LMFDB - Embedded newform 2025.2.b.l.649.4

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [2025,2,Mod(649,2025)]

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

lf = mfeigenbasis(mf)

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")

//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);

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

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

Self dual:	no
Analytic conductor:	$16.1697064093$
Analytic rank:	$0$
Dimension:	$6$
Coefficient field:	6.0.5089536.1
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{6} - 2x^{5} + 2x^{4} + 2x^{3} + 16x^{2} - 24x + 18 $ x^6 - 2x^5 + 2x^4 + 2x^3 + 16x^2 - 24*x + 18
Coefficient ring:	$\Z[a_1, \ldots, a_{7}]$
Coefficient ring index:	$ 2^{3} $
Twist minimal:	no (minimal twist has level 45)
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label			649.4
Root			$-1.33641 + 1.33641i$ of defining polynomial
Character	$\chi$	$=$	2025.649
Dual form			2025.2.b.l.649.3

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+0.571993i q^{2} +1.67282 q^{4} +1.42801i q^{7} +2.10083i q^{8} +O(q^{10})$	$q+0.571993i q^{2} +1.67282 q^{4} +1.42801i q^{7} +2.10083i q^{8} +2.67282 q^{11} -4.67282i q^{13} -0.816810 q^{14} +2.14399 q^{16} +2.67282i q^{17} -4.67282 q^{19} +1.52884i q^{22} +5.91764i q^{23} +2.67282 q^{26} +2.38880i q^{28} +9.48963 q^{29} +6.96080 q^{31} +5.42801i q^{32} -1.52884 q^{34} -1.81681i q^{37} -2.67282i q^{38} -1.47116 q^{41} -0.471163i q^{43} +4.47116 q^{44} -3.38485 q^{46} +6.95684i q^{47} +4.96080 q^{49} -7.81681i q^{52} +1.14399i q^{53} -3.00000 q^{56} +5.42801i q^{58} +1.14399 q^{59} -2.52884 q^{61} +3.98153i q^{62} +1.18319 q^{64} -6.59046i q^{67} +4.47116i q^{68} -12.8745 q^{71} +1.71203i q^{73} +1.03920 q^{74} -7.81681 q^{76} +3.81681i q^{77} +0.287973 q^{79} -0.841495i q^{82} +4.28402i q^{83} +0.269502 q^{86} +5.61515i q^{88} +3.00000 q^{89} +6.67282 q^{91} +9.89917i q^{92} -3.97927 q^{94} +7.83528i q^{97} +2.83754i q^{98} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 6 q - 10 q^{4}+O(q^{10}) $ 6 * q - 10 * q^4	$ 6 q - 10 q^{4} - 4 q^{11} + 18 q^{14} + 10 q^{16} - 8 q^{19} - 4 q^{26} + 14 q^{29} + 16 q^{31} + 8 q^{34} - 26 q^{41} + 44 q^{44} - 6 q^{46} + 4 q^{49} - 18 q^{56} + 4 q^{59} + 2 q^{61} + 30 q^{64} - 20 q^{71} + 32 q^{74} - 24 q^{76} - 4 q^{79} + 56 q^{86} + 18 q^{89} + 20 q^{91} + 62 q^{94}+O(q^{100}) $ 6 * q - 10 * q^4 - 4 * q^11 + 18 * q^14 + 10 * q^16 - 8 * q^19 - 4 * q^26 + 14 * q^29 + 16 * q^31 + 8 * q^34 - 26 * q^41 + 44 * q^44 - 6 * q^46 + 4 * q^49 - 18 * q^56 + 4 * q^59 + 2 * q^61 + 30 * q^64 - 20 * q^71 + 32 * q^74 - 24 * q^76 - 4 * q^79 + 56 * q^86 + 18 * q^89 + 20 * q^91 + 62 * q^94

Display 100 coefficients 10 coefficients

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 $

$2$	0.571993i	0.404460i	0.979338	+	0.202230i	$0.0648189\pi$
$2$	0.571993i	0.404460i	−0.979338	+	0.202230i	$0.935181\pi$
$3$	0	0
$3$	0	0
$4$	1.67282	0.836412
$5$	0	0
$5$	0	0
$6$	0	0
$7$	1.42801i	0.539736i	0.962897	+	0.269868i	$0.0869800\pi$
$7$	1.42801i	0.539736i	−0.962897	+	0.269868i	$0.913020\pi$
$8$	2.10083i	0.742756i
$9$	0	0
$10$	0	0
$11$	2.67282	0.805887	0.402943	−	0.915225i	$-0.367987\pi$
$11$	2.67282	0.805887	0.402943	+	0.915225i	$0.367987\pi$
$12$	0	0
$13$	− 4.67282i	− 1.29601i	−0.761637	−	0.648004i	$-0.775604\pi$
$13$	− 4.67282i	− 1.29601i	0.761637	−	0.648004i	$-0.224396\pi$
$14$	−0.816810	−0.218302
$15$	0	0
$16$	2.14399	0.535997
$17$	2.67282i	0.648255i	0.946013	+	0.324127i	$0.105071\pi$
$17$	2.67282i	0.648255i	−0.946013	+	0.324127i	$0.894929\pi$
$18$	0	0
$19$	−4.67282	−1.07202	−0.536010	−	0.844212i	$-0.680069\pi$
$19$	−4.67282	−1.07202	−0.536010	+	0.844212i	$0.680069\pi$
$20$	0	0
$21$	0	0
$22$	1.52884i	0.325949i
$23$	5.91764i	1.23391i	0.786997	+	0.616957i	$0.211635\pi$
$23$	5.91764i	1.23391i	−0.786997	+	0.616957i	$0.788365\pi$
$24$	0	0
$25$	0	0
$26$	2.67282	0.524184
$27$	0	0
$28$	2.38880i	0.451441i
$29$	9.48963	1.76218	0.881090	−	0.472948i	$-0.156810\pi$
$29$	9.48963	1.76218	0.881090	+	0.472948i	$0.156810\pi$
$30$	0	0
$31$	6.96080	1.25020	0.625098	−	0.780546i	$-0.285059\pi$
$31$	6.96080	1.25020	0.625098	+	0.780546i	$0.285059\pi$
$32$	5.42801i	0.959545i
$33$	0	0
$34$	−1.52884	−0.262193
$35$	0	0
$36$	0	0
$37$	− 1.81681i	− 0.298682i	−0.988786	−	0.149341i	$-0.952285\pi$
$37$	− 1.81681i	− 0.298682i	0.988786	−	0.149341i	$-0.0477152\pi$
$38$	− 2.67282i	− 0.433589i
$39$	0	0
$40$	0	0
$41$	−1.47116	−0.229757	−0.114879	−	0.993380i	$-0.536648\pi$
$41$	−1.47116	−0.229757	−0.114879	+	0.993380i	$0.536648\pi$
$42$	0	0
$43$	− 0.471163i	− 0.0718517i	−0.999354	−	0.0359258i	$-0.988562\pi$
$43$	− 0.471163i	− 0.0718517i	0.999354	−	0.0359258i	$-0.0114380\pi$
$44$	4.47116	0.674053
$45$	0	0
$46$	−3.38485	−0.499069
$47$	6.95684i	1.01476i	0.861722	+	0.507380i	$0.169386\pi$
$47$	6.95684i	1.01476i	−0.861722	+	0.507380i	$0.830614\pi$
$48$	0	0
$49$	4.96080	0.708685
$50$	0	0
$51$	0	0
$52$	− 7.81681i	− 1.08400i
$53$	1.14399i	0.157139i	0.996909	+	0.0785693i	$0.0250352\pi$
$53$	1.14399i	0.157139i	−0.996909	+	0.0785693i	$0.974965\pi$
$54$	0	0
$55$	0	0
$56$	−3.00000	−0.400892
$57$	0	0
$58$	5.42801i	0.712732i
$59$	1.14399	0.148934	0.0744672	−	0.997223i	$-0.476274\pi$
$59$	1.14399	0.148934	0.0744672	+	0.997223i	$0.476274\pi$
$60$	0	0
$61$	−2.52884	−0.323784	−0.161892	−	0.986808i	$-0.551760\pi$
$61$	−2.52884	−0.323784	−0.161892	+	0.986808i	$0.551760\pi$
$62$	3.98153i	0.505655i
$63$	0	0
$64$	1.18319	0.147899
$65$	0	0
$66$	0	0
$67$	− 6.59046i	− 0.805153i	−0.915386	−	0.402577i	$-0.868115\pi$
$67$	− 6.59046i	− 0.805153i	0.915386	−	0.402577i	$-0.131885\pi$
$68$	4.47116i	0.542208i
$69$	0	0
$70$	0	0
$71$	−12.8745	−1.52792	−0.763960	−	0.645263i	$-0.776748\pi$
$71$	−12.8745	−1.52792	−0.763960	+	0.645263i	$0.776748\pi$
$72$	0	0
$73$	1.71203i	0.200378i	0.994968	+	0.100189i	$0.0319447\pi$
$73$	1.71203i	0.200378i	−0.994968	+	0.100189i	$0.968055\pi$
$74$	1.03920	0.120805
$75$	0	0
$76$	−7.81681	−0.896650
$77$	3.81681i	0.434966i
$78$	0	0
$79$	0.287973	0.0323995	0.0161998	−	0.999869i	$-0.494843\pi$
$79$	0.287973	0.0323995	0.0161998	+	0.999869i	$0.494843\pi$
$80$	0	0
$81$	0	0
$82$	− 0.841495i	− 0.0929276i
$83$	4.28402i	0.470232i	0.971967	+	0.235116i	$0.0755471\pi$
$83$	4.28402i	0.470232i	−0.971967	+	0.235116i	$0.924453\pi$
$84$	0	0
$85$	0	0
$86$	0.269502	0.0290611
$87$	0	0
$88$	5.61515i	0.598577i
$89$	3.00000	0.317999	0.159000	−	0.987279i	$-0.449173\pi$
$89$	3.00000	0.317999	0.159000	+	0.987279i	$0.449173\pi$
$90$	0	0
$91$	6.67282	0.699502
$92$	9.89917i	1.03206i
$93$	0	0
$94$	−3.97927	−0.410430
$95$	0	0
$96$	0	0
$97$	7.83528i	0.795552i	0.917483	+	0.397776i	$0.130218\pi$
$97$	7.83528i	0.795552i	−0.917483	+	0.397776i	$0.869782\pi$
$98$	2.83754i	0.286635i
$99$	0	0
$100$	0	0
$101$	−4.20166	−0.418081	−0.209040	−	0.977907i	$-0.567034\pi$
$101$	−4.20166	−0.418081	−0.209040	+	0.977907i	$0.567034\pi$
$102$	0	0
$103$	1.81681i	0.179016i	0.995986	+	0.0895078i	$0.0285294\pi$
$103$	1.81681i	0.179016i	−0.995986	+	0.0895078i	$0.971471\pi$
$104$	9.81681	0.962617
$105$	0	0
$106$	−0.654353	−0.0635563
$107$	11.9176i	1.15212i	0.817407	+	0.576061i	$0.195411\pi$
$107$	11.9176i	1.15212i	−0.817407	+	0.576061i	$0.804589\pi$
$108$	0	0
$109$	16.6521	1.59498	0.797491	−	0.603331i	$-0.206160\pi$
$109$	16.6521	1.59498	0.797491	+	0.603331i	$0.206160\pi$
$110$	0	0
$111$	0	0
$112$	3.06163i	0.289297i
$113$	− 20.1233i	− 1.89304i	−0.322650	−	0.946518i	$-0.604574\pi$
$113$	− 20.1233i	− 1.89304i	0.322650	−	0.946518i	$-0.395426\pi$
$114$	0	0
$115$	0	0
$116$	15.8745	1.47391
$117$	0	0
$118$	0.654353i	0.0602380i
$119$	−3.81681	−0.349886
$120$	0	0
$121$	−3.85601	−0.350547
$122$	− 1.44648i	− 0.130958i
$123$	0	0
$124$	11.6442	1.04568
$125$	0	0
$126$	0	0
$127$	2.18714i	0.194078i	0.995281	+	0.0970388i	$0.0309371\pi$
$127$	2.18714i	0.194078i	−0.995281	+	0.0970388i	$0.969063\pi$
$128$	11.5328i	1.01936i
$129$	0	0
$130$	0	0
$131$	−6.00000	−0.524222	−0.262111	−	0.965038i	$-0.584419\pi$
$131$	−6.00000	−0.524222	−0.262111	+	0.965038i	$0.584419\pi$
$132$	0	0
$133$	− 6.67282i	− 0.578607i
$134$	3.76970	0.325653
$135$	0	0
$136$	−5.61515	−0.481495
$137$	− 10.2017i	− 0.871587i	−0.900047	−	0.435793i	$-0.856468\pi$
$137$	− 10.2017i	− 0.871587i	0.900047	−	0.435793i	$-0.143532\pi$
$138$	0	0
$139$	−8.00000	−0.678551	−0.339276	−	0.940687i	$-0.610182\pi$
$139$	−8.00000	−0.678551	−0.339276	+	0.940687i	$0.610182\pi$
$140$	0	0
$141$	0	0
$142$	− 7.36412i	− 0.617983i
$143$	− 12.4896i	− 1.04444i
$144$	0	0
$145$	0	0
$146$	−0.979268	−0.0810448
$147$	0	0
$148$	− 3.03920i	− 0.249821i
$149$	20.0761	1.64470	0.822351	−	0.568981i	$-0.192662\pi$
$149$	20.0761	1.64470	0.822351	+	0.568981i	$0.192662\pi$
$150$	0	0
$151$	3.03920	0.247327	0.123663	−	0.992324i	$-0.460536\pi$
$151$	3.03920	0.247327	0.123663	+	0.992324i	$0.460536\pi$
$152$	− 9.81681i	− 0.796248i
$153$	0	0
$154$	−2.18319	−0.175926
$155$	0	0
$156$	0	0
$157$	0.201661i	0.0160943i	0.999968	+	0.00804714i	$0.00256151\pi$
$157$	0.201661i	0.0160943i	−0.999968	+	0.00804714i	$0.997438\pi$
$158$	0.164719i	0.0131043i
$159$	0	0
$160$	0	0
$161$	−8.45043	−0.665987
$162$	0	0
$163$	− 17.8168i	− 1.39552i	−0.716331	−	0.697760i	$-0.754180\pi$
$163$	− 17.8168i	− 1.39552i	0.716331	−	0.697760i	$-0.245820\pi$
$164$	−2.46100	−0.192172
$165$	0	0
$166$	−2.45043	−0.190190
$167$	14.1008i	1.09116i	0.838060	+	0.545578i	$0.183690\pi$
$167$	14.1008i	1.09116i	−0.838060	+	0.545578i	$0.816310\pi$
$168$	0	0
$169$	−8.83528	−0.679637
$170$	0	0
$171$	0	0
$172$	− 0.788172i	− 0.0600976i
$173$	− 4.36638i	− 0.331970i	−0.986128	−	0.165985i	$-0.946920\pi$
$173$	− 4.36638i	− 0.331970i	0.986128	−	0.165985i	$-0.0530803\pi$
$174$	0	0
$175$	0	0
$176$	5.73050	0.431953
$177$	0	0
$178$	1.71598i	0.128618i
$179$	15.1625	1.13330	0.566648	−	0.823960i	$-0.308240\pi$
$179$	15.1625	1.13330	0.566648	+	0.823960i	$0.308240\pi$
$180$	0	0
$181$	3.20166	0.237978	0.118989	−	0.992896i	$-0.462035\pi$
$181$	3.20166	0.237978	0.118989	+	0.992896i	$0.462035\pi$
$182$	3.81681i	0.282921i
$183$	0	0
$184$	−12.4320	−0.916496
$185$	0	0
$186$	0	0
$187$	7.14399i	0.522420i
$188$	11.6376i	0.848757i
$189$	0	0
$190$	0	0
$191$	−2.83754	−0.205317	−0.102659	−	0.994717i	$-0.532735\pi$
$191$	−2.83754	−0.205317	−0.102659	+	0.994717i	$0.532735\pi$
$192$	0	0
$193$	18.7882i	1.35240i	0.736717	+	0.676201i	$0.236375\pi$
$193$	18.7882i	1.35240i	−0.736717	+	0.676201i	$0.763625\pi$
$194$	−4.48173	−0.321769
$195$	0	0
$196$	8.29854	0.592753
$197$	5.83528i	0.415747i	0.978156	+	0.207873i	$0.0666542\pi$
$197$	5.83528i	0.415747i	−0.978156	+	0.207873i	$0.933346\pi$
$198$	0	0
$199$	−13.0761	−0.926943	−0.463472	−	0.886112i	$-0.653396\pi$
$199$	−13.0761	−0.926943	−0.463472	+	0.886112i	$0.653396\pi$
$200$	0	0
$201$	0	0
$202$	− 2.40332i	− 0.169097i
$203$	13.5513i	0.951112i
$204$	0	0
$205$	0	0
$206$	−1.03920	−0.0724047
$207$	0	0
$208$	− 10.0185i	− 0.694656i
$209$	−12.4896	−0.863926
$210$	0	0
$211$	8.38485	0.577237	0.288618	−	0.957444i	$-0.406804\pi$
$211$	8.38485	0.577237	0.288618	+	0.957444i	$0.406804\pi$
$212$	1.91369i	0.131433i
$213$	0	0
$214$	−6.81681	−0.465988
$215$	0	0
$216$	0	0
$217$	9.94006i	0.674776i
$218$	9.52488i	0.645107i
$219$	0	0
$220$	0	0
$221$	12.4896	0.840144
$222$	0	0
$223$	− 9.16641i	− 0.613828i	−0.951737	−	0.306914i	$-0.900704\pi$
$223$	− 9.16641i	− 0.613828i	0.951737	−	0.306914i	$-0.0992964\pi$
$224$	−7.75123	−0.517901
$225$	0	0
$226$	11.5104	0.765658
$227$	− 2.67282i	− 0.177402i	−0.996058	−	0.0887008i	$-0.971729\pi$
$227$	− 2.67282i	− 0.177402i	0.996058	−	0.0887008i	$-0.0282715\pi$
$228$	0	0
$229$	−2.54731	−0.168331	−0.0841654	−	0.996452i	$-0.526822\pi$
$229$	−2.54731	−0.168331	−0.0841654	+	0.996452i	$0.526822\pi$
$230$	0	0
$231$	0	0
$232$	19.9361i	1.30887i
$233$	6.22013i	0.407494i	0.979024	+	0.203747i	$0.0653121\pi$
$233$	6.22013i	0.407494i	−0.979024	+	0.203747i	$0.934688\pi$
$234$	0	0
$235$	0	0
$236$	1.91369	0.124570
$237$	0	0
$238$	− 2.18319i	− 0.141515i
$239$	−8.12325	−0.525450	−0.262725	−	0.964871i	$-0.584621\pi$
$239$	−8.12325	−0.525450	−0.262725	+	0.964871i	$0.584621\pi$
$240$	0	0
$241$	−26.3641	−1.69826	−0.849131	−	0.528182i	$-0.822874\pi$
$241$	−26.3641	−1.69826	−0.849131	+	0.528182i	$0.822874\pi$
$242$	− 2.20561i	− 0.141782i
$243$	0	0
$244$	−4.23030	−0.270817
$245$	0	0
$246$	0	0
$247$	21.8353i	1.38935i
$248$	14.6235i	0.928590i
$249$	0	0
$250$	0	0
$251$	−0.549569	−0.0346885	−0.0173443	−	0.999850i	$-0.505521\pi$
$251$	−0.549569	−0.0346885	−0.0173443	+	0.999850i	$0.505521\pi$
$252$	0	0
$253$	15.8168i	0.994394i
$254$	−1.25103	−0.0784967
$255$	0	0
$256$	−4.23030	−0.264394
$257$	− 18.0000i	− 1.12281i	−0.827541	−	0.561405i	$-0.810261\pi$
$257$	− 18.0000i	− 1.12281i	0.827541	−	0.561405i	$-0.189739\pi$
$258$	0	0
$259$	2.59442	0.161209
$260$	0	0
$261$	0	0
$262$	− 3.43196i	− 0.212027i
$263$	− 11.8952i	− 0.733490i	−0.930321	−	0.366745i	$-0.880472\pi$
$263$	− 11.8952i	− 0.733490i	0.930321	−	0.366745i	$-0.119528\pi$
$264$	0	0
$265$	0	0
$266$	3.81681	0.234024
$267$	0	0
$268$	− 11.0247i	− 0.673440i
$269$	−28.5737	−1.74217	−0.871084	−	0.491134i	$-0.836583\pi$
$269$	−28.5737	−1.74217	−0.871084	+	0.491134i	$0.836583\pi$
$270$	0	0
$271$	−23.3641	−1.41927	−0.709635	−	0.704570i	$-0.751140\pi$
$271$	−23.3641	−1.41927	−0.709635	+	0.704570i	$0.751140\pi$
$272$	5.73050i	0.347462i
$273$	0	0
$274$	5.83528	0.352522
$275$	0	0
$276$	0	0
$277$	− 15.0761i	− 0.905838i	−0.891552	−	0.452919i	$-0.850383\pi$
$277$	− 15.0761i	− 0.905838i	0.891552	−	0.452919i	$-0.149617\pi$
$278$	− 4.57595i	− 0.274447i
$279$	0	0
$280$	0	0
$281$	6.65209	0.396831	0.198415	−	0.980118i	$-0.436421\pi$
$281$	6.65209	0.396831	0.198415	+	0.980118i	$0.436421\pi$
$282$	0	0
$283$	− 26.8969i	− 1.59886i	−0.600762	−	0.799428i	$-0.705136\pi$
$283$	− 26.8969i	− 1.59886i	0.600762	−	0.799428i	$-0.294864\pi$
$284$	−21.5367	−1.27797
$285$	0	0
$286$	7.14399	0.422433
$287$	− 2.10083i	− 0.124008i
$288$	0	0
$289$	9.85601	0.579765
$290$	0	0
$291$	0	0
$292$	2.86392i	0.167598i
$293$	− 12.3849i	− 0.723531i	−0.932269	−	0.361765i	$-0.882174\pi$
$293$	− 12.3849i	− 0.723531i	0.932269	−	0.361765i	$-0.117826\pi$
$294$	0	0
$295$	0	0
$296$	3.81681	0.221848
$297$	0	0
$298$	11.4834i	0.665217i
$299$	27.6521	1.59916
$300$	0	0
$301$	0.672824	0.0387809
$302$	1.73840i	0.100034i
$303$	0	0
$304$	−10.0185	−0.574599
$305$	0	0
$306$	0	0
$307$	− 2.49359i	− 0.142317i	−0.997465	−	0.0711583i	$-0.977330\pi$
$307$	− 2.49359i	− 0.142317i	0.997465	−	0.0711583i	$-0.0226695\pi$
$308$	6.38485i	0.363811i
$309$	0	0
$310$	0	0
$311$	−24.2201	−1.37340	−0.686699	−	0.726942i	$-0.740941\pi$
$311$	−24.2201	−1.37340	−0.686699	+	0.726942i	$0.740941\pi$
$312$	0	0
$313$	35.0841i	1.98307i	0.129848	+	0.991534i	$0.458551\pi$
$313$	35.0841i	1.98307i	−0.129848	+	0.991534i	$0.541449\pi$
$314$	−0.115349	−0.00650950
$315$	0	0
$316$	0.481728	0.0270993
$317$	− 10.4712i	− 0.588119i	−0.955787	−	0.294060i	$-0.904994\pi$
$317$	− 10.4712i	− 0.588119i	0.955787	−	0.294060i	$-0.0950064\pi$
$318$	0	0
$319$	25.3641	1.42012
$320$	0	0
$321$	0	0
$322$	− 4.83359i	− 0.269365i
$323$	− 12.4896i	− 0.694942i
$324$	0	0
$325$	0	0
$326$	10.1911	0.564433
$327$	0	0
$328$	− 3.09066i	− 0.170653i
$329$	−9.93442	−0.547702
$330$	0	0
$331$	16.7776	0.922181	0.461090	−	0.887353i	$-0.347458\pi$
$331$	16.7776	0.922181	0.461090	+	0.887353i	$0.347458\pi$
$332$	7.16641i	0.393308i
$333$	0	0
$334$	−8.06558	−0.441329
$335$	0	0
$336$	0	0
$337$	− 27.1809i	− 1.48064i	−0.672255	−	0.740320i	$-0.734674\pi$
$337$	− 27.1809i	− 1.48064i	0.672255	−	0.740320i	$-0.265326\pi$
$338$	− 5.05372i	− 0.274886i
$339$	0	0
$340$	0	0
$341$	18.6050	1.00752
$342$	0	0
$343$	17.0801i	0.922239i
$344$	0.989833	0.0533682
$345$	0	0
$346$	2.49754	0.134269
$347$	− 23.5658i	− 1.26508i	−0.774529	−	0.632539i	$-0.782013\pi$
$347$	− 23.5658i	− 1.26508i	0.774529	−	0.632539i	$-0.217987\pi$
$348$	0	0
$349$	10.7120	0.573402	0.286701	−	0.958020i	$-0.407441\pi$
$349$	10.7120	0.573402	0.286701	+	0.958020i	$0.407441\pi$
$350$	0	0
$351$	0	0
$352$	14.5081i	0.773285i
$353$	− 27.2672i	− 1.45129i	−0.688070	−	0.725644i	$-0.741542\pi$
$353$	− 27.2672i	− 1.45129i	0.688070	−	0.725644i	$-0.258458\pi$
$354$	0	0
$355$	0	0
$356$	5.01847	0.265978
$357$	0	0
$358$	8.67282i	0.458373i
$359$	10.6807	0.563707	0.281854	−	0.959457i	$-0.409051\pi$
$359$	10.6807	0.563707	0.281854	+	0.959457i	$0.409051\pi$
$360$	0	0
$361$	2.83528	0.149225
$362$	1.83133i	0.0962525i
$363$	0	0
$364$	11.1625	0.585072
$365$	0	0
$366$	0	0
$367$	− 8.47116i	− 0.442191i	−0.975252	−	0.221096i	$-0.929037\pi$
$367$	− 8.47116i	− 0.442191i	0.975252	−	0.221096i	$-0.0709633\pi$
$368$	12.6873i	0.661373i
$369$	0	0
$370$	0	0
$371$	−1.63362	−0.0848133
$372$	0	0
$373$	− 10.1233i	− 0.524162i	−0.965046	−	0.262081i	$-0.915591\pi$
$373$	− 10.1233i	− 0.524162i	0.965046	−	0.262081i	$-0.0844088\pi$
$374$	−4.08631	−0.211298
$375$	0	0
$376$	−14.6151	−0.753719
$377$	− 44.3434i	− 2.28380i
$378$	0	0
$379$	−11.9216	−0.612371	−0.306186	−	0.951972i	$-0.599053\pi$
$379$	−11.9216	−0.612371	−0.306186	+	0.951972i	$0.599053\pi$
$380$	0	0
$381$	0	0
$382$	− 1.62306i	− 0.0830427i
$383$	− 9.81681i	− 0.501616i	−0.968037	−	0.250808i	$-0.919304\pi$
$383$	− 9.81681i	− 0.501616i	0.968037	−	0.250808i	$-0.0806962\pi$
$384$	0	0
$385$	0	0
$386$	−10.7467	−0.546993
$387$	0	0
$388$	13.1070i	0.665409i
$389$	−9.22013	−0.467479	−0.233740	−	0.972299i	$-0.575096\pi$
$389$	−9.22013	−0.467479	−0.233740	+	0.972299i	$0.575096\pi$
$390$	0	0
$391$	−15.8168	−0.799890
$392$	10.4218i	0.526380i
$393$	0	0
$394$	−3.33774	−0.168153
$395$	0	0
$396$	0	0
$397$	− 22.9793i	− 1.15330i	−0.816993	−	0.576648i	$-0.804360\pi$
$397$	− 22.9793i	− 1.15330i	0.816993	−	0.576648i	$-0.195640\pi$
$398$	− 7.47947i	− 0.374912i
$399$	0	0
$400$	0	0
$401$	−11.0656	−0.552589	−0.276294	−	0.961073i	$-0.589106\pi$
$401$	−11.0656	−0.552589	−0.276294	+	0.961073i	$0.589106\pi$
$402$	0	0
$403$	− 32.5266i	− 1.62026i
$404$	−7.02864	−0.349688
$405$	0	0
$406$	−7.75123	−0.384687
$407$	− 4.85601i	− 0.240704i
$408$	0	0
$409$	17.6336	0.871926	0.435963	−	0.899964i	$-0.356408\pi$
$409$	17.6336	0.871926	0.435963	+	0.899964i	$0.356408\pi$
$410$	0	0
$411$	0	0
$412$	3.03920i	0.149731i
$413$	1.63362i	0.0803852i
$414$	0	0
$415$	0	0
$416$	25.3641	1.24358
$417$	0	0
$418$	− 7.14399i	− 0.349424i
$419$	−37.0347	−1.80926	−0.904631	−	0.426195i	$-0.859854\pi$
$419$	−37.0347	−1.80926	−0.904631	+	0.426195i	$0.859854\pi$
$420$	0	0
$421$	5.05767	0.246496	0.123248	−	0.992376i	$-0.460669\pi$
$421$	5.05767	0.246496	0.123248	+	0.992376i	$0.460669\pi$
$422$	4.79608i	0.233469i
$423$	0	0
$424$	−2.40332	−0.116716
$425$	0	0
$426$	0	0
$427$	− 3.61120i	− 0.174758i
$428$	19.9361i	0.963648i
$429$	0	0
$430$	0	0
$431$	−5.23030	−0.251935	−0.125967	−	0.992034i	$-0.540203\pi$
$431$	−5.23030	−0.251935	−0.125967	+	0.992034i	$0.540203\pi$
$432$	0	0
$433$	− 34.3434i	− 1.65044i	−0.564813	−	0.825219i	$-0.691052\pi$
$433$	− 34.3434i	− 1.65044i	0.564813	−	0.825219i	$-0.308948\pi$
$434$	−5.68565	−0.272920
$435$	0	0
$436$	27.8560	1.33406
$437$	− 27.6521i	− 1.32278i
$438$	0	0
$439$	19.5473	0.932942	0.466471	−	0.884536i	$-0.345525\pi$
$439$	19.5473	0.932942	0.466471	+	0.884536i	$0.345525\pi$
$440$	0	0
$441$	0	0
$442$	7.14399i	0.339805i
$443$	− 10.5042i	− 0.499067i	−0.968366	−	0.249534i	$-0.919723\pi$
$443$	− 10.5042i	− 0.499067i	0.968366	−	0.249534i	$-0.0802773\pi$
$444$	0	0
$445$	0	0
$446$	5.24313	0.248269
$447$	0	0
$448$	1.68960i	0.0798262i
$449$	−22.8560	−1.07864	−0.539321	−	0.842100i	$-0.681319\pi$
$449$	−22.8560	−1.07864	−0.539321	+	0.842100i	$0.681319\pi$
$450$	0	0
$451$	−3.93216	−0.185158
$452$	− 33.6627i	− 1.58336i
$453$	0	0
$454$	1.52884	0.0717519
$455$	0	0
$456$	0	0
$457$	− 14.0264i	− 0.656126i	−0.944656	−	0.328063i	$-0.893604\pi$
$457$	− 14.0264i	− 0.656126i	0.944656	−	0.328063i	$-0.106396\pi$
$458$	− 1.45704i	− 0.0680832i
$459$	0	0
$460$	0	0
$461$	−24.1025	−1.12257	−0.561283	−	0.827624i	$-0.689692\pi$
$461$	−24.1025	−1.12257	−0.561283	+	0.827624i	$0.689692\pi$
$462$	0	0
$463$	33.9401i	1.57733i	0.614824	+	0.788664i	$0.289227\pi$
$463$	33.9401i	1.57733i	−0.614824	+	0.788664i	$0.710773\pi$
$464$	20.3456	0.944523
$465$	0	0
$466$	−3.55787	−0.164815
$467$	− 27.3720i	− 1.26663i	−0.773896	−	0.633313i	$-0.781695\pi$
$467$	− 27.3720i	− 1.26663i	0.773896	−	0.633313i	$-0.218305\pi$
$468$	0	0
$469$	9.41123	0.434570
$470$	0	0
$471$	0	0
$472$	2.40332i	0.110622i
$473$	− 1.25934i	− 0.0579043i
$474$	0	0
$475$	0	0
$476$	−6.38485	−0.292649
$477$	0	0
$478$	− 4.64645i	− 0.212524i
$479$	−5.23030	−0.238978	−0.119489	−	0.992835i	$-0.538126\pi$
$479$	−5.23030	−0.238978	−0.119489	+	0.992835i	$0.538126\pi$
$480$	0	0
$481$	−8.48963	−0.387094
$482$	− 15.0801i	− 0.686880i
$483$	0	0
$484$	−6.45043	−0.293201
$485$	0	0
$486$	0	0
$487$	− 24.0185i	− 1.08838i	−0.838962	−	0.544190i	$-0.816837\pi$
$487$	− 24.0185i	− 1.08838i	0.838962	−	0.544190i	$-0.183163\pi$
$488$	− 5.31266i	− 0.240493i
$489$	0	0
$490$	0	0
$491$	14.7776	0.666904	0.333452	−	0.942767i	$-0.391786\pi$
$491$	14.7776	0.666904	0.333452	+	0.942767i	$0.391786\pi$
$492$	0	0
$493$	25.3641i	1.14234i
$494$	−12.4896	−0.561935
$495$	0	0
$496$	14.9239	0.670101
$497$	− 18.3849i	− 0.824673i
$498$	0	0
$499$	−24.8560	−1.11271	−0.556354	−	0.830945i	$-0.687800\pi$
$499$	−24.8560	−1.11271	−0.556354	+	0.830945i	$0.687800\pi$
$500$	0	0
$501$	0	0
$502$	− 0.314350i	− 0.0140301i
$503$	− 38.9154i	− 1.73515i	−0.497305	−	0.867576i	$-0.665677\pi$
$503$	− 38.9154i	− 1.73515i	0.497305	−	0.867576i	$-0.334323\pi$
$504$	0	0
$505$	0	0
$506$	−9.04711	−0.402193
$507$	0	0
$508$	3.65870i	0.162329i
$509$	−2.02073	−0.0895674	−0.0447837	−	0.998997i	$-0.514260\pi$
$509$	−2.02073	−0.0895674	−0.0447837	+	0.998997i	$0.514260\pi$
$510$	0	0
$511$	−2.44479	−0.108151
$512$	20.6459i	0.912428i
$513$	0	0
$514$	10.2959	0.454132
$515$	0	0
$516$	0	0
$517$	18.5944i	0.817782i
$518$	1.48399i	0.0652027i
$519$	0	0
$520$	0	0
$521$	−23.0290	−1.00892	−0.504460	−	0.863435i	$-0.668308\pi$
$521$	−23.0290	−1.00892	−0.504460	+	0.863435i	$0.668308\pi$
$522$	0	0
$523$	41.1170i	1.79792i	0.438028	+	0.898961i	$0.355677\pi$
$523$	41.1170i	1.79792i	−0.438028	+	0.898961i	$0.644323\pi$
$524$	−10.0369	−0.438466
$525$	0	0
$526$	6.80398	0.296668
$527$	18.6050i	0.810446i
$528$	0	0
$529$	−12.0185	−0.522542
$530$	0	0
$531$	0	0
$532$	− 11.1625i	− 0.483954i
$533$	6.87448i	0.297767i
$534$	0	0
$535$	0	0
$536$	13.8454	0.598032
$537$	0	0
$538$	− 16.3440i	− 0.704638i
$539$	13.2593	0.571120
$540$	0	0
$541$	5.20957	0.223977	0.111988	−	0.993710i	$-0.464278\pi$
$541$	5.20957	0.223977	0.111988	+	0.993710i	$0.464278\pi$
$542$	− 13.3641i	− 0.574038i
$543$	0	0
$544$	−14.5081	−0.622030
$545$	0	0
$546$	0	0
$547$	40.0409i	1.71203i	0.516955	+	0.856013i	$0.327065\pi$
$547$	40.0409i	1.71203i	−0.516955	+	0.856013i	$0.672935\pi$
$548$	− 17.0656i	− 0.729005i
$549$	0	0
$550$	0	0
$551$	−44.3434	−1.88909
$552$	0	0
$553$	0.411227i	0.0174872i
$554$	8.62345	0.366375
$555$	0	0
$556$	−13.3826	−0.567548
$557$	− 14.4033i	− 0.610288i	−0.952306	−	0.305144i	$-0.901295\pi$
$557$	− 14.4033i	− 0.610288i	0.952306	−	0.305144i	$-0.0987047\pi$
$558$	0	0
$559$	−2.20166	−0.0931203
$560$	0	0
$561$	0	0
$562$	3.80495i	0.160502i
$563$	− 29.3681i	− 1.23772i	−0.785503	−	0.618858i	$-0.787595\pi$
$563$	− 29.3681i	− 1.23772i	0.785503	−	0.618858i	$-0.212405\pi$
$564$	0	0
$565$	0	0
$566$	15.3849	0.646674
$567$	0	0
$568$	− 27.0471i	− 1.13487i
$569$	−46.8066	−1.96224	−0.981118	−	0.193409i	$-0.938046\pi$
$569$	−46.8066	−1.96224	−0.981118	+	0.193409i	$0.938046\pi$
$570$	0	0
$571$	20.0000	0.836974	0.418487	−	0.908223i	$-0.362561\pi$
$571$	20.0000	0.836974	0.418487	+	0.908223i	$0.362561\pi$
$572$	− 20.8930i	− 0.873578i
$573$	0	0
$574$	1.20166	0.0501564
$575$	0	0
$576$	0	0
$577$	28.2386i	1.17559i	0.809010	+	0.587794i	$0.200004\pi$
$577$	28.2386i	1.17559i	−0.809010	+	0.587794i	$0.799996\pi$
$578$	5.63757i	0.234492i
$579$	0	0
$580$	0	0
$581$	−6.11761	−0.253801
$582$	0	0
$583$	3.05767i	0.126636i
$584$	−3.59668	−0.148832
$585$	0	0
$586$	7.08405	0.292639
$587$	18.0824i	0.746339i	0.927763	+	0.373169i	$0.121729\pi$
$587$	18.0824i	0.746339i	−0.927763	+	0.373169i	$0.878271\pi$
$588$	0	0
$589$	−32.5266	−1.34023
$590$	0	0
$591$	0	0
$592$	− 3.89522i	− 0.160092i
$593$	7.73840i	0.317778i	0.987296	+	0.158889i	$0.0507912\pi$
$593$	7.73840i	0.317778i	−0.987296	+	0.158889i	$0.949209\pi$
$594$	0	0
$595$	0	0
$596$	33.5839	1.37565
$597$	0	0
$598$	15.8168i	0.646797i
$599$	27.9216	1.14085	0.570423	−	0.821351i	$-0.306779\pi$
$599$	27.9216	1.14085	0.570423	+	0.821351i	$0.306779\pi$
$600$	0	0
$601$	38.4403	1.56801	0.784006	−	0.620754i	$-0.213173\pi$
$601$	38.4403	1.56801	0.784006	+	0.620754i	$0.213173\pi$
$602$	0.384851i	0.0156853i
$603$	0	0
$604$	5.08405	0.206867
$605$	0	0
$606$	0	0
$607$	− 0.639834i	− 0.0259701i	−0.999916	−	0.0129850i	$-0.995867\pi$
$607$	− 0.639834i	− 0.0259701i	0.999916	−	0.0129850i	$-0.00413338\pi$
$608$	− 25.3641i	− 1.02865i
$609$	0	0
$610$	0	0
$611$	32.5081	1.31514
$612$	0	0
$613$	− 42.7467i	− 1.72652i	−0.504757	−	0.863262i	$-0.668418\pi$
$613$	− 42.7467i	− 1.72652i	0.504757	−	0.863262i	$-0.331582\pi$
$614$	1.42631	0.0575614
$615$	0	0
$616$	−8.01847	−0.323073
$617$	21.1025i	0.849556i	0.905298	+	0.424778i	$0.139648\pi$
$617$	21.1025i	0.849556i	−0.905298	+	0.424778i	$0.860352\pi$
$618$	0	0
$619$	13.6521	0.548724	0.274362	−	0.961626i	$-0.411533\pi$
$619$	13.6521	0.548724	0.274362	+	0.961626i	$0.411533\pi$
$620$	0	0
$621$	0	0
$622$	− 13.8538i	− 0.555485i
$623$	4.28402i	0.171636i
$624$	0	0
$625$	0	0
$626$	−20.0678	−0.802072
$627$	0	0
$628$	0.337343i	0.0134615i
$629$	4.85601	0.193622
$630$	0	0
$631$	33.2593	1.32403	0.662017	−	0.749489i	$-0.269701\pi$
$631$	33.2593	1.32403	0.662017	+	0.749489i	$0.269701\pi$
$632$	0.604983i	0.0240649i
$633$	0	0
$634$	5.98943	0.237871
$635$	0	0
$636$	0	0
$637$	− 23.1809i	− 0.918462i
$638$	14.5081i	0.574381i
$639$	0	0
$640$	0	0
$641$	26.2857	1.03822	0.519112	−	0.854706i	$-0.326263\pi$
$641$	26.2857	1.03822	0.519112	+	0.854706i	$0.326263\pi$
$642$	0	0
$643$	− 20.5826i	− 0.811697i	−0.913940	−	0.405848i	$-0.866976\pi$
$643$	− 20.5826i	− 0.811697i	0.913940	−	0.405848i	$-0.133024\pi$
$644$	−14.1361	−0.557040
$645$	0	0
$646$	7.14399	0.281076
$647$	23.2527i	0.914159i	0.889426	+	0.457079i	$0.151104\pi$
$647$	23.2527i	0.914159i	−0.889426	+	0.457079i	$0.848896\pi$
$648$	0	0
$649$	3.05767	0.120024
$650$	0	0
$651$	0	0
$652$	− 29.8044i	− 1.16723i
$653$	− 18.7591i	− 0.734102i	−0.930201	−	0.367051i	$-0.880367\pi$
$653$	− 18.7591i	− 0.734102i	0.930201	−	0.367051i	$-0.119633\pi$
$654$	0	0
$655$	0	0
$656$	−3.15415	−0.123149
$657$	0	0
$658$	− 5.68242i	− 0.221524i
$659$	0.280067	0.0109099	0.00545494	−	0.999985i	$-0.498264\pi$
$659$	0.280067	0.0109099	0.00545494	+	0.999985i	$0.498264\pi$
$660$	0	0
$661$	−39.7859	−1.54749	−0.773746	−	0.633496i	$-0.781619\pi$
$661$	−39.7859	−1.54749	−0.773746	+	0.633496i	$0.781619\pi$
$662$	9.59668i	0.372985i
$663$	0	0
$664$	−9.00000	−0.349268
$665$	0	0
$666$	0	0
$667$	56.1562i	2.17438i
$668$	23.5882i	0.912655i
$669$	0	0
$670$	0	0
$671$	−6.75914	−0.260934
$672$	0	0
$673$	− 33.5288i	− 1.29244i	−0.763150	−	0.646221i	$-0.776348\pi$
$673$	− 33.5288i	− 1.29244i	0.763150	−	0.646221i	$-0.223652\pi$
$674$	15.5473	0.598860
$675$	0	0
$676$	−14.7799	−0.568456
$677$	27.4874i	1.05643i	0.849112	+	0.528213i	$0.177138\pi$
$677$	27.4874i	1.05643i	−0.849112	+	0.528213i	$0.822862\pi$
$678$	0	0
$679$	−11.1888	−0.429388
$680$	0	0
$681$	0	0
$682$	10.6419i	0.407500i
$683$	34.5865i	1.32342i	0.749762	+	0.661708i	$0.230168\pi$
$683$	34.5865i	1.32342i	−0.749762	+	0.661708i	$0.769832\pi$
$684$	0	0
$685$	0	0
$686$	−9.76970	−0.373009
$687$	0	0
$688$	− 1.01017i	− 0.0385122i
$689$	5.34565	0.203653
$690$	0	0
$691$	40.7282	1.54938	0.774688	−	0.632344i	$-0.217907\pi$
$691$	40.7282	1.54938	0.774688	+	0.632344i	$0.217907\pi$
$692$	− 7.30418i	− 0.277663i
$693$	0	0
$694$	13.4795	0.511674
$695$	0	0
$696$	0	0
$697$	− 3.93216i	− 0.148941i
$698$	6.12721i	0.231918i
$699$	0	0
$700$	0	0
$701$	−19.4712	−0.735416	−0.367708	−	0.929941i	$-0.619857\pi$
$701$	−19.4712	−0.735416	−0.367708	+	0.929941i	$0.619857\pi$
$702$	0	0
$703$	8.48963i	0.320193i
$704$	3.16246	0.119190
$705$	0	0
$706$	15.5967	0.586989
$707$	− 6.00000i	− 0.225653i
$708$	0	0
$709$	−15.0863	−0.566578	−0.283289	−	0.959035i	$-0.591426\pi$
$709$	−15.0863	−0.566578	−0.283289	+	0.959035i	$0.591426\pi$
$710$	0	0
$711$	0	0
$712$	6.30249i	0.236196i
$713$	41.1915i	1.54263i
$714$	0	0
$715$	0	0
$716$	25.3641	0.947902
$717$	0	0
$718$	6.10931i	0.227997i
$719$	3.43196	0.127990	0.0639952	−	0.997950i	$-0.479616\pi$
$719$	3.43196	0.127990	0.0639952	+	0.997950i	$0.479616\pi$
$720$	0	0
$721$	−2.59442	−0.0966211
$722$	1.62176i	0.0603557i
$723$	0	0
$724$	5.35581	0.199047
$725$	0	0
$726$	0	0
$727$	− 35.7714i	− 1.32669i	−0.748315	−	0.663344i	$-0.769137\pi$
$727$	− 35.7714i	− 1.32669i	0.748315	−	0.663344i	$-0.230863\pi$
$728$	14.0185i	0.519559i
$729$	0	0
$730$	0	0
$731$	1.25934	0.0465782
$732$	0	0
$733$	22.0000i	0.812589i	0.913742	+	0.406294i	$0.133179\pi$
$733$	22.0000i	0.812589i	−0.913742	+	0.406294i	$0.866821\pi$
$734$	4.84545	0.178849
$735$	0	0
$736$	−32.1210	−1.18400
$737$	− 17.6151i	− 0.648862i
$738$	0	0
$739$	−6.08631	−0.223889	−0.111944	−	0.993714i	$-0.535708\pi$
$739$	−6.08631	−0.223889	−0.111944	+	0.993714i	$0.535708\pi$
$740$	0	0
$741$	0	0
$742$	− 0.934420i	− 0.0343036i
$743$	25.5019i	0.935574i	0.883841	+	0.467787i	$0.154949\pi$
$743$	25.5019i	0.935574i	−0.883841	+	0.467787i	$0.845051\pi$
$744$	0	0
$745$	0	0
$746$	5.79043	0.212003
$747$	0	0
$748$	11.9506i	0.436958i
$749$	−17.0185	−0.621841
$750$	0	0
$751$	−18.3928	−0.671161	−0.335581	−	0.942011i	$-0.608932\pi$
$751$	−18.3928	−0.671161	−0.335581	+	0.942011i	$0.608932\pi$
$752$	14.9154i	0.543908i
$753$	0	0
$754$	25.3641	0.923707
$755$	0	0
$756$	0	0
$757$	− 41.8986i	− 1.52283i	−0.648264	−	0.761415i	$-0.724505\pi$
$757$	− 41.8986i	− 1.52283i	0.648264	−	0.761415i	$-0.275495\pi$
$758$	− 6.81907i	− 0.247680i
$759$	0	0
$760$	0	0
$761$	−7.97136	−0.288962	−0.144481	−	0.989508i	$-0.546151\pi$
$761$	−7.97136	−0.288962	−0.144481	+	0.989508i	$0.546151\pi$
$762$	0	0
$763$	23.7793i	0.860868i
$764$	−4.74671	−0.171730
$765$	0	0
$766$	5.61515	0.202884
$767$	− 5.34565i	− 0.193020i
$768$	0	0
$769$	−6.02864	−0.217398	−0.108699	−	0.994075i	$-0.534669\pi$
$769$	−6.02864	−0.217398	−0.108699	+	0.994075i	$0.534669\pi$
$770$	0	0
$771$	0	0
$772$	31.4293i	1.13117i
$773$	− 44.4033i	− 1.59708i	−0.601944	−	0.798538i	$-0.705607\pi$
$773$	− 44.4033i	− 1.59708i	0.601944	−	0.798538i	$-0.294393\pi$
$774$	0	0
$775$	0	0
$776$	−16.4606	−0.590901
$777$	0	0
$778$	− 5.27385i	− 0.189077i
$779$	6.87448	0.246304
$780$	0	0
$781$	−34.4112	−1.23133
$782$	− 9.04711i	− 0.323524i
$783$	0	0
$784$	10.6359	0.379853
$785$	0	0
$786$	0	0
$787$	16.8375i	0.600194i	0.953909	+	0.300097i	$0.0970190\pi$
$787$	16.8375i	0.600194i	−0.953909	+	0.300097i	$0.902981\pi$
$788$	9.76140i	0.347735i
$789$	0	0
$790$	0	0
$791$	28.7361	1.02174
$792$	0	0
$793$	11.8168i	0.419627i
$794$	13.1440	0.466463
$795$	0	0
$796$	−21.8741	−0.775306
$797$	− 33.3720i	− 1.18210i	−0.806636	−	0.591049i	$-0.798714\pi$
$797$	− 33.3720i	− 1.18210i	0.806636	−	0.591049i	$-0.201286\pi$
$798$	0	0
$799$	−18.5944	−0.657823
$800$	0	0
$801$	0	0
$802$	− 6.32944i	− 0.223500i
$803$	4.57595i	0.161482i
$804$	0	0
$805$	0	0
$806$	18.6050	0.655333
$807$	0	0
$808$	− 8.82698i	− 0.310532i
$809$	−29.1809	−1.02595	−0.512973	−	0.858404i	$-0.671456\pi$
$809$	−29.1809	−1.02595	−0.512973	+	0.858404i	$0.671456\pi$
$810$	0	0
$811$	−15.5552	−0.546217	−0.273109	−	0.961983i	$-0.588052\pi$
$811$	−15.5552	−0.546217	−0.273109	+	0.961983i	$0.588052\pi$
$812$	22.6689i	0.795521i
$813$	0	0
$814$	2.77761	0.0973551
$815$	0	0
$816$	0	0
$817$	2.20166i	0.0770264i
$818$	10.0863i	0.352660i
$819$	0	0
$820$	0	0
$821$	−23.7177	−0.827752	−0.413876	−	0.910333i	$-0.635825\pi$
$821$	−23.7177	−0.827752	−0.413876	+	0.910333i	$0.635825\pi$
$822$	0	0
$823$	− 19.3681i	− 0.675129i	−0.941302	−	0.337564i	$-0.890397\pi$
$823$	− 19.3681i	− 0.675129i	0.941302	−	0.337564i	$-0.109603\pi$
$824$	−3.81681	−0.132965
$825$	0	0
$826$	−0.934420	−0.0325126
$827$	52.2241i	1.81601i	0.418960	+	0.908005i	$0.362395\pi$
$827$	52.2241i	1.81601i	−0.418960	+	0.908005i	$0.637605\pi$
$828$	0	0
$829$	26.6442	0.925391	0.462695	−	0.886517i	$-0.346882\pi$
$829$	26.6442	0.925391	0.462695	+	0.886517i	$0.346882\pi$
$830$	0	0
$831$	0	0
$832$	− 5.52884i	− 0.191678i
$833$	13.2593i	0.459409i
$834$	0	0
$835$	0	0
$836$	−20.8930	−0.722598
$837$	0	0
$838$	− 21.1836i	− 0.731775i
$839$	25.0392	0.864449	0.432225	−	0.901766i	$-0.357729\pi$
$839$	25.0392	0.864449	0.432225	+	0.901766i	$0.357729\pi$
$840$	0	0
$841$	61.0532	2.10528
$842$	2.89296i	0.0996978i
$843$	0	0
$844$	14.0264	0.482808
$845$	0	0
$846$	0	0
$847$	− 5.50641i	− 0.189203i
$848$	2.45269i	0.0842258i
$849$	0	0
$850$	0	0
$851$	10.7512	0.368547
$852$	0	0
$853$	10.8745i	0.372335i	0.982518	+	0.186168i	$0.0596068\pi$
$853$	10.8745i	0.372335i	−0.982518	+	0.186168i	$0.940393\pi$
$854$	2.06558	0.0706827
$855$	0	0
$856$	−25.0369	−0.855745
$857$	− 18.5944i	− 0.635173i	−0.948229	−	0.317587i	$-0.897128\pi$
$857$	− 18.5944i	− 0.635173i	0.948229	−	0.317587i	$-0.102872\pi$
$858$	0	0
$859$	4.66492	0.159165	0.0795825	−	0.996828i	$-0.474641\pi$
$859$	4.66492	0.159165	0.0795825	+	0.996828i	$0.474641\pi$
$860$	0	0
$861$	0	0
$862$	− 2.99170i	− 0.101898i
$863$	28.0594i	0.955152i	0.878590	+	0.477576i	$0.158484\pi$
$863$	28.0594i	0.955152i	−0.878590	+	0.477576i	$0.841516\pi$
$864$	0	0
$865$	0	0
$866$	19.6442	0.667537
$867$	0	0
$868$	16.6280i	0.564390i
$869$	0.769701	0.0261103
$870$	0	0
$871$	−30.7961	−1.04349
$872$	34.9832i	1.18468i
$873$	0	0
$874$	15.8168	0.535012
$875$	0	0
$876$	0	0
$877$	34.6683i	1.17067i	0.810793	+	0.585333i	$0.199036\pi$
$877$	34.6683i	1.17067i	−0.810793	+	0.585333i	$0.800964\pi$
$878$	11.1809i	0.377338i
$879$	0	0
$880$	0	0
$881$	−5.29854	−0.178512	−0.0892561	−	0.996009i	$-0.528449\pi$
$881$	−5.29854	−0.178512	−0.0892561	+	0.996009i	$0.528449\pi$
$882$	0	0
$883$	10.3025i	0.346706i	0.984860	+	0.173353i	$0.0554602\pi$
$883$	10.3025i	0.346706i	−0.984860	+	0.173353i	$0.944540\pi$
$884$	20.8930	0.702706
$885$	0	0
$886$	6.00830	0.201853
$887$	41.4504i	1.39177i	0.718154	+	0.695885i	$0.244988\pi$
$887$	41.4504i	1.39177i	−0.718154	+	0.695885i	$0.755012\pi$
$888$	0	0
$889$	−3.12325	−0.104751
$890$	0	0
$891$	0	0
$892$	− 15.3338i	− 0.513413i
$893$	− 32.5081i	− 1.08784i
$894$	0	0
$895$	0	0
$896$	−16.4689	−0.550187
$897$	0	0
$898$	− 13.0735i	− 0.436268i
$899$	66.0554	2.20307
$900$	0	0
$901$	−3.05767	−0.101866
$902$	− 2.24917i	− 0.0748891i
$903$	0	0
$904$	42.2755	1.40606
$905$	0	0
$906$	0	0
$907$	− 16.7921i	− 0.557573i	−0.960353	−	0.278787i	$-0.910068\pi$
$907$	− 16.7921i	− 0.557573i	0.960353	−	0.278787i	$-0.0899322\pi$
$908$	− 4.47116i	− 0.148381i
$909$	0	0
$910$	0	0
$911$	37.3536	1.23758	0.618789	−	0.785557i	$-0.287623\pi$
$911$	37.3536	1.23758	0.618789	+	0.785557i	$0.287623\pi$
$912$	0	0
$913$	11.4504i	0.378954i
$914$	8.02299	0.265377
$915$	0	0
$916$	−4.26120	−0.140794
$917$	− 8.56804i	− 0.282942i
$918$	0	0
$919$	−37.1316	−1.22486	−0.612429	−	0.790526i	$-0.709807\pi$
$919$	−37.1316	−1.22486	−0.612429	+	0.790526i	$0.709807\pi$
$920$	0	0
$921$	0	0
$922$	− 13.7865i	− 0.454034i
$923$	60.1602i	1.98020i
$924$	0	0
$925$	0	0
$926$	−19.4135	−0.637967
$927$	0	0
$928$	51.5098i	1.69089i
$929$	47.9955	1.57468	0.787340	−	0.616519i	$-0.211458\pi$
$929$	47.9955	1.57468	0.787340	+	0.616519i	$0.211458\pi$
$930$	0	0
$931$	−23.1809	−0.759724
$932$	10.4052i	0.340833i
$933$	0	0
$934$	15.6566	0.512300
$935$	0	0
$936$	0	0
$937$	22.0079i	0.718967i	0.933151	+	0.359483i	$0.117047\pi$
$937$	22.0079i	0.718967i	−0.933151	+	0.359483i	$0.882953\pi$
$938$	5.38316i	0.175766i
$939$	0	0
$940$	0	0
$941$	40.5843	1.32301	0.661504	−	0.749941i	$-0.269918\pi$
$941$	40.5843	1.32301	0.661504	+	0.749941i	$0.269918\pi$
$942$	0	0
$943$	− 8.70581i	− 0.283500i
$944$	2.45269	0.0798283
$945$	0	0
$946$	0.720331	0.0234200
$947$	30.2320i	0.982408i	0.871045	+	0.491204i	$0.163443\pi$
$947$	30.2320i	0.982408i	−0.871045	+	0.491204i	$0.836557\pi$
$948$	0	0
$949$	8.00000	0.259691
$950$	0	0
$951$	0	0
$952$	− 8.01847i	− 0.259880i
$953$	− 2.50811i	− 0.0812455i	−0.999175	−	0.0406227i	$-0.987066\pi$
$953$	− 2.50811i	− 0.0812455i	0.999175	−	0.0406227i	$-0.0129342\pi$
$954$	0	0
$955$	0	0
$956$	−13.5888	−0.439492
$957$	0	0
$958$	− 2.99170i	− 0.0966573i
$959$	14.5680	0.470427
$960$	0	0
$961$	17.4527	0.562990
$962$	− 4.85601i	− 0.156564i
$963$	0	0
$964$	−44.1025	−1.42045
$965$	0	0
$966$	0	0
$967$	− 21.7529i	− 0.699527i	−0.936838	−	0.349763i	$-0.886262\pi$
$967$	− 21.7529i	− 0.699527i	0.936838	−	0.349763i	$-0.113738\pi$
$968$	− 8.10083i	− 0.260371i
$969$	0	0
$970$	0	0
$971$	22.7512	0.730122	0.365061	−	0.930984i	$-0.381048\pi$
$971$	22.7512	0.730122	0.365061	+	0.930984i	$0.381048\pi$
$972$	0	0
$973$	− 11.4241i	− 0.366238i
$974$	13.7384	0.440207
$975$	0	0
$976$	−5.42179	−0.173547
$977$	39.2778i	1.25661i	0.777968	+	0.628304i	$0.216251\pi$
$977$	39.2778i	1.25661i	−0.777968	+	0.628304i	$0.783749\pi$
$978$	0	0
$979$	8.01847	0.256271
$980$	0	0
$981$	0	0
$982$	8.45269i	0.269736i
$983$	− 33.7899i	− 1.07773i	−0.842392	−	0.538865i	$-0.818853\pi$
$983$	− 33.7899i	− 1.07773i	0.842392	−	0.538865i	$-0.181147\pi$
$984$	0	0
$985$	0	0
$986$	−14.5081	−0.462032
$987$	0	0
$988$	36.5266i	1.16207i
$989$	2.78817	0.0886587
$990$	0	0
$991$	−23.7983	−0.755979	−0.377990	−	0.925810i	$-0.623384\pi$
$991$	−23.7983	−0.755979	−0.377990	+	0.925810i	$0.623384\pi$
$992$	37.7833i	1.19962i
$993$	0	0
$994$	10.5160	0.333548
$995$	0	0
$996$	0	0
$997$	51.6785i	1.63667i	0.574739	+	0.818337i	$0.305104\pi$
$997$	51.6785i	1.63667i	−0.574739	+	0.818337i	$0.694896\pi$
$998$	− 14.2175i	− 0.450046i
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	2025.2.b.l.649.4		6
3.2	odd	2		2025.2.b.m.649.3		6
5.2	odd	4		2025.2.a.n.1.2		3
5.3	odd	4		405.2.a.j.1.2		3
5.4	even	2	inner	2025.2.b.l.649.3		6
9.2	odd	6		675.2.k.b.199.3		12
9.4	even	3		225.2.k.b.124.3		12
9.5	odd	6		675.2.k.b.424.4		12
9.7	even	3		225.2.k.b.49.4		12
15.2	even	4		2025.2.a.o.1.2		3
15.8	even	4		405.2.a.i.1.2		3
15.14	odd	2		2025.2.b.m.649.4		6
20.3	even	4		6480.2.a.bv.1.2		3
45.2	even	12		675.2.e.b.226.2		6
45.4	even	6		225.2.k.b.124.4		12
45.7	odd	12		225.2.e.b.76.2		6
45.13	odd	12		45.2.e.b.16.2	✓	6
45.14	odd	6		675.2.k.b.424.3		12
45.22	odd	12		225.2.e.b.151.2		6
45.23	even	12		135.2.e.b.46.2		6
45.29	odd	6		675.2.k.b.199.4		12
45.32	even	12		675.2.e.b.451.2		6
45.34	even	6		225.2.k.b.49.3		12
45.38	even	12		135.2.e.b.91.2		6
45.43	odd	12		45.2.e.b.31.2	yes	6
60.23	odd	4		6480.2.a.bs.1.2		3
180.23	odd	12		2160.2.q.k.721.2		6
180.43	even	12		720.2.q.i.481.2		6
180.83	odd	12		2160.2.q.k.1441.2		6
180.103	even	12		720.2.q.i.241.2		6

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
45.2.e.b.16.2	✓	6	45.13	odd	12
45.2.e.b.31.2	yes	6	45.43	odd	12
135.2.e.b.46.2		6	45.23	even	12
135.2.e.b.91.2		6	45.38	even	12
225.2.e.b.76.2		6	45.7	odd	12
225.2.e.b.151.2		6	45.22	odd	12
225.2.k.b.49.3		12	45.34	even	6
225.2.k.b.49.4		12	9.7	even	3
225.2.k.b.124.3		12	9.4	even	3
225.2.k.b.124.4		12	45.4	even	6
405.2.a.i.1.2		3	15.8	even	4
405.2.a.j.1.2		3	5.3	odd	4
675.2.e.b.226.2		6	45.2	even	12
675.2.e.b.451.2		6	45.32	even	12
675.2.k.b.199.3		12	9.2	odd	6
675.2.k.b.199.4		12	45.29	odd	6
675.2.k.b.424.3		12	45.14	odd	6
675.2.k.b.424.4		12	9.5	odd	6
720.2.q.i.241.2		6	180.103	even	12
720.2.q.i.481.2		6	180.43	even	12
2025.2.a.n.1.2		3	5.2	odd	4
2025.2.a.o.1.2		3	15.2	even	4
2025.2.b.l.649.3		6	5.4	even	2	inner
2025.2.b.l.649.4		6	1.1	even	1	trivial
2025.2.b.m.649.3		6	3.2	odd	2
2025.2.b.m.649.4		6	15.14	odd	2
2160.2.q.k.721.2		6	180.23	odd	12
2160.2.q.k.1441.2		6	180.83	odd	12
6480.2.a.bs.1.2		3	60.23	odd	4
6480.2.a.bv.1.2		3	20.3	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:	\( N \)	\(=\)	\( 2025 = 3^{4} \cdot 5^{2} \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	2025.b (of order \(2\), degree \(1\), not minimal)

\(f(q)\)	\(=\)	\(q+0.571993i q^{2} +1.67282 q^{4} +1.42801i q^{7} +2.10083i q^{8} +O(q^{10})\)	\(q+0.571993i q^{2} +1.67282 q^{4} +1.42801i q^{7} +2.10083i q^{8} +2.67282 q^{11} -4.67282i q^{13} -0.816810 q^{14} +2.14399 q^{16} +2.67282i q^{17} -4.67282 q^{19} +1.52884i q^{22} +5.91764i q^{23} +2.67282 q^{26} +2.38880i q^{28} +9.48963 q^{29} +6.96080 q^{31} +5.42801i q^{32} -1.52884 q^{34} -1.81681i q^{37} -2.67282i q^{38} -1.47116 q^{41} -0.471163i q^{43} +4.47116 q^{44} -3.38485 q^{46} +6.95684i q^{47} +4.96080 q^{49} -7.81681i q^{52} +1.14399i q^{53} -3.00000 q^{56} +5.42801i q^{58} +1.14399 q^{59} -2.52884 q^{61} +3.98153i q^{62} +1.18319 q^{64} -6.59046i q^{67} +4.47116i q^{68} -12.8745 q^{71} +1.71203i q^{73} +1.03920 q^{74} -7.81681 q^{76} +3.81681i q^{77} +0.287973 q^{79} -0.841495i q^{82} +4.28402i q^{83} +0.269502 q^{86} +5.61515i q^{88} +3.00000 q^{89} +6.67282 q^{91} +9.89917i q^{92} -3.97927 q^{94} +7.83528i q^{97} +2.83754i q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 6 q - 10 q^{4}+O(q^{10}) \) 6 * q - 10 * q^4	\( 6 q - 10 q^{4} - 4 q^{11} + 18 q^{14} + 10 q^{16} - 8 q^{19} - 4 q^{26} + 14 q^{29} + 16 q^{31} + 8 q^{34} - 26 q^{41} + 44 q^{44} - 6 q^{46} + 4 q^{49} - 18 q^{56} + 4 q^{59} + 2 q^{61} + 30 q^{64} - 20 q^{71} + 32 q^{74} - 24 q^{76} - 4 q^{79} + 56 q^{86} + 18 q^{89} + 20 q^{91} + 62 q^{94}+O(q^{100}) \) 6 * q - 10 * q^4 - 4 * q^11 + 18 * q^14 + 10 * q^16 - 8 * q^19 - 4 * q^26 + 14 * q^29 + 16 * q^31 + 8 * q^34 - 26 * q^41 + 44 * q^44 - 6 * q^46 + 4 * q^49 - 18 * q^56 + 4 * q^59 + 2 * q^61 + 30 * q^64 - 20 * q^71 + 32 * q^74 - 24 * q^76 - 4 * q^79 + 56 * q^86 + 18 * q^89 + 20 * q^91 + 62 * q^94

Introduction

L-functions

Modular forms

Varieties

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Representations

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Database

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