LMFDB - Embedded newform 2025.2.b.n.649.1

Show commands: Magma / Pari/GP / SageMath

Newspace parameters

comment:Compute space of new eigenforms

gp:[N,k,chi] = [2025,2,Mod(649,2025)] mf = mfinit([N,k,chi],0) lf = mfeigenbasis(mf)

magma://Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code chi := DirichletCharacter("2025.649"); S:= CuspForms(chi, 2); N := Newforms(S);

sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(2025, base_ring=CyclotomicField(2)) chi = DirichletCharacter(H, H._module([0, 1])) N = Newforms(chi, 2, names="a")

Level:	$ N $	$=$	$ 2025 = 3^{4} \cdot 5^{2} $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	2025.b (of order $2$, degree $1$, not minimal)

Newform invariants

comment:select newform

sage:traces = [8,0,0,-8,0,0,0,0,0,0,-2] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(11)] == traces)

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

Self dual:	no
Analytic conductor:	$16.1697064093$
Analytic rank:	$0$
Dimension:	$8$
Coefficient field:	8.0.34810603776.6
comment:defining polynomial gp:f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{8} + 12x^{6} + 42x^{4} + 49x^{2} + 9 $ x^8 + 12x^6 + 42x^4 + 49*x^2 + 9
Coefficient ring:	$\Z[a_1, \ldots, a_{11}]$
Coefficient ring index:	$ 1 $
Twist minimal:	no (minimal twist has level 225)
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label			649.1
Root			$-2.63372i$ of defining polynomial
Character	$\chi$	$=$	2025.649
Dual form			2025.2.b.n.649.8

$q$-expansion

comment:q-expansion

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

gp:mfcoefs(f, 20)

$f(q)$	$=$	$q-2.63372i q^{2} -4.93650 q^{4} +1.79743i q^{7} +7.73393i q^{8} +1.80812 q^{11} +1.97183i q^{13} +4.73393 q^{14} +10.4960 q^{16} -4.80812i q^{17} -2.96467 q^{19} -4.76210i q^{22} -1.73393i q^{23} +5.19325 q^{26} -8.87300i q^{28} +7.36765 q^{29} -2.62303 q^{31} -12.1758i q^{32} -12.6633 q^{34} -11.6351i q^{37} +7.80812i q^{38} -2.46648 q^{41} -7.27814i q^{43} -8.92580 q^{44} -4.56668 q^{46} -6.29208i q^{47} +3.76926 q^{49} -9.73393i q^{52} +1.72540i q^{53} -13.9012 q^{56} -19.4044i q^{58} -11.0260 q^{59} -12.6704 q^{61} +6.90833i q^{62} -11.0756 q^{64} -9.10374i q^{67} +23.7353i q^{68} +1.27460 q^{71} -3.58770i q^{73} -30.6436 q^{74} +14.6351 q^{76} +3.24997i q^{77} -2.11090 q^{79} +6.49602i q^{82} +1.09883i q^{83} -19.1686 q^{86} +13.9839i q^{88} +13.2935 q^{89} -3.54422 q^{91} +8.55953i q^{92} -16.5716 q^{94} +3.83276i q^{97} -9.92718i q^{98} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 8 q - 8 q^{4} - 2 q^{11} - 6 q^{14} + 8 q^{16} - 4 q^{19} - 20 q^{26} - 2 q^{29} - 8 q^{31} - 18 q^{34} - 10 q^{41} - 44 q^{44} + 6 q^{49} - 60 q^{56} - 34 q^{59} - 26 q^{61} - 38 q^{64} - 16 q^{71} - 80 q^{74}+ \cdots - 6 q^{94}+O(q^{100}) $ 8 * q - 8 * q^4 - 2 * q^11 - 6 * q^14 + 8 * q^16 - 4 * q^19 - 20 * q^26 - 2 * q^29 - 8 * q^31 - 18 * q^34 - 10 * q^41 - 44 * q^44 + 6 * q^49 - 60 * q^56 - 34 * q^59 - 26 * q^61 - 38 * q^64 - 16 * q^71 - 80 * q^74 + 22 * q^76 + 14 * q^79 - 68 * q^86 + 18 * q^89 - 34 * q^91 - 6 * q^94

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/2025\mathbb{Z}\right)^\times$.

$n$	$326$	$1702$
$\chi(n)$	$1$	$-1$

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 $

$2$	− 2.63372i	− 1.86232i	−0.364606	−	0.931162i	$-0.618796\pi$
$2$	− 2.63372i	− 1.86232i	0.364606	−	0.931162i	$-0.381204\pi$
$3$	0	0
$3$	0	0
$4$	−4.93650	−2.46825
$5$	0	0
$5$	0	0
$6$	0	0
$7$	1.79743i	0.679364i	0.940540	+	0.339682i	$0.110319\pi$
$7$	1.79743i	0.679364i	−0.940540	+	0.339682i	$0.889681\pi$
$8$	7.73393i	2.73436i
$9$	0	0
$10$	0	0
$11$	1.80812	0.545170	0.272585	−	0.962132i	$-0.412121\pi$
$11$	1.80812	0.545170	0.272585	+	0.962132i	$0.412121\pi$
$12$	0	0
$13$	1.97183i	0.546887i	0.961888	+	0.273443i	$0.0881626\pi$
$13$	1.97183i	0.546887i	−0.961888	+	0.273443i	$0.911837\pi$
$14$	4.73393	1.26520
$15$	0	0
$16$	10.4960	2.62401
$17$	− 4.80812i	− 1.16614i	−0.812421	−	0.583071i	$-0.801851\pi$
$17$	− 4.80812i	− 1.16614i	0.812421	−	0.583071i	$-0.198149\pi$
$18$	0	0
$19$	−2.96467	−0.680142	−0.340071	−	0.940400i	$-0.610451\pi$
$19$	−2.96467	−0.680142	−0.340071	+	0.940400i	$0.610451\pi$
$20$	0	0
$21$	0	0
$22$	− 4.76210i	− 1.01528i
$23$	− 1.73393i	− 0.361549i	−0.983525	−	0.180774i	$-0.942140\pi$
$23$	− 1.73393i	− 0.361549i	0.983525	−	0.180774i	$-0.0578604\pi$
$24$	0	0
$25$	0	0
$26$	5.19325	1.01848
$27$	0	0
$28$	− 8.87300i	− 1.67684i
$29$	7.36765	1.36814	0.684069	−	0.729417i	$-0.260209\pi$
$29$	7.36765	1.36814	0.684069	+	0.729417i	$0.260209\pi$
$30$	0	0
$31$	−2.62303	−0.471110	−0.235555	−	0.971861i	$-0.575691\pi$
$31$	−2.62303	−0.471110	−0.235555	+	0.971861i	$0.575691\pi$
$32$	− 12.1758i	− 2.15239i
$33$	0	0
$34$	−12.6633	−2.17173
$35$	0	0
$36$	0	0
$37$	− 11.6351i	− 1.91280i	−0.292063	−	0.956399i	$-0.594342\pi$
$37$	− 11.6351i	− 1.91280i	0.292063	−	0.956399i	$-0.405658\pi$
$38$	7.80812i	1.26664i
$39$	0	0
$40$	0	0
$41$	−2.46648	−0.385199	−0.192600	−	0.981277i	$-0.561692\pi$
$41$	−2.46648	−0.385199	−0.192600	+	0.981277i	$0.561692\pi$
$42$	0	0
$43$	− 7.27814i	− 1.10991i	−0.831882	−	0.554953i	$-0.812736\pi$
$43$	− 7.27814i	− 1.10991i	0.831882	−	0.554953i	$-0.187264\pi$
$44$	−8.92580	−1.34562
$45$	0	0
$46$	−4.56668	−0.673321
$47$	− 6.29208i	− 0.917794i	−0.888489	−	0.458897i	$-0.848245\pi$
$47$	− 6.29208i	− 0.917794i	0.888489	−	0.458897i	$-0.151755\pi$
$48$	0	0
$49$	3.76926	0.538465
$50$	0	0
$51$	0	0
$52$	− 9.73393i	− 1.34985i
$53$	1.72540i	0.237001i	0.992954	+	0.118501i	$0.0378088\pi$
$53$	1.72540i	0.237001i	−0.992954	+	0.118501i	$0.962191\pi$
$54$	0	0
$55$	0	0
$56$	−13.9012	−1.85762
$57$	0	0
$58$	− 19.4044i	− 2.54792i
$59$	−11.0260	−1.43546	−0.717732	−	0.696320i	$-0.754820\pi$
$59$	−11.0260	−1.43546	−0.717732	+	0.696320i	$0.754820\pi$
$60$	0	0
$61$	−12.6704	−1.62228	−0.811141	−	0.584851i	$-0.801153\pi$
$61$	−12.6704	−1.62228	−0.811141	+	0.584851i	$0.801153\pi$
$62$	6.90833i	0.877358i
$63$	0	0
$64$	−11.0756	−1.38445
$65$	0	0
$66$	0	0
$67$	− 9.10374i	− 1.11220i	−0.831116	−	0.556100i	$-0.812297\pi$
$67$	− 9.10374i	− 1.11220i	0.831116	−	0.556100i	$-0.187703\pi$
$68$	23.7353i	2.87833i
$69$	0	0
$70$	0	0
$71$	1.27460	0.151268	0.0756338	−	0.997136i	$-0.475902\pi$
$71$	1.27460	0.151268	0.0756338	+	0.997136i	$0.475902\pi$
$72$	0	0
$73$	− 3.58770i	− 0.419908i	−0.977711	−	0.209954i	$-0.932669\pi$
$73$	− 3.58770i	− 0.419908i	0.977711	−	0.209954i	$-0.0673315\pi$
$74$	−30.6436	−3.56225
$75$	0	0
$76$	14.6351	1.67876
$77$	3.24997i	0.370369i
$78$	0	0
$79$	−2.11090	−0.237495	−0.118747	−	0.992924i	$-0.537888\pi$
$79$	−2.11090	−0.237495	−0.118747	+	0.992924i	$0.537888\pi$
$80$	0	0
$81$	0	0
$82$	6.49602i	0.717366i
$83$	1.09883i	0.120612i	0.998180	+	0.0603061i	$0.0192077\pi$
$83$	1.09883i	0.120612i	−0.998180	+	0.0603061i	$0.980792\pi$
$84$	0	0
$85$	0	0
$86$	−19.1686	−2.06701
$87$	0	0
$88$	13.9839i	1.49069i
$89$	13.2935	1.40910	0.704552	−	0.709653i	$-0.251148\pi$
$89$	13.2935	1.40910	0.704552	+	0.709653i	$0.251148\pi$
$90$	0	0
$91$	−3.54422	−0.371535
$92$	8.55953i	0.892392i
$93$	0	0
$94$	−16.5716	−1.70923
$95$	0	0
$96$	0	0
$97$	3.83276i	0.389157i	0.980887	+	0.194579i	$0.0623340\pi$
$97$	3.83276i	0.389157i	−0.980887	+	0.194579i	$0.937666\pi$
$98$	− 9.92718i	− 1.00280i
$99$	0	0
$100$	0	0
$101$	6.55237	0.651985	0.325993	−	0.945372i	$-0.394302\pi$
$101$	6.55237	0.651985	0.325993	+	0.945372i	$0.394302\pi$
$102$	0	0
$103$	− 8.07557i	− 0.795710i	−0.917448	−	0.397855i	$-0.869755\pi$
$103$	− 8.07557i	− 0.795710i	0.917448	−	0.397855i	$-0.130245\pi$
$104$	−15.2500	−1.49538
$105$	0	0
$106$	4.54422	0.441373
$107$	8.97674i	0.867814i	0.900958	+	0.433907i	$0.142865\pi$
$107$	8.97674i	0.867814i	−0.900958	+	0.433907i	$0.857135\pi$
$108$	0	0
$109$	6.34164	0.607419	0.303710	−	0.952765i	$-0.401775\pi$
$109$	6.34164	0.607419	0.303710	+	0.952765i	$0.401775\pi$
$110$	0	0
$111$	0	0
$112$	18.8658i	1.78265i
$113$	− 14.9025i	− 1.40191i	−0.713204	−	0.700957i	$-0.752757\pi$
$113$	− 14.9025i	− 1.40191i	0.713204	−	0.700957i	$-0.247243\pi$
$114$	0	0
$115$	0	0
$116$	−36.3704	−3.37691
$117$	0	0
$118$	29.0394i	2.67330i
$119$	8.64225	0.792234
$120$	0	0
$121$	−7.73069	−0.702790
$122$	33.3704i	3.02121i
$123$	0	0
$124$	12.9486	1.16282
$125$	0	0
$126$	0	0
$127$	− 3.62303i	− 0.321492i	−0.986996	−	0.160746i	$-0.948610\pi$
$127$	− 3.62303i	− 0.321492i	0.986996	−	0.160746i	$-0.0513899\pi$
$128$	4.81844i	0.425894i
$129$	0	0
$130$	0	0
$131$	7.29345	0.637232	0.318616	−	0.947884i	$-0.396782\pi$
$131$	7.29345	0.637232	0.318616	+	0.947884i	$0.396782\pi$
$132$	0	0
$133$	− 5.32878i	− 0.462064i
$134$	−23.9767	−2.07127
$135$	0	0
$136$	37.1857	3.18865
$137$	7.12621i	0.608833i	0.952539	+	0.304417i	$0.0984615\pi$
$137$	7.12621i	0.608833i	−0.952539	+	0.304417i	$0.901539\pi$
$138$	0	0
$139$	14.7107	1.24774	0.623871	−	0.781527i	$-0.285559\pi$
$139$	14.7107	1.24774	0.623871	+	0.781527i	$0.285559\pi$
$140$	0	0
$141$	0	0
$142$	− 3.35695i	− 0.281709i
$143$	3.56531i	0.298146i
$144$	0	0
$145$	0	0
$146$	−9.44900	−0.782005
$147$	0	0
$148$	57.4366i	4.72126i
$149$	0.565309	0.0463119	0.0231560	−	0.999732i	$-0.492629\pi$
$149$	0.565309	0.0463119	0.0231560	+	0.999732i	$0.492629\pi$
$150$	0	0
$151$	0.153385	0.0124823	0.00624115	−	0.999981i	$-0.498013\pi$
$151$	0.153385	0.0124823	0.00624115	+	0.999981i	$0.498013\pi$
$152$	− 22.9285i	− 1.85975i
$153$	0	0
$154$	8.55953	0.689746
$155$	0	0
$156$	0	0
$157$	11.4607i	0.914663i	0.889296	+	0.457332i	$0.151195\pi$
$157$	11.4607i	0.914663i	−0.889296	+	0.457332i	$0.848805\pi$
$158$	5.55953i	0.442292i
$159$	0	0
$160$	0	0
$161$	3.11661	0.245623
$162$	0	0
$163$	− 22.0595i	− 1.72783i	−0.503637	−	0.863915i	$-0.668005\pi$
$163$	− 22.0595i	− 1.72783i	0.503637	−	0.863915i	$-0.331995\pi$
$164$	12.1758	0.950768
$165$	0	0
$166$	2.89401	0.224619
$167$	− 17.0684i	− 1.32079i	−0.750917	−	0.660397i	$-0.770388\pi$
$167$	− 17.0684i	− 1.32079i	0.750917	−	0.660397i	$-0.229612\pi$
$168$	0	0
$169$	9.11189	0.700915
$170$	0	0
$171$	0	0
$172$	35.9285i	2.73953i
$173$	11.9447i	0.908135i	0.890967	+	0.454067i	$0.150028\pi$
$173$	11.9447i	0.908135i	−0.890967	+	0.454067i	$0.849972\pi$
$174$	0	0
$175$	0	0
$176$	18.9781	1.43053
$177$	0	0
$178$	− 35.0113i	− 2.62421i
$179$	−8.54921	−0.638998	−0.319499	−	0.947587i	$-0.603515\pi$
$179$	−8.54921	−0.638998	−0.319499	+	0.947587i	$0.603515\pi$
$180$	0	0
$181$	−10.5524	−0.784351	−0.392176	−	0.919890i	$-0.628277\pi$
$181$	−10.5524	−0.784351	−0.392176	+	0.919890i	$0.628277\pi$
$182$	9.33449i	0.691918i
$183$	0	0
$184$	13.4101	0.988603
$185$	0	0
$186$	0	0
$187$	− 8.69368i	− 0.635745i
$188$	31.0608i	2.26534i
$189$	0	0
$190$	0	0
$191$	−17.3372	−1.25448	−0.627239	−	0.778827i	$-0.715815\pi$
$191$	−17.3372	−1.25448	−0.627239	+	0.778827i	$0.715815\pi$
$192$	0	0
$193$	1.55953i	0.112257i	0.998424	+	0.0561286i	$0.0178757\pi$
$193$	1.55953i	0.112257i	−0.998424	+	0.0561286i	$0.982124\pi$
$194$	10.0944	0.724737
$195$	0	0
$196$	−18.6069	−1.32907
$197$	− 17.9767i	− 1.28079i	−0.768046	−	0.640395i	$-0.778771\pi$
$197$	− 17.9767i	− 1.28079i	0.768046	−	0.640395i	$-0.221229\pi$
$198$	0	0
$199$	−11.0225	−0.781362	−0.390681	−	0.920526i	$-0.627760\pi$
$199$	−11.0225	−0.781362	−0.390681	+	0.920526i	$0.627760\pi$
$200$	0	0
$201$	0	0
$202$	− 17.2571i	− 1.21421i
$203$	13.2428i	0.929463i
$204$	0	0
$205$	0	0
$206$	−21.2688	−1.48187
$207$	0	0
$208$	20.6964i	1.43503i
$209$	−5.36049	−0.370793
$210$	0	0
$211$	−23.9285	−1.64731	−0.823655	−	0.567092i	$-0.808068\pi$
$211$	−23.9285	−1.64731	−0.823655	+	0.567092i	$0.808068\pi$
$212$	− 8.51742i	− 0.584979i
$213$	0	0
$214$	23.6423	1.61615
$215$	0	0
$216$	0	0
$217$	− 4.71470i	− 0.320055i
$218$	− 16.7021i	− 1.13121i
$219$	0	0
$220$	0	0
$221$	9.48079	0.637747
$222$	0	0
$223$	21.7107i	1.45385i	0.686715	+	0.726927i	$0.259052\pi$
$223$	21.7107i	1.45385i	−0.686715	+	0.726927i	$0.740948\pi$
$224$	21.8851	1.46226
$225$	0	0
$226$	−39.2492	−2.61082
$227$	− 14.1002i	− 0.935863i	−0.883765	−	0.467932i	$-0.844999\pi$
$227$	− 14.1002i	− 0.935863i	0.883765	−	0.467932i	$-0.155001\pi$
$228$	0	0
$229$	−3.67758	−0.243021	−0.121511	−	0.992590i	$-0.538774\pi$
$229$	−3.67758	−0.243021	−0.121511	+	0.992590i	$0.538774\pi$
$230$	0	0
$231$	0	0
$232$	56.9809i	3.74098i
$233$	− 5.34164i	− 0.349943i	−0.984574	−	0.174971i	$-0.944017\pi$
$233$	− 5.34164i	− 0.349943i	0.984574	−	0.174971i	$-0.0559833\pi$
$234$	0	0
$235$	0	0
$236$	54.4299	3.54308
$237$	0	0
$238$	− 22.7613i	− 1.47540i
$239$	22.0335	1.42523	0.712613	−	0.701557i	$-0.247512\pi$
$239$	22.0335	1.42523	0.712613	+	0.701557i	$0.247512\pi$
$240$	0	0
$241$	−18.6472	−1.20117	−0.600585	−	0.799561i	$-0.705066\pi$
$241$	−18.6472	−1.20117	−0.600585	+	0.799561i	$0.705066\pi$
$242$	20.3605i	1.30882i
$243$	0	0
$244$	62.5475	4.00420
$245$	0	0
$246$	0	0
$247$	− 5.84582i	− 0.371961i
$248$	− 20.2863i	− 1.28818i
$249$	0	0
$250$	0	0
$251$	14.6929	0.927407	0.463704	−	0.885990i	$-0.346520\pi$
$251$	14.6929	0.927407	0.463704	+	0.885990i	$0.346520\pi$
$252$	0	0
$253$	− 3.13515i	− 0.197105i
$254$	−9.54205	−0.598721
$255$	0	0
$256$	−9.46070	−0.591294
$257$	− 22.2089i	− 1.38536i	−0.721247	−	0.692678i	$-0.756431\pi$
$257$	− 22.2089i	− 1.38536i	0.721247	−	0.692678i	$-0.243569\pi$
$258$	0	0
$259$	20.9132	1.29949
$260$	0	0
$261$	0	0
$262$	− 19.2089i	− 1.18673i
$263$	5.74001i	0.353944i	0.984216	+	0.176972i	$0.0566302\pi$
$263$	5.74001i	0.353944i	−0.984216	+	0.176972i	$0.943370\pi$
$264$	0	0
$265$	0	0
$266$	−14.0345	−0.860513
$267$	0	0
$268$	44.9406i	2.74519i
$269$	15.6162	0.952139	0.476070	−	0.879408i	$-0.342061\pi$
$269$	15.6162	0.952139	0.476070	+	0.879408i	$0.342061\pi$
$270$	0	0
$271$	−6.75315	−0.410225	−0.205112	−	0.978738i	$-0.565756\pi$
$271$	−6.75315	−0.410225	−0.205112	+	0.978738i	$0.565756\pi$
$272$	− 50.4662i	− 3.05996i
$273$	0	0
$274$	18.7685	1.13384
$275$	0	0
$276$	0	0
$277$	− 30.2966i	− 1.82034i	−0.414230	−	0.910172i	$-0.635949\pi$
$277$	− 30.2966i	− 1.82034i	0.414230	−	0.910172i	$-0.364051\pi$
$278$	− 38.7438i	− 2.32370i
$279$	0	0
$280$	0	0
$281$	18.6351	1.11168	0.555838	−	0.831290i	$-0.312397\pi$
$281$	18.6351	1.11168	0.555838	+	0.831290i	$0.312397\pi$
$282$	0	0
$283$	− 5.67366i	− 0.337264i	−0.985679	−	0.168632i	$-0.946065\pi$
$283$	− 5.67366i	− 0.337264i	0.985679	−	0.168632i	$-0.0539350\pi$
$284$	−6.29208	−0.373366
$285$	0	0
$286$	9.39004	0.555245
$287$	− 4.43332i	− 0.261690i
$288$	0	0
$289$	−6.11806	−0.359886
$290$	0	0
$291$	0	0
$292$	17.7107i	1.03644i
$293$	18.2773i	1.06777i	0.845556	+	0.533887i	$0.179269\pi$
$293$	18.2773i	1.06777i	−0.845556	+	0.533887i	$0.820731\pi$
$294$	0	0
$295$	0	0
$296$	89.9850	5.23027
$297$	0	0
$298$	− 1.48887i	− 0.0862478i
$299$	3.41900	0.197726
$300$	0	0
$301$	13.0819	0.754030
$302$	− 0.403974i	− 0.0232461i
$303$	0	0
$304$	−31.1173	−1.78470
$305$	0	0
$306$	0	0
$307$	15.5050i	0.884915i	0.896789	+	0.442458i	$0.145893\pi$
$307$	15.5050i	0.884915i	−0.896789	+	0.442458i	$0.854107\pi$
$308$	− 16.0435i	− 0.914162i
$309$	0	0
$310$	0	0
$311$	30.4464	1.72646	0.863228	−	0.504814i	$-0.168439\pi$
$311$	30.4464	1.72646	0.863228	+	0.504814i	$0.168439\pi$
$312$	0	0
$313$	− 6.94936i	− 0.392801i	−0.980524	−	0.196401i	$-0.937075\pi$
$313$	− 6.94936i	− 0.392801i	0.980524	−	0.196401i	$-0.0629253\pi$
$314$	30.1843	1.70340
$315$	0	0
$316$	10.4205	0.586196
$317$	16.1451i	0.906797i	0.891308	+	0.453398i	$0.149788\pi$
$317$	16.1451i	0.906797i	−0.891308	+	0.453398i	$0.850212\pi$
$318$	0	0
$319$	13.3216	0.745868
$320$	0	0
$321$	0	0
$322$	− 8.20828i	− 0.457430i
$323$	14.2545i	0.793142i
$324$	0	0
$325$	0	0
$326$	−58.0985	−3.21778
$327$	0	0
$328$	− 19.0756i	− 1.05327i
$329$	11.3096	0.623516
$330$	0	0
$331$	12.6222	0.693781	0.346890	−	0.937906i	$-0.387238\pi$
$331$	12.6222	0.693781	0.346890	+	0.937906i	$0.387238\pi$
$332$	− 5.42437i	− 0.297701i
$333$	0	0
$334$	−44.9535	−2.45975
$335$	0	0
$336$	0	0
$337$	6.92040i	0.376978i	0.982075	+	0.188489i	$0.0603590\pi$
$337$	6.92040i	0.376978i	−0.982075	+	0.188489i	$0.939641\pi$
$338$	− 23.9982i	− 1.30533i
$339$	0	0
$340$	0	0
$341$	−4.74276	−0.256835
$342$	0	0
$343$	19.3570i	1.04518i
$344$	56.2886	3.03488
$345$	0	0
$346$	31.4589	1.69124
$347$	− 13.8063i	− 0.741163i	−0.928800	−	0.370581i	$-0.879158\pi$
$347$	− 13.8063i	− 0.741163i	0.928800	−	0.370581i	$-0.120842\pi$
$348$	0	0
$349$	−6.56768	−0.351560	−0.175780	−	0.984429i	$-0.556245\pi$
$349$	−6.56768	−0.351560	−0.175780	+	0.984429i	$0.556245\pi$
$350$	0	0
$351$	0	0
$352$	− 22.0153i	− 1.17342i
$353$	− 3.52499i	− 0.187616i	−0.995590	−	0.0938082i	$-0.970096\pi$
$353$	− 3.52499i	− 0.187616i	0.995590	−	0.0938082i	$-0.0299040\pi$
$354$	0	0
$355$	0	0
$356$	−65.6231	−3.47802
$357$	0	0
$358$	22.5162i	1.19002i
$359$	−22.9285	−1.21012	−0.605061	−	0.796179i	$-0.706851\pi$
$359$	−22.9285	−1.21012	−0.605061	+	0.796179i	$0.706851\pi$
$360$	0	0
$361$	−10.2107	−0.537407
$362$	27.7920i	1.46072i
$363$	0	0
$364$	17.4960	0.917041
$365$	0	0
$366$	0	0
$367$	− 4.17931i	− 0.218158i	−0.994033	−	0.109079i	$-0.965210\pi$
$367$	− 4.17931i	− 0.218158i	0.994033	−	0.109079i	$-0.0347902\pi$
$368$	− 18.1993i	− 0.948706i
$369$	0	0
$370$	0	0
$371$	−3.10127	−0.161010
$372$	0	0
$373$	6.84091i	0.354209i	0.984192	+	0.177104i	$0.0566730\pi$
$373$	6.84091i	0.354209i	−0.984192	+	0.177104i	$0.943327\pi$
$374$	−22.8968	−1.18396
$375$	0	0
$376$	48.6625	2.50958
$377$	14.5277i	0.748217i
$378$	0	0
$379$	12.7764	0.656280	0.328140	−	0.944629i	$-0.393578\pi$
$379$	12.7764	0.656280	0.328140	+	0.944629i	$0.393578\pi$
$380$	0	0
$381$	0	0
$382$	45.6615i	2.33624i
$383$	7.53459i	0.385000i	0.981297	+	0.192500i	$0.0616595\pi$
$383$	7.53459i	0.385000i	−0.981297	+	0.192500i	$0.938341\pi$
$384$	0	0
$385$	0	0
$386$	4.10736	0.209059
$387$	0	0
$388$	− 18.9204i	− 0.960538i
$389$	−5.45175	−0.276415	−0.138207	−	0.990403i	$-0.544134\pi$
$389$	−5.45175	−0.276415	−0.138207	+	0.990403i	$0.544134\pi$
$390$	0	0
$391$	−8.33693	−0.421617
$392$	29.1511i	1.47235i
$393$	0	0
$394$	−47.3458	−2.38525
$395$	0	0
$396$	0	0
$397$	5.64549i	0.283339i	0.989914	+	0.141670i	$0.0452470\pi$
$397$	5.64549i	0.283339i	−0.989914	+	0.141670i	$0.954753\pi$
$398$	29.0301i	1.45515i
$399$	0	0
$400$	0	0
$401$	−5.50418	−0.274865	−0.137433	−	0.990511i	$-0.543885\pi$
$401$	−5.50418	−0.274865	−0.137433	+	0.990511i	$0.543885\pi$
$402$	0	0
$403$	− 5.17216i	− 0.257643i
$404$	−32.3458	−1.60926
$405$	0	0
$406$	34.8779	1.73096
$407$	− 21.0377i	− 1.04280i
$408$	0	0
$409$	−32.8530	−1.62448	−0.812238	−	0.583327i	$-0.801751\pi$
$409$	−32.8530	−1.62448	−0.812238	+	0.583327i	$0.801751\pi$
$410$	0	0
$411$	0	0
$412$	39.8650i	1.96401i
$413$	− 19.8184i	− 0.975202i
$414$	0	0
$415$	0	0
$416$	24.0085	1.17712
$417$	0	0
$418$	14.1181i	0.690537i
$419$	−22.8591	−1.11674	−0.558369	−	0.829593i	$-0.688573\pi$
$419$	−22.8591	−1.11674	−0.558369	+	0.829593i	$0.688573\pi$
$420$	0	0
$421$	17.9414	0.874411	0.437205	−	0.899362i	$-0.355968\pi$
$421$	17.9414	0.874411	0.437205	+	0.899362i	$0.355968\pi$
$422$	63.0212i	3.06782i
$423$	0	0
$424$	−13.3441	−0.648046
$425$	0	0
$426$	0	0
$427$	− 22.7742i	− 1.10212i
$428$	− 44.3137i	− 2.14198i
$429$	0	0
$430$	0	0
$431$	−6.18871	−0.298100	−0.149050	−	0.988830i	$-0.547622\pi$
$431$	−6.18871	−0.298100	−0.149050	+	0.988830i	$0.547622\pi$
$432$	0	0
$433$	− 3.11806i	− 0.149844i	−0.997189	−	0.0749221i	$-0.976129\pi$
$433$	− 3.11806i	− 0.149844i	0.997189	−	0.0749221i	$-0.0238708\pi$
$434$	−12.4172	−0.596045
$435$	0	0
$436$	−31.3055	−1.49926
$437$	5.14052i	0.245904i
$438$	0	0
$439$	−13.5099	−0.644792	−0.322396	−	0.946605i	$-0.604488\pi$
$439$	−13.5099	−0.644792	−0.322396	+	0.946605i	$0.604488\pi$
$440$	0	0
$441$	0	0
$442$	− 24.9698i	− 1.18769i
$443$	− 24.2773i	− 1.15345i	−0.816938	−	0.576726i	$-0.804330\pi$
$443$	− 24.2773i	− 1.15345i	0.816938	−	0.576726i	$-0.195670\pi$
$444$	0	0
$445$	0	0
$446$	57.1799	2.70755
$447$	0	0
$448$	− 19.9075i	− 0.940543i
$449$	24.1437	1.13941	0.569705	−	0.821849i	$-0.307057\pi$
$449$	24.1437	1.13941	0.569705	+	0.821849i	$0.307057\pi$
$450$	0	0
$451$	−4.45970	−0.209999
$452$	73.5664i	3.46027i
$453$	0	0
$454$	−37.1360	−1.74288
$455$	0	0
$456$	0	0
$457$	2.82157i	0.131987i	0.997820	+	0.0659937i	$0.0210217\pi$
$457$	2.82157i	0.131987i	−0.997820	+	0.0659937i	$0.978978\pi$
$458$	9.68573i	0.452585i
$459$	0	0
$460$	0	0
$461$	21.4572	0.999363	0.499681	−	0.866209i	$-0.333450\pi$
$461$	21.4572	0.999363	0.499681	+	0.866209i	$0.333450\pi$
$462$	0	0
$463$	− 19.8033i	− 0.920339i	−0.887831	−	0.460170i	$-0.847789\pi$
$463$	− 19.8033i	− 0.920339i	0.887831	−	0.460170i	$-0.152211\pi$
$464$	77.3310	3.59000
$465$	0	0
$466$	−14.0684	−0.651707
$467$	− 22.7210i	− 1.05140i	−0.850669	−	0.525701i	$-0.823803\pi$
$467$	− 22.7210i	− 1.05140i	0.850669	−	0.525701i	$-0.176197\pi$
$468$	0	0
$469$	16.3633	0.755588
$470$	0	0
$471$	0	0
$472$	− 85.2743i	− 3.92507i
$473$	− 13.1598i	− 0.605088i
$474$	0	0
$475$	0	0
$476$	−42.6625	−1.95543
$477$	0	0
$478$	− 58.0300i	− 2.65423i
$479$	−21.2880	−0.972672	−0.486336	−	0.873772i	$-0.661667\pi$
$479$	−21.2880	−0.972672	−0.486336	+	0.873772i	$0.661667\pi$
$480$	0	0
$481$	22.9424	1.04608
$482$	49.1115i	2.23697i
$483$	0	0
$484$	38.1625	1.73466
$485$	0	0
$486$	0	0
$487$	9.58690i	0.434424i	0.976124	+	0.217212i	$0.0696963\pi$
$487$	9.58690i	0.434424i	−0.976124	+	0.217212i	$0.930304\pi$
$488$	− 97.9921i	− 4.43590i
$489$	0	0
$490$	0	0
$491$	−37.8443	−1.70789	−0.853945	−	0.520363i	$-0.825797\pi$
$491$	−37.8443	−1.70789	−0.853945	+	0.520363i	$0.825797\pi$
$492$	0	0
$493$	− 35.4246i	− 1.59544i
$494$	−15.3963	−0.692711
$495$	0	0
$496$	−27.5314	−1.23619
$497$	2.29101i	0.102766i
$498$	0	0
$499$	−16.9253	−0.757681	−0.378840	−	0.925462i	$-0.623677\pi$
$499$	−16.9253	−0.757681	−0.378840	+	0.925462i	$0.623677\pi$
$500$	0	0
$501$	0	0
$502$	− 38.6970i	− 1.72713i
$503$	40.4168i	1.80210i	0.433719	+	0.901048i	$0.357201\pi$
$503$	40.4168i	1.80210i	−0.433719	+	0.901048i	$0.642799\pi$
$504$	0	0
$505$	0	0
$506$	−8.25713	−0.367074
$507$	0	0
$508$	17.8851i	0.793522i
$509$	41.4067	1.83532	0.917660	−	0.397366i	$-0.130076\pi$
$509$	41.4067	1.83532	0.917660	+	0.397366i	$0.130076\pi$
$510$	0	0
$511$	6.44863	0.285270
$512$	34.5537i	1.52707i
$513$	0	0
$514$	−58.4922	−2.57998
$515$	0	0
$516$	0	0
$517$	− 11.3769i	− 0.500354i
$518$	− 55.0797i	− 2.42006i
$519$	0	0
$520$	0	0
$521$	−17.0301	−0.746103	−0.373052	−	0.927811i	$-0.621689\pi$
$521$	−17.0301	−0.746103	−0.373052	+	0.927811i	$0.621689\pi$
$522$	0	0
$523$	9.57651i	0.418751i	0.977835	+	0.209376i	$0.0671432\pi$
$523$	9.57651i	0.418751i	−0.977835	+	0.209376i	$0.932857\pi$
$524$	−36.0041	−1.57285
$525$	0	0
$526$	15.1176	0.659159
$527$	12.6118i	0.549380i
$528$	0	0
$529$	19.9935	0.869283
$530$	0	0
$531$	0	0
$532$	26.3055i	1.14049i
$533$	− 4.86347i	− 0.210660i
$534$	0	0
$535$	0	0
$536$	70.4077	3.04115
$537$	0	0
$538$	− 41.1289i	− 1.77319i
$539$	6.81528	0.293555
$540$	0	0
$541$	−0.833751	−0.0358458	−0.0179229	−	0.999839i	$-0.505705\pi$
$541$	−0.833751	−0.0358458	−0.0179229	+	0.999839i	$0.505705\pi$
$542$	17.7859i	0.763971i
$543$	0	0
$544$	−58.5426	−2.50999
$545$	0	0
$546$	0	0
$547$	28.3270i	1.21117i	0.795779	+	0.605587i	$0.207062\pi$
$547$	28.3270i	1.21117i	−0.795779	+	0.605587i	$0.792938\pi$
$548$	− 35.1785i	− 1.50275i
$549$	0	0
$550$	0	0
$551$	−21.8427	−0.930529
$552$	0	0
$553$	− 3.79419i	− 0.161345i
$554$	−79.7928	−3.39007
$555$	0	0
$556$	−72.6192	−3.07974
$557$	− 11.5042i	− 0.487448i	−0.969845	−	0.243724i	$-0.921631\pi$
$557$	− 11.5042i	− 0.487448i	0.969845	−	0.243724i	$-0.0783690\pi$
$558$	0	0
$559$	14.3512	0.606993
$560$	0	0
$561$	0	0
$562$	− 49.0797i	− 2.07030i
$563$	33.0059i	1.39103i	0.718510	+	0.695517i	$0.244824\pi$
$563$	33.0059i	1.39103i	−0.718510	+	0.695517i	$0.755176\pi$
$564$	0	0
$565$	0	0
$566$	−14.9429	−0.628095
$567$	0	0
$568$	9.85769i	0.413619i
$569$	27.0088	1.13227	0.566135	−	0.824313i	$-0.308438\pi$
$569$	27.0088	1.13227	0.566135	+	0.824313i	$0.308438\pi$
$570$	0	0
$571$	−24.4244	−1.02213	−0.511064	−	0.859543i	$-0.670749\pi$
$571$	−24.4244	−1.02213	−0.511064	+	0.859543i	$0.670749\pi$
$572$	− 17.6001i	− 0.735899i
$573$	0	0
$574$	−11.6761	−0.487352
$575$	0	0
$576$	0	0
$577$	− 14.7976i	− 0.616033i	−0.951381	−	0.308017i	$-0.900335\pi$
$577$	− 14.7976i	− 0.616033i	0.951381	−	0.308017i	$-0.0996653\pi$
$578$	16.1133i	0.670224i
$579$	0	0
$580$	0	0
$581$	−1.97507	−0.0819396
$582$	0	0
$583$	3.11973i	0.129206i
$584$	27.7470	1.14818
$585$	0	0
$586$	48.1375	1.98854
$587$	− 30.5780i	− 1.26209i	−0.775747	−	0.631044i	$-0.782627\pi$
$587$	− 30.5780i	− 1.26209i	0.775747	−	0.631044i	$-0.217373\pi$
$588$	0	0
$589$	7.77641	0.320421
$590$	0	0
$591$	0	0
$592$	− 122.122i	− 5.01919i
$593$	− 5.09990i	− 0.209428i	−0.994502	−	0.104714i	$-0.966607\pi$
$593$	− 5.09990i	− 0.209428i	0.994502	−	0.104714i	$-0.0333927\pi$
$594$	0	0
$595$	0	0
$596$	−2.79065	−0.114309
$597$	0	0
$598$	− 9.00471i	− 0.368230i
$599$	0.565309	0.0230979	0.0115490	−	0.999933i	$-0.496324\pi$
$599$	0.565309	0.0230979	0.0115490	+	0.999933i	$0.496324\pi$
$600$	0	0
$601$	−11.0096	−0.449091	−0.224546	−	0.974464i	$-0.572090\pi$
$601$	−11.0096	−0.449091	−0.224546	+	0.974464i	$0.572090\pi$
$602$	− 34.4542i	− 1.40425i
$603$	0	0
$604$	−0.757185	−0.0308094
$605$	0	0
$606$	0	0
$607$	19.0983i	0.775174i	0.921833	+	0.387587i	$0.126691\pi$
$607$	19.0983i	0.775174i	−0.921833	+	0.387587i	$0.873309\pi$
$608$	36.0972i	1.46393i
$609$	0	0
$610$	0	0
$611$	12.4069	0.501929
$612$	0	0
$613$	9.33918i	0.377206i	0.982053	+	0.188603i	$0.0603959\pi$
$613$	9.33918i	0.377206i	−0.982053	+	0.188603i	$0.939604\pi$
$614$	40.8358	1.64800
$615$	0	0
$616$	−25.1350	−1.01272
$617$	24.4154i	0.982928i	0.870898	+	0.491464i	$0.163538\pi$
$617$	24.4154i	0.982928i	−0.870898	+	0.491464i	$0.836462\pi$
$618$	0	0
$619$	−39.4863	−1.58709	−0.793544	−	0.608513i	$-0.791766\pi$
$619$	−39.4863	−1.58709	−0.793544	+	0.608513i	$0.791766\pi$
$620$	0	0
$621$	0	0
$622$	− 80.1874i	− 3.21522i
$623$	23.8940i	0.957293i
$624$	0	0
$625$	0	0
$626$	−18.3027	−0.731523
$627$	0	0
$628$	− 56.5757i	− 2.25762i
$629$	−55.9430	−2.23059
$630$	0	0
$631$	42.1634	1.67850	0.839249	−	0.543747i	$-0.182995\pi$
$631$	42.1634	1.67850	0.839249	+	0.543747i	$0.182995\pi$
$632$	− 16.3255i	− 0.649395i
$633$	0	0
$634$	42.5216	1.68875
$635$	0	0
$636$	0	0
$637$	7.43232i	0.294479i
$638$	− 35.0855i	− 1.38905i
$639$	0	0
$640$	0	0
$641$	35.3155	1.39488	0.697438	−	0.716645i	$-0.254323\pi$
$641$	35.3155	1.39488	0.697438	+	0.716645i	$0.254323\pi$
$642$	0	0
$643$	14.1954i	0.559813i	0.960027	+	0.279906i	$0.0903034\pi$
$643$	14.1954i	0.559813i	−0.960027	+	0.279906i	$0.909697\pi$
$644$	−15.3851	−0.606259
$645$	0	0
$646$	37.5424	1.47709
$647$	− 17.4897i	− 0.687593i	−0.939044	−	0.343796i	$-0.888287\pi$
$647$	− 17.4897i	− 0.687593i	0.939044	−	0.343796i	$-0.111713\pi$
$648$	0	0
$649$	−19.9364	−0.782572
$650$	0	0
$651$	0	0
$652$	108.897i	4.26472i
$653$	− 10.9772i	− 0.429569i	−0.976661	−	0.214785i	$-0.931095\pi$
$653$	− 10.9772i	− 0.429569i	0.976661	−	0.214785i	$-0.0689050\pi$
$654$	0	0
$655$	0	0
$656$	−25.8882	−1.01077
$657$	0	0
$658$	− 29.7862i	− 1.16119i
$659$	15.7876	0.614998	0.307499	−	0.951548i	$-0.400508\pi$
$659$	15.7876	0.614998	0.307499	+	0.951548i	$0.400508\pi$
$660$	0	0
$661$	49.8932	1.94062	0.970311	−	0.241862i	$-0.0777581\pi$
$661$	49.8932	1.94062	0.970311	+	0.241862i	$0.0777581\pi$
$662$	− 33.2435i	− 1.29204i
$663$	0	0
$664$	−8.49827	−0.329797
$665$	0	0
$666$	0	0
$667$	− 12.7750i	− 0.494649i
$668$	84.2582i	3.26005i
$669$	0	0
$670$	0	0
$671$	−22.9097	−0.884419
$672$	0	0
$673$	28.8395i	1.11168i	0.831289	+	0.555840i	$0.187603\pi$
$673$	28.8395i	1.11168i	−0.831289	+	0.555840i	$0.812397\pi$
$674$	18.2264	0.702055
$675$	0	0
$676$	−44.9809	−1.73003
$677$	− 10.4636i	− 0.402150i	−0.979576	−	0.201075i	$-0.935557\pi$
$677$	− 10.4636i	− 0.402150i	0.979576	−	0.201075i	$-0.0644434\pi$
$678$	0	0
$679$	−6.88910	−0.264379
$680$	0	0
$681$	0	0
$682$	12.4911i	0.478309i
$683$	− 16.1875i	− 0.619396i	−0.950835	−	0.309698i	$-0.899772\pi$
$683$	− 16.1875i	− 0.619396i	0.950835	−	0.309698i	$-0.100228\pi$
$684$	0	0
$685$	0	0
$686$	50.9809	1.94646
$687$	0	0
$688$	− 76.3916i	− 2.91240i
$689$	−3.40218	−0.129613
$690$	0	0
$691$	9.88362	0.375991	0.187995	−	0.982170i	$-0.439801\pi$
$691$	9.88362	0.375991	0.187995	+	0.982170i	$0.439801\pi$
$692$	− 58.9648i	− 2.24150i
$693$	0	0
$694$	−36.3621	−1.38029
$695$	0	0
$696$	0	0
$697$	11.8591i	0.449197i
$698$	17.2974i	0.654718i
$699$	0	0
$700$	0	0
$701$	43.9692	1.66069	0.830346	−	0.557248i	$-0.188143\pi$
$701$	43.9692	1.66069	0.830346	+	0.557248i	$0.188143\pi$
$702$	0	0
$703$	34.4942i	1.30097i
$704$	−20.0260	−0.754758
$705$	0	0
$706$	−9.28385	−0.349402
$707$	11.7774i	0.442935i
$708$	0	0
$709$	−25.2260	−0.947384	−0.473692	−	0.880691i	$-0.657079\pi$
$709$	−25.2260	−0.947384	−0.473692	+	0.880691i	$0.657079\pi$
$710$	0	0
$711$	0	0
$712$	102.811i	3.85299i
$713$	4.54814i	0.170329i
$714$	0	0
$715$	0	0
$716$	42.2032	1.57721
$717$	0	0
$718$	60.3875i	2.25364i
$719$	36.8600	1.37465	0.687323	−	0.726352i	$-0.258786\pi$
$719$	36.8600	1.37465	0.687323	+	0.726352i	$0.258786\pi$
$720$	0	0
$721$	14.5153	0.540576
$722$	26.8922i	1.00082i
$723$	0	0
$724$	52.0918	1.93598
$725$	0	0
$726$	0	0
$727$	38.2451i	1.41843i	0.704990	+	0.709217i	$0.250951\pi$
$727$	38.2451i	1.41843i	−0.704990	+	0.709217i	$0.749049\pi$
$728$	− 27.4107i	− 1.01591i
$729$	0	0
$730$	0	0
$731$	−34.9942	−1.29431
$732$	0	0
$733$	39.5832i	1.46204i	0.682356	+	0.731020i	$0.260955\pi$
$733$	39.5832i	1.46204i	−0.682356	+	0.731020i	$0.739045\pi$
$734$	−11.0072	−0.406282
$735$	0	0
$736$	−21.1119	−0.778195
$737$	− 16.4607i	− 0.606338i
$738$	0	0
$739$	8.24773	0.303398	0.151699	−	0.988427i	$-0.451526\pi$
$739$	8.24773	0.303398	0.151699	+	0.988427i	$0.451526\pi$
$740$	0	0
$741$	0	0
$742$	8.16790i	0.299853i
$743$	1.30818i	0.0479925i	0.999712	+	0.0239963i	$0.00763898\pi$
$743$	1.30818i	0.0479925i	−0.999712	+	0.0239963i	$0.992361\pi$
$744$	0	0
$745$	0	0
$746$	18.0171	0.659651
$747$	0	0
$748$	42.9164i	1.56918i
$749$	−16.1350	−0.589562
$750$	0	0
$751$	28.4468	1.03804	0.519020	−	0.854762i	$-0.326297\pi$
$751$	28.4468	1.03804	0.519020	+	0.854762i	$0.326297\pi$
$752$	− 66.0418i	− 2.40830i
$753$	0	0
$754$	38.2620	1.39342
$755$	0	0
$756$	0	0
$757$	38.2012i	1.38845i	0.719760	+	0.694223i	$0.244252\pi$
$757$	38.2012i	1.38845i	−0.719760	+	0.694223i	$0.755748\pi$
$758$	− 33.6495i	− 1.22221i
$759$	0	0
$760$	0	0
$761$	22.1904	0.804401	0.402200	−	0.915552i	$-0.368245\pi$
$761$	22.1904	0.804401	0.402200	+	0.915552i	$0.368245\pi$
$762$	0	0
$763$	11.3986i	0.412659i
$764$	85.5852	3.09637
$765$	0	0
$766$	19.8440	0.716994
$767$	− 21.7414i	− 0.785036i
$768$	0	0
$769$	16.9130	0.609900	0.304950	−	0.952368i	$-0.401360\pi$
$769$	16.9130	0.609900	0.304950	+	0.952368i	$0.401360\pi$
$770$	0	0
$771$	0	0
$772$	− 7.69860i	− 0.277079i
$773$	− 38.6464i	− 1.39001i	−0.719003	−	0.695007i	$-0.755401\pi$
$773$	− 38.6464i	− 1.39001i	0.719003	−	0.695007i	$-0.244599\pi$
$774$	0	0
$775$	0	0
$776$	−29.6423	−1.06409
$777$	0	0
$778$	14.3584i	0.514774i
$779$	7.31230	0.261990
$780$	0	0
$781$	2.30464	0.0824665
$782$	21.9572i	0.785187i
$783$	0	0
$784$	39.5622	1.41294
$785$	0	0
$786$	0	0
$787$	− 10.9147i	− 0.389067i	−0.980896	−	0.194533i	$-0.937681\pi$
$787$	− 10.9147i	− 0.389067i	0.980896	−	0.194533i	$-0.0623192\pi$
$788$	88.7422i	3.16131i
$789$	0	0
$790$	0	0
$791$	26.7862	0.952409
$792$	0	0
$793$	− 24.9839i	− 0.887204i
$794$	14.8687	0.527669
$795$	0	0
$796$	54.4124	1.92860
$797$	2.92550i	0.103627i	0.998657	+	0.0518133i	$0.0165001\pi$
$797$	2.92550i	0.103627i	−0.998657	+	0.0518133i	$0.983500\pi$
$798$	0	0
$799$	−30.2531	−1.07028
$800$	0	0
$801$	0	0
$802$	14.4965i	0.511888i
$803$	− 6.48700i	− 0.228921i
$804$	0	0
$805$	0	0
$806$	−13.6220	−0.479816
$807$	0	0
$808$	50.6755i	1.78276i
$809$	37.9241	1.33334	0.666671	−	0.745352i	$-0.267719\pi$
$809$	37.9241	1.33334	0.666671	+	0.745352i	$0.267719\pi$
$810$	0	0
$811$	19.5050	0.684912	0.342456	−	0.939534i	$-0.388741\pi$
$811$	19.5050	0.684912	0.342456	+	0.939534i	$0.388741\pi$
$812$	− 65.3731i	− 2.29415i
$813$	0	0
$814$	−55.4075	−1.94203
$815$	0	0
$816$	0	0
$817$	21.5773i	0.754894i
$818$	86.5257i	3.02530i
$819$	0	0
$820$	0	0
$821$	44.7524	1.56187	0.780934	−	0.624613i	$-0.214743\pi$
$821$	44.7524	1.56187	0.780934	+	0.624613i	$0.214743\pi$
$822$	0	0
$823$	− 35.8994i	− 1.25137i	−0.780074	−	0.625687i	$-0.784819\pi$
$823$	− 35.8994i	− 1.25137i	0.780074	−	0.625687i	$-0.215181\pi$
$824$	62.4559	2.17575
$825$	0	0
$826$	−52.1963	−1.81614
$827$	16.2717i	0.565822i	0.959146	+	0.282911i	$0.0913001\pi$
$827$	16.2717i	0.565822i	−0.959146	+	0.282911i	$0.908700\pi$
$828$	0	0
$829$	20.6592	0.717525	0.358762	−	0.933429i	$-0.383199\pi$
$829$	20.6592	0.717525	0.358762	+	0.933429i	$0.383199\pi$
$830$	0	0
$831$	0	0
$832$	− 21.8391i	− 0.757135i
$833$	− 18.1230i	− 0.627926i
$834$	0	0
$835$	0	0
$836$	26.4621	0.915210
$837$	0	0
$838$	60.2044i	2.07973i
$839$	12.2367	0.422459	0.211229	−	0.977437i	$-0.432253\pi$
$839$	12.2367	0.422459	0.211229	+	0.977437i	$0.432253\pi$
$840$	0	0
$841$	25.2823	0.871802
$842$	− 47.2527i	− 1.62844i
$843$	0	0
$844$	118.123	4.06597
$845$	0	0
$846$	0	0
$847$	− 13.8953i	− 0.477450i
$848$	18.1098i	0.621893i
$849$	0	0
$850$	0	0
$851$	−20.1744	−0.691570
$852$	0	0
$853$	16.4293i	0.562528i	0.959630	+	0.281264i	$0.0907537\pi$
$853$	16.4293i	0.562528i	−0.959630	+	0.281264i	$0.909246\pi$
$854$	−59.9809	−2.05250
$855$	0	0
$856$	−69.4255	−2.37291
$857$	− 49.0897i	− 1.67687i	−0.545000	−	0.838436i	$-0.683470\pi$
$857$	− 49.0897i	− 1.67687i	0.545000	−	0.838436i	$-0.316530\pi$
$858$	0	0
$859$	−41.0908	−1.40200	−0.700999	−	0.713162i	$-0.747262\pi$
$859$	−41.0908	−1.40200	−0.700999	+	0.713162i	$0.747262\pi$
$860$	0	0
$861$	0	0
$862$	16.2994i	0.555158i
$863$	47.8366i	1.62838i	0.580601	+	0.814188i	$0.302817\pi$
$863$	47.8366i	1.62838i	−0.580601	+	0.814188i	$0.697183\pi$
$864$	0	0
$865$	0	0
$866$	−8.21210	−0.279058
$867$	0	0
$868$	23.2741i	0.789975i
$869$	−3.81677	−0.129475
$870$	0	0
$871$	17.9510	0.608247
$872$	49.0458i	1.66090i
$873$	0	0
$874$	13.5387	0.457954
$875$	0	0
$876$	0	0
$877$	11.3772i	0.384180i	0.981377	+	0.192090i	$0.0615265\pi$
$877$	11.3772i	0.384180i	−0.981377	+	0.192090i	$0.938473\pi$
$878$	35.5813i	1.20081i
$879$	0	0
$880$	0	0
$881$	−43.9924	−1.48214	−0.741071	−	0.671426i	$-0.765682\pi$
$881$	−43.9924	−1.48214	−0.741071	+	0.671426i	$0.765682\pi$
$882$	0	0
$883$	37.6820i	1.26810i	0.773291	+	0.634051i	$0.218609\pi$
$883$	37.6820i	1.26810i	−0.773291	+	0.634051i	$0.781391\pi$
$884$	−46.8019	−1.57412
$885$	0	0
$886$	−63.9398	−2.14810
$887$	33.2833i	1.11754i	0.829322	+	0.558771i	$0.188727\pi$
$887$	33.2833i	1.11754i	−0.829322	+	0.558771i	$0.811273\pi$
$888$	0	0
$889$	6.51213	0.218410
$890$	0	0
$891$	0	0
$892$	− 107.175i	− 3.58847i
$893$	18.6539i	0.624230i
$894$	0	0
$895$	0	0
$896$	−8.66080	−0.289337
$897$	0	0
$898$	− 63.5878i	− 2.12195i
$899$	−19.3255	−0.644543
$900$	0	0
$901$	8.29592	0.276377
$902$	11.7456i	0.391086i
$903$	0	0
$904$	115.255	3.83333
$905$	0	0
$906$	0	0
$907$	14.0784i	0.467464i	0.972301	+	0.233732i	$0.0750939\pi$
$907$	14.0784i	0.467464i	−0.972301	+	0.233732i	$0.924906\pi$
$908$	69.6056i	2.30994i
$909$	0	0
$910$	0	0
$911$	7.31522	0.242364	0.121182	−	0.992630i	$-0.461332\pi$
$911$	7.31522	0.242364	0.121182	+	0.992630i	$0.461332\pi$
$912$	0	0
$913$	1.98682i	0.0657542i
$914$	7.43123	0.245803
$915$	0	0
$916$	18.1544	0.599838
$917$	13.1094i	0.432912i
$918$	0	0
$919$	−4.44684	−0.146688	−0.0733438	−	0.997307i	$-0.523367\pi$
$919$	−4.44684	−0.146688	−0.0733438	+	0.997307i	$0.523367\pi$
$920$	0	0
$921$	0	0
$922$	− 56.5124i	− 1.86114i
$923$	2.51330i	0.0827262i
$924$	0	0
$925$	0	0
$926$	−52.1565	−1.71397
$927$	0	0
$928$	− 89.7068i	− 2.94477i
$929$	−13.1133	−0.430232	−0.215116	−	0.976588i	$-0.569013\pi$
$929$	−13.1133	−0.430232	−0.215116	+	0.976588i	$0.569013\pi$
$930$	0	0
$931$	−11.1746	−0.366233
$932$	26.3690i	0.863746i
$933$	0	0
$934$	−59.8408	−1.95805
$935$	0	0
$936$	0	0
$937$	33.5187i	1.09501i	0.836803	+	0.547504i	$0.184422\pi$
$937$	33.5187i	1.09501i	−0.836803	+	0.547504i	$0.815578\pi$
$938$	− 43.0964i	− 1.40715i
$939$	0	0
$940$	0	0
$941$	1.79076	0.0583772	0.0291886	−	0.999574i	$-0.490708\pi$
$941$	1.79076	0.0583772	0.0291886	+	0.999574i	$0.490708\pi$
$942$	0	0
$943$	4.27669i	0.139268i
$944$	−115.729	−3.76667
$945$	0	0
$946$	−34.6592	−1.12687
$947$	52.6350i	1.71041i	0.518292	+	0.855204i	$0.326568\pi$
$947$	52.6350i	1.71041i	−0.518292	+	0.855204i	$0.673432\pi$
$948$	0	0
$949$	7.07432	0.229642
$950$	0	0
$951$	0	0
$952$	66.8386i	2.16625i
$953$	− 18.4072i	− 0.596268i	−0.954524	−	0.298134i	$-0.903636\pi$
$953$	− 18.4072i	− 0.596268i	0.954524	−	0.298134i	$-0.0963642\pi$
$954$	0	0
$955$	0	0
$956$	−108.768	−3.51781
$957$	0	0
$958$	56.0666i	1.81143i
$959$	−12.8088	−0.413619
$960$	0	0
$961$	−24.1197	−0.778056
$962$	− 60.4240i	− 1.94815i
$963$	0	0
$964$	92.0517	2.96479
$965$	0	0
$966$	0	0
$967$	6.09579i	0.196027i	0.995185	+	0.0980137i	$0.0312489\pi$
$967$	6.09579i	0.196027i	−0.995185	+	0.0980137i	$0.968751\pi$
$968$	− 59.7886i	− 1.92168i
$969$	0	0
$970$	0	0
$971$	−34.1077	−1.09457	−0.547283	−	0.836947i	$-0.684338\pi$
$971$	−34.1077	−1.09457	−0.547283	+	0.836947i	$0.684338\pi$
$972$	0	0
$973$	26.4414i	0.847671i
$974$	25.2493	0.809038
$975$	0	0
$976$	−132.989	−4.25688
$977$	− 48.7914i	− 1.56098i	−0.625171	−	0.780488i	$-0.714971\pi$
$977$	− 48.7914i	− 1.56098i	0.625171	−	0.780488i	$-0.285029\pi$
$978$	0	0
$979$	24.0362	0.768201
$980$	0	0
$981$	0	0
$982$	99.6715i	3.18065i
$983$	33.4172i	1.06584i	0.846164	+	0.532922i	$0.178906\pi$
$983$	33.4172i	1.06584i	−0.846164	+	0.532922i	$0.821094\pi$
$984$	0	0
$985$	0	0
$986$	−93.2985	−2.97123
$987$	0	0
$988$	28.8579i	0.918092i
$989$	−12.6198	−0.401285
$990$	0	0
$991$	−19.3512	−0.614713	−0.307356	−	0.951595i	$-0.599444\pi$
$991$	−19.3512	−0.614713	−0.307356	+	0.951595i	$0.599444\pi$
$992$	31.9374i	1.01401i
$993$	0	0
$994$	6.03388	0.191383
$995$	0	0
$996$	0	0
$997$	− 39.5442i	− 1.25238i	−0.779671	−	0.626189i	$-0.784614\pi$
$997$	− 39.5442i	− 1.25238i	0.779671	−	0.626189i	$-0.215386\pi$
$998$	44.5766i	1.41105i
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	2025.2.b.n.649.1		8
3.2	odd	2		2025.2.b.o.649.8		8
5.2	odd	4		2025.2.a.y.1.4		4
5.3	odd	4		2025.2.a.q.1.1		4
5.4	even	2	inner	2025.2.b.n.649.8		8
9.2	odd	6		675.2.k.c.199.8		16
9.4	even	3		225.2.k.c.124.8		16
9.5	odd	6		675.2.k.c.424.1		16
9.7	even	3		225.2.k.c.49.1		16
15.2	even	4		2025.2.a.p.1.1		4
15.8	even	4		2025.2.a.z.1.4		4
15.14	odd	2		2025.2.b.o.649.1		8
45.2	even	12		675.2.e.e.226.4		8
45.4	even	6		225.2.k.c.124.1		16
45.7	odd	12		225.2.e.c.76.1	✓	8
45.13	odd	12		225.2.e.e.151.4	yes	8
45.14	odd	6		675.2.k.c.424.8		16
45.22	odd	12		225.2.e.c.151.1	yes	8
45.23	even	12		675.2.e.c.451.1		8
45.29	odd	6		675.2.k.c.199.1		16
45.32	even	12		675.2.e.e.451.4		8
45.34	even	6		225.2.k.c.49.8		16
45.38	even	12		675.2.e.c.226.1		8
45.43	odd	12		225.2.e.e.76.4	yes	8

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
225.2.e.c.76.1	✓	8	45.7	odd	12
225.2.e.c.151.1	yes	8	45.22	odd	12
225.2.e.e.76.4	yes	8	45.43	odd	12
225.2.e.e.151.4	yes	8	45.13	odd	12
225.2.k.c.49.1		16	9.7	even	3
225.2.k.c.49.8		16	45.34	even	6
225.2.k.c.124.1		16	45.4	even	6
225.2.k.c.124.8		16	9.4	even	3
675.2.e.c.226.1		8	45.38	even	12
675.2.e.c.451.1		8	45.23	even	12
675.2.e.e.226.4		8	45.2	even	12
675.2.e.e.451.4		8	45.32	even	12
675.2.k.c.199.1		16	45.29	odd	6
675.2.k.c.199.8		16	9.2	odd	6
675.2.k.c.424.1		16	9.5	odd	6
675.2.k.c.424.8		16	45.14	odd	6
2025.2.a.p.1.1		4	15.2	even	4
2025.2.a.q.1.1		4	5.3	odd	4
2025.2.a.y.1.4		4	5.2	odd	4
2025.2.a.z.1.4		4	15.8	even	4
2025.2.b.n.649.1		8	1.1	even	1	trivial
2025.2.b.n.649.8		8	5.4	even	2	inner
2025.2.b.o.649.1		8	15.14	odd	2
2025.2.b.o.649.8		8	3.2	odd	2

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 $\F_{q}$
Belyi maps

Level:	\( N \)	\(=\)	\( 2025 = 3^{4} \cdot 5^{2} \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	2025.b (of order \(2\), degree \(1\), not minimal)

\(f(q)\)	\(=\)	\(q-2.63372i q^{2} -4.93650 q^{4} +1.79743i q^{7} +7.73393i q^{8} +1.80812 q^{11} +1.97183i q^{13} +4.73393 q^{14} +10.4960 q^{16} -4.80812i q^{17} -2.96467 q^{19} -4.76210i q^{22} -1.73393i q^{23} +5.19325 q^{26} -8.87300i q^{28} +7.36765 q^{29} -2.62303 q^{31} -12.1758i q^{32} -12.6633 q^{34} -11.6351i q^{37} +7.80812i q^{38} -2.46648 q^{41} -7.27814i q^{43} -8.92580 q^{44} -4.56668 q^{46} -6.29208i q^{47} +3.76926 q^{49} -9.73393i q^{52} +1.72540i q^{53} -13.9012 q^{56} -19.4044i q^{58} -11.0260 q^{59} -12.6704 q^{61} +6.90833i q^{62} -11.0756 q^{64} -9.10374i q^{67} +23.7353i q^{68} +1.27460 q^{71} -3.58770i q^{73} -30.6436 q^{74} +14.6351 q^{76} +3.24997i q^{77} -2.11090 q^{79} +6.49602i q^{82} +1.09883i q^{83} -19.1686 q^{86} +13.9839i q^{88} +13.2935 q^{89} -3.54422 q^{91} +8.55953i q^{92} -16.5716 q^{94} +3.83276i q^{97} -9.92718i q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 8 q - 8 q^{4} - 2 q^{11} - 6 q^{14} + 8 q^{16} - 4 q^{19} - 20 q^{26} - 2 q^{29} - 8 q^{31} - 18 q^{34} - 10 q^{41} - 44 q^{44} + 6 q^{49} - 60 q^{56} - 34 q^{59} - 26 q^{61} - 38 q^{64} - 16 q^{71} - 80 q^{74}+ \cdots - 6 q^{94}+O(q^{100}) \) 8 * q - 8 * q^4 - 2 * q^11 - 6 * q^14 + 8 * q^16 - 4 * q^19 - 20 * q^26 - 2 * q^29 - 8 * q^31 - 18 * q^34 - 10 * q^41 - 44 * q^44 + 6 * q^49 - 60 * q^56 - 34 * q^59 - 26 * q^61 - 38 * q^64 - 16 * q^71 - 80 * q^74 + 22 * q^76 + 14 * q^79 - 68 * q^86 + 18 * q^89 - 34 * q^91 - 6 * q^94

Introduction

L-functions

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