LMFDB - Embedded newform 315.10.a.b.1.1

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [315,10,Mod(1,315)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(315, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([0, 0, 0]))

N = Newforms(chi, 10, names="a")

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

chi := DirichletCharacter("315.1");

S:= CuspForms(chi, 10);

N := Newforms(S);

Level:	$ N $	$=$	$ 315 = 3^{2} \cdot 5 \cdot 7 $
Weight:	$ k $	$=$	$ 10 $
Character orbit:	$[\chi]$	$=$	315.a (trivial)

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:	yes
Analytic conductor:	$162.236288392$
Analytic rank:	$1$
Dimension:	$2$
Coefficient field:	$\Q(\zeta_{8})^+$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{2} - 2 $ x^2 - 2
Coefficient ring:	$\Z[a_1, a_2]$
Coefficient ring index:	$ 2 $
Twist minimal:	no (minimal twist has level 35)
Fricke sign:	$+1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.1
Root			$-1.41421$ of defining polynomial
Character	$\chi$	$=$	315.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	\(q+9.17157 q^{2} -427.882 q^{4} -625.000 q^{5} +2401.00 q^{7} -8620.20 q^{8} -5732.23 q^{10} -35089.6 q^{11} -77401.4 q^{13} +22020.9 q^{14} +140015. q^{16} +229907. q^{17} +16433.6 q^{19} +267426. q^{20} -321827. q^{22} +2.57284e6 q^{23} +390625. q^{25} -709892. q^{26} -1.02735e6 q^{28} +6.62817e6 q^{29} -8.17416e6 q^{31} +5.69770e6 q^{32} +2.10861e6 q^{34} -1.50062e6 q^{35} +9.70272e6 q^{37} +150722. q^{38} +5.38762e6 q^{40} -2.98108e7 q^{41} -1.95343e7 q^{43} +1.50142e7 q^{44} +2.35970e7 q^{46} -5.93794e6 q^{47} +5.76480e6 q^{49} +3.58265e6 q^{50} +3.31187e7 q^{52} +2.74263e7 q^{53} +2.19310e7 q^{55} -2.06971e7 q^{56} +6.07908e7 q^{58} -5.24915e7 q^{59} +2.23282e7 q^{61} -7.49699e7 q^{62} -1.94308e7 q^{64} +4.83759e7 q^{65} +2.74351e8 q^{67} -9.83733e7 q^{68} -1.37631e7 q^{70} +3.63673e8 q^{71} +2.09245e7 q^{73} +8.89892e7 q^{74} -7.03163e6 q^{76} -8.42501e7 q^{77} -2.65896e8 q^{79} -8.75093e7 q^{80} -2.73412e8 q^{82} +9.43764e6 q^{83} -1.43692e8 q^{85} -1.79160e8 q^{86} +3.02479e8 q^{88} +6.64876e8 q^{89} -1.85841e8 q^{91} -1.10087e9 q^{92} -5.44603e7 q^{94} -1.02710e7 q^{95} -1.20731e9 q^{97} +5.28723e7 q^{98} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 2 q + 24 q^{2} - 720 q^{4} - 1250 q^{5} + 4802 q^{7} - 20544 q^{8} - 15000 q^{10} - 18566 q^{11} - 51090 q^{13} + 57624 q^{14} + 112768 q^{16} + 373910 q^{17} - 143276 q^{19} + 450000 q^{20} - 76808 q^{22}+ \cdots + 138355224 q^{98}+O(q^{100}) $ 2 * q + 24 * q^2 - 720 * q^4 - 1250 * q^5 + 4802 * q^7 - 20544 * q^8 - 15000 * q^10 - 18566 * q^11 - 51090 * q^13 + 57624 * q^14 + 112768 * q^16 + 373910 * q^17 - 143276 * q^19 + 450000 * q^20 - 76808 * q^22 + 498908 * q^23 + 781250 * q^25 - 319736 * q^26 - 1728720 * q^28 + 11577554 * q^29 - 3953760 * q^31 + 11398656 * q^32 + 4243944 * q^34 - 3001250 * q^35 - 3205412 * q^37 - 2217520 * q^38 + 12840000 * q^40 - 1058992 * q^41 + 15948180 * q^43 + 10187376 * q^44 - 7156176 * q^46 - 65501290 * q^47 + 11529602 * q^49 + 9375000 * q^50 + 25432656 * q^52 + 25114688 * q^53 + 11603750 * q^55 - 49326144 * q^56 + 134182296 * q^58 + 116159208 * q^59 - 44688544 * q^61 - 12388000 * q^62 + 79055872 * q^64 + 31931250 * q^65 + 118092496 * q^67 - 140439024 * q^68 - 36015000 * q^70 + 294165824 * q^71 - 57419332 * q^73 - 102418064 * q^74 + 39622368 * q^76 - 44576966 * q^77 - 692852854 * q^79 - 70480000 * q^80 + 152932128 * q^82 + 540679928 * q^83 - 233693750 * q^85 + 346989008 * q^86 + 105455296 * q^88 + 779043704 * q^89 - 122667090 * q^91 - 495040608 * q^92 - 937691032 * q^94 + 89547500 * q^95 - 2673039406 * q^97 + 138355224 * q^98

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 $

$2$	9.17157	0.405330	0.202665	−	0.979248i	$-0.435040\pi$
$2$	9.17157	0.405330	0.202665	+	0.979248i	$0.435040\pi$
$3$	0	0
$3$	0	0
$4$	−427.882	−0.835708
$5$	−625.000	−0.447214
$5$	−625.000	−0.447214
$6$	0	0
$7$	2401.00	0.377964
$7$	2401.00	0.377964
$8$	−8620.20	−0.744067
$9$	0	0
$10$	−5732.23	−0.181269
$11$	−35089.6	−0.722622	−0.361311	−	0.932445i	$-0.617671\pi$
$11$	−35089.6	−0.722622	−0.361311	+	0.932445i	$0.617671\pi$
$12$	0	0
$13$	−77401.4	−0.751629	−0.375815	−	0.926695i	$-0.622637\pi$
$13$	−77401.4	−0.751629	−0.375815	+	0.926695i	$0.622637\pi$
$14$	22020.9	0.153200
$15$	0	0
$16$	140015.	0.534115
$17$	229907.	0.667626	0.333813	−	0.942639i	$-0.391665\pi$
$17$	229907.	0.667626	0.333813	+	0.942639i	$0.391665\pi$
$18$	0	0
$19$	16433.6	0.0289295	0.0144647	−	0.999895i	$-0.495396\pi$
$19$	16433.6	0.0289295	0.0144647	+	0.999895i	$0.495396\pi$
$20$	267426.	0.373740
$21$	0	0
$22$	−321827.	−0.292900
$23$	2.57284e6	1.91707	0.958535	−	0.284975i	$-0.0919852\pi$
$23$	2.57284e6	1.91707	0.958535	+	0.284975i	$0.0919852\pi$
$24$	0	0
$25$	390625.	0.200000
$26$	−709892.	−0.304658
$27$	0	0
$28$	−1.02735e6	−0.315868
$29$	6.62817e6	1.74022	0.870108	−	0.492862i	$-0.164049\pi$
$29$	6.62817e6	1.74022	0.870108	+	0.492862i	$0.164049\pi$
$30$	0	0
$31$	−8.17416e6	−1.58970	−0.794851	−	0.606805i	$-0.792451\pi$
$31$	−8.17416e6	−1.58970	−0.794851	+	0.606805i	$0.792451\pi$
$32$	5.69770e6	0.960560
$33$	0	0
$34$	2.10861e6	0.270609
$35$	−1.50062e6	−0.169031
$36$	0	0
$37$	9.70272e6	0.851110	0.425555	−	0.904933i	$-0.360079\pi$
$37$	9.70272e6	0.851110	0.425555	+	0.904933i	$0.360079\pi$
$38$	150722.	0.0117260
$39$	0	0
$40$	5.38762e6	0.332757
$41$	−2.98108e7	−1.64758	−0.823789	−	0.566896i	$-0.808144\pi$
$41$	−2.98108e7	−1.64758	−0.823789	+	0.566896i	$0.808144\pi$
$42$	0	0
$43$	−1.95343e7	−0.871343	−0.435672	−	0.900106i	$-0.643489\pi$
$43$	−1.95343e7	−0.871343	−0.435672	+	0.900106i	$0.643489\pi$
$44$	1.50142e7	0.603900
$45$	0	0
$46$	2.35970e7	0.777046
$47$	−5.93794e6	−0.177499	−0.0887494	−	0.996054i	$-0.528287\pi$
$47$	−5.93794e6	−0.177499	−0.0887494	+	0.996054i	$0.528287\pi$
$48$	0	0
$49$	5.76480e6	0.142857
$50$	3.58265e6	0.0810660
$51$	0	0
$52$	3.31187e7	0.628142
$53$	2.74263e7	0.477448	0.238724	−	0.971088i	$-0.423271\pi$
$53$	2.74263e7	0.477448	0.238724	+	0.971088i	$0.423271\pi$
$54$	0	0
$55$	2.19310e7	0.323166
$56$	−2.06971e7	−0.281231
$57$	0	0
$58$	6.07908e7	0.705362
$59$	−5.24915e7	−0.563969	−0.281984	−	0.959419i	$-0.590993\pi$
$59$	−5.24915e7	−0.563969	−0.281984	+	0.959419i	$0.590993\pi$
$60$	0	0
$61$	2.23282e7	0.206476	0.103238	−	0.994657i	$-0.467080\pi$
$61$	2.23282e7	0.206476	0.103238	+	0.994657i	$0.467080\pi$
$62$	−7.49699e7	−0.644354
$63$	0	0
$64$	−1.94308e7	−0.144771
$65$	4.83759e7	0.336139
$66$	0	0
$67$	2.74351e8	1.66330	0.831649	−	0.555302i	$-0.187397\pi$
$67$	2.74351e8	1.66330	0.831649	+	0.555302i	$0.187397\pi$
$68$	−9.83733e7	−0.557940
$69$	0	0
$70$	−1.37631e7	−0.0685133
$71$	3.63673e8	1.69843	0.849216	−	0.528046i	$-0.177075\pi$
$71$	3.63673e8	1.69843	0.849216	+	0.528046i	$0.177075\pi$
$72$	0	0
$73$	2.09245e7	0.0862387	0.0431193	−	0.999070i	$-0.486270\pi$
$73$	2.09245e7	0.0862387	0.0431193	+	0.999070i	$0.486270\pi$
$74$	8.89892e7	0.344980
$75$	0	0
$76$	−7.03163e6	−0.0241766
$77$	−8.42501e7	−0.273125
$78$	0	0
$79$	−2.65896e8	−0.768051	−0.384025	−	0.923323i	$-0.625462\pi$
$79$	−2.65896e8	−0.768051	−0.384025	+	0.923323i	$0.625462\pi$
$80$	−8.75093e7	−0.238863
$81$	0	0
$82$	−2.73412e8	−0.667813
$83$	9.43764e6	0.0218279	0.0109140	−	0.999940i	$-0.496526\pi$
$83$	9.43764e6	0.0218279	0.0109140	+	0.999940i	$0.496526\pi$
$84$	0	0
$85$	−1.43692e8	−0.298571
$86$	−1.79160e8	−0.353182
$87$	0	0
$88$	3.02479e8	0.537679
$89$	6.64876e8	1.12327	0.561637	−	0.827384i	$-0.310172\pi$
$89$	6.64876e8	1.12327	0.561637	+	0.827384i	$0.310172\pi$
$90$	0	0
$91$	−1.85841e8	−0.284089
$92$	−1.10087e9	−1.60211
$93$	0	0
$94$	−5.44603e7	−0.0719456
$95$	−1.02710e7	−0.0129377
$96$	0	0
$97$	−1.20731e9	−1.38467	−0.692336	−	0.721575i	$-0.743418\pi$
$97$	−1.20731e9	−1.38467	−0.692336	+	0.721575i	$0.743418\pi$
$98$	5.28723e7	0.0579043
$99$	0	0
$100$	−1.67142e8	−0.167142
$101$	−1.18204e9	−1.13028	−0.565139	−	0.824996i	$-0.691177\pi$
$101$	−1.18204e9	−1.13028	−0.565139	+	0.824996i	$0.691177\pi$
$102$	0	0
$103$	1.97811e9	1.73174	0.865870	−	0.500268i	$-0.166765\pi$
$103$	1.97811e9	1.73174	0.865870	+	0.500268i	$0.166765\pi$
$104$	6.67215e8	0.559263
$105$	0	0
$106$	2.51542e8	0.193524
$107$	−1.67828e8	−0.123776	−0.0618881	−	0.998083i	$-0.519712\pi$
$107$	−1.67828e8	−0.123776	−0.0618881	+	0.998083i	$0.519712\pi$
$108$	0	0
$109$	−1.02540e9	−0.695784	−0.347892	−	0.937535i	$-0.613102\pi$
$109$	−1.02540e9	−0.695784	−0.347892	+	0.937535i	$0.613102\pi$
$110$	2.01142e8	0.130989
$111$	0	0
$112$	3.36176e8	0.201876
$113$	−1.27533e9	−0.735814	−0.367907	−	0.929863i	$-0.619926\pi$
$113$	−1.27533e9	−0.735814	−0.367907	+	0.929863i	$0.619926\pi$
$114$	0	0
$115$	−1.60803e9	−0.857340
$116$	−2.83608e9	−1.45431
$117$	0	0
$118$	−4.81430e8	−0.228594
$119$	5.52008e8	0.252339
$120$	0	0
$121$	−1.12667e9	−0.477818
$122$	2.04785e8	0.0836909
$123$	0	0
$124$	3.49758e9	1.32853
$125$	−2.44141e8	−0.0894427
$126$	0	0
$127$	−2.90339e9	−0.990349	−0.495174	−	0.868794i	$-0.664896\pi$
$127$	−2.90339e9	−0.990349	−0.495174	+	0.868794i	$0.664896\pi$
$128$	−3.09543e9	−1.01924
$129$	0	0
$130$	4.43683e8	0.136247
$131$	−2.05173e9	−0.608694	−0.304347	−	0.952561i	$-0.598438\pi$
$131$	−2.05173e9	−0.608694	−0.304347	+	0.952561i	$0.598438\pi$
$132$	0	0
$133$	3.94570e7	0.0109343
$134$	2.51623e9	0.674185
$135$	0	0
$136$	−1.98185e9	−0.496759
$137$	−3.25539e9	−0.789514	−0.394757	−	0.918786i	$-0.629171\pi$
$137$	−3.25539e9	−0.789514	−0.394757	+	0.918786i	$0.629171\pi$
$138$	0	0
$139$	−8.26776e9	−1.87854	−0.939272	−	0.343173i	$-0.888498\pi$
$139$	−8.26776e9	−1.87854	−0.939272	+	0.343173i	$0.888498\pi$
$140$	6.42091e8	0.141260
$141$	0	0
$142$	3.33545e9	0.688425
$143$	2.71598e9	0.543143
$144$	0	0
$145$	−4.14261e9	−0.778248
$146$	1.91910e8	0.0349551
$147$	0	0
$148$	−4.15162e9	−0.711279
$149$	−1.07127e9	−0.178058	−0.0890289	−	0.996029i	$-0.528376\pi$
$149$	−1.07127e9	−0.178058	−0.0890289	+	0.996029i	$0.528376\pi$
$150$	0	0
$151$	1.97304e9	0.308844	0.154422	−	0.988005i	$-0.450649\pi$
$151$	1.97304e9	0.308844	0.154422	+	0.988005i	$0.450649\pi$
$152$	−1.41661e8	−0.0215255
$153$	0	0
$154$	−7.72706e8	−0.110706
$155$	5.10885e9	0.710936
$156$	0	0
$157$	−4.61623e9	−0.606372	−0.303186	−	0.952931i	$-0.598050\pi$
$157$	−4.61623e9	−0.606372	−0.303186	+	0.952931i	$0.598050\pi$
$158$	−2.43868e9	−0.311314
$159$	0	0
$160$	−3.56106e9	−0.429576
$161$	6.17740e9	0.724584
$162$	0	0
$163$	6.26525e9	0.695175	0.347588	−	0.937648i	$-0.387001\pi$
$163$	6.26525e9	0.695175	0.347588	+	0.937648i	$0.387001\pi$
$164$	1.27555e10	1.37689
$165$	0	0
$166$	8.65580e7	0.00884751
$167$	6.21672e9	0.618496	0.309248	−	0.950981i	$-0.399923\pi$
$167$	6.21672e9	0.618496	0.309248	+	0.950981i	$0.399923\pi$
$168$	0	0
$169$	−4.61353e9	−0.435054
$170$	−1.31788e9	−0.121020
$171$	0	0
$172$	8.35837e9	0.728188
$173$	−8.97209e9	−0.761528	−0.380764	−	0.924672i	$-0.624339\pi$
$173$	−8.97209e9	−0.761528	−0.380764	+	0.924672i	$0.624339\pi$
$174$	0	0
$175$	9.37891e8	0.0755929
$176$	−4.91306e9	−0.385963
$177$	0	0
$178$	6.09796e9	0.455297
$179$	−1.76242e10	−1.28313	−0.641565	−	0.767069i	$-0.721714\pi$
$179$	−1.76242e10	−1.28313	−0.641565	+	0.767069i	$0.721714\pi$
$180$	0	0
$181$	1.62250e9	0.112365	0.0561824	−	0.998421i	$-0.482107\pi$
$181$	1.62250e9	0.112365	0.0561824	+	0.998421i	$0.482107\pi$
$182$	−1.70445e9	−0.115150
$183$	0	0
$184$	−2.21784e10	−1.42643
$185$	−6.06420e9	−0.380628
$186$	0	0
$187$	−8.06735e9	−0.482441
$188$	2.54074e9	0.148337
$189$	0	0
$190$	−9.42010e7	−0.00524402
$191$	−1.66601e10	−0.905788	−0.452894	−	0.891564i	$-0.649608\pi$
$191$	−1.66601e10	−0.905788	−0.452894	+	0.891564i	$0.649608\pi$
$192$	0	0
$193$	2.41341e10	1.25206	0.626028	−	0.779801i	$-0.284680\pi$
$193$	2.41341e10	1.25206	0.626028	+	0.779801i	$0.284680\pi$
$194$	−1.10730e10	−0.561249
$195$	0	0
$196$	−2.46666e9	−0.119387
$197$	3.89843e9	0.184413	0.0922066	−	0.995740i	$-0.470608\pi$
$197$	3.89843e9	0.184413	0.0922066	+	0.995740i	$0.470608\pi$
$198$	0	0
$199$	−1.87489e10	−0.847493	−0.423747	−	0.905781i	$-0.639285\pi$
$199$	−1.87489e10	−0.847493	−0.423747	+	0.905781i	$0.639285\pi$
$200$	−3.36727e9	−0.148813
$201$	0	0
$202$	−1.08411e10	−0.458135
$203$	1.59142e10	0.657740
$204$	0	0
$205$	1.86317e10	0.736820
$206$	1.81424e10	0.701927
$207$	0	0
$208$	−1.08373e10	−0.401456
$209$	−5.76647e8	−0.0209051
$210$	0	0
$211$	−2.20489e9	−0.0765801	−0.0382900	−	0.999267i	$-0.512191\pi$
$211$	−2.20489e9	−0.0765801	−0.0382900	+	0.999267i	$0.512191\pi$
$212$	−1.17352e10	−0.399007
$213$	0	0
$214$	−1.53925e9	−0.0501702
$215$	1.22089e10	0.389677
$216$	0	0
$217$	−1.96262e10	−0.600851
$218$	−9.40454e9	−0.282022
$219$	0	0
$220$	−9.38388e9	−0.270072
$221$	−1.77952e10	−0.501807
$222$	0	0
$223$	−2.65324e10	−0.718463	−0.359231	−	0.933249i	$-0.616961\pi$
$223$	−2.65324e10	−0.718463	−0.359231	+	0.933249i	$0.616961\pi$
$224$	1.36802e10	0.363058
$225$	0	0
$226$	−1.16967e10	−0.298248
$227$	−7.78091e10	−1.94498	−0.972488	−	0.232955i	$-0.925161\pi$
$227$	−7.78091e10	−1.94498	−0.972488	+	0.232955i	$0.925161\pi$
$228$	0	0
$229$	4.84637e10	1.16455	0.582274	−	0.812993i	$-0.302163\pi$
$229$	4.84637e10	1.16455	0.582274	+	0.812993i	$0.302163\pi$
$230$	−1.47481e10	−0.347506
$231$	0	0
$232$	−5.71362e10	−1.29484
$233$	2.38429e10	0.529978	0.264989	−	0.964251i	$-0.414632\pi$
$233$	2.38429e10	0.529978	0.264989	+	0.964251i	$0.414632\pi$
$234$	0	0
$235$	3.71121e9	0.0793799
$236$	2.24602e10	0.471313
$237$	0	0
$238$	5.06278e9	0.102280
$239$	−6.25895e10	−1.24083	−0.620413	−	0.784275i	$-0.713035\pi$
$239$	−6.25895e10	−1.24083	−0.620413	+	0.784275i	$0.713035\pi$
$240$	0	0
$241$	−7.96605e10	−1.52113	−0.760565	−	0.649262i	$-0.775078\pi$
$241$	−7.96605e10	−1.52113	−0.760565	+	0.649262i	$0.775078\pi$
$242$	−1.03333e10	−0.193674
$243$	0	0
$244$	−9.55384e9	−0.172554
$245$	−3.60300e9	−0.0638877
$246$	0	0
$247$	−1.27198e9	−0.0217442
$248$	7.04629e10	1.18285
$249$	0	0
$250$	−2.23915e9	−0.0362538
$251$	5.44549e10	0.865975	0.432988	−	0.901400i	$-0.357459\pi$
$251$	5.44549e10	0.865975	0.432988	+	0.901400i	$0.357459\pi$
$252$	0	0
$253$	−9.02799e10	−1.38532
$254$	−2.66286e10	−0.401418
$255$	0	0
$256$	−1.84414e10	−0.268358
$257$	5.35278e10	0.765385	0.382693	−	0.923876i	$-0.374997\pi$
$257$	5.35278e10	0.765385	0.382693	+	0.923876i	$0.374997\pi$
$258$	0	0
$259$	2.32962e10	0.321689
$260$	−2.06992e10	−0.280914
$261$	0	0
$262$	−1.88176e10	−0.246722
$263$	5.81425e10	0.749364	0.374682	−	0.927153i	$-0.377752\pi$
$263$	5.81425e10	0.749364	0.374682	+	0.927153i	$0.377752\pi$
$264$	0	0
$265$	−1.71414e10	−0.213521
$266$	3.61883e8	0.00443201
$267$	0	0
$268$	−1.17390e11	−1.39003
$269$	−4.67380e10	−0.544233	−0.272116	−	0.962264i	$-0.587724\pi$
$269$	−4.67380e10	−0.544233	−0.272116	+	0.962264i	$0.587724\pi$
$270$	0	0
$271$	2.68147e10	0.302003	0.151001	−	0.988534i	$-0.451750\pi$
$271$	2.68147e10	0.302003	0.151001	+	0.988534i	$0.451750\pi$
$272$	3.21905e10	0.356589
$273$	0	0
$274$	−2.98570e10	−0.320014
$275$	−1.37069e10	−0.144524
$276$	0	0
$277$	1.12549e11	1.14863	0.574316	−	0.818633i	$-0.305268\pi$
$277$	1.12549e11	1.14863	0.574316	+	0.818633i	$0.305268\pi$
$278$	−7.58284e10	−0.761431
$279$	0	0
$280$	1.29357e10	0.125770
$281$	4.60761e10	0.440857	0.220428	−	0.975403i	$-0.429254\pi$
$281$	4.60761e10	0.440857	0.220428	+	0.975403i	$0.429254\pi$
$282$	0	0
$283$	7.94071e10	0.735902	0.367951	−	0.929845i	$-0.380059\pi$
$283$	7.94071e10	0.735902	0.367951	+	0.929845i	$0.380059\pi$
$284$	−1.55609e11	−1.41939
$285$	0	0
$286$	2.49098e10	0.220152
$287$	−7.15757e10	−0.622726
$288$	0	0
$289$	−6.57304e10	−0.554276
$290$	−3.79942e10	−0.315447
$291$	0	0
$292$	−8.95322e9	−0.0720703
$293$	1.41265e11	1.11977	0.559887	−	0.828569i	$-0.310844\pi$
$293$	1.41265e11	1.11977	0.559887	+	0.828569i	$0.310844\pi$
$294$	0	0
$295$	3.28072e10	0.252215
$296$	−8.36393e10	−0.633283
$297$	0	0
$298$	−9.82523e9	−0.0721721
$299$	−1.99142e11	−1.44093
$300$	0	0
$301$	−4.69018e10	−0.329337
$302$	1.80958e10	0.125184
$303$	0	0
$304$	2.30094e9	0.0154517
$305$	−1.39551e10	−0.0923389
$306$	0	0
$307$	−5.58349e10	−0.358742	−0.179371	−	0.983781i	$-0.557406\pi$
$307$	−5.58349e10	−0.358742	−0.179371	+	0.983781i	$0.557406\pi$
$308$	3.60491e10	0.228253
$309$	0	0
$310$	4.68562e10	0.288164
$311$	−5.26501e10	−0.319137	−0.159569	−	0.987187i	$-0.551010\pi$
$311$	−5.26501e10	−0.319137	−0.159569	+	0.987187i	$0.551010\pi$
$312$	0	0
$313$	−2.51256e11	−1.47968	−0.739838	−	0.672785i	$-0.765098\pi$
$313$	−2.51256e11	−1.47968	−0.739838	+	0.672785i	$0.765098\pi$
$314$	−4.23381e10	−0.245781
$315$	0	0
$316$	1.13772e11	0.641866
$317$	1.16999e11	0.650749	0.325375	−	0.945585i	$-0.394510\pi$
$317$	1.16999e11	0.650749	0.325375	+	0.945585i	$0.394510\pi$
$318$	0	0
$319$	−2.32580e11	−1.25752
$320$	1.21442e10	0.0647434
$321$	0	0
$322$	5.66564e10	0.293696
$323$	3.77820e9	0.0193141
$324$	0	0
$325$	−3.02349e10	−0.150326
$326$	5.74622e10	0.281775
$327$	0	0
$328$	2.56975e11	1.22591
$329$	−1.42570e10	−0.0670883
$330$	0	0
$331$	−2.51419e11	−1.15126	−0.575629	−	0.817711i	$-0.695243\pi$
$331$	−2.51419e11	−1.15126	−0.575629	+	0.817711i	$0.695243\pi$
$332$	−4.03820e9	−0.0182417
$333$	0	0
$334$	5.70171e10	0.250695
$335$	−1.71469e11	−0.743849
$336$	0	0
$337$	6.11427e10	0.258232	0.129116	−	0.991630i	$-0.458786\pi$
$337$	6.11427e10	0.258232	0.129116	+	0.991630i	$0.458786\pi$
$338$	−4.23133e10	−0.176340
$339$	0	0
$340$	6.14833e10	0.249518
$341$	2.86828e11	1.14875
$342$	0	0
$343$	1.38413e10	0.0539949
$344$	1.68389e11	0.648338
$345$	0	0
$346$	−8.22882e10	−0.308670
$347$	1.68668e11	0.624524	0.312262	−	0.949996i	$-0.398913\pi$
$347$	1.68668e11	0.624524	0.312262	+	0.949996i	$0.398913\pi$
$348$	0	0
$349$	−3.31182e11	−1.19496	−0.597479	−	0.801885i	$-0.703831\pi$
$349$	−3.31182e11	−1.19496	−0.597479	+	0.801885i	$0.703831\pi$
$350$	8.60193e9	0.0306401
$351$	0	0
$352$	−1.99930e11	−0.694122
$353$	−3.78560e11	−1.29762	−0.648811	−	0.760949i	$-0.724734\pi$
$353$	−3.78560e11	−1.29762	−0.648811	+	0.760949i	$0.724734\pi$
$354$	0	0
$355$	−2.27295e11	−0.759562
$356$	−2.84489e11	−0.938728
$357$	0	0
$358$	−1.61641e11	−0.520091
$359$	1.60137e11	0.508822	0.254411	−	0.967096i	$-0.418118\pi$
$359$	1.60137e11	0.508822	0.254411	+	0.967096i	$0.418118\pi$
$360$	0	0
$361$	−3.22418e11	−0.999163
$362$	1.48809e10	0.0455449
$363$	0	0
$364$	7.95179e10	0.237415
$365$	−1.30778e10	−0.0385671
$366$	0	0
$367$	5.13837e11	1.47852	0.739261	−	0.673419i	$-0.235175\pi$
$367$	5.13837e11	1.47852	0.739261	+	0.673419i	$0.235175\pi$
$368$	3.60236e11	1.02394
$369$	0	0
$370$	−5.56182e10	−0.154280
$371$	6.58505e10	0.180458
$372$	0	0
$373$	−6.70900e10	−0.179460	−0.0897301	−	0.995966i	$-0.528600\pi$
$373$	−6.70900e10	−0.179460	−0.0897301	+	0.995966i	$0.528600\pi$
$374$	−7.39903e10	−0.195548
$375$	0	0
$376$	5.11862e10	0.132071
$377$	−5.13030e11	−1.30800
$378$	0	0
$379$	4.15471e11	1.03434	0.517171	−	0.855882i	$-0.326985\pi$
$379$	4.15471e11	1.03434	0.517171	+	0.855882i	$0.326985\pi$
$380$	4.39477e9	0.0108121
$381$	0	0
$382$	−1.52799e11	−0.367143
$383$	3.51976e11	0.835831	0.417915	−	0.908486i	$-0.362761\pi$
$383$	3.51976e11	0.835831	0.417915	+	0.908486i	$0.362761\pi$
$384$	0	0
$385$	5.26563e10	0.122145
$386$	2.21348e11	0.507496
$387$	0	0
$388$	5.16588e11	1.15718
$389$	2.60061e11	0.575840	0.287920	−	0.957654i	$-0.407036\pi$
$389$	2.60061e11	0.575840	0.287920	+	0.957654i	$0.407036\pi$
$390$	0	0
$391$	5.91516e11	1.27989
$392$	−4.96937e10	−0.106295
$393$	0	0
$394$	3.57548e10	0.0747482
$395$	1.66185e11	0.343483
$396$	0	0
$397$	7.34338e11	1.48367	0.741837	−	0.670580i	$-0.233955\pi$
$397$	7.34338e11	1.48367	0.741837	+	0.670580i	$0.233955\pi$
$398$	−1.71957e11	−0.343514
$399$	0	0
$400$	5.46933e10	0.106823
$401$	−8.08296e11	−1.56106	−0.780532	−	0.625116i	$-0.785052\pi$
$401$	−8.08296e11	−1.56106	−0.780532	+	0.625116i	$0.785052\pi$
$402$	0	0
$403$	6.32691e11	1.19487
$404$	5.05773e11	0.944581
$405$	0	0
$406$	1.45959e11	0.266602
$407$	−3.40464e11	−0.615030
$408$	0	0
$409$	−9.11153e11	−1.61004	−0.805020	−	0.593248i	$-0.797845\pi$
$409$	−9.11153e11	−1.61004	−0.805020	+	0.593248i	$0.797845\pi$
$410$	1.70882e11	0.298655
$411$	0	0
$412$	−8.46398e11	−1.44723
$413$	−1.26032e11	−0.213160
$414$	0	0
$415$	−5.89853e9	−0.00976174
$416$	−4.41010e11	−0.721985
$417$	0	0
$418$	−5.28876e9	−0.00847345
$419$	−4.94109e11	−0.783177	−0.391589	−	0.920140i	$-0.628074\pi$
$419$	−4.94109e11	−0.783177	−0.391589	+	0.920140i	$0.628074\pi$
$420$	0	0
$421$	−1.15145e10	−0.0178639	−0.00893197	−	0.999960i	$-0.502843\pi$
$421$	−1.15145e10	−0.0178639	−0.00893197	+	0.999960i	$0.502843\pi$
$422$	−2.02223e10	−0.0310402
$423$	0	0
$424$	−2.36420e11	−0.355253
$425$	8.98076e10	0.133525
$426$	0	0
$427$	5.36100e10	0.0780406
$428$	7.18106e10	0.103441
$429$	0	0
$430$	1.11975e11	0.157948
$431$	9.42534e11	1.31568	0.657839	−	0.753159i	$-0.271471\pi$
$431$	9.42534e11	1.31568	0.657839	+	0.753159i	$0.271471\pi$
$432$	0	0
$433$	1.01849e12	1.39239	0.696196	−	0.717852i	$-0.254875\pi$
$433$	1.01849e12	1.39239	0.696196	+	0.717852i	$0.254875\pi$
$434$	−1.80003e11	−0.243543
$435$	0	0
$436$	4.38751e11	0.581472
$437$	4.22810e10	0.0554598
$438$	0	0
$439$	−7.89357e11	−1.01434	−0.507169	−	0.861847i	$-0.669308\pi$
$439$	−7.89357e11	−1.01434	−0.507169	+	0.861847i	$0.669308\pi$
$440$	−1.89049e11	−0.240457
$441$	0	0
$442$	−1.63210e11	−0.203397
$443$	−1.06770e12	−1.31714	−0.658572	−	0.752518i	$-0.728839\pi$
$443$	−1.06770e12	−1.31714	−0.658572	+	0.752518i	$0.728839\pi$
$444$	0	0
$445$	−4.15547e11	−0.502343
$446$	−2.43344e11	−0.291215
$447$	0	0
$448$	−4.66533e10	−0.0547182
$449$	3.31695e9	0.00385150	0.00192575	−	0.999998i	$-0.499387\pi$
$449$	3.31695e9	0.00385150	0.00192575	+	0.999998i	$0.499387\pi$
$450$	0	0
$451$	1.04605e12	1.19058
$452$	5.45689e11	0.614926
$453$	0	0
$454$	−7.13632e11	−0.788357
$455$	1.16150e11	0.127049
$456$	0	0
$457$	−7.03146e11	−0.754089	−0.377045	−	0.926195i	$-0.623060\pi$
$457$	−7.03146e11	−0.754089	−0.377045	+	0.926195i	$0.623060\pi$
$458$	4.44489e11	0.472026
$459$	0	0
$460$	6.88046e11	0.716485
$461$	1.86192e12	1.92003	0.960015	−	0.279950i	$-0.0903178\pi$
$461$	1.86192e12	1.92003	0.960015	+	0.279950i	$0.0903178\pi$
$462$	0	0
$463$	−1.06950e11	−0.108160	−0.0540799	−	0.998537i	$-0.517223\pi$
$463$	−1.06950e11	−0.108160	−0.0540799	+	0.998537i	$0.517223\pi$
$464$	9.28043e11	0.929474
$465$	0	0
$466$	2.18677e11	0.214816
$467$	−4.13997e11	−0.402783	−0.201392	−	0.979511i	$-0.564546\pi$
$467$	−4.13997e11	−0.402783	−0.201392	+	0.979511i	$0.564546\pi$
$468$	0	0
$469$	6.58717e11	0.628667
$470$	3.40377e10	0.0321751
$471$	0	0
$472$	4.52487e11	0.419631
$473$	6.85450e11	0.629652
$474$	0	0
$475$	6.41936e9	0.00578589
$476$	−2.36194e11	−0.210881
$477$	0	0
$478$	−5.74044e11	−0.502944
$479$	8.54131e10	0.0741336	0.0370668	−	0.999313i	$-0.488199\pi$
$479$	8.54131e10	0.0741336	0.0370668	+	0.999313i	$0.488199\pi$
$480$	0	0
$481$	−7.51004e11	−0.639719
$482$	−7.30612e11	−0.616560
$483$	0	0
$484$	4.82082e11	0.399316
$485$	7.54571e11	0.619244
$486$	0	0
$487$	4.94789e11	0.398602	0.199301	−	0.979938i	$-0.436133\pi$
$487$	4.94789e11	0.398602	0.199301	+	0.979938i	$0.436133\pi$
$488$	−1.92474e11	−0.153632
$489$	0	0
$490$	−3.30452e10	−0.0258956
$491$	−1.11163e12	−0.863168	−0.431584	−	0.902073i	$-0.642045\pi$
$491$	−1.11163e12	−0.863168	−0.431584	+	0.902073i	$0.642045\pi$
$492$	0	0
$493$	1.52387e12	1.16181
$494$	−1.16661e10	−0.00881359
$495$	0	0
$496$	−1.14450e12	−0.849083
$497$	8.73178e11	0.641947
$498$	0	0
$499$	8.18377e11	0.590882	0.295441	−	0.955361i	$-0.404533\pi$
$499$	8.18377e11	0.590882	0.295441	+	0.955361i	$0.404533\pi$
$500$	1.04463e11	0.0747480
$501$	0	0
$502$	4.99437e11	0.351006
$503$	−3.13384e11	−0.218284	−0.109142	−	0.994026i	$-0.534810\pi$
$503$	−3.13384e11	−0.218284	−0.109142	+	0.994026i	$0.534810\pi$
$504$	0	0
$505$	7.38773e11	0.505475
$506$	−8.28009e11	−0.561510
$507$	0	0
$508$	1.24231e12	0.827642
$509$	1.02554e12	0.677211	0.338606	−	0.940928i	$-0.390045\pi$
$509$	1.02554e12	0.677211	0.338606	+	0.940928i	$0.390045\pi$
$510$	0	0
$511$	5.02397e10	0.0325951
$512$	1.41572e12	0.910467
$513$	0	0
$514$	4.90934e11	0.310234
$515$	−1.23632e12	−0.774458
$516$	0	0
$517$	2.08360e11	0.128265
$518$	2.13663e11	0.130390
$519$	0	0
$520$	−4.17010e11	−0.250110
$521$	3.18952e12	1.89651	0.948255	−	0.317510i	$-0.102847\pi$
$521$	3.18952e12	1.89651	0.948255	+	0.317510i	$0.102847\pi$
$522$	0	0
$523$	9.43708e11	0.551544	0.275772	−	0.961223i	$-0.411067\pi$
$523$	9.43708e11	0.551544	0.275772	+	0.961223i	$0.411067\pi$
$524$	8.77898e11	0.508690
$525$	0	0
$526$	5.33258e11	0.303740
$527$	−1.87930e12	−1.06133
$528$	0	0
$529$	4.81837e12	2.67516
$530$	−1.57214e11	−0.0865465
$531$	0	0
$532$	−1.68829e10	−0.00913789
$533$	2.30740e12	1.23837
$534$	0	0
$535$	1.04892e11	0.0553544
$536$	−2.36496e12	−1.23761
$537$	0	0
$538$	−4.28661e11	−0.220594
$539$	−2.02284e11	−0.103232
$540$	0	0
$541$	−8.78618e11	−0.440973	−0.220487	−	0.975390i	$-0.570765\pi$
$541$	−8.78618e11	−0.440973	−0.220487	+	0.975390i	$0.570765\pi$
$542$	2.45933e11	0.122411
$543$	0	0
$544$	1.30994e12	0.641295
$545$	6.40875e11	0.311164
$546$	0	0
$547$	4.56216e11	0.217885	0.108943	−	0.994048i	$-0.465254\pi$
$547$	4.56216e11	0.217885	0.108943	+	0.994048i	$0.465254\pi$
$548$	1.39292e12	0.659803
$549$	0	0
$550$	−1.25713e11	−0.0585801
$551$	1.08925e11	0.0503435
$552$	0	0
$553$	−6.38416e11	−0.290296
$554$	1.03225e12	0.465575
$555$	0	0
$556$	3.53763e12	1.56991
$557$	−1.63980e12	−0.721842	−0.360921	−	0.932596i	$-0.617538\pi$
$557$	−1.63980e12	−0.721842	−0.360921	+	0.932596i	$0.617538\pi$
$558$	0	0
$559$	1.51198e12	0.654927
$560$	−2.10110e11	−0.0902818
$561$	0	0
$562$	4.22590e11	0.178692
$563$	−4.36151e11	−0.182957	−0.0914786	−	0.995807i	$-0.529159\pi$
$563$	−4.36151e11	−0.182957	−0.0914786	+	0.995807i	$0.529159\pi$
$564$	0	0
$565$	7.97079e11	0.329066
$566$	7.28288e11	0.298283
$567$	0	0
$568$	−3.13493e12	−1.26375
$569$	−1.76284e12	−0.705029	−0.352514	−	0.935806i	$-0.614673\pi$
$569$	−1.76284e12	−0.705029	−0.352514	+	0.935806i	$0.614673\pi$
$570$	0	0
$571$	2.37232e11	0.0933922	0.0466961	−	0.998909i	$-0.485131\pi$
$571$	2.37232e11	0.0933922	0.0466961	+	0.998909i	$0.485131\pi$
$572$	−1.16212e12	−0.453909
$573$	0	0
$574$	−6.56462e11	−0.252410
$575$	1.00502e12	0.383414
$576$	0	0
$577$	−3.72080e12	−1.39748	−0.698739	−	0.715377i	$-0.746255\pi$
$577$	−3.72080e12	−1.39748	−0.698739	+	0.715377i	$0.746255\pi$
$578$	−6.02851e11	−0.224665
$579$	0	0
$580$	1.77255e12	0.650388
$581$	2.26598e10	0.00825017
$582$	0	0
$583$	−9.62377e11	−0.345014
$584$	−1.80373e11	−0.0641674
$585$	0	0
$586$	1.29562e12	0.453878
$587$	6.46176e11	0.224636	0.112318	−	0.993672i	$-0.464172\pi$
$587$	6.46176e11	0.224636	0.112318	+	0.993672i	$0.464172\pi$
$588$	0	0
$589$	−1.34331e11	−0.0459892
$590$	3.00894e11	0.102230
$591$	0	0
$592$	1.35853e12	0.454590
$593$	5.05774e12	1.67962	0.839808	−	0.542883i	$-0.182667\pi$
$593$	5.05774e12	1.67962	0.839808	+	0.542883i	$0.182667\pi$
$594$	0	0
$595$	−3.45005e11	−0.112849
$596$	4.58377e11	0.148804
$597$	0	0
$598$	−1.82644e12	−0.584051
$599$	4.61588e11	0.146499	0.0732494	−	0.997314i	$-0.476663\pi$
$599$	4.61588e11	0.146499	0.0732494	+	0.997314i	$0.476663\pi$
$600$	0	0
$601$	−6.31800e12	−1.97535	−0.987677	−	0.156509i	$-0.949976\pi$
$601$	−6.31800e12	−1.97535	−0.987677	+	0.156509i	$0.949976\pi$
$602$	−4.30163e11	−0.133490
$603$	0	0
$604$	−8.44227e11	−0.258103
$605$	7.04169e11	0.213687
$606$	0	0
$607$	−1.45276e12	−0.434356	−0.217178	−	0.976132i	$-0.569685\pi$
$607$	−1.45276e12	−0.434356	−0.217178	+	0.976132i	$0.569685\pi$
$608$	9.36335e10	0.0277885
$609$	0	0
$610$	−1.27990e11	−0.0374277
$611$	4.59605e11	0.133413
$612$	0	0
$613$	1.20124e12	0.343603	0.171802	−	0.985132i	$-0.445041\pi$
$613$	1.20124e12	0.343603	0.171802	+	0.985132i	$0.445041\pi$
$614$	−5.12094e11	−0.145409
$615$	0	0
$616$	7.26252e11	0.203224
$617$	−3.13545e12	−0.870997	−0.435498	−	0.900189i	$-0.643428\pi$
$617$	−3.13545e12	−0.870997	−0.435498	+	0.900189i	$0.643428\pi$
$618$	0	0
$619$	−5.17127e12	−1.41576	−0.707880	−	0.706333i	$-0.750348\pi$
$619$	−5.17127e12	−1.41576	−0.707880	+	0.706333i	$0.750348\pi$
$620$	−2.18599e12	−0.594135
$621$	0	0
$622$	−4.82884e11	−0.129356
$623$	1.59637e12	0.424557
$624$	0	0
$625$	1.52588e11	0.0400000
$626$	−2.30441e12	−0.599757
$627$	0	0
$628$	1.97520e12	0.506749
$629$	2.23073e12	0.568223
$630$	0	0
$631$	−6.37331e12	−1.60042	−0.800208	−	0.599723i	$-0.795278\pi$
$631$	−6.37331e12	−1.60042	−0.800208	+	0.599723i	$0.795278\pi$
$632$	2.29208e12	0.571481
$633$	0	0
$634$	1.07306e12	0.263768
$635$	1.81462e12	0.442897
$636$	0	0
$637$	−4.46204e11	−0.107376
$638$	−2.13312e12	−0.509710
$639$	0	0
$640$	1.93465e12	0.455818
$641$	5.74174e12	1.34333	0.671665	−	0.740855i	$-0.265579\pi$
$641$	5.74174e12	1.34333	0.671665	+	0.740855i	$0.265579\pi$
$642$	0	0
$643$	−5.85135e11	−0.134992	−0.0674958	−	0.997720i	$-0.521501\pi$
$643$	−5.85135e11	−0.134992	−0.0674958	+	0.997720i	$0.521501\pi$
$644$	−2.64320e12	−0.605541
$645$	0	0
$646$	3.46520e10	0.00782857
$647$	1.80915e12	0.405887	0.202943	−	0.979190i	$-0.434949\pi$
$647$	1.80915e12	0.405887	0.202943	+	0.979190i	$0.434949\pi$
$648$	0	0
$649$	1.84191e12	0.407536
$650$	−2.77302e11	−0.0609316
$651$	0	0
$652$	−2.68079e12	−0.580963
$653$	−2.43900e12	−0.524932	−0.262466	−	0.964941i	$-0.584536\pi$
$653$	−2.43900e12	−0.524932	−0.262466	+	0.964941i	$0.584536\pi$
$654$	0	0
$655$	1.28233e12	0.272216
$656$	−4.17396e12	−0.879996
$657$	0	0
$658$	−1.30759e11	−0.0271929
$659$	−7.42836e12	−1.53429	−0.767147	−	0.641471i	$-0.778324\pi$
$659$	−7.42836e12	−1.53429	−0.767147	+	0.641471i	$0.778324\pi$
$660$	0	0
$661$	−6.34861e12	−1.29352	−0.646759	−	0.762695i	$-0.723876\pi$
$661$	−6.34861e12	−1.29352	−0.646759	+	0.762695i	$0.723876\pi$
$662$	−2.30591e12	−0.466640
$663$	0	0
$664$	−8.13544e10	−0.0162414
$665$	−2.46606e10	−0.00488997
$666$	0	0
$667$	1.70533e13	3.33611
$668$	−2.66002e12	−0.516882
$669$	0	0
$670$	−1.57264e12	−0.301505
$671$	−7.83487e11	−0.149204
$672$	0	0
$673$	3.17186e12	0.596000	0.298000	−	0.954566i	$-0.403680\pi$
$673$	3.17186e12	0.596000	0.298000	+	0.954566i	$0.403680\pi$
$674$	5.60774e11	0.104669
$675$	0	0
$676$	1.97405e12	0.363578
$677$	2.32240e12	0.424901	0.212450	−	0.977172i	$-0.431856\pi$
$677$	2.32240e12	0.424901	0.212450	+	0.977172i	$0.431856\pi$
$678$	0	0
$679$	−2.89876e12	−0.523357
$680$	1.23866e12	0.222157
$681$	0	0
$682$	2.63066e12	0.465624
$683$	−3.78639e12	−0.665782	−0.332891	−	0.942965i	$-0.608024\pi$
$683$	−3.78639e12	−0.665782	−0.332891	+	0.942965i	$0.608024\pi$
$684$	0	0
$685$	2.03462e12	0.353081
$686$	1.26946e11	0.0218858
$687$	0	0
$688$	−2.73509e12	−0.465397
$689$	−2.12283e12	−0.358864
$690$	0	0
$691$	−5.70532e12	−0.951982	−0.475991	−	0.879450i	$-0.657911\pi$
$691$	−5.70532e12	−0.951982	−0.475991	+	0.879450i	$0.657911\pi$
$692$	3.83900e12	0.636415
$693$	0	0
$694$	1.54695e12	0.253138
$695$	5.16735e12	0.840111
$696$	0	0
$697$	−6.85372e12	−1.09997
$698$	−3.03746e12	−0.484352
$699$	0	0
$700$	−4.01307e11	−0.0631736
$701$	−6.25160e12	−0.977823	−0.488912	−	0.872333i	$-0.662606\pi$
$701$	−6.25160e12	−0.977823	−0.488912	+	0.872333i	$0.662606\pi$
$702$	0	0
$703$	1.59450e11	0.0246222
$704$	6.81818e11	0.104614
$705$	0	0
$706$	−3.47199e12	−0.525965
$707$	−2.83807e12	−0.427205
$708$	0	0
$709$	−8.01996e12	−1.19197	−0.595983	−	0.802997i	$-0.703238\pi$
$709$	−8.01996e12	−1.19197	−0.595983	+	0.802997i	$0.703238\pi$
$710$	−2.08466e12	−0.307873
$711$	0	0
$712$	−5.73136e12	−0.835791
$713$	−2.10308e13	−3.04757
$714$	0	0
$715$	−1.69749e12	−0.242901
$716$	7.54107e12	1.07232
$717$	0	0
$718$	1.46871e12	0.206241
$719$	−1.35002e12	−0.188391	−0.0941953	−	0.995554i	$-0.530028\pi$
$719$	−1.35002e12	−0.188391	−0.0941953	+	0.995554i	$0.530028\pi$
$720$	0	0
$721$	4.74944e12	0.654537
$722$	−2.95708e12	−0.404991
$723$	0	0
$724$	−6.94238e11	−0.0939042
$725$	2.58913e12	0.348043
$726$	0	0
$727$	−1.47778e13	−1.96203	−0.981013	−	0.193941i	$-0.937873\pi$
$727$	−1.47778e13	−1.96203	−0.981013	+	0.193941i	$0.937873\pi$
$728$	1.60198e12	0.211381
$729$	0	0
$730$	−1.19944e11	−0.0156324
$731$	−4.49108e12	−0.581731
$732$	0	0
$733$	−6.70116e12	−0.857398	−0.428699	−	0.903447i	$-0.641028\pi$
$733$	−6.70116e12	−0.857398	−0.428699	+	0.903447i	$0.641028\pi$
$734$	4.71269e12	0.599290
$735$	0	0
$736$	1.46593e13	1.84146
$737$	−9.62686e12	−1.20193
$738$	0	0
$739$	−1.39054e13	−1.71508	−0.857540	−	0.514417i	$-0.828008\pi$
$739$	−1.39054e13	−1.71508	−0.857540	+	0.514417i	$0.828008\pi$
$740$	2.59476e12	0.318094
$741$	0	0
$742$	6.03953e11	0.0731452
$743$	3.43953e12	0.414047	0.207024	−	0.978336i	$-0.433622\pi$
$743$	3.43953e12	0.414047	0.207024	+	0.978336i	$0.433622\pi$
$744$	0	0
$745$	6.69544e11	0.0796298
$746$	−6.15321e11	−0.0727407
$747$	0	0
$748$	3.45188e12	0.403179
$749$	−4.02955e11	−0.0467830
$750$	0	0
$751$	1.00823e13	1.15660	0.578298	−	0.815826i	$-0.303717\pi$
$751$	1.00823e13	1.15660	0.578298	+	0.815826i	$0.303717\pi$
$752$	−8.31401e11	−0.0948047
$753$	0	0
$754$	−4.70529e12	−0.530170
$755$	−1.23315e12	−0.138119
$756$	0	0
$757$	−1.02703e13	−1.13672	−0.568360	−	0.822780i	$-0.692422\pi$
$757$	−1.02703e13	−1.13672	−0.568360	+	0.822780i	$0.692422\pi$
$758$	3.81052e12	0.419250
$759$	0	0
$760$	8.85379e10	0.00962649
$761$	3.20840e12	0.346783	0.173391	−	0.984853i	$-0.444527\pi$
$761$	3.20840e12	0.346783	0.173391	+	0.984853i	$0.444527\pi$
$762$	0	0
$763$	−2.46199e12	−0.262982
$764$	7.12855e12	0.756974
$765$	0	0
$766$	3.22817e12	0.338787
$767$	4.06292e12	0.423896
$768$	0	0
$769$	1.52349e13	1.57098	0.785491	−	0.618872i	$-0.212410\pi$
$769$	1.52349e13	1.57098	0.785491	+	0.618872i	$0.212410\pi$
$770$	4.82941e11	0.0495092
$771$	0	0
$772$	−1.03266e13	−1.04635
$773$	8.54195e11	0.0860497	0.0430249	−	0.999074i	$-0.486301\pi$
$773$	8.54195e11	0.0860497	0.0430249	+	0.999074i	$0.486301\pi$
$774$	0	0
$775$	−3.19303e12	−0.317940
$776$	1.04073e13	1.03029
$777$	0	0
$778$	2.38517e12	0.233405
$779$	−4.89898e11	−0.0476636
$780$	0	0
$781$	−1.27611e13	−1.22732
$782$	5.42513e12	0.518776
$783$	0	0
$784$	8.07158e11	0.0763021
$785$	2.88514e12	0.271178
$786$	0	0
$787$	4.25016e12	0.394929	0.197465	−	0.980310i	$-0.436729\pi$
$787$	4.25016e12	0.394929	0.197465	+	0.980310i	$0.436729\pi$
$788$	−1.66807e12	−0.154116
$789$	0	0
$790$	1.52418e12	0.139224
$791$	−3.06206e12	−0.278112
$792$	0	0
$793$	−1.72823e12	−0.155193
$794$	6.73503e12	0.601378
$795$	0	0
$796$	8.02231e12	0.708256
$797$	−2.02157e13	−1.77470	−0.887352	−	0.461093i	$-0.847458\pi$
$797$	−2.02157e13	−1.77470	−0.887352	+	0.461093i	$0.847458\pi$
$798$	0	0
$799$	−1.36518e12	−0.118503
$800$	2.22566e12	0.192112
$801$	0	0
$802$	−7.41334e12	−0.632746
$803$	−7.34231e11	−0.0623179
$804$	0	0
$805$	−3.86087e12	−0.324044
$806$	5.80278e12	0.484315
$807$	0	0
$808$	1.01894e13	0.841002
$809$	6.65781e12	0.546466	0.273233	−	0.961948i	$-0.411907\pi$
$809$	6.65781e12	0.546466	0.273233	+	0.961948i	$0.411907\pi$
$810$	0	0
$811$	−9.35525e12	−0.759384	−0.379692	−	0.925113i	$-0.623970\pi$
$811$	−9.35525e12	−0.759384	−0.379692	+	0.925113i	$0.623970\pi$
$812$	−6.80942e12	−0.549678
$813$	0	0
$814$	−3.12259e12	−0.249290
$815$	−3.91578e12	−0.310892
$816$	0	0
$817$	−3.21018e11	−0.0252075
$818$	−8.35671e12	−0.652597
$819$	0	0
$820$	−7.97219e12	−0.615766
$821$	−1.61631e13	−1.24160	−0.620798	−	0.783971i	$-0.713191\pi$
$821$	−1.61631e13	−1.24160	−0.620798	+	0.783971i	$0.713191\pi$
$822$	0	0
$823$	5.97042e12	0.453634	0.226817	−	0.973937i	$-0.427168\pi$
$823$	5.97042e12	0.453634	0.226817	+	0.973937i	$0.427168\pi$
$824$	−1.70517e13	−1.28853
$825$	0	0
$826$	−1.15591e12	−0.0864003
$827$	−7.76424e12	−0.577197	−0.288599	−	0.957450i	$-0.593189\pi$
$827$	−7.76424e12	−0.577197	−0.288599	+	0.957450i	$0.593189\pi$
$828$	0	0
$829$	−4.09585e12	−0.301195	−0.150598	−	0.988595i	$-0.548120\pi$
$829$	−4.09585e12	−0.301195	−0.150598	+	0.988595i	$0.548120\pi$
$830$	−5.40988e10	−0.00395673
$831$	0	0
$832$	1.50397e12	0.108814
$833$	1.32537e12	0.0953751
$834$	0	0
$835$	−3.88545e12	−0.276600
$836$	2.46737e11	0.0174705
$837$	0	0
$838$	−4.53176e12	−0.317445
$839$	−4.46741e11	−0.0311263	−0.0155631	−	0.999879i	$-0.504954\pi$
$839$	−4.46741e11	−0.0311263	−0.0155631	+	0.999879i	$0.504954\pi$
$840$	0	0
$841$	2.94256e13	2.02835
$842$	−1.05606e11	−0.00724079
$843$	0	0
$844$	9.43433e11	0.0639985
$845$	2.88345e12	0.194562
$846$	0	0
$847$	−2.70513e12	−0.180598
$848$	3.84009e12	0.255012
$849$	0	0
$850$	8.23677e11	0.0541217
$851$	2.49636e13	1.63164
$852$	0	0
$853$	2.72968e13	1.76539	0.882696	−	0.469945i	$-0.155726\pi$
$853$	2.72968e13	1.76539	0.882696	+	0.469945i	$0.155726\pi$
$854$	4.91688e11	0.0316322
$855$	0	0
$856$	1.44671e12	0.0920979
$857$	−7.67571e12	−0.486077	−0.243038	−	0.970017i	$-0.578144\pi$
$857$	−7.67571e12	−0.486077	−0.243038	+	0.970017i	$0.578144\pi$
$858$	0	0
$859$	−1.67172e13	−1.04759	−0.523797	−	0.851843i	$-0.675485\pi$
$859$	−1.67172e13	−1.04759	−0.523797	+	0.851843i	$0.675485\pi$
$860$	−5.22398e12	−0.325656
$861$	0	0
$862$	8.64452e12	0.533284
$863$	2.85615e13	1.75280	0.876401	−	0.481582i	$-0.159938\pi$
$863$	2.85615e13	1.75280	0.876401	+	0.481582i	$0.159938\pi$
$864$	0	0
$865$	5.60756e12	0.340566
$866$	9.34116e12	0.564378
$867$	0	0
$868$	8.39769e12	0.502135
$869$	9.33018e12	0.555010
$870$	0	0
$871$	−2.12351e13	−1.25018
$872$	8.83916e12	0.517710
$873$	0	0
$874$	3.87783e11	0.0224795
$875$	−5.86182e11	−0.0338062
$876$	0	0
$877$	−8.40714e12	−0.479899	−0.239950	−	0.970785i	$-0.577131\pi$
$877$	−8.40714e12	−0.479899	−0.239950	+	0.970785i	$0.577131\pi$
$878$	−7.23964e12	−0.411142
$879$	0	0
$880$	3.07066e12	0.172608
$881$	1.99694e13	1.11680	0.558398	−	0.829573i	$-0.311416\pi$
$881$	1.99694e13	1.11680	0.558398	+	0.829573i	$0.311416\pi$
$882$	0	0
$883$	−2.36498e13	−1.30919	−0.654597	−	0.755978i	$-0.727162\pi$
$883$	−2.36498e13	−1.30919	−0.654597	+	0.755978i	$0.727162\pi$
$884$	7.61423e12	0.419364
$885$	0	0
$886$	−9.79250e12	−0.533878
$887$	3.83859e12	0.208217	0.104108	−	0.994566i	$-0.466801\pi$
$887$	3.83859e12	0.208217	0.104108	+	0.994566i	$0.466801\pi$
$888$	0	0
$889$	−6.97103e12	−0.374317
$890$	−3.81122e12	−0.203615
$891$	0	0
$892$	1.13527e13	0.600425
$893$	−9.75816e10	−0.00513495
$894$	0	0
$895$	1.10151e13	0.573833
$896$	−7.43213e12	−0.385237
$897$	0	0
$898$	3.04216e10	0.00156113
$899$	−5.41798e13	−2.76642
$900$	0	0
$901$	6.30551e12	0.318756
$902$	9.59390e12	0.482576
$903$	0	0
$904$	1.09936e13	0.547495
$905$	−1.01406e12	−0.0502511
$906$	0	0
$907$	−2.31944e13	−1.13802	−0.569009	−	0.822331i	$-0.692673\pi$
$907$	−2.31944e13	−1.13802	−0.569009	+	0.822331i	$0.692673\pi$
$908$	3.32931e13	1.62543
$909$	0	0
$910$	1.06528e12	0.0514966
$911$	−1.70743e13	−0.821317	−0.410659	−	0.911789i	$-0.634701\pi$
$911$	−1.70743e13	−0.821317	−0.410659	+	0.911789i	$0.634701\pi$
$912$	0	0
$913$	−3.31163e11	−0.0157733
$914$	−6.44896e12	−0.305655
$915$	0	0
$916$	−2.07368e13	−0.973221
$917$	−4.92620e12	−0.230065
$918$	0	0
$919$	1.49265e13	0.690301	0.345151	−	0.938547i	$-0.387828\pi$
$919$	1.49265e13	0.690301	0.345151	+	0.938547i	$0.387828\pi$
$920$	1.38615e13	0.637919
$921$	0	0
$922$	1.70768e13	0.778246
$923$	−2.81488e13	−1.27659
$924$	0	0
$925$	3.79012e12	0.170222
$926$	−9.80898e11	−0.0438404
$927$	0	0
$928$	3.77653e13	1.67158
$929$	−2.92103e12	−0.128666	−0.0643332	−	0.997928i	$-0.520492\pi$
$929$	−2.92103e12	−0.128666	−0.0643332	+	0.997928i	$0.520492\pi$
$930$	0	0
$931$	9.47362e10	0.00413278
$932$	−1.02020e13	−0.442906
$933$	0	0
$934$	−3.79701e12	−0.163260
$935$	5.04210e12	0.215754
$936$	0	0
$937$	−3.53996e13	−1.50027	−0.750135	−	0.661284i	$-0.770012\pi$
$937$	−3.53996e13	−1.50027	−0.750135	+	0.661284i	$0.770012\pi$
$938$	6.04147e12	0.254818
$939$	0	0
$940$	−1.58796e12	−0.0663384
$941$	4.67286e13	1.94280	0.971402	−	0.237439i	$-0.0763080\pi$
$941$	4.67286e13	1.94280	0.971402	+	0.237439i	$0.0763080\pi$
$942$	0	0
$943$	−7.66985e13	−3.15852
$944$	−7.34960e12	−0.301224
$945$	0	0
$946$	6.28665e12	0.255217
$947$	5.22799e12	0.211232	0.105616	−	0.994407i	$-0.466319\pi$
$947$	5.22799e12	0.211232	0.105616	+	0.994407i	$0.466319\pi$
$948$	0	0
$949$	−1.61958e12	−0.0648195
$950$	5.88756e10	0.00234520
$951$	0	0
$952$	−4.75842e12	−0.187757
$953$	2.46017e13	0.966156	0.483078	−	0.875577i	$-0.339519\pi$
$953$	2.46017e13	0.966156	0.483078	+	0.875577i	$0.339519\pi$
$954$	0	0
$955$	1.04125e13	0.405081
$956$	2.67809e13	1.03697
$957$	0	0
$958$	7.83373e11	0.0300486
$959$	−7.81618e12	−0.298408
$960$	0	0
$961$	4.03773e13	1.52715
$962$	−6.88788e12	−0.259297
$963$	0	0
$964$	3.40853e13	1.27122
$965$	−1.50838e13	−0.559936
$966$	0	0
$967$	1.24062e13	0.456267	0.228134	−	0.973630i	$-0.426738\pi$
$967$	1.24062e13	0.456267	0.228134	+	0.973630i	$0.426738\pi$
$968$	9.71212e12	0.355529
$969$	0	0
$970$	6.92060e12	0.250998
$971$	−5.30324e12	−0.191450	−0.0957248	−	0.995408i	$-0.530517\pi$
$971$	−5.30324e12	−0.191450	−0.0957248	+	0.995408i	$0.530517\pi$
$972$	0	0
$973$	−1.98509e13	−0.710023
$974$	4.53799e12	0.161565
$975$	0	0
$976$	3.12628e12	0.110282
$977$	1.45131e13	0.509606	0.254803	−	0.966993i	$-0.417989\pi$
$977$	1.45131e13	0.509606	0.254803	+	0.966993i	$0.417989\pi$
$978$	0	0
$979$	−2.33302e13	−0.811702
$980$	1.54166e12	0.0533914
$981$	0	0
$982$	−1.01954e13	−0.349868
$983$	3.61534e13	1.23498	0.617488	−	0.786580i	$-0.288150\pi$
$983$	3.61534e13	1.23498	0.617488	+	0.786580i	$0.288150\pi$
$984$	0	0
$985$	−2.43652e12	−0.0824721
$986$	1.39763e13	0.470917
$987$	0	0
$988$	5.44258e11	0.0181718
$989$	−5.02586e13	−1.67043
$990$	0	0
$991$	4.94162e13	1.62756	0.813782	−	0.581170i	$-0.197405\pi$
$991$	4.94162e13	1.62756	0.813782	+	0.581170i	$0.197405\pi$
$992$	−4.65739e13	−1.52700
$993$	0	0
$994$	8.00841e12	0.260200
$995$	1.17180e13	0.379010
$996$	0	0
$997$	−3.03522e13	−0.972886	−0.486443	−	0.873712i	$-0.661706\pi$
$997$	−3.03522e13	−0.972886	−0.486443	+	0.873712i	$0.661706\pi$
$998$	7.50580e12	0.239502
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	315.10.a.b.1.1		2
3.2	odd	2		35.10.a.b.1.2	✓	2
15.2	even	4		175.10.b.c.99.2		4
15.8	even	4		175.10.b.c.99.3		4
15.14	odd	2		175.10.a.c.1.1		2
21.20	even	2		245.10.a.c.1.2		2

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
35.10.a.b.1.2	✓	2	3.2	odd	2
175.10.a.c.1.1		2	15.14	odd	2
175.10.b.c.99.2		4	15.2	even	4
175.10.b.c.99.3		4	15.8	even	4
245.10.a.c.1.2		2	21.20	even	2
315.10.a.b.1.1		2	1.1	even	1	trivial

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:	\( N \)	\(=\)	\( 315 = 3^{2} \cdot 5 \cdot 7 \)
Weight:	\( k \)	\(=\)	\( 10 \)
Character orbit:	\([\chi]\)	\(=\)	315.a (trivial)

\(f(q)\)	\(=\)	\(q+9.17157 q^{2} -427.882 q^{4} -625.000 q^{5} +2401.00 q^{7} -8620.20 q^{8} -5732.23 q^{10} -35089.6 q^{11} -77401.4 q^{13} +22020.9 q^{14} +140015. q^{16} +229907. q^{17} +16433.6 q^{19} +267426. q^{20} -321827. q^{22} +2.57284e6 q^{23} +390625. q^{25} -709892. q^{26} -1.02735e6 q^{28} +6.62817e6 q^{29} -8.17416e6 q^{31} +5.69770e6 q^{32} +2.10861e6 q^{34} -1.50062e6 q^{35} +9.70272e6 q^{37} +150722. q^{38} +5.38762e6 q^{40} -2.98108e7 q^{41} -1.95343e7 q^{43} +1.50142e7 q^{44} +2.35970e7 q^{46} -5.93794e6 q^{47} +5.76480e6 q^{49} +3.58265e6 q^{50} +3.31187e7 q^{52} +2.74263e7 q^{53} +2.19310e7 q^{55} -2.06971e7 q^{56} +6.07908e7 q^{58} -5.24915e7 q^{59} +2.23282e7 q^{61} -7.49699e7 q^{62} -1.94308e7 q^{64} +4.83759e7 q^{65} +2.74351e8 q^{67} -9.83733e7 q^{68} -1.37631e7 q^{70} +3.63673e8 q^{71} +2.09245e7 q^{73} +8.89892e7 q^{74} -7.03163e6 q^{76} -8.42501e7 q^{77} -2.65896e8 q^{79} -8.75093e7 q^{80} -2.73412e8 q^{82} +9.43764e6 q^{83} -1.43692e8 q^{85} -1.79160e8 q^{86} +3.02479e8 q^{88} +6.64876e8 q^{89} -1.85841e8 q^{91} -1.10087e9 q^{92} -5.44603e7 q^{94} -1.02710e7 q^{95} -1.20731e9 q^{97} +5.28723e7 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 2 q + 24 q^{2} - 720 q^{4} - 1250 q^{5} + 4802 q^{7} - 20544 q^{8} - 15000 q^{10} - 18566 q^{11} - 51090 q^{13} + 57624 q^{14} + 112768 q^{16} + 373910 q^{17} - 143276 q^{19} + 450000 q^{20} - 76808 q^{22}+ \cdots + 138355224 q^{98}+O(q^{100}) \) 2 * q + 24 * q^2 - 720 * q^4 - 1250 * q^5 + 4802 * q^7 - 20544 * q^8 - 15000 * q^10 - 18566 * q^11 - 51090 * q^13 + 57624 * q^14 + 112768 * q^16 + 373910 * q^17 - 143276 * q^19 + 450000 * q^20 - 76808 * q^22 + 498908 * q^23 + 781250 * q^25 - 319736 * q^26 - 1728720 * q^28 + 11577554 * q^29 - 3953760 * q^31 + 11398656 * q^32 + 4243944 * q^34 - 3001250 * q^35 - 3205412 * q^37 - 2217520 * q^38 + 12840000 * q^40 - 1058992 * q^41 + 15948180 * q^43 + 10187376 * q^44 - 7156176 * q^46 - 65501290 * q^47 + 11529602 * q^49 + 9375000 * q^50 + 25432656 * q^52 + 25114688 * q^53 + 11603750 * q^55 - 49326144 * q^56 + 134182296 * q^58 + 116159208 * q^59 - 44688544 * q^61 - 12388000 * q^62 + 79055872 * q^64 + 31931250 * q^65 + 118092496 * q^67 - 140439024 * q^68 - 36015000 * q^70 + 294165824 * q^71 - 57419332 * q^73 - 102418064 * q^74 + 39622368 * q^76 - 44576966 * q^77 - 692852854 * q^79 - 70480000 * q^80 + 152932128 * q^82 + 540679928 * q^83 - 233693750 * q^85 + 346989008 * q^86 + 105455296 * q^88 + 779043704 * q^89 - 122667090 * q^91 - 495040608 * q^92 - 937691032 * q^94 + 89547500 * q^95 - 2673039406 * q^97 + 138355224 * q^98

Introduction

L-functions

Modular forms

Varieties

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Database

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