LMFDB - Embedded newform 2025.2.b.m.649.5

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 = [6,0,0,-10,0,0,0,0,0,0,4] 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:	$6$
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			649.5
Root			$0.675970 + 0.675970i$ of defining polynomial
Character	$\chi$	$=$	2025.649
Dual form			2025.2.b.m.649.2

$q$-expansion

comment:q-expansion

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

gp:mfcoefs(f, 20)

$f(q)$	$=$	$q+2.08613i q^{2} -2.35194 q^{4} +4.08613i q^{7} -0.734191i q^{8} +1.35194 q^{11} -0.648061i q^{13} -8.52420 q^{14} -3.17226 q^{16} +1.35194i q^{17} -0.648061 q^{19} +2.82032i q^{22} +4.79001i q^{23} +1.35194 q^{26} -9.61033i q^{28} +3.87614 q^{29} -7.69646 q^{31} -8.08613i q^{32} -2.82032 q^{34} +7.52420i q^{37} -1.35194i q^{38} +0.179679 q^{41} +0.820321i q^{43} -3.17968 q^{44} -9.99258 q^{46} -10.9065i q^{47} -9.69646 q^{49} +1.52420i q^{52} +4.17226i q^{53} +3.00000 q^{56} +8.08613i q^{58} +4.17226 q^{59} -3.82032 q^{61} -16.0558i q^{62} +10.5242 q^{64} +8.14195i q^{67} -3.17968i q^{68} +6.11644 q^{71} +12.3445i q^{73} -15.6965 q^{74} +1.52420 q^{76} +5.52420i q^{77} -10.3445 q^{79} +0.374833i q^{82} -12.2584i q^{83} -1.71130 q^{86} -0.992582i q^{88} -3.00000 q^{89} +2.64806 q^{91} -11.2658i q^{92} +22.7523 q^{94} -13.5800i q^{97} -20.2281i q^{98} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 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}+ \cdots + 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

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 $

$2$	2.08613i	1.47512i	0.675283	+	0.737558i	$0.264021\pi$
$2$	2.08613i	1.47512i	−0.675283	+	0.737558i	$0.735979\pi$
$3$	0	0
$3$	0	0
$4$	−2.35194	−1.17597
$5$	0	0
$5$	0	0
$6$	0	0
$7$	4.08613i	1.54441i	0.635372	+	0.772206i	$0.280847\pi$
$7$	4.08613i	1.54441i	−0.635372	+	0.772206i	$0.719153\pi$
$8$	− 0.734191i	− 0.259576i
$9$	0	0
$10$	0	0
$11$	1.35194	0.407625	0.203813	−	0.979010i	$-0.434667\pi$
$11$	1.35194	0.407625	0.203813	+	0.979010i	$0.434667\pi$
$12$	0	0
$13$	− 0.648061i	− 0.179740i	−0.995954	−	0.0898699i	$-0.971355\pi$
$13$	− 0.648061i	− 0.179740i	0.995954	−	0.0898699i	$-0.0286451\pi$
$14$	−8.52420	−2.27819
$15$	0	0
$16$	−3.17226	−0.793065
$17$	1.35194i	0.327893i	0.986469	+	0.163947i	$0.0524225\pi$
$17$	1.35194i	0.327893i	−0.986469	+	0.163947i	$0.947577\pi$
$18$	0	0
$19$	−0.648061	−0.148675	−0.0743377	−	0.997233i	$-0.523684\pi$
$19$	−0.648061	−0.148675	−0.0743377	+	0.997233i	$0.523684\pi$
$20$	0	0
$21$	0	0
$22$	2.82032i	0.601294i
$23$	4.79001i	0.998786i	0.866376	+	0.499393i	$0.166444\pi$
$23$	4.79001i	0.998786i	−0.866376	+	0.499393i	$0.833556\pi$
$24$	0	0
$25$	0	0
$26$	1.35194	0.265137
$27$	0	0
$28$	− 9.61033i	− 1.81618i
$29$	3.87614	0.719781	0.359890	−	0.932995i	$-0.382814\pi$
$29$	3.87614	0.719781	0.359890	+	0.932995i	$0.382814\pi$
$30$	0	0
$31$	−7.69646	−1.38233	−0.691163	−	0.722699i	$-0.742901\pi$
$31$	−7.69646	−1.38233	−0.691163	+	0.722699i	$0.742901\pi$
$32$	− 8.08613i	− 1.42944i
$33$	0	0
$34$	−2.82032	−0.483681
$35$	0	0
$36$	0	0
$37$	7.52420i	1.23697i	0.785796	+	0.618485i	$0.212253\pi$
$37$	7.52420i	1.23697i	−0.785796	+	0.618485i	$0.787747\pi$
$38$	− 1.35194i	− 0.219313i
$39$	0	0
$40$	0	0
$41$	0.179679	0.0280611	0.0140306	−	0.999902i	$-0.495534\pi$
$41$	0.179679	0.0280611	0.0140306	+	0.999902i	$0.495534\pi$
$42$	0	0
$43$	0.820321i	0.125098i	0.998042	+	0.0625489i	$0.0199229\pi$
$43$	0.820321i	0.125098i	−0.998042	+	0.0625489i	$0.980077\pi$
$44$	−3.17968	−0.479355
$45$	0	0
$46$	−9.99258	−1.47333
$47$	− 10.9065i	− 1.59087i	−0.606039	−	0.795435i	$-0.707243\pi$
$47$	− 10.9065i	− 1.59087i	0.606039	−	0.795435i	$-0.292757\pi$
$48$	0	0
$49$	−9.69646	−1.38521
$50$	0	0
$51$	0	0
$52$	1.52420i	0.211368i
$53$	4.17226i	0.573104i	0.958065	+	0.286552i	$0.0925091\pi$
$53$	4.17226i	0.573104i	−0.958065	+	0.286552i	$0.907491\pi$
$54$	0	0
$55$	0	0
$56$	3.00000	0.400892
$57$	0	0
$58$	8.08613i	1.06176i
$59$	4.17226	0.543182	0.271591	−	0.962413i	$-0.412450\pi$
$59$	4.17226	0.543182	0.271591	+	0.962413i	$0.412450\pi$
$60$	0	0
$61$	−3.82032	−0.489142	−0.244571	−	0.969631i	$-0.578647\pi$
$61$	−3.82032	−0.489142	−0.244571	+	0.969631i	$0.578647\pi$
$62$	− 16.0558i	− 2.03909i
$63$	0	0
$64$	10.5242	1.31552
$65$	0	0
$66$	0	0
$67$	8.14195i	0.994697i	0.867551	+	0.497349i	$0.165693\pi$
$67$	8.14195i	0.994697i	−0.867551	+	0.497349i	$0.834307\pi$
$68$	− 3.17968i	− 0.385593i
$69$	0	0
$70$	0	0
$71$	6.11644	0.725888	0.362944	−	0.931811i	$-0.381772\pi$
$71$	6.11644	0.725888	0.362944	+	0.931811i	$0.381772\pi$
$72$	0	0
$73$	12.3445i	1.44482i	0.691467	+	0.722408i	$0.256965\pi$
$73$	12.3445i	1.44482i	−0.691467	+	0.722408i	$0.743035\pi$
$74$	−15.6965	−1.82468
$75$	0	0
$76$	1.52420	0.174838
$77$	5.52420i	0.629541i
$78$	0	0
$79$	−10.3445	−1.16385	−0.581925	−	0.813243i	$-0.697700\pi$
$79$	−10.3445	−1.16385	−0.581925	+	0.813243i	$0.697700\pi$
$80$	0	0
$81$	0	0
$82$	0.374833i	0.0413934i
$83$	− 12.2584i	− 1.34553i	−0.739855	−	0.672767i	$-0.765106\pi$
$83$	− 12.2584i	− 1.34553i	0.739855	−	0.672767i	$-0.234894\pi$
$84$	0	0
$85$	0	0
$86$	−1.71130	−0.184534
$87$	0	0
$88$	− 0.992582i	− 0.105810i
$89$	−3.00000	−0.317999	−0.159000	−	0.987279i	$-0.550827\pi$
$89$	−3.00000	−0.317999	−0.159000	+	0.987279i	$0.550827\pi$
$90$	0	0
$91$	2.64806	0.277592
$92$	− 11.2658i	− 1.17454i
$93$	0	0
$94$	22.7523	2.34672
$95$	0	0
$96$	0	0
$97$	− 13.5800i	− 1.37884i	−0.724361	−	0.689421i	$-0.757865\pi$
$97$	− 13.5800i	− 1.37884i	0.724361	−	0.689421i	$-0.242135\pi$
$98$	− 20.2281i	− 2.04334i
$99$	0	0
$100$	0	0
$101$	1.46838	0.146109	0.0730547	−	0.997328i	$-0.476725\pi$
$101$	1.46838	0.146109	0.0730547	+	0.997328i	$0.476725\pi$
$102$	0	0
$103$	− 7.52420i	− 0.741381i	−0.928756	−	0.370691i	$-0.879121\pi$
$103$	− 7.52420i	− 0.741381i	0.928756	−	0.370691i	$-0.120879\pi$
$104$	−0.475800	−0.0466561
$105$	0	0
$106$	−8.70388	−0.845395
$107$	− 1.20999i	− 0.116974i	−0.998288	−	0.0584871i	$-0.981372\pi$
$107$	− 1.20999i	− 0.116974i	0.998288	−	0.0584871i	$-0.0186277\pi$
$108$	0	0
$109$	−14.1042	−1.35094	−0.675469	−	0.737388i	$-0.736059\pi$
$109$	−14.1042	−1.35094	−0.675469	+	0.737388i	$0.736059\pi$
$110$	0	0
$111$	0	0
$112$	− 12.9623i	− 1.22482i
$113$	− 11.9245i	− 1.12177i	−0.827895	−	0.560883i	$-0.810462\pi$
$113$	− 11.9245i	− 1.12177i	0.827895	−	0.560883i	$-0.189538\pi$
$114$	0	0
$115$	0	0
$116$	−9.11644	−0.846440
$117$	0	0
$118$	8.70388i	0.801257i
$119$	−5.52420	−0.506403
$120$	0	0
$121$	−9.17226	−0.833842
$122$	− 7.96969i	− 0.721542i
$123$	0	0
$124$	18.1016	1.62557
$125$	0	0
$126$	0	0
$127$	− 7.07871i	− 0.628134i	−0.949401	−	0.314067i	$-0.898308\pi$
$127$	− 7.07871i	− 0.628134i	0.949401	−	0.314067i	$-0.101692\pi$
$128$	5.78259i	0.511114i
$129$	0	0
$130$	0	0
$131$	6.00000	0.524222	0.262111	−	0.965038i	$-0.415581\pi$
$131$	6.00000	0.524222	0.262111	+	0.965038i	$0.415581\pi$
$132$	0	0
$133$	− 2.64806i	− 0.229616i
$134$	−16.9852	−1.46729
$135$	0	0
$136$	0.992582	0.0851132
$137$	7.46838i	0.638067i	0.947743	+	0.319033i	$0.103358\pi$
$137$	7.46838i	0.638067i	−0.947743	+	0.319033i	$0.896642\pi$
$138$	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
$142$	12.7597i	1.07077i
$143$	− 0.876139i	− 0.0732664i
$144$	0	0
$145$	0	0
$146$	−25.7523	−2.13127
$147$	0	0
$148$	− 17.6965i	− 1.45464i
$149$	−10.5848	−0.867143	−0.433571	−	0.901119i	$-0.642747\pi$
$149$	−10.5848	−0.867143	−0.433571	+	0.901119i	$0.642747\pi$
$150$	0	0
$151$	17.6965	1.44012	0.720059	−	0.693913i	$-0.244115\pi$
$151$	17.6965	1.44012	0.720059	+	0.693913i	$0.244115\pi$
$152$	0.475800i	0.0385925i
$153$	0	0
$154$	−11.5242	−0.928646
$155$	0	0
$156$	0	0
$157$	− 2.53162i	− 0.202045i	−0.994884	−	0.101023i	$-0.967789\pi$
$157$	− 2.53162i	− 0.202045i	0.994884	−	0.101023i	$-0.0322114\pi$
$158$	− 21.5800i	− 1.71681i
$159$	0	0
$160$	0	0
$161$	−19.5726	−1.54254
$162$	0	0
$163$	− 8.47580i	− 0.663876i	−0.943301	−	0.331938i	$-0.892298\pi$
$163$	− 8.47580i	− 0.663876i	0.943301	−	0.331938i	$-0.107702\pi$
$164$	−0.422594	−0.0329990
$165$	0	0
$166$	25.5726	1.98482
$167$	− 12.7342i	− 0.985401i	−0.870199	−	0.492701i	$-0.836010\pi$
$167$	− 12.7342i	− 0.985401i	0.870199	−	0.492701i	$-0.163990\pi$
$168$	0	0
$169$	12.5800	0.967694
$170$	0	0
$171$	0	0
$172$	− 1.92935i	− 0.147111i
$173$	23.0484i	1.75234i	0.482005	+	0.876169i	$0.339909\pi$
$173$	23.0484i	1.75234i	−0.482005	+	0.876169i	$0.660091\pi$
$174$	0	0
$175$	0	0
$176$	−4.28870	−0.323273
$177$	0	0
$178$	− 6.25839i	− 0.469086i
$179$	2.22808	0.166534	0.0832672	−	0.996527i	$-0.473465\pi$
$179$	2.22808	0.166534	0.0832672	+	0.996527i	$0.473465\pi$
$180$	0	0
$181$	0.468382	0.0348146	0.0174073	−	0.999848i	$-0.494459\pi$
$181$	0.468382	0.0348146	0.0174073	+	0.999848i	$0.494459\pi$
$182$	5.52420i	0.409481i
$183$	0	0
$184$	3.51678	0.259261
$185$	0	0
$186$	0	0
$187$	1.82774i	0.133658i
$188$	25.6513i	1.87081i
$189$	0	0
$190$	0	0
$191$	20.2281	1.46365	0.731826	−	0.681491i	$-0.238668\pi$
$191$	20.2281	1.46365	0.731826	+	0.681491i	$0.238668\pi$
$192$	0	0
$193$	19.9293i	1.43455i	0.696792	+	0.717273i	$0.254610\pi$
$193$	19.9293i	1.43455i	−0.696792	+	0.717273i	$0.745390\pi$
$194$	28.3297	2.03395
$195$	0	0
$196$	22.8055	1.62896
$197$	15.5800i	1.11003i	0.831840	+	0.555015i	$0.187288\pi$
$197$	15.5800i	1.11003i	−0.831840	+	0.555015i	$0.812712\pi$
$198$	0	0
$199$	−3.58482	−0.254121	−0.127061	−	0.991895i	$-0.540554\pi$
$199$	−3.58482	−0.254121	−0.127061	+	0.991895i	$0.540554\pi$
$200$	0	0
$201$	0	0
$202$	3.06324i	0.215529i
$203$	15.8384i	1.11164i
$204$	0	0
$205$	0	0
$206$	15.6965	1.09362
$207$	0	0
$208$	2.05582i	0.142545i
$209$	−0.876139	−0.0606038
$210$	0	0
$211$	14.9926	1.03213	0.516066	−	0.856549i	$-0.327396\pi$
$211$	14.9926	1.03213	0.516066	+	0.856549i	$0.327396\pi$
$212$	− 9.81290i	− 0.673953i
$213$	0	0
$214$	2.52420	0.172551
$215$	0	0
$216$	0	0
$217$	− 31.4487i	− 2.13488i
$218$	− 29.4232i	− 1.99279i
$219$	0	0
$220$	0	0
$221$	0.876139	0.0589355
$222$	0	0
$223$	26.8310i	1.79674i	0.439244	+	0.898368i	$0.355246\pi$
$223$	26.8310i	1.79674i	−0.439244	+	0.898368i	$0.644754\pi$
$224$	33.0410	2.20764
$225$	0	0
$226$	24.8761	1.65474
$227$	− 1.35194i	− 0.0897314i	−0.998993	−	0.0448657i	$-0.985714\pi$
$227$	− 1.35194i	− 0.0897314i	0.998993	−	0.0448657i	$-0.0142860\pi$
$228$	0	0
$229$	8.23550	0.544217	0.272108	−	0.962267i	$-0.412279\pi$
$229$	8.23550	0.544217	0.272108	+	0.962267i	$0.412279\pi$
$230$	0	0
$231$	0	0
$232$	− 2.84583i	− 0.186838i
$233$	8.58744i	0.562582i	0.959623	+	0.281291i	$0.0907626\pi$
$233$	8.58744i	0.562582i	−0.959623	+	0.281291i	$0.909237\pi$
$234$	0	0
$235$	0	0
$236$	−9.81290	−0.638766
$237$	0	0
$238$	− 11.5242i	− 0.747003i
$239$	−23.9245	−1.54755	−0.773775	−	0.633461i	$-0.781634\pi$
$239$	−23.9245	−1.54755	−0.773775	+	0.633461i	$0.781634\pi$
$240$	0	0
$241$	−6.24030	−0.401973	−0.200987	−	0.979594i	$-0.564415\pi$
$241$	−6.24030	−0.401973	−0.200987	+	0.979594i	$0.564415\pi$
$242$	− 19.1345i	− 1.23001i
$243$	0	0
$244$	8.98516	0.575216
$245$	0	0
$246$	0	0
$247$	0.419983i	0.0267229i
$248$	5.65067i	0.358818i
$249$	0	0
$250$	0	0
$251$	28.5726	1.80349	0.901743	−	0.432272i	$-0.142288\pi$
$251$	28.5726	1.80349	0.901743	+	0.432272i	$0.142288\pi$
$252$	0	0
$253$	6.47580i	0.407130i
$254$	14.7671	0.926571
$255$	0	0
$256$	8.98516	0.561573
$257$	18.0000i	1.12281i	0.827541	+	0.561405i	$0.189739\pi$
$257$	18.0000i	1.12281i	−0.827541	+	0.561405i	$0.810261\pi$
$258$	0	0
$259$	−30.7449	−1.91039
$260$	0	0
$261$	0	0
$262$	12.5168i	0.773289i
$263$	31.8687i	1.96511i	0.185974	+	0.982555i	$0.440456\pi$
$263$	31.8687i	1.96511i	−0.185974	+	0.982555i	$0.559544\pi$
$264$	0	0
$265$	0	0
$266$	5.52420	0.338710
$267$	0	0
$268$	− 19.1494i	− 1.16973i
$269$	−31.4971	−1.92041	−0.960207	−	0.279289i	$-0.909901\pi$
$269$	−31.4971	−1.92041	−0.960207	+	0.279289i	$0.909901\pi$
$270$	0	0
$271$	−3.24030	−0.196834	−0.0984172	−	0.995145i	$-0.531378\pi$
$271$	−3.24030	−0.196834	−0.0984172	+	0.995145i	$0.531378\pi$
$272$	− 4.28870i	− 0.260041i
$273$	0	0
$274$	−15.5800	−0.941223
$275$	0	0
$276$	0	0
$277$	− 5.58482i	− 0.335560i	−0.985824	−	0.167780i	$-0.946340\pi$
$277$	− 5.58482i	− 0.335560i	0.985824	−	0.167780i	$-0.0536598\pi$
$278$	− 16.6890i	− 1.00094i
$279$	0	0
$280$	0	0
$281$	24.1042	1.43794	0.718969	−	0.695043i	$-0.244615\pi$
$281$	24.1042	1.43794	0.718969	+	0.695043i	$0.244615\pi$
$282$	0	0
$283$	10.5423i	0.626674i	0.949642	+	0.313337i	$0.101447\pi$
$283$	10.5423i	0.626674i	−0.949642	+	0.313337i	$0.898553\pi$
$284$	−14.3855	−0.853622
$285$	0	0
$286$	1.82774	0.108077
$287$	0.734191i	0.0433379i
$288$	0	0
$289$	15.1723	0.892486
$290$	0	0
$291$	0	0
$292$	− 29.0336i	− 1.69906i
$293$	18.9926i	1.10956i	0.831998	+	0.554779i	$0.187197\pi$
$293$	18.9926i	1.10956i	−0.831998	+	0.554779i	$0.812803\pi$
$294$	0	0
$295$	0	0
$296$	5.52420	0.321088
$297$	0	0
$298$	− 22.0813i	− 1.27914i
$299$	3.10422	0.179521
$300$	0	0
$301$	−3.35194	−0.193203
$302$	36.9171i	2.12434i
$303$	0	0
$304$	2.05582	0.117909
$305$	0	0
$306$	0	0
$307$	29.4791i	1.68246i	0.540679	+	0.841229i	$0.318167\pi$
$307$	29.4791i	1.68246i	−0.540679	+	0.841229i	$0.681833\pi$
$308$	− 12.9926i	− 0.740321i
$309$	0	0
$310$	0	0
$311$	9.41256	0.533738	0.266869	−	0.963733i	$-0.414011\pi$
$311$	9.41256	0.533738	0.266869	+	0.963733i	$0.414011\pi$
$312$	0	0
$313$	− 11.6210i	− 0.656858i	−0.944529	−	0.328429i	$-0.893481\pi$
$313$	− 11.6210i	− 0.656858i	0.944529	−	0.328429i	$-0.106519\pi$
$314$	5.28128	0.298040
$315$	0	0
$316$	24.3297	1.36865
$317$	9.17968i	0.515582i	0.966201	+	0.257791i	$0.0829946\pi$
$317$	9.17968i	0.515582i	−0.966201	+	0.257791i	$0.917005\pi$
$318$	0	0
$319$	5.24030	0.293401
$320$	0	0
$321$	0	0
$322$	− 40.8310i	− 2.27542i
$323$	− 0.876139i	− 0.0487497i
$324$	0	0
$325$	0	0
$326$	17.6816	0.979295
$327$	0	0
$328$	− 0.131919i	− 0.00728398i
$329$	44.5652	2.45696
$330$	0	0
$331$	−7.22066	−0.396883	−0.198442	−	0.980113i	$-0.563588\pi$
$331$	−7.22066	−0.396883	−0.198442	+	0.980113i	$0.563588\pi$
$332$	28.8310i	1.58231i
$333$	0	0
$334$	26.5652	1.45358
$335$	0	0
$336$	0	0
$337$	2.28390i	0.124412i	0.998063	+	0.0622059i	$0.0198135\pi$
$337$	2.28390i	0.124412i	−0.998063	+	0.0622059i	$0.980186\pi$
$338$	26.2436i	1.42746i
$339$	0	0
$340$	0	0
$341$	−10.4051	−0.563470
$342$	0	0
$343$	− 11.0181i	− 0.594921i
$344$	0.602272	0.0324724
$345$	0	0
$346$	−48.0820	−2.58490
$347$	0.708686i	0.0380443i	0.999819	+	0.0190221i	$0.00605530\pi$
$347$	0.708686i	0.0380443i	−0.999819	+	0.0190221i	$0.993945\pi$
$348$	0	0
$349$	21.3445	1.14255	0.571273	−	0.820760i	$-0.306450\pi$
$349$	21.3445	1.14255	0.571273	+	0.820760i	$0.306450\pi$
$350$	0	0
$351$	0	0
$352$	− 10.9320i	− 0.582675i
$353$	− 10.0968i	− 0.537398i	−0.963224	−	0.268699i	$-0.913406\pi$
$353$	− 10.0968i	− 0.537398i	0.963224	−	0.268699i	$-0.0865937\pi$
$354$	0	0
$355$	0	0
$356$	7.05582	0.373958
$357$	0	0
$358$	4.64806i	0.245658i
$359$	30.5578	1.61278	0.806388	−	0.591386i	$-0.201419\pi$
$359$	30.5578	1.61278	0.806388	+	0.591386i	$0.201419\pi$
$360$	0	0
$361$	−18.5800	−0.977896
$362$	0.977106i	0.0513555i
$363$	0	0
$364$	−6.22808	−0.326440
$365$	0	0
$366$	0	0
$367$	− 7.17968i	− 0.374776i	−0.982286	−	0.187388i	$-0.939998\pi$
$367$	− 7.17968i	− 0.374776i	0.982286	−	0.187388i	$-0.0600022\pi$
$368$	− 15.1952i	− 0.792102i
$369$	0	0
$370$	0	0
$371$	−17.0484	−0.885109
$372$	0	0
$373$	21.9245i	1.13521i	0.823301	+	0.567605i	$0.192130\pi$
$373$	21.9245i	1.13521i	−0.823301	+	0.567605i	$0.807870\pi$
$374$	−3.81290	−0.197161
$375$	0	0
$376$	−8.00742	−0.412951
$377$	− 2.51197i	− 0.129373i
$378$	0	0
$379$	17.3929	0.893414	0.446707	−	0.894680i	$-0.352597\pi$
$379$	17.3929	0.893414	0.446707	+	0.894680i	$0.352597\pi$
$380$	0	0
$381$	0	0
$382$	42.1984i	2.15906i
$383$	0.475800i	0.0243123i	0.999926	+	0.0121561i	$0.00386951\pi$
$383$	0.475800i	0.0243123i	−0.999926	+	0.0121561i	$0.996130\pi$
$384$	0	0
$385$	0	0
$386$	−41.5752	−2.11612
$387$	0	0
$388$	31.9394i	1.62148i
$389$	−5.58744	−0.283294	−0.141647	−	0.989917i	$-0.545240\pi$
$389$	−5.58744	−0.283294	−0.141647	+	0.989917i	$0.545240\pi$
$390$	0	0
$391$	−6.47580	−0.327495
$392$	7.11905i	0.359567i
$393$	0	0
$394$	−32.5019	−1.63742
$395$	0	0
$396$	0	0
$397$	3.75228i	0.188321i	0.995557	+	0.0941607i	$0.0300168\pi$
$397$	3.75228i	0.188321i	−0.995557	+	0.0941607i	$0.969983\pi$
$398$	− 7.47841i	− 0.374859i
$399$	0	0
$400$	0	0
$401$	−23.5652	−1.17679	−0.588394	−	0.808574i	$-0.700240\pi$
$401$	−23.5652	−1.17679	−0.588394	+	0.808574i	$0.700240\pi$
$402$	0	0
$403$	4.98777i	0.248459i
$404$	−3.45355	−0.171820
$405$	0	0
$406$	−33.0410	−1.63980
$407$	10.1723i	0.504220i
$408$	0	0
$409$	−1.04840	−0.0518400	−0.0259200	−	0.999664i	$-0.508252\pi$
$409$	−1.04840	−0.0518400	−0.0259200	+	0.999664i	$0.508252\pi$
$410$	0	0
$411$	0	0
$412$	17.6965i	0.871842i
$413$	17.0484i	0.838897i
$414$	0	0
$415$	0	0
$416$	−5.24030	−0.256927
$417$	0	0
$418$	− 1.82774i	− 0.0893977i
$419$	−25.9197	−1.26626	−0.633131	−	0.774045i	$-0.718231\pi$
$419$	−25.9197	−1.26626	−0.633131	+	0.774045i	$0.718231\pi$
$420$	0	0
$421$	7.64064	0.372382	0.186191	−	0.982514i	$-0.440386\pi$
$421$	7.64064	0.372382	0.186191	+	0.982514i	$0.440386\pi$
$422$	31.2765i	1.52252i
$423$	0	0
$424$	3.06324	0.148764
$425$	0	0
$426$	0	0
$427$	− 15.6103i	− 0.755437i
$428$	2.84583i	0.137558i
$429$	0	0
$430$	0	0
$431$	−7.98516	−0.384632	−0.192316	−	0.981333i	$-0.561600\pi$
$431$	−7.98516	−0.384632	−0.192316	+	0.981333i	$0.561600\pi$
$432$	0	0
$433$	12.5120i	0.601287i	0.953737	+	0.300644i	$0.0972014\pi$
$433$	12.5120i	0.601287i	−0.953737	+	0.300644i	$0.902799\pi$
$434$	65.6062	3.14920
$435$	0	0
$436$	33.1723	1.58866
$437$	− 3.10422i	− 0.148495i
$438$	0	0
$439$	8.76450	0.418307	0.209153	−	0.977883i	$-0.432929\pi$
$439$	8.76450	0.418307	0.209153	+	0.977883i	$0.432929\pi$
$440$	0	0
$441$	0	0
$442$	1.82774i	0.0869367i
$443$	3.67095i	0.174412i	0.996190	+	0.0872062i	$0.0277939\pi$
$443$	3.67095i	0.174412i	−0.996190	+	0.0872062i	$0.972206\pi$
$444$	0	0
$445$	0	0
$446$	−55.9729	−2.65040
$447$	0	0
$448$	43.0032i	2.03171i
$449$	28.1723	1.32953	0.664766	−	0.747052i	$-0.268531\pi$
$449$	28.1723	1.32953	0.664766	+	0.747052i	$0.268531\pi$
$450$	0	0
$451$	0.242915	0.0114384
$452$	28.0458i	1.31916i
$453$	0	0
$454$	2.82032	0.132364
$455$	0	0
$456$	0	0
$457$	35.2616i	1.64947i	0.565519	+	0.824735i	$0.308676\pi$
$457$	35.2616i	1.64947i	−0.565519	+	0.824735i	$0.691324\pi$
$458$	17.1803i	0.802784i
$459$	0	0
$460$	0	0
$461$	−34.6768	−1.61506	−0.807530	−	0.589826i	$-0.799196\pi$
$461$	−34.6768	−1.61506	−0.807530	+	0.589826i	$0.799196\pi$
$462$	0	0
$463$	− 7.44874i	− 0.346172i	−0.984907	−	0.173086i	$-0.944626\pi$
$463$	− 7.44874i	− 0.346172i	0.984907	−	0.173086i	$-0.0553739\pi$
$464$	−12.2961	−0.570833
$465$	0	0
$466$	−17.9145	−0.829874
$467$	− 29.9655i	− 1.38664i	−0.720630	−	0.693319i	$-0.756148\pi$
$467$	− 29.9655i	− 1.38664i	0.720630	−	0.693319i	$-0.243852\pi$
$468$	0	0
$469$	−33.2691	−1.53622
$470$	0	0
$471$	0	0
$472$	− 3.06324i	− 0.140997i
$473$	1.10902i	0.0509930i
$474$	0	0
$475$	0	0
$476$	12.9926	0.595514
$477$	0	0
$478$	− 49.9097i	− 2.28282i
$479$	−7.98516	−0.364851	−0.182426	−	0.983220i	$-0.558395\pi$
$479$	−7.98516	−0.364851	−0.182426	+	0.983220i	$0.558395\pi$
$480$	0	0
$481$	4.87614	0.222333
$482$	− 13.0181i	− 0.592958i
$483$	0	0
$484$	21.5726	0.980573
$485$	0	0
$486$	0	0
$487$	− 11.9442i	− 0.541243i	−0.962686	−	0.270621i	$-0.912771\pi$
$487$	− 11.9442i	− 0.541243i	0.962686	−	0.270621i	$-0.0872291\pi$
$488$	2.80485i	0.126969i
$489$	0	0
$490$	0	0
$491$	9.22066	0.416123	0.208061	−	0.978116i	$-0.433285\pi$
$491$	9.22066	0.416123	0.208061	+	0.978116i	$0.433285\pi$
$492$	0	0
$493$	5.24030i	0.236011i
$494$	−0.876139	−0.0394193
$495$	0	0
$496$	24.4152	1.09627
$497$	24.9926i	1.12107i
$498$	0	0
$499$	−30.1723	−1.35070	−0.675348	−	0.737499i	$-0.736007\pi$
$499$	−30.1723	−1.35070	−0.675348	+	0.737499i	$0.736007\pi$
$500$	0	0
$501$	0	0
$502$	59.6062i	2.66035i
$503$	− 10.5981i	− 0.472546i	−0.971687	−	0.236273i	$-0.924074\pi$
$503$	− 10.5981i	− 0.472546i	0.971687	−	0.236273i	$-0.0759260\pi$
$504$	0	0
$505$	0	0
$506$	−13.5094	−0.600564
$507$	0	0
$508$	16.6487i	0.738667i
$509$	28.7523	1.27442	0.637211	−	0.770689i	$-0.280088\pi$
$509$	28.7523	1.27442	0.637211	+	0.770689i	$0.280088\pi$
$510$	0	0
$511$	−50.4413	−2.23139
$512$	30.3094i	1.33950i
$513$	0	0
$514$	−37.5503	−1.65627
$515$	0	0
$516$	0	0
$517$	− 14.7449i	− 0.648478i
$518$	− 64.1378i	− 2.81805i
$519$	0	0
$520$	0	0
$521$	36.0942	1.58132	0.790658	−	0.612259i	$-0.209739\pi$
$521$	36.0942	1.58132	0.790658	+	0.612259i	$0.209739\pi$
$522$	0	0
$523$	− 11.1297i	− 0.486669i	−0.969942	−	0.243334i	$-0.921759\pi$
$523$	− 11.1297i	− 0.486669i	0.969942	−	0.243334i	$-0.0782412\pi$
$524$	−14.1116	−0.616470
$525$	0	0
$526$	−66.4823	−2.89877
$527$	− 10.4051i	− 0.453255i
$528$	0	0
$529$	0.0558176	0.00242685
$530$	0	0
$531$	0	0
$532$	6.22808i	0.270021i
$533$	− 0.116443i	− 0.00504370i
$534$	0	0
$535$	0	0
$536$	5.97774	0.258199
$537$	0	0
$538$	− 65.7071i	− 2.83284i
$539$	−13.1090	−0.564646
$540$	0	0
$541$	−34.7374	−1.49348	−0.746740	−	0.665116i	$-0.768382\pi$
$541$	−34.7374	−1.49348	−0.746740	+	0.665116i	$0.768382\pi$
$542$	− 6.75970i	− 0.290354i
$543$	0	0
$544$	10.9320	0.468704
$545$	0	0
$546$	0	0
$547$	− 2.71455i	− 0.116066i	−0.998315	−	0.0580328i	$-0.981517\pi$
$547$	− 2.71455i	− 0.116066i	0.998315	−	0.0580328i	$-0.0184828\pi$
$548$	− 17.5652i	− 0.750347i
$549$	0	0
$550$	0	0
$551$	−2.51197	−0.107014
$552$	0	0
$553$	− 42.2691i	− 1.79746i
$554$	11.6507	0.494990
$555$	0	0
$556$	18.8155	0.797956
$557$	8.93676i	0.378663i	0.981913	+	0.189331i	$0.0606321\pi$
$557$	8.93676i	0.378663i	−0.981913	+	0.189331i	$0.939368\pi$
$558$	0	0
$559$	0.531618	0.0224850
$560$	0	0
$561$	0	0
$562$	50.2845i	2.12113i
$563$	− 9.36261i	− 0.394587i	−0.980344	−	0.197293i	$-0.936785\pi$
$563$	− 9.36261i	− 0.394587i	0.980344	−	0.197293i	$-0.0632152\pi$
$564$	0	0
$565$	0	0
$566$	−21.9926	−0.924417
$567$	0	0
$568$	− 4.49064i	− 0.188423i
$569$	35.8735	1.50390	0.751948	−	0.659222i	$-0.229114\pi$
$569$	35.8735	1.50390	0.751948	+	0.659222i	$0.229114\pi$
$570$	0	0
$571$	20.0000	0.836974	0.418487	−	0.908223i	$-0.362561\pi$
$571$	20.0000	0.836974	0.418487	+	0.908223i	$0.362561\pi$
$572$	2.06063i	0.0861591i
$573$	0	0
$574$	−1.53162	−0.0639285
$575$	0	0
$576$	0	0
$577$	1.35675i	0.0564821i	0.999601	+	0.0282411i	$0.00899060\pi$
$577$	1.35675i	0.0564821i	−0.999601	+	0.0282411i	$0.991009\pi$
$578$	31.6513i	1.31652i
$579$	0	0
$580$	0	0
$581$	50.0894	2.07806
$582$	0	0
$583$	5.64064i	0.233612i
$584$	9.06324	0.375039
$585$	0	0
$586$	−39.6210	−1.63673
$587$	− 28.7900i	− 1.18829i	−0.804358	−	0.594145i	$-0.797490\pi$
$587$	− 28.7900i	− 1.18829i	0.804358	−	0.594145i	$-0.202510\pi$
$588$	0	0
$589$	4.98777	0.205518
$590$	0	0
$591$	0	0
$592$	− 23.8687i	− 0.980998i
$593$	30.9171i	1.26961i	0.772671	+	0.634807i	$0.218920\pi$
$593$	30.9171i	1.26961i	−0.772671	+	0.634807i	$0.781080\pi$
$594$	0	0
$595$	0	0
$596$	24.8949	1.01973
$597$	0	0
$598$	6.47580i	0.264815i
$599$	1.39292	0.0569132	0.0284566	−	0.999595i	$-0.490941\pi$
$599$	1.39292	0.0569132	0.0284566	+	0.999595i	$0.490941\pi$
$600$	0	0
$601$	8.82513	0.359985	0.179992	−	0.983668i	$-0.442393\pi$
$601$	8.82513	0.359985	0.179992	+	0.983668i	$0.442393\pi$
$602$	− 6.99258i	− 0.284996i
$603$	0	0
$604$	−41.6210	−1.69353
$605$	0	0
$606$	0	0
$607$	− 2.15678i	− 0.0875412i	−0.999042	−	0.0437706i	$-0.986063\pi$
$607$	− 2.15678i	− 0.0875412i	0.999042	−	0.0437706i	$-0.0139371\pi$
$608$	5.24030i	0.212522i
$609$	0	0
$610$	0	0
$611$	−7.06804	−0.285942
$612$	0	0
$613$	9.57521i	0.386739i	0.981126	+	0.193370i	$0.0619416\pi$
$613$	9.57521i	0.386739i	−0.981126	+	0.193370i	$0.938058\pi$
$614$	−61.4971	−2.48182
$615$	0	0
$616$	4.05582	0.163414
$617$	37.6768i	1.51681i	0.651783	+	0.758406i	$0.274021\pi$
$617$	37.6768i	1.51681i	−0.651783	+	0.758406i	$0.725979\pi$
$618$	0	0
$619$	−17.1042	−0.687477	−0.343738	−	0.939065i	$-0.611693\pi$
$619$	−17.1042	−0.687477	−0.343738	+	0.939065i	$0.611693\pi$
$620$	0	0
$621$	0	0
$622$	19.6358i	0.787325i
$623$	− 12.2584i	− 0.491122i
$624$	0	0
$625$	0	0
$626$	24.2429	0.968942
$627$	0	0
$628$	5.95421i	0.237599i
$629$	−10.1723	−0.405595
$630$	0	0
$631$	33.1090	1.31805	0.659025	−	0.752121i	$-0.270969\pi$
$631$	33.1090	1.31805	0.659025	+	0.752121i	$0.270969\pi$
$632$	7.59485i	0.302107i
$633$	0	0
$634$	−19.1500	−0.760544
$635$	0	0
$636$	0	0
$637$	6.28390i	0.248977i
$638$	10.9320i	0.432800i
$639$	0	0
$640$	0	0
$641$	23.1526	0.914473	0.457237	−	0.889345i	$-0.348839\pi$
$641$	23.1526	0.914473	0.457237	+	0.889345i	$0.348839\pi$
$642$	0	0
$643$	− 43.0639i	− 1.69827i	−0.528173	−	0.849137i	$-0.677123\pi$
$643$	− 43.0639i	− 1.69827i	0.528173	−	0.849137i	$-0.322877\pi$
$644$	46.0336	1.81398
$645$	0	0
$646$	1.82774	0.0719115
$647$	20.6439i	0.811595i	0.913963	+	0.405798i	$0.133006\pi$
$647$	20.6439i	0.811595i	−0.913963	+	0.405798i	$0.866994\pi$
$648$	0	0
$649$	5.64064	0.221415
$650$	0	0
$651$	0	0
$652$	19.9346i	0.780698i
$653$	6.83516i	0.267480i	0.991016	+	0.133740i	$0.0426988\pi$
$653$	6.83516i	0.267480i	−0.991016	+	0.133740i	$0.957301\pi$
$654$	0	0
$655$	0	0
$656$	−0.569988	−0.0222543
$657$	0	0
$658$	92.9688i	3.62430i
$659$	−26.8613	−1.04637	−0.523184	−	0.852220i	$-0.675256\pi$
$659$	−26.8613	−1.04637	−0.523184	+	0.852220i	$0.675256\pi$
$660$	0	0
$661$	−2.12125	−0.0825071	−0.0412535	−	0.999149i	$-0.513135\pi$
$661$	−2.12125	−0.0825071	−0.0412535	+	0.999149i	$0.513135\pi$
$662$	− 15.0632i	− 0.585449i
$663$	0	0
$664$	−9.00000	−0.349268
$665$	0	0
$666$	0	0
$667$	18.5667i	0.718907i
$668$	29.9500i	1.15880i
$669$	0	0
$670$	0	0
$671$	−5.16484	−0.199387
$672$	0	0
$673$	− 34.8203i	− 1.34222i	−0.741356	−	0.671112i	$-0.765817\pi$
$673$	− 34.8203i	− 1.34222i	0.741356	−	0.671112i	$-0.234183\pi$
$674$	−4.76450	−0.183522
$675$	0	0
$676$	−29.5874	−1.13798
$677$	24.6842i	0.948692i	0.880338	+	0.474346i	$0.157315\pi$
$677$	24.6842i	0.948692i	−0.880338	+	0.474346i	$0.842685\pi$
$678$	0	0
$679$	55.4897	2.12950
$680$	0	0
$681$	0	0
$682$	− 21.7065i	− 0.831184i
$683$	− 38.4610i	− 1.47167i	−0.677162	−	0.735834i	$-0.736790\pi$
$683$	− 38.4610i	− 1.47167i	0.677162	−	0.735834i	$-0.263210\pi$
$684$	0	0
$685$	0	0
$686$	22.9852	0.877578
$687$	0	0
$688$	− 2.60227i	− 0.0992107i
$689$	2.70388	0.103010
$690$	0	0
$691$	0.480608	0.0182832	0.00914159	−	0.999958i	$-0.497090\pi$
$691$	0.480608	0.0182832	0.00914159	+	0.999958i	$0.497090\pi$
$692$	− 54.2084i	− 2.06070i
$693$	0	0
$694$	−1.47841	−0.0561197
$695$	0	0
$696$	0	0
$697$	0.242915i	0.00920105i
$698$	44.5274i	1.68539i
$699$	0	0
$700$	0	0
$701$	18.1797	0.686637	0.343318	−	0.939219i	$-0.388449\pi$
$701$	18.1797	0.686637	0.343318	+	0.939219i	$0.388449\pi$
$702$	0	0
$703$	− 4.87614i	− 0.183907i
$704$	14.2281	0.536241
$705$	0	0
$706$	21.0632	0.792725
$707$	6.00000i	0.225653i
$708$	0	0
$709$	−7.18710	−0.269917	−0.134959	−	0.990851i	$-0.543090\pi$
$709$	−7.18710	−0.269917	−0.134959	+	0.990851i	$0.543090\pi$
$710$	0	0
$711$	0	0
$712$	2.20257i	0.0825449i
$713$	− 36.8661i	− 1.38065i
$714$	0	0
$715$	0	0
$716$	−5.24030	−0.195839
$717$	0	0
$718$	63.7475i	2.37903i
$719$	12.5168	0.466797	0.233399	−	0.972381i	$-0.425015\pi$
$719$	12.5168	0.466797	0.233399	+	0.972381i	$0.425015\pi$
$720$	0	0
$721$	30.7449	1.14500
$722$	− 38.7603i	− 1.44251i
$723$	0	0
$724$	−1.10161	−0.0409409
$725$	0	0
$726$	0	0
$727$	8.42584i	0.312497i	0.987718	+	0.156249i	$0.0499401\pi$
$727$	8.42584i	0.312497i	−0.987718	+	0.156249i	$0.950060\pi$
$728$	− 1.94418i	− 0.0720562i
$729$	0	0
$730$	0	0
$731$	−1.10902	−0.0410187
$732$	0	0
$733$	22.0000i	0.812589i	0.913742	+	0.406294i	$0.133179\pi$
$733$	22.0000i	0.812589i	−0.913742	+	0.406294i	$0.866821\pi$
$734$	14.9777	0.552839
$735$	0	0
$736$	38.7326	1.42770
$737$	11.0074i	0.405463i
$738$	0	0
$739$	1.81290	0.0666887	0.0333444	−	0.999444i	$-0.489384\pi$
$739$	1.81290	0.0666887	0.0333444	+	0.999444i	$0.489384\pi$
$740$	0	0
$741$	0	0
$742$	− 35.5652i	− 1.30564i
$743$	20.1371i	0.738760i	0.929278	+	0.369380i	$0.120430\pi$
$743$	20.1371i	0.738760i	−0.929278	+	0.369380i	$0.879570\pi$
$744$	0	0
$745$	0	0
$746$	−45.7374	−1.67457
$747$	0	0
$748$	− 4.29873i	− 0.157177i
$749$	4.94418	0.180656
$750$	0	0
$751$	12.2132	0.445668	0.222834	−	0.974856i	$-0.428469\pi$
$751$	12.2132	0.445668	0.222834	+	0.974856i	$0.428469\pi$
$752$	34.5981i	1.26166i
$753$	0	0
$754$	5.24030	0.190841
$755$	0	0
$756$	0	0
$757$	52.9533i	1.92462i	0.271955	+	0.962310i	$0.412330\pi$
$757$	52.9533i	1.92462i	−0.271955	+	0.962310i	$0.587670\pi$
$758$	36.2839i	1.31789i
$759$	0	0
$760$	0	0
$761$	18.4535	0.668940	0.334470	−	0.942406i	$-0.391443\pi$
$761$	18.4535	0.668940	0.334470	+	0.942406i	$0.391443\pi$
$762$	0	0
$763$	− 57.6317i	− 2.08641i
$764$	−47.5752	−1.72121
$765$	0	0
$766$	−0.992582	−0.0358634
$767$	− 2.70388i	− 0.0976314i
$768$	0	0
$769$	4.45355	0.160599	0.0802995	−	0.996771i	$-0.474412\pi$
$769$	4.45355	0.160599	0.0802995	+	0.996771i	$0.474412\pi$
$770$	0	0
$771$	0	0
$772$	− 46.8726i	− 1.68698i
$773$	38.9368i	1.40046i	0.713918	+	0.700229i	$0.246919\pi$
$773$	38.9368i	1.40046i	−0.713918	+	0.700229i	$0.753081\pi$
$774$	0	0
$775$	0	0
$776$	−9.97033	−0.357914
$777$	0	0
$778$	− 11.6561i	− 0.417892i
$779$	−0.116443	−0.00417200
$780$	0	0
$781$	8.26906	0.295890
$782$	− 13.5094i	− 0.483094i
$783$	0	0
$784$	30.7597	1.09856
$785$	0	0
$786$	0	0
$787$	34.2281i	1.22010i	0.792363	+	0.610050i	$0.208850\pi$
$787$	34.2281i	1.22010i	−0.792363	+	0.610050i	$0.791150\pi$
$788$	− 36.6433i	− 1.30536i
$789$	0	0
$790$	0	0
$791$	48.7252	1.73247
$792$	0	0
$793$	2.47580i	0.0879183i
$794$	−7.82774	−0.277796
$795$	0	0
$796$	8.43129	0.298839
$797$	− 23.9655i	− 0.848902i	−0.905451	−	0.424451i	$-0.860467\pi$
$797$	− 23.9655i	− 0.848902i	0.905451	−	0.424451i	$-0.139533\pi$
$798$	0	0
$799$	14.7449	0.521636
$800$	0	0
$801$	0	0
$802$	− 49.1600i	− 1.73590i
$803$	16.6890i	0.588943i
$804$	0	0
$805$	0	0
$806$	−10.4051	−0.366506
$807$	0	0
$808$	− 1.07807i	− 0.0379265i
$809$	−0.283896	−0.00998124	−0.00499062	−	0.999988i	$-0.501589\pi$
$809$	−0.283896	−0.00998124	−0.00499062	+	0.999988i	$0.501589\pi$
$810$	0	0
$811$	32.4413	1.13917	0.569584	−	0.821933i	$-0.307104\pi$
$811$	32.4413	1.13917	0.569584	+	0.821933i	$0.307104\pi$
$812$	− 37.2510i	− 1.30725i
$813$	0	0
$814$	−21.2207	−0.743784
$815$	0	0
$816$	0	0
$817$	− 0.531618i	− 0.0185990i
$818$	− 2.18710i	− 0.0764701i
$819$	0	0
$820$	0	0
$821$	−41.6694	−1.45427	−0.727136	−	0.686493i	$-0.759149\pi$
$821$	−41.6694	−1.45427	−0.727136	+	0.686493i	$0.759149\pi$
$822$	0	0
$823$	19.3626i	0.674938i	0.941337	+	0.337469i	$0.109571\pi$
$823$	19.3626i	0.674938i	−0.941337	+	0.337469i	$0.890429\pi$
$824$	−5.52420	−0.192445
$825$	0	0
$826$	−35.5652	−1.23747
$827$	− 18.8097i	− 0.654076i	−0.945011	−	0.327038i	$-0.893950\pi$
$827$	− 18.8097i	− 0.654076i	0.945011	−	0.327038i	$-0.106050\pi$
$828$	0	0
$829$	33.1016	1.14967	0.574833	−	0.818271i	$-0.305067\pi$
$829$	33.1016	1.14967	0.574833	+	0.818271i	$0.305067\pi$
$830$	0	0
$831$	0	0
$832$	− 6.82032i	− 0.236452i
$833$	− 13.1090i	− 0.454201i
$834$	0	0
$835$	0	0
$836$	2.06063	0.0712682
$837$	0	0
$838$	− 54.0719i	− 1.86788i
$839$	−39.6965	−1.37047	−0.685237	−	0.728320i	$-0.740301\pi$
$839$	−39.6965	−1.37047	−0.685237	+	0.728320i	$0.740301\pi$
$840$	0	0
$841$	−13.9755	−0.481915
$842$	15.9394i	0.549307i
$843$	0	0
$844$	−35.2616	−1.21376
$845$	0	0
$846$	0	0
$847$	− 37.4791i	− 1.28780i
$848$	− 13.2355i	− 0.454509i
$849$	0	0
$850$	0	0
$851$	−36.0410	−1.23547
$852$	0	0
$853$	4.11644i	0.140944i	0.997514	+	0.0704722i	$0.0224506\pi$
$853$	4.11644i	0.140944i	−0.997514	+	0.0704722i	$0.977549\pi$
$854$	32.5652	1.11436
$855$	0	0
$856$	−0.888365	−0.0303637
$857$	− 14.7449i	− 0.503675i	−0.967770	−	0.251837i	$-0.918965\pi$
$857$	− 14.7449i	− 0.503675i	0.967770	−	0.251837i	$-0.0810348\pi$
$858$	0	0
$859$	37.8539	1.29156	0.645779	−	0.763524i	$-0.276533\pi$
$859$	37.8539	1.29156	0.645779	+	0.763524i	$0.276533\pi$
$860$	0	0
$861$	0	0
$862$	− 16.6581i	− 0.567377i
$863$	26.7704i	0.911274i	0.890166	+	0.455637i	$0.150588\pi$
$863$	26.7704i	0.911274i	−0.890166	+	0.455637i	$0.849412\pi$
$864$	0	0
$865$	0	0
$866$	−26.1016	−0.886969
$867$	0	0
$868$	73.9655i	2.51055i
$869$	−13.9852	−0.474414
$870$	0	0
$871$	5.27648	0.178787
$872$	10.3552i	0.350671i
$873$	0	0
$874$	6.47580	0.219047
$875$	0	0
$876$	0	0
$877$	− 46.9681i	− 1.58600i	−0.609221	−	0.793001i	$-0.708518\pi$
$877$	− 46.9681i	− 1.58600i	0.609221	−	0.793001i	$-0.291482\pi$
$878$	18.2839i	0.617052i
$879$	0	0
$880$	0	0
$881$	19.8055	0.667264	0.333632	−	0.942703i	$-0.391726\pi$
$881$	19.8055	0.667264	0.333632	+	0.942703i	$0.391726\pi$
$882$	0	0
$883$	6.20257i	0.208733i	0.994539	+	0.104367i	$0.0332815\pi$
$883$	6.20257i	0.208733i	−0.994539	+	0.104367i	$0.966718\pi$
$884$	−2.06063	−0.0693063
$885$	0	0
$886$	−7.65809	−0.257279
$887$	− 13.4274i	− 0.450848i	−0.974261	−	0.225424i	$-0.927623\pi$
$887$	− 13.4274i	− 0.450848i	0.974261	−	0.225424i	$-0.0723767\pi$
$888$	0	0
$889$	28.9245	0.970098
$890$	0	0
$891$	0	0
$892$	− 63.1049i	− 2.11291i
$893$	7.06804i	0.236523i
$894$	0	0
$895$	0	0
$896$	−23.6284	−0.789370
$897$	0	0
$898$	58.7710i	1.96121i
$899$	−29.8325	−0.994971
$900$	0	0
$901$	−5.64064	−0.187917
$902$	0.506752i	0.0168730i
$903$	0	0
$904$	−8.75489	−0.291183
$905$	0	0
$906$	0	0
$907$	0.673566i	0.0223654i	0.999937	+	0.0111827i	$0.00355964\pi$
$907$	0.673566i	0.0223654i	−0.999937	+	0.0111827i	$0.996440\pi$
$908$	3.17968i	0.105521i
$909$	0	0
$910$	0	0
$911$	7.90970	0.262060	0.131030	−	0.991378i	$-0.458172\pi$
$911$	7.90970	0.262060	0.131030	+	0.991378i	$0.458172\pi$
$912$	0	0
$913$	− 16.5726i	− 0.548473i
$914$	−73.5604	−2.43316
$915$	0	0
$916$	−19.3694	−0.639983
$917$	24.5168i	0.809615i
$918$	0	0
$919$	8.58263	0.283115	0.141557	−	0.989930i	$-0.454789\pi$
$919$	8.58263	0.283115	0.141557	+	0.989930i	$0.454789\pi$
$920$	0	0
$921$	0	0
$922$	− 72.3404i	− 2.38240i
$923$	− 3.96383i	− 0.130471i
$924$	0	0
$925$	0	0
$926$	15.5390	0.510644
$927$	0	0
$928$	− 31.3430i	− 1.02888i
$929$	29.6162	0.971676	0.485838	−	0.874049i	$-0.338515\pi$
$929$	29.6162	0.971676	0.485838	+	0.874049i	$0.338515\pi$
$930$	0	0
$931$	6.28390	0.205946
$932$	− 20.1971i	− 0.661579i
$933$	0	0
$934$	62.5120	2.04545
$935$	0	0
$936$	0	0
$937$	− 15.2058i	− 0.496753i	−0.968664	−	0.248376i	$-0.920103\pi$
$937$	− 15.2058i	− 0.496753i	0.968664	−	0.248376i	$-0.0798969\pi$
$938$	− 69.4036i	− 2.26611i
$939$	0	0
$940$	0	0
$941$	−5.65287	−0.184278	−0.0921391	−	0.995746i	$-0.529370\pi$
$941$	−5.65287	−0.184278	−0.0921391	+	0.995746i	$0.529370\pi$
$942$	0	0
$943$	0.860663i	0.0280270i
$944$	−13.2355	−0.430779
$945$	0	0
$946$	−2.31357	−0.0752206
$947$	40.3962i	1.31270i	0.754457	+	0.656350i	$0.227900\pi$
$947$	40.3962i	1.31270i	−0.754457	+	0.656350i	$0.772100\pi$
$948$	0	0
$949$	8.00000	0.259691
$950$	0	0
$951$	0	0
$952$	4.05582i	0.131450i
$953$	− 22.9320i	− 0.742839i	−0.928465	−	0.371419i	$-0.878871\pi$
$953$	− 22.9320i	− 0.742839i	0.928465	−	0.371419i	$-0.121129\pi$
$954$	0	0
$955$	0	0
$956$	56.2691	1.81987
$957$	0	0
$958$	− 16.6581i	− 0.538198i
$959$	−30.5168	−0.985438
$960$	0	0
$961$	28.2355	0.910822
$962$	10.1723i	0.327967i
$963$	0	0
$964$	14.6768	0.472708
$965$	0	0
$966$	0	0
$967$	10.3700i	0.333478i	0.986001	+	0.166739i	$0.0533237\pi$
$967$	10.3700i	0.333478i	−0.986001	+	0.166739i	$0.946676\pi$
$968$	6.73419i	0.216445i
$969$	0	0
$970$	0	0
$971$	−48.0410	−1.54171	−0.770854	−	0.637012i	$-0.780170\pi$
$971$	−48.0410	−1.54171	−0.770854	+	0.637012i	$0.780170\pi$
$972$	0	0
$973$	− 32.6890i	− 1.04796i
$974$	24.9171	0.798396
$975$	0	0
$976$	12.1191	0.387921
$977$	− 27.0532i	− 0.865509i	−0.901512	−	0.432754i	$-0.857542\pi$
$977$	− 27.0532i	− 0.865509i	0.901512	−	0.432754i	$-0.142458\pi$
$978$	0	0
$979$	−4.05582	−0.129624
$980$	0	0
$981$	0	0
$982$	19.2355i	0.613829i
$983$	− 22.4817i	− 0.717054i	−0.933519	−	0.358527i	$-0.883279\pi$
$983$	− 22.4817i	− 0.717054i	0.933519	−	0.358527i	$-0.116721\pi$
$984$	0	0
$985$	0	0
$986$	−10.9320	−0.348144
$987$	0	0
$988$	− 0.987774i	− 0.0314253i
$989$	−3.92935	−0.124946
$990$	0	0
$991$	−26.5316	−0.842805	−0.421402	−	0.906874i	$-0.638462\pi$
$991$	−26.5316	−0.842805	−0.421402	+	0.906874i	$0.638462\pi$
$992$	62.2346i	1.97595i
$993$	0	0
$994$	−52.1378	−1.65371
$995$	0	0
$996$	0	0
$997$	− 28.3659i	− 0.898356i	−0.893442	−	0.449178i	$-0.851717\pi$
$997$	− 28.3659i	− 0.898356i	0.893442	−	0.449178i	$-0.148283\pi$
$998$	− 62.9433i	− 1.99243i
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	2025.2.b.m.649.5		6
3.2	odd	2		2025.2.b.l.649.2		6
5.2	odd	4		2025.2.a.o.1.1		3
5.3	odd	4		405.2.a.i.1.3		3
5.4	even	2	inner	2025.2.b.m.649.2		6
9.2	odd	6		225.2.k.b.49.2		12
9.4	even	3		675.2.k.b.424.2		12
9.5	odd	6		225.2.k.b.124.5		12
9.7	even	3		675.2.k.b.199.5		12
15.2	even	4		2025.2.a.n.1.3		3
15.8	even	4		405.2.a.j.1.1		3
15.14	odd	2		2025.2.b.l.649.5		6
20.3	even	4		6480.2.a.bs.1.1		3
45.2	even	12		225.2.e.b.76.1		6
45.4	even	6		675.2.k.b.424.5		12
45.7	odd	12		675.2.e.b.226.3		6
45.13	odd	12		135.2.e.b.46.1		6
45.14	odd	6		225.2.k.b.124.2		12
45.22	odd	12		675.2.e.b.451.3		6
45.23	even	12		45.2.e.b.16.3	✓	6
45.29	odd	6		225.2.k.b.49.5		12
45.32	even	12		225.2.e.b.151.1		6
45.34	even	6		675.2.k.b.199.2		12
45.38	even	12		45.2.e.b.31.3	yes	6
45.43	odd	12		135.2.e.b.91.1		6
60.23	odd	4		6480.2.a.bv.1.1		3
180.23	odd	12		720.2.q.i.241.3		6
180.43	even	12		2160.2.q.k.1441.3		6
180.83	odd	12		720.2.q.i.481.3		6
180.103	even	12		2160.2.q.k.721.3		6

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
45.2.e.b.16.3	✓	6	45.23	even	12
45.2.e.b.31.3	yes	6	45.38	even	12
135.2.e.b.46.1		6	45.13	odd	12
135.2.e.b.91.1		6	45.43	odd	12
225.2.e.b.76.1		6	45.2	even	12
225.2.e.b.151.1		6	45.32	even	12
225.2.k.b.49.2		12	9.2	odd	6
225.2.k.b.49.5		12	45.29	odd	6
225.2.k.b.124.2		12	45.14	odd	6
225.2.k.b.124.5		12	9.5	odd	6
405.2.a.i.1.3		3	5.3	odd	4
405.2.a.j.1.1		3	15.8	even	4
675.2.e.b.226.3		6	45.7	odd	12
675.2.e.b.451.3		6	45.22	odd	12
675.2.k.b.199.2		12	45.34	even	6
675.2.k.b.199.5		12	9.7	even	3
675.2.k.b.424.2		12	9.4	even	3
675.2.k.b.424.5		12	45.4	even	6
720.2.q.i.241.3		6	180.23	odd	12
720.2.q.i.481.3		6	180.83	odd	12
2025.2.a.n.1.3		3	15.2	even	4
2025.2.a.o.1.1		3	5.2	odd	4
2025.2.b.l.649.2		6	3.2	odd	2
2025.2.b.l.649.5		6	15.14	odd	2
2025.2.b.m.649.2		6	5.4	even	2	inner
2025.2.b.m.649.5		6	1.1	even	1	trivial
2160.2.q.k.721.3		6	180.103	even	12
2160.2.q.k.1441.3		6	180.43	even	12
6480.2.a.bs.1.1		3	20.3	even	4
6480.2.a.bv.1.1		3	60.23	odd	4

Overview	Random
Universe	Knowledge

Classical	Maass
Hilbert	Bianchi

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

\(f(q)\)	\(=\)	\(q+2.08613i q^{2} -2.35194 q^{4} +4.08613i q^{7} -0.734191i q^{8} +1.35194 q^{11} -0.648061i q^{13} -8.52420 q^{14} -3.17226 q^{16} +1.35194i q^{17} -0.648061 q^{19} +2.82032i q^{22} +4.79001i q^{23} +1.35194 q^{26} -9.61033i q^{28} +3.87614 q^{29} -7.69646 q^{31} -8.08613i q^{32} -2.82032 q^{34} +7.52420i q^{37} -1.35194i q^{38} +0.179679 q^{41} +0.820321i q^{43} -3.17968 q^{44} -9.99258 q^{46} -10.9065i q^{47} -9.69646 q^{49} +1.52420i q^{52} +4.17226i q^{53} +3.00000 q^{56} +8.08613i q^{58} +4.17226 q^{59} -3.82032 q^{61} -16.0558i q^{62} +10.5242 q^{64} +8.14195i q^{67} -3.17968i q^{68} +6.11644 q^{71} +12.3445i q^{73} -15.6965 q^{74} +1.52420 q^{76} +5.52420i q^{77} -10.3445 q^{79} +0.374833i q^{82} -12.2584i q^{83} -1.71130 q^{86} -0.992582i q^{88} -3.00000 q^{89} +2.64806 q^{91} -11.2658i q^{92} +22.7523 q^{94} -13.5800i q^{97} -20.2281i q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 6 q - 10 q^{4} + 4 q^{11} - 18 q^{14} + 10 q^{16} - 8 q^{19} + 4 q^{26} - 14 q^{29} + 16 q^{31} + 8 q^{34} + 26 q^{41} - 44 q^{44} - 6 q^{46} + 4 q^{49} + 18 q^{56} - 4 q^{59} + 2 q^{61} + 30 q^{64} + 20 q^{71}+ \cdots + 62 q^{94}+O(q^{100}) \) 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

Fields

Representations

Groups

Database

Properties

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