LMFDB - Embedded newform 3700.2.d.a.149.1

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [3700,2,Mod(149,3700)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(3700, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([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("3700.149");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 3700 = 2^{2} \cdot 5^{2} \cdot 37 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	3700.d (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:	$29.5446487479$
Analytic rank:	$0$
Dimension:	$2$
Coefficient field:	$\Q(i)$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{2} + 1 $ x^2 + 1
Coefficient ring:	$\Z[a_1, \ldots, a_{13}]$
Coefficient ring index:	$ 1 $
Twist minimal:	no (minimal twist has level 740)
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label			149.1
Root			$-1.00000i$ of defining polynomial
Character	$\chi$	$=$	3700.149
Dual form			3700.2.d.a.149.2

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-3.00000i q^{3} -3.00000i q^{7} -6.00000 q^{9} +O(q^{10})$	$q-3.00000i q^{3} -3.00000i q^{7} -6.00000 q^{9} +5.00000 q^{11} -2.00000i q^{13} +4.00000i q^{17} +4.00000 q^{19} -9.00000 q^{21} -6.00000i q^{23} +9.00000i q^{27} -6.00000 q^{29} -4.00000 q^{31} -15.0000i q^{33} -1.00000i q^{37} -6.00000 q^{39} -9.00000 q^{41} -10.0000i q^{43} -11.0000i q^{47} -2.00000 q^{49} +12.0000 q^{51} +11.0000i q^{53} -12.0000i q^{57} +8.00000 q^{59} -8.00000 q^{61} +18.0000i q^{63} -8.00000i q^{67} -18.0000 q^{69} +3.00000 q^{71} -7.00000i q^{73} -15.0000i q^{77} -8.00000 q^{79} +9.00000 q^{81} +9.00000i q^{83} +18.0000i q^{87} +16.0000 q^{89} -6.00000 q^{91} +12.0000i q^{93} +12.0000i q^{97} -30.0000 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 2 q - 12 q^{9}+O(q^{10}) $ 2 * q - 12 * q^9	$ 2 q - 12 q^{9} + 10 q^{11} + 8 q^{19} - 18 q^{21} - 12 q^{29} - 8 q^{31} - 12 q^{39} - 18 q^{41} - 4 q^{49} + 24 q^{51} + 16 q^{59} - 16 q^{61} - 36 q^{69} + 6 q^{71} - 16 q^{79} + 18 q^{81} + 32 q^{89} - 12 q^{91} - 60 q^{99}+O(q^{100}) $ 2 * q - 12 * q^9 + 10 * q^11 + 8 * q^19 - 18 * q^21 - 12 * q^29 - 8 * q^31 - 12 * q^39 - 18 * q^41 - 4 * q^49 + 24 * q^51 + 16 * q^59 - 16 * q^61 - 36 * q^69 + 6 * q^71 - 16 * q^79 + 18 * q^81 + 32 * q^89 - 12 * q^91 - 60 * q^99

Display 100 coefficients 10 coefficients

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/3700\mathbb{Z}\right)^\times$.

$n$	$1001$	$1777$	$1851$
$\chi(n)$	$1$	$-1$	$1$

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
$3$	− 3.00000i	− 1.73205i	−0.500000	−	0.866025i	$-0.666667\pi$
$3$	− 3.00000i	− 1.73205i	0.500000	−	0.866025i	$-0.333333\pi$
$4$	0	0
$5$	0	0
$5$	0	0
$6$	0	0
$7$	− 3.00000i	− 1.13389i	−0.823754	−	0.566947i	$-0.808125\pi$
$7$	− 3.00000i	− 1.13389i	0.823754	−	0.566947i	$-0.191875\pi$
$8$	0	0
$9$	−6.00000	−2.00000
$10$	0	0
$11$	5.00000	1.50756	0.753778	−	0.657129i	$-0.228229\pi$
$11$	5.00000	1.50756	0.753778	+	0.657129i	$0.228229\pi$
$12$	0	0
$13$	− 2.00000i	− 0.554700i	−0.960769	−	0.277350i	$-0.910544\pi$
$13$	− 2.00000i	− 0.554700i	0.960769	−	0.277350i	$-0.0894562\pi$
$14$	0	0
$15$	0	0
$16$	0	0
$17$	4.00000i	0.970143i	0.874475	+	0.485071i	$0.161206\pi$
$17$	4.00000i	0.970143i	−0.874475	+	0.485071i	$0.838794\pi$
$18$	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
$21$	−9.00000	−1.96396
$22$	0	0
$23$	− 6.00000i	− 1.25109i	−0.780189	−	0.625543i	$-0.784877\pi$
$23$	− 6.00000i	− 1.25109i	0.780189	−	0.625543i	$-0.215123\pi$
$24$	0	0
$25$	0	0
$26$	0	0
$27$	9.00000i	1.73205i
$28$	0	0
$29$	−6.00000	−1.11417	−0.557086	−	0.830455i	$-0.688081\pi$
$29$	−6.00000	−1.11417	−0.557086	+	0.830455i	$0.688081\pi$
$30$	0	0
$31$	−4.00000	−0.718421	−0.359211	−	0.933257i	$-0.616954\pi$
$31$	−4.00000	−0.718421	−0.359211	+	0.933257i	$0.616954\pi$
$32$	0	0
$33$	− 15.0000i	− 2.61116i
$34$	0	0
$35$	0	0
$36$	0	0
$37$	− 1.00000i	− 0.164399i
$37$	− 1.00000i	− 0.164399i
$38$	0	0
$39$	−6.00000	−0.960769
$40$	0	0
$41$	−9.00000	−1.40556	−0.702782	−	0.711405i	$-0.748059\pi$
$41$	−9.00000	−1.40556	−0.702782	+	0.711405i	$0.748059\pi$
$42$	0	0
$43$	− 10.0000i	− 1.52499i	−0.646997	−	0.762493i	$-0.723975\pi$
$43$	− 10.0000i	− 1.52499i	0.646997	−	0.762493i	$-0.276025\pi$
$44$	0	0
$45$	0	0
$46$	0	0
$47$	− 11.0000i	− 1.60451i	−0.596978	−	0.802257i	$-0.703632\pi$
$47$	− 11.0000i	− 1.60451i	0.596978	−	0.802257i	$-0.296368\pi$
$48$	0	0
$49$	−2.00000	−0.285714
$50$	0	0
$51$	12.0000	1.68034
$52$	0	0
$53$	11.0000i	1.51097i	0.655168	+	0.755483i	$0.272598\pi$
$53$	11.0000i	1.51097i	−0.655168	+	0.755483i	$0.727402\pi$
$54$	0	0
$55$	0	0
$56$	0	0
$57$	− 12.0000i	− 1.58944i
$58$	0	0
$59$	8.00000	1.04151	0.520756	−	0.853706i	$-0.325650\pi$
$59$	8.00000	1.04151	0.520756	+	0.853706i	$0.325650\pi$
$60$	0	0
$61$	−8.00000	−1.02430	−0.512148	−	0.858898i	$-0.671150\pi$
$61$	−8.00000	−1.02430	−0.512148	+	0.858898i	$0.671150\pi$
$62$	0	0
$63$	18.0000i	2.26779i
$64$	0	0
$65$	0	0
$66$	0	0
$67$	− 8.00000i	− 0.977356i	−0.872464	−	0.488678i	$-0.837479\pi$
$67$	− 8.00000i	− 0.977356i	0.872464	−	0.488678i	$-0.162521\pi$
$68$	0	0
$69$	−18.0000	−2.16695
$70$	0	0
$71$	3.00000	0.356034	0.178017	−	0.984027i	$-0.443032\pi$
$71$	3.00000	0.356034	0.178017	+	0.984027i	$0.443032\pi$
$72$	0	0
$73$	− 7.00000i	− 0.819288i	−0.912245	−	0.409644i	$-0.865653\pi$
$73$	− 7.00000i	− 0.819288i	0.912245	−	0.409644i	$-0.134347\pi$
$74$	0	0
$75$	0	0
$76$	0	0
$77$	− 15.0000i	− 1.70941i
$78$	0	0
$79$	−8.00000	−0.900070	−0.450035	−	0.893011i	$-0.648589\pi$
$79$	−8.00000	−0.900070	−0.450035	+	0.893011i	$0.648589\pi$
$80$	0	0
$81$	9.00000	1.00000
$82$	0	0
$83$	9.00000i	0.987878i	0.869496	+	0.493939i	$0.164443\pi$
$83$	9.00000i	0.987878i	−0.869496	+	0.493939i	$0.835557\pi$
$84$	0	0
$85$	0	0
$86$	0	0
$87$	18.0000i	1.92980i
$88$	0	0
$89$	16.0000	1.69600	0.847998	−	0.529999i	$-0.177808\pi$
$89$	16.0000	1.69600	0.847998	+	0.529999i	$0.177808\pi$
$90$	0	0
$91$	−6.00000	−0.628971
$92$	0	0
$93$	12.0000i	1.24434i
$94$	0	0
$95$	0	0
$96$	0	0
$97$	12.0000i	1.21842i	0.793011	+	0.609208i	$0.208512\pi$
$97$	12.0000i	1.21842i	−0.793011	+	0.609208i	$0.791488\pi$
$98$	0	0
$99$	−30.0000	−3.01511
$100$	0	0
$101$	−9.00000	−0.895533	−0.447767	−	0.894150i	$-0.647781\pi$
$101$	−9.00000	−0.895533	−0.447767	+	0.894150i	$0.647781\pi$
$102$	0	0
$103$	6.00000i	0.591198i	0.955312	+	0.295599i	$0.0955191\pi$
$103$	6.00000i	0.591198i	−0.955312	+	0.295599i	$0.904481\pi$
$104$	0	0
$105$	0	0
$106$	0	0
$107$	4.00000i	0.386695i	0.981130	+	0.193347i	$0.0619344\pi$
$107$	4.00000i	0.386695i	−0.981130	+	0.193347i	$0.938066\pi$
$108$	0	0
$109$	−12.0000	−1.14939	−0.574696	−	0.818367i	$-0.694880\pi$
$109$	−12.0000	−1.14939	−0.574696	+	0.818367i	$0.694880\pi$
$110$	0	0
$111$	−3.00000	−0.284747
$112$	0	0
$113$	− 14.0000i	− 1.31701i	−0.752577	−	0.658505i	$-0.771189\pi$
$113$	− 14.0000i	− 1.31701i	0.752577	−	0.658505i	$-0.228811\pi$
$114$	0	0
$115$	0	0
$116$	0	0
$117$	12.0000i	1.10940i
$118$	0	0
$119$	12.0000	1.10004
$120$	0	0
$121$	14.0000	1.27273
$122$	0	0
$123$	27.0000i	2.43451i
$124$	0	0
$125$	0	0
$126$	0	0
$127$	− 13.0000i	− 1.15356i	−0.816898	−	0.576782i	$-0.804308\pi$
$127$	− 13.0000i	− 1.15356i	0.816898	−	0.576782i	$-0.195692\pi$
$128$	0	0
$129$	−30.0000	−2.64135
$130$	0	0
$131$	0	0		−	1.00000i	$-0.5\pi$
$131$	0	0			1.00000i	$0.5\pi$
$132$	0	0
$133$	− 12.0000i	− 1.04053i
$134$	0	0
$135$	0	0
$136$	0	0
$137$	18.0000i	1.53784i	0.639343	+	0.768922i	$0.279207\pi$
$137$	18.0000i	1.53784i	−0.639343	+	0.768922i	$0.720793\pi$
$138$	0	0
$139$	4.00000	0.339276	0.169638	−	0.985506i	$-0.445740\pi$
$139$	4.00000	0.339276	0.169638	+	0.985506i	$0.445740\pi$
$140$	0	0
$141$	−33.0000	−2.77910
$142$	0	0
$143$	− 10.0000i	− 0.836242i
$144$	0	0
$145$	0	0
$146$	0	0
$147$	6.00000i	0.494872i
$148$	0	0
$149$	1.00000	0.0819232	0.0409616	−	0.999161i	$-0.486958\pi$
$149$	1.00000	0.0819232	0.0409616	+	0.999161i	$0.486958\pi$
$150$	0	0
$151$	0	0		−	1.00000i	$-0.5\pi$
$151$	0	0			1.00000i	$0.5\pi$
$152$	0	0
$153$	− 24.0000i	− 1.94029i
$154$	0	0
$155$	0	0
$156$	0	0
$157$	3.00000i	0.239426i	0.992809	+	0.119713i	$0.0381975\pi$
$157$	3.00000i	0.239426i	−0.992809	+	0.119713i	$0.961803\pi$
$158$	0	0
$159$	33.0000	2.61707
$160$	0	0
$161$	−18.0000	−1.41860
$162$	0	0
$163$	10.0000i	0.783260i	0.920123	+	0.391630i	$0.128089\pi$
$163$	10.0000i	0.783260i	−0.920123	+	0.391630i	$0.871911\pi$
$164$	0	0
$165$	0	0
$166$	0	0
$167$	8.00000i	0.619059i	0.950890	+	0.309529i	$0.100171\pi$
$167$	8.00000i	0.619059i	−0.950890	+	0.309529i	$0.899829\pi$
$168$	0	0
$169$	9.00000	0.692308
$170$	0	0
$171$	−24.0000	−1.83533
$172$	0	0
$173$	11.0000i	0.836315i	0.908375	+	0.418157i	$0.137324\pi$
$173$	11.0000i	0.836315i	−0.908375	+	0.418157i	$0.862676\pi$
$174$	0	0
$175$	0	0
$176$	0	0
$177$	− 24.0000i	− 1.80395i
$178$	0	0
$179$	2.00000	0.149487	0.0747435	−	0.997203i	$-0.476186\pi$
$179$	2.00000	0.149487	0.0747435	+	0.997203i	$0.476186\pi$
$180$	0	0
$181$	25.0000	1.85824	0.929118	−	0.369784i	$-0.120568\pi$
$181$	25.0000	1.85824	0.929118	+	0.369784i	$0.120568\pi$
$182$	0	0
$183$	24.0000i	1.77413i
$184$	0	0
$185$	0	0
$186$	0	0
$187$	20.0000i	1.46254i
$188$	0	0
$189$	27.0000	1.96396
$190$	0	0
$191$	−12.0000	−0.868290	−0.434145	−	0.900843i	$-0.642949\pi$
$191$	−12.0000	−0.868290	−0.434145	+	0.900843i	$0.642949\pi$
$192$	0	0
$193$	− 26.0000i	− 1.87152i	−0.352636	−	0.935760i	$-0.614715\pi$
$193$	− 26.0000i	− 1.87152i	0.352636	−	0.935760i	$-0.385285\pi$
$194$	0	0
$195$	0	0
$196$	0	0
$197$	23.0000i	1.63868i	0.573306	+	0.819341i	$0.305660\pi$
$197$	23.0000i	1.63868i	−0.573306	+	0.819341i	$0.694340\pi$
$198$	0	0
$199$	2.00000	0.141776	0.0708881	−	0.997484i	$-0.477417\pi$
$199$	2.00000	0.141776	0.0708881	+	0.997484i	$0.477417\pi$
$200$	0	0
$201$	−24.0000	−1.69283
$202$	0	0
$203$	18.0000i	1.26335i
$204$	0	0
$205$	0	0
$206$	0	0
$207$	36.0000i	2.50217i
$208$	0	0
$209$	20.0000	1.38343
$210$	0	0
$211$	−11.0000	−0.757271	−0.378636	−	0.925546i	$-0.623607\pi$
$211$	−11.0000	−0.757271	−0.378636	+	0.925546i	$0.623607\pi$
$212$	0	0
$213$	− 9.00000i	− 0.616670i
$214$	0	0
$215$	0	0
$216$	0	0
$217$	12.0000i	0.814613i
$218$	0	0
$219$	−21.0000	−1.41905
$220$	0	0
$221$	8.00000	0.538138
$222$	0	0
$223$	− 5.00000i	− 0.334825i	−0.985887	−	0.167412i	$-0.946459\pi$
$223$	− 5.00000i	− 0.334825i	0.985887	−	0.167412i	$-0.0535411\pi$
$224$	0	0
$225$	0	0
$226$	0	0
$227$	− 16.0000i	− 1.06196i	−0.847385	−	0.530979i	$-0.821824\pi$
$227$	− 16.0000i	− 1.06196i	0.847385	−	0.530979i	$-0.178176\pi$
$228$	0	0
$229$	21.0000	1.38772	0.693860	−	0.720110i	$-0.255909\pi$
$229$	21.0000	1.38772	0.693860	+	0.720110i	$0.255909\pi$
$230$	0	0
$231$	−45.0000	−2.96078
$232$	0	0
$233$	− 14.0000i	− 0.917170i	−0.888650	−	0.458585i	$-0.848356\pi$
$233$	− 14.0000i	− 0.917170i	0.888650	−	0.458585i	$-0.151644\pi$
$234$	0	0
$235$	0	0
$236$	0	0
$237$	24.0000i	1.55897i
$238$	0	0
$239$	−6.00000	−0.388108	−0.194054	−	0.980991i	$-0.562164\pi$
$239$	−6.00000	−0.388108	−0.194054	+	0.980991i	$0.562164\pi$
$240$	0	0
$241$	14.0000	0.901819	0.450910	−	0.892570i	$-0.351100\pi$
$241$	14.0000	0.901819	0.450910	+	0.892570i	$0.351100\pi$
$242$	0	0
$243$	0	0
$244$	0	0
$245$	0	0
$246$	0	0
$247$	− 8.00000i	− 0.509028i
$248$	0	0
$249$	27.0000	1.71106
$250$	0	0
$251$	−10.0000	−0.631194	−0.315597	−	0.948893i	$-0.602205\pi$
$251$	−10.0000	−0.631194	−0.315597	+	0.948893i	$0.602205\pi$
$252$	0	0
$253$	− 30.0000i	− 1.88608i
$254$	0	0
$255$	0	0
$256$	0	0
$257$	− 16.0000i	− 0.998053i	−0.866587	−	0.499026i	$-0.833691\pi$
$257$	− 16.0000i	− 0.998053i	0.866587	−	0.499026i	$-0.166309\pi$
$258$	0	0
$259$	−3.00000	−0.186411
$260$	0	0
$261$	36.0000	2.22834
$262$	0	0
$263$	23.0000i	1.41824i	0.705087	+	0.709120i	$0.250908\pi$
$263$	23.0000i	1.41824i	−0.705087	+	0.709120i	$0.749092\pi$
$264$	0	0
$265$	0	0
$266$	0	0
$267$	− 48.0000i	− 2.93755i
$268$	0	0
$269$	22.0000	1.34136	0.670682	−	0.741745i	$-0.266002\pi$
$269$	22.0000	1.34136	0.670682	+	0.741745i	$0.266002\pi$
$270$	0	0
$271$	−5.00000	−0.303728	−0.151864	−	0.988401i	$-0.548528\pi$
$271$	−5.00000	−0.303728	−0.151864	+	0.988401i	$0.548528\pi$
$272$	0	0
$273$	18.0000i	1.08941i
$274$	0	0
$275$	0	0
$276$	0	0
$277$	− 4.00000i	− 0.240337i	−0.992754	−	0.120168i	$-0.961657\pi$
$277$	− 4.00000i	− 0.240337i	0.992754	−	0.120168i	$-0.0383434\pi$
$278$	0	0
$279$	24.0000	1.43684
$280$	0	0
$281$	−4.00000	−0.238620	−0.119310	−	0.992857i	$-0.538068\pi$
$281$	−4.00000	−0.238620	−0.119310	+	0.992857i	$0.538068\pi$
$282$	0	0
$283$	− 4.00000i	− 0.237775i	−0.992908	−	0.118888i	$-0.962067\pi$
$283$	− 4.00000i	− 0.237775i	0.992908	−	0.118888i	$-0.0379328\pi$
$284$	0	0
$285$	0	0
$286$	0	0
$287$	27.0000i	1.59376i
$288$	0	0
$289$	1.00000	0.0588235
$290$	0	0
$291$	36.0000	2.11036
$292$	0	0
$293$	− 30.0000i	− 1.75262i	−0.481749	−	0.876309i	$-0.659998\pi$
$293$	− 30.0000i	− 1.75262i	0.481749	−	0.876309i	$-0.340002\pi$
$294$	0	0
$295$	0	0
$296$	0	0
$297$	45.0000i	2.61116i
$298$	0	0
$299$	−12.0000	−0.693978
$300$	0	0
$301$	−30.0000	−1.72917
$302$	0	0
$303$	27.0000i	1.55111i
$304$	0	0
$305$	0	0
$306$	0	0
$307$	− 23.0000i	− 1.31268i	−0.754466	−	0.656340i	$-0.772104\pi$
$307$	− 23.0000i	− 1.31268i	0.754466	−	0.656340i	$-0.227896\pi$
$308$	0	0
$309$	18.0000	1.02398
$310$	0	0
$311$	12.0000	0.680458	0.340229	−	0.940343i	$-0.389495\pi$
$311$	12.0000	0.680458	0.340229	+	0.940343i	$0.389495\pi$
$312$	0	0
$313$	22.0000i	1.24351i	0.783210	+	0.621757i	$0.213581\pi$
$313$	22.0000i	1.24351i	−0.783210	+	0.621757i	$0.786419\pi$
$314$	0	0
$315$	0	0
$316$	0	0
$317$	14.0000i	0.786318i	0.919470	+	0.393159i	$0.128618\pi$
$317$	14.0000i	0.786318i	−0.919470	+	0.393159i	$0.871382\pi$
$318$	0	0
$319$	−30.0000	−1.67968
$320$	0	0
$321$	12.0000	0.669775
$322$	0	0
$323$	16.0000i	0.890264i
$324$	0	0
$325$	0	0
$326$	0	0
$327$	36.0000i	1.99080i
$328$	0	0
$329$	−33.0000	−1.81935
$330$	0	0
$331$	−22.0000	−1.20923	−0.604615	−	0.796518i	$-0.706673\pi$
$331$	−22.0000	−1.20923	−0.604615	+	0.796518i	$0.706673\pi$
$332$	0	0
$333$	6.00000i	0.328798i
$334$	0	0
$335$	0	0
$336$	0	0
$337$	7.00000i	0.381314i	0.981657	+	0.190657i	$0.0610619\pi$
$337$	7.00000i	0.381314i	−0.981657	+	0.190657i	$0.938938\pi$
$338$	0	0
$339$	−42.0000	−2.28113
$340$	0	0
$341$	−20.0000	−1.08306
$342$	0	0
$343$	− 15.0000i	− 0.809924i
$344$	0	0
$345$	0	0
$346$	0	0
$347$	10.0000i	0.536828i	0.963304	+	0.268414i	$0.0864995\pi$
$347$	10.0000i	0.536828i	−0.963304	+	0.268414i	$0.913500\pi$
$348$	0	0
$349$	−30.0000	−1.60586	−0.802932	−	0.596071i	$-0.796728\pi$
$349$	−30.0000	−1.60586	−0.802932	+	0.596071i	$0.796728\pi$
$350$	0	0
$351$	18.0000	0.960769
$352$	0	0
$353$	12.0000i	0.638696i	0.947638	+	0.319348i	$0.103464\pi$
$353$	12.0000i	0.638696i	−0.947638	+	0.319348i	$0.896536\pi$
$354$	0	0
$355$	0	0
$356$	0	0
$357$	− 36.0000i	− 1.90532i
$358$	0	0
$359$	21.0000	1.10834	0.554169	−	0.832404i	$-0.313036\pi$
$359$	21.0000	1.10834	0.554169	+	0.832404i	$0.313036\pi$
$360$	0	0
$361$	−3.00000	−0.157895
$362$	0	0
$363$	− 42.0000i	− 2.20443i
$364$	0	0
$365$	0	0
$366$	0	0
$367$	8.00000i	0.417597i	0.977959	+	0.208798i	$0.0669552\pi$
$367$	8.00000i	0.417597i	−0.977959	+	0.208798i	$0.933045\pi$
$368$	0	0
$369$	54.0000	2.81113
$370$	0	0
$371$	33.0000	1.71327
$372$	0	0
$373$	− 1.00000i	− 0.0517780i	−0.999665	−	0.0258890i	$-0.991758\pi$
$373$	− 1.00000i	− 0.0517780i	0.999665	−	0.0258890i	$-0.00824165\pi$
$374$	0	0
$375$	0	0
$376$	0	0
$377$	12.0000i	0.618031i
$378$	0	0
$379$	−17.0000	−0.873231	−0.436616	−	0.899648i	$-0.643823\pi$
$379$	−17.0000	−0.873231	−0.436616	+	0.899648i	$0.643823\pi$
$380$	0	0
$381$	−39.0000	−1.99803
$382$	0	0
$383$	− 36.0000i	− 1.83951i	−0.392488	−	0.919757i	$-0.628386\pi$
$383$	− 36.0000i	− 1.83951i	0.392488	−	0.919757i	$-0.371614\pi$
$384$	0	0
$385$	0	0
$386$	0	0
$387$	60.0000i	3.04997i
$388$	0	0
$389$	−4.00000	−0.202808	−0.101404	−	0.994845i	$-0.532333\pi$
$389$	−4.00000	−0.202808	−0.101404	+	0.994845i	$0.532333\pi$
$390$	0	0
$391$	24.0000	1.21373
$392$	0	0
$393$	0	0
$394$	0	0
$395$	0	0
$396$	0	0
$397$	7.00000i	0.351320i	0.984451	+	0.175660i	$0.0562059\pi$
$397$	7.00000i	0.351320i	−0.984451	+	0.175660i	$0.943794\pi$
$398$	0	0
$399$	−36.0000	−1.80225
$400$	0	0
$401$	−10.0000	−0.499376	−0.249688	−	0.968326i	$-0.580328\pi$
$401$	−10.0000	−0.499376	−0.249688	+	0.968326i	$0.580328\pi$
$402$	0	0
$403$	8.00000i	0.398508i
$404$	0	0
$405$	0	0
$406$	0	0
$407$	− 5.00000i	− 0.247841i
$408$	0	0
$409$	4.00000	0.197787	0.0988936	−	0.995098i	$-0.468470\pi$
$409$	4.00000	0.197787	0.0988936	+	0.995098i	$0.468470\pi$
$410$	0	0
$411$	54.0000	2.66362
$412$	0	0
$413$	− 24.0000i	− 1.18096i
$414$	0	0
$415$	0	0
$416$	0	0
$417$	− 12.0000i	− 0.587643i
$418$	0	0
$419$	−1.00000	−0.0488532	−0.0244266	−	0.999702i	$-0.507776\pi$
$419$	−1.00000	−0.0488532	−0.0244266	+	0.999702i	$0.507776\pi$
$420$	0	0
$421$	36.0000	1.75453	0.877266	−	0.480004i	$-0.159365\pi$
$421$	36.0000	1.75453	0.877266	+	0.480004i	$0.159365\pi$
$422$	0	0
$423$	66.0000i	3.20903i
$424$	0	0
$425$	0	0
$426$	0	0
$427$	24.0000i	1.16144i
$428$	0	0
$429$	−30.0000	−1.44841
$430$	0	0
$431$	−14.0000	−0.674356	−0.337178	−	0.941441i	$-0.609472\pi$
$431$	−14.0000	−0.674356	−0.337178	+	0.941441i	$0.609472\pi$
$432$	0	0
$433$	23.0000i	1.10531i	0.833410	+	0.552655i	$0.186385\pi$
$433$	23.0000i	1.10531i	−0.833410	+	0.552655i	$0.813615\pi$
$434$	0	0
$435$	0	0
$436$	0	0
$437$	− 24.0000i	− 1.14808i
$438$	0	0
$439$	−28.0000	−1.33637	−0.668184	−	0.743996i	$-0.732928\pi$
$439$	−28.0000	−1.33637	−0.668184	+	0.743996i	$0.732928\pi$
$440$	0	0
$441$	12.0000	0.571429
$442$	0	0
$443$	9.00000i	0.427603i	0.976877	+	0.213801i	$0.0685846\pi$
$443$	9.00000i	0.427603i	−0.976877	+	0.213801i	$0.931415\pi$
$444$	0	0
$445$	0	0
$446$	0	0
$447$	− 3.00000i	− 0.141895i
$448$	0	0
$449$	24.0000	1.13263	0.566315	−	0.824189i	$-0.308369\pi$
$449$	24.0000	1.13263	0.566315	+	0.824189i	$0.308369\pi$
$450$	0	0
$451$	−45.0000	−2.11897
$452$	0	0
$453$	0	0
$454$	0	0
$455$	0	0
$456$	0	0
$457$	6.00000i	0.280668i	0.990104	+	0.140334i	$0.0448177\pi$
$457$	6.00000i	0.280668i	−0.990104	+	0.140334i	$0.955182\pi$
$458$	0	0
$459$	−36.0000	−1.68034
$460$	0	0
$461$	10.0000	0.465746	0.232873	−	0.972507i	$-0.425187\pi$
$461$	10.0000	0.465746	0.232873	+	0.972507i	$0.425187\pi$
$462$	0	0
$463$	− 14.0000i	− 0.650635i	−0.945605	−	0.325318i	$-0.894529\pi$
$463$	− 14.0000i	− 0.650635i	0.945605	−	0.325318i	$-0.105471\pi$
$464$	0	0
$465$	0	0
$466$	0	0
$467$	10.0000i	0.462745i	0.972865	+	0.231372i	$0.0743216\pi$
$467$	10.0000i	0.462745i	−0.972865	+	0.231372i	$0.925678\pi$
$468$	0	0
$469$	−24.0000	−1.10822
$470$	0	0
$471$	9.00000	0.414698
$472$	0	0
$473$	− 50.0000i	− 2.29900i
$474$	0	0
$475$	0	0
$476$	0	0
$477$	− 66.0000i	− 3.02193i
$478$	0	0
$479$	14.0000	0.639676	0.319838	−	0.947472i	$-0.396371\pi$
$479$	14.0000	0.639676	0.319838	+	0.947472i	$0.396371\pi$
$480$	0	0
$481$	−2.00000	−0.0911922
$482$	0	0
$483$	54.0000i	2.45709i
$484$	0	0
$485$	0	0
$486$	0	0
$487$	− 4.00000i	− 0.181257i	−0.995885	−	0.0906287i	$-0.971112\pi$
$487$	− 4.00000i	− 0.181257i	0.995885	−	0.0906287i	$-0.0288876\pi$
$488$	0	0
$489$	30.0000	1.35665
$490$	0	0
$491$	20.0000	0.902587	0.451294	−	0.892375i	$-0.350963\pi$
$491$	20.0000	0.902587	0.451294	+	0.892375i	$0.350963\pi$
$492$	0	0
$493$	− 24.0000i	− 1.08091i
$494$	0	0
$495$	0	0
$496$	0	0
$497$	− 9.00000i	− 0.403705i
$498$	0	0
$499$	44.0000	1.96971	0.984855	−	0.173379i	$-0.0554684\pi$
$499$	44.0000	1.96971	0.984855	+	0.173379i	$0.0554684\pi$
$500$	0	0
$501$	24.0000	1.07224
$502$	0	0
$503$	− 12.0000i	− 0.535054i	−0.963550	−	0.267527i	$-0.913794\pi$
$503$	− 12.0000i	− 0.535054i	0.963550	−	0.267527i	$-0.0862064\pi$
$504$	0	0
$505$	0	0
$506$	0	0
$507$	− 27.0000i	− 1.19911i
$508$	0	0
$509$	−21.0000	−0.930809	−0.465404	−	0.885098i	$-0.654091\pi$
$509$	−21.0000	−0.930809	−0.465404	+	0.885098i	$0.654091\pi$
$510$	0	0
$511$	−21.0000	−0.928985
$512$	0	0
$513$	36.0000i	1.58944i
$514$	0	0
$515$	0	0
$516$	0	0
$517$	− 55.0000i	− 2.41890i
$518$	0	0
$519$	33.0000	1.44854
$520$	0	0
$521$	−41.0000	−1.79624	−0.898121	−	0.439748i	$-0.855068\pi$
$521$	−41.0000	−1.79624	−0.898121	+	0.439748i	$0.855068\pi$
$522$	0	0
$523$	6.00000i	0.262362i	0.991358	+	0.131181i	$0.0418769\pi$
$523$	6.00000i	0.262362i	−0.991358	+	0.131181i	$0.958123\pi$
$524$	0	0
$525$	0	0
$526$	0	0
$527$	− 16.0000i	− 0.696971i
$528$	0	0
$529$	−13.0000	−0.565217
$530$	0	0
$531$	−48.0000	−2.08302
$532$	0	0
$533$	18.0000i	0.779667i
$534$	0	0
$535$	0	0
$536$	0	0
$537$	− 6.00000i	− 0.258919i
$538$	0	0
$539$	−10.0000	−0.430730
$540$	0	0
$541$	−36.0000	−1.54776	−0.773880	−	0.633332i	$-0.781687\pi$
$541$	−36.0000	−1.54776	−0.773880	+	0.633332i	$0.781687\pi$
$542$	0	0
$543$	− 75.0000i	− 3.21856i
$544$	0	0
$545$	0	0
$546$	0	0
$547$	16.0000i	0.684111i	0.939680	+	0.342055i	$0.111123\pi$
$547$	16.0000i	0.684111i	−0.939680	+	0.342055i	$0.888877\pi$
$548$	0	0
$549$	48.0000	2.04859
$550$	0	0
$551$	−24.0000	−1.02243
$552$	0	0
$553$	24.0000i	1.02058i
$554$	0	0
$555$	0	0
$556$	0	0
$557$	− 10.0000i	− 0.423714i	−0.977301	−	0.211857i	$-0.932049\pi$
$557$	− 10.0000i	− 0.423714i	0.977301	−	0.211857i	$-0.0679510\pi$
$558$	0	0
$559$	−20.0000	−0.845910
$560$	0	0
$561$	60.0000	2.53320
$562$	0	0
$563$	− 42.0000i	− 1.77009i	−0.465506	−	0.885044i	$-0.654128\pi$
$563$	− 42.0000i	− 1.77009i	0.465506	−	0.885044i	$-0.345872\pi$
$564$	0	0
$565$	0	0
$566$	0	0
$567$	− 27.0000i	− 1.13389i
$568$	0	0
$569$	0	0		−	1.00000i	$-0.5\pi$
$569$	0	0			1.00000i	$0.5\pi$
$570$	0	0
$571$	33.0000	1.38101	0.690504	−	0.723329i	$-0.257389\pi$
$571$	33.0000	1.38101	0.690504	+	0.723329i	$0.257389\pi$
$572$	0	0
$573$	36.0000i	1.50392i
$574$	0	0
$575$	0	0
$576$	0	0
$577$	28.0000i	1.16566i	0.812596	+	0.582828i	$0.198054\pi$
$577$	28.0000i	1.16566i	−0.812596	+	0.582828i	$0.801946\pi$
$578$	0	0
$579$	−78.0000	−3.24157
$580$	0	0
$581$	27.0000	1.12015
$582$	0	0
$583$	55.0000i	2.27787i
$584$	0	0
$585$	0	0
$586$	0	0
$587$	− 36.0000i	− 1.48588i	−0.669359	−	0.742940i	$-0.733431\pi$
$587$	− 36.0000i	− 1.48588i	0.669359	−	0.742940i	$-0.266569\pi$
$588$	0	0
$589$	−16.0000	−0.659269
$590$	0	0
$591$	69.0000	2.83828
$592$	0	0
$593$	− 43.0000i	− 1.76580i	−0.469563	−	0.882899i	$-0.655588\pi$
$593$	− 43.0000i	− 1.76580i	0.469563	−	0.882899i	$-0.344412\pi$
$594$	0	0
$595$	0	0
$596$	0	0
$597$	− 6.00000i	− 0.245564i
$598$	0	0
$599$	45.0000	1.83865	0.919325	−	0.393499i	$-0.128735\pi$
$599$	45.0000	1.83865	0.919325	+	0.393499i	$0.128735\pi$
$600$	0	0
$601$	10.0000	0.407909	0.203954	−	0.978980i	$-0.434621\pi$
$601$	10.0000	0.407909	0.203954	+	0.978980i	$0.434621\pi$
$602$	0	0
$603$	48.0000i	1.95471i
$604$	0	0
$605$	0	0
$606$	0	0
$607$	− 12.0000i	− 0.487065i	−0.969893	−	0.243532i	$-0.921694\pi$
$607$	− 12.0000i	− 0.487065i	0.969893	−	0.243532i	$-0.0783062\pi$
$608$	0	0
$609$	54.0000	2.18819
$610$	0	0
$611$	−22.0000	−0.890025
$612$	0	0
$613$	− 27.0000i	− 1.09052i	−0.838267	−	0.545260i	$-0.816431\pi$
$613$	− 27.0000i	− 1.09052i	0.838267	−	0.545260i	$-0.183569\pi$
$614$	0	0
$615$	0	0
$616$	0	0
$617$	9.00000i	0.362326i	0.983453	+	0.181163i	$0.0579862\pi$
$617$	9.00000i	0.362326i	−0.983453	+	0.181163i	$0.942014\pi$
$618$	0	0
$619$	−25.0000	−1.00483	−0.502417	−	0.864625i	$-0.667556\pi$
$619$	−25.0000	−1.00483	−0.502417	+	0.864625i	$0.667556\pi$
$620$	0	0
$621$	54.0000	2.16695
$622$	0	0
$623$	− 48.0000i	− 1.92308i
$624$	0	0
$625$	0	0
$626$	0	0
$627$	− 60.0000i	− 2.39617i
$628$	0	0
$629$	4.00000	0.159490
$630$	0	0
$631$	−48.0000	−1.91085	−0.955425	−	0.295234i	$-0.904602\pi$
$631$	−48.0000	−1.91085	−0.955425	+	0.295234i	$0.904602\pi$
$632$	0	0
$633$	33.0000i	1.31163i
$634$	0	0
$635$	0	0
$636$	0	0
$637$	4.00000i	0.158486i
$638$	0	0
$639$	−18.0000	−0.712069
$640$	0	0
$641$	47.0000	1.85639	0.928194	−	0.372096i	$-0.121361\pi$
$641$	47.0000	1.85639	0.928194	+	0.372096i	$0.121361\pi$
$642$	0	0
$643$	− 34.0000i	− 1.34083i	−0.741987	−	0.670415i	$-0.766116\pi$
$643$	− 34.0000i	− 1.34083i	0.741987	−	0.670415i	$-0.233884\pi$
$644$	0	0
$645$	0	0
$646$	0	0
$647$	− 28.0000i	− 1.10079i	−0.834903	−	0.550397i	$-0.814476\pi$
$647$	− 28.0000i	− 1.10079i	0.834903	−	0.550397i	$-0.185524\pi$
$648$	0	0
$649$	40.0000	1.57014
$650$	0	0
$651$	36.0000	1.41095
$652$	0	0
$653$	8.00000i	0.313064i	0.987673	+	0.156532i	$0.0500315\pi$
$653$	8.00000i	0.313064i	−0.987673	+	0.156532i	$0.949969\pi$
$654$	0	0
$655$	0	0
$656$	0	0
$657$	42.0000i	1.63858i
$658$	0	0
$659$	49.0000	1.90877	0.954384	−	0.298580i	$-0.0965131\pi$
$659$	49.0000	1.90877	0.954384	+	0.298580i	$0.0965131\pi$
$660$	0	0
$661$	8.00000	0.311164	0.155582	−	0.987823i	$-0.450275\pi$
$661$	8.00000	0.311164	0.155582	+	0.987823i	$0.450275\pi$
$662$	0	0
$663$	− 24.0000i	− 0.932083i
$664$	0	0
$665$	0	0
$666$	0	0
$667$	36.0000i	1.39393i
$668$	0	0
$669$	−15.0000	−0.579934
$670$	0	0
$671$	−40.0000	−1.54418
$672$	0	0
$673$	− 19.0000i	− 0.732396i	−0.930537	−	0.366198i	$-0.880659\pi$
$673$	− 19.0000i	− 0.732396i	0.930537	−	0.366198i	$-0.119341\pi$
$674$	0	0
$675$	0	0
$676$	0	0
$677$	− 7.00000i	− 0.269032i	−0.990911	−	0.134516i	$-0.957052\pi$
$677$	− 7.00000i	− 0.269032i	0.990911	−	0.134516i	$-0.0429479\pi$
$678$	0	0
$679$	36.0000	1.38155
$680$	0	0
$681$	−48.0000	−1.83936
$682$	0	0
$683$	− 18.0000i	− 0.688751i	−0.938832	−	0.344375i	$-0.888091\pi$
$683$	− 18.0000i	− 0.688751i	0.938832	−	0.344375i	$-0.111909\pi$
$684$	0	0
$685$	0	0
$686$	0	0
$687$	− 63.0000i	− 2.40360i
$688$	0	0
$689$	22.0000	0.838133
$690$	0	0
$691$	28.0000	1.06517	0.532585	−	0.846376i	$-0.321221\pi$
$691$	28.0000	1.06517	0.532585	+	0.846376i	$0.321221\pi$
$692$	0	0
$693$	90.0000i	3.41882i
$694$	0	0
$695$	0	0
$696$	0	0
$697$	− 36.0000i	− 1.36360i
$698$	0	0
$699$	−42.0000	−1.58859
$700$	0	0
$701$	20.0000	0.755390	0.377695	−	0.925930i	$-0.376717\pi$
$701$	20.0000	0.755390	0.377695	+	0.925930i	$0.376717\pi$
$702$	0	0
$703$	− 4.00000i	− 0.150863i
$704$	0	0
$705$	0	0
$706$	0	0
$707$	27.0000i	1.01544i
$708$	0	0
$709$	−36.0000	−1.35201	−0.676004	−	0.736898i	$-0.736290\pi$
$709$	−36.0000	−1.35201	−0.676004	+	0.736898i	$0.736290\pi$
$710$	0	0
$711$	48.0000	1.80014
$712$	0	0
$713$	24.0000i	0.898807i
$714$	0	0
$715$	0	0
$716$	0	0
$717$	18.0000i	0.672222i
$718$	0	0
$719$	19.0000	0.708580	0.354290	−	0.935136i	$-0.384723\pi$
$719$	19.0000	0.708580	0.354290	+	0.935136i	$0.384723\pi$
$720$	0	0
$721$	18.0000	0.670355
$722$	0	0
$723$	− 42.0000i	− 1.56200i
$724$	0	0
$725$	0	0
$726$	0	0
$727$	4.00000i	0.148352i	0.997245	+	0.0741759i	$0.0236326\pi$
$727$	4.00000i	0.148352i	−0.997245	+	0.0741759i	$0.976367\pi$
$728$	0	0
$729$	27.0000	1.00000
$730$	0	0
$731$	40.0000	1.47945
$732$	0	0
$733$	− 43.0000i	− 1.58824i	−0.607760	−	0.794121i	$-0.707932\pi$
$733$	− 43.0000i	− 1.58824i	0.607760	−	0.794121i	$-0.292068\pi$
$734$	0	0
$735$	0	0
$736$	0	0
$737$	− 40.0000i	− 1.47342i
$738$	0	0
$739$	15.0000	0.551784	0.275892	−	0.961189i	$-0.411027\pi$
$739$	15.0000	0.551784	0.275892	+	0.961189i	$0.411027\pi$
$740$	0	0
$741$	−24.0000	−0.881662
$742$	0	0
$743$	9.00000i	0.330178i	0.986279	+	0.165089i	$0.0527911\pi$
$743$	9.00000i	0.330178i	−0.986279	+	0.165089i	$0.947209\pi$
$744$	0	0
$745$	0	0
$746$	0	0
$747$	− 54.0000i	− 1.97576i
$748$	0	0
$749$	12.0000	0.438470
$750$	0	0
$751$	11.0000	0.401396	0.200698	−	0.979653i	$-0.435679\pi$
$751$	11.0000	0.401396	0.200698	+	0.979653i	$0.435679\pi$
$752$	0	0
$753$	30.0000i	1.09326i
$754$	0	0
$755$	0	0
$756$	0	0
$757$	− 2.00000i	− 0.0726912i	−0.999339	−	0.0363456i	$-0.988428\pi$
$757$	− 2.00000i	− 0.0726912i	0.999339	−	0.0363456i	$-0.0115717\pi$
$758$	0	0
$759$	−90.0000	−3.26679
$760$	0	0
$761$	−3.00000	−0.108750	−0.0543750	−	0.998521i	$-0.517317\pi$
$761$	−3.00000	−0.108750	−0.0543750	+	0.998521i	$0.517317\pi$
$762$	0	0
$763$	36.0000i	1.30329i
$764$	0	0
$765$	0	0
$766$	0	0
$767$	− 16.0000i	− 0.577727i
$768$	0	0
$769$	−42.0000	−1.51456	−0.757279	−	0.653091i	$-0.773472\pi$
$769$	−42.0000	−1.51456	−0.757279	+	0.653091i	$0.773472\pi$
$770$	0	0
$771$	−48.0000	−1.72868
$772$	0	0
$773$	− 35.0000i	− 1.25886i	−0.777056	−	0.629431i	$-0.783288\pi$
$773$	− 35.0000i	− 1.25886i	0.777056	−	0.629431i	$-0.216712\pi$
$774$	0	0
$775$	0	0
$776$	0	0
$777$	9.00000i	0.322873i
$778$	0	0
$779$	−36.0000	−1.28983
$780$	0	0
$781$	15.0000	0.536742
$782$	0	0
$783$	− 54.0000i	− 1.92980i
$784$	0	0
$785$	0	0
$786$	0	0
$787$	21.0000i	0.748569i	0.927314	+	0.374285i	$0.122112\pi$
$787$	21.0000i	0.748569i	−0.927314	+	0.374285i	$0.877888\pi$
$788$	0	0
$789$	69.0000	2.45647
$790$	0	0
$791$	−42.0000	−1.49335
$792$	0	0
$793$	16.0000i	0.568177i
$794$	0	0
$795$	0	0
$796$	0	0
$797$	12.0000i	0.425062i	0.977154	+	0.212531i	$0.0681706\pi$
$797$	12.0000i	0.425062i	−0.977154	+	0.212531i	$0.931829\pi$
$798$	0	0
$799$	44.0000	1.55661
$800$	0	0
$801$	−96.0000	−3.39199
$802$	0	0
$803$	− 35.0000i	− 1.23512i
$804$	0	0
$805$	0	0
$806$	0	0
$807$	− 66.0000i	− 2.32331i
$808$	0	0
$809$	46.0000	1.61727	0.808637	−	0.588308i	$-0.200206\pi$
$809$	46.0000	1.61727	0.808637	+	0.588308i	$0.200206\pi$
$810$	0	0
$811$	49.0000	1.72062	0.860311	−	0.509769i	$-0.170269\pi$
$811$	49.0000	1.72062	0.860311	+	0.509769i	$0.170269\pi$
$812$	0	0
$813$	15.0000i	0.526073i
$814$	0	0
$815$	0	0
$816$	0	0
$817$	− 40.0000i	− 1.39942i
$818$	0	0
$819$	36.0000	1.25794
$820$	0	0
$821$	21.0000	0.732905	0.366453	−	0.930437i	$-0.380572\pi$
$821$	21.0000	0.732905	0.366453	+	0.930437i	$0.380572\pi$
$822$	0	0
$823$	− 8.00000i	− 0.278862i	−0.990232	−	0.139431i	$-0.955473\pi$
$823$	− 8.00000i	− 0.278862i	0.990232	−	0.139431i	$-0.0445274\pi$
$824$	0	0
$825$	0	0
$826$	0	0
$827$	− 30.0000i	− 1.04320i	−0.853189	−	0.521601i	$-0.825335\pi$
$827$	− 30.0000i	− 1.04320i	0.853189	−	0.521601i	$-0.174665\pi$
$828$	0	0
$829$	−52.0000	−1.80603	−0.903017	−	0.429604i	$-0.858653\pi$
$829$	−52.0000	−1.80603	−0.903017	+	0.429604i	$0.858653\pi$
$830$	0	0
$831$	−12.0000	−0.416275
$832$	0	0
$833$	− 8.00000i	− 0.277184i
$834$	0	0
$835$	0	0
$836$	0	0
$837$	− 36.0000i	− 1.24434i
$838$	0	0
$839$	−12.0000	−0.414286	−0.207143	−	0.978311i	$-0.566417\pi$
$839$	−12.0000	−0.414286	−0.207143	+	0.978311i	$0.566417\pi$
$840$	0	0
$841$	7.00000	0.241379
$842$	0	0
$843$	12.0000i	0.413302i
$844$	0	0
$845$	0	0
$846$	0	0
$847$	− 42.0000i	− 1.44314i
$848$	0	0
$849$	−12.0000	−0.411839
$850$	0	0
$851$	−6.00000	−0.205677
$852$	0	0
$853$	− 10.0000i	− 0.342393i	−0.985237	−	0.171197i	$-0.945237\pi$
$853$	− 10.0000i	− 0.342393i	0.985237	−	0.171197i	$-0.0547634\pi$
$854$	0	0
$855$	0	0
$856$	0	0
$857$	− 12.0000i	− 0.409912i	−0.978771	−	0.204956i	$-0.934295\pi$
$857$	− 12.0000i	− 0.409912i	0.978771	−	0.204956i	$-0.0657052\pi$
$858$	0	0
$859$	−56.0000	−1.91070	−0.955348	−	0.295484i	$-0.904519\pi$
$859$	−56.0000	−1.91070	−0.955348	+	0.295484i	$0.904519\pi$
$860$	0	0
$861$	81.0000	2.76047
$862$	0	0
$863$	24.0000i	0.816970i	0.912765	+	0.408485i	$0.133943\pi$
$863$	24.0000i	0.816970i	−0.912765	+	0.408485i	$0.866057\pi$
$864$	0	0
$865$	0	0
$866$	0	0
$867$	− 3.00000i	− 0.101885i
$868$	0	0
$869$	−40.0000	−1.35691
$870$	0	0
$871$	−16.0000	−0.542139
$872$	0	0
$873$	− 72.0000i	− 2.43683i
$874$	0	0
$875$	0	0
$876$	0	0
$877$	− 54.0000i	− 1.82345i	−0.410801	−	0.911725i	$-0.634751\pi$
$877$	− 54.0000i	− 1.82345i	0.410801	−	0.911725i	$-0.365249\pi$
$878$	0	0
$879$	−90.0000	−3.03562
$880$	0	0
$881$	42.0000	1.41502	0.707508	−	0.706705i	$-0.249819\pi$
$881$	42.0000	1.41502	0.707508	+	0.706705i	$0.249819\pi$
$882$	0	0
$883$	16.0000i	0.538443i	0.963078	+	0.269221i	$0.0867663\pi$
$883$	16.0000i	0.538443i	−0.963078	+	0.269221i	$0.913234\pi$
$884$	0	0
$885$	0	0
$886$	0	0
$887$	− 37.0000i	− 1.24234i	−0.783676	−	0.621169i	$-0.786658\pi$
$887$	− 37.0000i	− 1.24234i	0.783676	−	0.621169i	$-0.213342\pi$
$888$	0	0
$889$	−39.0000	−1.30802
$890$	0	0
$891$	45.0000	1.50756
$892$	0	0
$893$	− 44.0000i	− 1.47240i
$894$	0	0
$895$	0	0
$896$	0	0
$897$	36.0000i	1.20201i
$898$	0	0
$899$	24.0000	0.800445
$900$	0	0
$901$	−44.0000	−1.46585
$902$	0	0
$903$	90.0000i	2.99501i
$904$	0	0
$905$	0	0
$906$	0	0
$907$	4.00000i	0.132818i	0.997792	+	0.0664089i	$0.0211542\pi$
$907$	4.00000i	0.132818i	−0.997792	+	0.0664089i	$0.978846\pi$
$908$	0	0
$909$	54.0000	1.79107
$910$	0	0
$911$	42.0000	1.39152	0.695761	−	0.718273i	$-0.255067\pi$
$911$	42.0000	1.39152	0.695761	+	0.718273i	$0.255067\pi$
$912$	0	0
$913$	45.0000i	1.48928i
$914$	0	0
$915$	0	0
$916$	0	0
$917$	0	0
$918$	0	0
$919$	−14.0000	−0.461817	−0.230909	−	0.972975i	$-0.574170\pi$
$919$	−14.0000	−0.461817	−0.230909	+	0.972975i	$0.574170\pi$
$920$	0	0
$921$	−69.0000	−2.27363
$922$	0	0
$923$	− 6.00000i	− 0.197492i
$924$	0	0
$925$	0	0
$926$	0	0
$927$	− 36.0000i	− 1.18240i
$928$	0	0
$929$	−18.0000	−0.590561	−0.295280	−	0.955411i	$-0.595413\pi$
$929$	−18.0000	−0.590561	−0.295280	+	0.955411i	$0.595413\pi$
$930$	0	0
$931$	−8.00000	−0.262189
$932$	0	0
$933$	− 36.0000i	− 1.17859i
$934$	0	0
$935$	0	0
$936$	0	0
$937$	13.0000i	0.424691i	0.977195	+	0.212346i	$0.0681103\pi$
$937$	13.0000i	0.424691i	−0.977195	+	0.212346i	$0.931890\pi$
$938$	0	0
$939$	66.0000	2.15383
$940$	0	0
$941$	−18.0000	−0.586783	−0.293392	−	0.955992i	$-0.594784\pi$
$941$	−18.0000	−0.586783	−0.293392	+	0.955992i	$0.594784\pi$
$942$	0	0
$943$	54.0000i	1.75848i
$944$	0	0
$945$	0	0
$946$	0	0
$947$	8.00000i	0.259965i	0.991516	+	0.129983i	$0.0414921\pi$
$947$	8.00000i	0.259965i	−0.991516	+	0.129983i	$0.958508\pi$
$948$	0	0
$949$	−14.0000	−0.454459
$950$	0	0
$951$	42.0000	1.36194
$952$	0	0
$953$	51.0000i	1.65205i	0.563632	+	0.826026i	$0.309404\pi$
$953$	51.0000i	1.65205i	−0.563632	+	0.826026i	$0.690596\pi$
$954$	0	0
$955$	0	0
$956$	0	0
$957$	90.0000i	2.90929i
$958$	0	0
$959$	54.0000	1.74375
$960$	0	0
$961$	−15.0000	−0.483871
$962$	0	0
$963$	− 24.0000i	− 0.773389i
$964$	0	0
$965$	0	0
$966$	0	0
$967$	− 2.00000i	− 0.0643157i	−0.999483	−	0.0321578i	$-0.989762\pi$
$967$	− 2.00000i	− 0.0643157i	0.999483	−	0.0321578i	$-0.0102379\pi$
$968$	0	0
$969$	48.0000	1.54198
$970$	0	0
$971$	−24.0000	−0.770197	−0.385098	−	0.922876i	$-0.625832\pi$
$971$	−24.0000	−0.770197	−0.385098	+	0.922876i	$0.625832\pi$
$972$	0	0
$973$	− 12.0000i	− 0.384702i
$974$	0	0
$975$	0	0
$976$	0	0
$977$	28.0000i	0.895799i	0.894084	+	0.447900i	$0.147828\pi$
$977$	28.0000i	0.895799i	−0.894084	+	0.447900i	$0.852172\pi$
$978$	0	0
$979$	80.0000	2.55681
$980$	0	0
$981$	72.0000	2.29878
$982$	0	0
$983$	− 27.0000i	− 0.861166i	−0.902551	−	0.430583i	$-0.858308\pi$
$983$	− 27.0000i	− 0.861166i	0.902551	−	0.430583i	$-0.141692\pi$
$984$	0	0
$985$	0	0
$986$	0	0
$987$	99.0000i	3.15120i
$988$	0	0
$989$	−60.0000	−1.90789
$990$	0	0
$991$	−34.0000	−1.08005	−0.540023	−	0.841650i	$-0.681584\pi$
$991$	−34.0000	−1.08005	−0.540023	+	0.841650i	$0.681584\pi$
$992$	0	0
$993$	66.0000i	2.09445i
$994$	0	0
$995$	0	0
$996$	0	0
$997$	46.0000i	1.45683i	0.685134	+	0.728417i	$0.259744\pi$
$997$	46.0000i	1.45683i	−0.685134	+	0.728417i	$0.740256\pi$
$998$	0	0
$999$	9.00000	0.284747

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	3700.2.d.a.149.1		2
5.2	odd	4		3700.2.a.a.1.1		1
5.3	odd	4		740.2.a.c.1.1	✓	1
5.4	even	2	inner	3700.2.d.a.149.2		2
15.8	even	4		6660.2.a.b.1.1		1
20.3	even	4		2960.2.a.a.1.1		1

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
740.2.a.c.1.1	✓	1	5.3	odd	4
2960.2.a.a.1.1		1	20.3	even	4
3700.2.a.a.1.1		1	5.2	odd	4
3700.2.d.a.149.1		2	1.1	even	1	trivial
3700.2.d.a.149.2		2	5.4	even	2	inner
6660.2.a.b.1.1		1	15.8	even	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:	\( N \)	\(=\)	\( 3700 = 2^{2} \cdot 5^{2} \cdot 37 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	3700.d (of order \(2\), degree \(1\), not minimal)

\(f(q)\)	\(=\)	\(q-3.00000i q^{3} -3.00000i q^{7} -6.00000 q^{9} +O(q^{10})\)	\(q-3.00000i q^{3} -3.00000i q^{7} -6.00000 q^{9} +5.00000 q^{11} -2.00000i q^{13} +4.00000i q^{17} +4.00000 q^{19} -9.00000 q^{21} -6.00000i q^{23} +9.00000i q^{27} -6.00000 q^{29} -4.00000 q^{31} -15.0000i q^{33} -1.00000i q^{37} -6.00000 q^{39} -9.00000 q^{41} -10.0000i q^{43} -11.0000i q^{47} -2.00000 q^{49} +12.0000 q^{51} +11.0000i q^{53} -12.0000i q^{57} +8.00000 q^{59} -8.00000 q^{61} +18.0000i q^{63} -8.00000i q^{67} -18.0000 q^{69} +3.00000 q^{71} -7.00000i q^{73} -15.0000i q^{77} -8.00000 q^{79} +9.00000 q^{81} +9.00000i q^{83} +18.0000i q^{87} +16.0000 q^{89} -6.00000 q^{91} +12.0000i q^{93} +12.0000i q^{97} -30.0000 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 2 q - 12 q^{9}+O(q^{10}) \) 2 * q - 12 * q^9	\( 2 q - 12 q^{9} + 10 q^{11} + 8 q^{19} - 18 q^{21} - 12 q^{29} - 8 q^{31} - 12 q^{39} - 18 q^{41} - 4 q^{49} + 24 q^{51} + 16 q^{59} - 16 q^{61} - 36 q^{69} + 6 q^{71} - 16 q^{79} + 18 q^{81} + 32 q^{89} - 12 q^{91} - 60 q^{99}+O(q^{100}) \) 2 * q - 12 * q^9 + 10 * q^11 + 8 * q^19 - 18 * q^21 - 12 * q^29 - 8 * q^31 - 12 * q^39 - 18 * q^41 - 4 * q^49 + 24 * q^51 + 16 * q^59 - 16 * q^61 - 36 * q^69 + 6 * q^71 - 16 * q^79 + 18 * q^81 + 32 * q^89 - 12 * q^91 - 60 * q^99

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