LMFDB - Embedded newform 1840.2.i.a.1471.5

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [1840,2,Mod(1471,1840)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(1840, base_ring=CyclotomicField(2))

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

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

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

chi := DirichletCharacter("1840.1471");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 1840 = 2^{4} \cdot 5 \cdot 23 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	1840.i (of order $2$, degree $1$, minimal)

Newform invariants

comment: select newform

sage: f = N[0] # Warning: the index may be different

gp: f = lf[1] \\ Warning: the index may be different

Self dual:	no
Analytic conductor:	$14.6924739719$
Analytic rank:	$0$
Dimension:	$16$
Coefficient field:	$\mathbb{Q}[x]/(x^{16} - \cdots)$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{16} - 32x^{14} + 400x^{12} - 2398x^{10} + 7128x^{8} - 9200x^{6} + 4705x^{4} + 2696x^{2} + 784 $ x^16 - 32x^14 + 400x^12 - 2398x^10 + 7128x^8 - 9200x^6 + 4705x^4 + 2696*x^2 + 784
Coefficient ring:	$\Z[a_1, \ldots, a_{7}]$
Coefficient ring index:	$ 2^{8} $
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label			1471.5
Root			$-1.19389 + 0.500000i$ of defining polynomial
Character	$\chi$	$=$	1840.1471
Dual form			1840.2.i.a.1471.12

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-2.05992i q^{3} -1.00000i q^{5} +0.327869 q^{7} -1.24327 q^{9} +O(q^{10})$	$q-2.05992i q^{3} -1.00000i q^{5} +0.327869 q^{7} -1.24327 q^{9} +0.575977 q^{11} +0.243269 q^{13} -2.05992 q^{15} +0.756731i q^{17} +5.67864 q^{19} -0.675384i q^{21} +(4.36795 - 1.98016i) q^{23} -1.00000 q^{25} -3.61872i q^{27} +3.18885 q^{29} -10.3882i q^{31} -1.18647i q^{33} -0.327869i q^{35} -6.24089i q^{37} -0.501115i q^{39} -9.32224 q^{41} -0.655738 q^{43} +1.24327i q^{45} -6.10000i q^{47} -6.89250 q^{49} +1.55880 q^{51} +5.75435i q^{53} -0.575977i q^{55} -11.6976i q^{57} -0.921690i q^{59} +12.1334i q^{61} -0.407629 q^{63} -0.243269i q^{65} -9.65759 q^{67} +(-4.07897 - 8.99762i) q^{69} +8.06643i q^{71} -3.91628 q^{73} +2.05992i q^{75} +0.188845 q^{77} +15.5046 q^{79} -11.1841 q^{81} +9.65759 q^{83} +0.756731 q^{85} -6.56876i q^{87} +2.48654i q^{89} +0.0797604 q^{91} -21.3988 q^{93} -5.67864i q^{95} +1.65161i q^{97} -0.716095 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 16 q - 32 q^{9}+O(q^{10}) $ 16 * q - 32 * q^9	$ 16 q - 32 q^{9} + 16 q^{13} - 16 q^{25} + 72 q^{29} - 4 q^{41} - 32 q^{49} + 92 q^{69} - 20 q^{73} + 24 q^{77} + 60 q^{93}+O(q^{100}) $ 16 * q - 32 * q^9 + 16 * q^13 - 16 * q^25 + 72 * q^29 - 4 * q^41 - 32 * q^49 + 92 * q^69 - 20 * q^73 + 24 * q^77 + 60 * q^93

Display 100 coefficients 10 coefficients

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/1840\mathbb{Z}\right)^\times$.

$n$	$737$	$1151$	$1201$	$1381$
$\chi(n)$	$1$	$-1$	$-1$	$1$

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
$3$		−	2.05992i		−	1.18930i	−0.803986	−	0.594648i	$-0.797292\pi$
$3$		−	2.05992i		−	1.18930i	0.803986	−	0.594648i	$-0.202708\pi$
$4$		0			0
$5$		−	1.00000i		−	0.447214i
$5$		−	1.00000i		−	0.447214i
$6$		0			0
$7$	0.327869			0.123923			0.0619614	−	0.998079i	$-0.480264\pi$
$7$	0.327869			0.123923			0.0619614	+	0.998079i	$0.480264\pi$
$8$		0			0
$9$	−1.24327			−0.414423
$10$		0			0
$11$	0.575977			0.173664			0.0868319	−	0.996223i	$-0.472326\pi$
$11$	0.575977			0.173664			0.0868319	+	0.996223i	$0.472326\pi$
$12$		0			0
$13$	0.243269			0.0674708			0.0337354	−	0.999431i	$-0.489260\pi$
$13$	0.243269			0.0674708			0.0337354	+	0.999431i	$0.489260\pi$
$14$		0			0
$15$	−2.05992			−0.531869
$16$		0			0
$17$			0.756731i			0.183534i	0.995780	+	0.0917671i	$0.0292515\pi$
$17$			0.756731i			0.183534i	−0.995780	+	0.0917671i	$0.970748\pi$
$18$		0			0
$19$	5.67864			1.30277			0.651385	−	0.758747i	$-0.274188\pi$
$19$	5.67864			1.30277			0.651385	+	0.758747i	$0.274188\pi$
$20$		0			0
$21$		−	0.675384i		−	0.147381i
$22$		0			0
$23$	4.36795	−	1.98016i	0.910780	−	0.412892i
$23$	4.36795	−	1.98016i	0.910780	−	0.412892i
$24$		0			0
$25$	−1.00000			−0.200000
$26$		0			0
$27$		−	3.61872i		−	0.696424i
$28$		0			0
$29$	3.18885			0.592154			0.296077	−	0.955164i	$-0.404322\pi$
$29$	3.18885			0.592154			0.296077	+	0.955164i	$0.404322\pi$
$30$		0			0
$31$		−	10.3882i		−	1.86577i	−0.360170	−	0.932887i	$-0.617281\pi$
$31$		−	10.3882i		−	1.86577i	0.360170	−	0.932887i	$-0.382719\pi$
$32$		0			0
$33$		−	1.18647i		−	0.206537i
$34$		0			0
$35$		−	0.327869i		−	0.0554200i
$36$		0			0
$37$		−	6.24089i		−	1.02600i	−0.858390	−	0.512998i	$-0.828535\pi$
$37$		−	6.24089i		−	1.02600i	0.858390	−	0.512998i	$-0.171465\pi$
$38$		0			0
$39$		−	0.501115i		−	0.0802427i
$40$		0			0
$41$	−9.32224			−1.45589			−0.727945	−	0.685636i	$-0.759524\pi$
$41$	−9.32224			−1.45589			−0.727945	+	0.685636i	$0.759524\pi$
$42$		0			0
$43$	−0.655738			−0.0999991			−0.0499995	−	0.998749i	$-0.515922\pi$
$43$	−0.655738			−0.0999991			−0.0499995	+	0.998749i	$0.515922\pi$
$44$		0			0
$45$			1.24327i			0.185336i
$46$		0			0
$47$		−	6.10000i		−	0.889776i	−0.895586	−	0.444888i	$-0.853243\pi$
$47$		−	6.10000i		−	0.889776i	0.895586	−	0.444888i	$-0.146757\pi$
$48$		0			0
$49$	−6.89250			−0.984643
$50$		0			0
$51$	1.55880			0.218276
$52$		0			0
$53$			5.75435i			0.790421i	0.918591	+	0.395211i	$0.129328\pi$
$53$			5.75435i			0.790421i	−0.918591	+	0.395211i	$0.870672\pi$
$54$		0			0
$55$		−	0.575977i		−	0.0776648i
$56$		0			0
$57$		−	11.6976i		−	1.54938i
$58$		0			0
$59$		−	0.921690i		−	0.119994i	−0.998199	−	0.0599969i	$-0.980891\pi$
$59$		−	0.921690i		−	0.119994i	0.998199	−	0.0599969i	$-0.0191091\pi$
$60$		0			0
$61$			12.1334i			1.55352i	0.629796	+	0.776761i	$0.283139\pi$
$61$			12.1334i			1.55352i	−0.629796	+	0.776761i	$0.716861\pi$
$62$		0			0
$63$	−0.407629			−0.0513565
$64$		0			0
$65$		−	0.243269i		−	0.0301738i
$66$		0			0
$67$	−9.65759			−1.17986			−0.589931	−	0.807454i	$-0.700845\pi$
$67$	−9.65759			−1.17986			−0.589931	+	0.807454i	$0.700845\pi$
$68$		0			0
$69$	−4.07897	−	8.99762i	−0.491050	−	1.08319i
$70$		0			0
$71$			8.06643i			0.957309i	0.878003	+	0.478655i	$0.158875\pi$
$71$			8.06643i			0.957309i	−0.878003	+	0.478655i	$0.841125\pi$
$72$		0			0
$73$	−3.91628			−0.458365			−0.229183	−	0.973383i	$-0.573605\pi$
$73$	−3.91628			−0.458365			−0.229183	+	0.973383i	$0.573605\pi$
$74$		0			0
$75$			2.05992i			0.237859i
$76$		0			0
$77$	0.188845			0.0215209
$78$		0			0
$79$	15.5046			1.74440			0.872201	−	0.489148i	$-0.162692\pi$
$79$	15.5046			1.74440			0.872201	+	0.489148i	$0.162692\pi$
$80$		0			0
$81$	−11.1841			−1.24268
$82$		0			0
$83$	9.65759			1.06006			0.530029	−	0.847980i	$-0.322181\pi$
$83$	9.65759			1.06006			0.530029	+	0.847980i	$0.322181\pi$
$84$		0			0
$85$	0.756731			0.0820790
$86$		0			0
$87$		−	6.56876i		−	0.704246i
$88$		0			0
$89$			2.48654i			0.263573i	0.991278	+	0.131786i	$0.0420713\pi$
$89$			2.48654i			0.263573i	−0.991278	+	0.131786i	$0.957929\pi$
$90$		0			0
$91$	0.0797604			0.00836117
$92$		0			0
$93$	−21.3988			−2.21896
$94$		0			0
$95$		−	5.67864i		−	0.582617i
$96$		0			0
$97$			1.65161i			0.167696i	0.996479	+	0.0838478i	$0.0267210\pi$
$97$			1.65161i			0.167696i	−0.996479	+	0.0838478i	$0.973279\pi$
$98$		0			0
$99$	−0.716095			−0.0719703
$100$		0			0
$101$	12.8901			1.28262			0.641308	−	0.767284i	$-0.278392\pi$
$101$	12.8901			1.28262			0.641308	+	0.767284i	$0.278392\pi$
$102$		0			0
$103$	−10.7813			−1.06231			−0.531157	−	0.847273i	$-0.678243\pi$
$103$	−10.7813			−1.06231			−0.531157	+	0.847273i	$0.678243\pi$
$104$		0			0
$105$	−0.675384			−0.0659107
$106$		0			0
$107$	−1.88667			−0.182392			−0.0911958	−	0.995833i	$-0.529069\pi$
$107$	−1.88667			−0.182392			−0.0911958	+	0.995833i	$0.529069\pi$
$108$		0			0
$109$		−	1.67301i		−	0.160245i	−0.996785	−	0.0801224i	$-0.974469\pi$
$109$		−	1.67301i		−	0.160245i	0.996785	−	0.0801224i	$-0.0255311\pi$
$110$		0			0
$111$	−12.8557			−1.22021
$112$		0			0
$113$		−	11.3767i		−	1.07023i	−0.844780	−	0.535113i	$-0.820269\pi$
$113$		−	11.3767i		−	1.07023i	0.844780	−	0.535113i	$-0.179731\pi$
$114$		0			0
$115$	−1.98016	−	4.36795i	−0.184651	−	0.407313i
$116$		0			0
$117$	−0.302449			−0.0279614
$118$		0			0
$119$			0.248108i			0.0227441i
$120$		0			0
$121$	−10.6683			−0.969841
$122$		0			0
$123$			19.2031i			1.73148i
$124$		0			0
$125$			1.00000i			0.0894427i
$126$		0			0
$127$			0.172467i			0.0153039i	0.999971	+	0.00765197i	$0.00243572\pi$
$127$			0.172467i			0.0153039i	−0.999971	+	0.00765197i	$0.997564\pi$
$128$		0			0
$129$			1.35077i			0.118928i
$130$		0			0
$131$			1.98016i			0.173007i	0.996252	+	0.0865037i	$0.0275694\pi$
$131$			1.98016i			0.173007i	−0.996252	+	0.0865037i	$0.972431\pi$
$132$		0			0
$133$	1.86185			0.161443
$134$		0			0
$135$	−3.61872			−0.311450
$136$		0			0
$137$		−	17.4580i		−	1.49154i	−0.666204	−	0.745769i	$-0.732082\pi$
$137$		−	17.4580i		−	1.49154i	0.666204	−	0.745769i	$-0.267918\pi$
$138$		0			0
$139$			10.2988i			0.873534i	0.899575	+	0.436767i	$0.143877\pi$
$139$			10.2988i			0.873534i	−0.899575	+	0.436767i	$0.856123\pi$
$140$		0			0
$141$	−12.5655			−1.05821
$142$		0			0
$143$	0.140118			0.0117172
$144$		0			0
$145$		−	3.18885i		−	0.264819i
$146$		0			0
$147$			14.1980i			1.17103i
$148$		0			0
$149$			14.9715i			1.22651i	0.789885	+	0.613255i	$0.210140\pi$
$149$			14.9715i			1.22651i	−0.789885	+	0.613255i	$0.789860\pi$
$150$		0			0
$151$		−	12.1057i		−	0.985151i	−0.870270	−	0.492575i	$-0.836056\pi$
$151$		−	12.1057i		−	0.985151i	0.870270	−	0.492575i	$-0.163944\pi$
$152$		0			0
$153$		−	0.940820i		−	0.0760608i
$154$		0			0
$155$	−10.3882			−0.834399
$156$		0			0
$157$		−	17.3767i		−	1.38681i	−0.720549	−	0.693404i	$-0.756110\pi$
$157$		−	17.3767i		−	1.38681i	0.720549	−	0.693404i	$-0.243890\pi$
$158$		0			0
$159$	11.8535			0.940044
$160$		0			0
$161$	1.43211	−	0.649233i	0.112866	−	0.0511667i
$162$		0			0
$163$		−	8.00530i		−	0.627023i	−0.949584	−	0.313512i	$-0.898495\pi$
$163$		−	8.00530i		−	0.627023i	0.949584	−	0.313512i	$-0.101505\pi$
$164$		0			0
$165$	−1.18647			−0.0923663
$166$		0			0
$167$		−	1.99466i		−	0.154352i	−0.997017	−	0.0771759i	$-0.975410\pi$
$167$		−	1.99466i		−	0.154352i	0.997017	−	0.0771759i	$-0.0245903\pi$
$168$		0			0
$169$	−12.9408			−0.995448
$170$		0			0
$171$	−7.06008			−0.539898
$172$		0			0
$173$	8.61993			0.655361			0.327681	−	0.944789i	$-0.393733\pi$
$173$	8.61993			0.655361			0.327681	+	0.944789i	$0.393733\pi$
$174$		0			0
$175$	−0.327869			−0.0247846
$176$		0			0
$177$	−1.89861			−0.142708
$178$		0			0
$179$			3.29163i			0.246028i	0.992405	+	0.123014i	$0.0392561\pi$
$179$			3.29163i			0.246028i	−0.992405	+	0.123014i	$0.960744\pi$
$180$		0			0
$181$		−	0.0506960i		−	0.00376820i	−0.999998	−	0.00188410i	$-0.999400\pi$
$181$		−	0.0506960i		−	0.00376820i	0.999998	−	0.00188410i	$-0.000599728\pi$
$182$		0			0
$183$	24.9938			1.84760
$184$		0			0
$185$	−6.24089			−0.458839
$186$		0			0
$187$			0.435860i			0.0318732i
$188$		0			0
$189$		−	1.18647i		−	0.0863028i
$190$		0			0
$191$	2.64884			0.191664			0.0958318	−	0.995398i	$-0.469449\pi$
$191$	2.64884			0.191664			0.0958318	+	0.995398i	$0.469449\pi$
$192$		0			0
$193$	−9.05205			−0.651581			−0.325790	−	0.945442i	$-0.605630\pi$
$193$	−9.05205			−0.651581			−0.325790	+	0.945442i	$0.605630\pi$
$194$		0			0
$195$	−0.501115			−0.0358856
$196$		0			0
$197$	4.69890			0.334783			0.167391	−	0.985891i	$-0.446466\pi$
$197$	4.69890			0.334783			0.167391	+	0.985891i	$0.446466\pi$
$198$		0			0
$199$	10.8885			0.771867			0.385933	−	0.922527i	$-0.373879\pi$
$199$	10.8885			0.771867			0.385933	+	0.922527i	$0.373879\pi$
$200$		0			0
$201$			19.8939i			1.40320i
$202$		0			0
$203$	1.04552			0.0733813
$204$		0			0
$205$			9.32224i			0.651093i
$206$		0			0
$207$	−5.43054	+	2.46187i	−0.377448	+	0.171112i
$208$		0			0
$209$	3.27077			0.226244
$210$		0			0
$211$			6.66225i			0.458648i	0.973350	+	0.229324i	$0.0736516\pi$
$211$			6.66225i			0.458648i	−0.973350	+	0.229324i	$0.926348\pi$
$212$		0			0
$213$	16.6162			1.13852
$214$		0			0
$215$			0.655738i			0.0447209i
$216$		0			0
$217$		−	3.40596i		−	0.231212i
$218$		0			0
$219$			8.06721i			0.545132i
$220$		0			0
$221$			0.184089i			0.0123832i
$222$		0			0
$223$		−	12.2000i		−	0.816972i	−0.912765	−	0.408486i	$-0.866057\pi$
$223$		−	12.2000i		−	0.816972i	0.912765	−	0.408486i	$-0.133943\pi$
$224$		0			0
$225$	1.24327			0.0828846
$226$		0			0
$227$	−2.31215			−0.153463			−0.0767313	−	0.997052i	$-0.524448\pi$
$227$	−2.31215			−0.153463			−0.0767313	+	0.997052i	$0.524448\pi$
$228$		0			0
$229$		−	3.62231i		−	0.239369i	−0.992812	−	0.119684i	$-0.961812\pi$
$229$		−	3.62231i		−	0.239369i	0.992812	−	0.119684i	$-0.0381883\pi$
$230$		0			0
$231$		−	0.389006i		−	0.0255947i
$232$		0			0
$233$	9.42498			0.617451			0.308726	−	0.951151i	$-0.400098\pi$
$233$	9.42498			0.617451			0.308726	+	0.951151i	$0.400098\pi$
$234$		0			0
$235$	−6.10000			−0.397920
$236$		0			0
$237$		−	31.9382i		−	2.07461i
$238$		0			0
$239$		−	5.68276i		−	0.367587i	−0.982965	−	0.183794i	$-0.941162\pi$
$239$		−	5.68276i		−	0.367587i	0.982965	−	0.183794i	$-0.0588378\pi$
$240$		0			0
$241$			23.1263i			1.48969i	0.667236	+	0.744847i	$0.267477\pi$
$241$			23.1263i			1.48969i	−0.667236	+	0.744847i	$0.732523\pi$
$242$		0			0
$243$			12.1822i			0.781486i
$244$		0			0
$245$			6.89250i			0.440346i
$246$		0			0
$247$	1.38144			0.0878989
$248$		0			0
$249$		−	19.8939i		−	1.26072i
$250$		0			0
$251$	−10.6686			−0.673399			−0.336699	−	0.941612i	$-0.609311\pi$
$251$	−10.6686			−0.673399			−0.336699	+	0.941612i	$0.609311\pi$
$252$		0			0
$253$	2.51584	−	1.14053i	0.158169	−	0.0717043i
$254$		0			0
$255$		−	1.55880i		−	0.0976161i
$256$		0			0
$257$	15.0520			0.938921			0.469460	−	0.882954i	$-0.344448\pi$
$257$	15.0520			0.938921			0.469460	+	0.882954i	$0.344448\pi$
$258$		0			0
$259$		−	2.04619i		−	0.127144i
$260$		0			0
$261$	−3.96459			−0.245402
$262$		0			0
$263$	−12.1992			−0.752236			−0.376118	−	0.926572i	$-0.622741\pi$
$263$	−12.1992			−0.752236			−0.376118	+	0.926572i	$0.622741\pi$
$264$		0			0
$265$	5.75435			0.353487
$266$		0			0
$267$	5.12207			0.313466
$268$		0			0
$269$	9.18409			0.559964			0.279982	−	0.960005i	$-0.409672\pi$
$269$	9.18409			0.559964			0.279982	+	0.960005i	$0.409672\pi$
$270$		0			0
$271$		−	22.1065i		−	1.34287i	−0.741062	−	0.671436i	$-0.765678\pi$
$271$		−	22.1065i		−	1.34287i	0.741062	−	0.671436i	$-0.234322\pi$
$272$		0			0
$273$		−	0.164300i		−	0.00994389i
$274$		0			0
$275$	−0.575977			−0.0347327
$276$		0			0
$277$	26.5608			1.59588			0.797940	−	0.602736i	$-0.205923\pi$
$277$	26.5608			1.59588			0.797940	+	0.602736i	$0.205923\pi$
$278$		0			0
$279$			12.9153i			0.773220i
$280$		0			0
$281$			27.0222i			1.61201i	0.591911	+	0.806004i	$0.298374\pi$
$281$			27.0222i			1.61201i	−0.591911	+	0.806004i	$0.701626\pi$
$282$		0			0
$283$	9.44316			0.561338			0.280669	−	0.959805i	$-0.409444\pi$
$283$	9.44316			0.561338			0.280669	+	0.959805i	$0.409444\pi$
$284$		0			0
$285$	−11.6976			−0.692903
$286$		0			0
$287$	−3.05647			−0.180418
$288$		0			0
$289$	16.4274			0.966315
$290$		0			0
$291$	3.40219			0.199440
$292$		0			0
$293$			18.0211i			1.05281i	0.850235	+	0.526403i	$0.176460\pi$
$293$			18.0211i			1.05281i	−0.850235	+	0.526403i	$0.823540\pi$
$294$		0			0
$295$	−0.921690			−0.0536629
$296$		0			0
$297$		−	2.08430i		−	0.120944i
$298$		0			0
$299$	1.06259	−	0.481712i	0.0614510	−	0.0278581i
$300$		0			0
$301$	−0.214996			−0.0123922
$302$		0			0
$303$		−	26.5526i		−	1.52541i
$304$		0			0
$305$	12.1334			0.694756
$306$		0			0
$307$		−	20.7048i		−	1.18169i	−0.806786	−	0.590844i	$-0.798795\pi$
$307$		−	20.7048i		−	1.18169i	0.806786	−	0.590844i	$-0.201205\pi$
$308$		0			0
$309$			22.2086i			1.26341i
$310$		0			0
$311$		−	2.66335i		−	0.151025i	−0.997145	−	0.0755123i	$-0.975941\pi$
$311$		−	2.66335i		−	0.151025i	0.997145	−	0.0755123i	$-0.0240592\pi$
$312$		0			0
$313$		−	8.24404i		−	0.465981i	−0.972479	−	0.232990i	$-0.925149\pi$
$313$		−	8.24404i		−	0.465981i	0.972479	−	0.232990i	$-0.0748511\pi$
$314$		0			0
$315$			0.407629i			0.0229673i
$316$		0			0
$317$	15.8357			0.889422			0.444711	−	0.895674i	$-0.353306\pi$
$317$	15.8357			0.889422			0.444711	+	0.895674i	$0.353306\pi$
$318$		0			0
$319$	1.83670			0.102836
$320$		0			0
$321$			3.88640i			0.216917i
$322$		0			0
$323$			4.29720i			0.239103i
$324$		0			0
$325$	−0.243269			−0.0134942
$326$		0			0
$327$	−3.44626			−0.190578
$328$		0			0
$329$		−	2.00000i		−	0.110264i
$330$		0			0
$331$			7.30348i			0.401436i	0.979649	+	0.200718i	$0.0643275\pi$
$331$			7.30348i			0.401436i	−0.979649	+	0.200718i	$0.935673\pi$
$332$		0			0
$333$			7.75911i			0.425197i
$334$		0			0
$335$			9.65759i			0.527650i
$336$		0			0
$337$			24.4849i			1.33378i	0.745157	+	0.666890i	$0.232375\pi$
$337$			24.4849i			1.33378i	−0.745157	+	0.666890i	$0.767625\pi$
$338$		0			0
$339$	−23.4350			−1.27282
$340$		0			0
$341$		−	5.98336i		−	0.324017i
$342$		0			0
$343$	−4.55492			−0.245943
$344$		0			0
$345$	−8.99762	+	4.07897i	−0.484416	+	0.219604i
$346$		0			0
$347$			23.0088i			1.23517i	0.786502	+	0.617587i	$0.211890\pi$
$347$			23.0088i			1.23517i	−0.786502	+	0.617587i	$0.788110\pi$
$348$		0			0
$349$	7.83332			0.419308			0.209654	−	0.977776i	$-0.432766\pi$
$349$	7.83332			0.419308			0.209654	+	0.977776i	$0.432766\pi$
$350$		0			0
$351$		−	0.880324i		−	0.0469882i
$352$		0			0
$353$	31.2052			1.66089			0.830443	−	0.557103i	$-0.188087\pi$
$353$	31.2052			1.66089			0.830443	+	0.557103i	$0.188087\pi$
$354$		0			0
$355$	8.06643			0.428122
$356$		0			0
$357$	0.511084			0.0270494
$358$		0			0
$359$	27.3404			1.44297			0.721486	−	0.692429i	$-0.243459\pi$
$359$	27.3404			1.44297			0.721486	+	0.692429i	$0.243459\pi$
$360$		0			0
$361$	13.2470			0.697210
$362$		0			0
$363$			21.9757i			1.15343i
$364$		0			0
$365$			3.91628i			0.204987i
$366$		0			0
$367$	33.1874			1.73237			0.866185	−	0.499724i	$-0.166565\pi$
$367$	33.1874			1.73237			0.866185	+	0.499724i	$0.166565\pi$
$368$		0			0
$369$	11.5901			0.603354
$370$		0			0
$371$			1.88667i			0.0979512i
$372$		0			0
$373$			14.2668i			0.738706i	0.929289	+	0.369353i	$0.120421\pi$
$373$			14.2668i			0.738706i	−0.929289	+	0.369353i	$0.879579\pi$
$374$		0			0
$375$	2.05992			0.106374
$376$		0			0
$377$	0.775748			0.0399531
$378$		0			0
$379$	−29.2553			−1.50275			−0.751373	−	0.659878i	$-0.770608\pi$
$379$	−29.2553			−1.50275			−0.751373	+	0.659878i	$0.770608\pi$
$380$		0			0
$381$	0.355267			0.0182009
$382$		0			0
$383$	1.72715			0.0882534			0.0441267	−	0.999026i	$-0.485949\pi$
$383$	1.72715			0.0882534			0.0441267	+	0.999026i	$0.485949\pi$
$384$		0			0
$385$		−	0.188845i		−	0.00962444i
$386$		0			0
$387$	0.815259			0.0414419
$388$		0			0
$389$		−	24.6738i		−	1.25101i	−0.780220	−	0.625505i	$-0.784893\pi$
$389$		−	24.6738i		−	1.25101i	0.780220	−	0.625505i	$-0.215107\pi$
$390$		0			0
$391$	1.49845	+	3.30536i	0.0757797	+	0.167159i
$392$		0			0
$393$	4.07897			0.205757
$394$		0			0
$395$		−	15.5046i		−	0.780120i
$396$		0			0
$397$	−27.4020			−1.37527			−0.687633	−	0.726058i	$-0.741350\pi$
$397$	−27.4020			−1.37527			−0.687633	+	0.726058i	$0.741350\pi$
$398$		0			0
$399$		−	3.83526i		−	0.192003i
$400$		0			0
$401$			28.0993i			1.40321i	0.712564	+	0.701607i	$0.247534\pi$
$401$			28.0993i			1.40321i	−0.712564	+	0.701607i	$0.752466\pi$
$402$		0			0
$403$		−	2.52713i		−	0.125885i
$404$		0			0
$405$			11.1841i			0.555742i
$406$		0			0
$407$		−	3.59461i		−	0.178178i
$408$		0			0
$409$	5.83493			0.288519			0.144259	−	0.989540i	$-0.453920\pi$
$409$	5.83493			0.288519			0.144259	+	0.989540i	$0.453920\pi$
$410$		0			0
$411$	−35.9621			−1.77388
$412$		0			0
$413$		−	0.302194i		−	0.0148700i
$414$		0			0
$415$		−	9.65759i		−	0.474072i
$416$		0			0
$417$	21.2147			1.03889
$418$		0			0
$419$	17.1335			0.837028			0.418514	−	0.908210i	$-0.362551\pi$
$419$	17.1335			0.837028			0.418514	+	0.908210i	$0.362551\pi$
$420$		0			0
$421$			21.1865i			1.03257i	0.856418	+	0.516283i	$0.172685\pi$
$421$			21.1865i			1.03257i	−0.856418	+	0.516283i	$0.827315\pi$
$422$		0			0
$423$			7.58394i			0.368744i
$424$		0			0
$425$		−	0.756731i		−	0.0367068i
$426$		0			0
$427$			3.97816i			0.192517i
$428$		0			0
$429$		−	0.288631i		−	0.0139352i
$430$		0			0
$431$	−27.6771			−1.33316			−0.666580	−	0.745433i	$-0.732243\pi$
$431$	−27.6771			−1.33316			−0.666580	+	0.745433i	$0.732243\pi$
$432$		0			0
$433$		−	28.3957i		−	1.36461i	−0.731068	−	0.682305i	$-0.760978\pi$
$433$		−	28.3957i		−	1.36461i	0.731068	−	0.682305i	$-0.239022\pi$
$434$		0			0
$435$	−6.56876			−0.314948
$436$		0			0
$437$	24.8040	−	11.2446i	1.18654	−	0.537903i
$438$		0			0
$439$			0.0748622i			0.00357298i	0.999998	+	0.00178649i	$0.000568657\pi$
$439$			0.0748622i			0.00357298i	−0.999998	+	0.00178649i	$0.999431\pi$
$440$		0			0
$441$	8.56924			0.408059
$442$		0			0
$443$			0.945023i			0.0448994i	0.999748	+	0.0224497i	$0.00714656\pi$
$443$			0.945023i			0.0448994i	−0.999748	+	0.0224497i	$0.992853\pi$
$444$		0			0
$445$	2.48654			0.117873
$446$		0			0
$447$	30.8400			1.45868
$448$		0			0
$449$	21.7527			1.02658			0.513288	−	0.858217i	$-0.328427\pi$
$449$	21.7527			1.02658			0.513288	+	0.858217i	$0.328427\pi$
$450$		0			0
$451$	−5.36940			−0.252835
$452$		0			0
$453$	−24.9368			−1.17163
$454$		0			0
$455$		−	0.0797604i		−	0.00373923i
$456$		0			0
$457$		−	20.6138i		−	0.964274i	−0.876096	−	0.482137i	$-0.839861\pi$
$457$		−	20.6138i		−	0.964274i	0.876096	−	0.482137i	$-0.160139\pi$
$458$		0			0
$459$	2.73840			0.127818
$460$		0			0
$461$	−5.92103			−0.275770			−0.137885	−	0.990448i	$-0.544030\pi$
$461$	−5.92103			−0.275770			−0.137885	+	0.990448i	$0.544030\pi$
$462$		0			0
$463$			19.2862i			0.896304i	0.893957	+	0.448152i	$0.147918\pi$
$463$			19.2862i			0.896304i	−0.893957	+	0.448152i	$0.852082\pi$
$464$		0			0
$465$			21.3988i			0.992347i
$466$		0			0
$467$	32.7187			1.51404			0.757019	−	0.653392i	$-0.226655\pi$
$467$	32.7187			1.51404			0.757019	+	0.653392i	$0.226655\pi$
$468$		0			0
$469$	−3.16642			−0.146212
$470$		0			0
$471$	−35.7945			−1.64932
$472$		0			0
$473$	−0.377690			−0.0173662
$474$		0			0
$475$	−5.67864			−0.260554
$476$		0			0
$477$		−	7.15421i		−	0.327569i
$478$		0			0
$479$	−11.5427			−0.527399			−0.263700	−	0.964605i	$-0.584943\pi$
$479$	−11.5427			−0.527399			−0.263700	+	0.964605i	$0.584943\pi$
$480$		0			0
$481$		−	1.51822i		−	0.0692247i
$482$		0			0
$483$	−1.33737	−	2.95004i	−0.0608523	−	0.134231i
$484$		0			0
$485$	1.65161			0.0749958
$486$		0			0
$487$			25.6887i			1.16407i	0.813165	+	0.582034i	$0.197743\pi$
$487$			25.6887i			1.16407i	−0.813165	+	0.582034i	$0.802257\pi$
$488$		0			0
$489$	−16.4903			−0.745716
$490$		0			0
$491$			14.5699i			0.657532i	0.944411	+	0.328766i	$0.106633\pi$
$491$			14.5699i			0.657532i	−0.944411	+	0.328766i	$0.893367\pi$
$492$		0			0
$493$			2.41310i			0.108680i
$494$		0			0
$495$			0.716095i			0.0321861i
$496$		0			0
$497$			2.64473i			0.118632i
$498$		0			0
$499$		−	6.18054i		−	0.276679i	−0.990385	−	0.138339i	$-0.955824\pi$
$499$		−	6.18054i		−	0.276679i	0.990385	−	0.138339i	$-0.0441765\pi$
$500$		0			0
$501$	−4.10885			−0.183570
$502$		0			0
$503$	−29.9997			−1.33762			−0.668809	−	0.743434i	$-0.733196\pi$
$503$	−29.9997			−1.33762			−0.668809	+	0.743434i	$0.733196\pi$
$504$		0			0
$505$		−	12.8901i		−	0.573603i
$506$		0			0
$507$			26.6571i			1.18388i
$508$		0			0
$509$	26.3458			1.16776			0.583878	−	0.811842i	$-0.301535\pi$
$509$	26.3458			1.16776			0.583878	+	0.811842i	$0.301535\pi$
$510$		0			0
$511$	−1.28402			−0.0568019
$512$		0			0
$513$		−	20.5494i		−	0.907280i
$514$		0			0
$515$			10.7813i			0.475081i
$516$		0			0
$517$		−	3.51346i		−	0.154522i
$518$		0			0
$519$		−	17.7564i		−	0.779418i
$520$		0			0
$521$			28.7486i			1.25950i	0.776799	+	0.629749i	$0.216842\pi$
$521$			28.7486i			1.25950i	−0.776799	+	0.629749i	$0.783158\pi$
$522$		0			0
$523$	−16.1329			−0.705441			−0.352720	−	0.935729i	$-0.614743\pi$
$523$	−16.1329			−0.705441			−0.352720	+	0.935729i	$0.614743\pi$
$524$		0			0
$525$			0.675384i			0.0294762i
$526$		0			0
$527$	7.86106			0.342433
$528$		0			0
$529$	15.1579	−	17.2985i	0.659041	−	0.752107i
$530$		0			0
$531$			1.14591i			0.0497282i
$532$		0			0
$533$	−2.26781			−0.0982299
$534$		0			0
$535$			1.88667i			0.0815680i
$536$		0			0
$537$	6.78050			0.292600
$538$		0			0
$539$	−3.96993			−0.170997
$540$		0			0
$541$	−26.2369			−1.12801			−0.564006	−	0.825770i	$-0.690741\pi$
$541$	−26.2369			−1.12801			−0.564006	+	0.825770i	$0.690741\pi$
$542$		0			0
$543$	−0.104430			−0.00448150
$544$		0			0
$545$	−1.67301			−0.0716637
$546$		0			0
$547$			33.5298i			1.43363i	0.697263	+	0.716815i	$0.254401\pi$
$547$			33.5298i			1.43363i	−0.697263	+	0.716815i	$0.745599\pi$
$548$		0			0
$549$		−	15.0851i		−	0.643815i
$550$		0			0
$551$	18.1083			0.771440
$552$		0			0
$553$	5.08347			0.216171
$554$		0			0
$555$			12.8557i			0.545696i
$556$		0			0
$557$			35.2678i			1.49435i	0.664630	+	0.747173i	$0.268589\pi$
$557$			35.2678i			1.49435i	−0.664630	+	0.747173i	$0.731411\pi$
$558$		0			0
$559$	−0.159521			−0.00674701
$560$		0			0
$561$	0.897836			0.0379067
$562$		0			0
$563$	−12.7834			−0.538757			−0.269379	−	0.963034i	$-0.586818\pi$
$563$	−12.7834			−0.538757			−0.269379	+	0.963034i	$0.586818\pi$
$564$		0			0
$565$	−11.3767			−0.478620
$566$		0			0
$567$	−3.66692			−0.153996
$568$		0			0
$569$			27.3999i			1.14866i	0.818623	+	0.574331i	$0.194738\pi$
$569$			27.3999i			1.14866i	−0.818623	+	0.574331i	$0.805262\pi$
$570$		0			0
$571$	29.6003			1.23873			0.619366	−	0.785102i	$-0.287390\pi$
$571$	29.6003			1.23873			0.619366	+	0.785102i	$0.287390\pi$
$572$		0			0
$573$		−	5.45640i		−	0.227945i
$574$		0			0
$575$	−4.36795	+	1.98016i	−0.182156	+	0.0825784i
$576$		0			0
$577$	−23.4863			−0.977747			−0.488873	−	0.872355i	$-0.662592\pi$
$577$	−23.4863			−0.977747			−0.488873	+	0.872355i	$0.662592\pi$
$578$		0			0
$579$			18.6465i			0.774922i
$580$		0			0
$581$	3.16642			0.131365
$582$		0			0
$583$			3.31438i			0.137267i
$584$		0			0
$585$			0.302449i			0.0125047i
$586$		0			0
$587$			24.6440i			1.01717i	0.861013	+	0.508583i	$0.169831\pi$
$587$			24.6440i			1.01717i	−0.861013	+	0.508583i	$0.830169\pi$
$588$		0			0
$589$		−	58.9908i		−	2.43067i
$590$		0			0
$591$		−	9.67936i		−	0.398156i
$592$		0			0
$593$	33.6714			1.38272			0.691359	−	0.722511i	$-0.257012\pi$
$593$	33.6714			1.38272			0.691359	+	0.722511i	$0.257012\pi$
$594$		0			0
$595$	0.248108			0.0101715
$596$		0			0
$597$		−	22.4295i		−	0.917978i
$598$		0			0
$599$			32.1186i			1.31233i	0.754617	+	0.656165i	$0.227823\pi$
$599$			32.1186i			1.31233i	−0.754617	+	0.656165i	$0.772177\pi$
$600$		0			0
$601$	−20.2644			−0.826602			−0.413301	−	0.910594i	$-0.635624\pi$
$601$	−20.2644			−0.826602			−0.413301	+	0.910594i	$0.635624\pi$
$602$		0			0
$603$	12.0070			0.488962
$604$		0			0
$605$			10.6683i			0.433726i
$606$		0			0
$607$		−	28.0236i		−	1.13744i	−0.822530	−	0.568722i	$-0.807438\pi$
$607$		−	28.0236i		−	1.13744i	0.822530	−	0.568722i	$-0.192562\pi$
$608$		0			0
$609$		−	2.15369i		−	0.0872721i
$610$		0			0
$611$		−	1.48394i		−	0.0600339i
$612$		0			0
$613$			36.6492i			1.48025i	0.672470	+	0.740124i	$0.265233\pi$
$613$			36.6492i			1.48025i	−0.672470	+	0.740124i	$0.734767\pi$
$614$		0			0
$615$	19.2031			0.774342
$616$		0			0
$617$			8.31298i			0.334668i	0.985900	+	0.167334i	$0.0535158\pi$
$617$			8.31298i			0.334668i	−0.985900	+	0.167334i	$0.946484\pi$
$618$		0			0
$619$	−12.6068			−0.506712			−0.253356	−	0.967373i	$-0.581534\pi$
$619$	−12.6068			−0.506712			−0.253356	+	0.967373i	$0.581534\pi$
$620$		0			0
$621$	−7.16565	−	15.8064i	−0.287548	−	0.634289i
$622$		0			0
$623$			0.815259i			0.0326627i
$624$		0			0
$625$	1.00000			0.0400000
$626$		0			0
$627$		−	6.73752i		−	0.269071i
$628$		0			0
$629$	4.72267			0.188305
$630$		0			0
$631$	5.11895			0.203782			0.101891	−	0.994796i	$-0.467511\pi$
$631$	5.11895			0.203782			0.101891	+	0.994796i	$0.467511\pi$
$632$		0			0
$633$	13.7237			0.545468
$634$		0			0
$635$	0.172467			0.00684413
$636$		0			0
$637$	−1.67673			−0.0664346
$638$		0			0
$639$		−	10.0287i		−	0.396731i
$640$		0			0
$641$			34.8024i			1.37461i	0.726368	+	0.687306i	$0.241207\pi$
$641$			34.8024i			1.37461i	−0.726368	+	0.687306i	$0.758793\pi$
$642$		0			0
$643$	−10.2776			−0.405310			−0.202655	−	0.979250i	$-0.564957\pi$
$643$	−10.2776			−0.405310			−0.202655	+	0.979250i	$0.564957\pi$
$644$		0			0
$645$	1.35077			0.0531864
$646$		0			0
$647$			17.9633i			0.706210i	0.935584	+	0.353105i	$0.114874\pi$
$647$			17.9633i			0.706210i	−0.935584	+	0.353105i	$0.885126\pi$
$648$		0			0
$649$		−	0.530873i		−	0.0208386i
$650$		0			0
$651$	−7.01601			−0.274979
$652$		0			0
$653$	−11.7567			−0.460076			−0.230038	−	0.973182i	$-0.573885\pi$
$653$	−11.7567			−0.460076			−0.230038	+	0.973182i	$0.573885\pi$
$654$		0			0
$655$	1.98016			0.0773712
$656$		0			0
$657$	4.86898			0.189957
$658$		0			0
$659$	−31.6374			−1.23242			−0.616210	−	0.787582i	$-0.711333\pi$
$659$	−31.6374			−1.23242			−0.616210	+	0.787582i	$0.711333\pi$
$660$		0			0
$661$			10.4635i			0.406984i	0.979077	+	0.203492i	$0.0652292\pi$
$661$			10.4635i			0.406984i	−0.979077	+	0.203492i	$0.934771\pi$
$662$		0			0
$663$	0.379209			0.0147273
$664$		0			0
$665$		−	1.86185i		−	0.0721995i
$666$		0			0
$667$	13.9287	−	6.31442i	0.539322	−	0.244495i
$668$		0			0
$669$	−25.1310			−0.971621
$670$		0			0
$671$			6.98856i			0.269790i
$672$		0			0
$673$	−14.4614			−0.557447			−0.278723	−	0.960371i	$-0.589911\pi$
$673$	−14.4614			−0.557447			−0.278723	+	0.960371i	$0.589911\pi$
$674$		0			0
$675$			3.61872i			0.139285i
$676$		0			0
$677$		−	35.3719i		−	1.35945i	−0.733466	−	0.679726i	$-0.762099\pi$
$677$		−	35.3719i		−	1.35945i	0.733466	−	0.679726i	$-0.237901\pi$
$678$		0			0
$679$			0.541512i			0.0207813i
$680$		0			0
$681$			4.76284i			0.182512i
$682$		0			0
$683$			10.6541i			0.407669i	0.979005	+	0.203835i	$0.0653405\pi$
$683$			10.6541i			0.407669i	−0.979005	+	0.203835i	$0.934660\pi$
$684$		0			0
$685$	−17.4580			−0.667036
$686$		0			0
$687$	−7.46167			−0.284680
$688$		0			0
$689$			1.39986i			0.0533303i
$690$		0			0
$691$			3.46410i			0.131781i	0.997827	+	0.0658903i	$0.0209887\pi$
$691$			3.46410i			0.131781i	−0.997827	+	0.0658903i	$0.979011\pi$
$692$		0			0
$693$	−0.234785			−0.00891876
$694$		0			0
$695$	10.2988			0.390656
$696$		0			0
$697$		−	7.05442i		−	0.267205i
$698$		0			0
$699$		−	19.4147i		−	0.734332i
$700$		0			0
$701$		−	11.8880i		−	0.449004i	−0.974474	−	0.224502i	$-0.927925\pi$
$701$		−	11.8880i		−	0.449004i	0.974474	−	0.224502i	$-0.0720755\pi$
$702$		0			0
$703$		−	35.4398i		−	1.33664i
$704$		0			0
$705$			12.5655i			0.473244i
$706$		0			0
$707$	4.22627			0.158945
$708$		0			0
$709$		−	35.1033i		−	1.31833i	−0.751998	−	0.659165i	$-0.770910\pi$
$709$		−	35.1033i		−	1.31833i	0.751998	−	0.659165i	$-0.229090\pi$
$710$		0			0
$711$	−19.2764			−0.722920
$712$		0			0
$713$	−20.5703	−	45.3751i	−0.770362	−	1.69931i
$714$		0			0
$715$		−	0.140118i		−	0.00524010i
$716$		0			0
$717$	−11.7060			−0.437170
$718$		0			0
$719$			11.5629i			0.431223i	0.976479	+	0.215611i	$0.0691744\pi$
$719$			11.5629i			0.431223i	−0.976479	+	0.215611i	$0.930826\pi$
$720$		0			0
$721$	−3.53486			−0.131645
$722$		0			0
$723$	47.6382			1.77169
$724$		0			0
$725$	−3.18885			−0.118431
$726$		0			0
$727$	27.0222			1.00220			0.501098	−	0.865390i	$-0.332929\pi$
$727$	27.0222			1.00220			0.501098	+	0.865390i	$0.332929\pi$
$728$		0			0
$729$	−8.45801			−0.313260
$730$		0			0
$731$		−	0.496217i		−	0.0183532i
$732$		0			0
$733$			7.75705i			0.286513i	0.989686	+	0.143257i	$0.0457574\pi$
$733$			7.75705i			0.286513i	−0.989686	+	0.143257i	$0.954243\pi$
$734$		0			0
$735$	14.1980			0.523701
$736$		0			0
$737$	−5.56255			−0.204899
$738$		0			0
$739$		−	20.7103i		−	0.761842i	−0.924608	−	0.380921i	$-0.875607\pi$
$739$		−	20.7103i		−	0.761842i	0.924608	−	0.380921i	$-0.124393\pi$
$740$		0			0
$741$		−	2.84565i		−	0.104538i
$742$		0			0
$743$	−43.1115			−1.58161			−0.790805	−	0.612069i	$-0.790338\pi$
$743$	−43.1115			−1.58161			−0.790805	+	0.612069i	$0.790338\pi$
$744$		0			0
$745$	14.9715			0.548512
$746$		0			0
$747$	−12.0070			−0.439312
$748$		0			0
$749$	−0.618582			−0.0226025
$750$		0			0
$751$	−20.3092			−0.741092			−0.370546	−	0.928814i	$-0.620829\pi$
$751$	−20.3092			−0.741092			−0.370546	+	0.928814i	$0.620829\pi$
$752$		0			0
$753$			21.9765i			0.800870i
$754$		0			0
$755$	−12.1057			−0.440573
$756$		0			0
$757$			6.02590i			0.219015i	0.993986	+	0.109507i	$0.0349273\pi$
$757$			6.02590i			0.219015i	−0.993986	+	0.109507i	$0.965073\pi$
$758$		0			0
$759$	−2.34939	−	5.18243i	−0.0852776	−	0.188110i
$760$		0			0
$761$	11.3056			0.409828			0.204914	−	0.978780i	$-0.434309\pi$
$761$	11.3056			0.409828			0.204914	+	0.978780i	$0.434309\pi$
$762$		0			0
$763$		−	0.548527i		−	0.0198580i
$764$		0			0
$765$	−0.940820			−0.0340154
$766$		0			0
$767$		−	0.224219i		−	0.00809608i
$768$		0			0
$769$		−	1.99049i		−	0.0717789i	−0.999356	−	0.0358894i	$-0.988574\pi$
$769$		−	1.99049i		−	0.0717789i	0.999356	−	0.0358894i	$-0.0114264\pi$
$770$		0			0
$771$		−	31.0060i		−	1.11665i
$772$		0			0
$773$		−	20.7581i		−	0.746616i	−0.927707	−	0.373308i	$-0.878223\pi$
$773$		−	20.7581i		−	0.746616i	0.927707	−	0.373308i	$-0.121777\pi$
$774$		0			0
$775$			10.3882i			0.373155i
$776$		0			0
$777$	−4.21500			−0.151212
$778$		0			0
$779$	−52.9377			−1.89669
$780$		0			0
$781$			4.64608i			0.166250i
$782$		0			0
$783$		−	11.5396i		−	0.412390i
$784$		0			0
$785$	−17.3767			−0.620200
$786$		0			0
$787$	34.2446			1.22069			0.610343	−	0.792137i	$-0.291032\pi$
$787$	34.2446			1.22069			0.610343	+	0.792137i	$0.291032\pi$
$788$		0			0
$789$			25.1294i			0.894631i
$790$		0			0
$791$		−	3.73005i		−	0.132625i
$792$		0			0
$793$			2.95168i			0.104817i
$794$		0			0
$795$		−	11.8535i		−	0.420400i
$796$		0			0
$797$			45.8061i			1.62254i	0.584674	+	0.811268i	$0.301222\pi$
$797$			45.8061i			1.62254i	−0.584674	+	0.811268i	$0.698778\pi$
$798$		0			0
$799$	4.61606			0.163304
$800$		0			0
$801$		−	3.09144i		−	0.109231i
$802$		0			0
$803$	−2.25569			−0.0796014
$804$		0			0
$805$	−0.649233	−	1.43211i	−0.0228824	−	0.0504754i
$806$		0			0
$807$		−	18.9185i		−	0.665962i
$808$		0			0
$809$	−11.8877			−0.417951			−0.208975	−	0.977921i	$-0.567013\pi$
$809$	−11.8877			−0.417951			−0.208975	+	0.977921i	$0.567013\pi$
$810$		0			0
$811$			8.17520i			0.287070i	0.989645	+	0.143535i	$0.0458470\pi$
$811$			8.17520i			0.287070i	−0.989645	+	0.143535i	$0.954153\pi$
$812$		0			0
$813$	−45.5376			−1.59707
$814$		0			0
$815$	−8.00530			−0.280413
$816$		0			0
$817$	−3.72370			−0.130276
$818$		0			0
$819$	−0.0991637			−0.00346506
$820$		0			0
$821$	41.1923			1.43762			0.718811	−	0.695206i	$-0.244687\pi$
$821$	41.1923			1.43762			0.718811	+	0.695206i	$0.244687\pi$
$822$		0			0
$823$		−	54.3665i		−	1.89510i	−0.319611	−	0.947549i	$-0.603552\pi$
$823$		−	54.3665i		−	1.89510i	0.319611	−	0.947549i	$-0.396448\pi$
$824$		0			0
$825$			1.18647i			0.0413075i
$826$		0			0
$827$	50.6851			1.76249			0.881247	−	0.472656i	$-0.156705\pi$
$827$	50.6851			1.76249			0.881247	+	0.472656i	$0.156705\pi$
$828$		0			0
$829$	−5.59641			−0.194372			−0.0971858	−	0.995266i	$-0.530984\pi$
$829$	−5.59641			−0.194372			−0.0971858	+	0.995266i	$0.530984\pi$
$830$		0			0
$831$		−	54.7130i		−	1.89797i
$832$		0			0
$833$		−	5.21577i		−	0.180716i
$834$		0			0
$835$	−1.99466			−0.0690282
$836$		0			0
$837$	−37.5920			−1.29937
$838$		0			0
$839$	−15.5128			−0.535562			−0.267781	−	0.963480i	$-0.586290\pi$
$839$	−15.5128			−0.535562			−0.267781	+	0.963480i	$0.586290\pi$
$840$		0			0
$841$	−18.8313			−0.649354
$842$		0			0
$843$	55.6635			1.91715
$844$		0			0
$845$			12.9408i			0.445178i
$846$		0			0
$847$	−3.49779			−0.120185
$848$		0			0
$849$		−	19.4522i		−	0.667596i
$850$		0			0
$851$	−12.3580	−	27.2599i	−0.423625	−	0.934457i
$852$		0			0
$853$	23.0285			0.788482			0.394241	−	0.919007i	$-0.371008\pi$
$853$	23.0285			0.788482			0.394241	+	0.919007i	$0.371008\pi$
$854$		0			0
$855$			7.06008i			0.241450i
$856$		0			0
$857$	18.4994			0.631929			0.315965	−	0.948771i	$-0.397672\pi$
$857$	18.4994			0.631929			0.315965	+	0.948771i	$0.397672\pi$
$858$		0			0
$859$			6.20799i			0.211814i	0.994376	+	0.105907i	$0.0337746\pi$
$859$			6.20799i			0.211814i	−0.994376	+	0.105907i	$0.966225\pi$
$860$		0			0
$861$			6.29609i			0.214570i
$862$		0			0
$863$			2.13812i			0.0727825i	0.999338	+	0.0363912i	$0.0115863\pi$
$863$			2.13812i			0.0727825i	−0.999338	+	0.0363912i	$0.988414\pi$
$864$		0			0
$865$		−	8.61993i		−	0.293086i
$866$		0			0
$867$		−	33.8390i		−	1.14923i
$868$		0			0
$869$	8.93029			0.302939
$870$		0			0
$871$	−2.34939			−0.0796062
$872$		0			0
$873$		−	2.05340i		−	0.0694969i
$874$		0			0
$875$			0.327869i			0.0110840i
$876$		0			0
$877$	3.38007			0.114137			0.0570684	−	0.998370i	$-0.481825\pi$
$877$	3.38007			0.114137			0.0570684	+	0.998370i	$0.481825\pi$
$878$		0			0
$879$	37.1221			1.25210
$880$		0			0
$881$		−	44.5288i		−	1.50021i	−0.661316	−	0.750107i	$-0.730002\pi$
$881$		−	44.5288i		−	1.50021i	0.661316	−	0.750107i	$-0.269998\pi$
$882$		0			0
$883$			11.1441i			0.375028i	0.982262	+	0.187514i	$0.0600431\pi$
$883$			11.1441i			0.375028i	−0.982262	+	0.187514i	$0.939957\pi$
$884$		0			0
$885$			1.89861i			0.0638210i
$886$		0			0
$887$			42.0443i			1.41171i	0.708357	+	0.705854i	$0.249437\pi$
$887$			42.0443i			1.41171i	−0.708357	+	0.705854i	$0.750563\pi$
$888$		0			0
$889$			0.0565465i			0.00189651i
$890$		0			0
$891$	−6.44178			−0.215808
$892$		0			0
$893$		−	34.6397i		−	1.15917i
$894$		0			0
$895$	3.29163			0.110027
$896$		0			0
$897$	−0.992288	−	2.18885i	−0.0331315	−	0.0730834i
$898$		0			0
$899$		−	33.1263i		−	1.10482i
$900$		0			0
$901$	−4.35450			−0.145069
$902$		0			0
$903$			0.442875i			0.0147379i
$904$		0			0
$905$	−0.0506960			−0.00168519
$906$		0			0
$907$	30.5743			1.01520			0.507601	−	0.861592i	$-0.330532\pi$
$907$	30.5743			1.01520			0.507601	+	0.861592i	$0.330532\pi$
$908$		0			0
$909$	−16.0259			−0.531545
$910$		0			0
$911$	52.3962			1.73596			0.867981	−	0.496597i	$-0.165417\pi$
$911$	52.3962			1.73596			0.867981	+	0.496597i	$0.165417\pi$
$912$		0			0
$913$	5.56255			0.184094
$914$		0			0
$915$		−	24.9938i		−	0.826270i
$916$		0			0
$917$			0.649233i			0.0214396i
$918$		0			0
$919$	56.5709			1.86610			0.933051	−	0.359744i	$-0.117136\pi$
$919$	56.5709			1.86610			0.933051	+	0.359744i	$0.117136\pi$
$920$		0			0
$921$	−42.6503			−1.40538
$922$		0			0
$923$			1.96232i			0.0645904i
$924$		0			0
$925$			6.24089i			0.205199i
$926$		0			0
$927$	13.4041			0.440247
$928$		0			0
$929$	−21.8333			−0.716328			−0.358164	−	0.933659i	$-0.616597\pi$
$929$	−21.8333			−0.716328			−0.358164	+	0.933659i	$0.616597\pi$
$930$		0			0
$931$	−39.1401			−1.28276
$932$		0			0
$933$	−5.48628			−0.179613
$934$		0			0
$935$	0.435860			0.0142541
$936$		0			0
$937$			14.2794i			0.466489i	0.972418	+	0.233245i	$0.0749343\pi$
$937$			14.2794i			0.466489i	−0.972418	+	0.233245i	$0.925066\pi$
$938$		0			0
$939$	−16.9821			−0.554189
$940$		0			0
$941$			20.7213i			0.675496i	0.941237	+	0.337748i	$0.109665\pi$
$941$			20.7213i			0.675496i	−0.941237	+	0.337748i	$0.890335\pi$
$942$		0			0
$943$	−40.7191	+	18.4595i	−1.32599	+	0.601125i
$944$		0			0
$945$	−1.18647			−0.0385958
$946$		0			0
$947$		−	49.0634i		−	1.59435i	−0.603751	−	0.797173i	$-0.706328\pi$
$947$		−	49.0634i		−	1.59435i	0.603751	−	0.797173i	$-0.293672\pi$
$948$		0			0
$949$	−0.952709			−0.0309263
$950$		0			0
$951$		−	32.6203i		−	1.05778i
$952$		0			0
$953$			23.5380i			0.762471i	0.924478	+	0.381235i	$0.124501\pi$
$953$			23.5380i			0.762471i	−0.924478	+	0.381235i	$0.875499\pi$
$954$		0			0
$955$		−	2.64884i		−	0.0857145i
$956$		0			0
$957$		−	3.78346i		−	0.122302i
$958$		0			0
$959$		−	5.72394i		−	0.184836i
$960$		0			0
$961$	−76.9144			−2.48111
$962$		0			0
$963$	2.34564			0.0755873
$964$		0			0
$965$			9.05205i			0.291396i
$966$		0			0
$967$		−	15.5465i		−	0.499943i	−0.968253	−	0.249971i	$-0.919579\pi$
$967$		−	15.5465i		−	0.499943i	0.968253	−	0.249971i	$-0.0804212\pi$
$968$		0			0
$969$	8.85190			0.284364
$970$		0			0
$971$	23.0170			0.738651			0.369325	−	0.929300i	$-0.379589\pi$
$971$	23.0170			0.738651			0.369325	+	0.929300i	$0.379589\pi$
$972$		0			0
$973$			3.37666i			0.108251i
$974$		0			0
$975$			0.501115i			0.0160485i
$976$		0			0
$977$		−	24.8933i		−	0.796406i	−0.917297	−	0.398203i	$-0.869634\pi$
$977$		−	24.8933i		−	0.796406i	0.917297	−	0.398203i	$-0.130366\pi$
$978$		0			0
$979$			1.43219i			0.0457730i
$980$		0			0
$981$			2.08000i			0.0664092i
$982$		0			0
$983$	−25.0467			−0.798866			−0.399433	−	0.916762i	$-0.630793\pi$
$983$	−25.0467			−0.798866			−0.399433	+	0.916762i	$0.630793\pi$
$984$		0			0
$985$		−	4.69890i		−	0.149719i
$986$		0			0
$987$	−4.11984			−0.131136
$988$		0			0
$989$	−2.86423	+	1.29847i	−0.0910772	+	0.0412888i
$990$		0			0
$991$			43.6222i			1.38571i	0.721079	+	0.692853i	$0.243647\pi$
$991$			43.6222i			1.38571i	−0.721079	+	0.692853i	$0.756353\pi$
$992$		0			0
$993$	15.0446			0.477426
$994$		0			0
$995$		−	10.8885i		−	0.345189i
$996$		0			0
$997$	15.4549			0.489460			0.244730	−	0.969591i	$-0.421301\pi$
$997$	15.4549			0.489460			0.244730	+	0.969591i	$0.421301\pi$
$998$		0			0
$999$	−22.5841			−0.714528

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	1840.2.i.a.1471.5	✓	16
4.3	odd	2	inner	1840.2.i.a.1471.11	yes	16
23.22	odd	2	inner	1840.2.i.a.1471.6	yes	16
92.91	even	2	inner	1840.2.i.a.1471.12	yes	16

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
1840.2.i.a.1471.5	✓	16	1.1	even	1	trivial
1840.2.i.a.1471.6	yes	16	23.22	odd	2	inner
1840.2.i.a.1471.11	yes	16	4.3	odd	2	inner
1840.2.i.a.1471.12	yes	16	92.91	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:	\( N \)	\(=\)	\( 1840 = 2^{4} \cdot 5 \cdot 23 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	1840.i (of order \(2\), degree \(1\), minimal)

\(f(q)\)	\(=\)	\(q-2.05992i q^{3} -1.00000i q^{5} +0.327869 q^{7} -1.24327 q^{9} +O(q^{10})\)	\(q-2.05992i q^{3} -1.00000i q^{5} +0.327869 q^{7} -1.24327 q^{9} +0.575977 q^{11} +0.243269 q^{13} -2.05992 q^{15} +0.756731i q^{17} +5.67864 q^{19} -0.675384i q^{21} +(4.36795 - 1.98016i) q^{23} -1.00000 q^{25} -3.61872i q^{27} +3.18885 q^{29} -10.3882i q^{31} -1.18647i q^{33} -0.327869i q^{35} -6.24089i q^{37} -0.501115i q^{39} -9.32224 q^{41} -0.655738 q^{43} +1.24327i q^{45} -6.10000i q^{47} -6.89250 q^{49} +1.55880 q^{51} +5.75435i q^{53} -0.575977i q^{55} -11.6976i q^{57} -0.921690i q^{59} +12.1334i q^{61} -0.407629 q^{63} -0.243269i q^{65} -9.65759 q^{67} +(-4.07897 - 8.99762i) q^{69} +8.06643i q^{71} -3.91628 q^{73} +2.05992i q^{75} +0.188845 q^{77} +15.5046 q^{79} -11.1841 q^{81} +9.65759 q^{83} +0.756731 q^{85} -6.56876i q^{87} +2.48654i q^{89} +0.0797604 q^{91} -21.3988 q^{93} -5.67864i q^{95} +1.65161i q^{97} -0.716095 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 16 q - 32 q^{9}+O(q^{10}) \) 16 * q - 32 * q^9	\( 16 q - 32 q^{9} + 16 q^{13} - 16 q^{25} + 72 q^{29} - 4 q^{41} - 32 q^{49} + 92 q^{69} - 20 q^{73} + 24 q^{77} + 60 q^{93}+O(q^{100}) \) 16 * q - 32 * q^9 + 16 * q^13 - 16 * q^25 + 72 * q^29 - 4 * q^41 - 32 * q^49 + 92 * q^69 - 20 * q^73 + 24 * q^77 + 60 * q^93

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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