LMFDB - Embedded newform 2200.2.b.g.1849.4

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [2200,2,Mod(1849,2200)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(2200, 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("2200.1849");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 2200 = 2^{3} \cdot 5^{2} \cdot 11 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	2200.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:	$17.5670884447$
Analytic rank:	$0$
Dimension:	$4$
Coefficient field:	$\Q(i, \sqrt{17})$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{4} + 9x^{2} + 16 $ x^4 + 9*x^2 + 16
Coefficient ring:	$\Z[a_1, \ldots, a_{13}]$
Coefficient ring index:	$ 2 $
Twist minimal:	no (minimal twist has level 88)
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label			1849.4
Root			$2.56155i$ of defining polynomial
Character	$\chi$	$=$	2200.1849
Dual form			2200.2.b.g.1849.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+2.56155i q^{3} +5.12311i q^{7} -3.56155 q^{9} -1.00000 q^{11} +3.12311i q^{13} -2.00000i q^{17} +4.00000 q^{19} -13.1231 q^{21} +6.56155i q^{23} -1.43845i q^{27} -3.12311 q^{29} -1.43845 q^{31} -2.56155i q^{33} +3.43845i q^{37} -8.00000 q^{39} +7.12311 q^{41} +1.12311i q^{43} -8.00000i q^{47} -19.2462 q^{49} +5.12311 q^{51} -4.24621i q^{53} +10.2462i q^{57} +12.8078 q^{59} -7.12311 q^{61} -18.2462i q^{63} -5.43845i q^{67} -16.8078 q^{69} +3.68466 q^{71} -3.12311i q^{73} -5.12311i q^{77} +2.87689 q^{79} -7.00000 q^{81} +9.12311i q^{83} -8.00000i q^{87} +9.68466 q^{89} -16.0000 q^{91} -3.68466i q^{93} -11.4384i q^{97} +3.56155 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 4 q - 6 q^{9} - 4 q^{11} + 16 q^{19} - 36 q^{21} + 4 q^{29} - 14 q^{31} - 32 q^{39} + 12 q^{41} - 44 q^{49} + 4 q^{51} + 10 q^{59} - 12 q^{61} - 26 q^{69} - 10 q^{71} + 28 q^{79} - 28 q^{81} + 14 q^{89}+ \cdots + 6 q^{99}+O(q^{100}) $ 4 * q - 6 * q^9 - 4 * q^11 + 16 * q^19 - 36 * q^21 + 4 * q^29 - 14 * q^31 - 32 * q^39 + 12 * q^41 - 44 * q^49 + 4 * q^51 + 10 * q^59 - 12 * q^61 - 26 * q^69 - 10 * q^71 + 28 * q^79 - 28 * q^81 + 14 * q^89 - 64 * q^91 + 6 * q^99

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/2200\mathbb{Z}\right)^\times$.

$n$	$177$	$551$	$1101$	$1201$
$\chi(n)$	$-1$	$1$	$1$	$1$

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
$3$	2.56155i	1.47891i	0.673204	+	0.739457i	$0.264917\pi$
$3$	2.56155i	1.47891i	−0.673204	+	0.739457i	$0.735083\pi$
$4$	0	0
$5$	0	0
$5$	0	0
$6$	0	0
$7$	5.12311i	1.93635i	0.250270	+	0.968176i	$0.419480\pi$
$7$	5.12311i	1.93635i	−0.250270	+	0.968176i	$0.580520\pi$
$8$	0	0
$9$	−3.56155	−1.18718
$10$	0	0
$11$	−1.00000	−0.301511
$11$	−1.00000	−0.301511
$12$	0	0
$13$	3.12311i	0.866194i	0.901347	+	0.433097i	$0.142579\pi$
$13$	3.12311i	0.866194i	−0.901347	+	0.433097i	$0.857421\pi$
$14$	0	0
$15$	0	0
$16$	0	0
$17$	− 2.00000i	− 0.485071i	−0.970143	−	0.242536i	$-0.922021\pi$
$17$	− 2.00000i	− 0.485071i	0.970143	−	0.242536i	$-0.0779791\pi$
$18$	0	0
$19$	4.00000	0.917663	0.458831	−	0.888523i	$-0.348268\pi$
$19$	4.00000	0.917663	0.458831	+	0.888523i	$0.348268\pi$
$20$	0	0
$21$	−13.1231	−2.86370
$22$	0	0
$23$	6.56155i	1.36818i	0.729398	+	0.684089i	$0.239800\pi$
$23$	6.56155i	1.36818i	−0.729398	+	0.684089i	$0.760200\pi$
$24$	0	0
$25$	0	0
$26$	0	0
$27$	− 1.43845i	− 0.276829i
$28$	0	0
$29$	−3.12311	−0.579946	−0.289973	−	0.957035i	$-0.593646\pi$
$29$	−3.12311	−0.579946	−0.289973	+	0.957035i	$0.593646\pi$
$30$	0	0
$31$	−1.43845	−0.258353	−0.129176	−	0.991622i	$-0.541233\pi$
$31$	−1.43845	−0.258353	−0.129176	+	0.991622i	$0.541233\pi$
$32$	0	0
$33$	− 2.56155i	− 0.445909i
$34$	0	0
$35$	0	0
$36$	0	0
$37$	3.43845i	0.565277i	0.959226	+	0.282639i	$0.0912097\pi$
$37$	3.43845i	0.565277i	−0.959226	+	0.282639i	$0.908790\pi$
$38$	0	0
$39$	−8.00000	−1.28103
$40$	0	0
$41$	7.12311	1.11244	0.556221	−	0.831034i	$-0.312251\pi$
$41$	7.12311	1.11244	0.556221	+	0.831034i	$0.312251\pi$
$42$	0	0
$43$	1.12311i	0.171272i	0.996326	+	0.0856360i	$0.0272922\pi$
$43$	1.12311i	0.171272i	−0.996326	+	0.0856360i	$0.972708\pi$
$44$	0	0
$45$	0	0
$46$	0	0
$47$	− 8.00000i	− 1.16692i	−0.812142	−	0.583460i	$-0.801699\pi$
$47$	− 8.00000i	− 1.16692i	0.812142	−	0.583460i	$-0.198301\pi$
$48$	0	0
$49$	−19.2462	−2.74946
$50$	0	0
$51$	5.12311	0.717378
$52$	0	0
$53$	− 4.24621i	− 0.583262i	−0.956531	−	0.291631i	$-0.905802\pi$
$53$	− 4.24621i	− 0.583262i	0.956531	−	0.291631i	$-0.0941979\pi$
$54$	0	0
$55$	0	0
$56$	0	0
$57$	10.2462i	1.35714i
$58$	0	0
$59$	12.8078	1.66743	0.833714	−	0.552196i	$-0.186210\pi$
$59$	12.8078	1.66743	0.833714	+	0.552196i	$0.186210\pi$
$60$	0	0
$61$	−7.12311	−0.912020	−0.456010	−	0.889975i	$-0.650722\pi$
$61$	−7.12311	−0.912020	−0.456010	+	0.889975i	$0.650722\pi$
$62$	0	0
$63$	− 18.2462i	− 2.29881i
$64$	0	0
$65$	0	0
$66$	0	0
$67$	− 5.43845i	− 0.664412i	−0.943207	−	0.332206i	$-0.892207\pi$
$67$	− 5.43845i	− 0.664412i	0.943207	−	0.332206i	$-0.107793\pi$
$68$	0	0
$69$	−16.8078	−2.02342
$70$	0	0
$71$	3.68466	0.437289	0.218644	−	0.975805i	$-0.429837\pi$
$71$	3.68466	0.437289	0.218644	+	0.975805i	$0.429837\pi$
$72$	0	0
$73$	− 3.12311i	− 0.365532i	−0.983156	−	0.182766i	$-0.941495\pi$
$73$	− 3.12311i	− 0.365532i	0.983156	−	0.182766i	$-0.0585051\pi$
$74$	0	0
$75$	0	0
$76$	0	0
$77$	− 5.12311i	− 0.583832i
$78$	0	0
$79$	2.87689	0.323676	0.161838	−	0.986817i	$-0.448258\pi$
$79$	2.87689	0.323676	0.161838	+	0.986817i	$0.448258\pi$
$80$	0	0
$81$	−7.00000	−0.777778
$82$	0	0
$83$	9.12311i	1.00139i	0.865624	+	0.500695i	$0.166922\pi$
$83$	9.12311i	1.00139i	−0.865624	+	0.500695i	$0.833078\pi$
$84$	0	0
$85$	0	0
$86$	0	0
$87$	− 8.00000i	− 0.857690i
$88$	0	0
$89$	9.68466	1.02657	0.513286	−	0.858218i	$-0.328428\pi$
$89$	9.68466	1.02657	0.513286	+	0.858218i	$0.328428\pi$
$90$	0	0
$91$	−16.0000	−1.67726
$92$	0	0
$93$	− 3.68466i	− 0.382081i
$94$	0	0
$95$	0	0
$96$	0	0
$97$	− 11.4384i	− 1.16140i	−0.814118	−	0.580699i	$-0.802779\pi$
$97$	− 11.4384i	− 1.16140i	0.814118	−	0.580699i	$-0.197221\pi$
$98$	0	0
$99$	3.56155	0.357950
$100$	0	0
$101$	−2.00000	−0.199007	−0.0995037	−	0.995037i	$-0.531726\pi$
$101$	−2.00000	−0.199007	−0.0995037	+	0.995037i	$0.531726\pi$
$102$	0	0
$103$	0	0	1.00000			$0$
$103$	0	0	−1.00000			$\pi$
$104$	0	0
$105$	0	0
$106$	0	0
$107$	− 11.3693i	− 1.09911i	−0.835456	−	0.549557i	$-0.814797\pi$
$107$	− 11.3693i	− 1.09911i	0.835456	−	0.549557i	$-0.185203\pi$
$108$	0	0
$109$	4.24621	0.406713	0.203357	−	0.979105i	$-0.434815\pi$
$109$	4.24621	0.406713	0.203357	+	0.979105i	$0.434815\pi$
$110$	0	0
$111$	−8.80776	−0.835996
$112$	0	0
$113$	− 4.56155i	− 0.429115i	−0.976711	−	0.214557i	$-0.931169\pi$
$113$	− 4.56155i	− 0.429115i	0.976711	−	0.214557i	$-0.0688309\pi$
$114$	0	0
$115$	0	0
$116$	0	0
$117$	− 11.1231i	− 1.02833i
$118$	0	0
$119$	10.2462	0.939269
$120$	0	0
$121$	1.00000	0.0909091
$122$	0	0
$123$	18.2462i	1.64521i
$124$	0	0
$125$	0	0
$126$	0	0
$127$	10.2462i	0.909204i	0.890695	+	0.454602i	$0.150219\pi$
$127$	10.2462i	0.909204i	−0.890695	+	0.454602i	$0.849781\pi$
$128$	0	0
$129$	−2.87689	−0.253296
$130$	0	0
$131$	−11.3693	−0.993342	−0.496671	−	0.867939i	$-0.665444\pi$
$131$	−11.3693	−0.993342	−0.496671	+	0.867939i	$0.665444\pi$
$132$	0	0
$133$	20.4924i	1.77692i
$134$	0	0
$135$	0	0
$136$	0	0
$137$	12.5616i	1.07321i	0.843835	+	0.536603i	$0.180293\pi$
$137$	12.5616i	1.07321i	−0.843835	+	0.536603i	$0.819707\pi$
$138$	0	0
$139$	−6.87689	−0.583291	−0.291645	−	0.956527i	$-0.594203\pi$
$139$	−6.87689	−0.583291	−0.291645	+	0.956527i	$0.594203\pi$
$140$	0	0
$141$	20.4924	1.72577
$142$	0	0
$143$	− 3.12311i	− 0.261167i
$144$	0	0
$145$	0	0
$146$	0	0
$147$	− 49.3002i	− 4.06621i
$148$	0	0
$149$	12.2462	1.00325	0.501624	−	0.865086i	$-0.332736\pi$
$149$	12.2462	1.00325	0.501624	+	0.865086i	$0.332736\pi$
$150$	0	0
$151$	15.3693	1.25074	0.625369	−	0.780329i	$-0.284949\pi$
$151$	15.3693	1.25074	0.625369	+	0.780329i	$0.284949\pi$
$152$	0	0
$153$	7.12311i	0.575869i
$154$	0	0
$155$	0	0
$156$	0	0
$157$	8.56155i	0.683286i	0.939830	+	0.341643i	$0.110983\pi$
$157$	8.56155i	0.683286i	−0.939830	+	0.341643i	$0.889017\pi$
$158$	0	0
$159$	10.8769	0.862594
$160$	0	0
$161$	−33.6155	−2.64927
$162$	0	0
$163$	− 4.00000i	− 0.313304i	−0.987654	−	0.156652i	$-0.949930\pi$
$163$	− 4.00000i	− 0.313304i	0.987654	−	0.156652i	$-0.0500701\pi$
$164$	0	0
$165$	0	0
$166$	0	0
$167$	− 8.00000i	− 0.619059i	−0.950890	−	0.309529i	$-0.899829\pi$
$167$	− 8.00000i	− 0.619059i	0.950890	−	0.309529i	$-0.100171\pi$
$168$	0	0
$169$	3.24621	0.249709
$170$	0	0
$171$	−14.2462	−1.08944
$172$	0	0
$173$	− 4.24621i	− 0.322833i	−0.986886	−	0.161417i	$-0.948394\pi$
$173$	− 4.24621i	− 0.322833i	0.986886	−	0.161417i	$-0.0516063\pi$
$174$	0	0
$175$	0	0
$176$	0	0
$177$	32.8078i	2.46598i
$178$	0	0
$179$	10.5616	0.789408	0.394704	−	0.918808i	$-0.370847\pi$
$179$	10.5616	0.789408	0.394704	+	0.918808i	$0.370847\pi$
$180$	0	0
$181$	−13.6847	−1.01717	−0.508586	−	0.861011i	$-0.669832\pi$
$181$	−13.6847	−1.01717	−0.508586	+	0.861011i	$0.669832\pi$
$182$	0	0
$183$	− 18.2462i	− 1.34880i
$184$	0	0
$185$	0	0
$186$	0	0
$187$	2.00000i	0.146254i
$188$	0	0
$189$	7.36932	0.536039
$190$	0	0
$191$	−14.5616	−1.05364	−0.526818	−	0.849978i	$-0.676615\pi$
$191$	−14.5616	−1.05364	−0.526818	+	0.849978i	$0.676615\pi$
$192$	0	0
$193$	− 0.876894i	− 0.0631202i	−0.999502	−	0.0315601i	$-0.989952\pi$
$193$	− 0.876894i	− 0.0631202i	0.999502	−	0.0315601i	$-0.0100476\pi$
$194$	0	0
$195$	0	0
$196$	0	0
$197$	− 18.4924i	− 1.31753i	−0.752349	−	0.658765i	$-0.771079\pi$
$197$	− 18.4924i	− 1.31753i	0.752349	−	0.658765i	$-0.228921\pi$
$198$	0	0
$199$	−20.4924	−1.45267	−0.726335	−	0.687341i	$-0.758778\pi$
$199$	−20.4924	−1.45267	−0.726335	+	0.687341i	$0.758778\pi$
$200$	0	0
$201$	13.9309	0.982608
$202$	0	0
$203$	− 16.0000i	− 1.12298i
$204$	0	0
$205$	0	0
$206$	0	0
$207$	− 23.3693i	− 1.62428i
$208$	0	0
$209$	−4.00000	−0.276686
$210$	0	0
$211$	−24.4924	−1.68613	−0.843064	−	0.537813i	$-0.819251\pi$
$211$	−24.4924	−1.68613	−0.843064	+	0.537813i	$0.819251\pi$
$212$	0	0
$213$	9.43845i	0.646712i
$214$	0	0
$215$	0	0
$216$	0	0
$217$	− 7.36932i	− 0.500262i
$218$	0	0
$219$	8.00000	0.540590
$220$	0	0
$221$	6.24621	0.420166
$222$	0	0
$223$	8.80776i	0.589812i	0.955526	+	0.294906i	$0.0952883\pi$
$223$	8.80776i	0.589812i	−0.955526	+	0.294906i	$0.904712\pi$
$224$	0	0
$225$	0	0
$226$	0	0
$227$	− 14.8769i	− 0.987414i	−0.869628	−	0.493707i	$-0.835642\pi$
$227$	− 14.8769i	− 0.987414i	0.869628	−	0.493707i	$-0.164358\pi$
$228$	0	0
$229$	−2.31534	−0.153002	−0.0765010	−	0.997070i	$-0.524375\pi$
$229$	−2.31534	−0.153002	−0.0765010	+	0.997070i	$0.524375\pi$
$230$	0	0
$231$	13.1231	0.863437
$232$	0	0
$233$	17.3693i	1.13790i	0.822371	+	0.568951i	$0.192651\pi$
$233$	17.3693i	1.13790i	−0.822371	+	0.568951i	$0.807349\pi$
$234$	0	0
$235$	0	0
$236$	0	0
$237$	7.36932i	0.478689i
$238$	0	0
$239$	13.1231	0.848863	0.424432	−	0.905460i	$-0.360474\pi$
$239$	13.1231	0.848863	0.424432	+	0.905460i	$0.360474\pi$
$240$	0	0
$241$	20.8769	1.34480	0.672399	−	0.740188i	$-0.265264\pi$
$241$	20.8769	1.34480	0.672399	+	0.740188i	$0.265264\pi$
$242$	0	0
$243$	− 22.2462i	− 1.42710i
$244$	0	0
$245$	0	0
$246$	0	0
$247$	12.4924i	0.794874i
$248$	0	0
$249$	−23.3693	−1.48097
$250$	0	0
$251$	−2.56155	−0.161684	−0.0808419	−	0.996727i	$-0.525761\pi$
$251$	−2.56155	−0.161684	−0.0808419	+	0.996727i	$0.525761\pi$
$252$	0	0
$253$	− 6.56155i	− 0.412521i
$254$	0	0
$255$	0	0
$256$	0	0
$257$	− 28.2462i	− 1.76195i	−0.473163	−	0.880975i	$-0.656888\pi$
$257$	− 28.2462i	− 1.76195i	0.473163	−	0.880975i	$-0.343112\pi$
$258$	0	0
$259$	−17.6155	−1.09458
$260$	0	0
$261$	11.1231	0.688503
$262$	0	0
$263$	10.8769i	0.670698i	0.942094	+	0.335349i	$0.108854\pi$
$263$	10.8769i	0.670698i	−0.942094	+	0.335349i	$0.891146\pi$
$264$	0	0
$265$	0	0
$266$	0	0
$267$	24.8078i	1.51821i
$268$	0	0
$269$	−28.7386	−1.75223	−0.876113	−	0.482106i	$-0.839872\pi$
$269$	−28.7386	−1.75223	−0.876113	+	0.482106i	$0.839872\pi$
$270$	0	0
$271$	4.49242	0.272895	0.136448	−	0.990647i	$-0.456431\pi$
$271$	4.49242	0.272895	0.136448	+	0.990647i	$0.456431\pi$
$272$	0	0
$273$	− 40.9848i	− 2.48052i
$274$	0	0
$275$	0	0
$276$	0	0
$277$	18.0000i	1.08152i	0.841178	+	0.540758i	$0.181862\pi$
$277$	18.0000i	1.08152i	−0.841178	+	0.540758i	$0.818138\pi$
$278$	0	0
$279$	5.12311	0.306712
$280$	0	0
$281$	−0.246211	−0.0146877	−0.00734387	−	0.999973i	$-0.502338\pi$
$281$	−0.246211	−0.0146877	−0.00734387	+	0.999973i	$0.502338\pi$
$282$	0	0
$283$	20.0000i	1.18888i	0.804141	+	0.594438i	$0.202626\pi$
$283$	20.0000i	1.18888i	−0.804141	+	0.594438i	$0.797374\pi$
$284$	0	0
$285$	0	0
$286$	0	0
$287$	36.4924i	2.15408i
$288$	0	0
$289$	13.0000	0.764706
$290$	0	0
$291$	29.3002	1.71761
$292$	0	0
$293$	21.3693i	1.24841i	0.781261	+	0.624204i	$0.214577\pi$
$293$	21.3693i	1.24841i	−0.781261	+	0.624204i	$0.785423\pi$
$294$	0	0
$295$	0	0
$296$	0	0
$297$	1.43845i	0.0834672i
$298$	0	0
$299$	−20.4924	−1.18511
$300$	0	0
$301$	−5.75379	−0.331643
$302$	0	0
$303$	− 5.12311i	− 0.294315i
$304$	0	0
$305$	0	0
$306$	0	0
$307$	− 0.492423i	− 0.0281040i	−0.999901	−	0.0140520i	$-0.995527\pi$
$307$	− 0.492423i	− 0.0281040i	0.999901	−	0.0140520i	$-0.00447304\pi$
$308$	0	0
$309$	0	0
$310$	0	0
$311$	26.2462	1.48829	0.744143	−	0.668020i	$-0.232858\pi$
$311$	26.2462	1.48829	0.744143	+	0.668020i	$0.232858\pi$
$312$	0	0
$313$	10.8078i	0.610891i	0.952210	+	0.305445i	$0.0988054\pi$
$313$	10.8078i	0.610891i	−0.952210	+	0.305445i	$0.901195\pi$
$314$	0	0
$315$	0	0
$316$	0	0
$317$	34.8078i	1.95500i	0.210938	+	0.977499i	$0.432348\pi$
$317$	34.8078i	1.95500i	−0.210938	+	0.977499i	$0.567652\pi$
$318$	0	0
$319$	3.12311	0.174860
$320$	0	0
$321$	29.1231	1.62549
$322$	0	0
$323$	− 8.00000i	− 0.445132i
$324$	0	0
$325$	0	0
$326$	0	0
$327$	10.8769i	0.601494i
$328$	0	0
$329$	40.9848	2.25957
$330$	0	0
$331$	6.06913	0.333590	0.166795	−	0.985992i	$-0.446658\pi$
$331$	6.06913	0.333590	0.166795	+	0.985992i	$0.446658\pi$
$332$	0	0
$333$	− 12.2462i	− 0.671088i
$334$	0	0
$335$	0	0
$336$	0	0
$337$	− 32.7386i	− 1.78339i	−0.452640	−	0.891694i	$-0.649518\pi$
$337$	− 32.7386i	− 1.78339i	0.452640	−	0.891694i	$-0.350482\pi$
$338$	0	0
$339$	11.6847	0.634624
$340$	0	0
$341$	1.43845	0.0778963
$342$	0	0
$343$	− 62.7386i	− 3.38757i
$344$	0	0
$345$	0	0
$346$	0	0
$347$	26.7386i	1.43541i	0.696350	+	0.717703i	$0.254806\pi$
$347$	26.7386i	1.43541i	−0.696350	+	0.717703i	$0.745194\pi$
$348$	0	0
$349$	15.7538	0.843281	0.421640	−	0.906763i	$-0.361454\pi$
$349$	15.7538	0.843281	0.421640	+	0.906763i	$0.361454\pi$
$350$	0	0
$351$	4.49242	0.239788
$352$	0	0
$353$	13.0540i	0.694793i	0.937718	+	0.347397i	$0.112934\pi$
$353$	13.0540i	0.694793i	−0.937718	+	0.347397i	$0.887066\pi$
$354$	0	0
$355$	0	0
$356$	0	0
$357$	26.2462i	1.38910i
$358$	0	0
$359$	28.4924	1.50377	0.751886	−	0.659293i	$-0.229144\pi$
$359$	28.4924	1.50377	0.751886	+	0.659293i	$0.229144\pi$
$360$	0	0
$361$	−3.00000	−0.157895
$362$	0	0
$363$	2.56155i	0.134447i
$364$	0	0
$365$	0	0
$366$	0	0
$367$	5.93087i	0.309589i	0.987947	+	0.154794i	$0.0494715\pi$
$367$	5.93087i	0.309589i	−0.987947	+	0.154794i	$0.950528\pi$
$368$	0	0
$369$	−25.3693	−1.32067
$370$	0	0
$371$	21.7538	1.12940
$372$	0	0
$373$	8.24621i	0.426973i	0.976946	+	0.213486i	$0.0684819\pi$
$373$	8.24621i	0.426973i	−0.976946	+	0.213486i	$0.931518\pi$
$374$	0	0
$375$	0	0
$376$	0	0
$377$	− 9.75379i	− 0.502346i
$378$	0	0
$379$	−20.8078	−1.06882	−0.534412	−	0.845224i	$-0.679467\pi$
$379$	−20.8078	−1.06882	−0.534412	+	0.845224i	$0.679467\pi$
$380$	0	0
$381$	−26.2462	−1.34463
$382$	0	0
$383$	35.0540i	1.79117i	0.444886	+	0.895587i	$0.353244\pi$
$383$	35.0540i	1.79117i	−0.444886	+	0.895587i	$0.646756\pi$
$384$	0	0
$385$	0	0
$386$	0	0
$387$	− 4.00000i	− 0.203331i
$388$	0	0
$389$	0.561553	0.0284719	0.0142359	−	0.999899i	$-0.495468\pi$
$389$	0.561553	0.0284719	0.0142359	+	0.999899i	$0.495468\pi$
$390$	0	0
$391$	13.1231	0.663664
$392$	0	0
$393$	− 29.1231i	− 1.46907i
$394$	0	0
$395$	0	0
$396$	0	0
$397$	22.4924i	1.12886i	0.825480	+	0.564431i	$0.190904\pi$
$397$	22.4924i	1.12886i	−0.825480	+	0.564431i	$0.809096\pi$
$398$	0	0
$399$	−52.4924	−2.62791
$400$	0	0
$401$	−2.49242	−0.124466	−0.0622328	−	0.998062i	$-0.519822\pi$
$401$	−2.49242	−0.124466	−0.0622328	+	0.998062i	$0.519822\pi$
$402$	0	0
$403$	− 4.49242i	− 0.223784i
$404$	0	0
$405$	0	0
$406$	0	0
$407$	− 3.43845i	− 0.170437i
$408$	0	0
$409$	10.4924	0.518817	0.259408	−	0.965768i	$-0.416472\pi$
$409$	10.4924	0.518817	0.259408	+	0.965768i	$0.416472\pi$
$410$	0	0
$411$	−32.1771	−1.58718
$412$	0	0
$413$	65.6155i	3.22873i
$414$	0	0
$415$	0	0
$416$	0	0
$417$	− 17.6155i	− 0.862636i
$418$	0	0
$419$	−0.492423	−0.0240564	−0.0120282	−	0.999928i	$-0.503829\pi$
$419$	−0.492423	−0.0240564	−0.0120282	+	0.999928i	$0.503829\pi$
$420$	0	0
$421$	−30.4924	−1.48611	−0.743055	−	0.669230i	$-0.766624\pi$
$421$	−30.4924	−1.48611	−0.743055	+	0.669230i	$0.766624\pi$
$422$	0	0
$423$	28.4924i	1.38535i
$424$	0	0
$425$	0	0
$426$	0	0
$427$	− 36.4924i	− 1.76599i
$428$	0	0
$429$	8.00000	0.386244
$430$	0	0
$431$	18.8769	0.909268	0.454634	−	0.890678i	$-0.349770\pi$
$431$	18.8769	0.909268	0.454634	+	0.890678i	$0.349770\pi$
$432$	0	0
$433$	− 10.3153i	− 0.495724i	−0.968795	−	0.247862i	$-0.920272\pi$
$433$	− 10.3153i	− 0.495724i	0.968795	−	0.247862i	$-0.0797279\pi$
$434$	0	0
$435$	0	0
$436$	0	0
$437$	26.2462i	1.25553i
$438$	0	0
$439$	28.4924	1.35987	0.679935	−	0.733273i	$-0.262008\pi$
$439$	28.4924	1.35987	0.679935	+	0.733273i	$0.262008\pi$
$440$	0	0
$441$	68.5464	3.26411
$442$	0	0
$443$	23.6847i	1.12529i	0.826698	+	0.562646i	$0.190217\pi$
$443$	23.6847i	1.12529i	−0.826698	+	0.562646i	$0.809783\pi$
$444$	0	0
$445$	0	0
$446$	0	0
$447$	31.3693i	1.48372i
$448$	0	0
$449$	−33.5464	−1.58315	−0.791576	−	0.611070i	$-0.790739\pi$
$449$	−33.5464	−1.58315	−0.791576	+	0.611070i	$0.790739\pi$
$450$	0	0
$451$	−7.12311	−0.335414
$452$	0	0
$453$	39.3693i	1.84973i
$454$	0	0
$455$	0	0
$456$	0	0
$457$	33.8617i	1.58399i	0.610531	+	0.791993i	$0.290956\pi$
$457$	33.8617i	1.58399i	−0.610531	+	0.791993i	$0.709044\pi$
$458$	0	0
$459$	−2.87689	−0.134282
$460$	0	0
$461$	−7.12311	−0.331756	−0.165878	−	0.986146i	$-0.553046\pi$
$461$	−7.12311	−0.331756	−0.165878	+	0.986146i	$0.553046\pi$
$462$	0	0
$463$	27.6847i	1.28662i	0.765608	+	0.643308i	$0.222438\pi$
$463$	27.6847i	1.28662i	−0.765608	+	0.643308i	$0.777562\pi$
$464$	0	0
$465$	0	0
$466$	0	0
$467$	25.3002i	1.17075i	0.810762	+	0.585377i	$0.199053\pi$
$467$	25.3002i	1.17075i	−0.810762	+	0.585377i	$0.800947\pi$
$468$	0	0
$469$	27.8617	1.28654
$470$	0	0
$471$	−21.9309	−1.01052
$472$	0	0
$473$	− 1.12311i	− 0.0516405i
$474$	0	0
$475$	0	0
$476$	0	0
$477$	15.1231i	0.692439i
$478$	0	0
$479$	16.0000	0.731059	0.365529	−	0.930800i	$-0.380888\pi$
$479$	16.0000	0.731059	0.365529	+	0.930800i	$0.380888\pi$
$480$	0	0
$481$	−10.7386	−0.489640
$482$	0	0
$483$	− 86.1080i	− 3.91805i
$484$	0	0
$485$	0	0
$486$	0	0
$487$	13.9309i	0.631268i	0.948881	+	0.315634i	$0.102217\pi$
$487$	13.9309i	0.631268i	−0.948881	+	0.315634i	$0.897783\pi$
$488$	0	0
$489$	10.2462	0.463350
$490$	0	0
$491$	30.2462	1.36499	0.682496	−	0.730889i	$-0.260894\pi$
$491$	30.2462	1.36499	0.682496	+	0.730889i	$0.260894\pi$
$492$	0	0
$493$	6.24621i	0.281315i
$494$	0	0
$495$	0	0
$496$	0	0
$497$	18.8769i	0.846744i
$498$	0	0
$499$	−36.9848	−1.65567	−0.827835	−	0.560972i	$-0.810427\pi$
$499$	−36.9848	−1.65567	−0.827835	+	0.560972i	$0.810427\pi$
$500$	0	0
$501$	20.4924	0.915534
$502$	0	0
$503$	9.61553i	0.428735i	0.976753	+	0.214368i	$0.0687690\pi$
$503$	9.61553i	0.428735i	−0.976753	+	0.214368i	$0.931231\pi$
$504$	0	0
$505$	0	0
$506$	0	0
$507$	8.31534i	0.369297i
$508$	0	0
$509$	−35.3002	−1.56465	−0.782327	−	0.622868i	$-0.785967\pi$
$509$	−35.3002	−1.56465	−0.782327	+	0.622868i	$0.785967\pi$
$510$	0	0
$511$	16.0000	0.707798
$512$	0	0
$513$	− 5.75379i	− 0.254036i
$514$	0	0
$515$	0	0
$516$	0	0
$517$	8.00000i	0.351840i
$518$	0	0
$519$	10.8769	0.477443
$520$	0	0
$521$	13.6847	0.599536	0.299768	−	0.954012i	$-0.403091\pi$
$521$	13.6847	0.599536	0.299768	+	0.954012i	$0.403091\pi$
$522$	0	0
$523$	− 12.0000i	− 0.524723i	−0.964970	−	0.262362i	$-0.915499\pi$
$523$	− 12.0000i	− 0.524723i	0.964970	−	0.262362i	$-0.0845013\pi$
$524$	0	0
$525$	0	0
$526$	0	0
$527$	2.87689i	0.125319i
$528$	0	0
$529$	−20.0540	−0.871912
$530$	0	0
$531$	−45.6155	−1.97955
$532$	0	0
$533$	22.2462i	0.963590i
$534$	0	0
$535$	0	0
$536$	0	0
$537$	27.0540i	1.16747i
$538$	0	0
$539$	19.2462	0.828993
$540$	0	0
$541$	33.8617	1.45583	0.727915	−	0.685667i	$-0.240490\pi$
$541$	33.8617	1.45583	0.727915	+	0.685667i	$0.240490\pi$
$542$	0	0
$543$	− 35.0540i	− 1.50431i
$544$	0	0
$545$	0	0
$546$	0	0
$547$	25.7538i	1.10115i	0.834785	+	0.550576i	$0.185592\pi$
$547$	25.7538i	1.10115i	−0.834785	+	0.550576i	$0.814408\pi$
$548$	0	0
$549$	25.3693	1.08274
$550$	0	0
$551$	−12.4924	−0.532195
$552$	0	0
$553$	14.7386i	0.626750i
$554$	0	0
$555$	0	0
$556$	0	0
$557$	20.2462i	0.857860i	0.903338	+	0.428930i	$0.141109\pi$
$557$	20.2462i	0.857860i	−0.903338	+	0.428930i	$0.858891\pi$
$558$	0	0
$559$	−3.50758	−0.148355
$560$	0	0
$561$	−5.12311	−0.216298
$562$	0	0
$563$	− 8.49242i	− 0.357913i	−0.983857	−	0.178956i	$-0.942728\pi$
$563$	− 8.49242i	− 0.357913i	0.983857	−	0.178956i	$-0.0572721\pi$
$564$	0	0
$565$	0	0
$566$	0	0
$567$	− 35.8617i	− 1.50605i
$568$	0	0
$569$	35.1231	1.47244	0.736219	−	0.676744i	$-0.236610\pi$
$569$	35.1231	1.47244	0.736219	+	0.676744i	$0.236610\pi$
$570$	0	0
$571$	−16.4924	−0.690186	−0.345093	−	0.938568i	$-0.612153\pi$
$571$	−16.4924	−0.690186	−0.345093	+	0.938568i	$0.612153\pi$
$572$	0	0
$573$	− 37.3002i	− 1.55824i
$574$	0	0
$575$	0	0
$576$	0	0
$577$	− 11.4384i	− 0.476189i	−0.971242	−	0.238094i	$-0.923477\pi$
$577$	− 11.4384i	− 0.476189i	0.971242	−	0.238094i	$-0.0765228\pi$
$578$	0	0
$579$	2.24621	0.0933494
$580$	0	0
$581$	−46.7386	−1.93905
$582$	0	0
$583$	4.24621i	0.175860i
$584$	0	0
$585$	0	0
$586$	0	0
$587$	− 8.49242i	− 0.350520i	−0.984522	−	0.175260i	$-0.943923\pi$
$587$	− 8.49242i	− 0.350520i	0.984522	−	0.175260i	$-0.0560766\pi$
$588$	0	0
$589$	−5.75379	−0.237081
$590$	0	0
$591$	47.3693	1.94851
$592$	0	0
$593$	− 21.3693i	− 0.877533i	−0.898601	−	0.438766i	$-0.855416\pi$
$593$	− 21.3693i	− 0.877533i	0.898601	−	0.438766i	$-0.144584\pi$
$594$	0	0
$595$	0	0
$596$	0	0
$597$	− 52.4924i	− 2.14837i
$598$	0	0
$599$	16.0000	0.653742	0.326871	−	0.945069i	$-0.394006\pi$
$599$	16.0000	0.653742	0.326871	+	0.945069i	$0.394006\pi$
$600$	0	0
$601$	20.2462	0.825860	0.412930	−	0.910763i	$-0.364505\pi$
$601$	20.2462	0.825860	0.412930	+	0.910763i	$0.364505\pi$
$602$	0	0
$603$	19.3693i	0.788780i
$604$	0	0
$605$	0	0
$606$	0	0
$607$	− 11.8617i	− 0.481453i	−0.970593	−	0.240727i	$-0.922614\pi$
$607$	− 11.8617i	− 0.481453i	0.970593	−	0.240727i	$-0.0773857\pi$
$608$	0	0
$609$	40.9848	1.66079
$610$	0	0
$611$	24.9848	1.01078
$612$	0	0
$613$	− 45.8617i	− 1.85234i	−0.377108	−	0.926169i	$-0.623082\pi$
$613$	− 45.8617i	− 1.85234i	0.377108	−	0.926169i	$-0.376918\pi$
$614$	0	0
$615$	0	0
$616$	0	0
$617$	− 30.4924i	− 1.22758i	−0.789470	−	0.613789i	$-0.789644\pi$
$617$	− 30.4924i	− 1.22758i	0.789470	−	0.613789i	$-0.210356\pi$
$618$	0	0
$619$	9.93087	0.399155	0.199578	−	0.979882i	$-0.436043\pi$
$619$	9.93087	0.399155	0.199578	+	0.979882i	$0.436043\pi$
$620$	0	0
$621$	9.43845	0.378752
$622$	0	0
$623$	49.6155i	1.98780i
$624$	0	0
$625$	0	0
$626$	0	0
$627$	− 10.2462i	− 0.409194i
$628$	0	0
$629$	6.87689	0.274200
$630$	0	0
$631$	−4.94602	−0.196898	−0.0984491	−	0.995142i	$-0.531388\pi$
$631$	−4.94602	−0.196898	−0.0984491	+	0.995142i	$0.531388\pi$
$632$	0	0
$633$	− 62.7386i	− 2.49364i
$634$	0	0
$635$	0	0
$636$	0	0
$637$	− 60.1080i	− 2.38156i
$638$	0	0
$639$	−13.1231	−0.519142
$640$	0	0
$641$	7.30019	0.288340	0.144170	−	0.989553i	$-0.453949\pi$
$641$	7.30019	0.288340	0.144170	+	0.989553i	$0.453949\pi$
$642$	0	0
$643$	36.1771i	1.42668i	0.700816	+	0.713342i	$0.252820\pi$
$643$	36.1771i	1.42668i	−0.700816	+	0.713342i	$0.747180\pi$
$644$	0	0
$645$	0	0
$646$	0	0
$647$	− 9.43845i	− 0.371064i	−0.982638	−	0.185532i	$-0.940599\pi$
$647$	− 9.43845i	− 0.371064i	0.982638	−	0.185532i	$-0.0594008\pi$
$648$	0	0
$649$	−12.8078	−0.502749
$650$	0	0
$651$	18.8769	0.739844
$652$	0	0
$653$	− 10.1771i	− 0.398260i	−0.979973	−	0.199130i	$-0.936188\pi$
$653$	− 10.1771i	− 0.398260i	0.979973	−	0.199130i	$-0.0638116\pi$
$654$	0	0
$655$	0	0
$656$	0	0
$657$	11.1231i	0.433954i
$658$	0	0
$659$	−29.6155	−1.15366	−0.576829	−	0.816865i	$-0.695710\pi$
$659$	−29.6155	−1.15366	−0.576829	+	0.816865i	$0.695710\pi$
$660$	0	0
$661$	21.1922	0.824282	0.412141	−	0.911120i	$-0.364781\pi$
$661$	21.1922	0.824282	0.412141	+	0.911120i	$0.364781\pi$
$662$	0	0
$663$	16.0000i	0.621389i
$664$	0	0
$665$	0	0
$666$	0	0
$667$	− 20.4924i	− 0.793470i
$668$	0	0
$669$	−22.5616	−0.872280
$670$	0	0
$671$	7.12311	0.274984
$672$	0	0
$673$	− 49.2311i	− 1.89772i	−0.315701	−	0.948859i	$-0.602239\pi$
$673$	− 49.2311i	− 1.89772i	0.315701	−	0.948859i	$-0.397761\pi$
$674$	0	0
$675$	0	0
$676$	0	0
$677$	− 28.7386i	− 1.10452i	−0.833673	−	0.552258i	$-0.813766\pi$
$677$	− 28.7386i	− 1.10452i	0.833673	−	0.552258i	$-0.186234\pi$
$678$	0	0
$679$	58.6004	2.24888
$680$	0	0
$681$	38.1080	1.46030
$682$	0	0
$683$	− 42.7386i	− 1.63535i	−0.575681	−	0.817674i	$-0.695263\pi$
$683$	− 42.7386i	− 1.63535i	0.575681	−	0.817674i	$-0.304737\pi$
$684$	0	0
$685$	0	0
$686$	0	0
$687$	− 5.93087i	− 0.226277i
$688$	0	0
$689$	13.2614	0.505218
$690$	0	0
$691$	−47.0540	−1.79002	−0.895009	−	0.446049i	$-0.852831\pi$
$691$	−47.0540	−1.79002	−0.895009	+	0.446049i	$0.852831\pi$
$692$	0	0
$693$	18.2462i	0.693116i
$694$	0	0
$695$	0	0
$696$	0	0
$697$	− 14.2462i	− 0.539614i
$698$	0	0
$699$	−44.4924	−1.68286
$700$	0	0
$701$	17.5076	0.661252	0.330626	−	0.943762i	$-0.392740\pi$
$701$	17.5076	0.661252	0.330626	+	0.943762i	$0.392740\pi$
$702$	0	0
$703$	13.7538i	0.518734i
$704$	0	0
$705$	0	0
$706$	0	0
$707$	− 10.2462i	− 0.385348i
$708$	0	0
$709$	−22.8078	−0.856564	−0.428282	−	0.903645i	$-0.640881\pi$
$709$	−22.8078	−0.856564	−0.428282	+	0.903645i	$0.640881\pi$
$710$	0	0
$711$	−10.2462	−0.384263
$712$	0	0
$713$	− 9.43845i	− 0.353473i
$714$	0	0
$715$	0	0
$716$	0	0
$717$	33.6155i	1.25539i
$718$	0	0
$719$	26.4233	0.985423	0.492711	−	0.870193i	$-0.336006\pi$
$719$	26.4233	0.985423	0.492711	+	0.870193i	$0.336006\pi$
$720$	0	0
$721$	0	0
$722$	0	0
$723$	53.4773i	1.98884i
$724$	0	0
$725$	0	0
$726$	0	0
$727$	35.6847i	1.32347i	0.749737	+	0.661735i	$0.230180\pi$
$727$	35.6847i	1.32347i	−0.749737	+	0.661735i	$0.769820\pi$
$728$	0	0
$729$	35.9848	1.33277
$730$	0	0
$731$	2.24621	0.0830791
$732$	0	0
$733$	− 7.12311i	− 0.263098i	−0.991310	−	0.131549i	$-0.958005\pi$
$733$	− 7.12311i	− 0.263098i	0.991310	−	0.131549i	$-0.0419951\pi$
$734$	0	0
$735$	0	0
$736$	0	0
$737$	5.43845i	0.200328i
$738$	0	0
$739$	27.3693	1.00680	0.503398	−	0.864054i	$-0.332083\pi$
$739$	27.3693	1.00680	0.503398	+	0.864054i	$0.332083\pi$
$740$	0	0
$741$	−32.0000	−1.17555
$742$	0	0
$743$	− 38.7386i	− 1.42118i	−0.703605	−	0.710591i	$-0.748428\pi$
$743$	− 38.7386i	− 1.42118i	0.703605	−	0.710591i	$-0.251572\pi$
$744$	0	0
$745$	0	0
$746$	0	0
$747$	− 32.4924i	− 1.18884i
$748$	0	0
$749$	58.2462	2.12827
$750$	0	0
$751$	1.43845	0.0524897	0.0262448	−	0.999656i	$-0.491645\pi$
$751$	1.43845	0.0524897	0.0262448	+	0.999656i	$0.491645\pi$
$752$	0	0
$753$	− 6.56155i	− 0.239116i
$754$	0	0
$755$	0	0
$756$	0	0
$757$	− 32.2462i	− 1.17201i	−0.810308	−	0.586004i	$-0.800700\pi$
$757$	− 32.2462i	− 1.17201i	0.810308	−	0.586004i	$-0.199300\pi$
$758$	0	0
$759$	16.8078	0.610083
$760$	0	0
$761$	−3.12311	−0.113212	−0.0566062	−	0.998397i	$-0.518028\pi$
$761$	−3.12311	−0.113212	−0.0566062	+	0.998397i	$0.518028\pi$
$762$	0	0
$763$	21.7538i	0.787540i
$764$	0	0
$765$	0	0
$766$	0	0
$767$	40.0000i	1.44432i
$768$	0	0
$769$	15.6155	0.563110	0.281555	−	0.959545i	$-0.409150\pi$
$769$	15.6155	0.563110	0.281555	+	0.959545i	$0.409150\pi$
$770$	0	0
$771$	72.3542	2.60577
$772$	0	0
$773$	− 8.73863i	− 0.314307i	−0.987574	−	0.157153i	$-0.949768\pi$
$773$	− 8.73863i	− 0.314307i	0.987574	−	0.157153i	$-0.0502317\pi$
$774$	0	0
$775$	0	0
$776$	0	0
$777$	− 45.1231i	− 1.61878i
$778$	0	0
$779$	28.4924	1.02085
$780$	0	0
$781$	−3.68466	−0.131847
$782$	0	0
$783$	4.49242i	0.160546i
$784$	0	0
$785$	0	0
$786$	0	0
$787$	46.2462i	1.64850i	0.566226	+	0.824250i	$0.308403\pi$
$787$	46.2462i	1.64850i	−0.566226	+	0.824250i	$0.691597\pi$
$788$	0	0
$789$	−27.8617	−0.991904
$790$	0	0
$791$	23.3693	0.830917
$792$	0	0
$793$	− 22.2462i	− 0.789986i
$794$	0	0
$795$	0	0
$796$	0	0
$797$	34.8078i	1.23295i	0.787373	+	0.616477i	$0.211441\pi$
$797$	34.8078i	1.23295i	−0.787373	+	0.616477i	$0.788559\pi$
$798$	0	0
$799$	−16.0000	−0.566039
$800$	0	0
$801$	−34.4924	−1.21873
$802$	0	0
$803$	3.12311i	0.110212i
$804$	0	0
$805$	0	0
$806$	0	0
$807$	− 73.6155i	− 2.59139i
$808$	0	0
$809$	4.38447	0.154150	0.0770749	−	0.997025i	$-0.475442\pi$
$809$	4.38447	0.154150	0.0770749	+	0.997025i	$0.475442\pi$
$810$	0	0
$811$	1.12311	0.0394376	0.0197188	−	0.999806i	$-0.493723\pi$
$811$	1.12311	0.0394376	0.0197188	+	0.999806i	$0.493723\pi$
$812$	0	0
$813$	11.5076i	0.403588i
$814$	0	0
$815$	0	0
$816$	0	0
$817$	4.49242i	0.157170i
$818$	0	0
$819$	56.9848	1.99121
$820$	0	0
$821$	22.9848	0.802177	0.401088	−	0.916039i	$-0.368632\pi$
$821$	22.9848	0.802177	0.401088	+	0.916039i	$0.368632\pi$
$822$	0	0
$823$	15.5464i	0.541913i	0.962592	+	0.270957i	$0.0873400\pi$
$823$	15.5464i	0.541913i	−0.962592	+	0.270957i	$0.912660\pi$
$824$	0	0
$825$	0	0
$826$	0	0
$827$	10.7386i	0.373419i	0.982415	+	0.186709i	$0.0597823\pi$
$827$	10.7386i	0.373419i	−0.982415	+	0.186709i	$0.940218\pi$
$828$	0	0
$829$	−43.9309	−1.52578	−0.762891	−	0.646527i	$-0.776221\pi$
$829$	−43.9309	−1.52578	−0.762891	+	0.646527i	$0.776221\pi$
$830$	0	0
$831$	−46.1080	−1.59947
$832$	0	0
$833$	38.4924i	1.33368i
$834$	0	0
$835$	0	0
$836$	0	0
$837$	2.06913i	0.0715196i
$838$	0	0
$839$	−0.807764	−0.0278871	−0.0139436	−	0.999903i	$-0.504439\pi$
$839$	−0.807764	−0.0278871	−0.0139436	+	0.999903i	$0.504439\pi$
$840$	0	0
$841$	−19.2462	−0.663662
$842$	0	0
$843$	− 0.630683i	− 0.0217219i
$844$	0	0
$845$	0	0
$846$	0	0
$847$	5.12311i	0.176032i
$848$	0	0
$849$	−51.2311	−1.75825
$850$	0	0
$851$	−22.5616	−0.773400
$852$	0	0
$853$	− 13.5076i	− 0.462491i	−0.972895	−	0.231245i	$-0.925720\pi$
$853$	− 13.5076i	− 0.462491i	0.972895	−	0.231245i	$-0.0742800\pi$
$854$	0	0
$855$	0	0
$856$	0	0
$857$	44.1080i	1.50670i	0.657620	+	0.753349i	$0.271563\pi$
$857$	44.1080i	1.50670i	−0.657620	+	0.753349i	$0.728437\pi$
$858$	0	0
$859$	7.05398	0.240679	0.120339	−	0.992733i	$-0.461602\pi$
$859$	7.05398	0.240679	0.120339	+	0.992733i	$0.461602\pi$
$860$	0	0
$861$	−93.4773	−3.18570
$862$	0	0
$863$	− 3.50758i	− 0.119399i	−0.998216	−	0.0596997i	$-0.980986\pi$
$863$	− 3.50758i	− 0.119399i	0.998216	−	0.0596997i	$-0.0190143\pi$
$864$	0	0
$865$	0	0
$866$	0	0
$867$	33.3002i	1.13093i
$868$	0	0
$869$	−2.87689	−0.0975920
$870$	0	0
$871$	16.9848	0.575510
$872$	0	0
$873$	40.7386i	1.37879i
$874$	0	0
$875$	0	0
$876$	0	0
$877$	− 30.6307i	− 1.03432i	−0.855887	−	0.517162i	$-0.826988\pi$
$877$	− 30.6307i	− 1.03432i	0.855887	−	0.517162i	$-0.173012\pi$
$878$	0	0
$879$	−54.7386	−1.84629
$880$	0	0
$881$	−19.3002	−0.650240	−0.325120	−	0.945673i	$-0.605405\pi$
$881$	−19.3002	−0.650240	−0.325120	+	0.945673i	$0.605405\pi$
$882$	0	0
$883$	− 24.4924i	− 0.824236i	−0.911131	−	0.412118i	$-0.864789\pi$
$883$	− 24.4924i	− 0.824236i	0.911131	−	0.412118i	$-0.135211\pi$
$884$	0	0
$885$	0	0
$886$	0	0
$887$	9.61553i	0.322858i	0.986884	+	0.161429i	$0.0516103\pi$
$887$	9.61553i	0.322858i	−0.986884	+	0.161429i	$0.948390\pi$
$888$	0	0
$889$	−52.4924	−1.76054
$890$	0	0
$891$	7.00000	0.234509
$892$	0	0
$893$	− 32.0000i	− 1.07084i
$894$	0	0
$895$	0	0
$896$	0	0
$897$	− 52.4924i	− 1.75267i
$898$	0	0
$899$	4.49242	0.149831
$900$	0	0
$901$	−8.49242	−0.282924
$902$	0	0
$903$	− 14.7386i	− 0.490471i
$904$	0	0
$905$	0	0
$906$	0	0
$907$	16.4924i	0.547622i	0.961784	+	0.273811i	$0.0882843\pi$
$907$	16.4924i	0.547622i	−0.961784	+	0.273811i	$0.911716\pi$
$908$	0	0
$909$	7.12311	0.236259
$910$	0	0
$911$	22.7386	0.753365	0.376682	−	0.926343i	$-0.377065\pi$
$911$	22.7386	0.753365	0.376682	+	0.926343i	$0.377065\pi$
$912$	0	0
$913$	− 9.12311i	− 0.301931i
$914$	0	0
$915$	0	0
$916$	0	0
$917$	− 58.2462i	− 1.92346i
$918$	0	0
$919$	−31.3693	−1.03478	−0.517389	−	0.855750i	$-0.673096\pi$
$919$	−31.3693	−1.03478	−0.517389	+	0.855750i	$0.673096\pi$
$920$	0	0
$921$	1.26137	0.0415634
$922$	0	0
$923$	11.5076i	0.378777i
$924$	0	0
$925$	0	0
$926$	0	0
$927$	0	0
$928$	0	0
$929$	−13.5076	−0.443169	−0.221585	−	0.975141i	$-0.571123\pi$
$929$	−13.5076	−0.443169	−0.221585	+	0.975141i	$0.571123\pi$
$930$	0	0
$931$	−76.9848	−2.52308
$932$	0	0
$933$	67.2311i	2.20105i
$934$	0	0
$935$	0	0
$936$	0	0
$937$	− 32.1080i	− 1.04892i	−0.851435	−	0.524461i	$-0.824267\pi$
$937$	− 32.1080i	− 1.04892i	0.851435	−	0.524461i	$-0.175733\pi$
$938$	0	0
$939$	−27.6847	−0.903455
$940$	0	0
$941$	−15.7538	−0.513559	−0.256779	−	0.966470i	$-0.582661\pi$
$941$	−15.7538	−0.513559	−0.256779	+	0.966470i	$0.582661\pi$
$942$	0	0
$943$	46.7386i	1.52202i
$944$	0	0
$945$	0	0
$946$	0	0
$947$	0.315342i	0.0102472i	0.999987	+	0.00512361i	$0.00163090\pi$
$947$	0.315342i	0.0102472i	−0.999987	+	0.00512361i	$0.998369\pi$
$948$	0	0
$949$	9.75379	0.316621
$950$	0	0
$951$	−89.1619	−2.89127
$952$	0	0
$953$	− 16.2462i	− 0.526266i	−0.964760	−	0.263133i	$-0.915244\pi$
$953$	− 16.2462i	− 0.526266i	0.964760	−	0.263133i	$-0.0847559\pi$
$954$	0	0
$955$	0	0
$956$	0	0
$957$	8.00000i	0.258603i
$958$	0	0
$959$	−64.3542	−2.07810
$960$	0	0
$961$	−28.9309	−0.933254
$962$	0	0
$963$	40.4924i	1.30485i
$964$	0	0
$965$	0	0
$966$	0	0
$967$	− 8.00000i	− 0.257263i	−0.991692	−	0.128631i	$-0.958942\pi$
$967$	− 8.00000i	− 0.257263i	0.991692	−	0.128631i	$-0.0410584\pi$
$968$	0	0
$969$	20.4924	0.658311
$970$	0	0
$971$	35.5464	1.14074	0.570369	−	0.821389i	$-0.306800\pi$
$971$	35.5464	1.14074	0.570369	+	0.821389i	$0.306800\pi$
$972$	0	0
$973$	− 35.2311i	− 1.12946i
$974$	0	0
$975$	0	0
$976$	0	0
$977$	− 33.1922i	− 1.06191i	−0.847399	−	0.530957i	$-0.821833\pi$
$977$	− 33.1922i	− 1.06191i	0.847399	−	0.530957i	$-0.178167\pi$
$978$	0	0
$979$	−9.68466	−0.309523
$980$	0	0
$981$	−15.1231	−0.482844
$982$	0	0
$983$	2.06913i	0.0659950i	0.999455	+	0.0329975i	$0.0105053\pi$
$983$	2.06913i	0.0659950i	−0.999455	+	0.0329975i	$0.989495\pi$
$984$	0	0
$985$	0	0
$986$	0	0
$987$	104.985i	3.34170i
$988$	0	0
$989$	−7.36932	−0.234331
$990$	0	0
$991$	28.4924	0.905092	0.452546	−	0.891741i	$-0.350516\pi$
$991$	28.4924	0.905092	0.452546	+	0.891741i	$0.350516\pi$
$992$	0	0
$993$	15.5464i	0.493350i
$994$	0	0
$995$	0	0
$996$	0	0
$997$	− 35.7538i	− 1.13233i	−0.824291	−	0.566167i	$-0.808426\pi$
$997$	− 35.7538i	− 1.13233i	0.824291	−	0.566167i	$-0.191574\pi$
$998$	0	0
$999$	4.94602	0.156485

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	2200.2.b.g.1849.4		4
4.3	odd	2		4400.2.b.v.4049.1		4
5.2	odd	4		88.2.a.b.1.2	✓	2
5.3	odd	4		2200.2.a.o.1.1		2
5.4	even	2	inner	2200.2.b.g.1849.1		4
15.2	even	4		792.2.a.h.1.2		2
20.3	even	4		4400.2.a.bp.1.2		2
20.7	even	4		176.2.a.d.1.1		2
20.19	odd	2		4400.2.b.v.4049.4		4
35.27	even	4		4312.2.a.n.1.1		2
40.27	even	4		704.2.a.p.1.2		2
40.37	odd	4		704.2.a.m.1.1		2
55.2	even	20		968.2.i.q.81.2		8
55.7	even	20		968.2.i.q.753.1		8
55.17	even	20		968.2.i.q.729.2		8
55.27	odd	20		968.2.i.r.729.2		8
55.32	even	4		968.2.a.j.1.2		2
55.37	odd	20		968.2.i.r.753.1		8
55.42	odd	20		968.2.i.r.81.2		8
55.47	odd	20		968.2.i.r.9.1		8
55.52	even	20		968.2.i.q.9.1		8
60.47	odd	4		1584.2.a.t.1.2		2
80.27	even	4		2816.2.c.p.1409.4		4
80.37	odd	4		2816.2.c.w.1409.1		4
80.67	even	4		2816.2.c.p.1409.1		4
80.77	odd	4		2816.2.c.w.1409.4		4
120.77	even	4		6336.2.a.cu.1.1		2
120.107	odd	4		6336.2.a.cx.1.1		2
140.27	odd	4		8624.2.a.cb.1.2		2
165.32	odd	4		8712.2.a.bb.1.2		2
220.87	odd	4		1936.2.a.r.1.1		2
440.197	even	4		7744.2.a.by.1.1		2
440.307	odd	4		7744.2.a.cl.1.2		2

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
88.2.a.b.1.2	✓	2	5.2	odd	4
176.2.a.d.1.1		2	20.7	even	4
704.2.a.m.1.1		2	40.37	odd	4
704.2.a.p.1.2		2	40.27	even	4
792.2.a.h.1.2		2	15.2	even	4
968.2.a.j.1.2		2	55.32	even	4
968.2.i.q.9.1		8	55.52	even	20
968.2.i.q.81.2		8	55.2	even	20
968.2.i.q.729.2		8	55.17	even	20
968.2.i.q.753.1		8	55.7	even	20
968.2.i.r.9.1		8	55.47	odd	20
968.2.i.r.81.2		8	55.42	odd	20
968.2.i.r.729.2		8	55.27	odd	20
968.2.i.r.753.1		8	55.37	odd	20
1584.2.a.t.1.2		2	60.47	odd	4
1936.2.a.r.1.1		2	220.87	odd	4
2200.2.a.o.1.1		2	5.3	odd	4
2200.2.b.g.1849.1		4	5.4	even	2	inner
2200.2.b.g.1849.4		4	1.1	even	1	trivial
2816.2.c.p.1409.1		4	80.67	even	4
2816.2.c.p.1409.4		4	80.27	even	4
2816.2.c.w.1409.1		4	80.37	odd	4
2816.2.c.w.1409.4		4	80.77	odd	4
4312.2.a.n.1.1		2	35.27	even	4
4400.2.a.bp.1.2		2	20.3	even	4
4400.2.b.v.4049.1		4	4.3	odd	2
4400.2.b.v.4049.4		4	20.19	odd	2
6336.2.a.cu.1.1		2	120.77	even	4
6336.2.a.cx.1.1		2	120.107	odd	4
7744.2.a.by.1.1		2	440.197	even	4
7744.2.a.cl.1.2		2	440.307	odd	4
8624.2.a.cb.1.2		2	140.27	odd	4
8712.2.a.bb.1.2		2	165.32	odd	4

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:	\( N \)	\(=\)	\( 2200 = 2^{3} \cdot 5^{2} \cdot 11 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	2200.b (of order \(2\), degree \(1\), not minimal)

\(f(q)\)	\(=\)	\(q+2.56155i q^{3} +5.12311i q^{7} -3.56155 q^{9} -1.00000 q^{11} +3.12311i q^{13} -2.00000i q^{17} +4.00000 q^{19} -13.1231 q^{21} +6.56155i q^{23} -1.43845i q^{27} -3.12311 q^{29} -1.43845 q^{31} -2.56155i q^{33} +3.43845i q^{37} -8.00000 q^{39} +7.12311 q^{41} +1.12311i q^{43} -8.00000i q^{47} -19.2462 q^{49} +5.12311 q^{51} -4.24621i q^{53} +10.2462i q^{57} +12.8078 q^{59} -7.12311 q^{61} -18.2462i q^{63} -5.43845i q^{67} -16.8078 q^{69} +3.68466 q^{71} -3.12311i q^{73} -5.12311i q^{77} +2.87689 q^{79} -7.00000 q^{81} +9.12311i q^{83} -8.00000i q^{87} +9.68466 q^{89} -16.0000 q^{91} -3.68466i q^{93} -11.4384i q^{97} +3.56155 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 4 q - 6 q^{9} - 4 q^{11} + 16 q^{19} - 36 q^{21} + 4 q^{29} - 14 q^{31} - 32 q^{39} + 12 q^{41} - 44 q^{49} + 4 q^{51} + 10 q^{59} - 12 q^{61} - 26 q^{69} - 10 q^{71} + 28 q^{79} - 28 q^{81} + 14 q^{89}+ \cdots + 6 q^{99}+O(q^{100}) \) 4 * q - 6 * q^9 - 4 * q^11 + 16 * q^19 - 36 * q^21 + 4 * q^29 - 14 * q^31 - 32 * q^39 + 12 * q^41 - 44 * q^49 + 4 * q^51 + 10 * q^59 - 12 * q^61 - 26 * q^69 - 10 * q^71 + 28 * q^79 - 28 * q^81 + 14 * q^89 - 64 * q^91 + 6 * q^99

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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