient ring index:	$ 2^{3} $
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label			881.1
Root			$-0.923880 + 0.382683i$ of defining polynomial
Character	$\chi$	$=$	4410.881
Dual form			4410.2.b.c.881.8

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-1.00000i q^{2} -1.00000 q^{4} -1.00000 q^{5} +1.00000i q^{8} +O(q^{10})$	$q-1.00000i q^{2} -1.00000 q^{4} -1.00000 q^{5} +1.00000i q^{8} +1.00000i q^{10} -1.37849i q^{11} -4.02734i q^{13} +1.00000 q^{16} +0.648847 q^{17} +5.77337i q^{19} +1.00000 q^{20} -1.37849 q^{22} +1.66818i q^{23} +1.00000 q^{25} -4.02734 q^{26} -2.53233i q^{29} -6.20692i q^{31} -1.00000i q^{32} -0.648847i q^{34} +10.9366 q^{37} +5.77337 q^{38} -1.00000i q^{40} -6.26998 q^{41} -7.21146 q^{43} +1.37849i q^{44} +1.66818 q^{46} +0.598330 q^{47} -1.00000i q^{50} +4.02734i q^{52} -7.48022i q^{53} +1.37849i q^{55} -2.53233 q^{58} -3.71870 q^{59} -4.94495i q^{61} -6.20692 q^{62} -1.00000 q^{64} +4.02734i q^{65} -3.76990 q^{67} -0.648847 q^{68} -3.97682i q^{71} +7.53490i q^{73} -10.9366i q^{74} -5.77337i q^{76} +10.4525 q^{79} -1.00000 q^{80} +6.26998i q^{82} -4.08746 q^{83} -0.648847 q^{85} +7.21146i q^{86} +1.37849 q^{88} -9.76696 q^{89} -1.66818i q^{92} -0.598330i q^{94} -5.77337i q^{95} -7.65685i q^{97} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 8 q - 8 q^{4} - 8 q^{5}+O(q^{10}) $ 8 * q - 8 * q^4 - 8 * q^5	$ 8 q - 8 q^{4} - 8 q^{5} + 8 q^{16} + 8 q^{20} + 16 q^{22} + 8 q^{25} + 16 q^{37} + 16 q^{41} - 16 q^{43} + 16 q^{46} - 16 q^{47} - 16 q^{58} - 48 q^{59} - 8 q^{64} - 32 q^{67} - 8 q^{80} + 16 q^{83} - 16 q^{88} - 80 q^{89}+O(q^{100}) $ 8 * q - 8 * q^4 - 8 * q^5 + 8 * q^16 + 8 * q^20 + 16 * q^22 + 8 * q^25 + 16 * q^37 + 16 * q^41 - 16 * q^43 + 16 * q^46 - 16 * q^47 - 16 * q^58 - 48 * q^59 - 8 * q^64 - 32 * q^67 - 8 * q^80 + 16 * q^83 - 16 * q^88 - 80 * q^89

Display 100 coefficients 10 coefficients

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/4410\mathbb{Z}\right)^\times$.

$n$	$1081$	$2647$	$3431$
$\chi(n)$	$-1$	$1$	$-1$

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 $

$2$	− 1.00000i	− 0.707107i
$2$	− 1.00000i	− 0.707107i
$3$	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
$8$	1.00000i	0.353553i
$9$	0	0
$10$	1.00000i	0.316228i
$11$	− 1.37849i	− 0.415631i	−0.978168	−	0.207816i	$-0.933365\pi$
$11$	− 1.37849i	− 0.415631i	0.978168	−	0.207816i	$-0.0666354\pi$
$12$	0	0
$13$	− 4.02734i	− 1.11698i	−0.829510	−	0.558492i	$-0.811380\pi$
$13$	− 4.02734i	− 1.11698i	0.829510	−	0.558492i	$-0.188620\pi$
$14$	0	0
$15$	0	0
$16$	1.00000	0.250000
$17$	0.648847	0.157368	0.0786842	−	0.996900i	$-0.474928\pi$
$17$	0.648847	0.157368	0.0786842	+	0.996900i	$0.474928\pi$
$18$	0	0
$19$	5.77337i	1.32450i	0.749282	+	0.662251i	$0.230399\pi$
$19$	5.77337i	1.32450i	−0.749282	+	0.662251i	$0.769601\pi$
$20$	1.00000	0.223607
$21$	0	0
$22$	−1.37849	−0.293896
$23$	1.66818i	0.347839i	0.984760	+	0.173920i	$0.0556433\pi$
$23$	1.66818i	0.347839i	−0.984760	+	0.173920i	$0.944357\pi$
$24$	0	0
$25$	1.00000	0.200000
$26$	−4.02734	−0.789826
$27$	0	0
$28$	0	0
$29$	− 2.53233i	− 0.470241i	−0.971966	−	0.235121i	$-0.924451\pi$
$29$	− 2.53233i	− 0.470241i	0.971966	−	0.235121i	$-0.0755485\pi$
$30$	0	0
$31$	− 6.20692i	− 1.11480i	−0.830246	−	0.557398i	$-0.811800\pi$
$31$	− 6.20692i	− 1.11480i	0.830246	−	0.557398i	$-0.188200\pi$
$32$	− 1.00000i	− 0.176777i
$33$	0	0
$34$	− 0.648847i	− 0.111276i
$35$	0	0
$36$	0	0
$37$	10.9366	1.79796	0.898980	−	0.437989i	$-0.144309\pi$
$37$	10.9366	1.79796	0.898980	+	0.437989i	$0.144309\pi$
$38$	5.77337	0.936565
$39$	0	0
$40$	− 1.00000i	− 0.158114i
$41$	−6.26998	−0.979206	−0.489603	−	0.871945i	$-0.662858\pi$
$41$	−6.26998	−0.979206	−0.489603	+	0.871945i	$0.662858\pi$
$42$	0	0
$43$	−7.21146	−1.09974	−0.549868	−	0.835251i	$-0.685322\pi$
$43$	−7.21146	−1.09974	−0.549868	+	0.835251i	$0.685322\pi$
$44$	1.37849i	0.207816i
$45$	0	0
$46$	1.66818	0.245960
$47$	0.598330	0.0872754	0.0436377	−	0.999047i	$-0.486105\pi$
$47$	0.598330	0.0872754	0.0436377	+	0.999047i	$0.486105\pi$
$48$	0	0
$49$	0	0
$50$	− 1.00000i	− 0.141421i
$51$	0	0
$52$	4.02734i	0.558492i
$53$	− 7.48022i	− 1.02749i	−0.857944	−	0.513743i	$-0.828258\pi$
$53$	− 7.48022i	− 1.02749i	0.857944	−	0.513743i	$-0.171742\pi$
$54$	0	0
$55$	1.37849i	0.185876i
$56$	0	0
$57$	0	0
$58$	−2.53233	−0.332511
$59$	−3.71870	−0.484133	−0.242066	−	0.970260i	$-0.577825\pi$
$59$	−3.71870	−0.484133	−0.242066	+	0.970260i	$0.577825\pi$
$60$	0	0
$61$	− 4.94495i	− 0.633136i	−0.948570	−	0.316568i	$-0.897470\pi$
$61$	− 4.94495i	− 0.633136i	0.948570	−	0.316568i	$-0.102530\pi$
$62$	−6.20692	−0.788280
$63$	0	0
$64$	−1.00000	−0.125000
$65$	4.02734i	0.499530i
$66$	0	0
$67$	−3.76990	−0.460567	−0.230283	−	0.973124i	$-0.573965\pi$
$67$	−3.76990	−0.460567	−0.230283	+	0.973124i	$0.573965\pi$
$68$	−0.648847	−0.0786842
$69$	0	0
$70$	0	0
$71$	− 3.97682i	− 0.471962i	−0.971758	−	0.235981i	$-0.924170\pi$
$71$	− 3.97682i	− 0.471962i	0.971758	−	0.235981i	$-0.0758303\pi$
$72$	0	0
$73$	7.53490i	0.881893i	0.897533	+	0.440946i	$0.145357\pi$
$73$	7.53490i	0.881893i	−0.897533	+	0.440946i	$0.854643\pi$
$74$	− 10.9366i	− 1.27135i
$75$	0	0
$76$	− 5.77337i	− 0.662251i
$77$	0	0
$78$	0	0
$79$	10.4525	1.17600	0.587999	−	0.808861i	$-0.299916\pi$
$79$	10.4525	1.17600	0.587999	+	0.808861i	$0.299916\pi$
$80$	−1.00000	−0.111803
$81$	0	0
$82$	6.26998i	0.692403i
$83$	−4.08746	−0.448657	−0.224328	−	0.974514i	$-0.572019\pi$
$83$	−4.08746	−0.448657	−0.224328	+	0.974514i	$0.572019\pi$
$84$	0	0
$85$	−0.648847	−0.0703773
$86$	7.21146i	0.777631i
$87$	0	0
$88$	1.37849	0.146948
$89$	−9.76696	−1.03530	−0.517648	−	0.855594i	$-0.673192\pi$
$89$	−9.76696	−1.03530	−0.517648	+	0.855594i	$0.673192\pi$
$90$	0	0
$91$	0	0
$92$	− 1.66818i	− 0.173920i
$93$	0	0
$94$	− 0.598330i	− 0.0617130i
$95$	− 5.77337i	− 0.592336i
$96$	0	0
$97$	− 7.65685i	− 0.777436i	−0.921357	−	0.388718i	$-0.872918\pi$
$97$	− 7.65685i	− 0.777436i	0.921357	−	0.388718i	$-0.127082\pi$
$98$	0	0
$99$	0	0
$100$	−1.00000	−0.100000
$101$	−7.45288	−0.741589	−0.370795	−	0.928715i	$-0.620915\pi$
$101$	−7.45288	−0.741589	−0.370795	+	0.928715i	$0.620915\pi$
$102$	0	0
$103$	2.21530i	0.218280i	0.994026	+	0.109140i	$0.0348097\pi$
$103$	2.21530i	0.218280i	−0.994026	+	0.109140i	$0.965190\pi$
$104$	4.02734	0.394913
$105$	0	0
$106$	−7.48022	−0.726543
$107$	13.6063i	1.31537i	0.753291	+	0.657687i	$0.228465\pi$
$107$	13.6063i	1.31537i	−0.753291	+	0.657687i	$0.771535\pi$
$108$	0	0
$109$	8.64272	0.827822	0.413911	−	0.910317i	$-0.364162\pi$
$109$	8.64272	0.827822	0.413911	+	0.910317i	$0.364162\pi$
$110$	1.37849	0.131434
$111$	0	0
$112$	0	0
$113$	− 7.59448i	− 0.714429i	−0.934022	−	0.357215i	$-0.883727\pi$
$113$	− 7.59448i	− 0.714429i	0.934022	−	0.357215i	$-0.116273\pi$
$114$	0	0
$115$	− 1.66818i	− 0.155558i
$116$	2.53233i	0.235121i
$117$	0	0
$118$	3.71870i	0.342334i
$119$	0	0
$120$	0	0
$121$	9.09976	0.827251
$122$	−4.94495	−0.447694
$123$	0	0
$124$	6.20692i	0.557398i
$125$	−1.00000	−0.0894427
$126$	0	0
$127$	−4.02509	−0.357169	−0.178584	−	0.983925i	$-0.557152\pi$
$127$	−4.02509	−0.357169	−0.178584	+	0.983925i	$0.557152\pi$
$128$	1.00000i	0.0883883i
$129$	0	0
$130$	4.02734	0.353221
$131$	−20.9314	−1.82878	−0.914391	−	0.404832i	$-0.867330\pi$
$131$	−20.9314	−1.82878	−0.914391	+	0.404832i	$0.867330\pi$
$132$	0	0
$133$	0	0
$134$	3.76990i	0.325670i
$135$	0	0
$136$	0.648847i	0.0556381i
$137$	2.20889i	0.188718i	0.995538	+	0.0943590i	$0.0300801\pi$
$137$	2.20889i	0.188718i	−0.995538	+	0.0943590i	$0.969920\pi$
$138$	0	0
$139$	− 7.34596i	− 0.623076i	−0.950234	−	0.311538i	$-0.899156\pi$
$139$	− 7.34596i	− 0.623076i	0.950234	−	0.311538i	$-0.100844\pi$
$140$	0	0
$141$	0	0
$142$	−3.97682	−0.333728
$143$	−5.55166	−0.464253
$144$	0	0
$145$	2.53233i	0.210298i
$146$	7.53490	0.623592
$147$	0	0
$148$	−10.9366	−0.898980
$149$	− 10.9807i	− 0.899571i	−0.893137	−	0.449786i	$-0.851500\pi$
$149$	− 10.9807i	− 0.899571i	0.893137	−	0.449786i	$-0.148500\pi$
$150$	0	0
$151$	−15.4006	−1.25329	−0.626643	−	0.779306i	$-0.715572\pi$
$151$	−15.4006	−1.25329	−0.626643	+	0.779306i	$0.715572\pi$
$152$	−5.77337	−0.468283
$153$	0	0
$154$	0	0
$155$	6.20692i	0.498552i
$156$	0	0
$157$	− 4.02959i	− 0.321596i	−0.986987	−	0.160798i	$-0.948593\pi$
$157$	− 4.02959i	− 0.321596i	0.986987	−	0.160798i	$-0.0514068\pi$
$158$	− 10.4525i	− 0.831557i
$159$	0	0
$160$	1.00000i	0.0790569i
$161$	0	0
$162$	0	0
$163$	1.33023	0.104192	0.0520958	−	0.998642i	$-0.483410\pi$
$163$	1.33023	0.104192	0.0520958	+	0.998642i	$0.483410\pi$
$164$	6.26998	0.489603
$165$	0	0
$166$	4.08746i	0.317248i
$167$	−18.6185	−1.44074	−0.720372	−	0.693588i	$-0.756029\pi$
$167$	−18.6185	−1.44074	−0.720372	+	0.693588i	$0.756029\pi$
$168$	0	0
$169$	−3.21946	−0.247651
$170$	0.648847i	0.0497643i
$171$	0	0
$172$	7.21146	0.549868
$173$	−25.9993	−1.97669	−0.988343	−	0.152242i	$-0.951351\pi$
$173$	−25.9993	−1.97669	−0.988343	+	0.152242i	$0.951351\pi$
$174$	0	0
$175$	0	0
$176$	− 1.37849i	− 0.103908i
$177$	0	0
$178$	9.76696i	0.732065i
$179$	− 13.3189i	− 0.995502i	−0.867320	−	0.497751i	$-0.834159\pi$
$179$	− 13.3189i	− 0.995502i	0.867320	−	0.497751i	$-0.165841\pi$
$180$	0	0
$181$	9.90857i	0.736498i	0.929727	+	0.368249i	$0.120043\pi$
$181$	9.90857i	0.736498i	−0.929727	+	0.368249i	$0.879957\pi$
$182$	0	0
$183$	0	0
$184$	−1.66818	−0.122980
$185$	−10.9366	−0.804072
$186$	0	0
$187$	− 0.894430i	− 0.0654072i
$188$	−0.598330	−0.0436377
$189$	0	0
$190$	−5.77337	−0.418845
$191$	− 7.68592i	− 0.556133i	−0.960562	−	0.278067i	$-0.910306\pi$
$191$	− 7.68592i	− 0.556133i	0.960562	−	0.278067i	$-0.0896936\pi$
$192$	0	0
$193$	−24.8695	−1.79015	−0.895074	−	0.445918i	$-0.852877\pi$
$193$	−24.8695	−1.79015	−0.895074	+	0.445918i	$0.852877\pi$
$194$	−7.65685	−0.549730
$195$	0	0
$196$	0	0
$197$	− 22.9259i	− 1.63341i	−0.577059	−	0.816703i	$-0.695800\pi$
$197$	− 22.9259i	− 1.63341i	0.577059	−	0.816703i	$-0.304200\pi$
$198$	0	0
$199$	− 15.3025i	− 1.08477i	−0.840130	−	0.542384i	$-0.817522\pi$
$199$	− 15.3025i	− 1.08477i	0.840130	−	0.542384i	$-0.182478\pi$
$200$	1.00000i	0.0707107i
$201$	0	0
$202$	7.45288i	0.524383i
$203$	0	0
$204$	0	0
$205$	6.26998	0.437914
$206$	2.21530	0.154347
$207$	0	0
$208$	− 4.02734i	− 0.279246i
$209$	7.95856	0.550505
$210$	0	0
$211$	−27.6610	−1.90426	−0.952131	−	0.305689i	$-0.901113\pi$
$211$	−27.6610	−1.90426	−0.952131	+	0.305689i	$0.901113\pi$
$212$	7.48022i	0.513743i
$213$	0	0
$214$	13.6063	0.930110
$215$	7.21146	0.491817
$216$	0	0
$217$	0	0
$218$	− 8.64272i	− 0.585359i
$219$	0	0
$220$	− 1.37849i	− 0.0929380i
$221$	− 2.61313i	− 0.175778i
$222$	0	0
$223$	− 15.3997i	− 1.03124i	−0.856817	−	0.515620i	$-0.827562\pi$
$223$	− 15.3997i	− 1.03124i	0.856817	−	0.515620i	$-0.172438\pi$
$224$	0	0
$225$	0	0
$226$	−7.59448	−0.505178
$227$	−8.84200	−0.586864	−0.293432	−	0.955980i	$-0.594797\pi$
$227$	−8.84200	−0.586864	−0.293432	+	0.955980i	$0.594797\pi$
$228$	0	0
$229$	21.6556i	1.43104i	0.698592	+	0.715520i	$0.253810\pi$
$229$	21.6556i	1.43104i	−0.698592	+	0.715520i	$0.746190\pi$
$230$	−1.66818	−0.109996
$231$	0	0
$232$	2.53233	0.166255
$233$	8.76606i	0.574284i	0.957888	+	0.287142i	$0.0927051\pi$
$233$	8.76606i	0.574284i	−0.957888	+	0.287142i	$0.907295\pi$
$234$	0	0
$235$	−0.598330	−0.0390307
$236$	3.71870	0.242066
$237$	0	0
$238$	0	0
$239$	− 28.2670i	− 1.82844i	−0.405219	−	0.914219i	$-0.632805\pi$
$239$	− 28.2670i	− 1.82844i	0.405219	−	0.914219i	$-0.367195\pi$
$240$	0	0
$241$	− 3.31196i	− 0.213342i	−0.994294	−	0.106671i	$-0.965981\pi$
$241$	− 3.31196i	− 0.213342i	0.994294	−	0.106671i	$-0.0340192\pi$
$242$	− 9.09976i	− 0.584955i
$243$	0	0
$244$	4.94495i	0.316568i
$245$	0	0
$246$	0	0
$247$	23.2513	1.47945
$248$	6.20692	0.394140
$249$	0	0
$250$	1.00000i	0.0632456i
$251$	−7.32278	−0.462210	−0.231105	−	0.972929i	$-0.574234\pi$
$251$	−7.32278	−0.462210	−0.231105	+	0.972929i	$0.574234\pi$
$252$	0	0
$253$	2.29957	0.144573
$254$	4.02509i	0.252556i
$255$	0	0
$256$	1.00000	0.0625000
$257$	−9.00385	−0.561644	−0.280822	−	0.959760i	$-0.590607\pi$
$257$	−9.00385	−0.561644	−0.280822	+	0.959760i	$0.590607\pi$
$258$	0	0
$259$	0	0
$260$	− 4.02734i	− 0.249765i
$261$	0	0
$262$	20.9314i	1.29314i
$263$	30.2036i	1.86244i	0.364463	+	0.931218i	$0.381253\pi$
$263$	30.2036i	1.86244i	−0.364463	+	0.931218i	$0.618747\pi$
$264$	0	0
$265$	7.48022i	0.459506i
$266$	0	0
$267$	0	0
$268$	3.76990	0.230283
$269$	−23.3775	−1.42535	−0.712674	−	0.701495i	$-0.752516\pi$
$269$	−23.3775	−1.42535	−0.712674	+	0.701495i	$0.752516\pi$
$270$	0	0
$271$	16.8368i	1.02276i	0.859354	+	0.511382i	$0.170866\pi$
$271$	16.8368i	1.02276i	−0.859354	+	0.511382i	$0.829134\pi$
$272$	0.648847	0.0393421
$273$	0	0
$274$	2.20889	0.133444
$275$	− 1.37849i	− 0.0831262i
$276$	0	0
$277$	7.65073	0.459688	0.229844	−	0.973228i	$-0.426178\pi$
$277$	7.65073	0.459688	0.229844	+	0.973228i	$0.426178\pi$
$278$	−7.34596	−0.440581
$279$	0	0
$280$	0	0
$281$	5.37555i	0.320678i	0.987062	+	0.160339i	$0.0512588\pi$
$281$	5.37555i	0.320678i	−0.987062	+	0.160339i	$0.948741\pi$
$282$	0	0
$283$	− 3.80788i	− 0.226355i	−0.993575	−	0.113177i	$-0.963897\pi$
$283$	− 3.80788i	− 0.226355i	0.993575	−	0.113177i	$-0.0361028\pi$
$284$	3.97682i	0.235981i
$285$	0	0
$286$	5.55166i	0.328276i
$287$	0	0
$288$	0	0
$289$	−16.5790	−0.975235
$290$	2.53233	0.148703
$291$	0	0
$292$	− 7.53490i	− 0.440946i
$293$	3.42063	0.199835	0.0999176	−	0.994996i	$-0.468142\pi$
$293$	3.42063	0.199835	0.0999176	+	0.994996i	$0.468142\pi$
$294$	0	0
$295$	3.71870	0.216511
$296$	10.9366i	0.635675i
$297$	0	0
$298$	−10.9807	−0.636093
$299$	6.71832	0.388531
$300$	0	0
$301$	0	0
$302$	15.4006i	0.886207i
$303$	0	0
$304$	5.77337i	0.331126i
$305$	4.94495i	0.283147i
$306$	0	0
$307$	19.0002i	1.08440i	0.840251	+	0.542198i	$0.182408\pi$
$307$	19.0002i	1.08440i	−0.840251	+	0.542198i	$0.817592\pi$
$308$	0	0
$309$	0	0
$310$	6.20692	0.352529
$311$	6.02190	0.341471	0.170735	−	0.985317i	$-0.445386\pi$
$311$	6.02190	0.341471	0.170735	+	0.985317i	$0.445386\pi$
$312$	0	0
$313$	− 6.20761i	− 0.350875i	−0.984491	−	0.175437i	$-0.943866\pi$
$313$	− 6.20761i	− 0.350875i	0.984491	−	0.175437i	$-0.0561340\pi$
$314$	−4.02959	−0.227403
$315$	0	0
$316$	−10.4525	−0.587999
$317$	12.2362i	0.687255i	0.939106	+	0.343627i	$0.111656\pi$
$317$	12.2362i	0.687255i	−0.939106	+	0.343627i	$0.888344\pi$
$318$	0	0
$319$	−3.49079	−0.195447
$320$	1.00000	0.0559017
$321$	0	0
$322$	0	0
$323$	3.74603i	0.208435i
$324$	0	0
$325$	− 4.02734i	− 0.223397i
$326$	− 1.33023i	− 0.0736745i
$327$	0	0
$328$	− 6.26998i	− 0.346202i
$329$	0	0
$330$	0	0
$331$	2.71554	0.149260	0.0746298	−	0.997211i	$-0.476222\pi$
$331$	2.71554	0.149260	0.0746298	+	0.997211i	$0.476222\pi$
$332$	4.08746	0.224328
$333$	0	0
$334$	18.6185i	1.01876i
$335$	3.76990	0.205972
$336$	0	0
$337$	20.4776	1.11549	0.557743	−	0.830014i	$-0.311668\pi$
$337$	20.4776	1.11549	0.557743	+	0.830014i	$0.311668\pi$
$338$	3.21946i	0.175116i
$339$	0	0
$340$	0.648847	0.0351887
$341$	−8.55619	−0.463344
$342$	0	0
$343$	0	0
$344$	− 7.21146i	− 0.388816i
$345$	0	0
$346$	25.9993i	1.39773i
$347$	− 2.59539i	− 0.139328i	−0.997571	−	0.0696638i	$-0.977807\pi$
$347$	− 2.59539i	− 0.139328i	0.997571	−	0.0696638i	$-0.0221927\pi$
$348$	0	0
$349$	9.87122i	0.528394i	0.964469	+	0.264197i	$0.0851070\pi$
$349$	9.87122i	0.528394i	−0.964469	+	0.264197i	$0.914893\pi$
$350$	0	0
$351$	0	0
$352$	−1.37849	−0.0734739
$353$	−6.44858	−0.343223	−0.171612	−	0.985165i	$-0.554897\pi$
$353$	−6.44858	−0.343223	−0.171612	+	0.985165i	$0.554897\pi$
$354$	0	0
$355$	3.97682i	0.211068i
$356$	9.76696	0.517648
$357$	0	0
$358$	−13.3189	−0.703926
$359$	− 3.66083i	− 0.193211i	−0.995323	−	0.0966056i	$-0.969201\pi$
$359$	− 3.66083i	− 0.193211i	0.995323	−	0.0966056i	$-0.0307986\pi$
$360$	0	0
$361$	−14.3319	−0.754308
$362$	9.90857	0.520783
$363$	0	0
$364$	0	0
$365$	− 7.53490i	− 0.394394i
$366$	0	0
$367$	25.5641i	1.33444i	0.744862	+	0.667219i	$0.232515\pi$
$367$	25.5641i	1.33444i	−0.744862	+	0.667219i	$0.767485\pi$
$368$	1.66818i	0.0869598i
$369$	0	0
$370$	10.9366i	0.568565i
$371$	0	0
$372$	0	0
$373$	16.5865	0.858815	0.429408	−	0.903111i	$-0.358722\pi$
$373$	16.5865	0.858815	0.429408	+	0.903111i	$0.358722\pi$
$374$	−0.894430	−0.0462499
$375$	0	0
$376$	0.598330i	0.0308565i
$377$	−10.1985	−0.525251
$378$	0	0
$379$	12.3173	0.632698	0.316349	−	0.948643i	$-0.397543\pi$
$379$	12.3173	0.632698	0.316349	+	0.948643i	$0.397543\pi$
$380$	5.77337i	0.296168i
$381$	0	0
$382$	−7.68592	−0.393246
$383$	−14.1559	−0.723331	−0.361666	−	0.932308i	$-0.617792\pi$
$383$	−14.1559	−0.723331	−0.361666	+	0.932308i	$0.617792\pi$
$384$	0	0
$385$	0	0
$386$	24.8695i	1.26583i
$387$	0	0
$388$	7.65685i	0.388718i
$389$	− 29.7795i	− 1.50988i	−0.655793	−	0.754940i	$-0.727666\pi$
$389$	− 29.7795i	− 1.50988i	0.655793	−	0.754940i	$-0.272334\pi$
$390$	0	0
$391$	1.08239i	0.0547389i
$392$	0	0
$393$	0	0
$394$	−22.9259	−1.15499
$395$	−10.4525	−0.525923
$396$	0	0
$397$	− 8.18080i	− 0.410583i	−0.978701	−	0.205291i	$-0.934186\pi$
$397$	− 8.18080i	− 0.410583i	0.978701	−	0.205291i	$-0.0658142\pi$
$398$	−15.3025	−0.767047
$399$	0	0
$400$	1.00000	0.0500000
$401$	14.3746i	0.717836i	0.933369	+	0.358918i	$0.116854\pi$
$401$	14.3746i	0.717836i	−0.933369	+	0.358918i	$0.883146\pi$
$402$	0	0
$403$	−24.9974	−1.24521
$404$	7.45288	0.370795
$405$	0	0
$406$	0	0
$407$	− 15.0760i	− 0.747288i
$408$	0	0
$409$	− 37.6767i	− 1.86300i	−0.363748	−	0.931498i	$-0.618503\pi$
$409$	− 37.6767i	− 1.86300i	0.363748	−	0.931498i	$-0.381497\pi$
$410$	− 6.26998i	− 0.309652i
$411$	0	0
$412$	− 2.21530i	− 0.109140i
$413$	0	0
$414$	0	0
$415$	4.08746	0.200645
$416$	−4.02734	−0.197457
$417$	0	0
$418$	− 7.95856i	− 0.389266i
$419$	34.9552	1.70767	0.853836	−	0.520542i	$-0.174270\pi$
$419$	34.9552	1.70767	0.853836	+	0.520542i	$0.174270\pi$
$420$	0	0
$421$	4.52492	0.220531	0.110266	−	0.993902i	$-0.464830\pi$
$421$	4.52492	0.220531	0.110266	+	0.993902i	$0.464830\pi$
$422$	27.6610i	1.34652i
$423$	0	0
$424$	7.48022	0.363271
$425$	0.648847	0.0314737
$426$	0	0
$427$	0	0
$428$	− 13.6063i	− 0.657687i
$429$	0	0
$430$	− 7.21146i	− 0.347767i
$431$	13.4689i	0.648774i	0.945925	+	0.324387i	$0.105158\pi$
$431$	13.4689i	0.648774i	−0.945925	+	0.324387i	$0.894842\pi$
$432$	0	0
$433$	− 28.0598i	− 1.34847i	−0.738517	−	0.674234i	$-0.764474\pi$
$433$	− 28.0598i	− 1.34847i	0.738517	−	0.674234i	$-0.235526\pi$
$434$	0	0
$435$	0	0
$436$	−8.64272	−0.413911
$437$	−9.63102	−0.460714
$438$	0	0
$439$	41.0841i	1.96084i	0.196930	+	0.980418i	$0.436903\pi$
$439$	41.0841i	1.96084i	−0.196930	+	0.980418i	$0.563097\pi$
$440$	−1.37849	−0.0657171
$441$	0	0
$442$	−2.61313	−0.124294
$443$	− 40.3448i	− 1.91684i	−0.285360	−	0.958421i	$-0.592113\pi$
$443$	− 40.3448i	− 1.91684i	0.285360	−	0.958421i	$-0.407887\pi$
$444$	0	0
$445$	9.76696	0.462998
$446$	−15.3997	−0.729197
$447$	0	0
$448$	0	0
$449$	− 23.7591i	− 1.12126i	−0.828067	−	0.560629i	$-0.810559\pi$
$449$	− 23.7591i	− 1.12126i	0.828067	−	0.560629i	$-0.189441\pi$
$450$	0	0
$451$	8.64312i	0.406989i
$452$	7.59448i	0.357215i
$453$	0	0
$454$	8.84200i	0.414976i
$455$	0	0
$456$	0	0
$457$	−3.93763	−0.184195	−0.0920973	−	0.995750i	$-0.529357\pi$
$457$	−3.93763	−0.184195	−0.0920973	+	0.995750i	$0.529357\pi$
$458$	21.6556	1.01190
$459$	0	0
$460$	1.66818i	0.0777792i
$461$	10.6218	0.494706	0.247353	−	0.968925i	$-0.420439\pi$
$461$	10.6218	0.494706	0.247353	+	0.968925i	$0.420439\pi$
$462$	0	0
$463$	31.5119	1.46448	0.732241	−	0.681045i	$-0.238474\pi$
$463$	31.5119	1.46448	0.732241	+	0.681045i	$0.238474\pi$
$464$	− 2.53233i	− 0.117560i
$465$	0	0
$466$	8.76606	0.406080
$467$	28.2278	1.30623	0.653113	−	0.757260i	$-0.273463\pi$
$467$	28.2278	1.30623	0.653113	+	0.757260i	$0.273463\pi$
$468$	0	0
$469$	0	0
$470$	0.598330i	0.0275989i
$471$	0	0
$472$	− 3.71870i	− 0.171167i
$473$	9.94094i	0.457085i
$474$	0	0
$475$	5.77337i	0.264901i
$476$	0	0
$477$	0	0
$478$	−28.2670	−1.29290
$479$	37.3001	1.70429	0.852143	−	0.523309i	$-0.175302\pi$
$479$	37.3001	1.70429	0.852143	+	0.523309i	$0.175302\pi$
$480$	0	0
$481$	− 44.0453i	− 2.00829i
$482$	−3.31196	−0.150856
$483$	0	0
$484$	−9.09976	−0.413625
$485$	7.65685i	0.347680i
$486$	0	0
$487$	−27.4129	−1.24220	−0.621099	−	0.783732i	$-0.713314\pi$
$487$	−27.4129	−1.24220	−0.621099	+	0.783732i	$0.713314\pi$
$488$	4.94495	0.223847
$489$	0	0
$490$	0	0
$491$	− 31.2211i	− 1.40899i	−0.709711	−	0.704493i	$-0.751174\pi$
$491$	− 31.2211i	− 1.40899i	0.709711	−	0.704493i	$-0.248826\pi$
$492$	0	0
$493$	− 1.64309i	− 0.0740011i
$494$	− 23.2513i	− 1.04613i
$495$	0	0
$496$	− 6.20692i	− 0.278699i
$497$	0	0
$498$	0	0
$499$	32.6744	1.46271	0.731354	−	0.681998i	$-0.238889\pi$
$499$	32.6744	1.46271	0.731354	+	0.681998i	$0.238889\pi$
$500$	1.00000	0.0447214
$501$	0	0
$502$	7.32278i	0.326832i
$503$	12.4663	0.555846	0.277923	−	0.960603i	$-0.410354\pi$
$503$	12.4663	0.555846	0.277923	+	0.960603i	$0.410354\pi$
$504$	0	0
$505$	7.45288	0.331649
$506$	− 2.29957i	− 0.102228i
$507$	0	0
$508$	4.02509	0.178584
$509$	2.89934	0.128511	0.0642555	−	0.997933i	$-0.479533\pi$
$509$	2.89934	0.128511	0.0642555	+	0.997933i	$0.479533\pi$
$510$	0	0
$511$	0	0
$512$	− 1.00000i	− 0.0441942i
$513$	0	0
$514$	9.00385i	0.397143i
$515$	− 2.21530i	− 0.0976178i
$516$	0	0
$517$	− 0.824793i	− 0.0362744i
$518$	0	0
$519$	0	0
$520$	−4.02734	−0.176611
$521$	23.9505	1.04929	0.524644	−	0.851322i	$-0.324198\pi$
$521$	23.9505	1.04929	0.524644	+	0.851322i	$0.324198\pi$
$522$	0	0
$523$	− 4.24680i	− 0.185700i	−0.995680	−	0.0928499i	$-0.970402\pi$
$523$	− 4.24680i	− 0.185700i	0.995680	−	0.0928499i	$-0.0295977\pi$
$524$	20.9314	0.914391
$525$	0	0
$526$	30.2036	1.31694
$527$	− 4.02734i	− 0.175434i
$528$	0	0
$529$	20.2172	0.879008
$530$	7.48022	0.324920
$531$	0	0
$532$	0	0
$533$	25.2513i	1.09376i
$534$	0	0
$535$	− 13.6063i	− 0.588253i
$536$	− 3.76990i	− 0.162835i
$537$	0	0
$538$	23.3775i	1.00787i
$539$	0	0
$540$	0	0
$541$	−35.6139	−1.53116	−0.765581	−	0.643340i	$-0.777548\pi$
$541$	−35.6139	−1.53116	−0.765581	+	0.643340i	$0.777548\pi$
$542$	16.8368	0.723203
$543$	0	0
$544$	− 0.648847i	− 0.0278191i
$545$	−8.64272	−0.370213
$546$	0	0
$547$	−14.0005	−0.598617	−0.299308	−	0.954156i	$-0.596756\pi$
$547$	−14.0005	−0.598617	−0.299308	+	0.954156i	$0.596756\pi$
$548$	− 2.20889i	− 0.0943590i
$549$	0	0
$550$	−1.37849	−0.0587791
$551$	14.6201	0.622836
$552$	0	0
$553$	0	0
$554$	− 7.65073i	− 0.325048i
$555$	0	0
$556$	7.34596i	0.311538i
$557$	36.6688i	1.55371i	0.629682	+	0.776853i	$0.283185\pi$
$557$	36.6688i	1.55371i	−0.629682	+	0.776853i	$0.716815\pi$
$558$	0	0
$559$	29.0430i	1.22839i
$560$	0	0
$561$	0	0
$562$	5.37555	0.226754
$563$	10.8162	0.455847	0.227924	−	0.973679i	$-0.426806\pi$
$563$	10.8162	0.455847	0.227924	+	0.973679i	$0.426806\pi$
$564$	0	0
$565$	7.59448i	0.319502i
$566$	−3.80788	−0.160057
$567$	0	0
$568$	3.97682	0.166864
$569$	− 26.9610i	− 1.13026i	−0.825001	−	0.565131i	$-0.808826\pi$
$569$	− 26.9610i	− 1.13026i	0.825001	−	0.565131i	$-0.191174\pi$
$570$	0	0
$571$	−28.0971	−1.17583	−0.587914	−	0.808924i	$-0.700050\pi$
$571$	−28.0971	−1.17583	−0.587914	+	0.808924i	$0.700050\pi$
$572$	5.55166	0.232127
$573$	0	0
$574$	0	0
$575$	1.66818i	0.0695679i
$576$	0	0
$577$	2.22543i	0.0926459i	0.998927	+	0.0463229i	$0.0147503\pi$
$577$	2.22543i	0.0926459i	−0.998927	+	0.0463229i	$0.985250\pi$
$578$	16.5790i	0.689595i
$579$	0	0
$580$	− 2.53233i	− 0.105149i
$581$	0	0
$582$	0	0
$583$	−10.3114	−0.427056
$584$	−7.53490	−0.311796
$585$	0	0
$586$	− 3.42063i	− 0.141305i
$587$	−15.5641	−0.642401	−0.321201	−	0.947011i	$-0.604086\pi$
$587$	−15.5641	−0.642401	−0.321201	+	0.947011i	$0.604086\pi$
$588$	0	0
$589$	35.8349	1.47655
$590$	− 3.71870i	− 0.153096i
$591$	0	0
$592$	10.9366	0.449490
$593$	−41.2831	−1.69529	−0.847647	−	0.530560i	$-0.821982\pi$
$593$	−41.2831	−1.69529	−0.847647	+	0.530560i	$0.821982\pi$
$594$	0	0
$595$	0	0
$596$	10.9807i	0.449786i
$597$	0	0
$598$	− 6.71832i	− 0.274733i
$599$	− 10.9641i	− 0.447983i	−0.974591	−	0.223991i	$-0.928091\pi$
$599$	− 10.9641i	− 0.447983i	0.974591	−	0.223991i	$-0.0719088\pi$
$600$	0	0
$601$	− 17.1378i	− 0.699064i	−0.936924	−	0.349532i	$-0.886341\pi$
$601$	− 17.1378i	− 0.699064i	0.936924	−	0.349532i	$-0.113659\pi$
$602$	0	0
$603$	0	0
$604$	15.4006	0.626643
$605$	−9.09976	−0.369958
$606$	0	0
$607$	− 0.0268110i	− 0.00108822i	−1.00000		0.000544112i	$-0.999827\pi$
$607$	− 0.0268110i	− 0.00108822i	1.00000		0.000544112i	$-0.000173196\pi$
$608$	5.77337	0.234141
$609$	0	0
$610$	4.94495	0.200215
$611$	− 2.40968i	− 0.0974851i
$612$	0	0
$613$	−42.2904	−1.70809	−0.854047	−	0.520197i	$-0.825859\pi$
$613$	−42.2904	−1.70809	−0.854047	+	0.520197i	$0.825859\pi$
$614$	19.0002	0.766784
$615$	0	0
$616$	0	0
$617$	29.3571i	1.18187i	0.806719	+	0.590935i	$0.201241\pi$
$617$	29.3571i	1.18187i	−0.806719	+	0.590935i	$0.798759\pi$
$618$	0	0
$619$	− 45.7636i	− 1.83940i	−0.392628	−	0.919698i	$-0.628434\pi$
$619$	− 45.7636i	− 1.83940i	0.392628	−	0.919698i	$-0.371566\pi$
$620$	− 6.20692i	− 0.249276i
$621$	0	0
$622$	− 6.02190i	− 0.241456i
$623$	0	0
$624$	0	0
$625$	1.00000	0.0400000
$626$	−6.20761	−0.248106
$627$	0	0
$628$	4.02959i	0.160798i
$629$	7.09615	0.282942
$630$	0	0
$631$	0.640465	0.0254965	0.0127483	−	0.999919i	$-0.495942\pi$
$631$	0.640465	0.0254965	0.0127483	+	0.999919i	$0.495942\pi$
$632$	10.4525i	0.415778i
$633$	0	0
$634$	12.2362	0.485963
$635$	4.02509	0.159731
$636$	0	0
$637$	0	0
$638$	3.49079i	0.138202i
$639$	0	0
$640$	− 1.00000i	− 0.0395285i
$641$	17.6235i	0.696088i	0.937478	+	0.348044i	$0.113154\pi$
$641$	17.6235i	0.696088i	−0.937478	+	0.348044i	$0.886846\pi$
$642$	0	0
$643$	− 12.3415i	− 0.486700i	−0.969939	−	0.243350i	$-0.921754\pi$
$643$	− 12.3415i	− 0.486700i	0.969939	−	0.243350i	$-0.0782463\pi$
$644$	0	0
$645$	0	0
$646$	3.74603	0.147386
$647$	−19.3873	−0.762195	−0.381098	−	0.924535i	$-0.624454\pi$
$647$	−19.3873	−0.762195	−0.381098	+	0.924535i	$0.624454\pi$
$648$	0	0
$649$	5.12619i	0.201221i
$650$	−4.02734	−0.157965
$651$	0	0
$652$	−1.33023	−0.0520958
$653$	42.5277i	1.66424i	0.554597	+	0.832119i	$0.312872\pi$
$653$	42.5277i	1.66424i	−0.554597	+	0.832119i	$0.687128\pi$
$654$	0	0
$655$	20.9314	0.817856
$656$	−6.26998	−0.244802
$657$	0	0
$658$	0	0
$659$	43.5871i	1.69791i	0.528464	+	0.848956i	$0.322768\pi$
$659$	43.5871i	1.69791i	−0.528464	+	0.848956i	$0.677232\pi$
$660$	0	0
$661$	− 36.6893i	− 1.42705i	−0.700631	−	0.713523i	$-0.747098\pi$
$661$	− 36.6893i	− 1.42705i	0.700631	−	0.713523i	$-0.252902\pi$
$662$	− 2.71554i	− 0.105542i
$663$	0	0
$664$	− 4.08746i	− 0.158624i
$665$	0	0
$666$	0	0
$667$	4.22437	0.163568
$668$	18.6185	0.720372
$669$	0	0
$670$	− 3.76990i	− 0.145644i
$671$	−6.81657	−0.263151
$672$	0	0
$673$	8.39119	0.323457	0.161728	−	0.986835i	$-0.448293\pi$
$673$	8.39119	0.323457	0.161728	+	0.986835i	$0.448293\pi$
$674$	− 20.4776i	− 0.788767i
$675$	0	0
$676$	3.21946	0.123826
$677$	34.8636	1.33992	0.669959	−	0.742398i	$-0.266312\pi$
$677$	34.8636	1.33992	0.669959	+	0.742398i	$0.266312\pi$
$678$	0	0
$679$	0	0
$680$	− 0.648847i	− 0.0248821i
$681$	0	0
$682$	8.55619i	0.327634i
$683$	32.7294i	1.25236i	0.779680	+	0.626178i	$0.215382\pi$
$683$	32.7294i	1.25236i	−0.779680	+	0.626178i	$0.784618\pi$
$684$	0	0
$685$	− 2.20889i	− 0.0843972i
$686$	0	0
$687$	0	0
$688$	−7.21146	−0.274934
$689$	−30.1254	−1.14769
$690$	0	0
$691$	− 34.9101i	− 1.32804i	−0.747714	−	0.664021i	$-0.768848\pi$
$691$	− 34.9101i	− 1.32804i	0.747714	−	0.664021i	$-0.231152\pi$
$692$	25.9993	0.988343
$693$	0	0
$694$	−2.59539	−0.0985195
$695$	7.34596i	0.278648i
$696$	0	0
$697$	−4.06826	−0.154096
$698$	9.87122	0.373631
$699$	0	0
$700$	0	0
$701$	− 25.2001i	− 0.951796i	−0.879501	−	0.475898i	$-0.842123\pi$
$701$	− 25.2001i	− 0.951796i	0.879501	−	0.475898i	$-0.157877\pi$
$702$	0	0
$703$	63.1409i	2.38140i
$704$	1.37849i	0.0519539i
$705$	0	0
$706$	6.44858i	0.242696i
$707$	0	0
$708$	0	0
$709$	−25.0506	−0.940795	−0.470397	−	0.882455i	$-0.655889\pi$
$709$	−25.0506	−0.940795	−0.470397	+	0.882455i	$0.655889\pi$
$710$	3.97682	0.149247
$711$	0	0
$712$	− 9.76696i	− 0.366032i
$713$	10.3543	0.387770
$714$	0	0
$715$	5.55166	0.207620
$716$	13.3189i	0.497751i
$717$	0	0
$718$	−3.66083	−0.136621
$719$	28.0794	1.04718	0.523592	−	0.851969i	$-0.324592\pi$
$719$	28.0794	1.04718	0.523592	+	0.851969i	$0.324592\pi$
$720$	0	0
$721$	0	0
$722$	14.3319i	0.533376i
$723$	0	0
$724$	− 9.90857i	− 0.368249i
$725$	− 2.53233i	− 0.0940482i
$726$	0	0
$727$	19.2464i	0.713811i	0.934141	+	0.356905i	$0.116168\pi$
$727$	19.2464i	0.713811i	−0.934141	+	0.356905i	$0.883832\pi$
$728$	0	0
$729$	0	0
$730$	−7.53490	−0.278879
$731$	−4.67913	−0.173064
$732$	0	0
$733$	− 32.8650i	− 1.21390i	−0.794742	−	0.606948i	$-0.792394\pi$
$733$	− 32.8650i	− 1.21390i	0.794742	−	0.606948i	$-0.207606\pi$
$734$	25.5641	0.943590
$735$	0	0
$736$	1.66818	0.0614899
$737$	5.19678i	0.191426i
$738$	0	0
$739$	47.9728	1.76471	0.882355	−	0.470585i	$-0.155957\pi$
$739$	47.9728	1.76471	0.882355	+	0.470585i	$0.155957\pi$
$740$	10.9366	0.402036
$741$	0	0
$742$	0	0
$743$	40.0867i	1.47064i	0.677721	+	0.735319i	$0.262968\pi$
$743$	40.0867i	1.47064i	−0.677721	+	0.735319i	$0.737032\pi$
$744$	0	0
$745$	10.9807i	0.402301i
$746$	− 16.5865i	− 0.607274i
$747$	0	0
$748$	0.894430i	0.0327036i
$749$	0	0
$750$	0	0
$751$	22.0953	0.806267	0.403134	−	0.915141i	$-0.367921\pi$
$751$	22.0953	0.806267	0.403134	+	0.915141i	$0.367921\pi$
$752$	0.598330	0.0218188
$753$	0	0
$754$	10.1985i	0.371409i
$755$	15.4006	0.560487
$756$	0	0
$757$	16.9713	0.616832	0.308416	−	0.951252i	$-0.400201\pi$
$757$	16.9713	0.616832	0.308416	+	0.951252i	$0.400201\pi$
$758$	− 12.3173i	− 0.447385i
$759$	0	0
$760$	5.77337	0.209422
$761$	28.2253	1.02317	0.511584	−	0.859233i	$-0.329059\pi$
$761$	28.2253	1.02317	0.511584	+	0.859233i	$0.329059\pi$
$762$	0	0
$763$	0	0
$764$	7.68592i	0.278067i
$765$	0	0
$766$	14.1559i	0.511473i
$767$	14.9764i	0.540768i
$768$	0	0
$769$	− 5.49497i	− 0.198154i	−0.995080	−	0.0990768i	$-0.968411\pi$
$769$	− 5.49497i	− 0.198154i	0.995080	−	0.0990768i	$-0.0315890\pi$
$770$	0	0
$771$	0	0
$772$	24.8695	0.895074
$773$	23.9672	0.862041	0.431021	−	0.902342i	$-0.358154\pi$
$773$	23.9672	0.862041	0.431021	+	0.902342i	$0.358154\pi$
$774$	0	0
$775$	− 6.20692i	− 0.222959i
$776$	7.65685	0.274865
$777$	0	0
$778$	−29.7795	−1.06765
$779$	− 36.1989i	− 1.29696i
$780$	0	0
$781$	−5.48202	−0.196162
$782$	1.08239	0.0387063
$783$	0	0
$784$	0	0
$785$	4.02959i	0.143822i
$786$	0	0
$787$	− 37.9571i	− 1.35303i	−0.736431	−	0.676513i	$-0.763490\pi$
$787$	− 37.9571i	− 1.35303i	0.736431	−	0.676513i	$-0.236510\pi$
$788$	22.9259i	0.816703i
$789$	0	0
$790$	10.4525i	0.371883i
$791$	0	0
$792$	0	0
$793$	−19.9150	−0.707202
$794$	−8.18080	−0.290326
$795$	0	0
$796$	15.3025i	0.542384i
$797$	−43.8139	−1.55197	−0.775985	−	0.630752i	$-0.782747\pi$
$797$	−43.8139	−1.55197	−0.775985	+	0.630752i	$0.782747\pi$
$798$	0	0
$799$	0.388224	0.0137344
$800$	− 1.00000i	− 0.0353553i
$801$	0	0
$802$	14.3746	0.507586
$803$	10.3868	0.366542
$804$	0	0
$805$	0	0
$806$	24.9974i	0.880495i
$807$	0	0
$808$	− 7.45288i	− 0.262191i
$809$	− 11.6150i	− 0.408360i	−0.978933	−	0.204180i	$-0.934547\pi$
$809$	− 11.6150i	− 0.408360i	0.978933	−	0.204180i	$-0.0654528\pi$
$810$	0	0
$811$	− 11.2083i	− 0.393578i	−0.980446	−	0.196789i	$-0.936949\pi$
$811$	− 11.2083i	− 0.393578i	0.980446	−	0.196789i	$-0.0630513\pi$
$812$	0	0
$813$	0	0
$814$	−15.0760	−0.528413
$815$	−1.33023	−0.0465959
$816$	0	0
$817$	− 41.6344i	− 1.45660i
$818$	−37.6767	−1.31734
$819$	0	0
$820$	−6.26998	−0.218957
$821$	− 37.6272i	− 1.31320i	−0.754240	−	0.656599i	$-0.771994\pi$
$821$	− 37.6272i	− 1.31320i	0.754240	−	0.656599i	$-0.228006\pi$
$822$	0	0
$823$	−16.2459	−0.566295	−0.283148	−	0.959076i	$-0.591379\pi$
$823$	−16.2459	−0.566295	−0.283148	+	0.959076i	$0.591379\pi$
$824$	−2.21530	−0.0771737
$825$	0	0
$826$	0	0
$827$	20.6197i	0.717018i	0.933526	+	0.358509i	$0.116715\pi$
$827$	20.6197i	0.717018i	−0.933526	+	0.358509i	$0.883285\pi$
$828$	0	0
$829$	− 11.3473i	− 0.394109i	−0.980392	−	0.197055i	$-0.936862\pi$
$829$	− 11.3473i	− 0.394109i	0.980392	−	0.197055i	$-0.0631377\pi$
$830$	− 4.08746i	− 0.141878i
$831$	0	0
$832$	4.02734i	0.139623i
$833$	0	0
$834$	0	0
$835$	18.6185	0.644320
$836$	−7.95856	−0.275252
$837$	0	0
$838$	− 34.9552i	− 1.20751i
$839$	−8.25772	−0.285088	−0.142544	−	0.989788i	$-0.545528\pi$
$839$	−8.25772	−0.285088	−0.142544	+	0.989788i	$0.545528\pi$
$840$	0	0
$841$	22.5873	0.778873
$842$	− 4.52492i	− 0.155939i
$843$	0	0
$844$	27.6610	0.952131
$845$	3.21946	0.110753
$846$	0	0
$847$	0	0
$848$	− 7.48022i	− 0.256872i
$849$	0	0
$850$	− 0.648847i	− 0.0222553i
$851$	18.2441i	0.625401i
$852$	0	0
$853$	− 18.5562i	− 0.635353i	−0.948199	−	0.317677i	$-0.897097\pi$
$853$	− 18.5562i	− 0.635353i	0.948199	−	0.317677i	$-0.102903\pi$
$854$	0	0
$855$	0	0
$856$	−13.6063	−0.465055
$857$	23.3873	0.798897	0.399448	−	0.916756i	$-0.369202\pi$
$857$	23.3873	0.798897	0.399448	+	0.916756i	$0.369202\pi$
$858$	0	0
$859$	30.8328i	1.05200i	0.850484	+	0.526000i	$0.176309\pi$
$859$	30.8328i	1.05200i	−0.850484	+	0.526000i	$0.823691\pi$
$860$	−7.21146	−0.245909
$861$	0	0
$862$	13.4689	0.458752
$863$	− 3.78811i	− 0.128949i	−0.997919	−	0.0644744i	$-0.979463\pi$
$863$	− 3.78811i	− 0.128949i	0.997919	−	0.0644744i	$-0.0205371\pi$
$864$	0	0
$865$	25.9993	0.884001
$866$	−28.0598	−0.953511
$867$	0	0
$868$	0	0
$869$	− 14.4087i	− 0.488782i
$870$	0	0
$871$	15.1827i	0.514445i
$872$	8.64272i	0.292679i
$873$	0	0
$874$	9.63102i	0.325774i
$875$	0	0
$876$	0	0
$877$	0.0534599	0.00180521	0.000902606	−	1.00000i	$-0.499713\pi$
$877$	0.0534599	0.00180521	0.000902606		1.00000i	$0.499713\pi$
$878$	41.0841	1.38652
$879$	0	0
$880$	1.37849i	0.0464690i
$881$	−10.8552	−0.365722	−0.182861	−	0.983139i	$-0.558536\pi$
$881$	−10.8552	−0.365722	−0.182861	+	0.983139i	$0.558536\pi$
$882$	0	0
$883$	−51.7511	−1.74156	−0.870781	−	0.491671i	$-0.836386\pi$
$883$	−51.7511	−1.74156	−0.870781	+	0.491671i	$0.836386\pi$
$884$	2.61313i	0.0878889i
$885$	0	0
$886$	−40.3448	−1.35541
$887$	0.138478	0.00464965	0.00232482	−	0.999997i	$-0.499260\pi$
$887$	0.138478	0.00464965	0.00232482	+	0.999997i	$0.499260\pi$
$888$	0	0
$889$	0	0
$890$	− 9.76696i	− 0.327389i
$891$	0	0
$892$	15.3997i	0.515620i
$893$	3.45438i	0.115597i
$894$	0	0
$895$	13.3189i	0.445202i
$896$	0	0
$897$	0	0
$898$	−23.7591	−0.792850
$899$	−15.7179	−0.524223
$900$	0	0
$901$	− 4.85351i	− 0.161694i
$902$	8.64312	0.287785
$903$	0	0
$904$	7.59448	0.252589
$905$	− 9.90857i	− 0.329372i
$906$	0	0
$907$	−7.82320	−0.259765	−0.129883	−	0.991529i	$-0.541460\pi$
$907$	−7.82320	−0.259765	−0.129883	+	0.991529i	$0.541460\pi$
$908$	8.84200	0.293432
$909$	0	0
$910$	0	0
$911$	40.0649i	1.32741i	0.747995	+	0.663705i	$0.231017\pi$
$911$	40.0649i	1.32741i	−0.747995	+	0.663705i	$0.768983\pi$
$912$	0	0
$913$	5.63453i	0.186476i
$914$	3.93763i	0.130245i
$915$	0	0
$916$	− 21.6556i	− 0.715520i
$917$	0	0
$918$	0	0
$919$	−31.9897	−1.05524	−0.527621	−	0.849480i	$-0.676916\pi$
$919$	−31.9897	−1.05524	−0.527621	+	0.849480i	$0.676916\pi$
$920$	1.66818	0.0549982
$921$	0	0
$922$	− 10.6218i	− 0.349810i
$923$	−16.0160	−0.527174
$924$	0	0
$925$	10.9366	0.359592
$926$	− 31.5119i	− 1.03555i
$927$	0	0
$928$	−2.53233	−0.0831277
$929$	54.6217	1.79208	0.896040	−	0.443974i	$-0.146432\pi$
$929$	54.6217	1.79208	0.896040	+	0.443974i	$0.146432\pi$
$930$	0	0
$931$	0	0
$932$	− 8.76606i	− 0.287142i
$933$	0	0
$934$	− 28.2278i	− 0.923641i
$935$	0.894430i	0.0292510i
$936$	0	0
$937$	38.9970i	1.27398i	0.770873	+	0.636989i	$0.219820\pi$
$937$	38.9970i	1.27398i	−0.770873	+	0.636989i	$0.780180\pi$
$938$	0	0
$939$	0	0
$940$	0.598330	0.0195154
$941$	26.8330	0.874732	0.437366	−	0.899284i	$-0.355911\pi$
$941$	26.8330	0.874732	0.437366	+	0.899284i	$0.355911\pi$
$942$	0	0
$943$	− 10.4594i	− 0.340606i
$944$	−3.71870	−0.121033
$945$	0	0
$946$	9.94094	0.323208
$947$	− 45.3254i	− 1.47288i	−0.676503	−	0.736440i	$-0.736506\pi$
$947$	− 45.3254i	− 1.47288i	0.676503	−	0.736440i	$-0.263494\pi$
$948$	0	0
$949$	30.3456	0.985059
$950$	5.77337	0.187313
$951$	0	0
$952$	0	0
$953$	17.2122i	0.557559i	0.960355	+	0.278779i	$0.0899298\pi$
$953$	17.2122i	0.557559i	−0.960355	+	0.278779i	$0.910070\pi$
$954$	0	0
$955$	7.68592i	0.248710i
$956$	28.2670i	0.914219i
$957$	0	0
$958$	− 37.3001i	− 1.20511i
$959$	0	0
$960$	0	0
$961$	−7.52585	−0.242770
$962$	−44.0453	−1.42008
$963$	0	0
$964$	3.31196i	0.106671i
$965$	24.8695	0.800578
$966$	0	0
$967$	−25.6150	−0.823723	−0.411862	−	0.911246i	$-0.635121\pi$
$967$	−25.6150	−0.823723	−0.411862	+	0.911246i	$0.635121\pi$
$968$	9.09976i	0.292477i
$969$	0	0
$970$	7.65685	0.245847
$971$	36.1840	1.16120	0.580600	−	0.814189i	$-0.302818\pi$
$971$	36.1840	1.16120	0.580600	+	0.814189i	$0.302818\pi$
$972$	0	0
$973$	0	0
$974$	27.4129i	0.878367i
$975$	0	0
$976$	− 4.94495i	− 0.158284i
$977$	− 32.3563i	− 1.03517i	−0.855631	−	0.517586i	$-0.826831\pi$
$977$	− 32.3563i	− 1.03517i	0.855631	−	0.517586i	$-0.173169\pi$
$978$	0	0
$979$	13.4637i	0.430301i
$980$	0	0
$981$	0	0
$982$	−31.2211	−0.996304
$983$	0.642960	0.0205073	0.0102536	−	0.999947i	$-0.496736\pi$
$983$	0.642960	0.0205073	0.0102536	+	0.999947i	$0.496736\pi$
$984$	0	0
$985$	22.9259i	0.730481i
$986$	−1.64309	−0.0523267
$987$	0	0
$988$	−23.2513	−0.739724
$989$	− 12.0300i	− 0.382532i
$990$	0	0
$991$	22.1382	0.703242	0.351621	−	0.936142i	$-0.385631\pi$
$991$	22.1382	0.703242	0.351621	+	0.936142i	$0.385631\pi$
$992$	−6.20692	−0.197070
$993$	0	0
$994$	0	0
$995$	15.3025i	0.485123i
$996$	0	0
$997$	− 8.27865i	− 0.262187i	−0.991370	−	0.131094i	$-0.958151\pi$
$997$	− 8.27865i	− 0.262187i	0.991370	−	0.131094i	$-0.0418489\pi$
$998$	− 32.6744i	− 1.03429i
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	4410.2.b.c.881.1	✓	8
3.2	odd	2		4410.2.b.f.881.8	yes	8
7.6	odd	2		4410.2.b.f.881.1	yes	8
21.20	even	2	inner	4410.2.b.c.881.8	yes	8

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
4410.2.b.c.881.1	✓	8	1.1	even	1	trivial
4410.2.b.c.881.8	yes	8	21.20	even	2	inner
4410.2.b.f.881.1	yes	8	7.6	odd	2
4410.2.b.f.881.8	yes	8	3.2	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:	\( N \)	\(=\)	\( 4410 = 2 \cdot 3^{2} \cdot 5 \cdot 7^{2} \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	4410.b (of order \(2\), degree \(1\), minimal)

\(f(q)\)	\(=\)	\(q-1.00000i q^{2} -1.00000 q^{4} -1.00000 q^{5} +1.00000i q^{8} +O(q^{10})\)	\(q-1.00000i q^{2} -1.00000 q^{4} -1.00000 q^{5} +1.00000i q^{8} +1.00000i q^{10} -1.37849i q^{11} -4.02734i q^{13} +1.00000 q^{16} +0.648847 q^{17} +5.77337i q^{19} +1.00000 q^{20} -1.37849 q^{22} +1.66818i q^{23} +1.00000 q^{25} -4.02734 q^{26} -2.53233i q^{29} -6.20692i q^{31} -1.00000i q^{32} -0.648847i q^{34} +10.9366 q^{37} +5.77337 q^{38} -1.00000i q^{40} -6.26998 q^{41} -7.21146 q^{43} +1.37849i q^{44} +1.66818 q^{46} +0.598330 q^{47} -1.00000i q^{50} +4.02734i q^{52} -7.48022i q^{53} +1.37849i q^{55} -2.53233 q^{58} -3.71870 q^{59} -4.94495i q^{61} -6.20692 q^{62} -1.00000 q^{64} +4.02734i q^{65} -3.76990 q^{67} -0.648847 q^{68} -3.97682i q^{71} +7.53490i q^{73} -10.9366i q^{74} -5.77337i q^{76} +10.4525 q^{79} -1.00000 q^{80} +6.26998i q^{82} -4.08746 q^{83} -0.648847 q^{85} +7.21146i q^{86} +1.37849 q^{88} -9.76696 q^{89} -1.66818i q^{92} -0.598330i q^{94} -5.77337i q^{95} -7.65685i q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 8 q - 8 q^{4} - 8 q^{5}+O(q^{10}) \) 8 * q - 8 * q^4 - 8 * q^5	\( 8 q - 8 q^{4} - 8 q^{5} + 8 q^{16} + 8 q^{20} + 16 q^{22} + 8 q^{25} + 16 q^{37} + 16 q^{41} - 16 q^{43} + 16 q^{46} - 16 q^{47} - 16 q^{58} - 48 q^{59} - 8 q^{64} - 32 q^{67} - 8 q^{80} + 16 q^{83} - 16 q^{88} - 80 q^{89}+O(q^{100}) \) 8 * q - 8 * q^4 - 8 * q^5 + 8 * q^16 + 8 * q^20 + 16 * q^22 + 8 * q^25 + 16 * q^37 + 16 * q^41 - 16 * q^43 + 16 * q^46 - 16 * q^47 - 16 * q^58 - 48 * q^59 - 8 * q^64 - 32 * q^67 - 8 * q^80 + 16 * q^83 - 16 * q^88 - 80 * q^89

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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