LMFDB - Embedded newform 4410.2.b.c.881.4

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			881.4
Root			$0.923880 - 0.382683i$ of defining polynomial
Character	$\chi$	$=$	4410.881
Dual form			4410.2.b.c.881.5

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-1.00000i q^{2} -1.00000 q^{4} -1.00000 q^{5} +1.00000i q^{8} +O(q^{10})$	$q-1.00000i q^{2} -1.00000 q^{4} -1.00000 q^{5} +1.00000i q^{8} +1.00000i q^{10} +5.37849i q^{11} +1.19891i q^{13} +1.00000 q^{16} +2.17958 q^{17} +2.71191i q^{19} +1.00000 q^{20} +5.37849 q^{22} -0.496606i q^{23} +1.00000 q^{25} +1.19891 q^{26} -7.12453i q^{29} +0.550066i q^{31} -1.00000i q^{32} -2.17958i q^{34} -4.10814 q^{37} +2.71191 q^{38} -1.00000i q^{40} -1.04373 q^{41} +6.03988 q^{43} -5.37849i q^{44} -0.496606 q^{46} -7.42676 q^{47} -1.00000i q^{50} -1.19891i q^{52} +5.13707i q^{53} -5.37849i q^{55} -7.12453 q^{58} -11.1097 q^{59} -1.88348i q^{61} +0.550066 q^{62} -1.00000 q^{64} -1.19891i q^{65} +4.25518 q^{67} -2.17958 q^{68} +10.8052i q^{71} -15.5349i q^{73} +4.10814i q^{74} -2.71191i q^{76} -10.4525 q^{79} -1.00000 q^{80} +1.04373i q^{82} -14.5400 q^{83} -2.17958 q^{85} -6.03988i q^{86} -5.37849 q^{88} -15.8899 q^{89} +0.496606i q^{92} +7.42676i q^{94} -2.71191i q^{95} -7.65685i q^{97} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 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
$11$	5.37849i	1.62168i	0.585270	+	0.810838i	$0.300988\pi$
$11$	5.37849i	1.62168i	−0.585270	+	0.810838i	$0.699012\pi$
$12$	0	0
$13$	1.19891i	0.332518i	0.986082	+	0.166259i	$0.0531689\pi$
$13$	1.19891i	0.332518i	−0.986082	+	0.166259i	$0.946831\pi$
$14$	0	0
$15$	0	0
$16$	1.00000	0.250000
$17$	2.17958	0.528626	0.264313	−	0.964437i	$-0.414855\pi$
$17$	2.17958	0.528626	0.264313	+	0.964437i	$0.414855\pi$
$18$	0	0
$19$	2.71191i	0.622154i	0.950385	+	0.311077i	$0.100690\pi$
$19$	2.71191i	0.622154i	−0.950385	+	0.311077i	$0.899310\pi$
$20$	1.00000	0.223607
$21$	0	0
$22$	5.37849	1.14670
$23$	− 0.496606i	− 0.103549i	−0.998659	−	0.0517747i	$-0.983512\pi$
$23$	− 0.496606i	− 0.103549i	0.998659	−	0.0517747i	$-0.0164878\pi$
$24$	0	0
$25$	1.00000	0.200000
$26$	1.19891	0.235126
$27$	0	0
$28$	0	0
$29$	− 7.12453i	− 1.32299i	−0.749949	−	0.661496i	$-0.769922\pi$
$29$	− 7.12453i	− 1.32299i	0.749949	−	0.661496i	$-0.230078\pi$
$30$	0	0
$31$	0.550066i	0.0987947i	0.998779	+	0.0493974i	$0.0157301\pi$
$31$	0.550066i	0.0987947i	−0.998779	+	0.0493974i	$0.984270\pi$
$32$	− 1.00000i	− 0.176777i
$33$	0	0
$34$	− 2.17958i	− 0.373795i
$35$	0	0
$36$	0	0
$37$	−4.10814	−0.675374	−0.337687	−	0.941258i	$-0.609645\pi$
$37$	−4.10814	−0.675374	−0.337687	+	0.941258i	$0.609645\pi$
$38$	2.71191	0.439929
$39$	0	0
$40$	− 1.00000i	− 0.158114i
$41$	−1.04373	−0.163003	−0.0815015	−	0.996673i	$-0.525972\pi$
$41$	−1.04373	−0.163003	−0.0815015	+	0.996673i	$0.525972\pi$
$42$	0	0
$43$	6.03988	0.921074	0.460537	−	0.887641i	$-0.347657\pi$
$43$	6.03988	0.921074	0.460537	+	0.887641i	$0.347657\pi$
$44$	− 5.37849i	− 0.810838i
$45$	0	0
$46$	−0.496606	−0.0732205
$47$	−7.42676	−1.08330	−0.541652	−	0.840603i	$-0.682201\pi$
$47$	−7.42676	−1.08330	−0.541652	+	0.840603i	$0.682201\pi$
$48$	0	0
$49$	0	0
$50$	− 1.00000i	− 0.141421i
$51$	0	0
$52$	− 1.19891i	− 0.166259i
$53$	5.13707i	0.705631i	0.935693	+	0.352815i	$0.114776\pi$
$53$	5.13707i	0.705631i	−0.935693	+	0.352815i	$0.885224\pi$
$54$	0	0
$55$	− 5.37849i	− 0.725236i
$56$	0	0
$57$	0	0
$58$	−7.12453	−0.935496
$59$	−11.1097	−1.44636	−0.723182	−	0.690658i	$-0.757321\pi$
$59$	−11.1097	−1.44636	−0.723182	+	0.690658i	$0.757321\pi$
$60$	0	0
$61$	− 1.88348i	− 0.241155i	−0.992704	−	0.120577i	$-0.961525\pi$
$61$	− 1.88348i	− 0.241155i	0.992704	−	0.120577i	$-0.0384746\pi$
$62$	0.550066	0.0698584
$63$	0	0
$64$	−1.00000	−0.125000
$65$	− 1.19891i	− 0.148707i
$66$	0	0
$67$	4.25518	0.519853	0.259927	−	0.965628i	$-0.416302\pi$
$67$	4.25518	0.519853	0.259927	+	0.965628i	$0.416302\pi$
$68$	−2.17958	−0.264313
$69$	0	0
$70$	0	0
$71$	10.8052i	1.28235i	0.767396	+	0.641174i	$0.221552\pi$
$71$	10.8052i	1.28235i	−0.767396	+	0.641174i	$0.778448\pi$
$72$	0	0
$73$	− 15.5349i	− 1.81822i	−0.416554	−	0.909111i	$-0.636762\pi$
$73$	− 15.5349i	− 1.81822i	0.416554	−	0.909111i	$-0.363238\pi$
$74$	4.10814i	0.477561i
$75$	0	0
$76$	− 2.71191i	− 0.311077i
$77$	0	0
$78$	0	0
$79$	−10.4525	−1.17600	−0.587999	−	0.808861i	$-0.700084\pi$
$79$	−10.4525	−1.17600	−0.587999	+	0.808861i	$0.700084\pi$
$80$	−1.00000	−0.111803
$81$	0	0
$82$	1.04373i	0.115261i
$83$	−14.5400	−1.59597	−0.797984	−	0.602679i	$-0.794100\pi$
$83$	−14.5400	−1.59597	−0.797984	+	0.602679i	$0.794100\pi$
$84$	0	0
$85$	−2.17958	−0.236409
$86$	− 6.03988i	− 0.651297i
$87$	0	0
$88$	−5.37849	−0.573349
$89$	−15.8899	−1.68433	−0.842163	−	0.539223i	$-0.818718\pi$
$89$	−15.8899	−1.68433	−0.842163	+	0.539223i	$0.818718\pi$
$90$	0	0
$91$	0	0
$92$	0.496606i	0.0517747i
$93$	0	0
$94$	7.42676i	0.766011i
$95$	− 2.71191i	− 0.278236i
$96$	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
$101$	−0.0618412	−0.00615343	−0.00307671	−	0.999995i	$-0.500979\pi$
$101$	−0.0618412	−0.00615343	−0.00307671	+	0.999995i	$0.500979\pi$
$102$	0	0
$103$	7.44155i	0.733238i	0.930371	+	0.366619i	$0.119485\pi$
$103$	7.44155i	0.733238i	−0.930371	+	0.366619i	$0.880515\pi$
$104$	−1.19891	−0.117563
$105$	0	0
$106$	5.13707	0.498956
$107$	4.05052i	0.391578i	0.980646	+	0.195789i	$0.0627268\pi$
$107$	4.05052i	0.391578i	−0.980646	+	0.195789i	$0.937273\pi$
$108$	0	0
$109$	9.01414	0.863398	0.431699	−	0.902018i	$-0.357914\pi$
$109$	9.01414	0.863398	0.431699	+	0.902018i	$0.357914\pi$
$110$	−5.37849	−0.512819
$111$	0	0
$112$	0	0
$113$	18.9082i	1.77873i	0.457195	+	0.889367i	$0.348854\pi$
$113$	18.9082i	1.77873i	−0.457195	+	0.889367i	$0.651146\pi$
$114$	0	0
$115$	0.496606i	0.0463087i
$116$	7.12453i	0.661496i
$117$	0	0
$118$	11.1097i	1.02273i
$119$	0	0
$120$	0	0
$121$	−17.9282	−1.62983
$122$	−1.88348	−0.170522
$123$	0	0
$124$	− 0.550066i	− 0.0493974i
$125$	−1.00000	−0.0894427
$126$	0	0
$127$	12.0251	1.06705	0.533527	−	0.845783i	$-0.320866\pi$
$127$	12.0251	1.06705	0.533527	+	0.845783i	$0.320866\pi$
$128$	1.00000i	0.0883883i
$129$	0	0
$130$	−1.19891	−0.105152
$131$	−9.21077	−0.804748	−0.402374	−	0.915475i	$-0.631815\pi$
$131$	−9.21077	−0.804748	−0.402374	+	0.915475i	$0.631815\pi$
$132$	0	0
$133$	0	0
$134$	− 4.25518i	− 0.367592i
$135$	0	0
$136$	2.17958i	0.186897i
$137$	16.6195i	1.41990i	0.704250	+	0.709952i	$0.251283\pi$
$137$	16.6195i	1.41990i	−0.704250	+	0.709952i	$0.748717\pi$
$138$	0	0
$139$	− 9.13932i	− 0.775187i	−0.921830	−	0.387594i	$-0.873306\pi$
$139$	− 9.13932i	− 0.775187i	0.921830	−	0.387594i	$-0.126694\pi$
$140$	0	0
$141$	0	0
$142$	10.8052	0.906756
$143$	−6.44834	−0.539237
$144$	0	0
$145$	7.12453i	0.591660i
$146$	−15.5349	−1.28568
$147$	0	0
$148$	4.10814	0.337687
$149$	− 14.6762i	− 1.20232i	−0.799129	−	0.601160i	$-0.794705\pi$
$149$	− 14.6762i	− 1.20232i	0.799129	−	0.601160i	$-0.205295\pi$
$150$	0	0
$151$	−6.74150	−0.548615	−0.274308	−	0.961642i	$-0.588449\pi$
$151$	−6.74150	−0.548615	−0.274308	+	0.961642i	$0.588449\pi$
$152$	−2.71191	−0.219965
$153$	0	0
$154$	0	0
$155$	− 0.550066i	− 0.0441823i
$156$	0	0
$157$	− 9.62726i	− 0.768339i	−0.923263	−	0.384170i	$-0.874488\pi$
$157$	− 9.62726i	− 0.768339i	0.923263	−	0.384170i	$-0.125512\pi$
$158$	10.4525i	0.831557i
$159$	0	0
$160$	1.00000i	0.0790569i
$161$	0	0
$162$	0	0
$163$	−4.15866	−0.325731	−0.162866	−	0.986648i	$-0.552074\pi$
$163$	−4.15866	−0.325731	−0.162866	+	0.986648i	$0.552074\pi$
$164$	1.04373	0.0815015
$165$	0	0
$166$	14.5400i	1.12852i
$167$	12.4764	0.965451	0.482725	−	0.875772i	$-0.339647\pi$
$167$	12.4764	0.965451	0.482725	+	0.875772i	$0.339647\pi$
$168$	0	0
$169$	11.5626	0.889431
$170$	2.17958i	0.167166i
$171$	0	0
$172$	−6.03988	−0.460537
$173$	1.02869	0.0782098	0.0391049	−	0.999235i	$-0.487549\pi$
$173$	1.02869	0.0782098	0.0391049	+	0.999235i	$0.487549\pi$
$174$	0	0
$175$	0	0
$176$	5.37849i	0.405419i
$177$	0	0
$178$	15.8899i	1.19100i
$179$	− 9.99480i	− 0.747047i	−0.927621	−	0.373523i	$-0.878150\pi$
$179$	− 9.99480i	− 0.747047i	0.927621	−	0.373523i	$-0.121850\pi$
$180$	0	0
$181$	− 3.08014i	− 0.228945i	−0.993426	−	0.114472i	$-0.963482\pi$
$181$	− 3.08014i	− 0.228945i	0.993426	−	0.114472i	$-0.0365178\pi$
$182$	0	0
$183$	0	0
$184$	0.496606	0.0366103
$185$	4.10814	0.302036
$186$	0	0
$187$	11.7229i	0.857260i
$188$	7.42676	0.541652
$189$	0	0
$190$	−2.71191	−0.196742
$191$	5.82805i	0.421703i	0.977518	+	0.210852i	$0.0676237\pi$
$191$	5.82805i	0.421703i	−0.977518	+	0.210852i	$0.932376\pi$
$192$	0	0
$193$	−5.75789	−0.414462	−0.207231	−	0.978292i	$-0.566445\pi$
$193$	−5.75789	−0.414462	−0.207231	+	0.978292i	$0.566445\pi$
$194$	−7.65685	−0.549730
$195$	0	0
$196$	0	0
$197$	14.9259i	1.06343i	0.846924	+	0.531714i	$0.178452\pi$
$197$	14.9259i	1.06343i	−0.846924	+	0.531714i	$0.821548\pi$
$198$	0	0
$199$	26.6163i	1.88678i	0.331691	+	0.943388i	$0.392381\pi$
$199$	26.6163i	1.88678i	−0.331691	+	0.943388i	$0.607619\pi$
$200$	1.00000i	0.0707107i
$201$	0	0
$202$	0.0618412i	0.00435113i
$203$	0	0
$204$	0	0
$205$	1.04373	0.0728971
$206$	7.44155	0.518478
$207$	0	0
$208$	1.19891i	0.0831296i
$209$	−14.5860	−1.00893
$210$	0	0
$211$	−7.65269	−0.526833	−0.263417	−	0.964682i	$-0.584849\pi$
$211$	−7.65269	−0.526833	−0.263417	+	0.964682i	$0.584849\pi$
$212$	− 5.13707i	− 0.352815i
$213$	0	0
$214$	4.05052	0.276888
$215$	−6.03988	−0.411917
$216$	0	0
$217$	0	0
$218$	− 9.01414i	− 0.610514i
$219$	0	0
$220$	5.37849i	0.362618i
$221$	2.61313i	0.175778i
$222$	0	0
$223$	− 2.25715i	− 0.151150i	−0.997140	−	0.0755750i	$-0.975921\pi$
$223$	− 2.25715i	− 0.151150i	0.997140	−	0.0755750i	$-0.0240792\pi$
$224$	0	0
$225$	0	0
$226$	18.9082	1.25775
$227$	−17.5011	−1.16159	−0.580796	−	0.814049i	$-0.697259\pi$
$227$	−17.5011	−1.16159	−0.580796	+	0.814049i	$0.697259\pi$
$228$	0	0
$229$	− 24.4840i	− 1.61795i	−0.587844	−	0.808974i	$-0.700023\pi$
$229$	− 24.4840i	− 1.61795i	0.587844	−	0.808974i	$-0.299977\pi$
$230$	0.496606	0.0327452
$231$	0	0
$232$	7.12453	0.467748
$233$	− 17.7366i	− 1.16196i	−0.813916	−	0.580982i	$-0.802669\pi$
$233$	− 17.7366i	− 1.16196i	0.813916	−	0.580982i	$-0.197331\pi$
$234$	0	0
$235$	7.42676	0.484468
$236$	11.1097	0.723182
$237$	0	0
$238$	0	0
$239$	14.8111i	0.958053i	0.877801	+	0.479026i	$0.159010\pi$
$239$	14.8111i	0.958053i	−0.877801	+	0.479026i	$0.840990\pi$
$240$	0	0
$241$	9.93938i	0.640252i	0.947375	+	0.320126i	$0.103725\pi$
$241$	9.93938i	0.640252i	−0.947375	+	0.320126i	$0.896275\pi$
$242$	17.9282i	1.15247i
$243$	0	0
$244$	1.88348i	0.120577i
$245$	0	0
$246$	0	0
$247$	−3.25134	−0.206878
$248$	−0.550066	−0.0349292
$249$	0	0
$250$	1.00000i	0.0632456i
$251$	5.66593	0.357630	0.178815	−	0.983883i	$-0.442774\pi$
$251$	5.66593	0.357630	0.178815	+	0.983883i	$0.442774\pi$
$252$	0	0
$253$	2.67099	0.167924
$254$	− 12.0251i	− 0.754521i
$255$	0	0
$256$	1.00000	0.0625000
$257$	−27.4814	−1.71424	−0.857122	−	0.515113i	$-0.827750\pi$
$257$	−27.4814	−1.71424	−0.857122	+	0.515113i	$0.827750\pi$
$258$	0	0
$259$	0	0
$260$	1.19891i	0.0743534i
$261$	0	0
$262$	9.21077i	0.569043i
$263$	− 4.06150i	− 0.250443i	−0.992129	−	0.125221i	$-0.960036\pi$
$263$	− 4.06150i	− 0.250443i	0.992129	−	0.125221i	$-0.0399641\pi$
$264$	0	0
$265$	− 5.13707i	− 0.315568i
$266$	0	0
$267$	0	0
$268$	−4.25518	−0.259927
$269$	0.0637508	0.00388696	0.00194348	−	0.999998i	$-0.499381\pi$
$269$	0.0637508	0.00388696	0.00194348	+	0.999998i	$0.499381\pi$
$270$	0	0
$271$	28.8200i	1.75069i	0.483496	+	0.875347i	$0.339367\pi$
$271$	28.8200i	1.75069i	−0.483496	+	0.875347i	$0.660633\pi$
$272$	2.17958	0.132156
$273$	0	0
$274$	16.6195	1.00402
$275$	5.37849i	0.324335i
$276$	0	0
$277$	6.49141	0.390031	0.195016	−	0.980800i	$-0.437524\pi$
$277$	6.49141	0.390031	0.195016	+	0.980800i	$0.437524\pi$
$278$	−9.13932	−0.548140
$279$	0	0
$280$	0	0
$281$	12.7666i	0.761591i	0.924659	+	0.380795i	$0.124350\pi$
$281$	12.7666i	0.761591i	−0.924659	+	0.380795i	$0.875650\pi$
$282$	0	0
$283$	− 13.3637i	− 0.794389i	−0.917734	−	0.397195i	$-0.869984\pi$
$283$	− 13.3637i	− 0.794389i	0.917734	−	0.397195i	$-0.130016\pi$
$284$	− 10.8052i	− 0.641174i
$285$	0	0
$286$	6.44834i	0.381298i
$287$	0	0
$288$	0	0
$289$	−12.2494	−0.720555
$290$	7.12453	0.418367
$291$	0	0
$292$	15.5349i	0.909111i
$293$	−5.76377	−0.336723	−0.168362	−	0.985725i	$-0.553848\pi$
$293$	−5.76377	−0.336723	−0.168362	+	0.985725i	$0.553848\pi$
$294$	0	0
$295$	11.1097	0.646834
$296$	− 4.10814i	− 0.238781i
$297$	0	0
$298$	−14.6762	−0.850169
$299$	0.595387	0.0344321
$300$	0	0
$301$	0	0
$302$	6.74150i	0.387930i
$303$	0	0
$304$	2.71191i	0.155539i
$305$	1.88348i	0.107848i
$306$	0	0
$307$	24.5978i	1.40387i	0.712240	+	0.701936i	$0.247681\pi$
$307$	24.5978i	1.40387i	−0.712240	+	0.701936i	$0.752319\pi$
$308$	0	0
$309$	0	0
$310$	−0.550066	−0.0312416
$311$	−25.3356	−1.43665	−0.718325	−	0.695707i	$-0.755091\pi$
$311$	−25.3356	−1.43665	−0.718325	+	0.695707i	$0.755091\pi$
$312$	0	0
$313$	25.5213i	1.44255i	0.692649	+	0.721275i	$0.256444\pi$
$313$	25.5213i	1.44255i	−0.692649	+	0.721275i	$0.743556\pi$
$314$	−9.62726	−0.543298
$315$	0	0
$316$	10.4525	0.587999
$317$	21.4206i	1.20310i	0.798834	+	0.601551i	$0.205450\pi$
$317$	21.4206i	1.20310i	−0.798834	+	0.601551i	$0.794550\pi$
$318$	0	0
$319$	38.3192	2.14546
$320$	1.00000	0.0559017
$321$	0	0
$322$	0	0
$323$	5.91082i	0.328887i
$324$	0	0
$325$	1.19891i	0.0665037i
$326$	4.15866i	0.230327i
$327$	0	0
$328$	− 1.04373i	− 0.0576303i
$329$	0	0
$330$	0	0
$331$	−33.3430	−1.83269	−0.916347	−	0.400384i	$-0.868877\pi$
$331$	−33.3430	−1.83269	−0.916347	+	0.400384i	$0.868877\pi$
$332$	14.5400	0.797984
$333$	0	0
$334$	− 12.4764i	− 0.682677i
$335$	−4.25518	−0.232486
$336$	0	0
$337$	−16.4776	−0.897592	−0.448796	−	0.893634i	$-0.648147\pi$
$337$	−16.4776	−0.897592	−0.448796	+	0.893634i	$0.648147\pi$
$338$	− 11.5626i	− 0.628923i
$339$	0	0
$340$	2.17958	0.118204
$341$	−2.95852	−0.160213
$342$	0	0
$343$	0	0
$344$	6.03988i	0.325649i
$345$	0	0
$346$	− 1.02869i	− 0.0553027i
$347$	− 8.71832i	− 0.468024i	−0.972234	−	0.234012i	$-0.924814\pi$
$347$	− 8.71832i	− 0.468024i	0.972234	−	0.234012i	$-0.0751855\pi$
$348$	0	0
$349$	10.6141i	0.568158i	0.958801	+	0.284079i	$0.0916877\pi$
$349$	10.6141i	0.568158i	−0.958801	+	0.284079i	$0.908312\pi$
$350$	0	0
$351$	0	0
$352$	5.37849	0.286675
$353$	−35.0073	−1.86325	−0.931624	−	0.363423i	$-0.881608\pi$
$353$	−35.0073	−1.86325	−0.931624	+	0.363423i	$0.881608\pi$
$354$	0	0
$355$	− 10.8052i	− 0.573483i
$356$	15.8899	0.842163
$357$	0	0
$358$	−9.99480	−0.528242
$359$	− 6.19703i	− 0.327067i	−0.986538	−	0.163533i	$-0.947711\pi$
$359$	− 6.19703i	− 0.327067i	0.986538	−	0.163533i	$-0.0522892\pi$
$360$	0	0
$361$	11.6456	0.612924
$362$	−3.08014	−0.161889
$363$	0	0
$364$	0	0
$365$	15.5349i	0.813134i
$366$	0	0
$367$	− 33.5641i	− 1.75203i	−0.482280	−	0.876017i	$-0.660191\pi$
$367$	− 33.5641i	− 1.75203i	0.482280	−	0.876017i	$-0.339809\pi$
$368$	− 0.496606i	− 0.0258874i
$369$	0	0
$370$	− 4.10814i	− 0.213572i
$371$	0	0
$372$	0	0
$373$	−8.38547	−0.434183	−0.217092	−	0.976151i	$-0.569657\pi$
$373$	−8.38547	−0.434183	−0.217092	+	0.976151i	$0.569657\pi$
$374$	11.7229	0.606175
$375$	0	0
$376$	− 7.42676i	− 0.383006i
$377$	8.54168	0.439919
$378$	0	0
$379$	1.33954	0.0688077	0.0344038	−	0.999408i	$-0.489047\pi$
$379$	1.33954	0.0688077	0.0344038	+	0.999408i	$0.489047\pi$
$380$	2.71191i	0.139118i
$381$	0	0
$382$	5.82805	0.298189
$383$	21.2686	1.08677	0.543387	−	0.839483i	$-0.317142\pi$
$383$	21.2686	1.08677	0.543387	+	0.839483i	$0.317142\pi$
$384$	0	0
$385$	0	0
$386$	5.75789i	0.293069i
$387$	0	0
$388$	7.65685i	0.388718i
$389$	− 27.8774i	− 1.41344i	−0.707494	−	0.706719i	$-0.750175\pi$
$389$	− 27.8774i	− 1.41344i	0.707494	−	0.706719i	$-0.249825\pi$
$390$	0	0
$391$	− 1.08239i	− 0.0547389i
$392$	0	0
$393$	0	0
$394$	14.9259	0.751958
$395$	10.4525	0.525923
$396$	0	0
$397$	− 0.789763i	− 0.0396371i	−0.999804	−	0.0198185i	$-0.993691\pi$
$397$	− 0.789763i	− 0.0396371i	0.999804	−	0.0198185i	$-0.00630885\pi$
$398$	26.6163	1.33415
$399$	0	0
$400$	1.00000	0.0500000
$401$	− 10.8599i	− 0.542319i	−0.962534	−	0.271159i	$-0.912593\pi$
$401$	− 10.8599i	− 0.542319i	0.962534	−	0.271159i	$-0.0874071\pi$
$402$	0	0
$403$	−0.659481	−0.0328511
$404$	0.0618412	0.00307671
$405$	0	0
$406$	0	0
$407$	− 22.0956i	− 1.09524i
$408$	0	0
$409$	− 24.9507i	− 1.23373i	−0.787068	−	0.616866i	$-0.788402\pi$
$409$	− 24.9507i	− 1.23373i	0.787068	−	0.616866i	$-0.211598\pi$
$410$	− 1.04373i	− 0.0515461i
$411$	0	0
$412$	− 7.44155i	− 0.366619i
$413$	0	0
$414$	0	0
$415$	14.5400	0.713739
$416$	1.19891	0.0587815
$417$	0	0
$418$	14.5860i	0.713423i
$419$	−38.9552	−1.90309	−0.951543	−	0.307517i	$-0.900502\pi$
$419$	−38.9552	−1.90309	−0.951543	+	0.307517i	$0.900502\pi$
$420$	0	0
$421$	−38.1818	−1.86087	−0.930433	−	0.366462i	$-0.880569\pi$
$421$	−38.1818	−1.86087	−0.930433	+	0.366462i	$0.880569\pi$
$422$	7.65269i	0.372527i
$423$	0	0
$424$	−5.13707	−0.249478
$425$	2.17958	0.105725
$426$	0	0
$427$	0	0
$428$	− 4.05052i	− 0.195789i
$429$	0	0
$430$	6.03988i	0.291269i
$431$	3.01639i	0.145294i	0.997358	+	0.0726472i	$0.0231447\pi$
$431$	3.01639i	0.145294i	−0.997358	+	0.0726472i	$0.976855\pi$
$432$	0	0
$433$	37.7167i	1.81255i	0.422691	+	0.906274i	$0.361086\pi$
$433$	37.7167i	1.81255i	−0.422691	+	0.906274i	$0.638914\pi$
$434$	0	0
$435$	0	0
$436$	−9.01414	−0.431699
$437$	1.34675	0.0644237
$438$	0	0
$439$	− 21.3683i	− 1.01985i	−0.860217	−	0.509927i	$-0.829672\pi$
$439$	− 21.3683i	− 1.01985i	0.860217	−	0.509927i	$-0.170328\pi$
$440$	5.37849	0.256410
$441$	0	0
$442$	2.61313	0.124294
$443$	− 1.59629i	− 0.0758420i	−0.999281	−	0.0379210i	$-0.987926\pi$
$443$	− 1.59629i	− 0.0758420i	0.999281	−	0.0379210i	$-0.0120735\pi$
$444$	0	0
$445$	15.8899	0.753253
$446$	−2.25715	−0.106879
$447$	0	0
$448$	0	0
$449$	14.9895i	0.707398i	0.935359	+	0.353699i	$0.115076\pi$
$449$	14.9895i	0.707398i	−0.935359	+	0.353699i	$0.884924\pi$
$450$	0	0
$451$	− 5.61369i	− 0.264338i
$452$	− 18.9082i	− 0.889367i
$453$	0	0
$454$	17.5011i	0.821369i
$455$	0	0
$456$	0	0
$457$	22.5650	1.05555	0.527774	−	0.849385i	$-0.323027\pi$
$457$	22.5650	1.05555	0.527774	+	0.849385i	$0.323027\pi$
$458$	−24.4840	−1.14406
$459$	0	0
$460$	− 0.496606i	− 0.0231544i
$461$	7.03506	0.327655	0.163828	−	0.986489i	$-0.447616\pi$
$461$	7.03506	0.327655	0.163828	+	0.986489i	$0.447616\pi$
$462$	0	0
$463$	−28.8845	−1.34238	−0.671188	−	0.741287i	$-0.734216\pi$
$463$	−28.8845	−1.34238	−0.671188	+	0.741287i	$0.734216\pi$
$464$	− 7.12453i	− 0.330748i
$465$	0	0
$466$	−17.7366	−0.821633
$467$	−26.5709	−1.22956	−0.614778	−	0.788700i	$-0.710754\pi$
$467$	−26.5709	−1.22956	−0.614778	+	0.788700i	$0.710754\pi$
$468$	0	0
$469$	0	0
$470$	− 7.42676i	− 0.342571i
$471$	0	0
$472$	− 11.1097i	− 0.511367i
$473$	32.4855i	1.49368i
$474$	0	0
$475$	2.71191i	0.124431i
$476$	0	0
$477$	0	0
$478$	14.8111	0.677446
$479$	28.6410	1.30864	0.654320	−	0.756218i	$-0.272955\pi$
$479$	28.6410	1.30864	0.654320	+	0.756218i	$0.272955\pi$
$480$	0	0
$481$	− 4.92530i	− 0.224574i
$482$	9.93938	0.452726
$483$	0	0
$484$	17.9282	0.814917
$485$	7.65685i	0.347680i
$486$	0	0
$487$	18.7266	0.848585	0.424293	−	0.905525i	$-0.360523\pi$
$487$	18.7266	0.848585	0.424293	+	0.905525i	$0.360523\pi$
$488$	1.88348	0.0852611
$489$	0	0
$490$	0	0
$491$	− 24.0927i	− 1.08729i	−0.839317	−	0.543643i	$-0.817045\pi$
$491$	− 24.0927i	− 1.08729i	0.839317	−	0.543643i	$-0.182955\pi$
$492$	0	0
$493$	− 15.5285i	− 0.699368i
$494$	3.25134i	0.146285i
$495$	0	0
$496$	0.550066i	0.0246987i
$497$	0	0
$498$	0	0
$499$	−14.7333	−0.659552	−0.329776	−	0.944059i	$-0.606973\pi$
$499$	−14.7333	−0.659552	−0.329776	+	0.944059i	$0.606973\pi$
$500$	1.00000	0.0447214
$501$	0	0
$502$	− 5.66593i	− 0.252883i
$503$	29.6758	1.32318	0.661590	−	0.749866i	$-0.269882\pi$
$503$	29.6758	1.32318	0.661590	+	0.749866i	$0.269882\pi$
$504$	0	0
$505$	0.0618412	0.00275190
$506$	− 2.67099i	− 0.118740i
$507$	0	0
$508$	−12.0251	−0.533527
$509$	−2.69833	−0.119601	−0.0598007	−	0.998210i	$-0.519047\pi$
$509$	−2.69833	−0.119601	−0.0598007	+	0.998210i	$0.519047\pi$
$510$	0	0
$511$	0	0
$512$	− 1.00000i	− 0.0441942i
$513$	0	0
$514$	27.4814i	1.21215i
$515$	− 7.44155i	− 0.327914i
$516$	0	0
$517$	− 39.9448i	− 1.75677i
$518$	0	0
$519$	0	0
$520$	1.19891	0.0525758
$521$	−20.9210	−0.916567	−0.458283	−	0.888806i	$-0.651535\pi$
$521$	−20.9210	−0.916567	−0.458283	+	0.888806i	$0.651535\pi$
$522$	0	0
$523$	15.7615i	0.689203i	0.938749	+	0.344602i	$0.111986\pi$
$523$	15.7615i	0.689203i	−0.938749	+	0.344602i	$0.888014\pi$
$524$	9.21077	0.402374
$525$	0	0
$526$	−4.06150	−0.177090
$527$	1.19891i	0.0522254i
$528$	0	0
$529$	22.7534	0.989278
$530$	−5.13707	−0.223140
$531$	0	0
$532$	0	0
$533$	− 1.25134i	− 0.0542015i
$534$	0	0
$535$	− 4.05052i	− 0.175119i
$536$	4.25518i	0.183796i
$537$	0	0
$538$	− 0.0637508i	− 0.00274849i
$539$	0	0
$540$	0	0
$541$	40.9865	1.76215	0.881074	−	0.472979i	$-0.156821\pi$
$541$	40.9865	1.76215	0.881074	+	0.472979i	$0.156821\pi$
$542$	28.8200	1.23793
$543$	0	0
$544$	− 2.17958i	− 0.0934487i
$545$	−9.01414	−0.386123
$546$	0	0
$547$	18.8878	0.807583	0.403791	−	0.914851i	$-0.367692\pi$
$547$	18.8878	0.807583	0.403791	+	0.914851i	$0.367692\pi$
$548$	− 16.6195i	− 0.709952i
$549$	0	0
$550$	5.37849	0.229340
$551$	19.3211	0.823105
$552$	0	0
$553$	0	0
$554$	− 6.49141i	− 0.275794i
$555$	0	0
$556$	9.13932i	0.387594i
$557$	− 14.3256i	− 0.606997i	−0.952832	−	0.303498i	$-0.901845\pi$
$557$	− 14.3256i	− 0.606997i	0.952832	−	0.303498i	$-0.0981547\pi$
$558$	0	0
$559$	7.24129i	0.306274i
$560$	0	0
$561$	0	0
$562$	12.7666	0.538526
$563$	8.49754	0.358129	0.179064	−	0.983837i	$-0.442693\pi$
$563$	8.49754	0.358129	0.179064	+	0.983837i	$0.442693\pi$
$564$	0	0
$565$	− 18.9082i	− 0.795474i
$566$	−13.3637	−0.561718
$567$	0	0
$568$	−10.8052	−0.453378
$569$	− 20.8380i	− 0.873576i	−0.899565	−	0.436788i	$-0.856116\pi$
$569$	− 20.8380i	− 0.873576i	0.899565	−	0.436788i	$-0.143884\pi$
$570$	0	0
$571$	−4.87346	−0.203948	−0.101974	−	0.994787i	$-0.532516\pi$
$571$	−4.87346	−0.203948	−0.101974	+	0.994787i	$0.532516\pi$
$572$	6.44834	0.269619
$573$	0	0
$574$	0	0
$575$	− 0.496606i	− 0.0207099i
$576$	0	0
$577$	32.6863i	1.36075i	0.732866	+	0.680373i	$0.238182\pi$
$577$	32.6863i	1.36075i	−0.732866	+	0.680373i	$0.761818\pi$
$578$	12.2494i	0.509509i
$579$	0	0
$580$	− 7.12453i	− 0.295830i
$581$	0	0
$582$	0	0
$583$	−27.6297	−1.14431
$584$	15.5349	0.642839
$585$	0	0
$586$	5.76377i	0.238099i
$587$	43.5641	1.79808	0.899042	−	0.437862i	$-0.144264\pi$
$587$	43.5641	1.79808	0.899042	+	0.437862i	$0.144264\pi$
$588$	0	0
$589$	−1.49173	−0.0614655
$590$	− 11.1097i	− 0.457380i
$591$	0	0
$592$	−4.10814	−0.168843
$593$	20.7978	0.854065	0.427033	−	0.904236i	$-0.359559\pi$
$593$	20.7978	0.854065	0.427033	+	0.904236i	$0.359559\pi$
$594$	0	0
$595$	0	0
$596$	14.6762i	0.601160i
$597$	0	0
$598$	− 0.595387i	− 0.0243472i
$599$	− 20.1486i	− 0.823247i	−0.911354	−	0.411624i	$-0.864962\pi$
$599$	− 20.1486i	− 0.823247i	0.911354	−	0.411624i	$-0.135038\pi$
$600$	0	0
$601$	20.4515i	0.834233i	0.908853	+	0.417116i	$0.136959\pi$
$601$	20.4515i	0.834233i	−0.908853	+	0.417116i	$0.863041\pi$
$602$	0	0
$603$	0	0
$604$	6.74150	0.274308
$605$	17.9282	0.728884
$606$	0	0
$607$	24.3111i	0.986756i	0.869815	+	0.493378i	$0.164238\pi$
$607$	24.3111i	0.986756i	−0.869815	+	0.493378i	$0.835762\pi$
$608$	2.71191	0.109982
$609$	0	0
$610$	1.88348	0.0762599
$611$	− 8.90403i	− 0.360219i
$612$	0	0
$613$	−43.4497	−1.75492	−0.877459	−	0.479652i	$-0.840763\pi$
$613$	−43.4497	−1.75492	−0.877459	+	0.479652i	$0.840763\pi$
$614$	24.5978	0.992687
$615$	0	0
$616$	0	0
$617$	21.0693i	0.848220i	0.905611	+	0.424110i	$0.139413\pi$
$617$	21.0693i	0.848220i	−0.905611	+	0.424110i	$0.860587\pi$
$618$	0	0
$619$	27.6215i	1.11020i	0.831783	+	0.555101i	$0.187320\pi$
$619$	27.6215i	1.11020i	−0.831783	+	0.555101i	$0.812680\pi$
$620$	0.550066i	0.0220912i
$621$	0	0
$622$	25.3356i	1.01587i
$623$	0	0
$624$	0	0
$625$	1.00000	0.0400000
$626$	25.5213	1.02004
$627$	0	0
$628$	9.62726i	0.384170i
$629$	−8.95402	−0.357020
$630$	0	0
$631$	−9.81204	−0.390611	−0.195306	−	0.980742i	$-0.562570\pi$
$631$	−9.81204	−0.390611	−0.195306	+	0.980742i	$0.562570\pi$
$632$	− 10.4525i	− 0.415778i
$633$	0	0
$634$	21.4206	0.850722
$635$	−12.0251	−0.477201
$636$	0	0
$637$	0	0
$638$	− 38.3192i	− 1.51707i
$639$	0	0
$640$	− 1.00000i	− 0.0395285i
$641$	− 22.3931i	− 0.884474i	−0.896898	−	0.442237i	$-0.854185\pi$
$641$	− 22.3931i	− 0.884474i	0.896898	−	0.442237i	$-0.145815\pi$
$642$	0	0
$643$	19.1699i	0.755987i	0.925808	+	0.377993i	$0.123386\pi$
$643$	19.1699i	0.755987i	−0.925808	+	0.377993i	$0.876614\pi$
$644$	0	0
$645$	0	0
$646$	5.91082	0.232558
$647$	8.27465	0.325310	0.162655	−	0.986683i	$-0.447994\pi$
$647$	8.27465	0.325310	0.162655	+	0.986683i	$0.447994\pi$
$648$	0	0
$649$	− 59.7536i	− 2.34553i
$650$	1.19891	0.0470252
$651$	0	0
$652$	4.15866	0.162866
$653$	29.7566i	1.16446i	0.813023	+	0.582232i	$0.197821\pi$
$653$	29.7566i	1.16446i	−0.813023	+	0.582232i	$0.802179\pi$
$654$	0	0
$655$	9.21077	0.359894
$656$	−1.04373	−0.0407507
$657$	0	0
$658$	0	0
$659$	− 30.2145i	− 1.17699i	−0.808501	−	0.588495i	$-0.799721\pi$
$659$	− 30.2145i	− 1.17699i	0.808501	−	0.588495i	$-0.200279\pi$
$660$	0	0
$661$	− 44.0803i	− 1.71452i	−0.514880	−	0.857262i	$-0.672163\pi$
$661$	− 44.0803i	− 1.71452i	0.514880	−	0.857262i	$-0.327837\pi$
$662$	33.3430i	1.29591i
$663$	0	0
$664$	− 14.5400i	− 0.564260i
$665$	0	0
$666$	0	0
$667$	−3.53808	−0.136995
$668$	−12.4764	−0.482725
$669$	0	0
$670$	4.25518i	0.164392i
$671$	10.1303	0.391075
$672$	0	0
$673$	−0.793211	−0.0305760	−0.0152880	−	0.999883i	$-0.504867\pi$
$673$	−0.793211	−0.0305760	−0.0152880	+	0.999883i	$0.504867\pi$
$674$	16.4776i	0.634693i
$675$	0	0
$676$	−11.5626	−0.444716
$677$	44.0480	1.69290	0.846452	−	0.532465i	$-0.178734\pi$
$677$	44.0480	1.69290	0.846452	+	0.532465i	$0.178734\pi$
$678$	0	0
$679$	0	0
$680$	− 2.17958i	− 0.0835831i
$681$	0	0
$682$	2.95852i	0.113288i
$683$	16.5254i	0.632327i	0.948705	+	0.316164i	$0.102395\pi$
$683$	16.5254i	0.632327i	−0.948705	+	0.316164i	$0.897605\pi$
$684$	0	0
$685$	− 16.6195i	− 0.635000i
$686$	0	0
$687$	0	0
$688$	6.03988	0.230268
$689$	−6.15890	−0.234635
$690$	0	0
$691$	22.4248i	0.853080i	0.904469	+	0.426540i	$0.140268\pi$
$691$	22.4248i	0.853080i	−0.904469	+	0.426540i	$0.859732\pi$
$692$	−1.02869	−0.0391049
$693$	0	0
$694$	−8.71832	−0.330943
$695$	9.13932i	0.346674i
$696$	0	0
$697$	−2.27489	−0.0861676
$698$	10.6141	0.401748
$699$	0	0
$700$	0	0
$701$	− 14.1136i	− 0.533062i	−0.963826	−	0.266531i	$-0.914122\pi$
$701$	− 14.1136i	− 0.533062i	0.963826	−	0.266531i	$-0.0858775\pi$
$702$	0	0
$703$	− 11.1409i	− 0.420187i
$704$	− 5.37849i	− 0.202710i
$705$	0	0
$706$	35.0073i	1.31752i
$707$	0	0
$708$	0	0
$709$	33.3349	1.25192	0.625959	−	0.779856i	$-0.284708\pi$
$709$	33.3349	1.25192	0.625959	+	0.779856i	$0.284708\pi$
$710$	−10.8052	−0.405514
$711$	0	0
$712$	− 15.8899i	− 0.595499i
$713$	0.273166	0.0102301
$714$	0	0
$715$	6.44834	0.241154
$716$	9.99480i	0.373523i
$717$	0	0
$718$	−6.19703	−0.231271
$719$	16.2049	0.604341	0.302171	−	0.953254i	$-0.402289\pi$
$719$	16.2049	0.604341	0.302171	+	0.953254i	$0.402289\pi$
$720$	0	0
$721$	0	0
$722$	− 11.6456i	− 0.433403i
$723$	0	0
$724$	3.08014i	0.114472i
$725$	− 7.12453i	− 0.264598i
$726$	0	0
$727$	− 14.2759i	− 0.529463i	−0.964322	−	0.264731i	$-0.914717\pi$
$727$	− 14.2759i	− 0.529463i	0.964322	−	0.264731i	$-0.0852833\pi$
$728$	0	0
$729$	0	0
$730$	15.5349	0.574972
$731$	13.1644	0.486903
$732$	0	0
$733$	− 48.3899i	− 1.78732i	−0.448744	−	0.893660i	$-0.648128\pi$
$733$	− 48.3899i	− 1.78732i	0.448744	−	0.893660i	$-0.351872\pi$
$734$	−33.5641	−1.23888
$735$	0	0
$736$	−0.496606	−0.0183051
$737$	22.8865i	0.843034i
$738$	0	0
$739$	30.6546	1.12765	0.563823	−	0.825895i	$-0.309330\pi$
$739$	30.6546	1.12765	0.563823	+	0.825895i	$0.309330\pi$
$740$	−4.10814	−0.151018
$741$	0	0
$742$	0	0
$743$	23.5113i	0.862545i	0.902222	+	0.431272i	$0.141935\pi$
$743$	23.5113i	0.862545i	−0.902222	+	0.431272i	$0.858065\pi$
$744$	0	0
$745$	14.6762i	0.537694i
$746$	8.38547i	0.307014i
$747$	0	0
$748$	− 11.7229i	− 0.428630i
$749$	0	0
$750$	0	0
$751$	−26.5805	−0.969937	−0.484969	−	0.874531i	$-0.661169\pi$
$751$	−26.5805	−0.969937	−0.484969	+	0.874531i	$0.661169\pi$
$752$	−7.42676	−0.270826
$753$	0	0
$754$	− 8.54168i	− 0.311070i
$755$	6.74150	0.245348
$756$	0	0
$757$	−47.7997	−1.73731	−0.868655	−	0.495417i	$-0.835015\pi$
$757$	−47.7997	−1.73731	−0.868655	+	0.495417i	$0.835015\pi$
$758$	− 1.33954i	− 0.0486544i
$759$	0	0
$760$	2.71191	0.0983712
$761$	−14.8528	−0.538412	−0.269206	−	0.963083i	$-0.586761\pi$
$761$	−14.8528	−0.538412	−0.269206	+	0.963083i	$0.586761\pi$
$762$	0	0
$763$	0	0
$764$	− 5.82805i	− 0.210852i
$765$	0	0
$766$	− 21.2686i	− 0.768465i
$767$	− 13.3196i	− 0.480943i
$768$	0	0
$769$	− 35.4756i	− 1.27928i	−0.768674	−	0.639641i	$-0.779083\pi$
$769$	− 35.4756i	− 1.27928i	0.768674	−	0.639641i	$-0.220917\pi$
$770$	0	0
$771$	0	0
$772$	5.75789	0.207231
$773$	3.06221	0.110140	0.0550701	−	0.998482i	$-0.482462\pi$
$773$	3.06221	0.110140	0.0550701	+	0.998482i	$0.482462\pi$
$774$	0	0
$775$	0.550066i	0.0197589i
$776$	7.65685	0.274865
$777$	0	0
$778$	−27.8774	−0.999451
$779$	− 2.83049i	− 0.101413i
$780$	0	0
$781$	−58.1160	−2.07955
$782$	−1.08239	−0.0387063
$783$	0	0
$784$	0	0
$785$	9.62726i	0.343612i
$786$	0	0
$787$	− 34.8957i	− 1.24390i	−0.783058	−	0.621948i	$-0.786341\pi$
$787$	− 34.8957i	− 1.24390i	0.783058	−	0.621948i	$-0.213659\pi$
$788$	− 14.9259i	− 0.531714i
$789$	0	0
$790$	− 10.4525i	− 0.371883i
$791$	0	0
$792$	0	0
$793$	2.25813	0.0801885
$794$	−0.789763	−0.0280276
$795$	0	0
$796$	− 26.6163i	− 0.943388i
$797$	−2.52920	−0.0895888	−0.0447944	−	0.998996i	$-0.514263\pi$
$797$	−2.52920	−0.0895888	−0.0447944	+	0.998996i	$0.514263\pi$
$798$	0	0
$799$	−16.1872	−0.572662
$800$	− 1.00000i	− 0.0353553i
$801$	0	0
$802$	−10.8599	−0.383477
$803$	83.5543	2.94857
$804$	0	0
$805$	0	0
$806$	0.659481i	0.0232292i
$807$	0	0
$808$	− 0.0618412i	− 0.00217557i
$809$	− 43.4977i	− 1.52930i	−0.644446	−	0.764649i	$-0.722912\pi$
$809$	− 43.4977i	− 1.52930i	0.644446	−	0.764649i	$-0.277088\pi$
$810$	0	0
$811$	− 34.6495i	− 1.21671i	−0.793665	−	0.608355i	$-0.791830\pi$
$811$	− 34.6495i	− 1.21671i	0.793665	−	0.608355i	$-0.208170\pi$
$812$	0	0
$813$	0	0
$814$	−22.0956	−0.774450
$815$	4.15866	0.145671
$816$	0	0
$817$	16.3796i	0.573050i
$818$	−24.9507	−0.872380
$819$	0	0
$820$	−1.04373	−0.0364486
$821$	19.9704i	0.696970i	0.937314	+	0.348485i	$0.113304\pi$
$821$	19.9704i	0.696970i	−0.937314	+	0.348485i	$0.886696\pi$
$822$	0	0
$823$	8.24587	0.287433	0.143716	−	0.989619i	$-0.454095\pi$
$823$	8.24587	0.287433	0.143716	+	0.989619i	$0.454095\pi$
$824$	−7.44155	−0.259239
$825$	0	0
$826$	0	0
$827$	− 16.3355i	− 0.568039i	−0.958818	−	0.284020i	$-0.908332\pi$
$827$	− 16.3355i	− 0.568039i	0.958818	−	0.284020i	$-0.0916681\pi$
$828$	0	0
$829$	36.8032i	1.27823i	0.769112	+	0.639114i	$0.220699\pi$
$829$	36.8032i	1.27823i	−0.769112	+	0.639114i	$0.779301\pi$
$830$	− 14.5400i	− 0.504689i
$831$	0	0
$832$	− 1.19891i	− 0.0415648i
$833$	0	0
$834$	0	0
$835$	−12.4764	−0.431763
$836$	14.5860	0.504466
$837$	0	0
$838$	38.9552i	1.34568i
$839$	−0.712840	−0.0246100	−0.0123050	−	0.999924i	$-0.503917\pi$
$839$	−0.712840	−0.0246100	−0.0123050	+	0.999924i	$0.503917\pi$
$840$	0	0
$841$	−21.7589	−0.750307
$842$	38.1818i	1.31583i
$843$	0	0
$844$	7.65269	0.263417
$845$	−11.5626	−0.397766
$846$	0	0
$847$	0	0
$848$	5.13707i	0.176408i
$849$	0	0
$850$	− 2.17958i	− 0.0747590i
$851$	2.04013i	0.0699346i
$852$	0	0
$853$	26.8405i	0.919002i	0.888177	+	0.459501i	$0.151972\pi$
$853$	26.8405i	0.919002i	−0.888177	+	0.459501i	$0.848028\pi$
$854$	0	0
$855$	0	0
$856$	−4.05052	−0.138444
$857$	−4.27465	−0.146019	−0.0730096	−	0.997331i	$-0.523260\pi$
$857$	−4.27465	−0.146019	−0.0730096	+	0.997331i	$0.523260\pi$
$858$	0	0
$859$	− 11.7201i	− 0.399884i	−0.979808	−	0.199942i	$-0.935925\pi$
$859$	− 11.7201i	− 0.399884i	0.979808	−	0.199942i	$-0.0640754\pi$
$860$	6.03988	0.205958
$861$	0	0
$862$	3.01639	0.102739
$863$	48.4744i	1.65009i	0.565069	+	0.825044i	$0.308850\pi$
$863$	48.4744i	1.65009i	−0.565069	+	0.825044i	$0.691150\pi$
$864$	0	0
$865$	−1.02869	−0.0349765
$866$	37.7167	1.28166
$867$	0	0
$868$	0	0
$869$	− 56.2187i	− 1.90709i
$870$	0	0
$871$	5.10159i	0.172861i
$872$	9.01414i	0.305257i
$873$	0	0
$874$	− 1.34675i	− 0.0455545i
$875$	0	0
$876$	0	0
$877$	−4.53874	−0.153262	−0.0766312	−	0.997060i	$-0.524416\pi$
$877$	−4.53874	−0.153262	−0.0766312	+	0.997060i	$0.524416\pi$
$878$	−21.3683	−0.721146
$879$	0	0
$880$	− 5.37849i	− 0.181309i
$881$	13.4827	0.454242	0.227121	−	0.973867i	$-0.427069\pi$
$881$	13.4827	0.454242	0.227121	+	0.973867i	$0.427069\pi$
$882$	0	0
$883$	13.6089	0.457977	0.228989	−	0.973429i	$-0.426458\pi$
$883$	13.6089	0.457977	0.228989	+	0.973429i	$0.426458\pi$
$884$	− 2.61313i	− 0.0878889i
$885$	0	0
$886$	−1.59629	−0.0536284
$887$	17.7194	0.594959	0.297479	−	0.954728i	$-0.403854\pi$
$887$	17.7194	0.594959	0.297479	+	0.954728i	$0.403854\pi$
$888$	0	0
$889$	0	0
$890$	− 15.8899i	− 0.532630i
$891$	0	0
$892$	2.25715i	0.0755750i
$893$	− 20.1407i	− 0.673982i
$894$	0	0
$895$	9.99480i	0.334089i
$896$	0	0
$897$	0	0
$898$	14.9895	0.500206
$899$	3.91896	0.130705
$900$	0	0
$901$	11.1967i	0.373015i
$902$	−5.61369	−0.186915
$903$	0	0
$904$	−18.9082	−0.628877
$905$	3.08014i	0.102387i
$906$	0	0
$907$	−39.2895	−1.30459	−0.652293	−	0.757967i	$-0.726193\pi$
$907$	−39.2895	−1.30459	−0.652293	+	0.757967i	$0.726193\pi$
$908$	17.5011	0.580796
$909$	0	0
$910$	0	0
$911$	− 41.2365i	− 1.36623i	−0.730313	−	0.683113i	$-0.760626\pi$
$911$	− 41.2365i	− 1.36623i	0.730313	−	0.683113i	$-0.239374\pi$
$912$	0	0
$913$	− 78.2031i	− 2.58814i
$914$	− 22.5650i	− 0.746385i
$915$	0	0
$916$	24.4840i	0.808974i
$917$	0	0
$918$	0	0
$919$	1.16123	0.0383053	0.0191526	−	0.999817i	$-0.493903\pi$
$919$	1.16123	0.0383053	0.0191526	+	0.999817i	$0.493903\pi$
$920$	−0.496606	−0.0163726
$921$	0	0
$922$	− 7.03506i	− 0.231687i
$923$	−12.9545	−0.426404
$924$	0	0
$925$	−4.10814	−0.135075
$926$	28.8845i	0.949203i
$927$	0	0
$928$	−7.12453	−0.233874
$929$	−52.9648	−1.73772	−0.868860	−	0.495058i	$-0.835147\pi$
$929$	−52.9648	−1.73772	−0.868860	+	0.495058i	$0.835147\pi$
$930$	0	0
$931$	0	0
$932$	17.7366i	0.580982i
$933$	0	0
$934$	26.5709i	0.869428i
$935$	− 11.7229i	− 0.383378i
$936$	0	0
$937$	− 26.9970i	− 0.881955i	−0.897518	−	0.440977i	$-0.854632\pi$
$937$	− 26.9970i	− 0.881955i	0.897518	−	0.440977i	$-0.145368\pi$
$938$	0	0
$939$	0	0
$940$	−7.42676	−0.242234
$941$	−47.6026	−1.55180	−0.775900	−	0.630856i	$-0.782704\pi$
$941$	−47.6026	−1.55180	−0.775900	+	0.630856i	$0.782704\pi$
$942$	0	0
$943$	0.518321i	0.0168789i
$944$	−11.1097	−0.361591
$945$	0	0
$946$	32.4855	1.05619
$947$	41.7275i	1.35596i	0.735080	+	0.677980i	$0.237145\pi$
$947$	41.7275i	1.35596i	−0.735080	+	0.677980i	$0.762855\pi$
$948$	0	0
$949$	18.6250	0.604592
$950$	2.71191	0.0879859
$951$	0	0
$952$	0	0
$953$	52.5279i	1.70155i	0.525534	+	0.850773i	$0.323866\pi$
$953$	52.5279i	1.70155i	−0.525534	+	0.850773i	$0.676134\pi$
$954$	0	0
$955$	− 5.82805i	− 0.188591i
$956$	− 14.8111i	− 0.479026i
$957$	0	0
$958$	− 28.6410i	− 0.925349i
$959$	0	0
$960$	0	0
$961$	30.6974	0.990240
$962$	−4.92530	−0.158798
$963$	0	0
$964$	− 9.93938i	− 0.320126i
$965$	5.75789	0.185353
$966$	0	0
$967$	−17.6987	−0.569152	−0.284576	−	0.958653i	$-0.591853\pi$
$967$	−17.6987	−0.569152	−0.284576	+	0.958653i	$0.591853\pi$
$968$	− 17.9282i	− 0.576234i
$969$	0	0
$970$	7.65685	0.245847
$971$	44.1003	1.41525	0.707623	−	0.706590i	$-0.249768\pi$
$971$	44.1003	1.41525	0.707623	+	0.706590i	$0.249768\pi$
$972$	0	0
$973$	0	0
$974$	− 18.7266i	− 0.600040i
$975$	0	0
$976$	− 1.88348i	− 0.0602887i
$977$	31.1015i	0.995026i	0.867457	+	0.497513i	$0.165753\pi$
$977$	31.1015i	0.995026i	−0.867457	+	0.497513i	$0.834247\pi$
$978$	0	0
$979$	− 85.4637i	− 2.73143i
$980$	0	0
$981$	0	0
$982$	−24.0927	−0.768827
$983$	30.4697	0.971834	0.485917	−	0.874005i	$-0.338486\pi$
$983$	30.4697	0.971834	0.485917	+	0.874005i	$0.338486\pi$
$984$	0	0
$985$	− 14.9259i	− 0.475580i
$986$	−15.5285	−0.494528
$987$	0	0
$988$	3.25134	0.103439
$989$	− 2.99944i	− 0.0953767i
$990$	0	0
$991$	19.6020	0.622677	0.311338	−	0.950299i	$-0.399223\pi$
$991$	19.6020	0.622677	0.311338	+	0.950299i	$0.399223\pi$
$992$	0.550066	0.0174646
$993$	0	0
$994$	0	0
$995$	− 26.6163i	− 0.843792i
$996$	0	0
$997$	− 4.69192i	− 0.148594i	−0.997236	−	0.0742972i	$-0.976329\pi$
$997$	− 4.69192i	− 0.148594i	0.997236	−	0.0742972i	$-0.0236714\pi$
$998$	14.7333i	0.466374i
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	4410.2.b.c.881.4	✓	8
3.2	odd	2		4410.2.b.f.881.5	yes	8
7.6	odd	2		4410.2.b.f.881.4	yes	8
21.20	even	2	inner	4410.2.b.c.881.5	yes	8

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
4410.2.b.c.881.4	✓	8	1.1	even	1	trivial
4410.2.b.c.881.5	yes	8	21.20	even	2	inner
4410.2.b.f.881.4	yes	8	7.6	odd	2
4410.2.b.f.881.5	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)

\(f(q)\)	\(=\)	\(q-1.00000i q^{2} -1.00000 q^{4} -1.00000 q^{5} +1.00000i q^{8} +O(q^{10})\)	\(q-1.00000i q^{2} -1.00000 q^{4} -1.00000 q^{5} +1.00000i q^{8} +1.00000i q^{10} +5.37849i q^{11} +1.19891i q^{13} +1.00000 q^{16} +2.17958 q^{17} +2.71191i q^{19} +1.00000 q^{20} +5.37849 q^{22} -0.496606i q^{23} +1.00000 q^{25} +1.19891 q^{26} -7.12453i q^{29} +0.550066i q^{31} -1.00000i q^{32} -2.17958i q^{34} -4.10814 q^{37} +2.71191 q^{38} -1.00000i q^{40} -1.04373 q^{41} +6.03988 q^{43} -5.37849i q^{44} -0.496606 q^{46} -7.42676 q^{47} -1.00000i q^{50} -1.19891i q^{52} +5.13707i q^{53} -5.37849i q^{55} -7.12453 q^{58} -11.1097 q^{59} -1.88348i q^{61} +0.550066 q^{62} -1.00000 q^{64} -1.19891i q^{65} +4.25518 q^{67} -2.17958 q^{68} +10.8052i q^{71} -15.5349i q^{73} +4.10814i q^{74} -2.71191i q^{76} -10.4525 q^{79} -1.00000 q^{80} +1.04373i q^{82} -14.5400 q^{83} -2.17958 q^{85} -6.03988i q^{86} -5.37849 q^{88} -15.8899 q^{89} +0.496606i q^{92} +7.42676i q^{94} -2.71191i q^{95} -7.65685i q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 8 q - 8 q^{4} - 8 q^{5}+O(q^{10}) \) 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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