LMFDB - Embedded newform 2600.2.d.i.1249.1

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [2600,2,Mod(1249,2600)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(2600, base_ring=CyclotomicField(2))

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

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

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

chi := DirichletCharacter("2600.1249");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 2600 = 2^{3} \cdot 5^{2} \cdot 13 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	2600.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:	$20.7611045255$
Analytic rank:	$0$
Dimension:	$4$
Coefficient field:	$\Q(\zeta_{8})$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{4} + 1 $ x^4 + 1
Coefficient ring:	$\Z[a_1, \ldots, a_{13}]$
Coefficient ring index:	$ 2 $
Twist minimal:	no (minimal twist has level 520)
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label			1249.1
Root			$-0.707107 + 0.707107i$ of defining polynomial
Character	$\chi$	$=$	2600.1249
Dual form			2600.2.d.i.1249.4

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-3.41421i q^{3} +2.00000i q^{7} -8.65685 q^{9} +O(q^{10})$	$q-3.41421i q^{3} +2.00000i q^{7} -8.65685 q^{9} +4.24264 q^{11} -1.00000i q^{13} +4.82843i q^{17} +8.24264 q^{19} +6.82843 q^{21} +5.07107i q^{23} +19.3137i q^{27} +9.65685 q^{29} -1.41421 q^{31} -14.4853i q^{33} +1.17157i q^{37} -3.41421 q^{39} -0.828427 q^{41} -1.75736i q^{43} -2.00000i q^{47} +3.00000 q^{49} +16.4853 q^{51} -3.17157i q^{53} -28.1421i q^{57} +5.41421 q^{59} -7.31371 q^{61} -17.3137i q^{63} -0.828427i q^{67} +17.3137 q^{69} -9.89949 q^{71} +6.82843i q^{73} +8.48528i q^{77} -6.82843 q^{79} +39.9706 q^{81} -2.00000i q^{83} -32.9706i q^{87} +10.0000 q^{89} +2.00000 q^{91} +4.82843i q^{93} -0.343146i q^{97} -36.7279 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 4 q - 12 q^{9}+O(q^{10}) $ 4 * q - 12 * q^9	$ 4 q - 12 q^{9} + 16 q^{19} + 16 q^{21} + 16 q^{29} - 8 q^{39} + 8 q^{41} + 12 q^{49} + 32 q^{51} + 16 q^{59} + 16 q^{61} + 24 q^{69} - 16 q^{79} + 92 q^{81} + 40 q^{89} + 8 q^{91} - 96 q^{99}+O(q^{100}) $ 4 * q - 12 * q^9 + 16 * q^19 + 16 * q^21 + 16 * q^29 - 8 * q^39 + 8 * q^41 + 12 * q^49 + 32 * q^51 + 16 * q^59 + 16 * q^61 + 24 * q^69 - 16 * q^79 + 92 * q^81 + 40 * q^89 + 8 * q^91 - 96 * q^99

Display 100 coefficients 10 coefficients

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/2600\mathbb{Z}\right)^\times$.

$n$	$1301$	$1601$	$1951$	$1977$
$\chi(n)$	$1$	$1$	$1$	$-1$

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
$3$	− 3.41421i	− 1.97120i	−0.169102	−	0.985599i	$-0.554087\pi$
$3$	− 3.41421i	− 1.97120i	0.169102	−	0.985599i	$-0.445913\pi$
$4$	0	0
$5$	0	0
$5$	0	0
$6$	0	0
$7$	2.00000i	0.755929i	0.925820	+	0.377964i	$0.123376\pi$
$7$	2.00000i	0.755929i	−0.925820	+	0.377964i	$0.876624\pi$
$8$	0	0
$9$	−8.65685	−2.88562
$10$	0	0
$11$	4.24264	1.27920	0.639602	−	0.768706i	$-0.279099\pi$
$11$	4.24264	1.27920	0.639602	+	0.768706i	$0.279099\pi$
$12$	0	0
$13$	− 1.00000i	− 0.277350i
$13$	− 1.00000i	− 0.277350i
$14$	0	0
$15$	0	0
$16$	0	0
$17$	4.82843i	1.17107i	0.810649	+	0.585533i	$0.199115\pi$
$17$	4.82843i	1.17107i	−0.810649	+	0.585533i	$0.800885\pi$
$18$	0	0
$19$	8.24264	1.89099	0.945496	−	0.325634i	$-0.105578\pi$
$19$	8.24264	1.89099	0.945496	+	0.325634i	$0.105578\pi$
$20$	0	0
$21$	6.82843	1.49008
$22$	0	0
$23$	5.07107i	1.05739i	0.848812	+	0.528695i	$0.177319\pi$
$23$	5.07107i	1.05739i	−0.848812	+	0.528695i	$0.822681\pi$
$24$	0	0
$25$	0	0
$26$	0	0
$27$	19.3137i	3.71692i
$28$	0	0
$29$	9.65685	1.79323	0.896616	−	0.442808i	$-0.146018\pi$
$29$	9.65685	1.79323	0.896616	+	0.442808i	$0.146018\pi$
$30$	0	0
$31$	−1.41421	−0.254000	−0.127000	−	0.991903i	$-0.540535\pi$
$31$	−1.41421	−0.254000	−0.127000	+	0.991903i	$0.540535\pi$
$32$	0	0
$33$	− 14.4853i	− 2.52156i
$34$	0	0
$35$	0	0
$36$	0	0
$37$	1.17157i	0.192605i	0.995352	+	0.0963027i	$0.0307017\pi$
$37$	1.17157i	0.192605i	−0.995352	+	0.0963027i	$0.969298\pi$
$38$	0	0
$39$	−3.41421	−0.546712
$40$	0	0
$41$	−0.828427	−0.129379	−0.0646893	−	0.997905i	$-0.520606\pi$
$41$	−0.828427	−0.129379	−0.0646893	+	0.997905i	$0.520606\pi$
$42$	0	0
$43$	− 1.75736i	− 0.267995i	−0.990982	−	0.133997i	$-0.957219\pi$
$43$	− 1.75736i	− 0.267995i	0.990982	−	0.133997i	$-0.0427814\pi$
$44$	0	0
$45$	0	0
$46$	0	0
$47$	− 2.00000i	− 0.291730i	−0.989305	−	0.145865i	$-0.953403\pi$
$47$	− 2.00000i	− 0.291730i	0.989305	−	0.145865i	$-0.0465965\pi$
$48$	0	0
$49$	3.00000	0.428571
$50$	0	0
$51$	16.4853	2.30840
$52$	0	0
$53$	− 3.17157i	− 0.435649i	−0.975988	−	0.217825i	$-0.930104\pi$
$53$	− 3.17157i	− 0.435649i	0.975988	−	0.217825i	$-0.0698960\pi$
$54$	0	0
$55$	0	0
$56$	0	0
$57$	− 28.1421i	− 3.72752i
$58$	0	0
$59$	5.41421	0.704871	0.352435	−	0.935836i	$-0.385354\pi$
$59$	5.41421	0.704871	0.352435	+	0.935836i	$0.385354\pi$
$60$	0	0
$61$	−7.31371	−0.936424	−0.468212	−	0.883616i	$-0.655102\pi$
$61$	−7.31371	−0.936424	−0.468212	+	0.883616i	$0.655102\pi$
$62$	0	0
$63$	− 17.3137i	− 2.18132i
$64$	0	0
$65$	0	0
$66$	0	0
$67$	− 0.828427i	− 0.101208i	−0.998719	−	0.0506042i	$-0.983885\pi$
$67$	− 0.828427i	− 0.101208i	0.998719	−	0.0506042i	$-0.0161147\pi$
$68$	0	0
$69$	17.3137	2.08433
$70$	0	0
$71$	−9.89949	−1.17485	−0.587427	−	0.809277i	$-0.699859\pi$
$71$	−9.89949	−1.17485	−0.587427	+	0.809277i	$0.699859\pi$
$72$	0	0
$73$	6.82843i	0.799207i	0.916688	+	0.399603i	$0.130852\pi$
$73$	6.82843i	0.799207i	−0.916688	+	0.399603i	$0.869148\pi$
$74$	0	0
$75$	0	0
$76$	0	0
$77$	8.48528i	0.966988i
$78$	0	0
$79$	−6.82843	−0.768258	−0.384129	−	0.923279i	$-0.625498\pi$
$79$	−6.82843	−0.768258	−0.384129	+	0.923279i	$0.625498\pi$
$80$	0	0
$81$	39.9706	4.44117
$82$	0	0
$83$	− 2.00000i	− 0.219529i	−0.993958	−	0.109764i	$-0.964990\pi$
$83$	− 2.00000i	− 0.219529i	0.993958	−	0.109764i	$-0.0350096\pi$
$84$	0	0
$85$	0	0
$86$	0	0
$87$	− 32.9706i	− 3.53482i
$88$	0	0
$89$	10.0000	1.06000	0.529999	−	0.847998i	$-0.322192\pi$
$89$	10.0000	1.06000	0.529999	+	0.847998i	$0.322192\pi$
$90$	0	0
$91$	2.00000	0.209657
$92$	0	0
$93$	4.82843i	0.500685i
$94$	0	0
$95$	0	0
$96$	0	0
$97$	− 0.343146i	− 0.0348412i	−0.999848	−	0.0174206i	$-0.994455\pi$
$97$	− 0.343146i	− 0.0348412i	0.999848	−	0.0174206i	$-0.00554543\pi$
$98$	0	0
$99$	−36.7279	−3.69130
$100$	0	0
$101$	7.65685	0.761885	0.380943	−	0.924599i	$-0.375599\pi$
$101$	7.65685	0.761885	0.380943	+	0.924599i	$0.375599\pi$
$102$	0	0
$103$	8.58579i	0.845983i	0.906134	+	0.422991i	$0.139020\pi$
$103$	8.58579i	0.845983i	−0.906134	+	0.422991i	$0.860980\pi$
$104$	0	0
$105$	0	0
$106$	0	0
$107$	9.07107i	0.876933i	0.898747	+	0.438467i	$0.144478\pi$
$107$	9.07107i	0.876933i	−0.898747	+	0.438467i	$0.855522\pi$
$108$	0	0
$109$	5.31371	0.508961	0.254480	−	0.967078i	$-0.418096\pi$
$109$	5.31371	0.508961	0.254480	+	0.967078i	$0.418096\pi$
$110$	0	0
$111$	4.00000	0.379663
$112$	0	0
$113$	− 3.17157i	− 0.298356i	−0.988810	−	0.149178i	$-0.952337\pi$
$113$	− 3.17157i	− 0.298356i	0.988810	−	0.149178i	$-0.0476628\pi$
$114$	0	0
$115$	0	0
$116$	0	0
$117$	8.65685i	0.800326i
$118$	0	0
$119$	−9.65685	−0.885242
$120$	0	0
$121$	7.00000	0.636364
$122$	0	0
$123$	2.82843i	0.255031i
$124$	0	0
$125$	0	0
$126$	0	0
$127$	3.89949i	0.346024i	0.984920	+	0.173012i	$0.0553500\pi$
$127$	3.89949i	0.346024i	−0.984920	+	0.173012i	$0.944650\pi$
$128$	0	0
$129$	−6.00000	−0.528271
$130$	0	0
$131$	3.31371	0.289520	0.144760	−	0.989467i	$-0.453759\pi$
$131$	3.31371	0.289520	0.144760	+	0.989467i	$0.453759\pi$
$132$	0	0
$133$	16.4853i	1.42946i
$134$	0	0
$135$	0	0
$136$	0	0
$137$	− 2.00000i	− 0.170872i	−0.996344	−	0.0854358i	$-0.972772\pi$
$137$	− 2.00000i	− 0.170872i	0.996344	−	0.0854358i	$-0.0272282\pi$
$138$	0	0
$139$	−8.48528	−0.719712	−0.359856	−	0.933008i	$-0.617174\pi$
$139$	−8.48528	−0.719712	−0.359856	+	0.933008i	$0.617174\pi$
$140$	0	0
$141$	−6.82843	−0.575057
$142$	0	0
$143$	− 4.24264i	− 0.354787i
$144$	0	0
$145$	0	0
$146$	0	0
$147$	− 10.2426i	− 0.844799i
$148$	0	0
$149$	−10.9706	−0.898744	−0.449372	−	0.893345i	$-0.648352\pi$
$149$	−10.9706	−0.898744	−0.449372	+	0.893345i	$0.648352\pi$
$150$	0	0
$151$	0.928932	0.0755954	0.0377977	−	0.999285i	$-0.487966\pi$
$151$	0.928932	0.0755954	0.0377977	+	0.999285i	$0.487966\pi$
$152$	0	0
$153$	− 41.7990i	− 3.37925i
$154$	0	0
$155$	0	0
$156$	0	0
$157$	− 5.31371i	− 0.424080i	−0.977261	−	0.212040i	$-0.931989\pi$
$157$	− 5.31371i	− 0.424080i	0.977261	−	0.212040i	$-0.0680107\pi$
$158$	0	0
$159$	−10.8284	−0.858750
$160$	0	0
$161$	−10.1421	−0.799312
$162$	0	0
$163$	− 22.4853i	− 1.76118i	−0.473876	−	0.880592i	$-0.657146\pi$
$163$	− 22.4853i	− 1.76118i	0.473876	−	0.880592i	$-0.342854\pi$
$164$	0	0
$165$	0	0
$166$	0	0
$167$	13.3137i	1.03025i	0.857116	+	0.515123i	$0.172254\pi$
$167$	13.3137i	1.03025i	−0.857116	+	0.515123i	$0.827746\pi$
$168$	0	0
$169$	−1.00000	−0.0769231
$170$	0	0
$171$	−71.3553	−5.45668
$172$	0	0
$173$	− 20.8284i	− 1.58356i	−0.610809	−	0.791778i	$-0.709156\pi$
$173$	− 20.8284i	− 1.58356i	0.610809	−	0.791778i	$-0.290844\pi$
$174$	0	0
$175$	0	0
$176$	0	0
$177$	− 18.4853i	− 1.38944i
$178$	0	0
$179$	11.3137	0.845626	0.422813	−	0.906217i	$-0.361043\pi$
$179$	11.3137	0.845626	0.422813	+	0.906217i	$0.361043\pi$
$180$	0	0
$181$	23.3137	1.73289	0.866447	−	0.499269i	$-0.166398\pi$
$181$	23.3137	1.73289	0.866447	+	0.499269i	$0.166398\pi$
$182$	0	0
$183$	24.9706i	1.84588i
$184$	0	0
$185$	0	0
$186$	0	0
$187$	20.4853i	1.49803i
$188$	0	0
$189$	−38.6274	−2.80973
$190$	0	0
$191$	−21.6569	−1.56703	−0.783517	−	0.621370i	$-0.786577\pi$
$191$	−21.6569	−1.56703	−0.783517	+	0.621370i	$0.786577\pi$
$192$	0	0
$193$	− 8.34315i	− 0.600553i	−0.953852	−	0.300276i	$-0.902921\pi$
$193$	− 8.34315i	− 0.600553i	0.953852	−	0.300276i	$-0.0970789\pi$
$194$	0	0
$195$	0	0
$196$	0	0
$197$	0.343146i	0.0244481i	0.999925	+	0.0122241i	$0.00389114\pi$
$197$	0.343146i	0.0244481i	−0.999925	+	0.0122241i	$0.996109\pi$
$198$	0	0
$199$	−6.34315	−0.449654	−0.224827	−	0.974399i	$-0.572182\pi$
$199$	−6.34315	−0.449654	−0.224827	+	0.974399i	$0.572182\pi$
$200$	0	0
$201$	−2.82843	−0.199502
$202$	0	0
$203$	19.3137i	1.35556i
$204$	0	0
$205$	0	0
$206$	0	0
$207$	− 43.8995i	− 3.05123i
$208$	0	0
$209$	34.9706	2.41896
$210$	0	0
$211$	5.65685	0.389434	0.194717	−	0.980859i	$-0.437621\pi$
$211$	5.65685	0.389434	0.194717	+	0.980859i	$0.437621\pi$
$212$	0	0
$213$	33.7990i	2.31587i
$214$	0	0
$215$	0	0
$216$	0	0
$217$	− 2.82843i	− 0.192006i
$218$	0	0
$219$	23.3137	1.57539
$220$	0	0
$221$	4.82843	0.324795
$222$	0	0
$223$	25.3137i	1.69513i	0.530691	+	0.847566i	$0.321933\pi$
$223$	25.3137i	1.69513i	−0.530691	+	0.847566i	$0.678067\pi$
$224$	0	0
$225$	0	0
$226$	0	0
$227$	− 20.8284i	− 1.38243i	−0.722649	−	0.691216i	$-0.757075\pi$
$227$	− 20.8284i	− 1.38243i	0.722649	−	0.691216i	$-0.242925\pi$
$228$	0	0
$229$	13.7990	0.911863	0.455931	−	0.890015i	$-0.349306\pi$
$229$	13.7990	0.911863	0.455931	+	0.890015i	$0.349306\pi$
$230$	0	0
$231$	28.9706	1.90612
$232$	0	0
$233$	− 22.0000i	− 1.44127i	−0.693316	−	0.720634i	$-0.743851\pi$
$233$	− 22.0000i	− 1.44127i	0.693316	−	0.720634i	$-0.256149\pi$
$234$	0	0
$235$	0	0
$236$	0	0
$237$	23.3137i	1.51439i
$238$	0	0
$239$	−8.72792	−0.564562	−0.282281	−	0.959332i	$-0.591091\pi$
$239$	−8.72792	−0.564562	−0.282281	+	0.959332i	$0.591091\pi$
$240$	0	0
$241$	7.17157	0.461962	0.230981	−	0.972958i	$-0.425807\pi$
$241$	7.17157	0.461962	0.230981	+	0.972958i	$0.425807\pi$
$242$	0	0
$243$	− 78.5269i	− 5.03750i
$244$	0	0
$245$	0	0
$246$	0	0
$247$	− 8.24264i	− 0.524467i
$248$	0	0
$249$	−6.82843	−0.432734
$250$	0	0
$251$	10.1421	0.640166	0.320083	−	0.947390i	$-0.396289\pi$
$251$	10.1421	0.640166	0.320083	+	0.947390i	$0.396289\pi$
$252$	0	0
$253$	21.5147i	1.35262i
$254$	0	0
$255$	0	0
$256$	0	0
$257$	− 14.9706i	− 0.933838i	−0.884300	−	0.466919i	$-0.845364\pi$
$257$	− 14.9706i	− 0.933838i	0.884300	−	0.466919i	$-0.154636\pi$
$258$	0	0
$259$	−2.34315	−0.145596
$260$	0	0
$261$	−83.5980	−5.17459
$262$	0	0
$263$	13.7574i	0.848315i	0.905588	+	0.424158i	$0.139430\pi$
$263$	13.7574i	0.848315i	−0.905588	+	0.424158i	$0.860570\pi$
$264$	0	0
$265$	0	0
$266$	0	0
$267$	− 34.1421i	− 2.08946i
$268$	0	0
$269$	−16.6274	−1.01379	−0.506896	−	0.862007i	$-0.669207\pi$
$269$	−16.6274	−1.01379	−0.506896	+	0.862007i	$0.669207\pi$
$270$	0	0
$271$	9.89949	0.601351	0.300676	−	0.953726i	$-0.402788\pi$
$271$	9.89949	0.601351	0.300676	+	0.953726i	$0.402788\pi$
$272$	0	0
$273$	− 6.82843i	− 0.413275i
$274$	0	0
$275$	0	0
$276$	0	0
$277$	− 4.14214i	− 0.248877i	−0.992227	−	0.124438i	$-0.960287\pi$
$277$	− 4.14214i	− 0.248877i	0.992227	−	0.124438i	$-0.0397129\pi$
$278$	0	0
$279$	12.2426	0.732948
$280$	0	0
$281$	−22.4853	−1.34136	−0.670680	−	0.741747i	$-0.733997\pi$
$281$	−22.4853	−1.34136	−0.670680	+	0.741747i	$0.733997\pi$
$282$	0	0
$283$	3.89949i	0.231801i	0.993261	+	0.115900i	$0.0369754\pi$
$283$	3.89949i	0.231801i	−0.993261	+	0.115900i	$0.963025\pi$
$284$	0	0
$285$	0	0
$286$	0	0
$287$	− 1.65685i	− 0.0978010i
$288$	0	0
$289$	−6.31371	−0.371395
$290$	0	0
$291$	−1.17157	−0.0686788
$292$	0	0
$293$	2.82843i	0.165238i	0.996581	+	0.0826192i	$0.0263285\pi$
$293$	2.82843i	0.165238i	−0.996581	+	0.0826192i	$0.973671\pi$
$294$	0	0
$295$	0	0
$296$	0	0
$297$	81.9411i	4.75471i
$298$	0	0
$299$	5.07107	0.293267
$300$	0	0
$301$	3.51472	0.202585
$302$	0	0
$303$	− 26.1421i	− 1.50183i
$304$	0	0
$305$	0	0
$306$	0	0
$307$	− 15.6569i	− 0.893584i	−0.894638	−	0.446792i	$-0.852566\pi$
$307$	− 15.6569i	− 0.893584i	0.894638	−	0.446792i	$-0.147434\pi$
$308$	0	0
$309$	29.3137	1.66760
$310$	0	0
$311$	−10.1421	−0.575108	−0.287554	−	0.957764i	$-0.592842\pi$
$311$	−10.1421	−0.575108	−0.287554	+	0.957764i	$0.592842\pi$
$312$	0	0
$313$	− 7.17157i	− 0.405361i	−0.979245	−	0.202681i	$-0.935035\pi$
$313$	− 7.17157i	− 0.405361i	0.979245	−	0.202681i	$-0.0649653\pi$
$314$	0	0
$315$	0	0
$316$	0	0
$317$	− 24.4853i	− 1.37523i	−0.726075	−	0.687615i	$-0.758658\pi$
$317$	− 24.4853i	− 1.37523i	0.726075	−	0.687615i	$-0.241342\pi$
$318$	0	0
$319$	40.9706	2.29391
$320$	0	0
$321$	30.9706	1.72861
$322$	0	0
$323$	39.7990i	2.21448i
$324$	0	0
$325$	0	0
$326$	0	0
$327$	− 18.1421i	− 1.00326i
$328$	0	0
$329$	4.00000	0.220527
$330$	0	0
$331$	−0.928932	−0.0510587	−0.0255294	−	0.999674i	$-0.508127\pi$
$331$	−0.928932	−0.0510587	−0.0255294	+	0.999674i	$0.508127\pi$
$332$	0	0
$333$	− 10.1421i	− 0.555786i
$334$	0	0
$335$	0	0
$336$	0	0
$337$	36.1421i	1.96879i	0.175980	+	0.984394i	$0.443691\pi$
$337$	36.1421i	1.96879i	−0.175980	+	0.984394i	$0.556309\pi$
$338$	0	0
$339$	−10.8284	−0.588119
$340$	0	0
$341$	−6.00000	−0.324918
$342$	0	0
$343$	20.0000i	1.07990i
$344$	0	0
$345$	0	0
$346$	0	0
$347$	− 4.38478i	− 0.235387i	−0.993050	−	0.117694i	$-0.962450\pi$
$347$	− 4.38478i	− 0.235387i	0.993050	−	0.117694i	$-0.0375501\pi$
$348$	0	0
$349$	−35.4558	−1.89791	−0.948954	−	0.315415i	$-0.897856\pi$
$349$	−35.4558	−1.89791	−0.948954	+	0.315415i	$0.897856\pi$
$350$	0	0
$351$	19.3137	1.03089
$352$	0	0
$353$	5.17157i	0.275255i	0.990484	+	0.137628i	$0.0439477\pi$
$353$	5.17157i	0.275255i	−0.990484	+	0.137628i	$0.956052\pi$
$354$	0	0
$355$	0	0
$356$	0	0
$357$	32.9706i	1.74499i
$358$	0	0
$359$	27.5563	1.45437	0.727184	−	0.686442i	$-0.240829\pi$
$359$	27.5563	1.45437	0.727184	+	0.686442i	$0.240829\pi$
$360$	0	0
$361$	48.9411	2.57585
$362$	0	0
$363$	− 23.8995i	− 1.25440i
$364$	0	0
$365$	0	0
$366$	0	0
$367$	22.7279i	1.18639i	0.805060	+	0.593194i	$0.202133\pi$
$367$	22.7279i	1.18639i	−0.805060	+	0.593194i	$0.797867\pi$
$368$	0	0
$369$	7.17157	0.373337
$370$	0	0
$371$	6.34315	0.329320
$372$	0	0
$373$	10.0000i	0.517780i	0.965907	+	0.258890i	$0.0833568\pi$
$373$	10.0000i	0.517780i	−0.965907	+	0.258890i	$0.916643\pi$
$374$	0	0
$375$	0	0
$376$	0	0
$377$	− 9.65685i	− 0.497353i
$378$	0	0
$379$	−16.2426	−0.834328	−0.417164	−	0.908831i	$-0.636976\pi$
$379$	−16.2426	−0.834328	−0.417164	+	0.908831i	$0.636976\pi$
$380$	0	0
$381$	13.3137	0.682082
$382$	0	0
$383$	22.0000i	1.12415i	0.827087	+	0.562074i	$0.189996\pi$
$383$	22.0000i	1.12415i	−0.827087	+	0.562074i	$0.810004\pi$
$384$	0	0
$385$	0	0
$386$	0	0
$387$	15.2132i	0.773331i
$388$	0	0
$389$	13.3137	0.675032	0.337516	−	0.941320i	$-0.390413\pi$
$389$	13.3137	0.675032	0.337516	+	0.941320i	$0.390413\pi$
$390$	0	0
$391$	−24.4853	−1.23827
$392$	0	0
$393$	− 11.3137i	− 0.570701i
$394$	0	0
$395$	0	0
$396$	0	0
$397$	− 1.17157i	− 0.0587996i	−0.999568	−	0.0293998i	$-0.990640\pi$
$397$	− 1.17157i	− 0.0587996i	0.999568	−	0.0293998i	$-0.00935959\pi$
$398$	0	0
$399$	56.2843	2.81774
$400$	0	0
$401$	28.6274	1.42958	0.714792	−	0.699337i	$-0.246521\pi$
$401$	28.6274	1.42958	0.714792	+	0.699337i	$0.246521\pi$
$402$	0	0
$403$	1.41421i	0.0704470i
$404$	0	0
$405$	0	0
$406$	0	0
$407$	4.97056i	0.246382i
$408$	0	0
$409$	−4.82843	−0.238750	−0.119375	−	0.992849i	$-0.538089\pi$
$409$	−4.82843	−0.238750	−0.119375	+	0.992849i	$0.538089\pi$
$410$	0	0
$411$	−6.82843	−0.336821
$412$	0	0
$413$	10.8284i	0.532832i
$414$	0	0
$415$	0	0
$416$	0	0
$417$	28.9706i	1.41869i
$418$	0	0
$419$	34.1421	1.66795	0.833976	−	0.551800i	$-0.186059\pi$
$419$	34.1421	1.66795	0.833976	+	0.551800i	$0.186059\pi$
$420$	0	0
$421$	35.6569	1.73781	0.868904	−	0.494980i	$-0.164825\pi$
$421$	35.6569	1.73781	0.868904	+	0.494980i	$0.164825\pi$
$422$	0	0
$423$	17.3137i	0.841821i
$424$	0	0
$425$	0	0
$426$	0	0
$427$	− 14.6274i	− 0.707870i
$428$	0	0
$429$	−14.4853	−0.699356
$430$	0	0
$431$	13.2132	0.636458	0.318229	−	0.948014i	$-0.396912\pi$
$431$	13.2132	0.636458	0.318229	+	0.948014i	$0.396912\pi$
$432$	0	0
$433$	− 6.97056i	− 0.334984i	−0.985873	−	0.167492i	$-0.946433\pi$
$433$	− 6.97056i	− 0.334984i	0.985873	−	0.167492i	$-0.0535668\pi$
$434$	0	0
$435$	0	0
$436$	0	0
$437$	41.7990i	1.99952i
$438$	0	0
$439$	−28.2843	−1.34993	−0.674967	−	0.737848i	$-0.735842\pi$
$439$	−28.2843	−1.34993	−0.674967	+	0.737848i	$0.735842\pi$
$440$	0	0
$441$	−25.9706	−1.23669
$442$	0	0
$443$	− 3.41421i	− 0.162214i	−0.996705	−	0.0811071i	$-0.974154\pi$
$443$	− 3.41421i	− 0.162214i	0.996705	−	0.0811071i	$-0.0258456\pi$
$444$	0	0
$445$	0	0
$446$	0	0
$447$	37.4558i	1.77160i
$448$	0	0
$449$	−7.85786	−0.370836	−0.185418	−	0.982660i	$-0.559364\pi$
$449$	−7.85786	−0.370836	−0.185418	+	0.982660i	$0.559364\pi$
$450$	0	0
$451$	−3.51472	−0.165502
$452$	0	0
$453$	− 3.17157i	− 0.149013i
$454$	0	0
$455$	0	0
$456$	0	0
$457$	32.6274i	1.52625i	0.646253	+	0.763123i	$0.276335\pi$
$457$	32.6274i	1.52625i	−0.646253	+	0.763123i	$0.723665\pi$
$458$	0	0
$459$	−93.2548	−4.35276
$460$	0	0
$461$	−20.1421	−0.938113	−0.469056	−	0.883168i	$-0.655406\pi$
$461$	−20.1421	−0.938113	−0.469056	+	0.883168i	$0.655406\pi$
$462$	0	0
$463$	22.4853i	1.04498i	0.852646	+	0.522490i	$0.174997\pi$
$463$	22.4853i	1.04498i	−0.852646	+	0.522490i	$0.825003\pi$
$464$	0	0
$465$	0	0
$466$	0	0
$467$	− 14.2426i	− 0.659071i	−0.944143	−	0.329535i	$-0.893108\pi$
$467$	− 14.2426i	− 0.659071i	0.944143	−	0.329535i	$-0.106892\pi$
$468$	0	0
$469$	1.65685	0.0765064
$470$	0	0
$471$	−18.1421	−0.835945
$472$	0	0
$473$	− 7.45584i	− 0.342820i
$474$	0	0
$475$	0	0
$476$	0	0
$477$	27.4558i	1.25712i
$478$	0	0
$479$	32.2426	1.47320	0.736602	−	0.676327i	$-0.236429\pi$
$479$	32.2426	1.47320	0.736602	+	0.676327i	$0.236429\pi$
$480$	0	0
$481$	1.17157	0.0534191
$482$	0	0
$483$	34.6274i	1.57560i
$484$	0	0
$485$	0	0
$486$	0	0
$487$	− 6.48528i	− 0.293876i	−0.989146	−	0.146938i	$-0.953058\pi$
$487$	− 6.48528i	− 0.293876i	0.989146	−	0.146938i	$-0.0469418\pi$
$488$	0	0
$489$	−76.7696	−3.47164
$490$	0	0
$491$	−19.1127	−0.862544	−0.431272	−	0.902222i	$-0.641935\pi$
$491$	−19.1127	−0.862544	−0.431272	+	0.902222i	$0.641935\pi$
$492$	0	0
$493$	46.6274i	2.09999i
$494$	0	0
$495$	0	0
$496$	0	0
$497$	− 19.7990i	− 0.888106i
$498$	0	0
$499$	26.5858	1.19014	0.595072	−	0.803673i	$-0.297124\pi$
$499$	26.5858	1.19014	0.595072	+	0.803673i	$0.297124\pi$
$500$	0	0
$501$	45.4558	2.03082
$502$	0	0
$503$	− 27.4142i	− 1.22234i	−0.791500	−	0.611170i	$-0.790699\pi$
$503$	− 27.4142i	− 1.22234i	0.791500	−	0.611170i	$-0.209301\pi$
$504$	0	0
$505$	0	0
$506$	0	0
$507$	3.41421i	0.151631i
$508$	0	0
$509$	−17.5147	−0.776326	−0.388163	−	0.921591i	$-0.626890\pi$
$509$	−17.5147	−0.776326	−0.388163	+	0.921591i	$0.626890\pi$
$510$	0	0
$511$	−13.6569	−0.604144
$512$	0	0
$513$	159.196i	7.02867i
$514$	0	0
$515$	0	0
$516$	0	0
$517$	− 8.48528i	− 0.373182i
$518$	0	0
$519$	−71.1127	−3.12150
$520$	0	0
$521$	−12.6863	−0.555797	−0.277898	−	0.960611i	$-0.589638\pi$
$521$	−12.6863	−0.555797	−0.277898	+	0.960611i	$0.589638\pi$
$522$	0	0
$523$	− 23.6985i	− 1.03626i	−0.855301	−	0.518131i	$-0.826628\pi$
$523$	− 23.6985i	− 1.03626i	0.855301	−	0.518131i	$-0.173372\pi$
$524$	0	0
$525$	0	0
$526$	0	0
$527$	− 6.82843i	− 0.297451i
$528$	0	0
$529$	−2.71573	−0.118075
$530$	0	0
$531$	−46.8701	−2.03399
$532$	0	0
$533$	0.828427i	0.0358832i
$534$	0	0
$535$	0	0
$536$	0	0
$537$	− 38.6274i	− 1.66690i
$538$	0	0
$539$	12.7279	0.548230
$540$	0	0
$541$	14.4853	0.622771	0.311385	−	0.950284i	$-0.399207\pi$
$541$	14.4853	0.622771	0.311385	+	0.950284i	$0.399207\pi$
$542$	0	0
$543$	− 79.5980i	− 3.41588i
$544$	0	0
$545$	0	0
$546$	0	0
$547$	33.0711i	1.41402i	0.707205	+	0.707008i	$0.249956\pi$
$547$	33.0711i	1.41402i	−0.707205	+	0.707008i	$0.750044\pi$
$548$	0	0
$549$	63.3137	2.70216
$550$	0	0
$551$	79.5980	3.39099
$552$	0	0
$553$	− 13.6569i	− 0.580749i
$554$	0	0
$555$	0	0
$556$	0	0
$557$	− 42.1421i	− 1.78562i	−0.450434	−	0.892810i	$-0.648731\pi$
$557$	− 42.1421i	− 1.78562i	0.450434	−	0.892810i	$-0.351269\pi$
$558$	0	0
$559$	−1.75736	−0.0743284
$560$	0	0
$561$	69.9411	2.95292
$562$	0	0
$563$	17.5563i	0.739912i	0.929049	+	0.369956i	$0.120627\pi$
$563$	17.5563i	0.739912i	−0.929049	+	0.369956i	$0.879373\pi$
$564$	0	0
$565$	0	0
$566$	0	0
$567$	79.9411i	3.35721i
$568$	0	0
$569$	31.3137	1.31274	0.656369	−	0.754440i	$-0.272091\pi$
$569$	31.3137	1.31274	0.656369	+	0.754440i	$0.272091\pi$
$570$	0	0
$571$	−34.8284	−1.45752	−0.728762	−	0.684767i	$-0.759904\pi$
$571$	−34.8284	−1.45752	−0.728762	+	0.684767i	$0.759904\pi$
$572$	0	0
$573$	73.9411i	3.08893i
$574$	0	0
$575$	0	0
$576$	0	0
$577$	− 34.1421i	− 1.42136i	−0.703518	−	0.710678i	$-0.748388\pi$
$577$	− 34.1421i	− 1.42136i	0.703518	−	0.710678i	$-0.251612\pi$
$578$	0	0
$579$	−28.4853	−1.18381
$580$	0	0
$581$	4.00000	0.165948
$582$	0	0
$583$	− 13.4558i	− 0.557284i
$584$	0	0
$585$	0	0
$586$	0	0
$587$	14.4853i	0.597872i	0.954273	+	0.298936i	$0.0966316\pi$
$587$	14.4853i	0.597872i	−0.954273	+	0.298936i	$0.903368\pi$
$588$	0	0
$589$	−11.6569	−0.480312
$590$	0	0
$591$	1.17157	0.0481921
$592$	0	0
$593$	− 26.0000i	− 1.06769i	−0.845582	−	0.533846i	$-0.820746\pi$
$593$	− 26.0000i	− 1.06769i	0.845582	−	0.533846i	$-0.179254\pi$
$594$	0	0
$595$	0	0
$596$	0	0
$597$	21.6569i	0.886356i
$598$	0	0
$599$	28.4853	1.16388	0.581939	−	0.813233i	$-0.302294\pi$
$599$	28.4853	1.16388	0.581939	+	0.813233i	$0.302294\pi$
$600$	0	0
$601$	−31.9411	−1.30291	−0.651453	−	0.758689i	$-0.725840\pi$
$601$	−31.9411	−1.30291	−0.651453	+	0.758689i	$0.725840\pi$
$602$	0	0
$603$	7.17157i	0.292049i
$604$	0	0
$605$	0	0
$606$	0	0
$607$	37.5563i	1.52437i	0.647362	+	0.762183i	$0.275872\pi$
$607$	37.5563i	1.52437i	−0.647362	+	0.762183i	$0.724128\pi$
$608$	0	0
$609$	65.9411	2.67207
$610$	0	0
$611$	−2.00000	−0.0809113
$612$	0	0
$613$	22.0000i	0.888572i	0.895885	+	0.444286i	$0.146543\pi$
$613$	22.0000i	0.888572i	−0.895885	+	0.444286i	$0.853457\pi$
$614$	0	0
$615$	0	0
$616$	0	0
$617$	− 7.65685i	− 0.308253i	−0.988051	−	0.154127i	$-0.950744\pi$
$617$	− 7.65685i	− 0.308253i	0.988051	−	0.154127i	$-0.0492564\pi$
$618$	0	0
$619$	12.0416	0.483994	0.241997	−	0.970277i	$-0.422198\pi$
$619$	12.0416	0.483994	0.241997	+	0.970277i	$0.422198\pi$
$620$	0	0
$621$	−97.9411	−3.93024
$622$	0	0
$623$	20.0000i	0.801283i
$624$	0	0
$625$	0	0
$626$	0	0
$627$	− 119.397i	− 4.76826i
$628$	0	0
$629$	−5.65685	−0.225554
$630$	0	0
$631$	12.7279	0.506691	0.253345	−	0.967376i	$-0.418469\pi$
$631$	12.7279	0.506691	0.253345	+	0.967376i	$0.418469\pi$
$632$	0	0
$633$	− 19.3137i	− 0.767651i
$634$	0	0
$635$	0	0
$636$	0	0
$637$	− 3.00000i	− 0.118864i
$638$	0	0
$639$	85.6985	3.39018
$640$	0	0
$641$	39.6569	1.56635	0.783176	−	0.621800i	$-0.213598\pi$
$641$	39.6569	1.56635	0.783176	+	0.621800i	$0.213598\pi$
$642$	0	0
$643$	9.31371i	0.367297i	0.982992	+	0.183648i	$0.0587908\pi$
$643$	9.31371i	0.367297i	−0.982992	+	0.183648i	$0.941209\pi$
$644$	0	0
$645$	0	0
$646$	0	0
$647$	− 33.3553i	− 1.31133i	−0.755050	−	0.655667i	$-0.772388\pi$
$647$	− 33.3553i	− 1.31133i	0.755050	−	0.655667i	$-0.227612\pi$
$648$	0	0
$649$	22.9706	0.901673
$650$	0	0
$651$	−9.65685	−0.378482
$652$	0	0
$653$	− 42.9706i	− 1.68157i	−0.541371	−	0.840784i	$-0.682094\pi$
$653$	− 42.9706i	− 1.68157i	0.541371	−	0.840784i	$-0.317906\pi$
$654$	0	0
$655$	0	0
$656$	0	0
$657$	− 59.1127i	− 2.30621i
$658$	0	0
$659$	12.4853	0.486358	0.243179	−	0.969981i	$-0.421810\pi$
$659$	12.4853	0.486358	0.243179	+	0.969981i	$0.421810\pi$
$660$	0	0
$661$	−42.9706	−1.67136	−0.835681	−	0.549216i	$-0.814927\pi$
$661$	−42.9706	−1.67136	−0.835681	+	0.549216i	$0.814927\pi$
$662$	0	0
$663$	− 16.4853i	− 0.640235i
$664$	0	0
$665$	0	0
$666$	0	0
$667$	48.9706i	1.89615i
$668$	0	0
$669$	86.4264	3.34144
$670$	0	0
$671$	−31.0294	−1.19788
$672$	0	0
$673$	− 5.79899i	− 0.223535i	−0.993734	−	0.111767i	$-0.964349\pi$
$673$	− 5.79899i	− 0.223535i	0.993734	−	0.111767i	$-0.0356511\pi$
$674$	0	0
$675$	0	0
$676$	0	0
$677$	− 8.14214i	− 0.312928i	−0.987684	−	0.156464i	$-0.949991\pi$
$677$	− 8.14214i	− 0.312928i	0.987684	−	0.156464i	$-0.0500095\pi$
$678$	0	0
$679$	0.686292	0.0263375
$680$	0	0
$681$	−71.1127	−2.72504
$682$	0	0
$683$	− 25.1127i	− 0.960911i	−0.877019	−	0.480455i	$-0.840471\pi$
$683$	− 25.1127i	− 0.960911i	0.877019	−	0.480455i	$-0.159529\pi$
$684$	0	0
$685$	0	0
$686$	0	0
$687$	− 47.1127i	− 1.79746i
$688$	0	0
$689$	−3.17157	−0.120827
$690$	0	0
$691$	−46.5858	−1.77221	−0.886103	−	0.463488i	$-0.846598\pi$
$691$	−46.5858	−1.77221	−0.886103	+	0.463488i	$0.846598\pi$
$692$	0	0
$693$	− 73.4558i	− 2.79036i
$694$	0	0
$695$	0	0
$696$	0	0
$697$	− 4.00000i	− 0.151511i
$698$	0	0
$699$	−75.1127	−2.84102
$700$	0	0
$701$	24.6274	0.930165	0.465082	−	0.885267i	$-0.346025\pi$
$701$	24.6274	0.930165	0.465082	+	0.885267i	$0.346025\pi$
$702$	0	0
$703$	9.65685i	0.364215i
$704$	0	0
$705$	0	0
$706$	0	0
$707$	15.3137i	0.575931i
$708$	0	0
$709$	−31.4558	−1.18135	−0.590675	−	0.806910i	$-0.701138\pi$
$709$	−31.4558	−1.18135	−0.590675	+	0.806910i	$0.701138\pi$
$710$	0	0
$711$	59.1127	2.21690
$712$	0	0
$713$	− 7.17157i	− 0.268578i
$714$	0	0
$715$	0	0
$716$	0	0
$717$	29.7990i	1.11286i
$718$	0	0
$719$	−26.6274	−0.993035	−0.496518	−	0.868027i	$-0.665388\pi$
$719$	−26.6274	−0.993035	−0.496518	+	0.868027i	$0.665388\pi$
$720$	0	0
$721$	−17.1716	−0.639503
$722$	0	0
$723$	− 24.4853i	− 0.910617i
$724$	0	0
$725$	0	0
$726$	0	0
$727$	− 6.24264i	− 0.231527i	−0.993277	−	0.115763i	$-0.963069\pi$
$727$	− 6.24264i	− 0.231527i	0.993277	−	0.115763i	$-0.0369314\pi$
$728$	0	0
$729$	−148.196	−5.48874
$730$	0	0
$731$	8.48528	0.313839
$732$	0	0
$733$	− 12.6274i	− 0.466404i	−0.972428	−	0.233202i	$-0.925080\pi$
$733$	− 12.6274i	− 0.466404i	0.972428	−	0.233202i	$-0.0749204\pi$
$734$	0	0
$735$	0	0
$736$	0	0
$737$	− 3.51472i	− 0.129466i
$738$	0	0
$739$	−21.4142	−0.787735	−0.393867	−	0.919167i	$-0.628863\pi$
$739$	−21.4142	−0.787735	−0.393867	+	0.919167i	$0.628863\pi$
$740$	0	0
$741$	−28.1421	−1.03383
$742$	0	0
$743$	− 2.97056i	− 0.108979i	−0.998514	−	0.0544897i	$-0.982647\pi$
$743$	− 2.97056i	− 0.108979i	0.998514	−	0.0544897i	$-0.0173532\pi$
$744$	0	0
$745$	0	0
$746$	0	0
$747$	17.3137i	0.633475i
$748$	0	0
$749$	−18.1421	−0.662899
$750$	0	0
$751$	−8.48528	−0.309632	−0.154816	−	0.987943i	$-0.549479\pi$
$751$	−8.48528	−0.309632	−0.154816	+	0.987943i	$0.549479\pi$
$752$	0	0
$753$	− 34.6274i	− 1.26189i
$754$	0	0
$755$	0	0
$756$	0	0
$757$	− 20.1421i	− 0.732078i	−0.930599	−	0.366039i	$-0.880714\pi$
$757$	− 20.1421i	− 0.732078i	0.930599	−	0.366039i	$-0.119286\pi$
$758$	0	0
$759$	73.4558	2.66628
$760$	0	0
$761$	−18.6863	−0.677378	−0.338689	−	0.940898i	$-0.609983\pi$
$761$	−18.6863	−0.677378	−0.338689	+	0.940898i	$0.609983\pi$
$762$	0	0
$763$	10.6274i	0.384738i
$764$	0	0
$765$	0	0
$766$	0	0
$767$	− 5.41421i	− 0.195496i
$768$	0	0
$769$	8.34315	0.300862	0.150431	−	0.988621i	$-0.451934\pi$
$769$	8.34315	0.300862	0.150431	+	0.988621i	$0.451934\pi$
$770$	0	0
$771$	−51.1127	−1.84078
$772$	0	0
$773$	− 10.1421i	− 0.364787i	−0.983226	−	0.182394i	$-0.941615\pi$
$773$	− 10.1421i	− 0.364787i	0.983226	−	0.182394i	$-0.0583845\pi$
$774$	0	0
$775$	0	0
$776$	0	0
$777$	8.00000i	0.286998i
$778$	0	0
$779$	−6.82843	−0.244654
$780$	0	0
$781$	−42.0000	−1.50288
$782$	0	0
$783$	186.510i	6.66531i
$784$	0	0
$785$	0	0
$786$	0	0
$787$	38.0000i	1.35455i	0.735728	+	0.677277i	$0.236840\pi$
$787$	38.0000i	1.35455i	−0.735728	+	0.677277i	$0.763160\pi$
$788$	0	0
$789$	46.9706	1.67220
$790$	0	0
$791$	6.34315	0.225536
$792$	0	0
$793$	7.31371i	0.259717i
$794$	0	0
$795$	0	0
$796$	0	0
$797$	29.5980i	1.04841i	0.851591	+	0.524207i	$0.175638\pi$
$797$	29.5980i	1.04841i	−0.851591	+	0.524207i	$0.824362\pi$
$798$	0	0
$799$	9.65685	0.341635
$800$	0	0
$801$	−86.5685	−3.05875
$802$	0	0
$803$	28.9706i	1.02235i
$804$	0	0
$805$	0	0
$806$	0	0
$807$	56.7696i	1.99838i
$808$	0	0
$809$	30.3431	1.06681	0.533404	−	0.845861i	$-0.320913\pi$
$809$	30.3431	1.06681	0.533404	+	0.845861i	$0.320913\pi$
$810$	0	0
$811$	−9.89949	−0.347618	−0.173809	−	0.984779i	$-0.555608\pi$
$811$	−9.89949	−0.347618	−0.173809	+	0.984779i	$0.555608\pi$
$812$	0	0
$813$	− 33.7990i	− 1.18538i
$814$	0	0
$815$	0	0
$816$	0	0
$817$	− 14.4853i	− 0.506776i
$818$	0	0
$819$	−17.3137	−0.604990
$820$	0	0
$821$	29.3137	1.02306	0.511528	−	0.859267i	$-0.329080\pi$
$821$	29.3137	1.02306	0.511528	+	0.859267i	$0.329080\pi$
$822$	0	0
$823$	36.8701i	1.28521i	0.766198	+	0.642605i	$0.222146\pi$
$823$	36.8701i	1.28521i	−0.766198	+	0.642605i	$0.777854\pi$
$824$	0	0
$825$	0	0
$826$	0	0
$827$	24.6274i	0.856379i	0.903689	+	0.428190i	$0.140848\pi$
$827$	24.6274i	0.856379i	−0.903689	+	0.428190i	$0.859152\pi$
$828$	0	0
$829$	−25.6569	−0.891099	−0.445550	−	0.895257i	$-0.646992\pi$
$829$	−25.6569	−0.891099	−0.445550	+	0.895257i	$0.646992\pi$
$830$	0	0
$831$	−14.1421	−0.490585
$832$	0	0
$833$	14.4853i	0.501885i
$834$	0	0
$835$	0	0
$836$	0	0
$837$	− 27.3137i	− 0.944100i
$838$	0	0
$839$	18.1005	0.624899	0.312449	−	0.949934i	$-0.398851\pi$
$839$	18.1005	0.624899	0.312449	+	0.949934i	$0.398851\pi$
$840$	0	0
$841$	64.2548	2.21568
$842$	0	0
$843$	76.7696i	2.64408i
$844$	0	0
$845$	0	0
$846$	0	0
$847$	14.0000i	0.481046i
$848$	0	0
$849$	13.3137	0.456925
$850$	0	0
$851$	−5.94113	−0.203659
$852$	0	0
$853$	15.1127i	0.517449i	0.965951	+	0.258724i	$0.0833022\pi$
$853$	15.1127i	0.517449i	−0.965951	+	0.258724i	$0.916698\pi$
$854$	0	0
$855$	0	0
$856$	0	0
$857$	− 48.9117i	− 1.67079i	−0.549649	−	0.835396i	$-0.685239\pi$
$857$	− 48.9117i	− 1.67079i	0.549649	−	0.835396i	$-0.314761\pi$
$858$	0	0
$859$	−49.1716	−1.67771	−0.838856	−	0.544353i	$-0.816775\pi$
$859$	−49.1716	−1.67771	−0.838856	+	0.544353i	$0.816775\pi$
$860$	0	0
$861$	−5.65685	−0.192785
$862$	0	0
$863$	− 25.5980i	− 0.871365i	−0.900100	−	0.435683i	$-0.856507\pi$
$863$	− 25.5980i	− 0.871365i	0.900100	−	0.435683i	$-0.143493\pi$
$864$	0	0
$865$	0	0
$866$	0	0
$867$	21.5563i	0.732092i
$868$	0	0
$869$	−28.9706	−0.982759
$870$	0	0
$871$	−0.828427	−0.0280702
$872$	0	0
$873$	2.97056i	0.100538i
$874$	0	0
$875$	0	0
$876$	0	0
$877$	6.68629i	0.225780i	0.993607	+	0.112890i	$0.0360108\pi$
$877$	6.68629i	0.225780i	−0.993607	+	0.112890i	$0.963989\pi$
$878$	0	0
$879$	9.65685	0.325718
$880$	0	0
$881$	−41.9411	−1.41303	−0.706516	−	0.707697i	$-0.749734\pi$
$881$	−41.9411	−1.41303	−0.706516	+	0.707697i	$0.749734\pi$
$882$	0	0
$883$	− 9.75736i	− 0.328361i	−0.986430	−	0.164181i	$-0.947502\pi$
$883$	− 9.75736i	− 0.328361i	0.986430	−	0.164181i	$-0.0524980\pi$
$884$	0	0
$885$	0	0
$886$	0	0
$887$	18.0416i	0.605779i	0.953026	+	0.302889i	$0.0979513\pi$
$887$	18.0416i	0.605779i	−0.953026	+	0.302889i	$0.902049\pi$
$888$	0	0
$889$	−7.79899	−0.261570
$890$	0	0
$891$	169.581	5.68117
$892$	0	0
$893$	− 16.4853i	− 0.551659i
$894$	0	0
$895$	0	0
$896$	0	0
$897$	− 17.3137i	− 0.578088i
$898$	0	0
$899$	−13.6569	−0.455482
$900$	0	0
$901$	15.3137	0.510174
$902$	0	0
$903$	− 12.0000i	− 0.399335i
$904$	0	0
$905$	0	0
$906$	0	0
$907$	13.0711i	0.434018i	0.976170	+	0.217009i	$0.0696300\pi$
$907$	13.0711i	0.434018i	−0.976170	+	0.217009i	$0.930370\pi$
$908$	0	0
$909$	−66.2843	−2.19851
$910$	0	0
$911$	−27.5980	−0.914362	−0.457181	−	0.889374i	$-0.651141\pi$
$911$	−27.5980	−0.914362	−0.457181	+	0.889374i	$0.651141\pi$
$912$	0	0
$913$	− 8.48528i	− 0.280822i
$914$	0	0
$915$	0	0
$916$	0	0
$917$	6.62742i	0.218857i
$918$	0	0
$919$	−59.1127	−1.94995	−0.974974	−	0.222319i	$-0.928637\pi$
$919$	−59.1127	−1.94995	−0.974974	+	0.222319i	$0.928637\pi$
$920$	0	0
$921$	−53.4558	−1.76143
$922$	0	0
$923$	9.89949i	0.325846i
$924$	0	0
$925$	0	0
$926$	0	0
$927$	− 74.3259i	− 2.44118i
$928$	0	0
$929$	−21.1127	−0.692685	−0.346343	−	0.938108i	$-0.612577\pi$
$929$	−21.1127	−0.692685	−0.346343	+	0.938108i	$0.612577\pi$
$930$	0	0
$931$	24.7279	0.810425
$932$	0	0
$933$	34.6274i	1.13365i
$934$	0	0
$935$	0	0
$936$	0	0
$937$	10.2843i	0.335972i	0.985789	+	0.167986i	$0.0537264\pi$
$937$	10.2843i	0.335972i	−0.985789	+	0.167986i	$0.946274\pi$
$938$	0	0
$939$	−24.4853	−0.799047
$940$	0	0
$941$	−1.51472	−0.0493784	−0.0246892	−	0.999695i	$-0.507860\pi$
$941$	−1.51472	−0.0493784	−0.0246892	+	0.999695i	$0.507860\pi$
$942$	0	0
$943$	− 4.20101i	− 0.136804i
$944$	0	0
$945$	0	0
$946$	0	0
$947$	− 13.0294i	− 0.423400i	−0.977335	−	0.211700i	$-0.932100\pi$
$947$	− 13.0294i	− 0.423400i	0.977335	−	0.211700i	$-0.0678999\pi$
$948$	0	0
$949$	6.82843	0.221660
$950$	0	0
$951$	−83.5980	−2.71085
$952$	0	0
$953$	47.9411i	1.55297i	0.630139	+	0.776483i	$0.282998\pi$
$953$	47.9411i	1.55297i	−0.630139	+	0.776483i	$0.717002\pi$
$954$	0	0
$955$	0	0
$956$	0	0
$957$	− 139.882i	− 4.52175i
$958$	0	0
$959$	4.00000	0.129167
$960$	0	0
$961$	−29.0000	−0.935484
$962$	0	0
$963$	− 78.5269i	− 2.53049i
$964$	0	0
$965$	0	0
$966$	0	0
$967$	39.1716i	1.25967i	0.776728	+	0.629836i	$0.216878\pi$
$967$	39.1716i	1.25967i	−0.776728	+	0.629836i	$0.783122\pi$
$968$	0	0
$969$	135.882	4.36517
$970$	0	0
$971$	38.6274	1.23961	0.619806	−	0.784755i	$-0.287211\pi$
$971$	38.6274	1.23961	0.619806	+	0.784755i	$0.287211\pi$
$972$	0	0
$973$	− 16.9706i	− 0.544051i
$974$	0	0
$975$	0	0
$976$	0	0
$977$	22.4264i	0.717484i	0.933437	+	0.358742i	$0.116794\pi$
$977$	22.4264i	0.717484i	−0.933437	+	0.358742i	$0.883206\pi$
$978$	0	0
$979$	42.4264	1.35595
$980$	0	0
$981$	−46.0000	−1.46867
$982$	0	0
$983$	− 17.5147i	− 0.558633i	−0.960199	−	0.279316i	$-0.909892\pi$
$983$	− 17.5147i	− 0.558633i	0.960199	−	0.279316i	$-0.0901078\pi$
$984$	0	0
$985$	0	0
$986$	0	0
$987$	− 13.6569i	− 0.434702i
$988$	0	0
$989$	8.91169	0.283375
$990$	0	0
$991$	−23.5980	−0.749615	−0.374807	−	0.927103i	$-0.622291\pi$
$991$	−23.5980	−0.749615	−0.374807	+	0.927103i	$0.622291\pi$
$992$	0	0
$993$	3.17157i	0.100647i
$994$	0	0
$995$	0	0
$996$	0	0
$997$	8.82843i	0.279599i	0.990180	+	0.139800i	$0.0446458\pi$
$997$	8.82843i	0.279599i	−0.990180	+	0.139800i	$0.955354\pi$
$998$	0	0
$999$	−22.6274	−0.715900

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	2600.2.d.i.1249.1		4
5.2	odd	4		520.2.a.c.1.1	✓	2
5.3	odd	4		2600.2.a.w.1.2		2
5.4	even	2	inner	2600.2.d.i.1249.4		4
15.2	even	4		4680.2.a.w.1.1		2
20.3	even	4		5200.2.a.bl.1.1		2
20.7	even	4		1040.2.a.n.1.2		2
40.27	even	4		4160.2.a.u.1.1		2
40.37	odd	4		4160.2.a.bn.1.2		2
60.47	odd	4		9360.2.a.ck.1.2		2
65.12	odd	4		6760.2.a.n.1.1		2

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
520.2.a.c.1.1	✓	2	5.2	odd	4
1040.2.a.n.1.2		2	20.7	even	4
2600.2.a.w.1.2		2	5.3	odd	4
2600.2.d.i.1249.1		4	1.1	even	1	trivial
2600.2.d.i.1249.4		4	5.4	even	2	inner
4160.2.a.u.1.1		2	40.27	even	4
4160.2.a.bn.1.2		2	40.37	odd	4
4680.2.a.w.1.1		2	15.2	even	4
5200.2.a.bl.1.1		2	20.3	even	4
6760.2.a.n.1.1		2	65.12	odd	4
9360.2.a.ck.1.2		2	60.47	odd	4

Overview	Random
Universe	Knowledge

Classical	Maass
Hilbert	Bianchi

Elliptic curves over $\Q$
Elliptic curves over $\Q(\alpha)$
Genus 2 curves over $\Q$
Higher genus families
Abelian varieties over $\F_{q}$

Level:	\( N \)	\(=\)	\( 2600 = 2^{3} \cdot 5^{2} \cdot 13 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	2600.d (of order \(2\), degree \(1\), not minimal)

\(f(q)\)	\(=\)	\(q-3.41421i q^{3} +2.00000i q^{7} -8.65685 q^{9} +O(q^{10})\)	\(q-3.41421i q^{3} +2.00000i q^{7} -8.65685 q^{9} +4.24264 q^{11} -1.00000i q^{13} +4.82843i q^{17} +8.24264 q^{19} +6.82843 q^{21} +5.07107i q^{23} +19.3137i q^{27} +9.65685 q^{29} -1.41421 q^{31} -14.4853i q^{33} +1.17157i q^{37} -3.41421 q^{39} -0.828427 q^{41} -1.75736i q^{43} -2.00000i q^{47} +3.00000 q^{49} +16.4853 q^{51} -3.17157i q^{53} -28.1421i q^{57} +5.41421 q^{59} -7.31371 q^{61} -17.3137i q^{63} -0.828427i q^{67} +17.3137 q^{69} -9.89949 q^{71} +6.82843i q^{73} +8.48528i q^{77} -6.82843 q^{79} +39.9706 q^{81} -2.00000i q^{83} -32.9706i q^{87} +10.0000 q^{89} +2.00000 q^{91} +4.82843i q^{93} -0.343146i q^{97} -36.7279 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 4 q - 12 q^{9}+O(q^{10}) \) 4 * q - 12 * q^9	\( 4 q - 12 q^{9} + 16 q^{19} + 16 q^{21} + 16 q^{29} - 8 q^{39} + 8 q^{41} + 12 q^{49} + 32 q^{51} + 16 q^{59} + 16 q^{61} + 24 q^{69} - 16 q^{79} + 92 q^{81} + 40 q^{89} + 8 q^{91} - 96 q^{99}+O(q^{100}) \) 4 * q - 12 * q^9 + 16 * q^19 + 16 * q^21 + 16 * q^29 - 8 * q^39 + 8 * q^41 + 12 * q^49 + 32 * q^51 + 16 * q^59 + 16 * q^61 + 24 * q^69 - 16 * q^79 + 92 * q^81 + 40 * q^89 + 8 * q^91 - 96 * q^99

Introduction

L-functions

Modular forms

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Database

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