LMFDB - Embedded newform 225.5.d.c.224.6

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [225,5,Mod(224,225)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(225, base_ring=CyclotomicField(2))

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

N = Newforms(chi, 5, names="a")

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

chi := DirichletCharacter("225.224");

S:= CuspForms(chi, 5);

N := Newforms(S);

Level:	$ N $	$=$	$ 225 = 3^{2} \cdot 5^{2} $
Weight:	$ k $	$=$	$ 5 $
Character orbit:	$[\chi]$	$=$	225.d (of order $2$, degree $1$, not minimal)

Newform invariants

comment: select newform

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

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

Self dual:	no
Analytic conductor:	$23.2582416939$
Analytic rank:	$0$
Dimension:	$12$
Coefficient field:	$\mathbb{Q}[x]/(x^{12} + \cdots)$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{12} + 44 x^{10} - 24 x^{9} + 968 x^{8} - 132 x^{7} - 10486 x^{6} + 2904 x^{5} + 56980 x^{4} + \cdots + 162 $ x^12 + 44x^10 - 24x^9 + 968x^8 - 132x^7 - 10486x^6 + 2904x^5 + 56980x^4 - 31836x^3 + 13068x^2 - 2376x + 162
Coefficient ring:	$\Z[a_1, \ldots, a_{13}]$
Coefficient ring index:	$ 2^{12}\cdot 3^{10}\cdot 5^{4} $
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label			224.6
Root			$0.136829 + 0.330335i$ of defining polynomial
Character	$\chi$	$=$	225.224
Dual form			225.5.d.c.224.5

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-1.80123 q^{2} -12.7556 q^{4} +12.1136i q^{7} +51.7953 q^{8} +O(q^{10})$	$q-1.80123 q^{2} -12.7556 q^{4} +12.1136i q^{7} +51.7953 q^{8} -60.4434i q^{11} -168.896i q^{13} -21.8194i q^{14} +110.794 q^{16} -377.366 q^{17} +45.1263 q^{19} +108.872i q^{22} -1.37206 q^{23} +304.221i q^{26} -154.516i q^{28} +1401.24i q^{29} +1605.80 q^{31} -1028.29 q^{32} +679.721 q^{34} +1324.72i q^{37} -81.2826 q^{38} +2354.07i q^{41} -468.707i q^{43} +770.991i q^{44} +2.47139 q^{46} -3910.98 q^{47} +2254.26 q^{49} +2154.37i q^{52} +3228.98 q^{53} +627.429i q^{56} -2523.94i q^{58} +2543.15i q^{59} -254.070 q^{61} -2892.40 q^{62} +79.4731 q^{64} +7493.35i q^{67} +4813.52 q^{68} +4143.85i q^{71} +3711.28i q^{73} -2386.11i q^{74} -575.612 q^{76} +732.189 q^{77} -80.2118 q^{79} -4240.21i q^{82} -5159.51 q^{83} +844.248i q^{86} -3130.68i q^{88} +5870.07i q^{89} +2045.95 q^{91} +17.5015 q^{92} +7044.55 q^{94} -8921.02i q^{97} -4060.43 q^{98} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 12 q + 184 q^{4}+O(q^{10}) $ 12 * q + 184 * q^4	$ 12 q + 184 q^{4} + 3632 q^{16} + 500 q^{19} + 3692 q^{31} + 7384 q^{34} + 19512 q^{46} + 6464 q^{49} + 21676 q^{61} + 72288 q^{64} - 49560 q^{76} - 7808 q^{79} + 34612 q^{91} - 132728 q^{94}+O(q^{100}) $ 12 * q + 184 * q^4 + 3632 * q^16 + 500 * q^19 + 3692 * q^31 + 7384 * q^34 + 19512 * q^46 + 6464 * q^49 + 21676 * q^61 + 72288 * q^64 - 49560 * q^76 - 7808 * q^79 + 34612 * q^91 - 132728 * q^94

Display 100 coefficients 10 coefficients

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/225\mathbb{Z}\right)^\times$.

$n$	$101$	$127$
$\chi(n)$	$-1$	$-1$

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 $

$2$	−1.80123	−0.450306	−0.225153	−	0.974323i	$-0.572288\pi$
$2$	−1.80123	−0.450306	−0.225153	+	0.974323i	$0.572288\pi$
$3$	0	0
$3$	0	0
$4$	−12.7556	−0.797224
$5$	0	0
$5$	0	0
$6$	0	0
$7$	12.1136i	0.247217i	0.992331	+	0.123609i	$0.0394467\pi$
$7$	12.1136i	0.247217i	−0.992331	+	0.123609i	$0.960553\pi$
$8$	51.7953	0.809302
$9$	0	0
$10$	0	0
$11$	− 60.4434i	− 0.499532i	−0.968306	−	0.249766i	$-0.919646\pi$
$11$	− 60.4434i	− 0.499532i	0.968306	−	0.249766i	$-0.0803537\pi$
$12$	0	0
$13$	− 168.896i	− 0.999387i	−0.866202	−	0.499694i	$-0.833446\pi$
$13$	− 168.896i	− 0.999387i	0.866202	−	0.499694i	$-0.166554\pi$
$14$	− 21.8194i	− 0.111323i
$15$	0	0
$16$	110.794	0.432790
$17$	−377.366	−1.30576	−0.652882	−	0.757460i	$-0.726440\pi$
$17$	−377.366	−1.30576	−0.652882	+	0.757460i	$0.726440\pi$
$18$	0	0
$19$	45.1263	0.125004	0.0625018	−	0.998045i	$-0.480092\pi$
$19$	45.1263	0.125004	0.0625018	+	0.998045i	$0.480092\pi$
$20$	0	0
$21$	0	0
$22$	108.872i	0.224943i
$23$	−1.37206	−0.00259369	−0.00129685	−	0.999999i	$-0.500413\pi$
$23$	−1.37206	−0.00259369	−0.00129685	+	0.999999i	$0.500413\pi$
$24$	0	0
$25$	0	0
$26$	304.221i	0.450030i
$27$	0	0
$28$	− 154.516i	− 0.197087i
$29$	1401.24i	1.66615i	0.553156	+	0.833077i	$0.313423\pi$
$29$	1401.24i	1.66615i	−0.553156	+	0.833077i	$0.686577\pi$
$30$	0	0
$31$	1605.80	1.67097	0.835483	−	0.549517i	$-0.185188\pi$
$31$	1605.80	1.67097	0.835483	+	0.549517i	$0.185188\pi$
$32$	−1028.29	−1.00419
$33$	0	0
$34$	679.721	0.587994
$35$	0	0
$36$	0	0
$37$	1324.72i	0.967652i	0.875164	+	0.483826i	$0.160753\pi$
$37$	1324.72i	0.967652i	−0.875164	+	0.483826i	$0.839247\pi$
$38$	−81.2826	−0.0562899
$39$	0	0
$40$	0	0
$41$	2354.07i	1.40040i	0.713948	+	0.700199i	$0.246905\pi$
$41$	2354.07i	1.40040i	−0.713948	+	0.700199i	$0.753095\pi$
$42$	0	0
$43$	− 468.707i	− 0.253492i	−0.991935	−	0.126746i	$-0.959547\pi$
$43$	− 468.707i	− 0.253492i	0.991935	−	0.126746i	$-0.0404534\pi$
$44$	770.991i	0.398239i
$45$	0	0
$46$	2.47139	0.00116796
$47$	−3910.98	−1.77047	−0.885237	−	0.465141i	$-0.846004\pi$
$47$	−3910.98	−1.77047	−0.885237	+	0.465141i	$0.846004\pi$
$48$	0	0
$49$	2254.26	0.938884
$50$	0	0
$51$	0	0
$52$	2154.37i	0.796735i
$53$	3228.98	1.14951	0.574756	−	0.818325i	$-0.305097\pi$
$53$	3228.98	1.14951	0.574756	+	0.818325i	$0.305097\pi$
$54$	0	0
$55$	0	0
$56$	627.429i	0.200073i
$57$	0	0
$58$	− 2523.94i	− 0.750280i
$59$	2543.15i	0.730582i	0.930893	+	0.365291i	$0.119030\pi$
$59$	2543.15i	0.730582i	−0.930893	+	0.365291i	$0.880970\pi$
$60$	0	0
$61$	−254.070	−0.0682800	−0.0341400	−	0.999417i	$-0.510869\pi$
$61$	−254.070	−0.0682800	−0.0341400	+	0.999417i	$0.510869\pi$
$62$	−2892.40	−0.752446
$63$	0	0
$64$	79.4731	0.0194026
$65$	0	0
$66$	0	0
$67$	7493.35i	1.66927i	0.550805	+	0.834634i	$0.314321\pi$
$67$	7493.35i	1.66927i	−0.550805	+	0.834634i	$0.685679\pi$
$68$	4813.52	1.04099
$69$	0	0
$70$	0	0
$71$	4143.85i	0.822029i	0.911629	+	0.411014i	$0.134825\pi$
$71$	4143.85i	0.822029i	−0.911629	+	0.411014i	$0.865175\pi$
$72$	0	0
$73$	3711.28i	0.696431i	0.937415	+	0.348215i	$0.113212\pi$
$73$	3711.28i	0.696431i	−0.937415	+	0.348215i	$0.886788\pi$
$74$	− 2386.11i	− 0.435740i
$75$	0	0
$76$	−575.612	−0.0996559
$77$	732.189	0.123493
$78$	0	0
$79$	−80.2118	−0.0128524	−0.00642620	−	0.999979i	$-0.502046\pi$
$79$	−80.2118	−0.0128524	−0.00642620	+	0.999979i	$0.502046\pi$
$80$	0	0
$81$	0	0
$82$	− 4240.21i	− 0.630608i
$83$	−5159.51	−0.748949	−0.374474	−	0.927237i	$-0.622177\pi$
$83$	−5159.51	−0.748949	−0.374474	+	0.927237i	$0.622177\pi$
$84$	0	0
$85$	0	0
$86$	844.248i	0.114149i
$87$	0	0
$88$	− 3130.68i	− 0.404272i
$89$	5870.07i	0.741076i	0.928817	+	0.370538i	$0.120827\pi$
$89$	5870.07i	0.741076i	−0.928817	+	0.370538i	$0.879173\pi$
$90$	0	0
$91$	2045.95	0.247065
$92$	17.5015	0.00206775
$93$	0	0
$94$	7044.55	0.797256
$95$	0	0
$96$	0	0
$97$	− 8921.02i	− 0.948137i	−0.880488	−	0.474068i	$-0.842785\pi$
$97$	− 8921.02i	− 0.948137i	0.880488	−	0.474068i	$-0.157215\pi$
$98$	−4060.43	−0.422785
$99$	0	0
$100$	0	0
$101$	− 2618.57i	− 0.256697i	−0.991729	−	0.128349i	$-0.959032\pi$
$101$	− 2618.57i	− 0.256697i	0.991729	−	0.128349i	$-0.0409676\pi$
$102$	0	0
$103$	9021.12i	0.850327i	0.905116	+	0.425164i	$0.139783\pi$
$103$	9021.12i	0.850327i	−0.905116	+	0.425164i	$0.860217\pi$
$104$	− 8748.04i	− 0.808805i
$105$	0	0
$106$	−5816.12	−0.517632
$107$	17075.5	1.49144	0.745722	−	0.666257i	$-0.232105\pi$
$107$	17075.5	1.49144	0.745722	+	0.666257i	$0.232105\pi$
$108$	0	0
$109$	−11156.2	−0.938997	−0.469498	−	0.882933i	$-0.655565\pi$
$109$	−11156.2	−0.938997	−0.469498	+	0.882933i	$0.655565\pi$
$110$	0	0
$111$	0	0
$112$	1342.12i	0.106993i
$113$	−15403.5	−1.20632	−0.603160	−	0.797621i	$-0.706092\pi$
$113$	−15403.5	−1.20632	−0.603160	+	0.797621i	$0.706092\pi$
$114$	0	0
$115$	0	0
$116$	− 17873.6i	− 1.32830i
$117$	0	0
$118$	− 4580.80i	− 0.328986i
$119$	− 4571.27i	− 0.322807i
$120$	0	0
$121$	10987.6	0.750468
$122$	457.637	0.0307469
$123$	0	0
$124$	−20482.9	−1.33213
$125$	0	0
$126$	0	0
$127$	− 24976.5i	− 1.54855i	−0.632852	−	0.774273i	$-0.718116\pi$
$127$	− 24976.5i	− 1.54855i	0.632852	−	0.774273i	$-0.281884\pi$
$128$	16309.5	0.995453
$129$	0	0
$130$	0	0
$131$	− 17627.3i	− 1.02717i	−0.858039	−	0.513585i	$-0.828317\pi$
$131$	− 17627.3i	− 1.02717i	0.858039	−	0.513585i	$-0.171683\pi$
$132$	0	0
$133$	546.643i	0.0309030i
$134$	− 13497.2i	− 0.751682i
$135$	0	0
$136$	−19545.8	−1.05676
$137$	24784.3	1.32049	0.660247	−	0.751049i	$-0.270452\pi$
$137$	24784.3	1.32049	0.660247	+	0.751049i	$0.270452\pi$
$138$	0	0
$139$	241.690	0.0125092	0.00625460	−	0.999980i	$-0.498009\pi$
$139$	241.690	0.0125092	0.00625460	+	0.999980i	$0.498009\pi$
$140$	0	0
$141$	0	0
$142$	− 7464.00i	− 0.370165i
$143$	−10208.7	−0.499226
$144$	0	0
$145$	0	0
$146$	− 6684.85i	− 0.313607i
$147$	0	0
$148$	− 16897.5i	− 0.771436i
$149$	− 12802.4i	− 0.576657i	−0.957532	−	0.288328i	$-0.906901\pi$
$149$	− 12802.4i	− 0.576657i	0.957532	−	0.288328i	$-0.0930995\pi$
$150$	0	0
$151$	−14780.0	−0.648218	−0.324109	−	0.946020i	$-0.605065\pi$
$151$	−14780.0	−0.648218	−0.324109	+	0.946020i	$0.605065\pi$
$152$	2337.33	0.101166
$153$	0	0
$154$	−1318.84	−0.0556096
$155$	0	0
$156$	0	0
$157$	− 9720.94i	− 0.394375i	−0.980366	−	0.197187i	$-0.936819\pi$
$157$	− 9720.94i	− 0.394375i	0.980366	−	0.197187i	$-0.0631807\pi$
$158$	144.480	0.00578752
$159$	0	0
$160$	0	0
$161$	− 16.6207i	0 0.000641204i
$162$	0	0
$163$	34436.1i	1.29610i	0.761597	+	0.648051i	$0.224416\pi$
$163$	34436.1i	1.29610i	−0.761597	+	0.648051i	$0.775584\pi$
$164$	− 30027.5i	− 1.11643i
$165$	0	0
$166$	9293.44	0.337256
$167$	16921.4	0.606740	0.303370	−	0.952873i	$-0.401888\pi$
$167$	16921.4	0.606740	0.303370	+	0.952873i	$0.401888\pi$
$168$	0	0
$169$	35.0031	0.00122555
$170$	0	0
$171$	0	0
$172$	5978.64i	0.202090i
$173$	−39172.9	−1.30886	−0.654431	−	0.756122i	$-0.727092\pi$
$173$	−39172.9	−1.30886	−0.654431	+	0.756122i	$0.727092\pi$
$174$	0	0
$175$	0	0
$176$	− 6696.79i	− 0.216193i
$177$	0	0
$178$	− 10573.3i	− 0.333711i
$179$	51682.6i	1.61301i	0.591224	+	0.806507i	$0.298645\pi$
$179$	51682.6i	1.61301i	−0.591224	+	0.806507i	$0.701355\pi$
$180$	0	0
$181$	662.351	0.0202177	0.0101088	−	0.999949i	$-0.496782\pi$
$181$	662.351	0.0202177	0.0101088	+	0.999949i	$0.496782\pi$
$182$	−3685.22	−0.111255
$183$	0	0
$184$	−71.0664	−0.00209908
$185$	0	0
$186$	0	0
$187$	22809.3i	0.652271i
$188$	49886.8	1.41146
$189$	0	0
$190$	0	0
$191$	63272.8i	1.73440i	0.497956	+	0.867202i	$0.334084\pi$
$191$	63272.8i	1.73440i	−0.497956	+	0.867202i	$0.665916\pi$
$192$	0	0
$193$	51567.4i	1.38440i	0.721707	+	0.692198i	$0.243358\pi$
$193$	51567.4i	1.38440i	−0.721707	+	0.692198i	$0.756642\pi$
$194$	16068.8i	0.426952i
$195$	0	0
$196$	−28754.4	−0.748501
$197$	11639.8	0.299926	0.149963	−	0.988692i	$-0.452085\pi$
$197$	11639.8	0.299926	0.149963	+	0.988692i	$0.452085\pi$
$198$	0	0
$199$	44340.2	1.11967	0.559837	−	0.828603i	$-0.310864\pi$
$199$	44340.2	1.11967	0.559837	+	0.828603i	$0.310864\pi$
$200$	0	0
$201$	0	0
$202$	4716.63i	0.115592i
$203$	−16974.1	−0.411902
$204$	0	0
$205$	0	0
$206$	− 16249.1i	− 0.382908i
$207$	0	0
$208$	− 18712.8i	− 0.432525i
$209$	− 2727.59i	− 0.0624433i
$210$	0	0
$211$	34425.8	0.773249	0.386624	−	0.922237i	$-0.373641\pi$
$211$	34425.8	0.773249	0.386624	+	0.922237i	$0.373641\pi$
$212$	−41187.5	−0.916418
$213$	0	0
$214$	−30756.9	−0.671607
$215$	0	0
$216$	0	0
$217$	19452.0i	0.413091i
$218$	20094.9	0.422836
$219$	0	0
$220$	0	0
$221$	63735.7i	1.30496i
$222$	0	0
$223$	− 80710.8i	− 1.62301i	−0.584344	−	0.811506i	$-0.698648\pi$
$223$	− 80710.8i	− 1.62301i	0.584344	−	0.811506i	$-0.301352\pi$
$224$	− 12456.3i	− 0.248253i
$225$	0	0
$226$	27745.2	0.543213
$227$	−96827.1	−1.87908	−0.939539	−	0.342441i	$-0.888746\pi$
$227$	−96827.1	−1.87908	−0.939539	+	0.342441i	$0.888746\pi$
$228$	0	0
$229$	63732.2	1.21531	0.607656	−	0.794200i	$-0.292110\pi$
$229$	63732.2	1.21531	0.607656	+	0.794200i	$0.292110\pi$
$230$	0	0
$231$	0	0
$232$	72577.5i	1.34842i
$233$	22886.7	0.421571	0.210786	−	0.977532i	$-0.432398\pi$
$233$	22886.7	0.421571	0.210786	+	0.977532i	$0.432398\pi$
$234$	0	0
$235$	0	0
$236$	− 32439.4i	− 0.582437i
$237$	0	0
$238$	8233.89i	0.145362i
$239$	26181.9i	0.458358i	0.973384	+	0.229179i	$0.0736041\pi$
$239$	26181.9i	0.458358i	−0.973384	+	0.229179i	$0.926396\pi$
$240$	0	0
$241$	5080.09	0.0874657	0.0437328	−	0.999043i	$-0.486075\pi$
$241$	5080.09	0.0874657	0.0437328	+	0.999043i	$0.486075\pi$
$242$	−19791.1	−0.337940
$243$	0	0
$244$	3240.81	0.0544345
$245$	0	0
$246$	0	0
$247$	− 7621.67i	− 0.124927i
$248$	83172.8	1.35231
$249$	0	0
$250$	0	0
$251$	23074.8i	0.366260i	0.983089	+	0.183130i	$0.0586230\pi$
$251$	23074.8i	0.366260i	−0.983089	+	0.183130i	$0.941377\pi$
$252$	0	0
$253$	82.9321i	0.00129563i
$254$	44988.3i	0.697320i
$255$	0	0
$256$	−30648.7	−0.467661
$257$	−51974.3	−0.786905	−0.393452	−	0.919345i	$-0.628719\pi$
$257$	−51974.3	−0.786905	−0.393452	+	0.919345i	$0.628719\pi$
$258$	0	0
$259$	−16047.1	−0.239220
$260$	0	0
$261$	0	0
$262$	31750.7i	0.462541i
$263$	−24353.6	−0.352089	−0.176044	−	0.984382i	$-0.556330\pi$
$263$	−24353.6	−0.352089	−0.176044	+	0.984382i	$0.556330\pi$
$264$	0	0
$265$	0	0
$266$	− 984.628i	− 0.0139158i
$267$	0	0
$268$	− 95582.0i	− 1.33078i
$269$	98409.2i	1.35998i	0.733223	+	0.679988i	$0.238015\pi$
$269$	98409.2i	1.35998i	−0.733223	+	0.679988i	$0.761985\pi$
$270$	0	0
$271$	135833.	1.84956	0.924778	−	0.380508i	$-0.124251\pi$
$271$	135833.	1.84956	0.924778	+	0.380508i	$0.124251\pi$
$272$	−41810.0	−0.565122
$273$	0	0
$274$	−44642.2	−0.594627
$275$	0	0
$276$	0	0
$277$	19597.6i	0.255413i	0.991812	+	0.127706i	$0.0407615\pi$
$277$	19597.6i	0.255413i	−0.991812	+	0.127706i	$0.959238\pi$
$278$	−435.339	−0.00563297
$279$	0	0
$280$	0	0
$281$	− 25457.8i	− 0.322409i	−0.986921	−	0.161205i	$-0.948462\pi$
$281$	− 25457.8i	− 0.322409i	0.986921	−	0.161205i	$-0.0515379\pi$
$282$	0	0
$283$	88022.8i	1.09906i	0.835473	+	0.549531i	$0.185194\pi$
$283$	88022.8i	1.09906i	−0.835473	+	0.549531i	$0.814806\pi$
$284$	− 52857.2i	− 0.655341i
$285$	0	0
$286$	18388.1	0.224805
$287$	−28516.3	−0.346202
$288$	0	0
$289$	58883.9	0.705020
$290$	0	0
$291$	0	0
$292$	− 47339.5i	− 0.555211i
$293$	−91729.0	−1.06849	−0.534246	−	0.845329i	$-0.679404\pi$
$293$	−91729.0	−1.06849	−0.534246	+	0.845329i	$0.679404\pi$
$294$	0	0
$295$	0	0
$296$	68614.1i	0.783123i
$297$	0	0
$298$	23059.9i	0.259672i
$299$	231.736i	0.00259210i
$300$	0	0
$301$	5677.75	0.0626676
$302$	26622.2	0.291897
$303$	0	0
$304$	4999.74	0.0541004
$305$	0	0
$306$	0	0
$307$	1346.81i	0.0142899i	0.999974	+	0.00714494i	$0.00227433\pi$
$307$	1346.81i	0.0142899i	−0.999974	+	0.00714494i	$0.997726\pi$
$308$	−9339.50	−0.0984515
$309$	0	0
$310$	0	0
$311$	48663.9i	0.503137i	0.967839	+	0.251568i	$0.0809464\pi$
$311$	48663.9i	0.503137i	−0.967839	+	0.251568i	$0.919054\pi$
$312$	0	0
$313$	− 106387.i	− 1.08593i	−0.839757	−	0.542963i	$-0.817302\pi$
$313$	− 106387.i	− 1.08593i	0.839757	−	0.542963i	$-0.182698\pi$
$314$	17509.6i	0.177589i
$315$	0	0
$316$	1023.15	0.0102462
$317$	92241.5	0.917926	0.458963	−	0.888455i	$-0.348221\pi$
$317$	92241.5	0.917926	0.458963	+	0.888455i	$0.348221\pi$
$318$	0	0
$319$	84695.5	0.832298
$320$	0	0
$321$	0	0
$322$	29.9376i	0 0.000288738i
$323$	−17029.1	−0.163225
$324$	0	0
$325$	0	0
$326$	− 62027.2i	− 0.583643i
$327$	0	0
$328$	121930.i	1.13334i
$329$	− 47376.1i	− 0.437691i
$330$	0	0
$331$	−160354.	−1.46361	−0.731803	−	0.681516i	$-0.761321\pi$
$331$	−160354.	−1.46361	−0.731803	+	0.681516i	$0.761321\pi$
$332$	65812.5	0.597080
$333$	0	0
$334$	−30479.2	−0.273219
$335$	0	0
$336$	0	0
$337$	− 213689.i	− 1.88158i	−0.338996	−	0.940788i	$-0.610087\pi$
$337$	− 213689.i	− 1.88158i	0.338996	−	0.940788i	$-0.389913\pi$
$338$	−63.0484	−0.000551875 0
$339$	0	0
$340$	0	0
$341$	− 97059.9i	− 0.834701i
$342$	0	0
$343$	56392.1i	0.479325i
$344$	− 24276.8i	− 0.205152i
$345$	0	0
$346$	70559.3	0.589389
$347$	61402.4	0.509948	0.254974	−	0.966948i	$-0.417933\pi$
$347$	61402.4	0.509948	0.254974	+	0.966948i	$0.417933\pi$
$348$	0	0
$349$	−215730.	−1.77117	−0.885585	−	0.464477i	$-0.846242\pi$
$349$	−215730.	−1.77117	−0.885585	+	0.464477i	$0.846242\pi$
$350$	0	0
$351$	0	0
$352$	62153.4i	0.501625i
$353$	13643.8	0.109493	0.0547465	−	0.998500i	$-0.482565\pi$
$353$	13643.8	0.109493	0.0547465	+	0.998500i	$0.482565\pi$
$354$	0	0
$355$	0	0
$356$	− 74876.1i	− 0.590804i
$357$	0	0
$358$	− 93092.0i	− 0.726351i
$359$	− 130543.i	− 1.01290i	−0.862270	−	0.506449i	$-0.830958\pi$
$359$	− 130543.i	− 1.01290i	0.862270	−	0.506449i	$-0.169042\pi$
$360$	0	0
$361$	−128285.	−0.984374
$362$	−1193.04	−0.00910414
$363$	0	0
$364$	−26097.3	−0.196967
$365$	0	0
$366$	0	0
$367$	168011.i	1.24740i	0.781663	+	0.623701i	$0.214372\pi$
$367$	168011.i	1.24740i	−0.781663	+	0.623701i	$0.785628\pi$
$368$	−152.017	−0.00112252
$369$	0	0
$370$	0	0
$371$	39114.6i	0.284179i
$372$	0	0
$373$	158259.i	1.13750i	0.822510	+	0.568750i	$0.192573\pi$
$373$	158259.i	1.13750i	−0.822510	+	0.568750i	$0.807427\pi$
$374$	− 41084.6i	− 0.293722i
$375$	0	0
$376$	−202570.	−1.43285
$377$	236664.	1.66513
$378$	0	0
$379$	18061.8	0.125743	0.0628715	−	0.998022i	$-0.479974\pi$
$379$	18061.8	0.125743	0.0628715	+	0.998022i	$0.479974\pi$
$380$	0	0
$381$	0	0
$382$	− 113969.i	− 0.781013i
$383$	110035.	0.750126	0.375063	−	0.926999i	$-0.377621\pi$
$383$	110035.	0.750126	0.375063	+	0.926999i	$0.377621\pi$
$384$	0	0
$385$	0	0
$386$	− 92884.5i	− 0.623403i
$387$	0	0
$388$	113793.i	0.755877i
$389$	− 247517.i	− 1.63571i	−0.575425	−	0.817855i	$-0.695163\pi$
$389$	− 247517.i	− 1.63571i	0.575425	−	0.817855i	$-0.304837\pi$
$390$	0	0
$391$	517.769	0.00338675
$392$	116760.	0.759840
$393$	0	0
$394$	−20966.0	−0.135059
$395$	0	0
$396$	0	0
$397$	− 170927.i	− 1.08450i	−0.840218	−	0.542249i	$-0.817573\pi$
$397$	− 170927.i	− 1.08450i	0.840218	−	0.542249i	$-0.182427\pi$
$398$	−79866.7	−0.504196
$399$	0	0
$400$	0	0
$401$	− 209413.i	− 1.30231i	−0.758944	−	0.651156i	$-0.774284\pi$
$401$	− 209413.i	− 1.30231i	0.758944	−	0.651156i	$-0.225716\pi$
$402$	0	0
$403$	− 271213.i	− 1.66994i
$404$	33401.4i	0.204645i
$405$	0	0
$406$	30574.1	0.185482
$407$	80070.3	0.483374
$408$	0	0
$409$	−47045.0	−0.281233	−0.140617	−	0.990064i	$-0.544908\pi$
$409$	−47045.0	−0.281233	−0.140617	+	0.990064i	$0.544908\pi$
$410$	0	0
$411$	0	0
$412$	− 115070.i	− 0.677902i
$413$	−30806.8	−0.180612
$414$	0	0
$415$	0	0
$416$	173675.i	1.00357i
$417$	0	0
$418$	4913.00i	0.0281186i
$419$	88341.4i	0.503195i	0.967832	+	0.251597i	$0.0809559\pi$
$419$	88341.4i	0.503195i	−0.967832	+	0.251597i	$0.919044\pi$
$420$	0	0
$421$	−146004.	−0.823759	−0.411879	−	0.911238i	$-0.635127\pi$
$421$	−146004.	−0.823759	−0.411879	+	0.911238i	$0.635127\pi$
$422$	−62008.6	−0.348199
$423$	0	0
$424$	167246.	0.930301
$425$	0	0
$426$	0	0
$427$	− 3077.71i	− 0.0168800i
$428$	−217809.	−1.18901
$429$	0	0
$430$	0	0
$431$	− 38335.8i	− 0.206372i	−0.994662	−	0.103186i	$-0.967096\pi$
$431$	− 38335.8i	− 0.206372i	0.994662	−	0.103186i	$-0.0329036\pi$
$432$	0	0
$433$	236134.i	1.25946i	0.776815	+	0.629728i	$0.216834\pi$
$433$	236134.i	1.25946i	−0.776815	+	0.629728i	$0.783166\pi$
$434$	− 35037.5i	− 0.186018i
$435$	0	0
$436$	142304.	0.748591
$437$	−61.9161	−0.000324221 0
$438$	0	0
$439$	−240933.	−1.25017	−0.625083	−	0.780558i	$-0.714935\pi$
$439$	−240933.	−1.25017	−0.625083	+	0.780558i	$0.714935\pi$
$440$	0	0
$441$	0	0
$442$	− 114802.i	− 0.587633i
$443$	70261.5	0.358022	0.179011	−	0.983847i	$-0.442710\pi$
$443$	70261.5	0.358022	0.179011	+	0.983847i	$0.442710\pi$
$444$	0	0
$445$	0	0
$446$	145378.i	0.730853i
$447$	0	0
$448$	962.708i	0.00479666i
$449$	− 93152.3i	− 0.462062i	−0.972946	−	0.231031i	$-0.925790\pi$
$449$	− 93152.3i	− 0.462062i	0.972946	−	0.231031i	$-0.0742100\pi$
$450$	0	0
$451$	142288.	0.699544
$452$	196481.	0.961707
$453$	0	0
$454$	174407.	0.846161
$455$	0	0
$456$	0	0
$457$	145652.i	0.697406i	0.937233	+	0.348703i	$0.113378\pi$
$457$	145652.i	0.697406i	−0.937233	+	0.348703i	$0.886622\pi$
$458$	−114796.	−0.547263
$459$	0	0
$460$	0	0
$461$	− 128000.i	− 0.602294i	−0.953578	−	0.301147i	$-0.902630\pi$
$461$	− 128000.i	− 0.602294i	0.953578	−	0.301147i	$-0.0973695\pi$
$462$	0	0
$463$	112886.i	0.526597i	0.964714	+	0.263298i	$0.0848104\pi$
$463$	112886.i	0.526597i	−0.964714	+	0.263298i	$0.915190\pi$
$464$	155249.i	0.721096i
$465$	0	0
$466$	−41224.1	−0.189836
$467$	−317214.	−1.45452	−0.727258	−	0.686364i	$-0.759206\pi$
$467$	−317214.	−1.45452	−0.727258	+	0.686364i	$0.759206\pi$
$468$	0	0
$469$	−90771.7	−0.412672
$470$	0	0
$471$	0	0
$472$	131723.i	0.591261i
$473$	−28330.3	−0.126628
$474$	0	0
$475$	0	0
$476$	58309.2i	0.257350i
$477$	0	0
$478$	− 47159.4i	− 0.206402i
$479$	374765.i	1.63338i	0.577073	+	0.816692i	$0.304195\pi$
$479$	374765.i	1.63338i	−0.577073	+	0.816692i	$0.695805\pi$
$480$	0	0
$481$	223740.	0.967059
$482$	−9150.40	−0.0393864
$483$	0	0
$484$	−140153.	−0.598291
$485$	0	0
$486$	0	0
$487$	− 47680.4i	− 0.201040i	−0.994935	−	0.100520i	$-0.967949\pi$
$487$	− 47680.4i	− 0.201040i	0.994935	−	0.100520i	$-0.0320506\pi$
$488$	−13159.6	−0.0552591
$489$	0	0
$490$	0	0
$491$	− 153766.i	− 0.637817i	−0.947786	−	0.318908i	$-0.896684\pi$
$491$	− 153766.i	− 0.637817i	0.947786	−	0.318908i	$-0.103316\pi$
$492$	0	0
$493$	− 528779.i	− 2.17561i
$494$	13728.3i	0.0562554i
$495$	0	0
$496$	177913.	0.723178
$497$	−50197.0	−0.203220
$498$	0	0
$499$	348851.	1.40100	0.700501	−	0.713651i	$-0.252960\pi$
$499$	348851.	1.40100	0.700501	+	0.713651i	$0.252960\pi$
$500$	0	0
$501$	0	0
$502$	− 41562.9i	− 0.164929i
$503$	269552.	1.06539	0.532693	−	0.846308i	$-0.321180\pi$
$503$	269552.	1.06539	0.532693	+	0.846308i	$0.321180\pi$
$504$	0	0
$505$	0	0
$506$	− 149.379i	0 0.000583431i
$507$	0	0
$508$	318590.i	1.23454i
$509$	− 177376.i	− 0.684636i	−0.939584	−	0.342318i	$-0.888788\pi$
$509$	− 177376.i	− 0.684636i	0.939584	−	0.342318i	$-0.111212\pi$
$510$	0	0
$511$	−44957.1	−0.172170
$512$	−205747.	−0.784862
$513$	0	0
$514$	93617.4	0.354348
$515$	0	0
$516$	0	0
$517$	236393.i	0.884409i
$518$	28904.5	0.107722
$519$	0	0
$520$	0	0
$521$	233062.i	0.858609i	0.903160	+	0.429305i	$0.141241\pi$
$521$	233062.i	0.858609i	−0.903160	+	0.429305i	$0.858759\pi$
$522$	0	0
$523$	− 315841.i	− 1.15469i	−0.816501	−	0.577345i	$-0.804089\pi$
$523$	− 315841.i	− 1.15469i	0.816501	−	0.577345i	$-0.195911\pi$
$524$	224846.i	0.818885i
$525$	0	0
$526$	43866.4	0.158548
$527$	−605973.	−2.18189
$528$	0	0
$529$	−279839.	−0.999993
$530$	0	0
$531$	0	0
$532$	− 6972.76i	− 0.0246366i
$533$	397594.	1.39954
$534$	0	0
$535$	0	0
$536$	388120.i	1.35094i
$537$	0	0
$538$	− 177257.i	− 0.612406i
$539$	− 136255.i	− 0.469003i
$540$	0	0
$541$	−186404.	−0.636886	−0.318443	−	0.947942i	$-0.603160\pi$
$541$	−186404.	−0.636886	−0.318443	+	0.947942i	$0.603160\pi$
$542$	−244666.	−0.832867
$543$	0	0
$544$	388042.	1.31123
$545$	0	0
$546$	0	0
$547$	443355.i	1.48176i	0.671639	+	0.740879i	$0.265591\pi$
$547$	443355.i	1.48176i	−0.671639	+	0.740879i	$0.734409\pi$
$548$	−316139.	−1.05273
$549$	0	0
$550$	0	0
$551$	63232.6i	0.208275i
$552$	0	0
$553$	− 971.657i	− 0.00317733i
$554$	− 35299.7i	− 0.115014i
$555$	0	0
$556$	−3082.90	−0.00997263
$557$	82562.1	0.266116	0.133058	−	0.991108i	$-0.457520\pi$
$557$	82562.1	0.266116	0.133058	+	0.991108i	$0.457520\pi$
$558$	0	0
$559$	−79163.0	−0.253337
$560$	0	0
$561$	0	0
$562$	45855.2i	0.145183i
$563$	518910.	1.63710	0.818550	−	0.574435i	$-0.194778\pi$
$563$	518910.	1.63710	0.818550	+	0.574435i	$0.194778\pi$
$564$	0	0
$565$	0	0
$566$	− 158549.i	− 0.494915i
$567$	0	0
$568$	214632.i	0.665269i
$569$	152195.i	0.470085i	0.971985	+	0.235042i	$0.0755229\pi$
$569$	152195.i	0.470085i	−0.971985	+	0.235042i	$0.924477\pi$
$570$	0	0
$571$	414409.	1.27103	0.635516	−	0.772088i	$-0.280787\pi$
$571$	414409.	1.27103	0.635516	+	0.772088i	$0.280787\pi$
$572$	130218.	0.397995
$573$	0	0
$574$	51364.4	0.155897
$575$	0	0
$576$	0	0
$577$	− 31681.0i	− 0.0951585i	−0.998867	−	0.0475793i	$-0.984849\pi$
$577$	− 31681.0i	− 0.0951585i	0.998867	−	0.0475793i	$-0.0151507\pi$
$578$	−106063.	−0.317475
$579$	0	0
$580$	0	0
$581$	− 62500.4i	− 0.185153i
$582$	0	0
$583$	− 195170.i	− 0.574218i
$584$	192227.i	0.563622i
$585$	0	0
$586$	165225.	0.481149
$587$	−347152.	−1.00750	−0.503748	−	0.863850i	$-0.668046\pi$
$587$	−347152.	−1.00750	−0.503748	+	0.863850i	$0.668046\pi$
$588$	0	0
$589$	72463.7	0.208877
$590$	0	0
$591$	0	0
$592$	146771.i	0.418791i
$593$	27468.2	0.0781125	0.0390563	−	0.999237i	$-0.487565\pi$
$593$	27468.2	0.0781125	0.0390563	+	0.999237i	$0.487565\pi$
$594$	0	0
$595$	0	0
$596$	163302.i	0.459725i
$597$	0	0
$598$	− 417.409i	− 0.00116724i
$599$	600764.i	1.67436i	0.546924	+	0.837182i	$0.315799\pi$
$599$	600764.i	1.67436i	−0.546924	+	0.837182i	$0.684201\pi$
$600$	0	0
$601$	−494521.	−1.36910	−0.684551	−	0.728965i	$-0.740002\pi$
$601$	−494521.	−1.36910	−0.684551	+	0.728965i	$0.740002\pi$
$602$	−10226.9	−0.0282196
$603$	0	0
$604$	188528.	0.516775
$605$	0	0
$606$	0	0
$607$	578761.i	1.57080i	0.618987	+	0.785401i	$0.287543\pi$
$607$	578761.i	1.57080i	−0.618987	+	0.785401i	$0.712457\pi$
$608$	−46402.9	−0.125527
$609$	0	0
$610$	0	0
$611$	660550.i	1.76939i
$612$	0	0
$613$	538455.i	1.43294i	0.697616	+	0.716471i	$0.254244\pi$
$613$	538455.i	1.43294i	−0.697616	+	0.716471i	$0.745756\pi$
$614$	− 2425.90i	− 0.00643483i
$615$	0	0
$616$	37924.0	0.0999430
$617$	486339.	1.27752	0.638762	−	0.769404i	$-0.279447\pi$
$617$	486339.	1.27752	0.638762	+	0.769404i	$0.279447\pi$
$618$	0	0
$619$	163391.	0.426430	0.213215	−	0.977005i	$-0.431606\pi$
$619$	163391.	0.426430	0.213215	+	0.977005i	$0.431606\pi$
$620$	0	0
$621$	0	0
$622$	− 87654.7i	− 0.226566i
$623$	−71107.8	−0.183207
$624$	0	0
$625$	0	0
$626$	191627.i	0.488999i
$627$	0	0
$628$	123996.i	0.314405i
$629$	− 499903.i	− 1.26353i
$630$	0	0
$631$	−132675.	−0.333220	−0.166610	−	0.986023i	$-0.553282\pi$
$631$	−132675.	−0.333220	−0.166610	+	0.986023i	$0.553282\pi$
$632$	−4154.60	−0.0104015
$633$	0	0
$634$	−166148.	−0.413348
$635$	0	0
$636$	0	0
$637$	− 380736.i	− 0.938308i
$638$	−152556.	−0.374789
$639$	0	0
$640$	0	0
$641$	− 648011.i	− 1.57713i	−0.614954	−	0.788563i	$-0.710826\pi$
$641$	− 648011.i	− 1.57713i	0.614954	−	0.788563i	$-0.289174\pi$
$642$	0	0
$643$	62712.3i	0.151681i	0.997120	+	0.0758405i	$0.0241640\pi$
$643$	62712.3i	0.151681i	−0.997120	+	0.0758405i	$0.975836\pi$
$644$	212.006i	0 0.000511184i
$645$	0	0
$646$	30673.3	0.0735013
$647$	−472097.	−1.12778	−0.563888	−	0.825852i	$-0.690695\pi$
$647$	−472097.	−1.12778	−0.563888	+	0.825852i	$0.690695\pi$
$648$	0	0
$649$	153717.	0.364949
$650$	0	0
$651$	0	0
$652$	− 439253.i	− 1.03328i
$653$	−233253.	−0.547017	−0.273509	−	0.961870i	$-0.588184\pi$
$653$	−233253.	−0.547017	−0.273509	+	0.961870i	$0.588184\pi$
$654$	0	0
$655$	0	0
$656$	260818.i	0.606079i
$657$	0	0
$658$	85335.1i	0.197095i
$659$	− 399872.i	− 0.920768i	−0.887720	−	0.460384i	$-0.847712\pi$
$659$	− 399872.i	− 0.920768i	0.887720	−	0.460384i	$-0.152288\pi$
$660$	0	0
$661$	112993.	0.258613	0.129306	−	0.991605i	$-0.458725\pi$
$661$	112993.	0.258613	0.129306	+	0.991605i	$0.458725\pi$
$662$	288834.	0.659071
$663$	0	0
$664$	−267238.	−0.606125
$665$	0	0
$666$	0	0
$667$	− 1922.58i	− 0.00432149i
$668$	−215842.	−0.483708
$669$	0	0
$670$	0	0
$671$	15356.9i	0.0341081i
$672$	0	0
$673$	− 544760.i	− 1.20275i	−0.798967	−	0.601374i	$-0.794620\pi$
$673$	− 544760.i	− 1.20275i	0.798967	−	0.601374i	$-0.205380\pi$
$674$	384901.i	0.847285i
$675$	0	0
$676$	−446.485	−0.000977042 0
$677$	475103.	1.03660	0.518299	−	0.855199i	$-0.326565\pi$
$677$	475103.	1.03660	0.518299	+	0.855199i	$0.326565\pi$
$678$	0	0
$679$	108066.	0.234396
$680$	0	0
$681$	0	0
$682$	174827.i	0.375871i
$683$	53819.1	0.115371	0.0576853	−	0.998335i	$-0.481628\pi$
$683$	53819.1	0.115371	0.0576853	+	0.998335i	$0.481628\pi$
$684$	0	0
$685$	0	0
$686$	− 101575.i	− 0.215843i
$687$	0	0
$688$	− 51930.1i	− 0.109709i
$689$	− 545363.i	− 1.14881i
$690$	0	0
$691$	−400653.	−0.839097	−0.419548	−	0.907733i	$-0.637812\pi$
$691$	−400653.	−0.839097	−0.419548	+	0.907733i	$0.637812\pi$
$692$	499673.	1.04346
$693$	0	0
$694$	−110600.	−0.229633
$695$	0	0
$696$	0	0
$697$	− 888345.i	− 1.82859i
$698$	388579.	0.797569
$699$	0	0
$700$	0	0
$701$	− 860604.i	− 1.75133i	−0.482921	−	0.875664i	$-0.660424\pi$
$701$	− 860604.i	− 1.75133i	0.482921	−	0.875664i	$-0.339576\pi$
$702$	0	0
$703$	59779.5i	0.120960i
$704$	− 4803.63i	− 0.00969223i
$705$	0	0
$706$	−24575.6	−0.0493054
$707$	31720.4	0.0634599
$708$	0	0
$709$	−489913.	−0.974601	−0.487300	−	0.873234i	$-0.662018\pi$
$709$	−489913.	−0.974601	−0.487300	+	0.873234i	$0.662018\pi$
$710$	0	0
$711$	0	0
$712$	304042.i	0.599754i
$713$	−2203.25	−0.00433397
$714$	0	0
$715$	0	0
$716$	− 659242.i	− 1.28593i
$717$	0	0
$718$	235138.i	0.456114i
$719$	− 288471.i	− 0.558014i	−0.960289	−	0.279007i	$-0.909995\pi$
$719$	− 288471.i	− 0.558014i	0.960289	−	0.279007i	$-0.0900053\pi$
$720$	0	0
$721$	−109279.	−0.210215
$722$	231070.	0.443270
$723$	0	0
$724$	−8448.67	−0.0161180
$725$	0	0
$726$	0	0
$727$	75173.7i	0.142232i	0.997468	+	0.0711160i	$0.0226561\pi$
$727$	75173.7i	0.142232i	−0.997468	+	0.0711160i	$0.977344\pi$
$728$	105971.	0.199950
$729$	0	0
$730$	0	0
$731$	176874.i	0.331001i
$732$	0	0
$733$	108535.i	0.202005i	0.994886	+	0.101003i	$0.0322051\pi$
$733$	108535.i	0.202005i	−0.994886	+	0.101003i	$0.967795\pi$
$734$	− 302627.i	− 0.561714i
$735$	0	0
$736$	1410.88	0.00260456
$737$	452923.	0.833853
$738$	0	0
$739$	−654768.	−1.19894	−0.599472	−	0.800396i	$-0.704623\pi$
$739$	−654768.	−1.19894	−0.599472	+	0.800396i	$0.704623\pi$
$740$	0	0
$741$	0	0
$742$	− 70454.3i	− 0.127967i
$743$	−423118.	−0.766450	−0.383225	−	0.923655i	$-0.625187\pi$
$743$	−423118.	−0.766450	−0.383225	+	0.923655i	$0.625187\pi$
$744$	0	0
$745$	0	0
$746$	− 285061.i	− 0.512224i
$747$	0	0
$748$	− 290946.i	− 0.520006i
$749$	206847.i	0.368710i
$750$	0	0
$751$	93003.6	0.164900	0.0824498	−	0.996595i	$-0.473726\pi$
$751$	93003.6	0.164900	0.0824498	+	0.996595i	$0.473726\pi$
$752$	−433314.	−0.766244
$753$	0	0
$754$	−426285.	−0.749820
$755$	0	0
$756$	0	0
$757$	− 849613.i	− 1.48262i	−0.671164	−	0.741309i	$-0.734205\pi$
$757$	− 849613.i	− 1.48262i	0.671164	−	0.741309i	$-0.265795\pi$
$758$	−32533.4	−0.0566228
$759$	0	0
$760$	0	0
$761$	− 407917.i	− 0.704373i	−0.935930	−	0.352186i	$-0.885438\pi$
$761$	− 407917.i	− 0.704373i	0.935930	−	0.352186i	$-0.114562\pi$
$762$	0	0
$763$	− 135142.i	− 0.232136i
$764$	− 807082.i	− 1.38271i
$765$	0	0
$766$	−198198.	−0.337787
$767$	429530.	0.730134
$768$	0	0
$769$	−569500.	−0.963033	−0.481517	−	0.876437i	$-0.659914\pi$
$769$	−569500.	−0.963033	−0.481517	+	0.876437i	$0.659914\pi$
$770$	0	0
$771$	0	0
$772$	− 657772.i	− 1.10367i
$773$	−711311.	−1.19042	−0.595211	−	0.803570i	$-0.702931\pi$
$773$	−711311.	−1.19042	−0.595211	+	0.803570i	$0.702931\pi$
$774$	0	0
$775$	0	0
$776$	− 462067.i	− 0.767329i
$777$	0	0
$778$	445834.i	0.736571i
$779$	106230.i	0.175055i
$780$	0	0
$781$	250468.	0.410630
$782$	−932.619	−0.00152507
$783$	0	0
$784$	249759.	0.406340
$785$	0	0
$786$	0	0
$787$	− 215275.i	− 0.347572i	−0.984783	−	0.173786i	$-0.944400\pi$
$787$	− 215275.i	− 0.347572i	0.984783	−	0.173786i	$-0.0556001\pi$
$788$	−148473.	−0.239108
$789$	0	0
$790$	0	0
$791$	− 186592.i	− 0.298223i
$792$	0	0
$793$	42911.5i	0.0682382i
$794$	307878.i	0.488357i
$795$	0	0
$796$	−565585.	−0.892630
$797$	623058.	0.980871	0.490435	−	0.871478i	$-0.336838\pi$
$797$	623058.	0.980871	0.490435	+	0.871478i	$0.336838\pi$
$798$	0	0
$799$	1.47587e6	2.31182
$800$	0	0
$801$	0	0
$802$	377200.i	0.586440i
$803$	224322.	0.347890
$804$	0	0
$805$	0	0
$806$	488517.i	0.751985i
$807$	0	0
$808$	− 135630.i	− 0.207745i
$809$	− 385095.i	− 0.588398i	−0.955744	−	0.294199i	$-0.904947\pi$
$809$	− 385095.i	− 0.588398i	0.955744	−	0.294199i	$-0.0950528\pi$
$810$	0	0
$811$	586828.	0.892215	0.446107	−	0.894979i	$-0.352810\pi$
$811$	586828.	0.892215	0.446107	+	0.894979i	$0.352810\pi$
$812$	216514.	0.328378
$813$	0	0
$814$	−144225.	−0.217666
$815$	0	0
$816$	0	0
$817$	− 21151.0i	− 0.0316875i
$818$	84738.6	0.126641
$819$	0	0
$820$	0	0
$821$	879206.i	1.30438i	0.758055	+	0.652191i	$0.226150\pi$
$821$	879206.i	1.30438i	−0.758055	+	0.652191i	$0.773850\pi$
$822$	0	0
$823$	− 56403.1i	− 0.0832728i	−0.999133	−	0.0416364i	$-0.986743\pi$
$823$	− 56403.1i	− 0.0832728i	0.999133	−	0.0416364i	$-0.0132571\pi$
$824$	467252.i	0.688171i
$825$	0	0
$826$	55490.1	0.0813308
$827$	−201627.	−0.294807	−0.147403	−	0.989076i	$-0.547092\pi$
$827$	−201627.	−0.294807	−0.147403	+	0.989076i	$0.547092\pi$
$828$	0	0
$829$	85071.2	0.123787	0.0618933	−	0.998083i	$-0.480286\pi$
$829$	85071.2	0.123787	0.0618933	+	0.998083i	$0.480286\pi$
$830$	0	0
$831$	0	0
$832$	− 13422.7i	− 0.0193907i
$833$	−850681.	−1.22596
$834$	0	0
$835$	0	0
$836$	34792.0i	0.0497813i
$837$	0	0
$838$	− 159123.i	− 0.226592i
$839$	150877.i	0.214338i	0.994241	+	0.107169i	$0.0341785\pi$
$839$	150877.i	0.214338i	−0.994241	+	0.107169i	$0.965821\pi$
$840$	0	0
$841$	−1.25618e6	−1.77607
$842$	262986.	0.370944
$843$	0	0
$844$	−439121.	−0.616452
$845$	0	0
$846$	0	0
$847$	133100.i	0.185528i
$848$	357752.	0.497497
$849$	0	0
$850$	0	0
$851$	− 1817.59i	− 0.00250979i
$852$	0	0
$853$	1.17719e6i	1.61788i	0.587890	+	0.808941i	$0.299959\pi$
$853$	1.17719e6i	1.61788i	−0.587890	+	0.808941i	$0.700041\pi$
$854$	5543.65i	0.00760117i
$855$	0	0
$856$	884433.	1.20703
$857$	341140.	0.464485	0.232242	−	0.972658i	$-0.425394\pi$
$857$	341140.	0.464485	0.232242	+	0.972658i	$0.425394\pi$
$858$	0	0
$859$	−44556.7	−0.0603846	−0.0301923	−	0.999544i	$-0.509612\pi$
$859$	−44556.7	−0.0603846	−0.0301923	+	0.999544i	$0.509612\pi$
$860$	0	0
$861$	0	0
$862$	69051.4i	0.0929304i
$863$	5280.21	0.00708973	0.00354486	−	0.999994i	$-0.498872\pi$
$863$	5280.21	0.00708973	0.00354486	+	0.999994i	$0.498872\pi$
$864$	0	0
$865$	0	0
$866$	− 425331.i	− 0.567141i
$867$	0	0
$868$	− 248122.i	− 0.329326i
$869$	4848.27i	0.00642019i
$870$	0	0
$871$	1.26560e6	1.66825
$872$	−577840.	−0.759932
$873$	0	0
$874$	111.525	0.000145999 0
$875$	0	0
$876$	0	0
$877$	1228.71i	0.00159754i	1.00000		0.000798768i	$0.000254256\pi$
$877$	1228.71i	0.00159754i	−1.00000		0.000798768i	$0.999746\pi$
$878$	433975.	0.562958
$879$	0	0
$880$	0	0
$881$	513912.i	0.662120i	0.943610	+	0.331060i	$0.107406\pi$
$881$	513912.i	0.662120i	−0.943610	+	0.331060i	$0.892594\pi$
$882$	0	0
$883$	− 116500.i	− 0.149419i	−0.997205	−	0.0747093i	$-0.976197\pi$
$883$	− 116500.i	− 0.149419i	0.997205	−	0.0747093i	$-0.0238029\pi$
$884$	− 812987.i	− 1.04035i
$885$	0	0
$886$	−126557.	−0.161220
$887$	−1.23199e6	−1.56588	−0.782941	−	0.622096i	$-0.786281\pi$
$887$	−1.23199e6	−1.56588	−0.782941	+	0.622096i	$0.786281\pi$
$888$	0	0
$889$	302556.	0.382827
$890$	0	0
$891$	0	0
$892$	1.02951e6i	1.29390i
$893$	−176488.	−0.221316
$894$	0	0
$895$	0	0
$896$	197567.i	0.246093i
$897$	0	0
$898$	167788.i	0.208070i
$899$	2.25010e6i	2.78409i
$900$	0	0
$901$	−1.21851e6	−1.50099
$902$	−256293.	−0.315009
$903$	0	0
$904$	−797828.	−0.976276
$905$	0	0
$906$	0	0
$907$	1.02123e6i	1.24139i	0.784053	+	0.620694i	$0.213149\pi$
$907$	1.02123e6i	1.24139i	−0.784053	+	0.620694i	$0.786851\pi$
$908$	1.23509e6	1.49805
$909$	0	0
$910$	0	0
$911$	1.49023e6i	1.79562i	0.440378	+	0.897812i	$0.354844\pi$
$911$	1.49023e6i	1.79562i	−0.440378	+	0.897812i	$0.645156\pi$
$912$	0	0
$913$	311858.i	0.374124i
$914$	− 262353.i	− 0.314046i
$915$	0	0
$916$	−812942.	−0.968877
$917$	213530.	0.253934
$918$	0	0
$919$	1.16759e6	1.38248	0.691240	−	0.722625i	$-0.257065\pi$
$919$	1.16759e6	1.38248	0.691240	+	0.722625i	$0.257065\pi$
$920$	0	0
$921$	0	0
$922$	230557.i	0.271217i
$923$	699881.	0.821525
$924$	0	0
$925$	0	0
$926$	− 203333.i	− 0.237130i
$927$	0	0
$928$	− 1.44088e6i	− 1.67314i
$929$	− 709717.i	− 0.822345i	−0.911558	−	0.411172i	$-0.865119\pi$
$929$	− 709717.i	− 0.822345i	0.911558	−	0.411172i	$-0.134881\pi$
$930$	0	0
$931$	101726.	0.117364
$932$	−291933.	−0.336087
$933$	0	0
$934$	571374.	0.654978
$935$	0	0
$936$	0	0
$937$	294273.i	0.335175i	0.985857	+	0.167587i	$0.0535977\pi$
$937$	294273.i	0.335175i	−0.985857	+	0.167587i	$0.946402\pi$
$938$	163500.	0.185829
$939$	0	0
$940$	0	0
$941$	− 783701.i	− 0.885057i	−0.896755	−	0.442528i	$-0.854082\pi$
$941$	− 783701.i	− 0.885057i	0.896755	−	0.442528i	$-0.145918\pi$
$942$	0	0
$943$	− 3229.93i	− 0.00363220i
$944$	281767.i	0.316189i
$945$	0	0
$946$	51029.2	0.0570212
$947$	−1.35618e6	−1.51222	−0.756112	−	0.654442i	$-0.772904\pi$
$947$	−1.35618e6	−1.51222	−0.756112	+	0.654442i	$0.772904\pi$
$948$	0	0
$949$	626822.	0.696004
$950$	0	0
$951$	0	0
$952$	− 236770.i	− 0.261248i
$953$	226626.	0.249531	0.124766	−	0.992186i	$-0.460182\pi$
$953$	226626.	0.249531	0.124766	+	0.992186i	$0.460182\pi$
$954$	0	0
$955$	0	0
$956$	− 333965.i	− 0.365414i
$957$	0	0
$958$	− 675037.i	− 0.735524i
$959$	300228.i	0.326448i
$960$	0	0
$961$	1.65507e6	1.79212
$962$	−403006.	−0.435473
$963$	0	0
$964$	−64799.6	−0.0697298
$965$	0	0
$966$	0	0
$967$	688811.i	0.736626i	0.929702	+	0.368313i	$0.120065\pi$
$967$	688811.i	0.736626i	−0.929702	+	0.368313i	$0.879935\pi$
$968$	569106.	0.607355
$969$	0	0
$970$	0	0
$971$	1.30899e6i	1.38835i	0.719807	+	0.694174i	$0.244230\pi$
$971$	1.30899e6i	1.38835i	−0.719807	+	0.694174i	$0.755770\pi$
$972$	0	0
$973$	2927.75i	0.00309249i
$974$	85883.2i	0.0905295i
$975$	0	0
$976$	−28149.5	−0.0295509
$977$	−29454.1	−0.0308572	−0.0154286	−	0.999881i	$-0.504911\pi$
$977$	−29454.1	−0.0308572	−0.0154286	+	0.999881i	$0.504911\pi$
$978$	0	0
$979$	354807.	0.370192
$980$	0	0
$981$	0	0
$982$	276966.i	0.287213i
$983$	−369377.	−0.382264	−0.191132	−	0.981564i	$-0.561216\pi$
$983$	−369377.	−0.382264	−0.191132	+	0.981564i	$0.561216\pi$
$984$	0	0
$985$	0	0
$986$	952450.i	0.979689i
$987$	0	0
$988$	97218.9i	0.0995948i
$989$	643.096i	0 0.000657481i
$990$	0	0
$991$	121176.	0.123387	0.0616933	−	0.998095i	$-0.480350\pi$
$991$	121176.	0.123387	0.0616933	+	0.998095i	$0.480350\pi$
$992$	−1.65123e6	−1.67797
$993$	0	0
$994$	90416.2	0.0915110
$995$	0	0
$996$	0	0
$997$	277965.i	0.279640i	0.990177	+	0.139820i	$0.0446525\pi$
$997$	277965.i	0.279640i	−0.990177	+	0.139820i	$0.955348\pi$
$998$	−628359.	−0.630880
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	225.5.d.c.224.6		12
3.2	odd	2	inner	225.5.d.c.224.8		12
5.2	odd	4		225.5.c.d.26.3	yes	6
5.3	odd	4		225.5.c.c.26.4	yes	6
5.4	even	2	inner	225.5.d.c.224.7		12
15.2	even	4		225.5.c.d.26.4	yes	6
15.8	even	4		225.5.c.c.26.3	✓	6
15.14	odd	2	inner	225.5.d.c.224.5		12

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
225.5.c.c.26.3	✓	6	15.8	even	4
225.5.c.c.26.4	yes	6	5.3	odd	4
225.5.c.d.26.3	yes	6	5.2	odd	4
225.5.c.d.26.4	yes	6	15.2	even	4
225.5.d.c.224.5		12	15.14	odd	2	inner
225.5.d.c.224.6		12	1.1	even	1	trivial
225.5.d.c.224.7		12	5.4	even	2	inner
225.5.d.c.224.8		12	3.2	odd	2	inner

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:	\( N \)	\(=\)	\( 225 = 3^{2} \cdot 5^{2} \)
Weight:	\( k \)	\(=\)	\( 5 \)
Character orbit:	\([\chi]\)	\(=\)	225.d (of order \(2\), degree \(1\), not minimal)

\(f(q)\)	\(=\)	\(q-1.80123 q^{2} -12.7556 q^{4} +12.1136i q^{7} +51.7953 q^{8} +O(q^{10})\)	\(q-1.80123 q^{2} -12.7556 q^{4} +12.1136i q^{7} +51.7953 q^{8} -60.4434i q^{11} -168.896i q^{13} -21.8194i q^{14} +110.794 q^{16} -377.366 q^{17} +45.1263 q^{19} +108.872i q^{22} -1.37206 q^{23} +304.221i q^{26} -154.516i q^{28} +1401.24i q^{29} +1605.80 q^{31} -1028.29 q^{32} +679.721 q^{34} +1324.72i q^{37} -81.2826 q^{38} +2354.07i q^{41} -468.707i q^{43} +770.991i q^{44} +2.47139 q^{46} -3910.98 q^{47} +2254.26 q^{49} +2154.37i q^{52} +3228.98 q^{53} +627.429i q^{56} -2523.94i q^{58} +2543.15i q^{59} -254.070 q^{61} -2892.40 q^{62} +79.4731 q^{64} +7493.35i q^{67} +4813.52 q^{68} +4143.85i q^{71} +3711.28i q^{73} -2386.11i q^{74} -575.612 q^{76} +732.189 q^{77} -80.2118 q^{79} -4240.21i q^{82} -5159.51 q^{83} +844.248i q^{86} -3130.68i q^{88} +5870.07i q^{89} +2045.95 q^{91} +17.5015 q^{92} +7044.55 q^{94} -8921.02i q^{97} -4060.43 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 12 q + 184 q^{4}+O(q^{10}) \) 12 * q + 184 * q^4	\( 12 q + 184 q^{4} + 3632 q^{16} + 500 q^{19} + 3692 q^{31} + 7384 q^{34} + 19512 q^{46} + 6464 q^{49} + 21676 q^{61} + 72288 q^{64} - 49560 q^{76} - 7808 q^{79} + 34612 q^{91} - 132728 q^{94}+O(q^{100}) \) 12 * q + 184 * q^4 + 3632 * q^16 + 500 * q^19 + 3692 * q^31 + 7384 * q^34 + 19512 * q^46 + 6464 * q^49 + 21676 * q^61 + 72288 * q^64 - 49560 * q^76 - 7808 * q^79 + 34612 * q^91 - 132728 * q^94

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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