LMFDB - Embedded newform 825.2.c.c.199.2

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [825,2,Mod(199,825)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(825, 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("825.199");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 825 = 3 \cdot 5^{2} \cdot 11 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	825.c (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:	$6.58765816676$
Analytic rank:	$0$
Dimension:	$4$
Coefficient field:	$\Q(\zeta_{12})$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{4} - x^{2} + 1 $ x^4 - x^2 + 1
Coefficient ring:	$\Z[a_1, a_2, a_3]$
Coefficient ring index:	$ 2^{2} $
Twist minimal:	no (minimal twist has level 165)
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label			199.2
Root			$-0.866025 + 0.500000i$ of defining polynomial
Character	$\chi$	$=$	825.199
Dual form			825.2.c.c.199.3

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-1.73205i q^{2} +1.00000i q^{3} -1.00000 q^{4} +1.73205 q^{6} -2.00000i q^{7} -1.73205i q^{8} -1.00000 q^{9} +O(q^{10})$	\(q-1.73205i q^{2} +1.00000i q^{3} -1.00000 q^{4} +1.73205 q^{6} -2.00000i q^{7} -1.73205i q^{8} -1.00000 q^{9} -1.00000 q^{11} -1.00000i q^{12} -1.46410i q^{13} -3.46410 q^{14} -5.00000 q^{16} +1.73205i q^{18} +1.46410 q^{19} +2.00000 q^{21} +1.73205i q^{22} -6.92820i q^{23} +1.73205 q^{24} -2.53590 q^{26} -1.00000i q^{27} +2.00000i q^{28} -3.46410 q^{29} +2.92820 q^{31} +5.19615i q^{32} -1.00000i q^{33} +1.00000 q^{36} -8.92820i q^{37} -2.53590i q^{38} +1.46410 q^{39} -3.46410 q^{41} -3.46410i q^{42} +8.92820i q^{43} +1.00000 q^{44} -12.0000 q^{46} -6.92820i q^{47} -5.00000i q^{48} +3.00000 q^{49} +1.46410i q^{52} -12.9282i q^{53} -1.73205 q^{54} -3.46410 q^{56} +1.46410i q^{57} +6.00000i q^{58} -6.92820 q^{59} +2.00000 q^{61} -5.07180i q^{62} +2.00000i q^{63} -1.00000 q^{64} -1.73205 q^{66} -8.00000i q^{67} +6.92820 q^{69} -13.8564 q^{71} +1.73205i q^{72} +12.3923i q^{73} -15.4641 q^{74} -1.46410 q^{76} +2.00000i q^{77} -2.53590i q^{78} +13.4641 q^{79} +1.00000 q^{81} +6.00000i q^{82} +15.4641i q^{83} -2.00000 q^{84} +15.4641 q^{86} -3.46410i q^{87} +1.73205i q^{88} +12.9282 q^{89} -2.92820 q^{91} +6.92820i q^{92} +2.92820i q^{93} -12.0000 q^{94} -5.19615 q^{96} +10.0000i q^{97} -5.19615i q^{98} +1.00000 q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 4 q - 4 q^{4} - 4 q^{9}+O(q^{10}) $ 4 * q - 4 * q^4 - 4 * q^9	$ 4 q - 4 q^{4} - 4 q^{9} - 4 q^{11} - 20 q^{16} - 8 q^{19} + 8 q^{21} - 24 q^{26} - 16 q^{31} + 4 q^{36} - 8 q^{39} + 4 q^{44} - 48 q^{46} + 12 q^{49} + 8 q^{61} - 4 q^{64} - 48 q^{74} + 8 q^{76} + 40 q^{79} + 4 q^{81} - 8 q^{84} + 48 q^{86} + 24 q^{89} + 16 q^{91} - 48 q^{94} + 4 q^{99}+O(q^{100}) $ 4 * q - 4 * q^4 - 4 * q^9 - 4 * q^11 - 20 * q^16 - 8 * q^19 + 8 * q^21 - 24 * q^26 - 16 * q^31 + 4 * q^36 - 8 * q^39 + 4 * q^44 - 48 * q^46 + 12 * q^49 + 8 * q^61 - 4 * q^64 - 48 * q^74 + 8 * q^76 + 40 * q^79 + 4 * q^81 - 8 * q^84 + 48 * q^86 + 24 * q^89 + 16 * q^91 - 48 * q^94 + 4 * q^99

Display 100 coefficients 10 coefficients

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/825\mathbb{Z}\right)^\times$.

$n$	$376$	$551$	$727$
$\chi(n)$	$1$	$1$	$-1$

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 $

$2$	− 1.73205i	− 1.22474i	−0.790569	−	0.612372i	$-0.790215\pi$
$2$	− 1.73205i	− 1.22474i	0.790569	−	0.612372i	$-0.209785\pi$
$3$	1.00000i	0.577350i
$3$	1.00000i	0.577350i
$4$	−1.00000	−0.500000
$5$	0	0
$5$	0	0
$6$	1.73205	0.707107
$7$	− 2.00000i	− 0.755929i	−0.925820	−	0.377964i	$-0.876624\pi$
$7$	− 2.00000i	− 0.755929i	0.925820	−	0.377964i	$-0.123376\pi$
$8$	− 1.73205i	− 0.612372i
$9$	−1.00000	−0.333333
$10$	0	0
$11$	−1.00000	−0.301511
$11$	−1.00000	−0.301511
$12$	− 1.00000i	− 0.288675i
$13$	− 1.46410i	− 0.406069i	−0.979172	−	0.203034i	$-0.934920\pi$
$13$	− 1.46410i	− 0.406069i	0.979172	−	0.203034i	$-0.0650803\pi$
$14$	−3.46410	−0.925820
$15$	0	0
$16$	−5.00000	−1.25000
$17$	0	0	1.00000			$0$
$17$	0	0	−1.00000			$\pi$
$18$	1.73205i	0.408248i
$19$	1.46410	0.335888	0.167944	−	0.985797i	$-0.446287\pi$
$19$	1.46410	0.335888	0.167944	+	0.985797i	$0.446287\pi$
$20$	0	0
$21$	2.00000	0.436436
$22$	1.73205i	0.369274i
$23$	− 6.92820i	− 1.44463i	−0.691564	−	0.722315i	$-0.743078\pi$
$23$	− 6.92820i	− 1.44463i	0.691564	−	0.722315i	$-0.256922\pi$
$24$	1.73205	0.353553
$25$	0	0
$26$	−2.53590	−0.497331
$27$	− 1.00000i	− 0.192450i
$28$	2.00000i	0.377964i
$29$	−3.46410	−0.643268	−0.321634	−	0.946864i	$-0.604232\pi$
$29$	−3.46410	−0.643268	−0.321634	+	0.946864i	$0.604232\pi$
$30$	0	0
$31$	2.92820	0.525921	0.262960	−	0.964807i	$-0.415301\pi$
$31$	2.92820	0.525921	0.262960	+	0.964807i	$0.415301\pi$
$32$	5.19615i	0.918559i
$33$	− 1.00000i	− 0.174078i
$34$	0	0
$35$	0	0
$36$	1.00000	0.166667
$37$	− 8.92820i	− 1.46779i	−0.679264	−	0.733894i	$-0.737701\pi$
$37$	− 8.92820i	− 1.46779i	0.679264	−	0.733894i	$-0.262299\pi$
$38$	− 2.53590i	− 0.411377i
$39$	1.46410	0.234444
$40$	0	0
$41$	−3.46410	−0.541002	−0.270501	−	0.962720i	$-0.587189\pi$
$41$	−3.46410	−0.541002	−0.270501	+	0.962720i	$0.587189\pi$
$42$	− 3.46410i	− 0.534522i
$43$	8.92820i	1.36154i	0.732498	+	0.680769i	$0.238354\pi$
$43$	8.92820i	1.36154i	−0.732498	+	0.680769i	$0.761646\pi$
$44$	1.00000	0.150756
$45$	0	0
$46$	−12.0000	−1.76930
$47$	− 6.92820i	− 1.01058i	−0.862949	−	0.505291i	$-0.831385\pi$
$47$	− 6.92820i	− 1.01058i	0.862949	−	0.505291i	$-0.168615\pi$
$48$	− 5.00000i	− 0.721688i
$49$	3.00000	0.428571
$50$	0	0
$51$	0	0
$52$	1.46410i	0.203034i
$53$	− 12.9282i	− 1.77583i	−0.460012	−	0.887913i	$-0.652155\pi$
$53$	− 12.9282i	− 1.77583i	0.460012	−	0.887913i	$-0.347845\pi$
$54$	−1.73205	−0.235702
$55$	0	0
$56$	−3.46410	−0.462910
$57$	1.46410i	0.193925i
$58$	6.00000i	0.787839i
$59$	−6.92820	−0.901975	−0.450988	−	0.892530i	$-0.648928\pi$
$59$	−6.92820	−0.901975	−0.450988	+	0.892530i	$0.648928\pi$
$60$	0	0
$61$	2.00000	0.256074	0.128037	−	0.991769i	$-0.459132\pi$
$61$	2.00000	0.256074	0.128037	+	0.991769i	$0.459132\pi$
$62$	− 5.07180i	− 0.644119i
$63$	2.00000i	0.251976i
$64$	−1.00000	−0.125000
$65$	0	0
$66$	−1.73205	−0.213201
$67$	− 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
$69$	6.92820	0.834058
$70$	0	0
$71$	−13.8564	−1.64445	−0.822226	−	0.569160i	$-0.807268\pi$
$71$	−13.8564	−1.64445	−0.822226	+	0.569160i	$0.807268\pi$
$72$	1.73205i	0.204124i
$73$	12.3923i	1.45041i	0.688533	+	0.725205i	$0.258255\pi$
$73$	12.3923i	1.45041i	−0.688533	+	0.725205i	$0.741745\pi$
$74$	−15.4641	−1.79767
$75$	0	0
$76$	−1.46410	−0.167944
$77$	2.00000i	0.227921i
$78$	− 2.53590i	− 0.287134i
$79$	13.4641	1.51483	0.757415	−	0.652934i	$-0.226462\pi$
$79$	13.4641	1.51483	0.757415	+	0.652934i	$0.226462\pi$
$80$	0	0
$81$	1.00000	0.111111
$82$	6.00000i	0.662589i
$83$	15.4641i	1.69741i	0.528870	+	0.848703i	$0.322616\pi$
$83$	15.4641i	1.69741i	−0.528870	+	0.848703i	$0.677384\pi$
$84$	−2.00000	−0.218218
$85$	0	0
$86$	15.4641	1.66754
$87$	− 3.46410i	− 0.371391i
$88$	1.73205i	0.184637i
$89$	12.9282	1.37039	0.685193	−	0.728361i	$-0.259718\pi$
$89$	12.9282	1.37039	0.685193	+	0.728361i	$0.259718\pi$
$90$	0	0
$91$	−2.92820	−0.306959
$92$	6.92820i	0.722315i
$93$	2.92820i	0.303641i
$94$	−12.0000	−1.23771
$95$	0	0
$96$	−5.19615	−0.530330
$97$	10.0000i	1.01535i	0.861550	+	0.507673i	$0.169494\pi$
$97$	10.0000i	1.01535i	−0.861550	+	0.507673i	$0.830506\pi$
$98$	− 5.19615i	− 0.524891i
$99$	1.00000	0.100504
$100$	0	0
$101$	10.3923	1.03407	0.517036	−	0.855963i	$-0.327035\pi$
$101$	10.3923	1.03407	0.517036	+	0.855963i	$0.327035\pi$
$102$	0	0
$103$	8.00000i	0.788263i	0.919054	+	0.394132i	$0.128955\pi$
$103$	8.00000i	0.788263i	−0.919054	+	0.394132i	$0.871045\pi$
$104$	−2.53590	−0.248665
$105$	0	0
$106$	−22.3923	−2.17493
$107$	− 15.4641i	− 1.49497i	−0.664278	−	0.747486i	$-0.731261\pi$
$107$	− 15.4641i	− 1.49497i	0.664278	−	0.747486i	$-0.268739\pi$
$108$	1.00000i	0.0962250i
$109$	10.0000	0.957826	0.478913	−	0.877862i	$-0.341031\pi$
$109$	10.0000	0.957826	0.478913	+	0.877862i	$0.341031\pi$
$110$	0	0
$111$	8.92820	0.847428
$112$	10.0000i	0.944911i
$113$	0.928203i	0.0873180i	0.999046	+	0.0436590i	$0.0139015\pi$
$113$	0.928203i	0.0873180i	−0.999046	+	0.0436590i	$0.986098\pi$
$114$	2.53590	0.237509
$115$	0	0
$116$	3.46410	0.321634
$117$	1.46410i	0.135356i
$118$	12.0000i	1.10469i
$119$	0	0
$120$	0	0
$121$	1.00000	0.0909091
$122$	− 3.46410i	− 0.313625i
$123$	− 3.46410i	− 0.312348i
$124$	−2.92820	−0.262960
$125$	0	0
$126$	3.46410	0.308607
$127$	4.92820i	0.437307i	0.975803	+	0.218654i	$0.0701665\pi$
$127$	4.92820i	0.437307i	−0.975803	+	0.218654i	$0.929834\pi$
$128$	12.1244i	1.07165i
$129$	−8.92820	−0.786084
$130$	0	0
$131$	5.07180	0.443125	0.221562	−	0.975146i	$-0.428884\pi$
$131$	5.07180	0.443125	0.221562	+	0.975146i	$0.428884\pi$
$132$	1.00000i	0.0870388i
$133$	− 2.92820i	− 0.253907i
$134$	−13.8564	−1.19701
$135$	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	$0.720793\pi$
$138$	− 12.0000i	− 1.02151i
$139$	8.39230	0.711826	0.355913	−	0.934519i	$-0.384170\pi$
$139$	8.39230	0.711826	0.355913	+	0.934519i	$0.384170\pi$
$140$	0	0
$141$	6.92820	0.583460
$142$	24.0000i	2.01404i
$143$	1.46410i	0.122434i
$144$	5.00000	0.416667
$145$	0	0
$146$	21.4641	1.77638
$147$	3.00000i	0.247436i
$148$	8.92820i	0.733894i
$149$	8.53590	0.699288	0.349644	−	0.936883i	$-0.386303\pi$
$149$	8.53590	0.699288	0.349644	+	0.936883i	$0.386303\pi$
$150$	0	0
$151$	0.392305	0.0319253	0.0159627	−	0.999873i	$-0.494919\pi$
$151$	0.392305	0.0319253	0.0159627	+	0.999873i	$0.494919\pi$
$152$	− 2.53590i	− 0.205689i
$153$	0	0
$154$	3.46410	0.279145
$155$	0	0
$156$	−1.46410	−0.117222
$157$	16.9282i	1.35102i	0.737352	+	0.675509i	$0.236076\pi$
$157$	16.9282i	1.35102i	−0.737352	+	0.675509i	$0.763924\pi$
$158$	− 23.3205i	− 1.85528i
$159$	12.9282	1.02527
$160$	0	0
$161$	−13.8564	−1.09204
$162$	− 1.73205i	− 0.136083i
$163$	− 17.8564i	− 1.39862i	−0.714818	−	0.699311i	$-0.753490\pi$
$163$	− 17.8564i	− 1.39862i	0.714818	−	0.699311i	$-0.246510\pi$
$164$	3.46410	0.270501
$165$	0	0
$166$	26.7846	2.07889
$167$	− 10.3923i	− 0.804181i	−0.915600	−	0.402090i	$-0.868284\pi$
$167$	− 10.3923i	− 0.804181i	0.915600	−	0.402090i	$-0.131716\pi$
$168$	− 3.46410i	− 0.267261i
$169$	10.8564	0.835108
$170$	0	0
$171$	−1.46410	−0.111963
$172$	− 8.92820i	− 0.680769i
$173$	− 12.0000i	− 0.912343i	−0.889892	−	0.456172i	$-0.849220\pi$
$173$	− 12.0000i	− 0.912343i	0.889892	−	0.456172i	$-0.150780\pi$
$174$	−6.00000	−0.454859
$175$	0	0
$176$	5.00000	0.376889
$177$	− 6.92820i	− 0.520756i
$178$	− 22.3923i	− 1.67837i
$179$	6.92820	0.517838	0.258919	−	0.965899i	$-0.416634\pi$
$179$	6.92820	0.517838	0.258919	+	0.965899i	$0.416634\pi$
$180$	0	0
$181$	−11.8564	−0.881280	−0.440640	−	0.897684i	$-0.645248\pi$
$181$	−11.8564	−0.881280	−0.440640	+	0.897684i	$0.645248\pi$
$182$	5.07180i	0.375947i
$183$	2.00000i	0.147844i
$184$	−12.0000	−0.884652
$185$	0	0
$186$	5.07180	0.371882
$187$	0	0
$188$	6.92820i	0.505291i
$189$	−2.00000	−0.145479
$190$	0	0
$191$	−5.07180	−0.366982	−0.183491	−	0.983021i	$-0.558740\pi$
$191$	−5.07180	−0.366982	−0.183491	+	0.983021i	$0.558740\pi$
$192$	− 1.00000i	− 0.0721688i
$193$	3.60770i	0.259688i	0.991534	+	0.129844i	$0.0414476\pi$
$193$	3.60770i	0.259688i	−0.991534	+	0.129844i	$0.958552\pi$
$194$	17.3205	1.24354
$195$	0	0
$196$	−3.00000	−0.214286
$197$	12.0000i	0.854965i	0.904024	+	0.427482i	$0.140599\pi$
$197$	12.0000i	0.854965i	−0.904024	+	0.427482i	$0.859401\pi$
$198$	− 1.73205i	− 0.123091i
$199$	−16.7846	−1.18983	−0.594915	−	0.803789i	$-0.702814\pi$
$199$	−16.7846	−1.18983	−0.594915	+	0.803789i	$0.702814\pi$
$200$	0	0
$201$	8.00000	0.564276
$202$	− 18.0000i	− 1.26648i
$203$	6.92820i	0.486265i
$204$	0	0
$205$	0	0
$206$	13.8564	0.965422
$207$	6.92820i	0.481543i
$208$	7.32051i	0.507586i
$209$	−1.46410	−0.101274
$210$	0	0
$211$	12.3923	0.853121	0.426561	−	0.904459i	$-0.359725\pi$
$211$	12.3923	0.853121	0.426561	+	0.904459i	$0.359725\pi$
$212$	12.9282i	0.887913i
$213$	− 13.8564i	− 0.949425i
$214$	−26.7846	−1.83096
$215$	0	0
$216$	−1.73205	−0.117851
$217$	− 5.85641i	− 0.397559i
$218$	− 17.3205i	− 1.17309i
$219$	−12.3923	−0.837394
$220$	0	0
$221$	0	0
$222$	− 15.4641i	− 1.03788i
$223$	− 17.8564i	− 1.19575i	−0.801588	−	0.597877i	$-0.796011\pi$
$223$	− 17.8564i	− 1.19575i	0.801588	−	0.597877i	$-0.203989\pi$
$224$	10.3923	0.694365
$225$	0	0
$226$	1.60770	0.106942
$227$	− 8.53590i	− 0.566547i	−0.959039	−	0.283274i	$-0.908580\pi$
$227$	− 8.53590i	− 0.566547i	0.959039	−	0.283274i	$-0.0914205\pi$
$228$	− 1.46410i	− 0.0969625i
$229$	−3.85641	−0.254839	−0.127419	−	0.991849i	$-0.540669\pi$
$229$	−3.85641	−0.254839	−0.127419	+	0.991849i	$0.540669\pi$
$230$	0	0
$231$	−2.00000	−0.131590
$232$	6.00000i	0.393919i
$233$	12.0000i	0.786146i	0.919507	+	0.393073i	$0.128588\pi$
$233$	12.0000i	0.786146i	−0.919507	+	0.393073i	$0.871412\pi$
$234$	2.53590	0.165777
$235$	0	0
$236$	6.92820	0.450988
$237$	13.4641i	0.874587i
$238$	0	0
$239$	12.0000	0.776215	0.388108	−	0.921614i	$-0.373129\pi$
$239$	12.0000	0.776215	0.388108	+	0.921614i	$0.373129\pi$
$240$	0	0
$241$	27.8564	1.79439	0.897194	−	0.441636i	$-0.145602\pi$
$241$	27.8564	1.79439	0.897194	+	0.441636i	$0.145602\pi$
$242$	− 1.73205i	− 0.111340i
$243$	1.00000i	0.0641500i
$244$	−2.00000	−0.128037
$245$	0	0
$246$	−6.00000	−0.382546
$247$	− 2.14359i	− 0.136394i
$248$	− 5.07180i	− 0.322059i
$249$	−15.4641	−0.979998
$250$	0	0
$251$	25.8564	1.63204	0.816021	−	0.578022i	$-0.196175\pi$
$251$	25.8564	1.63204	0.816021	+	0.578022i	$0.196175\pi$
$252$	− 2.00000i	− 0.125988i
$253$	6.92820i	0.435572i
$254$	8.53590	0.535590
$255$	0	0
$256$	19.0000	1.18750
$257$	− 7.85641i	− 0.490069i	−0.969514	−	0.245035i	$-0.921201\pi$
$257$	− 7.85641i	− 0.490069i	0.969514	−	0.245035i	$-0.0787993\pi$
$258$	15.4641i	0.962753i
$259$	−17.8564	−1.10954
$260$	0	0
$261$	3.46410	0.214423
$262$	− 8.78461i	− 0.542715i
$263$	27.4641i	1.69351i	0.531984	+	0.846755i	$0.321447\pi$
$263$	27.4641i	1.69351i	−0.531984	+	0.846755i	$0.678553\pi$
$264$	−1.73205	−0.106600
$265$	0	0
$266$	−5.07180	−0.310972
$267$	12.9282i	0.791193i
$268$	8.00000i	0.488678i
$269$	7.85641	0.479014	0.239507	−	0.970895i	$-0.423014\pi$
$269$	7.85641	0.479014	0.239507	+	0.970895i	$0.423014\pi$
$270$	0	0
$271$	−32.3923	−1.96769	−0.983846	−	0.179016i	$-0.942709\pi$
$271$	−32.3923	−1.96769	−0.983846	+	0.179016i	$0.942709\pi$
$272$	0	0
$273$	− 2.92820i	− 0.177223i
$274$	31.1769	1.88347
$275$	0	0
$276$	−6.92820	−0.417029
$277$	− 22.5359i	− 1.35405i	−0.735960	−	0.677025i	$-0.763269\pi$
$277$	− 22.5359i	− 1.35405i	0.735960	−	0.677025i	$-0.236731\pi$
$278$	− 14.5359i	− 0.871805i
$279$	−2.92820	−0.175307
$280$	0	0
$281$	−3.46410	−0.206651	−0.103325	−	0.994648i	$-0.532948\pi$
$281$	−3.46410	−0.206651	−0.103325	+	0.994648i	$0.532948\pi$
$282$	− 12.0000i	− 0.714590i
$283$	8.92820i	0.530727i	0.964148	+	0.265363i	$0.0854919\pi$
$283$	8.92820i	0.530727i	−0.964148	+	0.265363i	$0.914508\pi$
$284$	13.8564	0.822226
$285$	0	0
$286$	2.53590	0.149951
$287$	6.92820i	0.408959i
$288$	− 5.19615i	− 0.306186i
$289$	17.0000	1.00000
$290$	0	0
$291$	−10.0000	−0.586210
$292$	− 12.3923i	− 0.725205i
$293$	13.8564i	0.809500i	0.914427	+	0.404750i	$0.132641\pi$
$293$	13.8564i	0.809500i	−0.914427	+	0.404750i	$0.867359\pi$
$294$	5.19615	0.303046
$295$	0	0
$296$	−15.4641	−0.898833
$297$	1.00000i	0.0580259i
$298$	− 14.7846i	− 0.856449i
$299$	−10.1436	−0.586619
$300$	0	0
$301$	17.8564	1.02923
$302$	− 0.679492i	− 0.0391004i
$303$	10.3923i	0.597022i
$304$	−7.32051	−0.419860
$305$	0	0
$306$	0	0
$307$	− 14.0000i	− 0.799022i	−0.916728	−	0.399511i	$-0.869180\pi$
$307$	− 14.0000i	− 0.799022i	0.916728	−	0.399511i	$-0.130820\pi$
$308$	− 2.00000i	− 0.113961i
$309$	−8.00000	−0.455104
$310$	0	0
$311$	18.9282	1.07332	0.536660	−	0.843799i	$-0.319686\pi$
$311$	18.9282	1.07332	0.536660	+	0.843799i	$0.319686\pi$
$312$	− 2.53590i	− 0.143567i
$313$	7.07180i	0.399722i	0.979824	+	0.199861i	$0.0640490\pi$
$313$	7.07180i	0.399722i	−0.979824	+	0.199861i	$0.935951\pi$
$314$	29.3205	1.65465
$315$	0	0
$316$	−13.4641	−0.757415
$317$	11.0718i	0.621854i	0.950434	+	0.310927i	$0.100639\pi$
$317$	11.0718i	0.621854i	−0.950434	+	0.310927i	$0.899361\pi$
$318$	− 22.3923i	− 1.25570i
$319$	3.46410	0.193952
$320$	0	0
$321$	15.4641	0.863122
$322$	24.0000i	1.33747i
$323$	0	0
$324$	−1.00000	−0.0555556
$325$	0	0
$326$	−30.9282	−1.71295
$327$	10.0000i	0.553001i
$328$	6.00000i	0.331295i
$329$	−13.8564	−0.763928
$330$	0	0
$331$	−17.8564	−0.981477	−0.490738	−	0.871307i	$-0.663273\pi$
$331$	−17.8564	−0.981477	−0.490738	+	0.871307i	$0.663273\pi$
$332$	− 15.4641i	− 0.848703i
$333$	8.92820i	0.489263i
$334$	−18.0000	−0.984916
$335$	0	0
$336$	−10.0000	−0.545545
$337$	29.1769i	1.58937i	0.607023	+	0.794684i	$0.292363\pi$
$337$	29.1769i	1.58937i	−0.607023	+	0.794684i	$0.707637\pi$
$338$	− 18.8038i	− 1.02279i
$339$	−0.928203	−0.0504131
$340$	0	0
$341$	−2.92820	−0.158571
$342$	2.53590i	0.137126i
$343$	− 20.0000i	− 1.07990i
$344$	15.4641	0.833768
$345$	0	0
$346$	−20.7846	−1.11739
$347$	− 1.60770i	− 0.0863056i	−0.999068	−	0.0431528i	$-0.986260\pi$
$347$	− 1.60770i	− 0.0863056i	0.999068	−	0.0431528i	$-0.0137402\pi$
$348$	3.46410i	0.185695i
$349$	35.8564	1.91935	0.959675	−	0.281113i	$-0.0907035\pi$
$349$	35.8564	1.91935	0.959675	+	0.281113i	$0.0907035\pi$
$350$	0	0
$351$	−1.46410	−0.0781480
$352$	− 5.19615i	− 0.276956i
$353$	− 0.928203i	− 0.0494033i	−0.999695	−	0.0247016i	$-0.992136\pi$
$353$	− 0.928203i	− 0.0494033i	0.999695	−	0.0247016i	$-0.00786357\pi$
$354$	−12.0000	−0.637793
$355$	0	0
$356$	−12.9282	−0.685193
$357$	0	0
$358$	− 12.0000i	− 0.634220i
$359$	−20.7846	−1.09697	−0.548485	−	0.836160i	$-0.684795\pi$
$359$	−20.7846	−1.09697	−0.548485	+	0.836160i	$0.684795\pi$
$360$	0	0
$361$	−16.8564	−0.887179
$362$	20.5359i	1.07934i
$363$	1.00000i	0.0524864i
$364$	2.92820	0.153480
$365$	0	0
$366$	3.46410	0.181071
$367$	− 20.0000i	− 1.04399i	−0.852948	−	0.521996i	$-0.825188\pi$
$367$	− 20.0000i	− 1.04399i	0.852948	−	0.521996i	$-0.174812\pi$
$368$	34.6410i	1.80579i
$369$	3.46410	0.180334
$370$	0	0
$371$	−25.8564	−1.34240
$372$	− 2.92820i	− 0.151820i
$373$	0.392305i	0.0203128i	0.999948	+	0.0101564i	$0.00323293\pi$
$373$	0.392305i	0.0203128i	−0.999948	+	0.0101564i	$0.996767\pi$
$374$	0	0
$375$	0	0
$376$	−12.0000	−0.618853
$377$	5.07180i	0.261211i
$378$	3.46410i	0.178174i
$379$	−9.85641	−0.506290	−0.253145	−	0.967428i	$-0.581465\pi$
$379$	−9.85641	−0.506290	−0.253145	+	0.967428i	$0.581465\pi$
$380$	0	0
$381$	−4.92820	−0.252479
$382$	8.78461i	0.449460i
$383$	13.8564i	0.708029i	0.935240	+	0.354015i	$0.115184\pi$
$383$	13.8564i	0.708029i	−0.935240	+	0.354015i	$0.884816\pi$
$384$	−12.1244	−0.618718
$385$	0	0
$386$	6.24871	0.318051
$387$	− 8.92820i	− 0.453846i
$388$	− 10.0000i	− 0.507673i
$389$	−24.9282	−1.26391	−0.631955	−	0.775005i	$-0.717747\pi$
$389$	−24.9282	−1.26391	−0.631955	+	0.775005i	$0.717747\pi$
$390$	0	0
$391$	0	0
$392$	− 5.19615i	− 0.262445i
$393$	5.07180i	0.255838i
$394$	20.7846	1.04711
$395$	0	0
$396$	−1.00000	−0.0502519
$397$	− 2.00000i	− 0.100377i	−0.998740	−	0.0501886i	$-0.984018\pi$
$397$	− 2.00000i	− 0.100377i	0.998740	−	0.0501886i	$-0.0159822\pi$
$398$	29.0718i	1.45724i
$399$	2.92820	0.146594
$400$	0	0
$401$	19.8564	0.991582	0.495791	−	0.868442i	$-0.334878\pi$
$401$	19.8564	0.991582	0.495791	+	0.868442i	$0.334878\pi$
$402$	− 13.8564i	− 0.691095i
$403$	− 4.28719i	− 0.213560i
$404$	−10.3923	−0.517036
$405$	0	0
$406$	12.0000	0.595550
$407$	8.92820i	0.442555i
$408$	0	0
$409$	−34.7846	−1.71999	−0.859994	−	0.510304i	$-0.829533\pi$
$409$	−34.7846	−1.71999	−0.859994	+	0.510304i	$0.829533\pi$
$410$	0	0
$411$	−18.0000	−0.887875
$412$	− 8.00000i	− 0.394132i
$413$	13.8564i	0.681829i
$414$	12.0000	0.589768
$415$	0	0
$416$	7.60770	0.372998
$417$	8.39230i	0.410973i
$418$	2.53590i	0.124035i
$419$	−17.0718	−0.834012	−0.417006	−	0.908904i	$-0.636921\pi$
$419$	−17.0718	−0.834012	−0.417006	+	0.908904i	$0.636921\pi$
$420$	0	0
$421$	2.00000	0.0974740	0.0487370	−	0.998812i	$-0.484480\pi$
$421$	2.00000	0.0974740	0.0487370	+	0.998812i	$0.484480\pi$
$422$	− 21.4641i	− 1.04486i
$423$	6.92820i	0.336861i
$424$	−22.3923	−1.08747
$425$	0	0
$426$	−24.0000	−1.16280
$427$	− 4.00000i	− 0.193574i
$428$	15.4641i	0.747486i
$429$	−1.46410	−0.0706875
$430$	0	0
$431$	−32.7846	−1.57918	−0.789590	−	0.613635i	$-0.789706\pi$
$431$	−32.7846	−1.57918	−0.789590	+	0.613635i	$0.789706\pi$
$432$	5.00000i	0.240563i
$433$	27.8564i	1.33869i	0.742950	+	0.669347i	$0.233426\pi$
$433$	27.8564i	1.33869i	−0.742950	+	0.669347i	$0.766574\pi$
$434$	−10.1436	−0.486908
$435$	0	0
$436$	−10.0000	−0.478913
$437$	− 10.1436i	− 0.485234i
$438$	21.4641i	1.02559i
$439$	29.1769	1.39254	0.696269	−	0.717781i	$-0.254842\pi$
$439$	29.1769	1.39254	0.696269	+	0.717781i	$0.254842\pi$
$440$	0	0
$441$	−3.00000	−0.142857
$442$	0	0
$443$	12.0000i	0.570137i	0.958507	+	0.285069i	$0.0920164\pi$
$443$	12.0000i	0.570137i	−0.958507	+	0.285069i	$0.907984\pi$
$444$	−8.92820	−0.423714
$445$	0	0
$446$	−30.9282	−1.46449
$447$	8.53590i	0.403734i
$448$	2.00000i	0.0944911i
$449$	−14.7846	−0.697729	−0.348864	−	0.937173i	$-0.613433\pi$
$449$	−14.7846	−0.697729	−0.348864	+	0.937173i	$0.613433\pi$
$450$	0	0
$451$	3.46410	0.163118
$452$	− 0.928203i	− 0.0436590i
$453$	0.392305i	0.0184321i
$454$	−14.7846	−0.693876
$455$	0	0
$456$	2.53590	0.118754
$457$	8.39230i	0.392575i	0.980546	+	0.196288i	$0.0628887\pi$
$457$	8.39230i	0.392575i	−0.980546	+	0.196288i	$0.937111\pi$
$458$	6.67949i	0.312112i
$459$	0	0
$460$	0	0
$461$	−12.2487	−0.570479	−0.285240	−	0.958456i	$-0.592073\pi$
$461$	−12.2487	−0.570479	−0.285240	+	0.958456i	$0.592073\pi$
$462$	3.46410i	0.161165i
$463$	− 28.0000i	− 1.30127i	−0.759390	−	0.650635i	$-0.774503\pi$
$463$	− 28.0000i	− 1.30127i	0.759390	−	0.650635i	$-0.225497\pi$
$464$	17.3205	0.804084
$465$	0	0
$466$	20.7846	0.962828
$467$	− 18.9282i	− 0.875893i	−0.899001	−	0.437946i	$-0.855706\pi$
$467$	− 18.9282i	− 0.875893i	0.899001	−	0.437946i	$-0.144294\pi$
$468$	− 1.46410i	− 0.0676781i
$469$	−16.0000	−0.738811
$470$	0	0
$471$	−16.9282	−0.780010
$472$	12.0000i	0.552345i
$473$	− 8.92820i	− 0.410519i
$474$	23.3205	1.07115
$475$	0	0
$476$	0	0
$477$	12.9282i	0.591942i
$478$	− 20.7846i	− 0.950666i
$479$	−12.0000	−0.548294	−0.274147	−	0.961688i	$-0.588395\pi$
$479$	−12.0000	−0.548294	−0.274147	+	0.961688i	$0.588395\pi$
$480$	0	0
$481$	−13.0718	−0.596023
$482$	− 48.2487i	− 2.19767i
$483$	− 13.8564i	− 0.630488i
$484$	−1.00000	−0.0454545
$485$	0	0
$486$	1.73205	0.0785674
$487$	− 23.7128i	− 1.07453i	−0.843413	−	0.537265i	$-0.819457\pi$
$487$	− 23.7128i	− 1.07453i	0.843413	−	0.537265i	$-0.180543\pi$
$488$	− 3.46410i	− 0.156813i
$489$	17.8564	0.807495
$490$	0	0
$491$	−17.0718	−0.770439	−0.385220	−	0.922825i	$-0.625874\pi$
$491$	−17.0718	−0.770439	−0.385220	+	0.922825i	$0.625874\pi$
$492$	3.46410i	0.156174i
$493$	0	0
$494$	−3.71281	−0.167047
$495$	0	0
$496$	−14.6410	−0.657401
$497$	27.7128i	1.24309i
$498$	26.7846i	1.20025i
$499$	12.7846	0.572318	0.286159	−	0.958182i	$-0.407621\pi$
$499$	12.7846	0.572318	0.286159	+	0.958182i	$0.407621\pi$
$500$	0	0
$501$	10.3923	0.464294
$502$	− 44.7846i	− 1.99883i
$503$	− 31.1769i	− 1.39011i	−0.718957	−	0.695055i	$-0.755380\pi$
$503$	− 31.1769i	− 1.39011i	0.718957	−	0.695055i	$-0.244620\pi$
$504$	3.46410	0.154303
$505$	0	0
$506$	12.0000	0.533465
$507$	10.8564i	0.482150i
$508$	− 4.92820i	− 0.218654i
$509$	7.85641	0.348229	0.174115	−	0.984725i	$-0.444294\pi$
$509$	7.85641	0.348229	0.174115	+	0.984725i	$0.444294\pi$
$510$	0	0
$511$	24.7846	1.09641
$512$	− 8.66025i	− 0.382733i
$513$	− 1.46410i	− 0.0646417i
$514$	−13.6077	−0.600210
$515$	0	0
$516$	8.92820	0.393042
$517$	6.92820i	0.304702i
$518$	30.9282i	1.35891i
$519$	12.0000	0.526742
$520$	0	0
$521$	6.00000	0.262865	0.131432	−	0.991325i	$-0.458042\pi$
$521$	6.00000	0.262865	0.131432	+	0.991325i	$0.458042\pi$
$522$	− 6.00000i	− 0.262613i
$523$	− 22.0000i	− 0.961993i	−0.876723	−	0.480996i	$-0.840275\pi$
$523$	− 22.0000i	− 0.961993i	0.876723	−	0.480996i	$-0.159725\pi$
$524$	−5.07180	−0.221562
$525$	0	0
$526$	47.5692	2.07412
$527$	0	0
$528$	5.00000i	0.217597i
$529$	−25.0000	−1.08696
$530$	0	0
$531$	6.92820	0.300658
$532$	2.92820i	0.126954i
$533$	5.07180i	0.219684i
$534$	22.3923	0.969010
$535$	0	0
$536$	−13.8564	−0.598506
$537$	6.92820i	0.298974i
$538$	− 13.6077i	− 0.586669i
$539$	−3.00000	−0.129219
$540$	0	0
$541$	0.143594	0.00617357	0.00308678	−	0.999995i	$-0.499017\pi$
$541$	0.143594	0.00617357	0.00308678	+	0.999995i	$0.499017\pi$
$542$	56.1051i	2.40992i
$543$	− 11.8564i	− 0.508807i
$544$	0	0
$545$	0	0
$546$	−5.07180	−0.217053
$547$	− 2.00000i	− 0.0855138i	−0.999086	−	0.0427569i	$-0.986386\pi$
$547$	− 2.00000i	− 0.0855138i	0.999086	−	0.0427569i	$-0.0136141\pi$
$548$	− 18.0000i	− 0.768922i
$549$	−2.00000	−0.0853579
$550$	0	0
$551$	−5.07180	−0.216066
$552$	− 12.0000i	− 0.510754i
$553$	− 26.9282i	− 1.14510i
$554$	−39.0333	−1.65837
$555$	0	0
$556$	−8.39230	−0.355913
$557$	− 44.7846i	− 1.89758i	−0.315900	−	0.948792i	$-0.602306\pi$
$557$	− 44.7846i	− 1.89758i	0.315900	−	0.948792i	$-0.397694\pi$
$558$	5.07180i	0.214706i
$559$	13.0718	0.552878
$560$	0	0
$561$	0	0
$562$	6.00000i	0.253095i
$563$	− 10.3923i	− 0.437983i	−0.975727	−	0.218992i	$-0.929723\pi$
$563$	− 10.3923i	− 0.437983i	0.975727	−	0.218992i	$-0.0702768\pi$
$564$	−6.92820	−0.291730
$565$	0	0
$566$	15.4641	0.650005
$567$	− 2.00000i	− 0.0839921i
$568$	24.0000i	1.00702i
$569$	29.3205	1.22918	0.614590	−	0.788847i	$-0.289322\pi$
$569$	29.3205	1.22918	0.614590	+	0.788847i	$0.289322\pi$
$570$	0	0
$571$	24.3923	1.02079	0.510393	−	0.859941i	$-0.329500\pi$
$571$	24.3923	1.02079	0.510393	+	0.859941i	$0.329500\pi$
$572$	− 1.46410i	− 0.0612172i
$573$	− 5.07180i	− 0.211877i
$574$	12.0000	0.500870
$575$	0	0
$576$	1.00000	0.0416667
$577$	− 22.7846i	− 0.948536i	−0.880381	−	0.474268i	$-0.842713\pi$
$577$	− 22.7846i	− 0.948536i	0.880381	−	0.474268i	$-0.157287\pi$
$578$	− 29.4449i	− 1.22474i
$579$	−3.60770	−0.149931
$580$	0	0
$581$	30.9282	1.28312
$582$	17.3205i	0.717958i
$583$	12.9282i	0.535431i
$584$	21.4641	0.888191
$585$	0	0
$586$	24.0000	0.991431
$587$	5.07180i	0.209335i	0.994507	+	0.104668i	$0.0333779\pi$
$587$	5.07180i	0.209335i	−0.994507	+	0.104668i	$0.966622\pi$
$588$	− 3.00000i	− 0.123718i
$589$	4.28719	0.176650
$590$	0	0
$591$	−12.0000	−0.493614
$592$	44.6410i	1.83473i
$593$	− 32.7846i	− 1.34630i	−0.739505	−	0.673151i	$-0.764940\pi$
$593$	− 32.7846i	− 1.34630i	0.739505	−	0.673151i	$-0.235060\pi$
$594$	1.73205	0.0710669
$595$	0	0
$596$	−8.53590	−0.349644
$597$	− 16.7846i	− 0.686948i
$598$	17.5692i	0.718459i
$599$	10.1436	0.414456	0.207228	−	0.978293i	$-0.433556\pi$
$599$	10.1436	0.414456	0.207228	+	0.978293i	$0.433556\pi$
$600$	0	0
$601$	36.6410	1.49462	0.747309	−	0.664477i	$-0.231345\pi$
$601$	36.6410	1.49462	0.747309	+	0.664477i	$0.231345\pi$
$602$	− 30.9282i	− 1.26054i
$603$	8.00000i	0.325785i
$604$	−0.392305	−0.0159627
$605$	0	0
$606$	18.0000	0.731200
$607$	− 22.7846i	− 0.924799i	−0.886672	−	0.462399i	$-0.846989\pi$
$607$	− 22.7846i	− 0.924799i	0.886672	−	0.462399i	$-0.153011\pi$
$608$	7.60770i	0.308533i
$609$	−6.92820	−0.280745
$610$	0	0
$611$	−10.1436	−0.410366
$612$	0	0
$613$	0.392305i	0.0158450i	0.999969	+	0.00792252i	$0.00252184\pi$
$613$	0.392305i	0.0158450i	−0.999969	+	0.00792252i	$0.997478\pi$
$614$	−24.2487	−0.978598
$615$	0	0
$616$	3.46410	0.139573
$617$	23.0718i	0.928836i	0.885616	+	0.464418i	$0.153736\pi$
$617$	23.0718i	0.928836i	−0.885616	+	0.464418i	$0.846264\pi$
$618$	13.8564i	0.557386i
$619$	−20.0000	−0.803868	−0.401934	−	0.915669i	$-0.631662\pi$
$619$	−20.0000	−0.803868	−0.401934	+	0.915669i	$0.631662\pi$
$620$	0	0
$621$	−6.92820	−0.278019
$622$	− 32.7846i	− 1.31454i
$623$	− 25.8564i	− 1.03592i
$624$	−7.32051	−0.293055
$625$	0	0
$626$	12.2487	0.489557
$627$	− 1.46410i	− 0.0584706i
$628$	− 16.9282i	− 0.675509i
$629$	0	0
$630$	0	0
$631$	−34.9282	−1.39047	−0.695235	−	0.718783i	$-0.744700\pi$
$631$	−34.9282	−1.39047	−0.695235	+	0.718783i	$0.744700\pi$
$632$	− 23.3205i	− 0.927640i
$633$	12.3923i	0.492550i
$634$	19.1769	0.761613
$635$	0	0
$636$	−12.9282	−0.512637
$637$	− 4.39230i	− 0.174029i
$638$	− 6.00000i	− 0.237542i
$639$	13.8564	0.548151
$640$	0	0
$641$	0.928203	0.0366618	0.0183309	−	0.999832i	$-0.494165\pi$
$641$	0.928203	0.0366618	0.0183309	+	0.999832i	$0.494165\pi$
$642$	− 26.7846i	− 1.05710i
$643$	− 45.5692i	− 1.79707i	−0.438897	−	0.898537i	$-0.644631\pi$
$643$	− 45.5692i	− 1.79707i	0.438897	−	0.898537i	$-0.355369\pi$
$644$	13.8564	0.546019
$645$	0	0
$646$	0	0
$647$	27.7128i	1.08950i	0.838597	+	0.544752i	$0.183376\pi$
$647$	27.7128i	1.08950i	−0.838597	+	0.544752i	$0.816624\pi$
$648$	− 1.73205i	− 0.0680414i
$649$	6.92820	0.271956
$650$	0	0
$651$	5.85641	0.229531
$652$	17.8564i	0.699311i
$653$	− 7.85641i	− 0.307445i	−0.988114	−	0.153722i	$-0.950874\pi$
$653$	− 7.85641i	− 0.307445i	0.988114	−	0.153722i	$-0.0491262\pi$
$654$	17.3205	0.677285
$655$	0	0
$656$	17.3205	0.676252
$657$	− 12.3923i	− 0.483470i
$658$	24.0000i	0.935617i
$659$	−39.7128	−1.54699	−0.773496	−	0.633801i	$-0.781494\pi$
$659$	−39.7128	−1.54699	−0.773496	+	0.633801i	$0.781494\pi$
$660$	0	0
$661$	14.0000	0.544537	0.272268	−	0.962221i	$-0.412226\pi$
$661$	14.0000	0.544537	0.272268	+	0.962221i	$0.412226\pi$
$662$	30.9282i	1.20206i
$663$	0	0
$664$	26.7846	1.03944
$665$	0	0
$666$	15.4641	0.599222
$667$	24.0000i	0.929284i
$668$	10.3923i	0.402090i
$669$	17.8564	0.690369
$670$	0	0
$671$	−2.00000	−0.0772091
$672$	10.3923i	0.400892i
$673$	31.3205i	1.20732i	0.797243	+	0.603658i	$0.206291\pi$
$673$	31.3205i	1.20732i	−0.797243	+	0.603658i	$0.793709\pi$
$674$	50.5359	1.94657
$675$	0	0
$676$	−10.8564	−0.417554
$677$	32.7846i	1.26001i	0.776589	+	0.630007i	$0.216948\pi$
$677$	32.7846i	1.26001i	−0.776589	+	0.630007i	$0.783052\pi$
$678$	1.60770i	0.0617432i
$679$	20.0000	0.767530
$680$	0	0
$681$	8.53590	0.327096
$682$	5.07180i	0.194209i
$683$	8.78461i	0.336134i	0.985776	+	0.168067i	$0.0537525\pi$
$683$	8.78461i	0.336134i	−0.985776	+	0.168067i	$0.946248\pi$
$684$	1.46410	0.0559813
$685$	0	0
$686$	−34.6410	−1.32260
$687$	− 3.85641i	− 0.147131i
$688$	− 44.6410i	− 1.70192i
$689$	−18.9282	−0.721107
$690$	0	0
$691$	−7.71281	−0.293409	−0.146705	−	0.989180i	$-0.546867\pi$
$691$	−7.71281	−0.293409	−0.146705	+	0.989180i	$0.546867\pi$
$692$	12.0000i	0.456172i
$693$	− 2.00000i	− 0.0759737i
$694$	−2.78461	−0.105702
$695$	0	0
$696$	−6.00000	−0.227429
$697$	0	0
$698$	− 62.1051i	− 2.35071i
$699$	−12.0000	−0.453882
$700$	0	0
$701$	32.5359	1.22886	0.614432	−	0.788970i	$-0.289385\pi$
$701$	32.5359	1.22886	0.614432	+	0.788970i	$0.289385\pi$
$702$	2.53590i	0.0957113i
$703$	− 13.0718i	− 0.493012i
$704$	1.00000	0.0376889
$705$	0	0
$706$	−1.60770	−0.0605064
$707$	− 20.7846i	− 0.781686i
$708$	6.92820i	0.260378i
$709$	−15.8564	−0.595500	−0.297750	−	0.954644i	$-0.596236\pi$
$709$	−15.8564	−0.595500	−0.297750	+	0.954644i	$0.596236\pi$
$710$	0	0
$711$	−13.4641	−0.504943
$712$	− 22.3923i	− 0.839187i
$713$	− 20.2872i	− 0.759761i
$714$	0	0
$715$	0	0
$716$	−6.92820	−0.258919
$717$	12.0000i	0.448148i
$718$	36.0000i	1.34351i
$719$	18.9282	0.705903	0.352951	−	0.935642i	$-0.385178\pi$
$719$	18.9282	0.705903	0.352951	+	0.935642i	$0.385178\pi$
$720$	0	0
$721$	16.0000	0.595871
$722$	29.1962i	1.08657i
$723$	27.8564i	1.03599i
$724$	11.8564	0.440640
$725$	0	0
$726$	1.73205	0.0642824
$727$	− 8.00000i	− 0.296704i	−0.988935	−	0.148352i	$-0.952603\pi$
$727$	− 8.00000i	− 0.296704i	0.988935	−	0.148352i	$-0.0473968\pi$
$728$	5.07180i	0.187973i
$729$	−1.00000	−0.0370370
$730$	0	0
$731$	0	0
$732$	− 2.00000i	− 0.0739221i
$733$	− 49.9615i	− 1.84537i	−0.385553	−	0.922686i	$-0.625989\pi$
$733$	− 49.9615i	− 1.84537i	0.385553	−	0.922686i	$-0.374011\pi$
$734$	−34.6410	−1.27862
$735$	0	0
$736$	36.0000	1.32698
$737$	8.00000i	0.294684i
$738$	− 6.00000i	− 0.220863i
$739$	−10.5359	−0.387569	−0.193785	−	0.981044i	$-0.562076\pi$
$739$	−10.5359	−0.387569	−0.193785	+	0.981044i	$0.562076\pi$
$740$	0	0
$741$	2.14359	0.0787469
$742$	44.7846i	1.64409i
$743$	46.3923i	1.70197i	0.525191	+	0.850984i	$0.323994\pi$
$743$	46.3923i	1.70197i	−0.525191	+	0.850984i	$0.676006\pi$
$744$	5.07180	0.185941
$745$	0	0
$746$	0.679492	0.0248780
$747$	− 15.4641i	− 0.565802i
$748$	0	0
$749$	−30.9282	−1.13009
$750$	0	0
$751$	13.0718	0.476997	0.238498	−	0.971143i	$-0.423345\pi$
$751$	13.0718	0.476997	0.238498	+	0.971143i	$0.423345\pi$
$752$	34.6410i	1.26323i
$753$	25.8564i	0.942260i
$754$	8.78461	0.319917
$755$	0	0
$756$	2.00000	0.0727393
$757$	6.78461i	0.246591i	0.992370	+	0.123295i	$0.0393463\pi$
$757$	6.78461i	0.246591i	−0.992370	+	0.123295i	$0.960654\pi$
$758$	17.0718i	0.620076i
$759$	−6.92820	−0.251478
$760$	0	0
$761$	−39.4641	−1.43057	−0.715286	−	0.698832i	$-0.753704\pi$
$761$	−39.4641	−1.43057	−0.715286	+	0.698832i	$0.753704\pi$
$762$	8.53590i	0.309223i
$763$	− 20.0000i	− 0.724049i
$764$	5.07180	0.183491
$765$	0	0
$766$	24.0000	0.867155
$767$	10.1436i	0.366264i
$768$	19.0000i	0.685603i
$769$	46.4974	1.67674	0.838370	−	0.545102i	$-0.183509\pi$
$769$	46.4974	1.67674	0.838370	+	0.545102i	$0.183509\pi$
$770$	0	0
$771$	7.85641	0.282942
$772$	− 3.60770i	− 0.129844i
$773$	− 31.8564i	− 1.14580i	−0.819627	−	0.572898i	$-0.805819\pi$
$773$	− 31.8564i	− 1.14580i	0.819627	−	0.572898i	$-0.194181\pi$
$774$	−15.4641	−0.555846
$775$	0	0
$776$	17.3205	0.621770
$777$	− 17.8564i	− 0.640595i
$778$	43.1769i	1.54797i
$779$	−5.07180	−0.181716
$780$	0	0
$781$	13.8564	0.495821
$782$	0	0
$783$	3.46410i	0.123797i
$784$	−15.0000	−0.535714
$785$	0	0
$786$	8.78461	0.313337
$787$	18.7846i	0.669599i	0.942289	+	0.334800i	$0.108669\pi$
$787$	18.7846i	0.669599i	−0.942289	+	0.334800i	$0.891331\pi$
$788$	− 12.0000i	− 0.427482i
$789$	−27.4641	−0.977748
$790$	0	0
$791$	1.85641	0.0660062
$792$	− 1.73205i	− 0.0615457i
$793$	− 2.92820i	− 0.103984i
$794$	−3.46410	−0.122936
$795$	0	0
$796$	16.7846	0.594915
$797$	− 16.6410i	− 0.589455i	−0.955581	−	0.294728i	$-0.904771\pi$
$797$	− 16.6410i	− 0.589455i	0.955581	−	0.294728i	$-0.0952289\pi$
$798$	− 5.07180i	− 0.179540i
$799$	0	0
$800$	0	0
$801$	−12.9282	−0.456796
$802$	− 34.3923i	− 1.21443i
$803$	− 12.3923i	− 0.437315i
$804$	−8.00000	−0.282138
$805$	0	0
$806$	−7.42563	−0.261557
$807$	7.85641i	0.276559i
$808$	− 18.0000i	− 0.633238i
$809$	8.53590	0.300106	0.150053	−	0.988678i	$-0.452056\pi$
$809$	8.53590	0.300106	0.150053	+	0.988678i	$0.452056\pi$
$810$	0	0
$811$	−8.39230	−0.294694	−0.147347	−	0.989085i	$-0.547073\pi$
$811$	−8.39230	−0.294694	−0.147347	+	0.989085i	$0.547073\pi$
$812$	− 6.92820i	− 0.243132i
$813$	− 32.3923i	− 1.13605i
$814$	15.4641	0.542016
$815$	0	0
$816$	0	0
$817$	13.0718i	0.457324i
$818$	60.2487i	2.10655i
$819$	2.92820	0.102320
$820$	0	0
$821$	27.4641	0.958504	0.479252	−	0.877677i	$-0.340908\pi$
$821$	27.4641	0.958504	0.479252	+	0.877677i	$0.340908\pi$
$822$	31.1769i	1.08742i
$823$	49.5692i	1.72787i	0.503600	+	0.863937i	$0.332009\pi$
$823$	49.5692i	1.72787i	−0.503600	+	0.863937i	$0.667991\pi$
$824$	13.8564	0.482711
$825$	0	0
$826$	24.0000	0.835067
$827$	− 1.60770i	− 0.0559050i	−0.999609	−	0.0279525i	$-0.991101\pi$
$827$	− 1.60770i	− 0.0559050i	0.999609	−	0.0279525i	$-0.00889872\pi$
$828$	− 6.92820i	− 0.240772i
$829$	25.7128	0.893043	0.446521	−	0.894773i	$-0.352663\pi$
$829$	25.7128	0.893043	0.446521	+	0.894773i	$0.352663\pi$
$830$	0	0
$831$	22.5359	0.781762
$832$	1.46410i	0.0507586i
$833$	0	0
$834$	14.5359	0.503337
$835$	0	0
$836$	1.46410	0.0506370
$837$	− 2.92820i	− 0.101214i
$838$	29.5692i	1.02145i
$839$	−15.2154	−0.525294	−0.262647	−	0.964892i	$-0.584595\pi$
$839$	−15.2154	−0.525294	−0.262647	+	0.964892i	$0.584595\pi$
$840$	0	0
$841$	−17.0000	−0.586207
$842$	− 3.46410i	− 0.119381i
$843$	− 3.46410i	− 0.119310i
$844$	−12.3923	−0.426561
$845$	0	0
$846$	12.0000	0.412568
$847$	− 2.00000i	− 0.0687208i
$848$	64.6410i	2.21978i
$849$	−8.92820	−0.306415
$850$	0	0
$851$	−61.8564	−2.12041
$852$	13.8564i	0.474713i
$853$	24.3923i	0.835177i	0.908636	+	0.417588i	$0.137125\pi$
$853$	24.3923i	0.835177i	−0.908636	+	0.417588i	$0.862875\pi$
$854$	−6.92820	−0.237078
$855$	0	0
$856$	−26.7846	−0.915479
$857$	10.1436i	0.346499i	0.984878	+	0.173249i	$0.0554266\pi$
$857$	10.1436i	0.346499i	−0.984878	+	0.173249i	$0.944573\pi$
$858$	2.53590i	0.0865741i
$859$	−47.7128	−1.62794	−0.813970	−	0.580907i	$-0.802698\pi$
$859$	−47.7128	−1.62794	−0.813970	+	0.580907i	$0.802698\pi$
$860$	0	0
$861$	−6.92820	−0.236113
$862$	56.7846i	1.93409i
$863$	− 10.1436i	− 0.345292i	−0.984984	−	0.172646i	$-0.944768\pi$
$863$	− 10.1436i	− 0.345292i	0.984984	−	0.172646i	$-0.0552317\pi$
$864$	5.19615	0.176777
$865$	0	0
$866$	48.2487	1.63956
$867$	17.0000i	0.577350i
$868$	5.85641i	0.198779i
$869$	−13.4641	−0.456738
$870$	0	0
$871$	−11.7128	−0.396874
$872$	− 17.3205i	− 0.586546i
$873$	− 10.0000i	− 0.338449i
$874$	−17.5692	−0.594288
$875$	0	0
$876$	12.3923	0.418697
$877$	− 14.2487i	− 0.481145i	−0.970631	−	0.240572i	$-0.922665\pi$
$877$	− 14.2487i	− 0.481145i	0.970631	−	0.240572i	$-0.0773351\pi$
$878$	− 50.5359i	− 1.70550i
$879$	−13.8564	−0.467365
$880$	0	0
$881$	12.9282	0.435562	0.217781	−	0.975998i	$-0.430118\pi$
$881$	12.9282	0.435562	0.217781	+	0.975998i	$0.430118\pi$
$882$	5.19615i	0.174964i
$883$	− 4.00000i	− 0.134611i	−0.997732	−	0.0673054i	$-0.978560\pi$
$883$	− 4.00000i	− 0.134611i	0.997732	−	0.0673054i	$-0.0214402\pi$
$884$	0	0
$885$	0	0
$886$	20.7846	0.698273
$887$	36.2487i	1.21711i	0.793511	+	0.608556i	$0.208251\pi$
$887$	36.2487i	1.21711i	−0.793511	+	0.608556i	$0.791749\pi$
$888$	− 15.4641i	− 0.518941i
$889$	9.85641	0.330573
$890$	0	0
$891$	−1.00000	−0.0335013
$892$	17.8564i	0.597877i
$893$	− 10.1436i	− 0.339442i
$894$	14.7846	0.494471
$895$	0	0
$896$	24.2487	0.810093
$897$	− 10.1436i	− 0.338685i
$898$	25.6077i	0.854540i
$899$	−10.1436	−0.338308
$900$	0	0
$901$	0	0
$902$	− 6.00000i	− 0.199778i
$903$	17.8564i	0.594224i
$904$	1.60770	0.0534711
$905$	0	0
$906$	0.679492	0.0225746
$907$	− 45.8564i	− 1.52264i	−0.648378	−	0.761318i	$-0.724552\pi$
$907$	− 45.8564i	− 1.52264i	0.648378	−	0.761318i	$-0.275448\pi$
$908$	8.53590i	0.283274i
$909$	−10.3923	−0.344691
$910$	0	0
$911$	5.07180	0.168036	0.0840181	−	0.996464i	$-0.473225\pi$
$911$	5.07180	0.168036	0.0840181	+	0.996464i	$0.473225\pi$
$912$	− 7.32051i	− 0.242406i
$913$	− 15.4641i	− 0.511787i
$914$	14.5359	0.480805
$915$	0	0
$916$	3.85641	0.127419
$917$	− 10.1436i	− 0.334971i
$918$	0	0
$919$	11.6077	0.382903	0.191451	−	0.981502i	$-0.438681\pi$
$919$	11.6077	0.382903	0.191451	+	0.981502i	$0.438681\pi$
$920$	0	0
$921$	14.0000	0.461316
$922$	21.2154i	0.698692i
$923$	20.2872i	0.667761i
$924$	2.00000	0.0657952
$925$	0	0
$926$	−48.4974	−1.59372
$927$	− 8.00000i	− 0.262754i
$928$	− 18.0000i	− 0.590879i
$929$	38.7846	1.27248	0.636241	−	0.771490i	$-0.280488\pi$
$929$	38.7846	1.27248	0.636241	+	0.771490i	$0.280488\pi$
$930$	0	0
$931$	4.39230	0.143952
$932$	− 12.0000i	− 0.393073i
$933$	18.9282i	0.619682i
$934$	−32.7846	−1.07275
$935$	0	0
$936$	2.53590	0.0828884
$937$	− 0.392305i	− 0.0128160i	−0.999979	−	0.00640802i	$-0.997960\pi$
$937$	− 0.392305i	− 0.0128160i	0.999979	−	0.00640802i	$-0.00203975\pi$
$938$	27.7128i	0.904855i
$939$	−7.07180	−0.230779
$940$	0	0
$941$	20.5359	0.669451	0.334726	−	0.942316i	$-0.391356\pi$
$941$	20.5359	0.669451	0.334726	+	0.942316i	$0.391356\pi$
$942$	29.3205i	0.955314i
$943$	24.0000i	0.781548i
$944$	34.6410	1.12747
$945$	0	0
$946$	−15.4641	−0.502781
$947$	5.07180i	0.164811i	0.996599	+	0.0824056i	$0.0262603\pi$
$947$	5.07180i	0.164811i	−0.996599	+	0.0824056i	$0.973740\pi$
$948$	− 13.4641i	− 0.437294i
$949$	18.1436	0.588966
$950$	0	0
$951$	−11.0718	−0.359028
$952$	0	0
$953$	− 44.7846i	− 1.45072i	−0.688372	−	0.725358i	$-0.741674\pi$
$953$	− 44.7846i	− 1.45072i	0.688372	−	0.725358i	$-0.258326\pi$
$954$	22.3923	0.724978
$955$	0	0
$956$	−12.0000	−0.388108
$957$	3.46410i	0.111979i
$958$	20.7846i	0.671520i
$959$	36.0000	1.16250
$960$	0	0
$961$	−22.4256	−0.723407
$962$	22.6410i	0.729976i
$963$	15.4641i	0.498324i
$964$	−27.8564	−0.897194
$965$	0	0
$966$	−24.0000	−0.772187
$967$	18.7846i	0.604072i	0.953296	+	0.302036i	$0.0976663\pi$
$967$	18.7846i	0.604072i	−0.953296	+	0.302036i	$0.902334\pi$
$968$	− 1.73205i	− 0.0556702i
$969$	0	0
$970$	0	0
$971$	−1.85641	−0.0595749	−0.0297875	−	0.999556i	$-0.509483\pi$
$971$	−1.85641	−0.0595749	−0.0297875	+	0.999556i	$0.509483\pi$
$972$	− 1.00000i	− 0.0320750i
$973$	− 16.7846i	− 0.538090i
$974$	−41.0718	−1.31603
$975$	0	0
$976$	−10.0000	−0.320092
$977$	35.5692i	1.13796i	0.822351	+	0.568980i	$0.192662\pi$
$977$	35.5692i	1.13796i	−0.822351	+	0.568980i	$0.807338\pi$
$978$	− 30.9282i	− 0.988975i
$979$	−12.9282	−0.413187
$980$	0	0
$981$	−10.0000	−0.319275
$982$	29.5692i	0.943592i
$983$	24.0000i	0.765481i	0.923856	+	0.382741i	$0.125020\pi$
$983$	24.0000i	0.765481i	−0.923856	+	0.382741i	$0.874980\pi$
$984$	−6.00000	−0.191273
$985$	0	0
$986$	0	0
$987$	− 13.8564i	− 0.441054i
$988$	2.14359i	0.0681968i
$989$	61.8564	1.96692
$990$	0	0
$991$	−48.7846	−1.54969	−0.774847	−	0.632149i	$-0.782173\pi$
$991$	−48.7846	−1.54969	−0.774847	+	0.632149i	$0.782173\pi$
$992$	15.2154i	0.483089i
$993$	− 17.8564i	− 0.566656i
$994$	48.0000	1.52247
$995$	0	0
$996$	15.4641	0.489999
$997$	− 48.3923i	− 1.53260i	−0.642483	−	0.766300i	$-0.722096\pi$
$997$	− 48.3923i	− 1.53260i	0.642483	−	0.766300i	$-0.277904\pi$
$998$	− 22.1436i	− 0.700943i
$999$	−8.92820	−0.282476

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	825.2.c.c.199.2		4
3.2	odd	2		2475.2.c.n.199.3		4
5.2	odd	4		165.2.a.b.1.2	✓	2
5.3	odd	4		825.2.a.e.1.1		2
5.4	even	2	inner	825.2.c.c.199.3		4
15.2	even	4		495.2.a.c.1.1		2
15.8	even	4		2475.2.a.r.1.2		2
15.14	odd	2		2475.2.c.n.199.2		4
20.7	even	4		2640.2.a.x.1.1		2
35.27	even	4		8085.2.a.bd.1.2		2
55.32	even	4		1815.2.a.i.1.1		2
55.43	even	4		9075.2.a.bh.1.2		2
60.47	odd	4		7920.2.a.bz.1.1		2
165.32	odd	4		5445.2.a.s.1.2		2

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
165.2.a.b.1.2	✓	2	5.2	odd	4
495.2.a.c.1.1		2	15.2	even	4
825.2.a.e.1.1		2	5.3	odd	4
825.2.c.c.199.2		4	1.1	even	1	trivial
825.2.c.c.199.3		4	5.4	even	2	inner
1815.2.a.i.1.1		2	55.32	even	4
2475.2.a.r.1.2		2	15.8	even	4
2475.2.c.n.199.2		4	15.14	odd	2
2475.2.c.n.199.3		4	3.2	odd	2
2640.2.a.x.1.1		2	20.7	even	4
5445.2.a.s.1.2		2	165.32	odd	4
7920.2.a.bz.1.1		2	60.47	odd	4
8085.2.a.bd.1.2		2	35.27	even	4
9075.2.a.bh.1.2		2	55.43	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:	\( N \)	\(=\)	\( 825 = 3 \cdot 5^{2} \cdot 11 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	825.c (of order \(2\), degree \(1\), not minimal)

\(f(q)\)	\(=\)	\(q-1.73205i q^{2} +1.00000i q^{3} -1.00000 q^{4} +1.73205 q^{6} -2.00000i q^{7} -1.73205i q^{8} -1.00000 q^{9} +O(q^{10})\)	\(q-1.73205i q^{2} +1.00000i q^{3} -1.00000 q^{4} +1.73205 q^{6} -2.00000i q^{7} -1.73205i q^{8} -1.00000 q^{9} -1.00000 q^{11} -1.00000i q^{12} -1.46410i q^{13} -3.46410 q^{14} -5.00000 q^{16} +1.73205i q^{18} +1.46410 q^{19} +2.00000 q^{21} +1.73205i q^{22} -6.92820i q^{23} +1.73205 q^{24} -2.53590 q^{26} -1.00000i q^{27} +2.00000i q^{28} -3.46410 q^{29} +2.92820 q^{31} +5.19615i q^{32} -1.00000i q^{33} +1.00000 q^{36} -8.92820i q^{37} -2.53590i q^{38} +1.46410 q^{39} -3.46410 q^{41} -3.46410i q^{42} +8.92820i q^{43} +1.00000 q^{44} -12.0000 q^{46} -6.92820i q^{47} -5.00000i q^{48} +3.00000 q^{49} +1.46410i q^{52} -12.9282i q^{53} -1.73205 q^{54} -3.46410 q^{56} +1.46410i q^{57} +6.00000i q^{58} -6.92820 q^{59} +2.00000 q^{61} -5.07180i q^{62} +2.00000i q^{63} -1.00000 q^{64} -1.73205 q^{66} -8.00000i q^{67} +6.92820 q^{69} -13.8564 q^{71} +1.73205i q^{72} +12.3923i q^{73} -15.4641 q^{74} -1.46410 q^{76} +2.00000i q^{77} -2.53590i q^{78} +13.4641 q^{79} +1.00000 q^{81} +6.00000i q^{82} +15.4641i q^{83} -2.00000 q^{84} +15.4641 q^{86} -3.46410i q^{87} +1.73205i q^{88} +12.9282 q^{89} -2.92820 q^{91} +6.92820i q^{92} +2.92820i q^{93} -12.0000 q^{94} -5.19615 q^{96} +10.0000i q^{97} -5.19615i q^{98} +1.00000 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 4 q - 4 q^{4} - 4 q^{9}+O(q^{10}) \) 4 * q - 4 * q^4 - 4 * q^9	\( 4 q - 4 q^{4} - 4 q^{9} - 4 q^{11} - 20 q^{16} - 8 q^{19} + 8 q^{21} - 24 q^{26} - 16 q^{31} + 4 q^{36} - 8 q^{39} + 4 q^{44} - 48 q^{46} + 12 q^{49} + 8 q^{61} - 4 q^{64} - 48 q^{74} + 8 q^{76} + 40 q^{79} + 4 q^{81} - 8 q^{84} + 48 q^{86} + 24 q^{89} + 16 q^{91} - 48 q^{94} + 4 q^{99}+O(q^{100}) \) 4 * q - 4 * q^4 - 4 * q^9 - 4 * q^11 - 20 * q^16 - 8 * q^19 + 8 * q^21 - 24 * q^26 - 16 * q^31 + 4 * q^36 - 8 * q^39 + 4 * q^44 - 48 * q^46 + 12 * q^49 + 8 * q^61 - 4 * q^64 - 48 * q^74 + 8 * q^76 + 40 * q^79 + 4 * q^81 - 8 * q^84 + 48 * q^86 + 24 * q^89 + 16 * q^91 - 48 * q^94 + 4 * q^99

Introduction

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