LMFDB - Embedded newform 4400.2.b.bb.4049.2

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [4400,2,Mod(4049,4400)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(4400, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([0, 0, 1, 0]))

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

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

chi := DirichletCharacter("4400.4049");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 4400 = 2^{4} \cdot 5^{2} \cdot 11 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	4400.b (of order $2$, degree $1$, not 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:	$35.1341768894$
Analytic rank:	$0$
Dimension:	$6$
Coefficient field:	6.0.96668224.1
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{6} + 15x^{4} + 61x^{2} + 36 $ x^6 + 15x^4 + 61x^2 + 36
Coefficient ring:	$\Z[a_1, \ldots, a_{13}]$
Coefficient ring index:	$ 1 $
Twist minimal:	no (minimal twist has level 2200)
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label			4049.2
Root			$-2.59261i$ of defining polynomial
Character	$\chi$	$=$	4400.4049
Dual form			4400.2.b.bb.4049.5

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-2.59261i q^{3} -2.72165i q^{7} -3.72165 q^{9} +O(q^{10})$	$q-2.59261i q^{3} -2.72165i q^{7} -3.72165 q^{9} -1.00000 q^{11} -1.00000i q^{13} -4.59261i q^{17} +8.18523 q^{19} -7.05619 q^{21} +0.407385i q^{23} +1.87096i q^{27} -7.46358 q^{29} +4.44330 q^{31} +2.59261i q^{33} -7.31427i q^{37} -2.59261 q^{39} -3.31427 q^{41} -7.49950i q^{43} -7.05619i q^{47} -0.407385 q^{49} -11.9069 q^{51} -0.979724i q^{53} -21.2211i q^{57} +7.05619 q^{59} -4.46358 q^{61} +10.1290i q^{63} +2.25807i q^{67} +1.05619 q^{69} -10.3143 q^{71} +12.1650i q^{73} +2.72165i q^{77} +3.14931 q^{79} -6.31427 q^{81} +16.6488i q^{83} +19.3502i q^{87} -8.27835 q^{89} -2.72165 q^{91} -11.5198i q^{93} +3.03592i q^{97} +3.72165 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 6 q - 12 q^{9}+O(q^{10}) $ 6 * q - 12 * q^9	$ 6 q - 12 q^{9} - 6 q^{11} + 14 q^{19} - 20 q^{29} + 6 q^{31} + 2 q^{39} + 8 q^{41} - 20 q^{49} - 26 q^{51} - 2 q^{61} - 36 q^{69} - 34 q^{71} + 22 q^{79} - 10 q^{81} - 60 q^{89} - 6 q^{91} + 12 q^{99}+O(q^{100}) $ 6 * q - 12 * q^9 - 6 * q^11 + 14 * q^19 - 20 * q^29 + 6 * q^31 + 2 * q^39 + 8 * q^41 - 20 * q^49 - 26 * q^51 - 2 * q^61 - 36 * q^69 - 34 * q^71 + 22 * q^79 - 10 * q^81 - 60 * q^89 - 6 * q^91 + 12 * q^99

Display 100 coefficients 10 coefficients

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/4400\mathbb{Z}\right)^\times$.

$n$	$177$	$1201$	$2751$	$3301$
$\chi(n)$	$-1$	$1$	$1$	$1$

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 $

$2$	0	0
$2$	0	0
$3$	− 2.59261i	− 1.49685i	−0.663221	−	0.748423i	$-0.730811\pi$
$3$	− 2.59261i	− 1.49685i	0.663221	−	0.748423i	$-0.269189\pi$
$4$	0	0
$5$	0	0
$5$	0	0
$6$	0	0
$7$	− 2.72165i	− 1.02869i	−0.857584	−	0.514344i	$-0.828036\pi$
$7$	− 2.72165i	− 1.02869i	0.857584	−	0.514344i	$-0.171964\pi$
$8$	0	0
$9$	−3.72165	−1.24055
$10$	0	0
$11$	−1.00000	−0.301511
$11$	−1.00000	−0.301511
$12$	0	0
$13$	− 1.00000i	− 0.277350i	−0.990338	−	0.138675i	$-0.955716\pi$
$13$	− 1.00000i	− 0.277350i	0.990338	−	0.138675i	$-0.0442844\pi$
$14$	0	0
$15$	0	0
$16$	0	0
$17$	− 4.59261i	− 1.11387i	−0.830555	−	0.556936i	$-0.811977\pi$
$17$	− 4.59261i	− 1.11387i	0.830555	−	0.556936i	$-0.188023\pi$
$18$	0	0
$19$	8.18523	1.87782	0.938910	−	0.344162i	$-0.111837\pi$
$19$	8.18523	1.87782	0.938910	+	0.344162i	$0.111837\pi$
$20$	0	0
$21$	−7.05619	−1.53979
$22$	0	0
$23$	0.407385i	0.0849457i	0.999098	+	0.0424728i	$0.0135236\pi$
$23$	0.407385i	0.0849457i	−0.999098	+	0.0424728i	$0.986476\pi$
$24$	0	0
$25$	0	0
$26$	0	0
$27$	1.87096i	0.360067i
$28$	0	0
$29$	−7.46358	−1.38595	−0.692976	−	0.720961i	$-0.743701\pi$
$29$	−7.46358	−1.38595	−0.692976	+	0.720961i	$0.743701\pi$
$30$	0	0
$31$	4.44330	0.798041	0.399020	−	0.916942i	$-0.369350\pi$
$31$	4.44330	0.798041	0.399020	+	0.916942i	$0.369350\pi$
$32$	0	0
$33$	2.59261i	0.451316i
$34$	0	0
$35$	0	0
$36$	0	0
$37$	− 7.31427i	− 1.20246i	−0.799077	−	0.601229i	$-0.794678\pi$
$37$	− 7.31427i	− 1.20246i	0.799077	−	0.601229i	$-0.205322\pi$
$38$	0	0
$39$	−2.59261	−0.415151
$40$	0	0
$41$	−3.31427	−0.517601	−0.258801	−	0.965931i	$-0.583327\pi$
$41$	−3.31427	−0.517601	−0.258801	+	0.965931i	$0.583327\pi$
$42$	0	0
$43$	− 7.49950i	− 1.14366i	−0.820371	−	0.571831i	$-0.806233\pi$
$43$	− 7.49950i	− 1.14366i	0.820371	−	0.571831i	$-0.193767\pi$
$44$	0	0
$45$	0	0
$46$	0	0
$47$	− 7.05619i	− 1.02925i	−0.857415	−	0.514626i	$-0.827931\pi$
$47$	− 7.05619i	− 1.02925i	0.857415	−	0.514626i	$-0.172069\pi$
$48$	0	0
$49$	−0.407385	−0.0581979
$50$	0	0
$51$	−11.9069	−1.66730
$52$	0	0
$53$	− 0.979724i	− 0.134575i	−0.997734	−	0.0672877i	$-0.978565\pi$
$53$	− 0.979724i	− 0.134575i	0.997734	−	0.0672877i	$-0.0214345\pi$
$54$	0	0
$55$	0	0
$56$	0	0
$57$	− 21.2211i	− 2.81081i
$58$	0	0
$59$	7.05619	0.918638	0.459319	−	0.888271i	$-0.348093\pi$
$59$	7.05619	0.918638	0.459319	+	0.888271i	$0.348093\pi$
$60$	0	0
$61$	−4.46358	−0.571503	−0.285751	−	0.958304i	$-0.592243\pi$
$61$	−4.46358	−0.571503	−0.285751	+	0.958304i	$0.592243\pi$
$62$	0	0
$63$	10.1290i	1.27614i
$64$	0	0
$65$	0	0
$66$	0	0
$67$	2.25807i	0.275868i	0.990441	+	0.137934i	$0.0440461\pi$
$67$	2.25807i	0.275868i	−0.990441	+	0.137934i	$0.955954\pi$
$68$	0	0
$69$	1.05619	0.127151
$70$	0	0
$71$	−10.3143	−1.22408	−0.612039	−	0.790828i	$-0.709650\pi$
$71$	−10.3143	−1.22408	−0.612039	+	0.790828i	$0.709650\pi$
$72$	0	0
$73$	12.1650i	1.42380i	0.702281	+	0.711900i	$0.252165\pi$
$73$	12.1650i	1.42380i	−0.702281	+	0.711900i	$0.747835\pi$
$74$	0	0
$75$	0	0
$76$	0	0
$77$	2.72165i	0.310161i
$78$	0	0
$79$	3.14931	0.354325	0.177163	−	0.984182i	$-0.443308\pi$
$79$	3.14931	0.354325	0.177163	+	0.984182i	$0.443308\pi$
$80$	0	0
$81$	−6.31427	−0.701585
$82$	0	0
$83$	16.6488i	1.82744i	0.406340	+	0.913722i	$0.366805\pi$
$83$	16.6488i	1.82744i	−0.406340	+	0.913722i	$0.633195\pi$
$84$	0	0
$85$	0	0
$86$	0	0
$87$	19.3502i	2.07456i
$88$	0	0
$89$	−8.27835	−0.877503	−0.438752	−	0.898608i	$-0.644579\pi$
$89$	−8.27835	−0.877503	−0.438752	+	0.898608i	$0.644579\pi$
$90$	0	0
$91$	−2.72165	−0.285307
$92$	0	0
$93$	− 11.5198i	− 1.19454i
$94$	0	0
$95$	0	0
$96$	0	0
$97$	3.03592i	0.308251i	0.988051	+	0.154125i	$0.0492560\pi$
$97$	3.03592i	0.308251i	−0.988051	+	0.154125i	$0.950744\pi$
$98$	0	0
$99$	3.72165	0.374040
$100$	0	0
$101$	5.59261	0.556486	0.278243	−	0.960511i	$-0.410248\pi$
$101$	5.59261	0.556486	0.278243	+	0.960511i	$0.410248\pi$
$102$	0	0
$103$	− 4.64881i	− 0.458061i	−0.973419	−	0.229030i	$-0.926445\pi$
$103$	− 4.64881i	− 0.458061i	0.973419	−	0.229030i	$-0.0735555\pi$
$104$	0	0
$105$	0	0
$106$	0	0
$107$	− 0.701375i	− 0.0678045i	−0.999425	−	0.0339023i	$-0.989207\pi$
$107$	− 0.701375i	− 0.0678045i	0.999425	−	0.0339023i	$-0.0107935\pi$
$108$	0	0
$109$	−4.27835	−0.409791	−0.204896	−	0.978784i	$-0.565686\pi$
$109$	−4.27835	−0.409791	−0.204896	+	0.978784i	$0.565686\pi$
$110$	0	0
$111$	−18.9631	−1.79990
$112$	0	0
$113$	− 11.6847i	− 1.09921i	−0.835426	−	0.549603i	$-0.814779\pi$
$113$	− 11.6847i	− 1.09921i	0.835426	−	0.549603i	$-0.185221\pi$
$114$	0	0
$115$	0	0
$116$	0	0
$117$	3.72165i	0.344067i
$118$	0	0
$119$	−12.4995	−1.14583
$120$	0	0
$121$	1.00000	0.0909091
$122$	0	0
$123$	8.59261i	0.774770i
$124$	0	0
$125$	0	0
$126$	0	0
$127$	− 9.51977i	− 0.844743i	−0.906423	−	0.422372i	$-0.861198\pi$
$127$	− 9.51977i	− 0.844743i	0.906423	−	0.422372i	$-0.138802\pi$
$128$	0	0
$129$	−19.4433	−1.71189
$130$	0	0
$131$	−22.4792	−1.96402	−0.982009	−	0.188833i	$-0.939530\pi$
$131$	−22.4792	−1.96402	−0.982009	+	0.188833i	$0.939530\pi$
$132$	0	0
$133$	− 22.2773i	− 1.93169i
$134$	0	0
$135$	0	0
$136$	0	0
$137$	15.5354i	1.32728i	0.748052	+	0.663640i	$0.230989\pi$
$137$	15.5354i	1.32728i	−0.748052	+	0.663640i	$0.769011\pi$
$138$	0	0
$139$	13.3299	1.13063	0.565314	−	0.824876i	$-0.308755\pi$
$139$	13.3299	1.13063	0.565314	+	0.824876i	$0.308755\pi$
$140$	0	0
$141$	−18.2940	−1.54063
$142$	0	0
$143$	1.00000i	0.0836242i
$144$	0	0
$145$	0	0
$146$	0	0
$147$	1.05619i	0.0871133i
$148$	0	0
$149$	0.628532	0.0514913	0.0257457	−	0.999669i	$-0.491804\pi$
$149$	0.628532	0.0514913	0.0257457	+	0.999669i	$0.491804\pi$
$150$	0	0
$151$	19.1280	1.55662	0.778308	−	0.627882i	$-0.216078\pi$
$151$	19.1280	1.55662	0.778308	+	0.627882i	$0.216078\pi$
$152$	0	0
$153$	17.0921i	1.38182i
$154$	0	0
$155$	0	0
$156$	0	0
$157$	8.88660i	0.709228i	0.935013	+	0.354614i	$0.115388\pi$
$157$	8.88660i	0.709228i	−0.935013	+	0.354614i	$0.884612\pi$
$158$	0	0
$159$	−2.54005	−0.201439
$160$	0	0
$161$	1.10876	0.0873826
$162$	0	0
$163$	20.8340i	1.63185i	0.578159	+	0.815924i	$0.303771\pi$
$163$	20.8340i	1.63185i	−0.578159	+	0.815924i	$0.696229\pi$
$164$	0	0
$165$	0	0
$166$	0	0
$167$	− 17.4267i	− 1.34851i	−0.738496	−	0.674257i	$-0.764464\pi$
$167$	− 17.4267i	− 1.34851i	0.738496	−	0.674257i	$-0.235536\pi$
$168$	0	0
$169$	12.0000	0.923077
$170$	0	0
$171$	−30.4626	−2.32953
$172$	0	0
$173$	8.18523i	0.622311i	0.950359	+	0.311156i	$0.100716\pi$
$173$	8.18523i	0.622311i	−0.950359	+	0.311156i	$0.899284\pi$
$174$	0	0
$175$	0	0
$176$	0	0
$177$	− 18.2940i	− 1.37506i
$178$	0	0
$179$	−10.7217	−0.801374	−0.400687	−	0.916215i	$-0.631228\pi$
$179$	−10.7217	−0.801374	−0.400687	+	0.916215i	$0.631228\pi$
$180$	0	0
$181$	23.4626	1.74396	0.871980	−	0.489542i	$-0.162836\pi$
$181$	23.4626	1.74396	0.871980	+	0.489542i	$0.162836\pi$
$182$	0	0
$183$	11.5723i	0.855452i
$184$	0	0
$185$	0	0
$186$	0	0
$187$	4.59261i	0.335845i
$188$	0	0
$189$	5.09211	0.370397
$190$	0	0
$191$	15.5354	1.12410	0.562052	−	0.827102i	$-0.310012\pi$
$191$	15.5354	1.12410	0.562052	+	0.827102i	$0.310012\pi$
$192$	0	0
$193$	1.62954i	0.117297i	0.998279	+	0.0586485i	$0.0186791\pi$
$193$	1.62954i	0.117297i	−0.998279	+	0.0586485i	$0.981321\pi$
$194$	0	0
$195$	0	0
$196$	0	0
$197$	− 2.77784i	− 0.197913i	−0.995092	−	0.0989566i	$-0.968449\pi$
$197$	− 2.77784i	− 0.197913i	0.995092	−	0.0989566i	$-0.0315505\pi$
$198$	0	0
$199$	−26.1639	−1.85471	−0.927356	−	0.374179i	$-0.877924\pi$
$199$	−26.1639	−1.85471	−0.927356	+	0.374179i	$0.877924\pi$
$200$	0	0
$201$	5.85431	0.412931
$202$	0	0
$203$	20.3133i	1.42571i
$204$	0	0
$205$	0	0
$206$	0	0
$207$	− 1.51615i	− 0.105379i
$208$	0	0
$209$	−8.18523	−0.566184
$210$	0	0
$211$	2.38711	0.164335	0.0821677	−	0.996619i	$-0.473816\pi$
$211$	2.38711	0.164335	0.0821677	+	0.996619i	$0.473816\pi$
$212$	0	0
$213$	26.7409i	1.83226i
$214$	0	0
$215$	0	0
$216$	0	0
$217$	− 12.0931i	− 0.820934i
$218$	0	0
$219$	31.5390	2.13121
$220$	0	0
$221$	−4.59261	−0.308933
$222$	0	0
$223$	25.4267i	1.70269i	0.524603	+	0.851347i	$0.324214\pi$
$223$	25.4267i	1.70269i	−0.524603	+	0.851347i	$0.675786\pi$
$224$	0	0
$225$	0	0
$226$	0	0
$227$	− 10.2783i	− 0.682198i	−0.940027	−	0.341099i	$-0.889201\pi$
$227$	− 10.2783i	− 0.682198i	0.940027	−	0.341099i	$-0.110799\pi$
$228$	0	0
$229$	−29.0349	−1.91868	−0.959340	−	0.282252i	$-0.908919\pi$
$229$	−29.0349	−1.91868	−0.959340	+	0.282252i	$0.908919\pi$
$230$	0	0
$231$	7.05619	0.464263
$232$	0	0
$233$	− 5.90688i	− 0.386973i	−0.981103	−	0.193486i	$-0.938020\pi$
$233$	− 5.90688i	− 0.386973i	0.981103	−	0.193486i	$-0.0619795\pi$
$234$	0	0
$235$	0	0
$236$	0	0
$237$	− 8.16495i	− 0.530371i
$238$	0	0
$239$	3.79449	0.245445	0.122723	−	0.992441i	$-0.460837\pi$
$239$	3.79449	0.245445	0.122723	+	0.992441i	$0.460837\pi$
$240$	0	0
$241$	−23.5916	−1.51967	−0.759834	−	0.650117i	$-0.774720\pi$
$241$	−23.5916	−1.51967	−0.759834	+	0.650117i	$0.774720\pi$
$242$	0	0
$243$	21.9833i	1.41023i
$244$	0	0
$245$	0	0
$246$	0	0
$247$	− 8.18523i	− 0.520814i
$248$	0	0
$249$	43.1639	2.73540
$250$	0	0
$251$	21.5916	1.36285	0.681425	−	0.731888i	$-0.261361\pi$
$251$	21.5916	1.36285	0.681425	+	0.731888i	$0.261361\pi$
$252$	0	0
$253$	− 0.407385i	− 0.0256121i
$254$	0	0
$255$	0	0
$256$	0	0
$257$	− 4.01564i	− 0.250489i	−0.992126	−	0.125244i	$-0.960028\pi$
$257$	− 4.01564i	− 0.250489i	0.992126	−	0.125244i	$-0.0399715\pi$
$258$	0	0
$259$	−19.9069	−1.23695
$260$	0	0
$261$	27.7768	1.71934
$262$	0	0
$263$	− 29.2404i	− 1.80304i	−0.432736	−	0.901521i	$-0.642452\pi$
$263$	− 29.2404i	− 1.80304i	0.432736	−	0.901521i	$-0.357548\pi$
$264$	0	0
$265$	0	0
$266$	0	0
$267$	21.4626i	1.31349i
$268$	0	0
$269$	8.03592	0.489959	0.244979	−	0.969528i	$-0.421219\pi$
$269$	8.03592	0.489959	0.244979	+	0.969528i	$0.421219\pi$
$270$	0	0
$271$	19.4267	1.18009	0.590043	−	0.807372i	$-0.299111\pi$
$271$	19.4267	1.18009	0.590043	+	0.807372i	$0.299111\pi$
$272$	0	0
$273$	7.05619i	0.427060i
$274$	0	0
$275$	0	0
$276$	0	0
$277$	− 1.83041i	− 0.109979i	−0.998487	−	0.0549894i	$-0.982487\pi$
$277$	− 1.83041i	− 0.109979i	0.998487	−	0.0549894i	$-0.0175125\pi$
$278$	0	0
$279$	−16.5364	−0.990010
$280$	0	0
$281$	17.6847	1.05498	0.527491	−	0.849561i	$-0.323133\pi$
$281$	17.6847	1.05498	0.527491	+	0.849561i	$0.323133\pi$
$282$	0	0
$283$	16.8507i	1.00167i	0.865543	+	0.500835i	$0.166974\pi$
$283$	16.8507i	1.00167i	−0.865543	+	0.500835i	$0.833026\pi$
$284$	0	0
$285$	0	0
$286$	0	0
$287$	9.02028i	0.532450i
$288$	0	0
$289$	−4.09211	−0.240712
$290$	0	0
$291$	7.87096	0.461404
$292$	0	0
$293$	28.3705i	1.65742i	0.559678	+	0.828710i	$0.310925\pi$
$293$	28.3705i	1.65742i	−0.559678	+	0.828710i	$0.689075\pi$
$294$	0	0
$295$	0	0
$296$	0	0
$297$	− 1.87096i	− 0.108564i
$298$	0	0
$299$	0.407385	0.0235597
$300$	0	0
$301$	−20.4110	−1.17647
$302$	0	0
$303$	− 14.4995i	− 0.832974i
$304$	0	0
$305$	0	0
$306$	0	0
$307$	2.29399i	0.130925i	0.997855	+	0.0654625i	$0.0208523\pi$
$307$	2.29399i	0.130925i	−0.997855	+	0.0654625i	$0.979148\pi$
$308$	0	0
$309$	−12.0526	−0.685647
$310$	0	0
$311$	−11.2581	−0.638387	−0.319193	−	0.947690i	$-0.603412\pi$
$311$	−11.2581	−0.638387	−0.319193	+	0.947690i	$0.603412\pi$
$312$	0	0
$313$	− 7.18523i	− 0.406133i	−0.979165	−	0.203067i	$-0.934909\pi$
$313$	− 7.18523i	− 0.406133i	0.979165	−	0.203067i	$-0.0650908\pi$
$314$	0	0
$315$	0	0
$316$	0	0
$317$	29.5198i	1.65800i	0.559252	+	0.828998i	$0.311088\pi$
$317$	29.5198i	1.65800i	−0.559252	+	0.828998i	$0.688912\pi$
$318$	0	0
$319$	7.46358	0.417880
$320$	0	0
$321$	−1.81840	−0.101493
$322$	0	0
$323$	− 37.5916i	− 2.09165i
$324$	0	0
$325$	0	0
$326$	0	0
$327$	11.0921i	0.613395i
$328$	0	0
$329$	−19.2045	−1.05878
$330$	0	0
$331$	6.12904	0.336882	0.168441	−	0.985712i	$-0.446127\pi$
$331$	6.12904	0.336882	0.168441	+	0.985712i	$0.446127\pi$
$332$	0	0
$333$	27.2211i	1.49171i
$334$	0	0
$335$	0	0
$336$	0	0
$337$	− 10.0000i	− 0.544735i	−0.962193	−	0.272367i	$-0.912193\pi$
$337$	− 10.0000i	− 0.544735i	0.962193	−	0.272367i	$-0.0878066\pi$
$338$	0	0
$339$	−30.2940	−1.64534
$340$	0	0
$341$	−4.44330	−0.240618
$342$	0	0
$343$	− 17.9428i	− 0.968820i
$344$	0	0
$345$	0	0
$346$	0	0
$347$	− 14.7217i	− 0.790300i	−0.918617	−	0.395150i	$-0.870693\pi$
$347$	− 14.7217i	− 0.790300i	0.918617	−	0.395150i	$-0.129307\pi$
$348$	0	0
$349$	−32.2976	−1.72885	−0.864426	−	0.502760i	$-0.832318\pi$
$349$	−32.2976	−1.72885	−0.864426	+	0.502760i	$0.832318\pi$
$350$	0	0
$351$	1.87096	0.0998646
$352$	0	0
$353$	− 3.37046i	− 0.179391i	−0.995969	−	0.0896957i	$-0.971411\pi$
$353$	− 3.37046i	− 0.179391i	0.995969	−	0.0896957i	$-0.0285895\pi$
$354$	0	0
$355$	0	0
$356$	0	0
$357$	32.4064i	1.71513i
$358$	0	0
$359$	−19.7419	−1.04194	−0.520970	−	0.853575i	$-0.674429\pi$
$359$	−19.7419	−1.04194	−0.520970	+	0.853575i	$0.674429\pi$
$360$	0	0
$361$	47.9980	2.52621
$362$	0	0
$363$	− 2.59261i	− 0.136077i
$364$	0	0
$365$	0	0
$366$	0	0
$367$	− 30.4792i	− 1.59100i	−0.605952	−	0.795501i	$-0.707208\pi$
$367$	− 30.4792i	− 1.59100i	0.605952	−	0.795501i	$-0.292792\pi$
$368$	0	0
$369$	12.3345	0.642111
$370$	0	0
$371$	−2.66647	−0.138436
$372$	0	0
$373$	27.4985i	1.42382i	0.702272	+	0.711909i	$0.252169\pi$
$373$	27.4985i	1.42382i	−0.702272	+	0.711909i	$0.747831\pi$
$374$	0	0
$375$	0	0
$376$	0	0
$377$	7.46358i	0.384394i
$378$	0	0
$379$	3.87096	0.198838	0.0994190	−	0.995046i	$-0.468302\pi$
$379$	3.87096	0.198838	0.0994190	+	0.995046i	$0.468302\pi$
$380$	0	0
$381$	−24.6811	−1.26445
$382$	0	0
$383$	− 33.2414i	− 1.69856i	−0.527945	−	0.849279i	$-0.677037\pi$
$383$	− 33.2414i	− 1.69856i	0.527945	−	0.849279i	$-0.322963\pi$
$384$	0	0
$385$	0	0
$386$	0	0
$387$	27.9105i	1.41877i
$388$	0	0
$389$	−33.9980	−1.72377	−0.861883	−	0.507107i	$-0.830715\pi$
$389$	−33.9980	−1.72377	−0.861883	+	0.507107i	$0.830715\pi$
$390$	0	0
$391$	1.87096	0.0946187
$392$	0	0
$393$	58.2800i	2.93983i
$394$	0	0
$395$	0	0
$396$	0	0
$397$	2.85069i	0.143072i	0.997438	+	0.0715360i	$0.0227901\pi$
$397$	2.85069i	0.143072i	−0.997438	+	0.0715360i	$0.977210\pi$
$398$	0	0
$399$	−57.7566	−2.89144
$400$	0	0
$401$	16.0967	0.803833	0.401917	−	0.915676i	$-0.368344\pi$
$401$	16.0967	0.803833	0.401917	+	0.915676i	$0.368344\pi$
$402$	0	0
$403$	− 4.44330i	− 0.221337i
$404$	0	0
$405$	0	0
$406$	0	0
$407$	7.31427i	0.362555i
$408$	0	0
$409$	−0.757568	−0.0374593	−0.0187297	−	0.999825i	$-0.505962\pi$
$409$	−0.757568	−0.0374593	−0.0187297	+	0.999825i	$0.505962\pi$
$410$	0	0
$411$	40.2773	1.98673
$412$	0	0
$413$	− 19.2045i	− 0.944991i
$414$	0	0
$415$	0	0
$416$	0	0
$417$	− 34.5593i	− 1.69238i
$418$	0	0
$419$	36.8700	1.80122	0.900608	−	0.434633i	$-0.143122\pi$
$419$	36.8700	1.80122	0.900608	+	0.434633i	$0.143122\pi$
$420$	0	0
$421$	−32.2055	−1.56960	−0.784800	−	0.619749i	$-0.787234\pi$
$421$	−32.2055	−1.56960	−0.784800	+	0.619749i	$0.787234\pi$
$422$	0	0
$423$	26.2607i	1.27684i
$424$	0	0
$425$	0	0
$426$	0	0
$427$	12.1483i	0.587898i
$428$	0	0
$429$	2.59261	0.125173
$430$	0	0
$431$	−17.9428	−0.864274	−0.432137	−	0.901808i	$-0.642240\pi$
$431$	−17.9428	−0.864274	−0.432137	+	0.901808i	$0.642240\pi$
$432$	0	0
$433$	14.2258i	0.683647i	0.939764	+	0.341824i	$0.111045\pi$
$433$	14.2258i	0.683647i	−0.939764	+	0.341824i	$0.888955\pi$
$434$	0	0
$435$	0	0
$436$	0	0
$437$	3.33454i	0.159513i
$438$	0	0
$439$	−9.70601	−0.463243	−0.231621	−	0.972806i	$-0.574403\pi$
$439$	−9.70601	−0.463243	−0.231621	+	0.972806i	$0.574403\pi$
$440$	0	0
$441$	1.51615	0.0721974
$442$	0	0
$443$	− 4.44431i	− 0.211156i	−0.994411	−	0.105578i	$-0.966331\pi$
$443$	− 4.44431i	− 0.211156i	0.994411	−	0.105578i	$-0.0336692\pi$
$444$	0	0
$445$	0	0
$446$	0	0
$447$	− 1.62954i	− 0.0770746i
$448$	0	0
$449$	12.4479	0.587454	0.293727	−	0.955889i	$-0.405104\pi$
$449$	12.4479	0.587454	0.293727	+	0.955889i	$0.405104\pi$
$450$	0	0
$451$	3.31427	0.156063
$452$	0	0
$453$	− 49.5916i	− 2.33002i
$454$	0	0
$455$	0	0
$456$	0	0
$457$	16.2368i	0.759525i	0.925084	+	0.379762i	$0.123994\pi$
$457$	16.2368i	0.759525i	−0.925084	+	0.379762i	$0.876006\pi$
$458$	0	0
$459$	8.59261	0.401069
$460$	0	0
$461$	−41.8689	−1.95003	−0.975016	−	0.222136i	$-0.928697\pi$
$461$	−41.8689	−1.95003	−0.975016	+	0.222136i	$0.928697\pi$
$462$	0	0
$463$	18.7014i	0.869127i	0.900641	+	0.434563i	$0.143097\pi$
$463$	18.7014i	0.869127i	−0.900641	+	0.434563i	$0.856903\pi$
$464$	0	0
$465$	0	0
$466$	0	0
$467$	− 12.2820i	− 0.568342i	−0.958774	−	0.284171i	$-0.908282\pi$
$467$	− 12.2820i	− 0.568342i	0.958774	−	0.284171i	$-0.0917183\pi$
$468$	0	0
$469$	6.14569	0.283781
$470$	0	0
$471$	23.0395	1.06161
$472$	0	0
$473$	7.49950i	0.344827i
$474$	0	0
$475$	0	0
$476$	0	0
$477$	3.64619i	0.166948i
$478$	0	0
$479$	6.75757	0.308761	0.154381	−	0.988011i	$-0.450662\pi$
$479$	6.75757	0.308761	0.154381	+	0.988011i	$0.450662\pi$
$480$	0	0
$481$	−7.31427	−0.333502
$482$	0	0
$483$	− 2.87459i	− 0.130798i
$484$	0	0
$485$	0	0
$486$	0	0
$487$	− 34.0395i	− 1.54248i	−0.636545	−	0.771239i	$-0.719637\pi$
$487$	− 34.0395i	− 1.54248i	0.636545	−	0.771239i	$-0.280363\pi$
$488$	0	0
$489$	54.0146	2.44263
$490$	0	0
$491$	17.3299	0.782088	0.391044	−	0.920372i	$-0.372114\pi$
$491$	17.3299	0.782088	0.391044	+	0.920372i	$0.372114\pi$
$492$	0	0
$493$	34.2773i	1.54377i
$494$	0	0
$495$	0	0
$496$	0	0
$497$	28.0718i	1.25919i
$498$	0	0
$499$	−14.0931	−0.630895	−0.315447	−	0.948943i	$-0.602155\pi$
$499$	−14.0931	−0.630895	−0.315447	+	0.948943i	$0.602155\pi$
$500$	0	0
$501$	−45.1806	−2.01852
$502$	0	0
$503$	15.7825i	0.703706i	0.936055	+	0.351853i	$0.114448\pi$
$503$	15.7825i	0.703706i	−0.936055	+	0.351853i	$0.885552\pi$
$504$	0	0
$505$	0	0
$506$	0	0
$507$	− 31.1114i	− 1.38170i
$508$	0	0
$509$	13.3907	0.593534	0.296767	−	0.954950i	$-0.404092\pi$
$509$	13.3907	0.593534	0.296767	+	0.954950i	$0.404092\pi$
$510$	0	0
$511$	33.1088	1.46465
$512$	0	0
$513$	15.3143i	0.676141i
$514$	0	0
$515$	0	0
$516$	0	0
$517$	7.05619i	0.310331i
$518$	0	0
$519$	21.2211	0.931505
$520$	0	0
$521$	11.6857	0.511961	0.255981	−	0.966682i	$-0.417602\pi$
$521$	11.6857	0.511961	0.255981	+	0.966682i	$0.417602\pi$
$522$	0	0
$523$	18.9022i	0.826538i	0.910609	+	0.413269i	$0.135613\pi$
$523$	18.9022i	0.826538i	−0.910609	+	0.413269i	$0.864387\pi$
$524$	0	0
$525$	0	0
$526$	0	0
$527$	− 20.4064i	− 0.888916i
$528$	0	0
$529$	22.8340	0.992784
$530$	0	0
$531$	−26.2607	−1.13962
$532$	0	0
$533$	3.31427i	0.143557i
$534$	0	0
$535$	0	0
$536$	0	0
$537$	27.7971i	1.19953i
$538$	0	0
$539$	0.407385	0.0175473
$540$	0	0
$541$	−10.8497	−0.466464	−0.233232	−	0.972421i	$-0.574930\pi$
$541$	−10.8497	−0.466464	−0.233232	+	0.972421i	$0.574930\pi$
$542$	0	0
$543$	− 60.8294i	− 2.61044i
$544$	0	0
$545$	0	0
$546$	0	0
$547$	44.9621i	1.92244i	0.275784	+	0.961220i	$0.411062\pi$
$547$	44.9621i	1.92244i	−0.275784	+	0.961220i	$0.588938\pi$
$548$	0	0
$549$	16.6119	0.708978
$550$	0	0
$551$	−61.0911	−2.60257
$552$	0	0
$553$	− 8.57133i	− 0.364490i
$554$	0	0
$555$	0	0
$556$	0	0
$557$	− 1.99899i	− 0.0847000i	−0.999103	−	0.0423500i	$-0.986516\pi$
$557$	− 1.99899i	− 0.0847000i	0.999103	−	0.0423500i	$-0.0134845\pi$
$558$	0	0
$559$	−7.49950	−0.317195
$560$	0	0
$561$	11.9069	0.502709
$562$	0	0
$563$	− 28.8340i	− 1.21521i	−0.794239	−	0.607605i	$-0.792130\pi$
$563$	− 28.8340i	− 1.21521i	0.794239	−	0.607605i	$-0.207870\pi$
$564$	0	0
$565$	0	0
$566$	0	0
$567$	17.1852i	0.721712i
$568$	0	0
$569$	−25.6894	−1.07695	−0.538477	−	0.842640i	$-0.681000\pi$
$569$	−25.6894	−1.07695	−0.538477	+	0.842640i	$0.681000\pi$
$570$	0	0
$571$	36.4792	1.52661	0.763304	−	0.646040i	$-0.223576\pi$
$571$	36.4792	1.52661	0.763304	+	0.646040i	$0.223576\pi$
$572$	0	0
$573$	− 40.2773i	− 1.68261i
$574$	0	0
$575$	0	0
$576$	0	0
$577$	− 36.1078i	− 1.50319i	−0.659628	−	0.751593i	$-0.729286\pi$
$577$	− 36.1078i	− 1.50319i	0.659628	−	0.751593i	$-0.270714\pi$
$578$	0	0
$579$	4.22477	0.175576
$580$	0	0
$581$	45.3122	1.87987
$582$	0	0
$583$	0.979724i	0.0405760i
$584$	0	0
$585$	0	0
$586$	0	0
$587$	5.77784i	0.238477i	0.992866	+	0.119239i	$0.0380453\pi$
$587$	5.77784i	0.238477i	−0.992866	+	0.119239i	$0.961955\pi$
$588$	0	0
$589$	36.3694	1.49858
$590$	0	0
$591$	−7.20188	−0.296246
$592$	0	0
$593$	− 32.9418i	− 1.35276i	−0.736554	−	0.676379i	$-0.763548\pi$
$593$	− 32.9418i	− 1.35276i	0.736554	−	0.676379i	$-0.236452\pi$
$594$	0	0
$595$	0	0
$596$	0	0
$597$	67.8330i	2.77622i
$598$	0	0
$599$	−8.14830	−0.332931	−0.166465	−	0.986047i	$-0.553235\pi$
$599$	−8.14830	−0.332931	−0.166465	+	0.986047i	$0.553235\pi$
$600$	0	0
$601$	7.39073	0.301474	0.150737	−	0.988574i	$-0.451835\pi$
$601$	7.39073	0.301474	0.150737	+	0.988574i	$0.451835\pi$
$602$	0	0
$603$	− 8.40376i	− 0.342228i
$604$	0	0
$605$	0	0
$606$	0	0
$607$	− 27.9980i	− 1.13640i	−0.822889	−	0.568202i	$-0.807639\pi$
$607$	− 27.9980i	− 1.13640i	0.822889	−	0.568202i	$-0.192361\pi$
$608$	0	0
$609$	52.6644	2.13407
$610$	0	0
$611$	−7.05619	−0.285463
$612$	0	0
$613$	− 39.0036i	− 1.57534i	−0.616096	−	0.787671i	$-0.711287\pi$
$613$	− 39.0036i	− 1.57534i	0.616096	−	0.787671i	$-0.288713\pi$
$614$	0	0
$615$	0	0
$616$	0	0
$617$	− 26.1686i	− 1.05351i	−0.850018	−	0.526754i	$-0.823409\pi$
$617$	− 26.1686i	− 1.05351i	0.850018	−	0.526754i	$-0.176591\pi$
$618$	0	0
$619$	−22.2820	−0.895588	−0.447794	−	0.894137i	$-0.647790\pi$
$619$	−22.2820	−0.895588	−0.447794	+	0.894137i	$0.647790\pi$
$620$	0	0
$621$	−0.762203	−0.0305862
$622$	0	0
$623$	22.5308i	0.902677i
$624$	0	0
$625$	0	0
$626$	0	0
$627$	21.2211i	0.847491i
$628$	0	0
$629$	−33.5916	−1.33939
$630$	0	0
$631$	−34.1629	−1.36000	−0.680002	−	0.733210i	$-0.738021\pi$
$631$	−34.1629	−1.36000	−0.680002	+	0.733210i	$0.738021\pi$
$632$	0	0
$633$	− 6.18885i	− 0.245985i
$634$	0	0
$635$	0	0
$636$	0	0
$637$	0.407385i	0.0161412i
$638$	0	0
$639$	38.3861	1.51853
$640$	0	0
$641$	−5.37046	−0.212120	−0.106060	−	0.994360i	$-0.533824\pi$
$641$	−5.37046	−0.212120	−0.106060	+	0.994360i	$0.533824\pi$
$642$	0	0
$643$	15.8856i	0.626467i	0.949676	+	0.313233i	$0.101412\pi$
$643$	15.8856i	0.626467i	−0.949676	+	0.313233i	$0.898588\pi$
$644$	0	0
$645$	0	0
$646$	0	0
$647$	− 4.87197i	− 0.191537i	−0.995404	−	0.0957685i	$-0.969469\pi$
$647$	− 4.87197i	− 0.191537i	0.995404	−	0.0957685i	$-0.0305309\pi$
$648$	0	0
$649$	−7.05619	−0.276980
$650$	0	0
$651$	−31.3528	−1.22881
$652$	0	0
$653$	16.8101i	0.657831i	0.944359	+	0.328916i	$0.106683\pi$
$653$	16.8101i	0.657831i	−0.944359	+	0.328916i	$0.893317\pi$
$654$	0	0
$655$	0	0
$656$	0	0
$657$	− 45.2737i	− 1.76630i
$658$	0	0
$659$	2.79449	0.108858	0.0544290	−	0.998518i	$-0.482666\pi$
$659$	2.79449	0.108858	0.0544290	+	0.998518i	$0.482666\pi$
$660$	0	0
$661$	33.4267	1.30015	0.650073	−	0.759872i	$-0.274738\pi$
$661$	33.4267	1.30015	0.650073	+	0.759872i	$0.274738\pi$
$662$	0	0
$663$	11.9069i	0.462425i
$664$	0	0
$665$	0	0
$666$	0	0
$667$	− 3.04055i	− 0.117731i
$668$	0	0
$669$	65.9215	2.54867
$670$	0	0
$671$	4.46358	0.172315
$672$	0	0
$673$	− 11.9115i	− 0.459155i	−0.973290	−	0.229578i	$-0.926266\pi$
$673$	− 11.9115i	− 0.459155i	0.973290	−	0.229578i	$-0.0737345\pi$
$674$	0	0
$675$	0	0
$676$	0	0
$677$	13.4277i	0.516067i	0.966136	+	0.258033i	$0.0830745\pi$
$677$	13.4277i	0.516067i	−0.966136	+	0.258033i	$0.916926\pi$
$678$	0	0
$679$	8.26271	0.317094
$680$	0	0
$681$	−26.6478	−1.02115
$682$	0	0
$683$	− 47.7160i	− 1.82580i	−0.408181	−	0.912901i	$-0.633837\pi$
$683$	− 47.7160i	− 1.82580i	0.408181	−	0.912901i	$-0.366163\pi$
$684$	0	0
$685$	0	0
$686$	0	0
$687$	75.2763i	2.87197i
$688$	0	0
$689$	−0.979724	−0.0373245
$690$	0	0
$691$	29.2450	1.11253	0.556267	−	0.831004i	$-0.312233\pi$
$691$	29.2450	1.11253	0.556267	+	0.831004i	$0.312233\pi$
$692$	0	0
$693$	− 10.1290i	− 0.384770i
$694$	0	0
$695$	0	0
$696$	0	0
$697$	15.2211i	0.576542i
$698$	0	0
$699$	−15.3143	−0.579239
$700$	0	0
$701$	18.6442	0.704181	0.352090	−	0.935966i	$-0.385471\pi$
$701$	18.6442	0.704181	0.352090	+	0.935966i	$0.385471\pi$
$702$	0	0
$703$	− 59.8689i	− 2.25800i
$704$	0	0
$705$	0	0
$706$	0	0
$707$	− 15.2211i	− 0.572450i
$708$	0	0
$709$	30.3705	1.14059	0.570293	−	0.821441i	$-0.306830\pi$
$709$	30.3705	1.14059	0.570293	+	0.821441i	$0.306830\pi$
$710$	0	0
$711$	−11.7206	−0.439558
$712$	0	0
$713$	1.81014i	0.0677901i
$714$	0	0
$715$	0	0
$716$	0	0
$717$	− 9.83766i	− 0.367394i
$718$	0	0
$719$	4.49950	0.167803	0.0839014	−	0.996474i	$-0.473262\pi$
$719$	4.49950	0.167803	0.0839014	+	0.996474i	$0.473262\pi$
$720$	0	0
$721$	−12.6524	−0.471201
$722$	0	0
$723$	61.1639i	2.27471i
$724$	0	0
$725$	0	0
$726$	0	0
$727$	− 38.7925i	− 1.43873i	−0.694631	−	0.719367i	$-0.744432\pi$
$727$	− 38.7925i	− 1.43873i	0.694631	−	0.719367i	$-0.255568\pi$
$728$	0	0
$729$	38.0516	1.40932
$730$	0	0
$731$	−34.4423	−1.27389
$732$	0	0
$733$	− 29.5547i	− 1.09163i	−0.837907	−	0.545813i	$-0.816221\pi$
$733$	− 29.5547i	− 1.09163i	0.837907	−	0.545813i	$-0.183779\pi$
$734$	0	0
$735$	0	0
$736$	0	0
$737$	− 2.25807i	− 0.0831772i
$738$	0	0
$739$	10.9079	0.401253	0.200627	−	0.979668i	$-0.435702\pi$
$739$	10.9079	0.401253	0.200627	+	0.979668i	$0.435702\pi$
$740$	0	0
$741$	−21.2211	−0.779578
$742$	0	0
$743$	7.94743i	0.291563i	0.989317	+	0.145782i	$0.0465697\pi$
$743$	7.94743i	0.291563i	−0.989317	+	0.145782i	$0.953430\pi$
$744$	0	0
$745$	0	0
$746$	0	0
$747$	− 61.9611i	− 2.26704i
$748$	0	0
$749$	−1.90890	−0.0697497
$750$	0	0
$751$	18.2783	0.666986	0.333493	−	0.942753i	$-0.391773\pi$
$751$	18.2783	0.666986	0.333493	+	0.942753i	$0.391773\pi$
$752$	0	0
$753$	− 55.9787i	− 2.03998i
$754$	0	0
$755$	0	0
$756$	0	0
$757$	− 30.2727i	− 1.10028i	−0.835072	−	0.550140i	$-0.814574\pi$
$757$	− 30.2727i	− 1.10028i	0.835072	−	0.550140i	$-0.185426\pi$
$758$	0	0
$759$	−1.05619	−0.0383374
$760$	0	0
$761$	14.7429	0.534431	0.267216	−	0.963637i	$-0.413896\pi$
$761$	14.7429	0.534431	0.267216	+	0.963637i	$0.413896\pi$
$762$	0	0
$763$	11.6442i	0.421547i
$764$	0	0
$765$	0	0
$766$	0	0
$767$	− 7.05619i	− 0.254784i
$768$	0	0
$769$	−3.94280	−0.142181	−0.0710905	−	0.997470i	$-0.522648\pi$
$769$	−3.94280	−0.142181	−0.0710905	+	0.997470i	$0.522648\pi$
$770$	0	0
$771$	−10.4110	−0.374943
$772$	0	0
$773$	52.6478i	1.89361i	0.321808	+	0.946805i	$0.395709\pi$
$773$	52.6478i	1.89361i	−0.321808	+	0.946805i	$0.604291\pi$
$774$	0	0
$775$	0	0
$776$	0	0
$777$	51.6109i	1.85153i
$778$	0	0
$779$	−27.1280	−0.971962
$780$	0	0
$781$	10.3143	0.369073
$782$	0	0
$783$	− 13.9641i	− 0.499036i
$784$	0	0
$785$	0	0
$786$	0	0
$787$	− 44.5713i	− 1.58880i	−0.607397	−	0.794398i	$-0.707786\pi$
$787$	− 44.5713i	− 1.58880i	0.607397	−	0.794398i	$-0.292214\pi$
$788$	0	0
$789$	−75.8091	−2.69888
$790$	0	0
$791$	−31.8017	−1.13074
$792$	0	0
$793$	4.46358i	0.158506i
$794$	0	0
$795$	0	0
$796$	0	0
$797$	9.03693i	0.320104i	0.987109	+	0.160052i	$0.0511662\pi$
$797$	9.03693i	0.320104i	−0.987109	+	0.160052i	$0.948834\pi$
$798$	0	0
$799$	−32.4064	−1.14646
$800$	0	0
$801$	30.8091	1.08859
$802$	0	0
$803$	− 12.1650i	− 0.429292i
$804$	0	0
$805$	0	0
$806$	0	0
$807$	− 20.8340i	− 0.733393i
$808$	0	0
$809$	−9.14468	−0.321510	−0.160755	−	0.986994i	$-0.551393\pi$
$809$	−9.14468	−0.321510	−0.160755	+	0.986994i	$0.551393\pi$
$810$	0	0
$811$	1.35482	0.0475741	0.0237870	−	0.999717i	$-0.492428\pi$
$811$	1.35482	0.0475741	0.0237870	+	0.999717i	$0.492428\pi$
$812$	0	0
$813$	− 50.3658i	− 1.76641i
$814$	0	0
$815$	0	0
$816$	0	0
$817$	− 61.3851i	− 2.14759i
$818$	0	0
$819$	10.1290	0.353937
$820$	0	0
$821$	−27.2414	−0.950732	−0.475366	−	0.879788i	$-0.657684\pi$
$821$	−27.2414	−0.950732	−0.475366	+	0.879788i	$0.657684\pi$
$822$	0	0
$823$	6.96771i	0.242879i	0.992599	+	0.121440i	$0.0387510\pi$
$823$	6.96771i	0.242879i	−0.992599	+	0.121440i	$0.961249\pi$
$824$	0	0
$825$	0	0
$826$	0	0
$827$	− 2.68472i	− 0.0933570i	−0.998910	−	0.0466785i	$-0.985136\pi$
$827$	− 2.68472i	− 0.0933570i	0.998910	−	0.0466785i	$-0.0148636\pi$
$828$	0	0
$829$	20.2222	0.702345	0.351172	−	0.936311i	$-0.385783\pi$
$829$	20.2222	0.702345	0.351172	+	0.936311i	$0.385783\pi$
$830$	0	0
$831$	−4.74555	−0.164621
$832$	0	0
$833$	1.87096i	0.0648250i
$834$	0	0
$835$	0	0
$836$	0	0
$837$	8.31326i	0.287348i
$838$	0	0
$839$	35.5344	1.22678	0.613392	−	0.789779i	$-0.289805\pi$
$839$	35.5344	1.22678	0.613392	+	0.789779i	$0.289805\pi$
$840$	0	0
$841$	26.7050	0.920862
$842$	0	0
$843$	− 45.8497i	− 1.57915i
$844$	0	0
$845$	0	0
$846$	0	0
$847$	− 2.72165i	− 0.0935170i
$848$	0	0
$849$	43.6873	1.49935
$850$	0	0
$851$	2.97972	0.102144
$852$	0	0
$853$	− 26.3253i	− 0.901360i	−0.892686	−	0.450680i	$-0.851182\pi$
$853$	− 26.3253i	− 0.901360i	0.892686	−	0.450680i	$-0.148818\pi$
$854$	0	0
$855$	0	0
$856$	0	0
$857$	− 48.1270i	− 1.64399i	−0.569496	−	0.821994i	$-0.692862\pi$
$857$	− 48.1270i	− 1.64399i	0.569496	−	0.821994i	$-0.307138\pi$
$858$	0	0
$859$	40.8148	1.39258	0.696291	−	0.717760i	$-0.254832\pi$
$859$	40.8148	1.39258	0.696291	+	0.717760i	$0.254832\pi$
$860$	0	0
$861$	23.3861	0.796996
$862$	0	0
$863$	− 39.8866i	− 1.35776i	−0.734251	−	0.678878i	$-0.762467\pi$
$863$	− 39.8866i	− 1.35776i	0.734251	−	0.678878i	$-0.237533\pi$
$864$	0	0
$865$	0	0
$866$	0	0
$867$	10.6093i	0.360310i
$868$	0	0
$869$	−3.14931	−0.106833
$870$	0	0
$871$	2.25807	0.0765119
$872$	0	0
$873$	− 11.2986i	− 0.382401i
$874$	0	0
$875$	0	0
$876$	0	0
$877$	− 11.7263i	− 0.395969i	−0.980205	−	0.197984i	$-0.936561\pi$
$877$	− 11.7263i	− 0.395969i	0.980205	−	0.197984i	$-0.0634395\pi$
$878$	0	0
$879$	73.5537	2.48090
$880$	0	0
$881$	−11.7363	−0.395405	−0.197703	−	0.980262i	$-0.563348\pi$
$881$	−11.7363	−0.395405	−0.197703	+	0.980262i	$0.563348\pi$
$882$	0	0
$883$	− 14.0146i	− 0.471630i	−0.971798	−	0.235815i	$-0.924224\pi$
$883$	− 14.0146i	− 0.471630i	0.971798	−	0.235815i	$-0.0757759\pi$
$884$	0	0
$885$	0	0
$886$	0	0
$887$	4.12904i	0.138639i	0.997594	+	0.0693197i	$0.0220829\pi$
$887$	4.12904i	0.138639i	−0.997594	+	0.0693197i	$0.977917\pi$
$888$	0	0
$889$	−25.9095	−0.868977
$890$	0	0
$891$	6.31427	0.211536
$892$	0	0
$893$	− 57.7566i	− 1.93275i
$894$	0	0
$895$	0	0
$896$	0	0
$897$	− 1.05619i	− 0.0352653i
$898$	0	0
$899$	−33.1629	−1.10605
$900$	0	0
$901$	−4.49950	−0.149900
$902$	0	0
$903$	52.9179i	1.76100i
$904$	0	0
$905$	0	0
$906$	0	0
$907$	25.2258i	0.837608i	0.908077	+	0.418804i	$0.137551\pi$
$907$	25.2258i	0.837608i	−0.908077	+	0.418804i	$0.862449\pi$
$908$	0	0
$909$	−20.8138	−0.690349
$910$	0	0
$911$	19.7419	0.654079	0.327040	−	0.945011i	$-0.393949\pi$
$911$	19.7419	0.654079	0.327040	+	0.945011i	$0.393949\pi$
$912$	0	0
$913$	− 16.6488i	− 0.550995i
$914$	0	0
$915$	0	0
$916$	0	0
$917$	61.1806i	2.02036i
$918$	0	0
$919$	39.1114	1.29017	0.645083	−	0.764113i	$-0.276823\pi$
$919$	39.1114	1.29017	0.645083	+	0.764113i	$0.276823\pi$
$920$	0	0
$921$	5.94743	0.195975
$922$	0	0
$923$	10.3143i	0.339498i
$924$	0	0
$925$	0	0
$926$	0	0
$927$	17.3012i	0.568247i
$928$	0	0
$929$	0.670093	0.0219850	0.0109925	−	0.999940i	$-0.496501\pi$
$929$	0.670093	0.0219850	0.0109925	+	0.999940i	$0.496501\pi$
$930$	0	0
$931$	−3.33454	−0.109285
$932$	0	0
$933$	29.1878i	0.955567i
$934$	0	0
$935$	0	0
$936$	0	0
$937$	− 31.3715i	− 1.02486i	−0.858729	−	0.512431i	$-0.828745\pi$
$937$	− 31.3715i	− 1.02486i	0.858729	−	0.512431i	$-0.171255\pi$
$938$	0	0
$939$	−18.6285	−0.607919
$940$	0	0
$941$	−39.5703	−1.28996	−0.644978	−	0.764201i	$-0.723133\pi$
$941$	−39.5703	−1.28996	−0.644978	+	0.764201i	$0.723133\pi$
$942$	0	0
$943$	− 1.35018i	− 0.0439680i
$944$	0	0
$945$	0	0
$946$	0	0
$947$	9.31427i	0.302673i	0.988482	+	0.151336i	$0.0483577\pi$
$947$	9.31427i	0.302673i	−0.988482	+	0.151336i	$0.951642\pi$
$948$	0	0
$949$	12.1650	0.394891
$950$	0	0
$951$	76.5334	2.48177
$952$	0	0
$953$	26.7409i	0.866223i	0.901340	+	0.433112i	$0.142584\pi$
$953$	26.7409i	0.866223i	−0.901340	+	0.433112i	$0.857416\pi$
$954$	0	0
$955$	0	0
$956$	0	0
$957$	− 19.3502i	− 0.625503i
$958$	0	0
$959$	42.2820	1.36536
$960$	0	0
$961$	−11.2571	−0.363131
$962$	0	0
$963$	2.61027i	0.0841149i
$964$	0	0
$965$	0	0
$966$	0	0
$967$	29.0442i	0.933998i	0.884258	+	0.466999i	$0.154665\pi$
$967$	29.0442i	0.933998i	−0.884258	+	0.466999i	$0.845335\pi$
$968$	0	0
$969$	−97.4606	−3.13088
$970$	0	0
$971$	41.3482	1.32693	0.663463	−	0.748209i	$-0.269086\pi$
$971$	41.3482	1.32693	0.663463	+	0.748209i	$0.269086\pi$
$972$	0	0
$973$	− 36.2794i	− 1.16306i
$974$	0	0
$975$	0	0
$976$	0	0
$977$	− 14.9105i	− 0.477029i	−0.971139	−	0.238515i	$-0.923339\pi$
$977$	− 14.9105i	− 0.477029i	0.971139	−	0.238515i	$-0.0766605\pi$
$978$	0	0
$979$	8.27835	0.264577
$980$	0	0
$981$	15.9225	0.508367
$982$	0	0
$983$	− 21.4682i	− 0.684730i	−0.939567	−	0.342365i	$-0.888772\pi$
$983$	− 21.4682i	− 0.684730i	0.939567	−	0.342365i	$-0.111228\pi$
$984$	0	0
$985$	0	0
$986$	0	0
$987$	49.7899i	1.58483i
$988$	0	0
$989$	3.05518	0.0971492
$990$	0	0
$991$	−3.47922	−0.110521	−0.0552605	−	0.998472i	$-0.517599\pi$
$991$	−3.47922	−0.110521	−0.0552605	+	0.998472i	$0.517599\pi$
$992$	0	0
$993$	− 15.8902i	− 0.504261i
$994$	0	0
$995$	0	0
$996$	0	0
$997$	− 23.4054i	− 0.741255i	−0.928782	−	0.370628i	$-0.879143\pi$
$997$	− 23.4054i	− 0.741255i	0.928782	−	0.370628i	$-0.120857\pi$
$998$	0	0
$999$	13.6847	0.432966

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	4400.2.b.bb.4049.2		6
4.3	odd	2		2200.2.b.m.1849.5		6
5.2	odd	4		4400.2.a.bz.1.1		3
5.3	odd	4		4400.2.a.by.1.3		3
5.4	even	2	inner	4400.2.b.bb.4049.5		6
20.3	even	4		2200.2.a.v.1.1	yes	3
20.7	even	4		2200.2.a.u.1.3	✓	3
20.19	odd	2		2200.2.b.m.1849.2		6

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
2200.2.a.u.1.3	✓	3	20.7	even	4
2200.2.a.v.1.1	yes	3	20.3	even	4
2200.2.b.m.1849.2		6	20.19	odd	2
2200.2.b.m.1849.5		6	4.3	odd	2
4400.2.a.by.1.3		3	5.3	odd	4
4400.2.a.bz.1.1		3	5.2	odd	4
4400.2.b.bb.4049.2		6	1.1	even	1	trivial
4400.2.b.bb.4049.5		6	5.4	even	2	inner

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:	\( N \)	\(=\)	\( 4400 = 2^{4} \cdot 5^{2} \cdot 11 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	4400.b (of order \(2\), degree \(1\), not minimal)

\(f(q)\)	\(=\)	\(q-2.59261i q^{3} -2.72165i q^{7} -3.72165 q^{9} +O(q^{10})\)	\(q-2.59261i q^{3} -2.72165i q^{7} -3.72165 q^{9} -1.00000 q^{11} -1.00000i q^{13} -4.59261i q^{17} +8.18523 q^{19} -7.05619 q^{21} +0.407385i q^{23} +1.87096i q^{27} -7.46358 q^{29} +4.44330 q^{31} +2.59261i q^{33} -7.31427i q^{37} -2.59261 q^{39} -3.31427 q^{41} -7.49950i q^{43} -7.05619i q^{47} -0.407385 q^{49} -11.9069 q^{51} -0.979724i q^{53} -21.2211i q^{57} +7.05619 q^{59} -4.46358 q^{61} +10.1290i q^{63} +2.25807i q^{67} +1.05619 q^{69} -10.3143 q^{71} +12.1650i q^{73} +2.72165i q^{77} +3.14931 q^{79} -6.31427 q^{81} +16.6488i q^{83} +19.3502i q^{87} -8.27835 q^{89} -2.72165 q^{91} -11.5198i q^{93} +3.03592i q^{97} +3.72165 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 6 q - 12 q^{9}+O(q^{10}) \) 6 * q - 12 * q^9	\( 6 q - 12 q^{9} - 6 q^{11} + 14 q^{19} - 20 q^{29} + 6 q^{31} + 2 q^{39} + 8 q^{41} - 20 q^{49} - 26 q^{51} - 2 q^{61} - 36 q^{69} - 34 q^{71} + 22 q^{79} - 10 q^{81} - 60 q^{89} - 6 q^{91} + 12 q^{99}+O(q^{100}) \) 6 * q - 12 * q^9 - 6 * q^11 + 14 * q^19 - 20 * q^29 + 6 * q^31 + 2 * q^39 + 8 * q^41 - 20 * q^49 - 26 * q^51 - 2 * q^61 - 36 * q^69 - 34 * q^71 + 22 * q^79 - 10 * q^81 - 60 * q^89 - 6 * q^91 + 12 * q^99

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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