LMFDB - Embedded newform 1100.6.b.b.749.2

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [1100,6,Mod(749,1100)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(1100, base_ring=CyclotomicField(2))

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

N = Newforms(chi, 6, names="a")

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

chi := DirichletCharacter("1100.749");

S:= CuspForms(chi, 6);

N := Newforms(S);

Level:	$ N $	$=$	$ 1100 = 2^{2} \cdot 5^{2} \cdot 11 $
Weight:	$ k $	$=$	$ 6 $
Character orbit:	$[\chi]$	$=$	1100.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:	$176.422201794$
Analytic rank:	$0$
Dimension:	$4$
Coefficient field:	$\Q(i, \sqrt{1761})$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{4} + 881x^{2} + 193600 $ x^4 + 881*x^2 + 193600
Coefficient ring:	$\Z[a_1, \ldots, a_{7}]$
Coefficient ring index:	$ 2 $
Twist minimal:	no (minimal twist has level 220)
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label			749.2
Root			$-20.4821i$ of defining polynomial
Character	$\chi$	$=$	1100.749
Dual form			1100.6.b.b.749.3

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-11.4821i q^{3} +39.4107i q^{7} +111.161 q^{9} -121.000 q^{11} +1182.68i q^{13} -2195.77i q^{17} -1866.37 q^{19} +452.519 q^{21} -337.644i q^{23} -4066.52i q^{27} +5432.02 q^{29} +139.481 q^{31} +1389.34i q^{33} +5943.41i q^{37} +13579.7 q^{39} +4630.43 q^{41} +1155.36i q^{43} +3413.28i q^{47} +15253.8 q^{49} -25212.1 q^{51} -5651.62i q^{53} +21430.0i q^{57} -13627.0 q^{59} -34279.1 q^{61} +4380.91i q^{63} -9963.03i q^{67} -3876.87 q^{69} -19869.0 q^{71} +51656.5i q^{73} -4768.69i q^{77} -50960.7 q^{79} -19680.3 q^{81} +8628.02i q^{83} -62371.2i q^{87} +29938.4 q^{89} -46610.2 q^{91} -1601.54i q^{93} +48871.1i q^{97} -13450.4 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 4 q - 1150 q^{9} - 484 q^{11} - 6878 q^{19} + 11294 q^{21} + 10062 q^{29} - 8926 q^{31} + 1108 q^{39} + 39336 q^{41} + 6042 q^{49} - 164718 q^{51} - 101676 q^{59} - 88354 q^{61} + 248196 q^{69} - 2178 q^{71}+ \cdots + 139150 q^{99}+O(q^{100}) $ 4 * q - 1150 * q^9 - 484 * q^11 - 6878 * q^19 + 11294 * q^21 + 10062 * q^29 - 8926 * q^31 + 1108 * q^39 + 39336 * q^41 + 6042 * q^49 - 164718 * q^51 - 101676 * q^59 - 88354 * q^61 + 248196 * q^69 - 2178 * q^71 + 32668 * q^79 + 450700 * q^81 + 161634 * q^89 + 52420 * q^91 + 139150 * q^99

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/1100\mathbb{Z}\right)^\times$.

$n$	$101$	$177$	$551$
$\chi(n)$	$1$	$-1$	$1$

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 $

$2$	0	0
$2$	0	0
$3$	− 11.4821i	− 0.736579i	−0.929711	−	0.368290i	$-0.879943\pi$
$3$	− 11.4821i	− 0.736579i	0.929711	−	0.368290i	$-0.120057\pi$
$4$	0	0
$5$	0	0
$5$	0	0
$6$	0	0
$7$	39.4107i	0.303997i	0.988381	+	0.151998i	$0.0485708\pi$
$7$	39.4107i	0.303997i	−0.988381	+	0.151998i	$0.951429\pi$
$8$	0	0
$9$	111.161	0.457451
$10$	0	0
$11$	−121.000	−0.301511
$11$	−121.000	−0.301511
$12$	0	0
$13$	1182.68i	1.94092i	0.241256	+	0.970461i	$0.422441\pi$
$13$	1182.68i	1.94092i	−0.241256	+	0.970461i	$0.577559\pi$
$14$	0	0
$15$	0	0
$16$	0	0
$17$	− 2195.77i	− 1.84274i	−0.388687	−	0.921370i	$-0.627071\pi$
$17$	− 2195.77i	− 1.84274i	0.388687	−	0.921370i	$-0.372929\pi$
$18$	0	0
$19$	−1866.37	−1.18608	−0.593041	−	0.805172i	$-0.702073\pi$
$19$	−1866.37	−1.18608	−0.593041	+	0.805172i	$0.702073\pi$
$20$	0	0
$21$	452.519	0.223918
$22$	0	0
$23$	− 337.644i	− 0.133088i	−0.997783	−	0.0665440i	$-0.978803\pi$
$23$	− 337.644i	− 0.133088i	0.997783	−	0.0665440i	$-0.0211973\pi$
$24$	0	0
$25$	0	0
$26$	0	0
$27$	− 4066.52i	− 1.07353i
$28$	0	0
$29$	5432.02	1.19941	0.599703	−	0.800223i	$-0.295285\pi$
$29$	5432.02	1.19941	0.599703	+	0.800223i	$0.295285\pi$
$30$	0	0
$31$	139.481	0.0260682	0.0130341	−	0.999915i	$-0.495851\pi$
$31$	139.481	0.0260682	0.0130341	+	0.999915i	$0.495851\pi$
$32$	0	0
$33$	1389.34i	0.222087i
$34$	0	0
$35$	0	0
$36$	0	0
$37$	5943.41i	0.713726i	0.934157	+	0.356863i	$0.116154\pi$
$37$	5943.41i	0.713726i	−0.934157	+	0.356863i	$0.883846\pi$
$38$	0	0
$39$	13579.7	1.42964
$40$	0	0
$41$	4630.43	0.430191	0.215096	−	0.976593i	$-0.430994\pi$
$41$	4630.43	0.430191	0.215096	+	0.976593i	$0.430994\pi$
$42$	0	0
$43$	1155.36i	0.0952897i	0.998864	+	0.0476449i	$0.0151716\pi$
$43$	1155.36i	0.0952897i	−0.998864	+	0.0476449i	$0.984828\pi$
$44$	0	0
$45$	0	0
$46$	0	0
$47$	3413.28i	0.225386i	0.993630	+	0.112693i	$0.0359477\pi$
$47$	3413.28i	0.225386i	−0.993630	+	0.112693i	$0.964052\pi$
$48$	0	0
$49$	15253.8	0.907586
$50$	0	0
$51$	−25212.1	−1.35732
$52$	0	0
$53$	− 5651.62i	− 0.276365i	−0.990407	−	0.138183i	$-0.955874\pi$
$53$	− 5651.62i	− 0.276365i	0.990407	−	0.138183i	$-0.0441261\pi$
$54$	0	0
$55$	0	0
$56$	0	0
$57$	21430.0i	0.873644i
$58$	0	0
$59$	−13627.0	−0.509649	−0.254825	−	0.966987i	$-0.582018\pi$
$59$	−13627.0	−0.509649	−0.254825	+	0.966987i	$0.582018\pi$
$60$	0	0
$61$	−34279.1	−1.17952	−0.589760	−	0.807579i	$-0.700778\pi$
$61$	−34279.1	−1.17952	−0.589760	+	0.807579i	$0.700778\pi$
$62$	0	0
$63$	4380.91i	0.139064i
$64$	0	0
$65$	0	0
$66$	0	0
$67$	− 9963.03i	− 0.271147i	−0.990767	−	0.135573i	$-0.956712\pi$
$67$	− 9963.03i	− 0.271147i	0.990767	−	0.135573i	$-0.0432877\pi$
$68$	0	0
$69$	−3876.87	−0.0980299
$70$	0	0
$71$	−19869.0	−0.467769	−0.233884	−	0.972264i	$-0.575144\pi$
$71$	−19869.0	−0.467769	−0.233884	+	0.972264i	$0.575144\pi$
$72$	0	0
$73$	51656.5i	1.13453i	0.823534	+	0.567267i	$0.191999\pi$
$73$	51656.5i	1.13453i	−0.823534	+	0.567267i	$0.808001\pi$
$74$	0	0
$75$	0	0
$76$	0	0
$77$	− 4768.69i	− 0.0916584i
$78$	0	0
$79$	−50960.7	−0.918686	−0.459343	−	0.888259i	$-0.651915\pi$
$79$	−50960.7	−0.918686	−0.459343	+	0.888259i	$0.651915\pi$
$80$	0	0
$81$	−19680.3	−0.333288
$82$	0	0
$83$	8628.02i	0.137472i	0.997635	+	0.0687362i	$0.0218967\pi$
$83$	8628.02i	0.137472i	−0.997635	+	0.0687362i	$0.978103\pi$
$84$	0	0
$85$	0	0
$86$	0	0
$87$	− 62371.2i	− 0.883457i
$88$	0	0
$89$	29938.4	0.400640	0.200320	−	0.979731i	$-0.435802\pi$
$89$	29938.4	0.400640	0.200320	+	0.979731i	$0.435802\pi$
$90$	0	0
$91$	−46610.2	−0.590034
$92$	0	0
$93$	− 1601.54i	− 0.0192013i
$94$	0	0
$95$	0	0
$96$	0	0
$97$	48871.1i	0.527379i	0.964608	+	0.263690i	$0.0849394\pi$
$97$	48871.1i	0.527379i	−0.964608	+	0.263690i	$0.915061\pi$
$98$	0	0
$99$	−13450.4	−0.137927
$100$	0	0
$101$	−46237.3	−0.451013	−0.225506	−	0.974242i	$-0.572404\pi$
$101$	−46237.3	−0.451013	−0.225506	+	0.974242i	$0.572404\pi$
$102$	0	0
$103$	156417.i	1.45275i	0.687297	+	0.726376i	$0.258797\pi$
$103$	156417.i	1.45275i	−0.687297	+	0.726376i	$0.741203\pi$
$104$	0	0
$105$	0	0
$106$	0	0
$107$	− 144650.i	− 1.22140i	−0.791861	−	0.610702i	$-0.790887\pi$
$107$	− 144650.i	− 1.22140i	0.791861	−	0.610702i	$-0.209113\pi$
$108$	0	0
$109$	15491.8	0.124892	0.0624462	−	0.998048i	$-0.480110\pi$
$109$	15491.8	0.124892	0.0624462	+	0.998048i	$0.480110\pi$
$110$	0	0
$111$	68243.0	0.525715
$112$	0	0
$113$	− 7502.99i	− 0.0552762i	−0.999618	−	0.0276381i	$-0.991201\pi$
$113$	− 7502.99i	− 0.0552762i	0.999618	−	0.0276381i	$-0.00879860\pi$
$114$	0	0
$115$	0	0
$116$	0	0
$117$	131467.i	0.887877i
$118$	0	0
$119$	86536.7	0.560187
$120$	0	0
$121$	14641.0	0.0909091
$122$	0	0
$123$	− 53167.2i	− 0.316870i
$124$	0	0
$125$	0	0
$126$	0	0
$127$	282664.i	1.55511i	0.628813	+	0.777556i	$0.283541\pi$
$127$	282664.i	1.55511i	−0.628813	+	0.777556i	$0.716459\pi$
$128$	0	0
$129$	13266.0	0.0701884
$130$	0	0
$131$	148305.	0.755052	0.377526	−	0.925999i	$-0.376775\pi$
$131$	148305.	0.755052	0.377526	+	0.925999i	$0.376775\pi$
$132$	0	0
$133$	− 73555.1i	− 0.360565i
$134$	0	0
$135$	0	0
$136$	0	0
$137$	94749.1i	0.431294i	0.976471	+	0.215647i	$0.0691861\pi$
$137$	94749.1i	0.431294i	−0.976471	+	0.215647i	$0.930814\pi$
$138$	0	0
$139$	−379277.	−1.66502	−0.832511	−	0.554009i	$-0.813097\pi$
$139$	−379277.	−1.66502	−0.832511	+	0.554009i	$0.813097\pi$
$140$	0	0
$141$	39191.7	0.166015
$142$	0	0
$143$	− 143104.i	− 0.585210i
$144$	0	0
$145$	0	0
$146$	0	0
$147$	− 175146.i	− 0.668509i
$148$	0	0
$149$	346333.	1.27799	0.638996	−	0.769210i	$-0.279350\pi$
$149$	346333.	1.27799	0.638996	+	0.769210i	$0.279350\pi$
$150$	0	0
$151$	−65850.9	−0.235028	−0.117514	−	0.993071i	$-0.537492\pi$
$151$	−65850.9	−0.235028	−0.117514	+	0.993071i	$0.537492\pi$
$152$	0	0
$153$	− 244083.i	− 0.842963i
$154$	0	0
$155$	0	0
$156$	0	0
$157$	367291.i	1.18922i	0.804015	+	0.594609i	$0.202693\pi$
$157$	367291.i	1.18922i	−0.804015	+	0.594609i	$0.797307\pi$
$158$	0	0
$159$	−64892.7	−0.203565
$160$	0	0
$161$	13306.8	0.0404583
$162$	0	0
$163$	481404.i	1.41919i	0.704609	+	0.709596i	$0.251122\pi$
$163$	481404.i	1.41919i	−0.704609	+	0.709596i	$0.748878\pi$
$164$	0	0
$165$	0	0
$166$	0	0
$167$	600777.i	1.66695i	0.552558	+	0.833475i	$0.313652\pi$
$167$	600777.i	1.66695i	−0.552558	+	0.833475i	$0.686348\pi$
$168$	0	0
$169$	−1.02744e6	−2.76718
$170$	0	0
$171$	−207467.	−0.542575
$172$	0	0
$173$	− 401433.i	− 1.01976i	−0.860245	−	0.509880i	$-0.829690\pi$
$173$	− 401433.i	− 1.01976i	0.860245	−	0.509880i	$-0.170310\pi$
$174$	0	0
$175$	0	0
$176$	0	0
$177$	156468.i	0.375397i
$178$	0	0
$179$	−547366.	−1.27687	−0.638433	−	0.769677i	$-0.720417\pi$
$179$	−547366.	−1.27687	−0.638433	+	0.769677i	$0.720417\pi$
$180$	0	0
$181$	605472.	1.37372	0.686859	−	0.726791i	$-0.258989\pi$
$181$	605472.	1.37372	0.686859	+	0.726791i	$0.258989\pi$
$182$	0	0
$183$	393597.i	0.868810i
$184$	0	0
$185$	0	0
$186$	0	0
$187$	265688.i	0.555607i
$188$	0	0
$189$	160264.	0.326349
$190$	0	0
$191$	49593.8	0.0983657	0.0491829	−	0.998790i	$-0.484338\pi$
$191$	49593.8	0.0983657	0.0491829	+	0.998790i	$0.484338\pi$
$192$	0	0
$193$	899272.i	1.73779i	0.494994	+	0.868896i	$0.335170\pi$
$193$	899272.i	1.73779i	−0.494994	+	0.868896i	$0.664830\pi$
$194$	0	0
$195$	0	0
$196$	0	0
$197$	− 279635.i	− 0.513365i	−0.966496	−	0.256683i	$-0.917371\pi$
$197$	− 279635.i	− 0.513365i	0.966496	−	0.256683i	$-0.0826295\pi$
$198$	0	0
$199$	822226.	1.47183	0.735917	−	0.677072i	$-0.236752\pi$
$199$	822226.	1.47183	0.735917	+	0.677072i	$0.236752\pi$
$200$	0	0
$201$	−114397.	−0.199721
$202$	0	0
$203$	214079.i	0.364615i
$204$	0	0
$205$	0	0
$206$	0	0
$207$	− 37532.7i	− 0.0608812i
$208$	0	0
$209$	225831.	0.357617
$210$	0	0
$211$	−534284.	−0.826164	−0.413082	−	0.910694i	$-0.635548\pi$
$211$	−534284.	−0.826164	−0.413082	+	0.910694i	$0.635548\pi$
$212$	0	0
$213$	228139.i	0.344549i
$214$	0	0
$215$	0	0
$216$	0	0
$217$	5497.05i	0.00792466i
$218$	0	0
$219$	593126.	0.835674
$220$	0	0
$221$	2.59689e6	3.57662
$222$	0	0
$223$	25689.6i	0.0345936i	0.999850	+	0.0172968i	$0.00550601\pi$
$223$	25689.6i	0.0345936i	−0.999850	+	0.0172968i	$0.994494\pi$
$224$	0	0
$225$	0	0
$226$	0	0
$227$	− 519615.i	− 0.669295i	−0.942343	−	0.334647i	$-0.891383\pi$
$227$	− 519615.i	− 0.669295i	0.942343	−	0.334647i	$-0.108617\pi$
$228$	0	0
$229$	−566856.	−0.714305	−0.357153	−	0.934046i	$-0.616252\pi$
$229$	−566856.	−0.714305	−0.357153	+	0.934046i	$0.616252\pi$
$230$	0	0
$231$	−54754.8	−0.0675137
$232$	0	0
$233$	250125.i	0.301834i	0.988546	+	0.150917i	$0.0482226\pi$
$233$	250125.i	0.301834i	−0.988546	+	0.150917i	$0.951777\pi$
$234$	0	0
$235$	0	0
$236$	0	0
$237$	585137.i	0.676685i
$238$	0	0
$239$	1.29750e6	1.46931	0.734654	−	0.678442i	$-0.237344\pi$
$239$	1.29750e6	1.46931	0.734654	+	0.678442i	$0.237344\pi$
$240$	0	0
$241$	582411.	0.645932	0.322966	−	0.946411i	$-0.395320\pi$
$241$	582411.	0.645932	0.322966	+	0.946411i	$0.395320\pi$
$242$	0	0
$243$	− 762192.i	− 0.828035i
$244$	0	0
$245$	0	0
$246$	0	0
$247$	− 2.20732e6i	− 2.30209i
$248$	0	0
$249$	99068.1	0.101259
$250$	0	0
$251$	1.75979e6	1.76310	0.881550	−	0.472091i	$-0.156501\pi$
$251$	1.75979e6	1.76310	0.881550	+	0.472091i	$0.156501\pi$
$252$	0	0
$253$	40854.9i	0.0401275i
$254$	0	0
$255$	0	0
$256$	0	0
$257$	− 539156.i	− 0.509192i	−0.967048	−	0.254596i	$-0.918058\pi$
$257$	− 539156.i	− 0.509192i	0.967048	−	0.254596i	$-0.0819424\pi$
$258$	0	0
$259$	−234234.	−0.216970
$260$	0	0
$261$	603826.	0.548669
$262$	0	0
$263$	2.20351e6i	1.96438i	0.187881	+	0.982192i	$0.439838\pi$
$263$	2.20351e6i	1.96438i	−0.187881	+	0.982192i	$0.560162\pi$
$264$	0	0
$265$	0	0
$266$	0	0
$267$	− 343757.i	− 0.295103i
$268$	0	0
$269$	731509.	0.616366	0.308183	−	0.951327i	$-0.400279\pi$
$269$	731509.	0.616366	0.308183	+	0.951327i	$0.400279\pi$
$270$	0	0
$271$	446543.	0.369352	0.184676	−	0.982799i	$-0.440876\pi$
$271$	446543.	0.369352	0.184676	+	0.982799i	$0.440876\pi$
$272$	0	0
$273$	535184.i	0.434607i
$274$	0	0
$275$	0	0
$276$	0	0
$277$	1.60933e6i	1.26022i	0.776507	+	0.630109i	$0.216990\pi$
$277$	1.60933e6i	1.26022i	−0.776507	+	0.630109i	$0.783010\pi$
$278$	0	0
$279$	15504.8	0.0119249
$280$	0	0
$281$	−836866.	−0.632252	−0.316126	−	0.948717i	$-0.602382\pi$
$281$	−836866.	−0.632252	−0.316126	+	0.948717i	$0.602382\pi$
$282$	0	0
$283$	1.43720e6i	1.06672i	0.845887	+	0.533362i	$0.179072\pi$
$283$	1.43720e6i	1.06672i	−0.845887	+	0.533362i	$0.820928\pi$
$284$	0	0
$285$	0	0
$286$	0	0
$287$	182488.i	0.130777i
$288$	0	0
$289$	−3.40154e6	−2.39569
$290$	0	0
$291$	561145.	0.388456
$292$	0	0
$293$	977430.i	0.665146i	0.943078	+	0.332573i	$0.107917\pi$
$293$	977430.i	0.665146i	−0.943078	+	0.332573i	$0.892083\pi$
$294$	0	0
$295$	0	0
$296$	0	0
$297$	492049.i	0.323681i
$298$	0	0
$299$	399324.	0.258314
$300$	0	0
$301$	−45533.5	−0.0289677
$302$	0	0
$303$	530903.i	0.332207i
$304$	0	0
$305$	0	0
$306$	0	0
$307$	242927.i	0.147106i	0.997291	+	0.0735528i	$0.0234337\pi$
$307$	242927.i	0.147106i	−0.997291	+	0.0735528i	$0.976566\pi$
$308$	0	0
$309$	1.79600e6	1.07007
$310$	0	0
$311$	−1.93118e6	−1.13220	−0.566098	−	0.824338i	$-0.691548\pi$
$311$	−1.93118e6	−1.13220	−0.566098	+	0.824338i	$0.691548\pi$
$312$	0	0
$313$	1.03171e6i	0.595249i	0.954683	+	0.297625i	$0.0961944\pi$
$313$	1.03171e6i	0.595249i	−0.954683	+	0.297625i	$0.903806\pi$
$314$	0	0
$315$	0	0
$316$	0	0
$317$	− 2.05142e6i	− 1.14659i	−0.819351	−	0.573293i	$-0.805666\pi$
$317$	− 2.05142e6i	− 1.14659i	0.819351	−	0.573293i	$-0.194334\pi$
$318$	0	0
$319$	−657274.	−0.361634
$320$	0	0
$321$	−1.66089e6	−0.899661
$322$	0	0
$323$	4.09812e6i	2.18564i
$324$	0	0
$325$	0	0
$326$	0	0
$327$	− 177879.i	− 0.0919931i
$328$	0	0
$329$	−134520.	−0.0685166
$330$	0	0
$331$	1.58685e6	0.796095	0.398047	−	0.917365i	$-0.369688\pi$
$331$	1.58685e6	0.796095	0.398047	+	0.917365i	$0.369688\pi$
$332$	0	0
$333$	660673.i	0.326494i
$334$	0	0
$335$	0	0
$336$	0	0
$337$	2.33349e6i	1.11926i	0.828742	+	0.559630i	$0.189057\pi$
$337$	2.33349e6i	1.11926i	−0.828742	+	0.559630i	$0.810943\pi$
$338$	0	0
$339$	−86150.3	−0.0407153
$340$	0	0
$341$	−16877.2	−0.00785987
$342$	0	0
$343$	1.26354e6i	0.579900i
$344$	0	0
$345$	0	0
$346$	0	0
$347$	− 194423.i	− 0.0866811i	−0.999060	−	0.0433406i	$-0.986200\pi$
$347$	− 194423.i	− 0.0866811i	0.999060	−	0.0433406i	$-0.0138000\pi$
$348$	0	0
$349$	744025.	0.326982	0.163491	−	0.986545i	$-0.447725\pi$
$349$	744025.	0.326982	0.163491	+	0.986545i	$0.447725\pi$
$350$	0	0
$351$	4.80938e6	2.08364
$352$	0	0
$353$	257236.i	0.109874i	0.998490	+	0.0549370i	$0.0174958\pi$
$353$	257236.i	0.109874i	−0.998490	+	0.0549370i	$0.982504\pi$
$354$	0	0
$355$	0	0
$356$	0	0
$357$	− 993626.i	− 0.412622i
$358$	0	0
$359$	1.04798e6	0.429157	0.214579	−	0.976707i	$-0.431162\pi$
$359$	1.04798e6	0.429157	0.214579	+	0.976707i	$0.431162\pi$
$360$	0	0
$361$	1.00726e6	0.406792
$362$	0	0
$363$	− 168110.i	− 0.0669618i
$364$	0	0
$365$	0	0
$366$	0	0
$367$	2.52808e6i	0.979772i	0.871787	+	0.489886i	$0.162962\pi$
$367$	2.52808e6i	0.979772i	−0.871787	+	0.489886i	$0.837038\pi$
$368$	0	0
$369$	514721.	0.196791
$370$	0	0
$371$	222734.	0.0840141
$372$	0	0
$373$	398673.i	0.148370i	0.997245	+	0.0741848i	$0.0236355\pi$
$373$	398673.i	0.148370i	−0.997245	+	0.0741848i	$0.976365\pi$
$374$	0	0
$375$	0	0
$376$	0	0
$377$	6.42433e6i	2.32795i
$378$	0	0
$379$	41188.8	0.0147293	0.00736463	−	0.999973i	$-0.497656\pi$
$379$	41188.8	0.0147293	0.00736463	+	0.999973i	$0.497656\pi$
$380$	0	0
$381$	3.24559e6	1.14546
$382$	0	0
$383$	2.53931e6i	0.884543i	0.896881	+	0.442271i	$0.145827\pi$
$383$	2.53931e6i	0.884543i	−0.896881	+	0.442271i	$0.854173\pi$
$384$	0	0
$385$	0	0
$386$	0	0
$387$	128430.i	0.0435904i
$388$	0	0
$389$	3.99822e6	1.33965	0.669827	−	0.742518i	$-0.266368\pi$
$389$	3.99822e6	1.33965	0.669827	+	0.742518i	$0.266368\pi$
$390$	0	0
$391$	−741387.	−0.245247
$392$	0	0
$393$	− 1.70286e6i	− 0.556156i
$394$	0	0
$395$	0	0
$396$	0	0
$397$	− 5.04861e6i	− 1.60766i	−0.594856	−	0.803832i	$-0.702791\pi$
$397$	− 5.04861e6i	− 1.60766i	0.594856	−	0.803832i	$-0.297209\pi$
$398$	0	0
$399$	−844570.	−0.265585
$400$	0	0
$401$	4.71689e6	1.46485	0.732427	−	0.680845i	$-0.238387\pi$
$401$	4.71689e6	1.46485	0.732427	+	0.680845i	$0.238387\pi$
$402$	0	0
$403$	164962.i	0.0505964i
$404$	0	0
$405$	0	0
$406$	0	0
$407$	− 719152.i	− 0.215196i
$408$	0	0
$409$	1.19007e6	0.351775	0.175887	−	0.984410i	$-0.443721\pi$
$409$	1.19007e6	0.351775	0.175887	+	0.984410i	$0.443721\pi$
$410$	0	0
$411$	1.08792e6	0.317682
$412$	0	0
$413$	− 537051.i	− 0.154932i
$414$	0	0
$415$	0	0
$416$	0	0
$417$	4.35491e6i	1.22642i
$418$	0	0
$419$	−6.85301e6	−1.90698	−0.953491	−	0.301422i	$-0.902539\pi$
$419$	−6.85301e6	−1.90698	−0.953491	+	0.301422i	$0.902539\pi$
$420$	0	0
$421$	6.62822e6	1.82260	0.911300	−	0.411743i	$-0.135080\pi$
$421$	6.62822e6	1.82260	0.911300	+	0.411743i	$0.135080\pi$
$422$	0	0
$423$	379422.i	0.103103i
$424$	0	0
$425$	0	0
$426$	0	0
$427$	− 1.35096e6i	− 0.358570i
$428$	0	0
$429$	−1.64314e6	−0.431054
$430$	0	0
$431$	−1.04575e6	−0.271165	−0.135582	−	0.990766i	$-0.543291\pi$
$431$	−1.04575e6	−0.271165	−0.135582	+	0.990766i	$0.543291\pi$
$432$	0	0
$433$	− 934093.i	− 0.239425i	−0.992809	−	0.119713i	$-0.961803\pi$
$433$	− 934093.i	− 0.239425i	0.992809	−	0.119713i	$-0.0381973\pi$
$434$	0	0
$435$	0	0
$436$	0	0
$437$	630169.i	0.157853i
$438$	0	0
$439$	−3.24978e6	−0.804809	−0.402404	−	0.915462i	$-0.631825\pi$
$439$	−3.24978e6	−0.804809	−0.402404	+	0.915462i	$0.631825\pi$
$440$	0	0
$441$	1.69562e6	0.415176
$442$	0	0
$443$	− 3.11830e6i	− 0.754933i	−0.926023	−	0.377466i	$-0.876795\pi$
$443$	− 3.11830e6i	− 0.754933i	0.926023	−	0.377466i	$-0.123205\pi$
$444$	0	0
$445$	0	0
$446$	0	0
$447$	− 3.97664e6i	− 0.941343i
$448$	0	0
$449$	−3.36184e6	−0.786976	−0.393488	−	0.919330i	$-0.628732\pi$
$449$	−3.36184e6	−0.786976	−0.393488	+	0.919330i	$0.628732\pi$
$450$	0	0
$451$	−560282.	−0.129708
$452$	0	0
$453$	756109.i	0.173117i
$454$	0	0
$455$	0	0
$456$	0	0
$457$	3.98953e6i	0.893575i	0.894640	+	0.446787i	$0.147432\pi$
$457$	3.98953e6i	0.893575i	−0.894640	+	0.446787i	$0.852568\pi$
$458$	0	0
$459$	−8.92913e6	−1.97823
$460$	0	0
$461$	−4.93053e6	−1.08054	−0.540271	−	0.841491i	$-0.681678\pi$
$461$	−4.93053e6	−1.08054	−0.540271	+	0.841491i	$0.681678\pi$
$462$	0	0
$463$	6.61725e6i	1.43458i	0.696774	+	0.717290i	$0.254618\pi$
$463$	6.61725e6i	1.43458i	−0.696774	+	0.717290i	$0.745382\pi$
$464$	0	0
$465$	0	0
$466$	0	0
$467$	3.80218e6i	0.806752i	0.915034	+	0.403376i	$0.132163\pi$
$467$	3.80218e6i	0.806752i	−0.915034	+	0.403376i	$0.867837\pi$
$468$	0	0
$469$	392650.	0.0824277
$470$	0	0
$471$	4.21729e6	0.875954
$472$	0	0
$473$	− 139798.i	− 0.0287309i
$474$	0	0
$475$	0	0
$476$	0	0
$477$	− 628237.i	− 0.126424i
$478$	0	0
$479$	−2.57373e6	−0.512536	−0.256268	−	0.966606i	$-0.582493\pi$
$479$	−2.57373e6	−0.512536	−0.256268	+	0.966606i	$0.582493\pi$
$480$	0	0
$481$	−7.02914e6	−1.38529
$482$	0	0
$483$	− 152790.i	− 0.0298007i
$484$	0	0
$485$	0	0
$486$	0	0
$487$	9.87913e6i	1.88754i	0.330603	+	0.943770i	$0.392748\pi$
$487$	9.87913e6i	1.88754i	−0.330603	+	0.943770i	$0.607252\pi$
$488$	0	0
$489$	5.52755e6	1.04535
$490$	0	0
$491$	−9.45632e6	−1.77018	−0.885092	−	0.465415i	$-0.845905\pi$
$491$	−9.45632e6	−1.77018	−0.885092	+	0.465415i	$0.845905\pi$
$492$	0	0
$493$	− 1.19274e7i	− 2.21019i
$494$	0	0
$495$	0	0
$496$	0	0
$497$	− 783053.i	− 0.142200i
$498$	0	0
$499$	1.64467e6	0.295684	0.147842	−	0.989011i	$-0.452767\pi$
$499$	1.64467e6	0.295684	0.147842	+	0.989011i	$0.452767\pi$
$500$	0	0
$501$	6.89821e6	1.22784
$502$	0	0
$503$	1.04146e7i	1.83536i	0.397315	+	0.917682i	$0.369942\pi$
$503$	1.04146e7i	1.83536i	−0.397315	+	0.917682i	$0.630058\pi$
$504$	0	0
$505$	0	0
$506$	0	0
$507$	1.17972e7i	2.03825i
$508$	0	0
$509$	8.06523e6	1.37982	0.689910	−	0.723896i	$-0.257650\pi$
$509$	8.06523e6	1.37982	0.689910	+	0.723896i	$0.257650\pi$
$510$	0	0
$511$	−2.03582e6	−0.344895
$512$	0	0
$513$	7.58965e6i	1.27329i
$514$	0	0
$515$	0	0
$516$	0	0
$517$	− 413007.i	− 0.0679565i
$518$	0	0
$519$	−4.60931e6	−0.751134
$520$	0	0
$521$	−7.28702e6	−1.17613	−0.588065	−	0.808814i	$-0.700110\pi$
$521$	−7.28702e6	−1.17613	−0.588065	+	0.808814i	$0.700110\pi$
$522$	0	0
$523$	− 5.73656e6i	− 0.917059i	−0.888679	−	0.458530i	$-0.848376\pi$
$523$	− 5.73656e6i	− 0.917059i	0.888679	−	0.458530i	$-0.151624\pi$
$524$	0	0
$525$	0	0
$526$	0	0
$527$	− 306268.i	− 0.0480370i
$528$	0	0
$529$	6.32234e6	0.982288
$530$	0	0
$531$	−1.51479e6	−0.233140
$532$	0	0
$533$	5.47631e6i	0.834968i
$534$	0	0
$535$	0	0
$536$	0	0
$537$	6.28493e6i	0.940513i
$538$	0	0
$539$	−1.84571e6	−0.273647
$540$	0	0
$541$	5.90660e6	0.867650	0.433825	−	0.900997i	$-0.357164\pi$
$541$	5.90660e6	0.867650	0.433825	+	0.900997i	$0.357164\pi$
$542$	0	0
$543$	− 6.95211e6i	− 1.01185i
$544$	0	0
$545$	0	0
$546$	0	0
$547$	− 5.14720e6i	− 0.735534i	−0.929918	−	0.367767i	$-0.880122\pi$
$547$	− 5.14720e6i	− 0.735534i	0.929918	−	0.367767i	$-0.119878\pi$
$548$	0	0
$549$	−3.81049e6	−0.539572
$550$	0	0
$551$	−1.01382e7	−1.42259
$552$	0	0
$553$	− 2.00839e6i	− 0.279278i
$554$	0	0
$555$	0	0
$556$	0	0
$557$	− 9.30363e6i	− 1.27062i	−0.772259	−	0.635308i	$-0.780873\pi$
$557$	− 9.30363e6i	− 1.27062i	0.772259	−	0.635308i	$-0.219127\pi$
$558$	0	0
$559$	−1.36642e6	−0.184950
$560$	0	0
$561$	3.05066e6	0.409249
$562$	0	0
$563$	1.00562e7i	1.33709i	0.743670	+	0.668547i	$0.233083\pi$
$563$	1.00562e7i	1.33709i	−0.743670	+	0.668547i	$0.766917\pi$
$564$	0	0
$565$	0	0
$566$	0	0
$567$	− 775614.i	− 0.101318i
$568$	0	0
$569$	−933127.	−0.120826	−0.0604130	−	0.998173i	$-0.519242\pi$
$569$	−933127.	−0.120826	−0.0604130	+	0.998173i	$0.519242\pi$
$570$	0	0
$571$	1.07508e7	1.37990	0.689952	−	0.723855i	$-0.257632\pi$
$571$	1.07508e7	1.37990	0.689952	+	0.723855i	$0.257632\pi$
$572$	0	0
$573$	− 569442.i	− 0.0724541i
$574$	0	0
$575$	0	0
$576$	0	0
$577$	− 387934.i	− 0.0485086i	−0.999706	−	0.0242543i	$-0.992279\pi$
$577$	− 387934.i	− 0.0485086i	0.999706	−	0.0242543i	$-0.00772114\pi$
$578$	0	0
$579$	1.03256e7	1.28002
$580$	0	0
$581$	−340036.	−0.0417912
$582$	0	0
$583$	683846.i	0.0833273i
$584$	0	0
$585$	0	0
$586$	0	0
$587$	− 1.14459e7i	− 1.37105i	−0.728049	−	0.685525i	$-0.759573\pi$
$587$	− 1.14459e7i	− 1.37105i	0.728049	−	0.685525i	$-0.240427\pi$
$588$	0	0
$589$	−260324.	−0.0309191
$590$	0	0
$591$	−3.21081e6	−0.378134
$592$	0	0
$593$	− 8.28060e6i	− 0.966997i	−0.875345	−	0.483499i	$-0.839366\pi$
$593$	− 8.28060e6i	− 0.966997i	0.875345	−	0.483499i	$-0.160634\pi$
$594$	0	0
$595$	0	0
$596$	0	0
$597$	− 9.44091e6i	− 1.08412i
$598$	0	0
$599$	133370.	0.0151876	0.00759382	−	0.999971i	$-0.497583\pi$
$599$	133370.	0.0151876	0.00759382	+	0.999971i	$0.497583\pi$
$600$	0	0
$601$	−8.78786e6	−0.992424	−0.496212	−	0.868201i	$-0.665276\pi$
$601$	−8.78786e6	−0.992424	−0.496212	+	0.868201i	$0.665276\pi$
$602$	0	0
$603$	− 1.10750e6i	− 0.124036i
$604$	0	0
$605$	0	0
$606$	0	0
$607$	− 1.50026e7i	− 1.65271i	−0.563151	−	0.826354i	$-0.690411\pi$
$607$	− 1.50026e7i	− 1.65271i	0.563151	−	0.826354i	$-0.309589\pi$
$608$	0	0
$609$	2.45809e6	0.268568
$610$	0	0
$611$	−4.03681e6	−0.437457
$612$	0	0
$613$	1.21629e7i	1.30733i	0.756785	+	0.653664i	$0.226769\pi$
$613$	1.21629e7i	1.30733i	−0.756785	+	0.653664i	$0.773231\pi$
$614$	0	0
$615$	0	0
$616$	0	0
$617$	2.77990e6i	0.293979i	0.989138	+	0.146989i	$0.0469583\pi$
$617$	2.77990e6i	0.293979i	−0.989138	+	0.146989i	$0.953042\pi$
$618$	0	0
$619$	2.84183e6	0.298107	0.149053	−	0.988829i	$-0.452377\pi$
$619$	2.84183e6	0.298107	0.149053	+	0.988829i	$0.452377\pi$
$620$	0	0
$621$	−1.37303e6	−0.142874
$622$	0	0
$623$	1.17989e6i	0.121793i
$624$	0	0
$625$	0	0
$626$	0	0
$627$	− 2.59303e6i	− 0.263414i
$628$	0	0
$629$	1.30503e7	1.31521
$630$	0	0
$631$	1.36584e7	1.36561	0.682806	−	0.730599i	$-0.260759\pi$
$631$	1.36584e7	1.36561	0.682806	+	0.730599i	$0.260759\pi$
$632$	0	0
$633$	6.13473e6i	0.608535i
$634$	0	0
$635$	0	0
$636$	0	0
$637$	1.80403e7i	1.76155i
$638$	0	0
$639$	−2.20865e6	−0.213981
$640$	0	0
$641$	1.47678e7	1.41961	0.709807	−	0.704396i	$-0.248782\pi$
$641$	1.47678e7	1.41961	0.709807	+	0.704396i	$0.248782\pi$
$642$	0	0
$643$	− 1.42372e7i	− 1.35799i	−0.734143	−	0.678995i	$-0.762416\pi$
$643$	− 1.42372e7i	− 1.35799i	0.734143	−	0.678995i	$-0.237584\pi$
$644$	0	0
$645$	0	0
$646$	0	0
$647$	− 8.49593e6i	− 0.797903i	−0.916972	−	0.398951i	$-0.869374\pi$
$647$	− 8.49593e6i	− 0.797903i	0.916972	−	0.398951i	$-0.130626\pi$
$648$	0	0
$649$	1.64887e6	0.153665
$650$	0	0
$651$	63117.9	0.00583714
$652$	0	0
$653$	1.41649e7i	1.29996i	0.759952	+	0.649979i	$0.225222\pi$
$653$	1.41649e7i	1.29996i	−0.759952	+	0.649979i	$0.774778\pi$
$654$	0	0
$655$	0	0
$656$	0	0
$657$	5.74216e6i	0.518994i
$658$	0	0
$659$	−1.39876e7	−1.25467	−0.627335	−	0.778749i	$-0.715855\pi$
$659$	−1.39876e7	−1.25467	−0.627335	+	0.778749i	$0.715855\pi$
$660$	0	0
$661$	−1.75134e7	−1.55908	−0.779538	−	0.626355i	$-0.784546\pi$
$661$	−1.75134e7	−1.55908	−0.779538	+	0.626355i	$0.784546\pi$
$662$	0	0
$663$	− 2.98178e7i	− 2.63446i
$664$	0	0
$665$	0	0
$666$	0	0
$667$	− 1.83409e6i	− 0.159626i
$668$	0	0
$669$	294972.	0.0254809
$670$	0	0
$671$	4.14777e6	0.355638
$672$	0	0
$673$	1.13007e7i	0.961758i	0.876787	+	0.480879i	$0.159682\pi$
$673$	1.13007e7i	0.961758i	−0.876787	+	0.480879i	$0.840318\pi$
$674$	0	0
$675$	0	0
$676$	0	0
$677$	− 9.15138e6i	− 0.767388i	−0.923460	−	0.383694i	$-0.874652\pi$
$677$	− 9.15138e6i	− 0.767388i	0.923460	−	0.383694i	$-0.125348\pi$
$678$	0	0
$679$	−1.92604e6	−0.160321
$680$	0	0
$681$	−5.96629e6	−0.492989
$682$	0	0
$683$	3.66856e6i	0.300915i	0.988616	+	0.150458i	$0.0480747\pi$
$683$	3.66856e6i	0.300915i	−0.988616	+	0.150458i	$0.951925\pi$
$684$	0	0
$685$	0	0
$686$	0	0
$687$	6.50871e6i	0.526142i
$688$	0	0
$689$	6.68405e6	0.536404
$690$	0	0
$691$	−7.24875e6	−0.577521	−0.288761	−	0.957401i	$-0.593243\pi$
$691$	−7.24875e6	−0.577521	−0.288761	+	0.957401i	$0.593243\pi$
$692$	0	0
$693$	− 530091.i	− 0.0419292i
$694$	0	0
$695$	0	0
$696$	0	0
$697$	− 1.01673e7i	− 0.792731i
$698$	0	0
$699$	2.87197e6	0.222325
$700$	0	0
$701$	−1.67191e7	−1.28505	−0.642523	−	0.766267i	$-0.722112\pi$
$701$	−1.67191e7	−1.28505	−0.642523	+	0.766267i	$0.722112\pi$
$702$	0	0
$703$	− 1.10926e7i	− 0.846537i
$704$	0	0
$705$	0	0
$706$	0	0
$707$	− 1.82224e6i	− 0.137106i
$708$	0	0
$709$	−9.11227e6	−0.680786	−0.340393	−	0.940283i	$-0.610560\pi$
$709$	−9.11227e6	−0.680786	−0.340393	+	0.940283i	$0.610560\pi$
$710$	0	0
$711$	−5.66482e6	−0.420254
$712$	0	0
$713$	− 47095.0i	− 0.00346937i
$714$	0	0
$715$	0	0
$716$	0	0
$717$	− 1.48981e7i	− 1.08226i
$718$	0	0
$719$	1.02074e7	0.736368	0.368184	−	0.929753i	$-0.379980\pi$
$719$	1.02074e7	0.736368	0.368184	+	0.929753i	$0.379980\pi$
$720$	0	0
$721$	−6.16451e6	−0.441632
$722$	0	0
$723$	− 6.68732e6i	− 0.475780i
$724$	0	0
$725$	0	0
$726$	0	0
$727$	− 3.96570e6i	− 0.278281i	−0.990273	−	0.139140i	$-0.955566\pi$
$727$	− 3.96570e6i	− 0.278281i	0.990273	−	0.139140i	$-0.0444340\pi$
$728$	0	0
$729$	−1.35339e7	−0.943201
$730$	0	0
$731$	2.53690e6	0.175594
$732$	0	0
$733$	3.49709e6i	0.240407i	0.992749	+	0.120203i	$0.0383547\pi$
$733$	3.49709e6i	0.240407i	−0.992749	+	0.120203i	$0.961645\pi$
$734$	0	0
$735$	0	0
$736$	0	0
$737$	1.20553e6i	0.0817539i
$738$	0	0
$739$	1.53549e7	1.03428	0.517138	−	0.855902i	$-0.326997\pi$
$739$	1.53549e7	1.03428	0.517138	+	0.855902i	$0.326997\pi$
$740$	0	0
$741$	−2.53448e7	−1.69568
$742$	0	0
$743$	− 1.99982e7i	− 1.32898i	−0.747296	−	0.664491i	$-0.768648\pi$
$743$	− 1.99982e7i	− 1.32898i	0.747296	−	0.664491i	$-0.231352\pi$
$744$	0	0
$745$	0	0
$746$	0	0
$747$	959095.i	0.0628869i
$748$	0	0
$749$	5.70076e6	0.371303
$750$	0	0
$751$	1.89895e7	1.22861	0.614305	−	0.789069i	$-0.289436\pi$
$751$	1.89895e7	1.22861	0.614305	+	0.789069i	$0.289436\pi$
$752$	0	0
$753$	− 2.02062e7i	− 1.29866i
$754$	0	0
$755$	0	0
$756$	0	0
$757$	2.47833e7i	1.57188i	0.618304	+	0.785939i	$0.287820\pi$
$757$	2.47833e7i	1.57188i	−0.618304	+	0.785939i	$0.712180\pi$
$758$	0	0
$759$	469101.	0.0295571
$760$	0	0
$761$	2.62584e7	1.64364	0.821820	−	0.569747i	$-0.192959\pi$
$761$	2.62584e7	1.64364	0.821820	+	0.569747i	$0.192959\pi$
$762$	0	0
$763$	610543.i	0.0379669i
$764$	0	0
$765$	0	0
$766$	0	0
$767$	− 1.61164e7i	− 0.989190i
$768$	0	0
$769$	−1.92359e7	−1.17300	−0.586500	−	0.809950i	$-0.699494\pi$
$769$	−1.92359e7	−1.17300	−0.586500	+	0.809950i	$0.699494\pi$
$770$	0	0
$771$	−6.19066e6	−0.375060
$772$	0	0
$773$	7.49718e6i	0.451283i	0.974210	+	0.225642i	$0.0724478\pi$
$773$	7.49718e6i	0.451283i	−0.974210	+	0.225642i	$0.927552\pi$
$774$	0	0
$775$	0	0
$776$	0	0
$777$	2.68950e6i	0.159816i
$778$	0	0
$779$	−8.64212e6	−0.510242
$780$	0	0
$781$	2.40415e6	0.141038
$782$	0	0
$783$	− 2.20894e7i	− 1.28760i
$784$	0	0
$785$	0	0
$786$	0	0
$787$	1.48570e7i	0.855053i	0.904003	+	0.427526i	$0.140615\pi$
$787$	1.48570e7i	0.855053i	−0.904003	+	0.427526i	$0.859385\pi$
$788$	0	0
$789$	2.53010e7	1.44692
$790$	0	0
$791$	295698.	0.0168038
$792$	0	0
$793$	− 4.05412e7i	− 2.28936i
$794$	0	0
$795$	0	0
$796$	0	0
$797$	− 2.89236e7i	− 1.61290i	−0.591303	−	0.806450i	$-0.701386\pi$
$797$	− 2.89236e7i	− 1.61290i	0.591303	−	0.806450i	$-0.298614\pi$
$798$	0	0
$799$	7.49477e6	0.415328
$800$	0	0
$801$	3.32797e6	0.183273
$802$	0	0
$803$	− 6.25043e6i	− 0.342075i
$804$	0	0
$805$	0	0
$806$	0	0
$807$	− 8.39928e6i	− 0.454003i
$808$	0	0
$809$	1.97152e7	1.05908	0.529541	−	0.848285i	$-0.322364\pi$
$809$	1.97152e7	1.05908	0.529541	+	0.848285i	$0.322364\pi$
$810$	0	0
$811$	−6.00642e6	−0.320674	−0.160337	−	0.987062i	$-0.551258\pi$
$811$	−6.00642e6	−0.320674	−0.160337	+	0.987062i	$0.551258\pi$
$812$	0	0
$813$	− 5.12727e6i	− 0.272057i
$814$	0	0
$815$	0	0
$816$	0	0
$817$	− 2.15633e6i	− 0.113021i
$818$	0	0
$819$	−5.18121e6	−0.269912
$820$	0	0
$821$	−3.12629e7	−1.61872	−0.809361	−	0.587312i	$-0.800186\pi$
$821$	−3.12629e7	−1.61872	−0.809361	+	0.587312i	$0.800186\pi$
$822$	0	0
$823$	− 1.43689e7i	− 0.739474i	−0.929137	−	0.369737i	$-0.879448\pi$
$823$	− 1.43689e7i	− 0.739474i	0.929137	−	0.369737i	$-0.120552\pi$
$824$	0	0
$825$	0	0
$826$	0	0
$827$	− 3.13803e7i	− 1.59549i	−0.602997	−	0.797744i	$-0.706027\pi$
$827$	− 3.13803e7i	− 1.59549i	0.602997	−	0.797744i	$-0.293973\pi$
$828$	0	0
$829$	−1.74548e6	−0.0882121	−0.0441060	−	0.999027i	$-0.514044\pi$
$829$	−1.74548e6	−0.0882121	−0.0441060	+	0.999027i	$0.514044\pi$
$830$	0	0
$831$	1.84785e7	0.928251
$832$	0	0
$833$	− 3.34938e7i	− 1.67244i
$834$	0	0
$835$	0	0
$836$	0	0
$837$	− 567203.i	− 0.0279850i
$838$	0	0
$839$	−3.04376e7	−1.49281	−0.746406	−	0.665491i	$-0.768222\pi$
$839$	−3.04376e7	−1.49281	−0.746406	+	0.665491i	$0.768222\pi$
$840$	0	0
$841$	8.99566e6	0.438574
$842$	0	0
$843$	9.60901e6i	0.465704i
$844$	0	0
$845$	0	0
$846$	0	0
$847$	577012.i	0.0276361i
$848$	0	0
$849$	1.65022e7	0.785726
$850$	0	0
$851$	2.00675e6	0.0949883
$852$	0	0
$853$	2.05924e6i	0.0969022i	0.998826	+	0.0484511i	$0.0154285\pi$
$853$	2.05924e6i	0.0969022i	−0.998826	+	0.0484511i	$0.984571\pi$
$854$	0	0
$855$	0	0
$856$	0	0
$857$	2.90177e7i	1.34962i	0.737993	+	0.674809i	$0.235774\pi$
$857$	2.90177e7i	1.34962i	−0.737993	+	0.674809i	$0.764226\pi$
$858$	0	0
$859$	−4.14336e7	−1.91589	−0.957943	−	0.286958i	$-0.907356\pi$
$859$	−4.14336e7	−1.91589	−0.957943	+	0.286958i	$0.907356\pi$
$860$	0	0
$861$	2.09536e6	0.0963274
$862$	0	0
$863$	3.45945e7i	1.58117i	0.612349	+	0.790587i	$0.290225\pi$
$863$	3.45945e7i	1.58117i	−0.612349	+	0.790587i	$0.709775\pi$
$864$	0	0
$865$	0	0
$866$	0	0
$867$	3.90569e7i	1.76462i
$868$	0	0
$869$	6.16624e6	0.276994
$870$	0	0
$871$	1.17831e7	0.526275
$872$	0	0
$873$	5.43254e6i	0.241250i
$874$	0	0
$875$	0	0
$876$	0	0
$877$	7.03277e6i	0.308764i	0.988011	+	0.154382i	$0.0493387\pi$
$877$	7.03277e6i	0.308764i	−0.988011	+	0.154382i	$0.950661\pi$
$878$	0	0
$879$	1.12230e7	0.489933
$880$	0	0
$881$	1.30800e7	0.567762	0.283881	−	0.958860i	$-0.408378\pi$
$881$	1.30800e7	0.567762	0.283881	+	0.958860i	$0.408378\pi$
$882$	0	0
$883$	− 1.87695e6i	− 0.0810125i	−0.999179	−	0.0405062i	$-0.987103\pi$
$883$	− 1.87695e6i	− 0.0810125i	0.999179	−	0.0405062i	$-0.0128971\pi$
$884$	0	0
$885$	0	0
$886$	0	0
$887$	− 5.77221e6i	− 0.246339i	−0.992386	−	0.123170i	$-0.960694\pi$
$887$	− 5.77221e6i	− 0.246339i	0.992386	−	0.123170i	$-0.0393059\pi$
$888$	0	0
$889$	−1.11400e7	−0.472749
$890$	0	0
$891$	2.38132e6	0.100490
$892$	0	0
$893$	− 6.37046e6i	− 0.267327i
$894$	0	0
$895$	0	0
$896$	0	0
$897$	− 4.58509e6i	− 0.190268i
$898$	0	0
$899$	757665.	0.0312664
$900$	0	0
$901$	−1.24096e7	−0.509269
$902$	0	0
$903$	522822.i	0.0213370i
$904$	0	0
$905$	0	0
$906$	0	0
$907$	− 3.37177e7i	− 1.36094i	−0.732774	−	0.680472i	$-0.761775\pi$
$907$	− 3.37177e7i	− 1.36094i	0.732774	−	0.680472i	$-0.238225\pi$
$908$	0	0
$909$	−5.13976e6	−0.206316
$910$	0	0
$911$	−1.06739e6	−0.0426116	−0.0213058	−	0.999773i	$-0.506782\pi$
$911$	−1.06739e6	−0.0426116	−0.0213058	+	0.999773i	$0.506782\pi$
$912$	0	0
$913$	− 1.04399e6i	− 0.0414495i
$914$	0	0
$915$	0	0
$916$	0	0
$917$	5.84479e6i	0.229533i
$918$	0	0
$919$	−3.23736e7	−1.26445	−0.632225	−	0.774785i	$-0.717858\pi$
$919$	−3.23736e7	−1.26445	−0.632225	+	0.774785i	$0.717858\pi$
$920$	0	0
$921$	2.78932e6	0.108355
$922$	0	0
$923$	− 2.34987e7i	− 0.907903i
$924$	0	0
$925$	0	0
$926$	0	0
$927$	1.73874e7i	0.664563i
$928$	0	0
$929$	4.75549e7	1.80782	0.903911	−	0.427721i	$-0.140683\pi$
$929$	4.75549e7	1.80782	0.903911	+	0.427721i	$0.140683\pi$
$930$	0	0
$931$	−2.84693e7	−1.07647
$932$	0	0
$933$	2.21740e7i	0.833952i
$934$	0	0
$935$	0	0
$936$	0	0
$937$	− 5.71529e6i	− 0.212662i	−0.994331	−	0.106331i	$-0.966090\pi$
$937$	− 5.71529e6i	− 0.212662i	0.994331	−	0.106331i	$-0.0339103\pi$
$938$	0	0
$939$	1.18463e7	0.438448
$940$	0	0
$941$	−4.66044e7	−1.71575	−0.857873	−	0.513862i	$-0.828214\pi$
$941$	−4.66044e7	−1.71575	−0.857873	+	0.513862i	$0.828214\pi$
$942$	0	0
$943$	− 1.56344e6i	− 0.0572533i
$944$	0	0
$945$	0	0
$946$	0	0
$947$	1.59416e7i	0.577641i	0.957383	+	0.288820i	$0.0932630\pi$
$947$	1.59416e7i	0.577641i	−0.957383	+	0.288820i	$0.906737\pi$
$948$	0	0
$949$	−6.10930e7	−2.20204
$950$	0	0
$951$	−2.35547e7	−0.844551
$952$	0	0
$953$	2.31743e6i	0.0826560i	0.999146	+	0.0413280i	$0.0131589\pi$
$953$	2.31743e6i	0.0826560i	−0.999146	+	0.0413280i	$0.986841\pi$
$954$	0	0
$955$	0	0
$956$	0	0
$957$	7.54691e6i	0.266372i
$958$	0	0
$959$	−3.73412e6	−0.131112
$960$	0	0
$961$	−2.86097e7	−0.999320
$962$	0	0
$963$	− 1.60794e7i	− 0.558732i
$964$	0	0
$965$	0	0
$966$	0	0
$967$	− 3.54405e7i	− 1.21880i	−0.792862	−	0.609401i	$-0.791410\pi$
$967$	− 3.54405e7i	− 1.21880i	0.792862	−	0.609401i	$-0.208590\pi$
$968$	0	0
$969$	4.70552e7	1.60990
$970$	0	0
$971$	−4.41218e7	−1.50178	−0.750888	−	0.660429i	$-0.770374\pi$
$971$	−4.41218e7	−1.50178	−0.750888	+	0.660429i	$0.770374\pi$
$972$	0	0
$973$	− 1.49476e7i	− 0.506161i
$974$	0	0
$975$	0	0
$976$	0	0
$977$	− 1.53807e7i	− 0.515513i	−0.966210	−	0.257757i	$-0.917017\pi$
$977$	− 1.53807e7i	− 0.515513i	0.966210	−	0.257757i	$-0.0829833\pi$
$978$	0	0
$979$	−3.62255e6	−0.120797
$980$	0	0
$981$	1.72208e6	0.0571321
$982$	0	0
$983$	− 3.82728e7i	− 1.26330i	−0.775253	−	0.631650i	$-0.782378\pi$
$983$	− 3.82728e7i	− 1.26330i	0.775253	−	0.631650i	$-0.217622\pi$
$984$	0	0
$985$	0	0
$986$	0	0
$987$	1.54457e6i	0.0504679i
$988$	0	0
$989$	390100.	0.0126819
$990$	0	0
$991$	−3.12166e6	−0.100972	−0.0504860	−	0.998725i	$-0.516077\pi$
$991$	−3.12166e6	−0.100972	−0.0504860	+	0.998725i	$0.516077\pi$
$992$	0	0
$993$	− 1.82204e7i	− 0.586387i
$994$	0	0
$995$	0	0
$996$	0	0
$997$	1.26050e7i	0.401610i	0.979631	+	0.200805i	$0.0643558\pi$
$997$	1.26050e7i	0.401610i	−0.979631	+	0.200805i	$0.935644\pi$
$998$	0	0
$999$	2.41690e7	0.766205

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	1100.6.b.b.749.2		4
5.2	odd	4		1100.6.a.c.1.1		2
5.3	odd	4		220.6.a.a.1.2	✓	2
5.4	even	2	inner	1100.6.b.b.749.3		4
20.3	even	4		880.6.a.h.1.1		2

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
220.6.a.a.1.2	✓	2	5.3	odd	4
880.6.a.h.1.1		2	20.3	even	4
1100.6.a.c.1.1		2	5.2	odd	4
1100.6.b.b.749.2		4	1.1	even	1	trivial
1100.6.b.b.749.3		4	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:	\( N \)	\(=\)	\( 1100 = 2^{2} \cdot 5^{2} \cdot 11 \)
Weight:	\( k \)	\(=\)	\( 6 \)
Character orbit:	\([\chi]\)	\(=\)	1100.b (of order \(2\), degree \(1\), not minimal)

\(f(q)\)	\(=\)	\(q-11.4821i q^{3} +39.4107i q^{7} +111.161 q^{9} -121.000 q^{11} +1182.68i q^{13} -2195.77i q^{17} -1866.37 q^{19} +452.519 q^{21} -337.644i q^{23} -4066.52i q^{27} +5432.02 q^{29} +139.481 q^{31} +1389.34i q^{33} +5943.41i q^{37} +13579.7 q^{39} +4630.43 q^{41} +1155.36i q^{43} +3413.28i q^{47} +15253.8 q^{49} -25212.1 q^{51} -5651.62i q^{53} +21430.0i q^{57} -13627.0 q^{59} -34279.1 q^{61} +4380.91i q^{63} -9963.03i q^{67} -3876.87 q^{69} -19869.0 q^{71} +51656.5i q^{73} -4768.69i q^{77} -50960.7 q^{79} -19680.3 q^{81} +8628.02i q^{83} -62371.2i q^{87} +29938.4 q^{89} -46610.2 q^{91} -1601.54i q^{93} +48871.1i q^{97} -13450.4 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 4 q - 1150 q^{9} - 484 q^{11} - 6878 q^{19} + 11294 q^{21} + 10062 q^{29} - 8926 q^{31} + 1108 q^{39} + 39336 q^{41} + 6042 q^{49} - 164718 q^{51} - 101676 q^{59} - 88354 q^{61} + 248196 q^{69} - 2178 q^{71}+ \cdots + 139150 q^{99}+O(q^{100}) \) 4 * q - 1150 * q^9 - 484 * q^11 - 6878 * q^19 + 11294 * q^21 + 10062 * q^29 - 8926 * q^31 + 1108 * q^39 + 39336 * q^41 + 6042 * q^49 - 164718 * q^51 - 101676 * q^59 - 88354 * q^61 + 248196 * q^69 - 2178 * q^71 + 32668 * q^79 + 450700 * q^81 + 161634 * q^89 + 52420 * q^91 + 139150 * q^99

Introduction

L-functions

Modular forms

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Database

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